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\begin{document}

\pagenumbering{Alph}
\begin{titlepage}

	\begin{center}
		\hfill\break

		\vspace{2.5cm}

		{\bf\Large{On Supersymmetric Interface Defects,\\ \vskip 0.2in Brane Parallel Transport, Order-Disorder Transition \\ \vskip 0.2in and Homological Mirror Symmetry}}
		\vskip 1cm
		%
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		%
		{Dmitry Galakhov\footnote[2]{e-mail: dmitrii.galakhov@ipmu.jp; galakhov@itep.ru}}
		\vskip 0.2in

		\renewcommand{\thefootnote}{\roman{footnote}}

		{\small{
				\textit{Kavli Institute for the Physics and Mathematics of the Universe (WPI),\\ }
				\textit{University of Tokyo, Kashiwa, Chiba 277-8583, Japan}\\
				%
				\vskip 0.4 cm
				\textit{Institute for Information Transmission Problems,\\}
				\textit{ Moscow, 127994, Russia}
		}}

	\end{center}

	\vskip 0.5in
	\baselineskip 16pt

	\begin{abstract}
		We concentrate on a treatment of a Higgs-Coulomb duality as an absence of manifest phase transition between ordered and disordered phases of 2d ${\cal N}=(2,2)$ theories.
		We consider these examples of QFTs in the Schr\"odinger picture and identify Hilbert spaces of BPS states with morphisms in triangulated Abelian categories of D-brane boundary conditions.
		As a result of Higgs-Coulomb duality D-brane categories on IR vacuum moduli spaces are equivalent, this resembles an analog of homological mirror symmetry.
		Following construction ideas behind the Gaiotto-Moore-Witten algebra of the infrared one is able to introduce interface defects in these theories and associate them to D-brane parallel transport functors.
		We concentrate on surveying simple examples, analytic when possible calculations, numerical estimates and simple physical picture behind curtains of geometric objects.
		Categorification of hypergeometric series analytic continuation is derived as an Atiyah flop of the conifold.
		Finally we arrive to an interpretation of the braid group action on the derived category of coherent sheaves on cotangent bundles to  flag varieties as a categorification of Berry connection on the Fayet-Illiopolous parameter space of a sigma-model with a quiver variety target space.

	\end{abstract}

	\date{\today}
\end{titlepage}
\pagenumbering{arabic}

\newpage
\setcounter{tocdepth}{2}
\tableofcontents
\newpage

\section{Introduction}
\subsection{Supersymmetric interfaces}
Since being discovered in the late 80's mirror symmetry has acquired a lot of attention of both physicist and mathematician communities.
We are not aiming to give a detailed profound review of this huge topic in the modern string theory referring the reader to canonical literature sources on this subject \cite{D-book_1,D_book_2,Strominger:1996it} as well as modern reviews of mirror symmetry and Langlands correspondence physical applications \cite{Bullimore:2016nji,Dimofte:2019zzj,Rimanyi:2019zyi,Aganagic:2016jmx,Braverman:2016pwk,Gu:2018fpm,Fan:2007ba} and references therein.
Instead we would like to narrow our current scope to a relation between homological mirror symmetry as it is understood in algebraic geometry and its physical avatar -- duality of D-brane boundary conditions in 2d ${\cal N}=(2,2)$ supersymmetric theories.
In practice, homological mirror symmetry \cite{Kontsevich:1994dn} relates certain triangulated categories on a mirror pair of Calabi-Yau manifolds.
A physical interpretation of this symmetry refers to a duality between observables of different phases of the same theory where the manifolds of the mirror pair represent vacuum moduli spaces.

Our aim is to look at this problem in a perspective of the algebra of the infrared discovered in \cite{GMW} and developed its mathematical counterpart in \cite{Kapranov:2014uwa} (see also \cite{Kapranov:2020zoa,Khan:2020hir,2018arXiv181008776S}).
A common proposal for physical categorification of various quantities indicates that such geometric objects as cohomology appear in consideration of topologically protected Hilbert subspaces of quantum theories with supersymmetry \cite{Witten:1982im}.
An approach of \cite{GMW} refers to a canonical consideration of a 2d ${\cal N}=(2,2)$ theory as a quantum system with a Hilbert space of physical states where the evolution is driven by a Hamiltonian evolution operator in the Schr\"odinger picture.
An alternative approach to $d$-dimensional quantum field theory is the standard path integral approach where the resulting partition function is calculated as a continual integral over maps from a $d$-dimensional world-volume ${\cal V}_d$ to the target space of fields ${\cal T}$.
To pass to the mechanics of quantum systems -- quantum mechanics -- one chooses a Killing vector in ${\cal V}_d$ as a temporal direction and splits the world-volume ${\cal V}_d={\mathbb{R}}_t\times {\cal V}_{d-1}$, then the quantum mechanics configuration space is given by field maps
\bar{e}\label{maps}
{\rm Map}\left({\cal V}_{d-1}\to {\cal T}\right).
 \end{equation}

In general, ${\cal V}_{d-1}$ could be both non-compact or have boundaries, in either case constraints on the field asymptotic behavior or boundary conditions are in order.
Naively, admissible boundary conditions form an abstract set, however relations in QFT could produce certain structures on these sets.
In particular, in 2d ${\cal N}=(2,2)$ theories we will be interested in the boundary conditions that will have a structure of a \emph{category}.
In more general situations the boundary conditions of $d$-dimensional topological field theory are expected to form a $d-1$-category \cite{Carqueville:2016kdq,Leinster,Gukov:2017kmk}.

In the concrete case of 2d ${\cal N}=(2,2)$ theories we are going to consider ${\cal V}_d$ given by a 2d strip, and ${\cal V}_{d-1}$ is a segment of length, say, $L$:
\bar{e}\nonumber
\begin{array}{c}
	\begin{tikzpicture}
	\draw[dashed,fill=blue, opacity=0.4] (0,0) -- (4,0) -- (4,2) -- (0,2) -- cycle;
	\node at (2,1) {interface $p(x^1)$};
	\draw[ultra thick] (0,0) -- (0,2) (4,0) -- (4,2);
	\node[left] at (0,1) {${\cal A}$};
	\node[right] at (4,1) {${\cal B}$};
	\draw[<->] (-1,2) -- (-1,-0.5) -- (4.5,-0.5);
	\node[right] at (4.5,-0.5) {$x^1$};
	\node[above] at (-1,2) {$x^0$};
	\draw[ultra thick] (0,-0.4) -- (0,-0.6) (4,-0.4) -- (4,-0.6);
	\node[below] at (0,-0.6) {$0$};
	\node[below] at (4,-0.6) {$L$};
	\end{tikzpicture}
\end{array}
 \end{equation}
Maps from 1d spatial segment $[0,L]$ to ${\cal T}$ may be called ``strings" in ${\cal T}$.
The standard reasoning \cite{D-book_1} leads to a conclusion that admissible boundary conditions can be reinterpreted as a permission for string ends to move only inside special loci in ${\cal T}$ called D(irichlet)-branes.
In addition D-branes will be allowed to carry additional information  -- Chan-Paton factors -- that in this case are complexes of vector bundles.
We will denote the corresponding category of D-branes as $\mathfrak{D}$.

In general, we would like to consider families of theories fibered over parameter space ${\cal P}$ spanned by fugacities and couplings.
In the literature ${\cal P}$ is often called a moduli space when ${\cal T}$ is Calabi-Yau and fugacity parameters are its K\"ahler moduli.
To a generic path $\wp$ in the parameter space ${\cal P}$ one is able to associate  parallel transport induced by a Berry connection on a fibration of Hilbert spaces over ${\cal P}$ \cite{Moore:2017byz}, in the case of 2d ${\cal N}=(2,2)$  Landau-Ginzburg models this connection is also known as $tt^*$-connection \cite{Cecotti:1992rm}, in the case of Gromov-Witten theory this connection is also known as a Casimir \cite{tarasov2002duality} or quantum connection \cite{MaulikOkounkov,Okounkov:2016sya}.
A physical counterpart of categorification for the Berry connection  is
an interface defect $\mathfrak{J}_{\wp}$ preserving a half of the initial supersymmetries.
In general, the category of D-branes $\mathfrak{D}_p$ depends on a choice of a point $p\in {\cal P}$.
If a path $\wp$ interpolates between points $p_1$ and $p_2$ in ${\cal P}$ then we choose D-brane boundary conditions for the left and the right strip edges as objects in corresponding categories:
\bar{e}
{\cal A}\in \mathfrak{D}_{p_1},\quad {\cal B}\in\mathfrak{D}_{p_2}.
 \end{equation}

Certain topological properties of theories in consideration follow form the superalgebra properties.
We will concentrate on topologically protected states.
These states saturate a Bogomolny-Prasad-Sommerfeld (BPS)  lower bound on the Hamiltonian eigenvalues and are annihilated by two supercharges ${\cal Q}$ and ${\cal Q}^{\dag}$ out of initial four:
\bar{e}
{\cal Q}|\Psi_{\rm BPS}\rangle={\cal Q}^{\dagger}|\Psi_{\rm BPS}\rangle=0.
 \end{equation}

For the Hamiltonian ${\cal H}$ eigenvalue we have:
\bar{e}
{\cal H}|\Psi_{\rm BPS}\rangle=|{\cal Z}|\cdot|\Psi_{\rm BPS}\rangle,
 \end{equation}
where ${\cal Z}$ is a central element of the superalgebra.

The BPS states saturating the BPS bound form a subspace of the Hilbert space of all states we call a BPS Hilbert space $\mathfrak{G}_{\rm BPS}$.
It depends on all incoming data including $\mathfrak{J}_{\wp}$ and ${\cal A}$, ${\cal B}$ and, in general, has a grading by various flavor charges preserved by the interface supersymmetry.
The main observation \cite{GMW,Aganagic:2020olg,aganagic2021knot} we are going to justify and exploit throughout this paper is the following relation between interface BPS Hilbert spaces and morphisms in D-brane categories:
\bar{e}\label{main}
\boxed{
	\mathfrak{G}_{\rm BPS}^{(*,*,\ldots)}\left(\mathfrak{J}_{\wp}|{\cal A},{\cal B}\right)\cong {\rm Hom}_{\mathfrak{D}_{p_2}}^{*,*,\ldots}\left(\beta_{\wp}({\cal A}),{\cal B}\right)},
 \end{equation}
where $\beta_{\wp}$ is a {\bf parallel transport} functor acting in D-brane categories:
$$
\beta_{\wp}:\quad \mathfrak{D}_{p_1}\longrightarrow \mathfrak{D}_{p_2},
$$
satisfying the parallel transport relation:
$$
\beta_{\wp_2}\circ\beta_{\wp_1}=\beta_{\wp_1\circ\wp_2}.
$$
Eigenvalues of physical charges grading the BPS Hilbert space in the l.h.s. of \eqref{main} are identified with various grading degrees of the r.h.s.

Another property we expect from functor $\beta_{\wp}$ is ``flatness" -- a reflection of a protection for the BPS Hilbert space on the supersymmetry side from small deformations, so that for two homotopic paths $\wp_1$ and $\wp_2$:
$$
\beta_{\wp_1}= \beta_{\wp_2}.
$$

\subsection{Higgs-Coulomb duality, order-disorder transition and mirror symmetry}\label{sec:Intro_mirror}

The quantum mechanics approach to the quantum field theory allows one to treat supercharge operators ${\cal Q}$ and ${\cal Q}^{\dagger}$ following \cite{Witten:1982im} as a differential and its Hodge dual on the cotangent bundle to the space of maps \eqref{maps}.
This, in turn, incorporates the QFT framework into the Morse and equivariant localization techniques \cite{Cordes:1994fc} reducing a generic problem of constructing wave-functions to a simplified counting of BPS classical field configurations and only few loop quantum corrections to them.
The localization techniques allow one to construct physical quantities that are invariant under the action of the renormalization group, in other words those quantities are  independent of the Plank constant $\hbar$, or, alternatively, of the Yang-Mills coupling constant $g_{\rm YM}^2$.
Therefore we can calculate them in the limit:
$$
\hbar\longrightarrow 0,
$$
where classical field configuration give the major contribution.  See \cite{Pestun:2016qko} for a review of the localization paradigm.

A basic model we will consider in this paper is a 2d ${\cal N}=(2,2)$ supersymmetric gauged sigma-model with a matter content encoded in a quiver diagram $\mathfrak{Q}$ \cite{Douglas:1996sw}.
A quiver is an oriented graph, we denote sets of quiver nodes and quiver arrows as $\mathfrak{Q}_0$ and $\mathfrak{Q}_1$ respectively.
Nodes of the quiver label gauge multiplets consisting of a gauge connection, complex scalar, fermion partners and an auxiliary field:
$$
A_v,\quad \sigma_v,\quad \lambda_{\alpha,v},\quad {\bf D}_v,\quad v\in\mathfrak{Q}_0,
$$
arrows label bi-fundamentally charged chiral multiplets consisting of a complex scalar, fermion partners and an auxiliary field:
$$
\phi_a,\quad \psi_{\alpha,a},\quad {\bf F}_a,\quad a\in\mathfrak{Q}_1.
$$
To fix the theory we have to choose parameters ${\cal P}$ that can be thought of as additional elements entering data associated to the quiver.
So to the quiver nodes $\mathfrak{Q}_0$ one associates the rank of the gauge group $U(n)$, a theta-angle and a Fayet-Illiopolous parameter:
$$
n_v\in \mathbb{N},\quad \theta_v\in {\mathbb{R}},\quad r_v\in{\mathbb{R}},\quad v\in \mathfrak{Q}_0
$$
To assign to chiral field multiplets non-trivial masses one could use a ``framing" procedure \cite{nakajima1996varieties}.
A quiver node with flavor group $U(n_f)$ is claimed to be a framing node, to construct flavor symmetry it suffices to ``freeze" gauge degrees of freedom to fixed expectation values:
$$
\sigma={\rm diag}(\mu_1,\ldots,\mu_{n_f}),
$$
here $\mu_i$ are complex masses assigned to the chiral multiplet, and the gauge group $U(n_f)$ becomes a flavor symmetry group of the chiral multiplet.

For example, a theory with $n_f$ chiral fields having charge +1 with respect to the gauge group $U(k)$ can be schematically depicted by the following quiver:
$$
\begin{array}{c}
\begin{tikzpicture}
\node at (0,0) {$k$};
\node at (2,0) {$n_f$};
\draw (0,0) circle (0.3);
\begin{scope}[shift={(2,0)}]
\draw (-0.3,-0.3) -- (-0.3,0.3) -- (0.3,0.3) -- (0.3,-0.3) -- cycle;
\end{scope}
\draw[->] (1.7,0) -- (0.3,0);
\end{tikzpicture}
\end{array}
$$

Models described by quivers with loops allow introducing a gauge-invariant superpotential function $W(\phi_a)$ that is a holomorphic function of complex fields $\phi_a$.

The localization procedure forces quantum field expectation values to approach classical vacua.
 Classical vacua form a moduli space -- zero locus of D-term and F-term constraints:
\bar{e}\label{D-term}
\begin{split}
	&{\bf D}_v=r_v-\sum\limits_{\substack{w\in \mathfrak{Q}_0\\ (a:v\to w)\in\mathfrak{Q}_1}}\phi_a\phi_a^{\dagger}+\sum\limits_{\substack{w\in \mathfrak{Q}_0\\ (a:w\to v)\in\mathfrak{Q}_1}}\phi_a\phi_a^{\dagger}=0,\quad \forall v\in \mathfrak{Q}_0\\
	&{\bf F}_a=-\partial_{\phi_a}W=0,\quad a\in \mathfrak{Q}_1
\end{split}
 \end{equation}
modulo the action of the gauge group.
Due to mixing of the gauge fields $A_v$ and the scalar fields $\sigma_v$ the initial unitary gauge group is complexified:
$$
\prod\limits_{v\in \mathfrak{Q}_0}U(n_v)\longrightarrow \prod\limits_{v\in \mathfrak{Q}_0}GL(n_v,\mathbb{C}).
$$
An enhancement of a Riemann manifold structure of the moduli space to a complex variety structure of stable quiver representations is usually referred to as Narashiman-Shishadri-Hitchin-Kobayashi correspondence \cite{donaldson1983new}.

The complex gauge field delivers a complex equivariant action on the moduli space with associated Killing vector fields:
$$
V\sim\sum\limits_{(a:v\to w)\in\mathfrak{Q}_1}{\rm Tr}\left[(\sigma_v\phi_a-\phi_a\sigma_w)\frac{\partial}{\partial q_a}\right].
$$

Equivariant localization demands the points of the moduli spaces contributing to the vacua to be fixed points of $V$.
This constraint in terms of field expectation values has two generic solutions usually referred to as a Higgs branch and a Coulomb branch in the literature \cite{deBoer:1996mp}:
\bar{e}
\mbox{Higgs}:\; \langle \phi\rangle\neq0,\; \langle\sigma\rangle=0,\quad \mbox{Coulomb}:\; \langle \phi\rangle=0,\; \langle\sigma\rangle\neq 0,
 \end{equation}
we will also identify these branches to physical phases of the theory as in \cite{Hori:2013ika,Witten:1993yc}.
Surely, in practice one may encounter all sorts of mixed branches.

Despite both the Higgs and Coulomb branches solve apparently the fixed point constraint they can not simultaneously solve the D-term constraint \eqref{D-term}.
If the FI parameters $r$ are non-zero we see the chiral field has to acquire an expectation value of order $\sim\sqrt{r}$, therefore the Higgs branch is realized in the IR.
On the other hand, if $r=0$ there is no vev for chiral fields, therefore the Coulomb branch is realized in the IR.
From the practical point of view there is no need to put strictly $r=0$ to get the Coulomb branch, rather it suffices to make it small of the order of quantum fluctuations $\sim \hbar^{\frac{1}{2}}$.
To mark the weak coupling region contingently let us assume:
$$
\hbar<\hbar_0\ll 1.
$$
In this region depending if $|r|>\sqrt{\hbar}$ or $|r|<\sqrt{\hbar}$ the BPS Hilbert space is described effectively in the IR by either the Higgs branch or the Coulomb branch.
So that one can depict the phase diagram as in Fig.\ref{fig_phase}.

\begin{figure}[h!]
	\begin{center}
		\begin{tikzpicture}
		\draw[<->,thick] (0,3.2) -- (0,0) -- (6.2,0);
		\draw[fill=white!80!blue] (0,0.5) -- (6,0.5) -- (6,3) -- (0,3) -- cycle;
		%
		\begin{scope}[shift={(0,3)}]
		\begin{scope}[xscale=0.8]
			\draw[fill=white!80!green] (0,0) -- ([shift=(270:2.5)]0,0) arc (270:360:2.5) -- cycle;
		\end{scope}
		\end{scope}
		%
		\begin{scope}[shift={(6,3)}]
			\begin{scope}[xscale=-0.8]
				\draw[fill=white!80!green] (0,0) -- ([shift=(270:2.5)]0,0) arc (270:360:2.5) -- cycle;
			\end{scope}
		\end{scope}
		%
		\draw[fill=white!80!gray] (0,0) -- (0,0.8) -- (6,0.8) -- (6,0) -- cycle;
		\draw[thick] (0,0.8) -- (6,0.8);
		%
		\begin{scope}[shift={(0,3)}]
			\begin{scope}[xscale=0.8]
				\draw[thick] ([shift=(270:2.5)]0,0) arc (270:360:2.5);
			\end{scope}
		\end{scope}
		%
		\begin{scope}[shift={(6,3)}]
			\begin{scope}[xscale=-0.8]
				\draw[thick] ([shift=(270:2.5)]0,0) arc (270:360:2.5);
			\end{scope}
		\end{scope}
		%
		\draw[dashed] (3,0) -- (3,3);
		\node at (3,2.5) {Coulomb};
		\node at (1,2.5) {Higgs I};
		\node at (5,2.5) {Higgs II};
		\node at (3,0.4) {???};
		\node[right] at (6.2,0) {$r$};
		\node[above] at (0,3.2) {$\hbar^{-1}$, $g_{\rm YM}^{-2}$};
		\node[left] at (0,0.8) {$\hbar_0^{-1}$};
		\node[left] at (0,0.4) {\color{orange} strong $\downarrow$};
		\node[left] at (0,1.9) {\color{orange} weak $\uparrow$};
		%
		\draw[purple,<->,thick,>=stealth'] (2,1) -- (1,2);
		\draw[purple,<->,thick,>=stealth'] (4,1) -- (5,2);
		\node[below left,purple] at (1.5,1.5) {\tiny cHC duality};
		\node[below right,purple] at (4.5,1.5) {\tiny cHC duality};
		\node[below] at (3,0) {$0$};
		\end{tikzpicture}
	\caption{Simple gauged sigma-model phase diagram.}\label{fig_phase}
	\end{center}
\end{figure}

Consider a path in $\hbar$ connecting Higgs branch and Coulomb branch regions as in Fig.\ref{fig_phase}.
The BPS Hilbert spaces for different values of $\hbar$ are isomorphic.
We promote this isomorphism to an isomorphism between Higgs and Coulomb branch descriptions:
\bar{e}\label{HC_du}
\mathfrak{G}_{\rm BPS,\; Clmb}\cong\mathfrak{G}_{\rm BPS,\; Higgs}
 \end{equation}
This relation is a direct analog of a Coulomb-Higgs duality for theories in other dimensions (see e.g. \cite{Manschot:2013sya,deBoer:1996mp}).

We should stress that the Higgs and Coulomb branch descriptions are quite different.
The Coulomb branch description reduces to an effective field theory of twisted chiral superfield $\Sigma=\sigma+\ldots$.
The description of ${\cal N}=(2,2)$ twisted chiral field is quite similar \cite{Hori:2000kt} to a description of a chiral field, only some fermion variables of opposite chiralities should be swapped, so this description is equivalent to a Landau-Ginzburg model of a chiral field spanning the IR target space $X_{\rm Clmb}$.
In the Higgs branch description when FI parameters are non-zero the D-term constraint can be substituted by a stability condition \cite{donaldson1983new}.
So that the Higgs branch is given by a stable quiver representation moduli space \cite{Alim:2011kw}.

Returning back to \eqref{main} we see that the Higgs-Coulomb duality applied to 2d gauged sigma-models provides two equivalent descriptions of the categories of the D-brane boundary conditions.

A D-brane category for the Landau-Ginzburg theory is usually identified with a Fukaya-Seidel category of $X_{\rm Coulomb}$, in general, with a superpotential $W$, and boundary conditions in the sigma-model are identified with a derived category of coherent sheaves on $X_{\rm Higgs}$, thus we will arrive to the following form of the Higgs-Coulomb duality for BPS Hilbert spaces:
\bar{e}\label{mirror}
\mbox{Fukaya-Seidel}(X_{\rm Clmb},W)\con
D^b{\rm Coh}(X_{\rm Higgs})
 \end{equation}

In the literature \cite{Hori:2000kt} authors also use an ``intermediate" model description of the Landau-Ginzburg theory in addition to twisted chiral field $\Sigma$ considering a twisted chiral field traditionally denoted $Y$.
We will identify the chiral field $\phi$ with an \emph{order} parameter on the Higgs branch.
Then the dual \emph{disorder} parameter field representing an insertion of a vortex singularity give rise in a supersymmetric theory to a twisted chiral field $Y$.
This duality is direct analog of an order-disorder transition appearing in the 2d Ising model \cite{Polyakov:1987ez}.

The order-disorder duality is also related to T-duality.
Applying T-duality \cite{Strominger:1996it,Ooguri:1996ck} one also exchanges twisted chiral fields in a B-twisted model to chiral fields in an A-twisted model \cite{Hori:2000kt}.
\emph{Mirror symmetry} relates B-twisted effective non-linear sigma-model of order parameter $\phi$ to an A-twisted Landau-Ginzburg model of disorder parameter $Y$.
A further elaborated analog of relation \eqref{mirror} for triangulated categories on corresponding varieties is usually referred to as \emph{homological mirror symmetry} \cite{Kontsevich:1994dn,Kontsevich:2008fj,Gukov:2010sw,2008arXiv0811.1228F}.

To return to the original model of fields $\Sigma$ one has to integrate out fields $Y$ to an effective theory in the IR, however this operation comes with its price.
In general, integration produces singularities in the effective action with a possibility to spoil the homotopy properties of interface observables and mirror symmetry.
These obstructions can be resolved either using a model with a UV completed superpotential without singularities -- in other words, a superpotential with both fields $\Sigma$ and $Y$ (see \cite[Section 7]{Galakhov:2016cji}) -- or dualizing the theory back to gauged sigma-model of field $\phi$.
Fortunately, in our examples we will not encounter these obstructions directly, so we leave them beyond the scope of this paper and will return to it elsewhere.
Nevertheless, keeping in mind this issue and the fact that $X_{\rm Clmb}$ and $X_{\rm Higgs}$  are not ideal mirrors in our construction in full generality we will refer to relation \eqref{mirror} as a \emph{categorified Higgs-Coulomb duality}, or simply a cHC duality for brevity.

\subsection{Decategorification, categorification and parallel transport}\label{sec:dec_cat_par}

A concept simpler than an Abelian category ${\cal C}$ capturing its partial behavior is its Grothendieck group $K_0({\cal C})$.
A transition from ${\cal C}$ to $K_0({\cal C})$ may be called a \emph{decategorification} process, and the inverse process is usually referred to as a \emph{categorification} process \cite{Chun:2015gda}.
Physically, a relation between a category and its Grothendieck group is similar to a relation between the graded BPS Hilbert space and its supersymmetric index:
\bar{e}
\begin{array}{c}
\begin{tikzpicture}
\node(A)[left] at (0,0) {$\mathfrak{G}_{\rm BPS}^{\left({\cal F},J_1,J_2,\ldots\right)}$};
\node(B)[right] at (4,0) {${\rm Tr}_{\mathfrak{G}_{\rm BPS}}(-1)^{{\cal F}}\cdot y_1^{J_1}\cdot y_2^{J_2}\cdot\ldots$};
\draw[-stealth] (0.2,0.06) -- (3.8,0.06);
\draw[stealth-] (0.2,-0.06) -- (3.8,-0.06);
\node[above] at (2,0.06) {\tiny decategorification};
\node[below] at (2,-0.06) {\tiny categorification};
\end{tikzpicture}
\end{array},
 \end{equation}
where ${\cal F}$ is a fermion number, $J_i$ are charges commuting with the supercharges and $y_i$ are introduced fugacities.
Apparently, the index does not distinguish a contribution of a boson-fermion pair of BPS states with other identical quantum numbers.

The supersymmetric index can be re-interpreted in the path integral formulation as a partition function.
To do so we substitute the temporal dimension ${\mathbb{R}}_t$ with a thermal circle $S^1_{\rm thrm}$, apply the Wick rotation to the theory, the fermionic fields are periodic on the thermal circle $S^1_{\rm thrm}$.
The resulting world-volume manifold is ${\cal V}^{\rm E}_d=S_{\rm thrm}^1\times {\cal V}_{d-1}$.
In our case ${\cal V}_2^{\rm E}$ is a cylinder of width $L$.

Let us make  a brief digression and review properties of partition functions of theories in question.
We start with a partition function on a disk.
To identify this partition function $Z$ we have to choose some parameter values $p\in{\cal P}$ and a D-brane boundary condition ${\cal A}\in\mathfrak{D}_p$.
Mathematically $Z$ forms a functor from the category of D-branes to a vector space -- its Grothendieck group:
\bar{e}
Z:\quad \mathfrak{D}_p\longrightarrow K_0\left(\mathfrak{D}_p\right)
 \end{equation}

The disk partition function can be calculated \cite{Hori:2013ika} for various choices of boundary conditions with the help of localization techniques.
Again, one has two options to localize on either the Higgs or the Coulomb branch, therefore as we mentioned in the previous section there are two dual choices of categories:
$$
\mathfrak{D}_p^{\rm Clmb},\quad \mathfrak{D}_p^{\rm Higgs},
$$
corresponding to the Fukaya-Seidel category and the derived category of coherent sheaves as well as two partition functions.

The Coulomb branch partition function is a partition function of a Landau-Ginzburg theory with a superpotential $W$. The D-branes are represented  by Lagrangian loci in $X_{\rm Clmb}$.
The corresponding expression reads:
\bar{e}\label{Clmb_br_pf}
Z_{\rm Clmb}[p,{\cal A}_{\rm Clmb}]=\int\limits_{{\cal A}_{\rm Clmb}}\Omega\; e^{2\pi{\rm i}\; W},\quad {\cal A}_{\rm Clmb}\in\mathfrak{D}_p^{\rm Clmb}.
 \end{equation}
where $\Omega$ is  a holomorphic form.

On the Higgs branch the effective theory is a sigma-model with the target space given by $X_{\rm Higgs}$, the D-brane boundary conditions correspond to a choice of a complex of holomorphic vector fields, or, in more precise mathematical formulation, of a coherent sheaf ${\cal A}_{\rm Higgs}$.
The corresponding partition function reads (see also \cite{Honda:2013uca}):
\bar{e}
Z_{\rm Higgs}[p,{\cal A}_{\rm Higgs}]=\int\limits_{X_{\rm Higgs}} e^{B+\frac{{\rm i}}{2\pi}\omega} {\rm ch}\left({\cal A}_{\rm Higgs}\right)\sqrt{{\rm Td}\left(X_{\rm Higgs}\right)},\quad {\cal A}_{\rm Higgs}\in\mathfrak{D}_p^{\rm Higgs},
 \end{equation}
where $B$ and $\omega$ are a B-field and a K\"ahler form correspondingly, $\rm Td$ denotes the Todd class, and ${\rm ch}(*)$ is a Chern class.

In these terms Higgs-Coulomb duality manifests itself as an equality between Higgs and Coulmb partition functions:
\bar{e}
Z_{\rm Clmb}[p,{\cal A}_{\rm Clmb}]\mathop{\scalebox{6}[1]{$=$}}\limits^{\rm HC} Z_{\rm Higgs}[p,{\cal A}_{\rm Higgs}],
 \end{equation}
for some choices of branes ${\cal A}_{\rm Clmb}$ and ${\cal A}_{\rm Higgs}$.
The categories we will consider are triangulated, one can choose a \emph{basis} among category objects.
Usually, such a basis is chosen to be a set of exceptional objects of the triangulated category \cite{Seidel_book}.
Similarly, a basis can be chosen in the vector space $K_0(\mathfrak{D})$.
It is rather simple to observe this phenomenon in the case of the partition function on the Coulomb branch.
Indeed according to \eqref{Clmb_br_pf} $Z_{\rm Clmb}$ is a holomorphic integral and depends only on the homotopy class of ${\cal A}_{\rm Clmb}$ preserving the asymptotic behavior of the integrand so that the integral converges.
A good choice of a basis of such integration cycles is Lefschetz thimbles.
On the other hand, $Z_{\rm Higgs}$ is resembling a matrix integral with an inserted operator defined by ${\cal A}_{\rm Higgs}$.
It is known that such a set of ``interesting" insertions is finite, has a linear structure and, therefore, a basis.
Moreover both are expected to be flat sections of the Berry connection on the parameter space ${\cal P}$ so that $Z_{\rm Clmb}[p,{\cal A}_{\rm Clmb}]$ and $Z_{\rm Higgs}[p,{\cal A}_{\rm Higgs}]$ are solutions of the same set of differential equations.

Over a generic points $p$ of ${\cal P}$ we expect $K_0(\mathfrak{D}_p)$ to be of some fixed dimension.
Therefore on ${\cal P}$ minus some singular loci $K_0(\mathfrak{D}_p)$ forms a vector bundle.
We would like to consider a \emph{parallel transport} induced by the Berry connection of a vector $Z$ along a path $\wp$ from some point $p_1$ to a new point $p_2$.
Geometrically, we employ the topological properties of the theory and bend a world-sheet disk in such a way that it resembles a vial, then extend its neck.
We assume that parameters are slowly varying along this neck, in other words we insert interface defect $\mathfrak{J}_\wp$ into this neck.
One can cut the resulting world-sheet into  a smaller vial and a long neck cylinder.
Accordingly, we glue the resulting disk partition function out a smaller disk partition function and the interface partition function:
\bar{e}\nonumber
\begin{split}
	\begin{array}{c}
		\begin{tikzpicture}
		\begin{scope}[xscale=0.5]
		\draw circle (0.3);
		\end{scope}
		\draw (0,0.3) to[out=180,in=90] (-1,0) to[out=270,in=180] (0,-0.3);
		\begin{scope}[shift={(0.5,0)}]
		\begin{scope}[xscale=0.5]
		\draw circle (0.3);
		\end{scope}
		\end{scope}
		\begin{scope}[shift={(2.5,0)}]
		\begin{scope}[xscale=0.5]
		\draw circle (0.3);
		\end{scope}
		\end{scope}
		\draw (0.5,0.3) -- (2.5,0.3) (0.5,-0.3) -- (2.5,-0.3);
		\draw[dashed] (0,0.3) -- (0.5,0.3) (0,-0.3) -- (0.5,-0.3);
		\begin{scope}[shift={(6,0)}]
		\begin{scope}[xscale=0.5]
		\draw circle (0.3);
		\end{scope}
		\draw (0,0.3) to[out=180,in=90] (-1,0) to[out=270,in=180] (0,-0.3);
		\end{scope}
		\draw[->] (3,0) -- (4.7,0);
		\node[above] at (-0.1,0.3) {${\cal A}$};
		\node[above] at (0.6,0.3) {${\cal A}$};
		\node[above] at (2.5,0.3) {${\cal A}'$};
		\node[above] at (6,0.3) {${\cal A}'$};
		\node[below] at (1.5,-0.3) {$\mathfrak{J}_\wp$};
		\node[below] at (0.25,-0.3) {$p_1$};
		\node[below] at (2.5,-0.3) {$p_2$};
		\node[below] at (6,-0.3) {$p_2$};
		\end{tikzpicture}
	\end{array}\\
	Z[p_2,{\cal A}']=\sum\limits_{{\cal A}'}Z_{\rm intf}[\mathfrak{J}_{\wp}|{\cal A},{\cal A}']Z[p_1,{\cal A}].
\end{split}
 \end{equation}
The role of the interface partition function -- Witten index -- $Z_{\rm intf}$ is a linear map between fibers of the partition function bundle associated to the parallel transport along $\wp$:\footnote{A tentative reader, especially familiar with the phenomenon of wall-crossing \cite{Kontsevich:2008fj,Aganagic:2009kf,Gaiotto:2008cd} may argue that dimensions entering this expression are only piece-wise constant functions of parameters ${\cal P}$, therefore the parallel transport will not be a parallel transport on a differentiable bundle governed by some connection.
This is not the case.
Indeed we are able to identify the categories and calculate some dimensions of the BPS Hilbert spaces by localization only in the IR limit $\hbar\to 0$, for partition functions this limit corresponds to a semi-classical or WKB limit, or a calculation of their asymptotic behavior.
Asymptotic develops natural discontinuities on certain loci of the parameter space even when the very function at all finite values of the parameters is continuous, this phenomenon is known as a Stokes phenomenon.}
\bar{e}
Z_{\rm intf}[\mathfrak{J}_{\wp}|{\cal A},{\cal B}]={\rm Tr}_{\mathfrak{G}_{\rm BPS}}(-1)^{{\cal F}}=\sum\limits_{{\cal F}}(-1)^{{\cal F}}{\rm dim}\,\mathfrak{G}_{\rm BPS}^{\left({\cal F},\ldots\right)}\left(\mathfrak{J}_{\wp}|{\cal A},{\cal B}\right)=\sum\limits_{{\cal F}}(-1)^{{\cal F}}{\rm dim}\,{\rm Hom}_{\mathfrak{D}_{p_2}}^{{\cal F},\ldots}\left(\beta_{\wp}({\cal A}),{\cal B}\right).
 \end{equation}
Therefore we call the functor $\beta_{\wp}$ \emph{categorified parallel transport} functor. Obviously, the only partition function does not have enough information to calculate this functor, this is why we are aiming to exploit QFT machinery to calculate explicitly $\mathfrak{G}_{\rm BPS}$ and to apply this information to a restoration of the $\beta_{\wp}$ action on categories of D-branes.

We should stress that examples of calculations of such categorified parallel transport functors or similar quantities are known in the literature \cite{HHP,GMW,Khan:2020hir,Galakhov:2017pod,Kerr:2017usa,Clingempeel:2018iub,Brunner:2020miu,Brunner:2021cga,Chen:2020iyo,Brunner:2009zt,Brunner:2008fa,Aspinwall:2004jr}.
Usually approaches to this subject tend to focus mostly on either Fukaya-Seidel categories and LG models or derived categories of coherent sheaves and gauged linear $\sigma$-models.
In this note we will make an attempt to synthesize the best options from the two worlds.
In particular, we will apply techniques of the algebra of the infrared \cite{GMW} to calculate \emph{Fourier-Mukai kernel} for associated categorified parallel transport functor.

\subsection{Atiyah flop and hypergeometric series}\label{sec:Atiyah_intro}
We would like to concentrate on a simple and yet non-trivial model of conifold transition.
Consider a $U(1)$ gauged sigma model with $n_f$ chirals.
We assume that chirals have charges $Q_i$ under this $U(1)$ and masses $\mu_i$ where the index runs over a set $i=1,\ldots,n_f$.
For the Higgs branch disk partition function the localization calculation produces the following result:
\bar{e}\label{Z_B}
Z[t,{\cal E}]=\int\limits_{-{\rm i}\infty}^{{\rm i}\infty}d\sigma \; f_{{\cal E}}\left(e^{2\pi{\rm i} \sigma}\right)\; e^{t\sigma}\prod\limits_{i=1}^{N_f}\Gamma(Q_i\sigma+\mu_i),
 \end{equation}
where
$$
t=r-{\rm i}\theta
$$
is a complexified combination of Fayet-Illiopolous parameter $r$ and theta-angle $\theta$, and $f_{{\cal E}}(z)$ is a polynomial in $z$ defining so called D-brane data of a coherent sheaf ${\cal E}\in\mathfrak{D}_{p}$. This expression is a well-known Barnes representation for generalized hypergeometric series (see Appendix \ref{sec:hypergeom}):
$$
{}_pF_q\left[\begin{array}{c}
a_1,\ldots,a_p\\
b_1,\ldots,b_q
\end{array}\right](z),\quad \mbox{where}\; p=\sum\limits_{i:\;Q_i>0}|Q_i|,\;q=\sum\limits_{i:\;Q_i<0}|Q_i|-1,
$$
where $z=e^{2\pi t}$ and parameters $a_i$ and $b_i$ are linearly related to masses
$\mu_i$. In other words partition function \eqref{Z_B} is annihilated by a differential operator:
\bar{e}
D_{p,q}\left(z,\frac{d}{dz}\right)=z\prod\limits_{i=1}^p\left(z\frac{d}{dz}+a_i\right)-z\frac{d}{dz}\prod\limits_{i=1}^q\left(z\frac{d}{dz}+b_i-1\right)
 \end{equation}
This differential operator has a mathematical interpretation of a quantum connection in Gromov-Witten theory \cite{MaulikOkounkov}, and the physical interpretation is an induced Berry connection on the parameter space.
The parallel transport induced by the supersymmetric interfaces is compatible with this connection.

The simplest non-trivial case is the ordinary hypergeometric series ${}_2F_1$.
The solutions of the hypergeometric equations are known to develop singularities in three marked points $0$, $1$ and $\infty$ on ${\cal P}$ that is a Riemann sphere in this case parameterized by $z$.

Corresponding theory is $U(1)$ gauged sigma-model with $n_f=4$ chiral multiplets with charges:
$$
(+1,+1,-1,-1).
$$
This model describes a conifold resolution. The conifold is a singular hypersurface in $\mathbb{C}^4$ defined by an algebraic equation as a set of degenerate 2 by 2 $\mathbb{C}$-valued matrices:
\bar{e}
S:=\left(\begin{array}{cc}
M_{11} & M_{12}\\
M_{21} & M_{22}
\end{array}\right),\quad {\rm det}\;S=M_{11}M_{22}-M_{12}M_{21}=0
 \end{equation}
To resolve it we rewrite $\mathbb{C}^4$ coordinates $X_i$ in terms of chiral fields:
\bar{e}
M_{11}=\phi_2\phi_4,\quad M_{12}=-\phi_1\phi_4,\quad M_{21}=-\phi_2\phi_3,\quad M_{22}=\phi_1\phi_3.
 \end{equation}
This assignment is invariant under the action of the complexified gauge group $\mathbb{C}^{\times}$:
$$
(\phi_1,\phi_2,\phi_3,\phi_4)\to \left(\lambda \phi_1,\lambda \phi_2,\lambda^{-1} \phi_3,\lambda^{-1} \phi_4\right).
$$
Consider an obvious consequence of this identification:
\bar{e}\label{res_+}
\left(\begin{array}{cc}
	M_{11} & M_{12}\\
	M_{21} & M_{22}
\end{array}\right)\left(\begin{array}{c}
\phi_1\\ \phi_2
\end{array}\right)=0.
 \end{equation}
If $|\phi_1|^2+|\phi_2|^2>0$ then a pair $(\phi_1:\phi_2)$ modulo the action of $\mathbb{C}^\times$ can be associated with a projective coordinate of $\mathbb{C}\mathbb{P}^1$.
In this case \eqref{res_+} represents a canonical \emph{blowup} resolution of the conic singularity \cite{Hartshorne}.
The D-term constraint in this theory reads:
\bar{e}
|\phi_1|^2+|\phi_2|^2-|\phi_3|^2-|\phi_4|^2=r.
 \end{equation}
In the case $r>0$ the moduli space is exactly confined to a locus $|\phi_1|^2+|\phi_2|^2>0$, we call this blowup resolution $X_+$ so that:
\bar{e}
X_{\rm Higgs}(r>0)=:X_+.
 \end{equation}

Another possible blowup resolution of the conic singularity is as follows:
\bar{e}\label{res_-}
\left(\begin{array}{cc}
\phi_3 & \phi_4
\end{array}\right)\left(\begin{array}{cc}
	M_{11} & M_{12}\\
	M_{21} & M_{22}
\end{array}\right)=0,\quad (\phi_3:\phi_4)\in \mathbb{C}\mathbb{P}^1
 \end{equation}
This resolution corresponds to the region $r<0$ and we call this variety $X_-$ correspondingly:
\bar{e}
X_{\rm Higgs}(r<0)=:X_-
 \end{equation}

The most intriguing situation is $r=0$, in this case the conifold remains unresolved.
However from the physical point of view as we discussed in the previous section it is not correct to talk about the Higgs branch in $\hbar$-neighborhood of point $r=0$, in this area the theory is in its Coulomb phase.

The transition between $X_\pm$ is known in the literature as a conifold transition or Atiyah flop \cite{Flop}.
Let us consider corresponding path $\wp$ connecting regions $r<0$ and $r>0$ in the parameter space and performing this transition (see Fig.\ref{fig:wp}(a)).
Naively a transition between Higgs and Coulomb phase descriptions of the theory occurring at $r=0$ makes this problem more complex.
However the disk partition function given by the hypergeometric series ${}_2F_1$ is a smooth function of $e^{2\pi t}$, therefore we do not expect any problems with parallel transport of the D-brane category itself.
On the Riemann sphere parameterized by a complex variable $z=e^{2\pi t}$ the locus $r=0$ corresponds to a unit circle (see Fig.\ref{fig:wp}(c)).
And parallel transport along path $\wp$ is a well-known and well-studied problem in the theory of hypergeometric series known as \emph{analytic continuation} of the hypergeometric series outside the unit circle.
As long as this path does not hit the singularity located at $e^{2\pi t}=1$ the parallel transport is smooth.
\begin{figure}[h!]
	\begin{center}
		\begin{tikzpicture}
		\begin{scope}[shift={(-2,-0.5)}]
		\draw[->] (0,0) -- (4,0);
		\node[right] at (4,0) {$r$};
		\draw[dashed] (1.5,0) -- (1.5,1);
		\draw[dashed] (2.5,0) -- (2.5,1);
		\draw[ultra thick, red] (2,-0.1) -- (2,0.1);
		\node[below] at (2,-0.1) {$0$};
		\node[below] at (1.5,-0.1) {$-\hbar$};
		\node[below] at (2.5,-0.1) {$\hbar$};
		\begin{scope}[shift={(2,0.5)}]
		\draw[thick] (-0.4,-0.4) -- (0.4,0.4) (0.4,-0.4) -- (-0.4,0.4);
		\end{scope}
		\begin{scope}[shift={(0.5,0.5)}]
		\draw[dashed] (-0.4,-0.4) -- (0.4,0.4) (0.4,-0.4) -- (-0.4,0.4);
		\draw[thick] (-0.4,0.4) to[out=315,in=180]  (0,0.15) to[out=0,in=225] (0.4,0.4);
		\draw[thick] (-0.4,-0.4) to[out=45,in=180]  (0,-0.15) to[out=0,in=135] (0.4,-0.4);
		\end{scope}
		\begin{scope}[shift={(3.5,0.5)}]
		\begin{scope}[rotate=90]
		\draw[dashed] (-0.4,-0.4) -- (0.4,0.4) (0.4,-0.4) -- (-0.4,0.4);
		\draw[thick] (-0.4,0.4) to[out=315,in=180]  (0,0.15) to[out=0,in=225] (0.4,0.4);
		\draw[thick] (-0.4,-0.4) to[out=45,in=180]  (0,-0.15) to[out=0,in=135] (0.4,-0.4);
		\end{scope}
		\end{scope}
		\node[above] at (0.5,1) {$X_-$};
		\node[above] at (3.5,1) {$X_+$};
		\node[above] at (2,1) {???};
		\end{scope}
		%
		\begin{scope}[shift={(3.5,-0.5)}]
		\draw[<->] (0,2) -- (0,0) -- (4,0);
		\node[right] at (4,0) {$r$};
		\node[above] at (0,2) {$\theta$};
		\draw[dashed, thick, red] (2,-0.1) -- (2,2.3);
		\begin{scope}[shift={(2,0)}]
		\draw[ultra thick] (-0.1,-0.1) -- (0.1,0.1) (0.1,-0.1) -- (-0.1,0.1);
		\end{scope}
		\begin{scope}[shift={(2,1)}]
		\draw[ultra thick] (-0.1,-0.1) -- (0.1,0.1) (0.1,-0.1) -- (-0.1,0.1);
		\end{scope}
		\begin{scope}[shift={(2,2)}]
		\draw[ultra thick] (-0.1,-0.1) -- (0.1,0.1) (0.1,-0.1) -- (-0.1,0.1);
		\end{scope}
		\node[below] at (0.5,-0.1) {$-\infty$};
		\node[below] at (2,-0.1) {$0$};
		\node[below] at (3.5,-0.1) {$+\infty$};
		\draw[->,thick,blue] (0.5,0.5) to[out=0,in=180] (2,1.5);
		\draw[thick,blue] (2,1.5) to[out=0,in=180] (3.5,0.5);
		\node[above,blue] at (3.5,0.5) {$\wp$};
		\draw (2,1) to[out=0,in=90] (2.2,0.75) to[out=270,in=180] (2.4,0.5) to[out=180,in=90] (2.2,0.25) to[out=270,in=0] (2,0);
		\node[right] at (2.4,0.5) {$2\pi$};
		\end{scope}
		%
		\begin{scope}[shift={(11,0.25)}]
		\begin{scope}[yscale=0.5]
		\draw(0,0) circle (1.5);
		\end{scope}
		\begin{scope}[xscale=0.5]
		\draw[thick, red, dashed] (0,0) circle (0.75);
		\draw[ultra thick, red] ([shift=(90:0.75)]0,0) arc (90:270:0.75);
		\end{scope}
		\draw[fill=black] (-1.4,0) circle (0.06) (1.4,0) circle (0.06) (-0.35,-0.3) circle (0.06);
		\node[left] at (-1.5,0) {$0$};
		\node[right] at (1.5,0) {$\infty$};
		\node[left] at (-0.35,-0.3) {$1$};
		\draw[->,blue,thick] (-1.3,0) to[out=0,in=180] (0,0.5);
		\draw[blue, thick] (0,0.5) to[out=0,in=180] (1.3,0);
		\node[blue] at (1.1,-0.2) {$\wp$};
		\draw (-2,1) -- (-1.5,1) -- (-1.5,1.5);
		\node[above left] at (-1.5,1) {$e^{2\pi t}$};
		\end{scope}
		\node at (0,-1.5) {(a)};
		\node at (5.5,-1.5) {(b)};
		\node at (11,-1.5) {(c)};
		\end{tikzpicture}
	\caption{Different representations of the path $\wp$ on the parameter space ${\cal P}$.}\label{fig:wp}
		\end{center}
\end{figure}

The corresponding parallel transport functor $\beta_{\wp}$ can be calculated by alternative means \cite{HHP} using a brane grade restriction rule. The result coincides with the mathematical description of the conifold transition functor \cite{Kapranov:2014uwa}:
\bar{e}
\beta_{\wp}:\quad D^b{\rm Coh}(X_-)\longrightarrow D^b{\rm Coh}(X_+)
 \end{equation}
This transition is given by Fourier-Mukai transform $\Phi_{{\cal K}}$ (see Appendix \ref{sec:App_FM} for details on this definition) with the kernel:
\bar{e}
{\cal K}={\cal O}_{\{S_{(-)}=S_{(+)}\}}
 \end{equation}
We will calculate this kernel and Fourier-Mukai transform explicitly using the algebra of the infrared \cite{GMW}.

\subsection{Braid group categorification, affine Grassmannians, crystal melting/solidifying}

A natural way to generalize the framework of analytic continuation of hypergeometric series and Atiyah flops discussed in the previous section is to consider applications to a categorification of the braid group.
Naturalness is dictated by observations that simple objects related to the braid group action like Drinfeld associator \cite{DuninBarkowski:2012na} and simple conformal blocks with a degenerate at level 2 vertex operator insertion in 2d conformal field theories \cite{Belavin:1984vu} are described by hypergeometric series.

\bigskip

To physicists the braid group is mostly known due to appearance in problems related to anyon statistics, FQHE, braiding vertex operators in 2d conformal field theory \cite{DiFrancesco:639405} as well as an impact on knotted Wilson line observables in 3d Chern-Simons theory \cite{Witten:1988hf} and its mathematical counterpart -- knot invariants and braided modular categories of representations of quantum groups \cite{Kirillov:191317,drinfeld1989quasi}.

\bigskip

A categorification of Chern-Simons link invariants \cite{Khovanov:1999qla,KR} opened a vast amount of opportunities for applications in both string theories \cite{Ooguri:1999bv,Gukov:2004hz,Gukov:2007tf,Haydys:2010dv,WItten:2011pz,Witten:2011zz,Anokhina:2017iui,Anokhina:2021lbh,Dolotin:2013osa,Arthamonov:2017oxw,2007arXiv0708.2228M} and pure mathematics \cite{2005math......8510R,2016arXiv160807308G,2019arXiv190208281G,2019arXiv190506511O,2007arXiv0710.4300O,Abouzaid:2017clg} as well as their profound synthesis to discover underlying structures in relations between physics and geometry.
Unfortunately, we are unable to cover literature for this popular topic even partially and indicate just few sources the reader could use to find a particular subject interesting for a concrete application.

Some approaches \cite{2008arXiv0803.4121K,2008arXiv0812.5023R,Seidel:2000ia,gorodentsev2004helix,2011arXiv1104.0352C,2012arXiv1212.6076L,CautisKamnitzerI,seidel2001braid,2013arXiv1306.6242M,2015arXiv150206011M,2014arXiv1409.4461W} consider as an intermediate step a categorification of braid group representations induced by quantum groups.

A geometric way to categorify braid group representations is to consider coherent sheaves on algebraic varieties with permutable elements, then to identify permutations with functor actions.
A physical setup of \cite{Gaiotto:2015zna,GMW} proposes to consider a braid as a defect operator in a 5d supersymmetric theory.
A consideration of the IR dynamics in this theory translates the problem to a language of a categorified Berry connection in a certain Landau-Ginzburg model.
A BPS Hilbert space for the corresponding interface will deliver the desired categorification \cite{Galakhov:2016cji}.
As it is explained in \cite{Aganagic:2020olg} the most suitable dual counterpart incorporating categories of coherent sheaves on algebraic varieties is a field theoretic description of affine Grassmannians related to moduli spaces of monopole-like solutions of \cite{Gaiotto:2015zna}.

\bigskip

We will consider a parallel transport in moduli spaces of cotangent bundles to flag varieties representing specific slices in affine Grassmannians.
Physically this parallel transport is represented by a supersymmetric interface defect $\mathfrak{J}_{\wp}$ in a 2d ${\cal N}=(2,2)$ gauged linear sigma-model with a target space given by a Nakajima quiver variety \cite{nakajima1994instantons,nakajima1998quiver,nakajima2001quiver,2009arXiv0905.0686G}.
The transport path $\wp$ represents a tangle in the parameter space spanned by Fayet-Illiopolous parameters complexified with the help of topological angles.
Boundary conditions on interval ends are defined by coherent sheaves ${\cal A}$ and ${\cal B}$.
Hilbert space $\mathfrak{G}_{\rm BPS}(\mathfrak{J}_{\wp}|{\cal A},{\cal B})$ is a categorification of the parallel transport along $\wp$ induced by a Berry connection  and categorifies a braid group representation associated to $\wp$.

Using our interpretation of cHC duality we will describe the Berry parallel transport of brane boundary conditions associated to simple braids as a Fourier-Mukai transform on the derived category of coherent sheaves.
Our Fourier-Mukai transform is in complete agreement with mathematical constructions of \cite{CautisKamnitzerI}.

\bigskip

We should stress that this phenomenon finds an intuitively natural description in a language of condensed matter physics.
So classical vacua in certain QFTs with quiver target spaces representing D-brane systems on Calabi-Yau 3-folds have a labeling by crystal lattices \cite{Ooguri:2008yb,Okounkov:2003sp,Yamazaki:2010fz,Li:2020rij,Aganagic:2010qr}.
Higgs branch varieties associated to initial and final points of path $\wp$ are isomorphic, so the Berry parallel transport along $\wp$ is in practice a \emph{Berry holonomy} from the crystal phase to itself.
However the very path $\wp$ overlaps inevitably with a parameter space region where the Higgs branch develops a conic singularity.
The crystal melts as the path enters this region to a partially ``liquid" state and solidifies back afterwards as the path exits the critical region.
An impact of this hysteresis on the BPS Hilbert space can be calculated explicitly due to localizing properties of supersymmetry.

\subsection{Outline of the paper}

The paper is organized as follows.

In Section \ref{sec:Preamble} we present a slow pace review of the cHC duality on a cylinder $S^1\times {\mathbb{R}}_t$ quantizing this system using the Hamiltonian evolution along temporal direction ${\mathbb{R}}_t$.
$S^1$ does not have boundaries, there is no need to consider complicated non-trivial boundary conditions except periodic ones.
Notions of order and disorder parameters are introduced.

In Section \ref{sec:Localization} we make all the preliminary work discussing localization to the Higgs, Coulomb and mixed phases.
We describe how the effects of the RG group flow on the BPS Hilbert space may be taken into account exactly at just few perturbation loops and how the notion of interface defects can be taken into consideration.
Eventually, we discuss the low energy dynamics of mentioned phases and give a brief review of a web formalism of \cite{GMW} used for the IR description of solitons and instantons in 2d ${\cal N}=(2,2)$ theories with massive vacua.

In Section \ref{sec:equiv_CP_1} we consider the IR dynamics in a GLSM describing equivariant $\mathbb{C}\mathbb{P}^1$.
This model is an elementary building block for more complicated models.
We compare the equivariant Higgs and Coulomb branch soliton spectra and find their agreement.
Some scattering soliton vertices contributing to instanton transitions are calculated.

In Section \ref{sec:cat_contin_hyper} we construct a categorification of the analytic continuation for hypergeometric series and calculate an associated Fourier-Mukai kernel.

In Section \ref{sec:quiver} we consider a categorified braid group action on Nakajima quiver varieties and cotangent bundles to flag varieties related by Maffei's isomorphism.
Associated Fourier-Mukai kernels are calculated.
An accompanying  physical intuition of melting and solidifying crystals during Berry parallel transport is discussed.

In Section \ref{sec:future} we discuss open problems and possible future directions.

\section{Preamble: a simple version of Higgs-Coulomb duality on a cylinder}\label{sec:Preamble}

\subsection{Order parameter}

We would like to start with a very simple setup of a single uncharged complex field with a complex mass $\mu=\mu_{{\mathbb{R}}}+{\rm i}\,\mu_{\mathbb{I}}$.
Rather than considering this theory on a strip world-sheet we put it on a cylinder. For this we just put periodic boundary conditions for fields:
\bar{e}\label{periodic}
\phi(x^1+L)=\phi(x^1).
 \end{equation}
There is no need to consider D-brane boundary conditions and D-brane category in this case.
The BPS Hilbert space is a vector space spanned by wave functions satisfying the BPS constraint:
\bar{e}
{\cal Q}_B|\Psi_{\rm BPS}\rangle=\bar {\cal Q}_B|\Psi_{\rm BPS}\rangle=0.
 \end{equation}
The supercharge $\bar {\cal Q}_B$ in this simple model reads (see Appendix \ref{sec:App_SUSY}):
\bar{e}\label{sc}
\begin{split}
	\bar {\cal Q}_B=\int\limits_0^L dx^1\Big[\bar\psi_1\left(-{\rm i}\delta_{\bar\phi}-{\rm i} \mu_{{\mathbb{R}}}\phi \right)+\bar\psi_2(-\partial_{1}\phi+\mu_{\mathbb{I}}\phi)\Big].
\end{split}
 \end{equation}

Using periodicity of the field functions \eqref{periodic} it is natural to decompose bosonic and fermionic fields over normalized Fourier modes:
\bar{e}
\phi(x^1)=\sum\limits_{n=-\infty}^{\infty}\phi_n\frac{ e^{{\rm i} \kappa_n x^1}}{\sqrt{L}},\quad \psi_a(x^1)=\sum\limits_{n=-\infty}^{\infty}\psi_{a,n}\frac{ e^{{\rm i} \kappa_n x^1}}{\sqrt{L}},\quad \kappa_n=\frac{2\pi n}{L}.
 \end{equation}
The variation operators are decomposed in a similar way:
\bar{e}
\frac{\delta}{\delta \phi(x^1)}=\sum\limits_{n=-\infty}^{\infty}\frac{ e^{-{\rm i} \kappa_n x^1}}{\sqrt{L}}\partial_{\phi_n},\quad \frac{\delta}{\delta \bar\phi(x^1)}=\sum\limits_{n=-\infty}^{\infty}\frac{ e^{{\rm i} \kappa_n x^1}}{\sqrt{L}}\partial_{\bar\phi_n}.
 \end{equation}
The supercharge expression in terms of normalized Fourier modes has the following form:
\bar{e}
\bar {\cal Q}_B=\sum\limits_{n=-\infty}^{\infty}\Big[\bar\psi_{1,n}\left(-{\rm i}\partial_{\bar\phi_n}-{\rm i}\mu_{{\mathbb{R}}}\phi_n\right)+\bar\psi_{2,n}\left(-{\rm i} \kappa_n+\mu_{\mathbb{I}}\right)\phi_n\Big].
 \end{equation}

There are two fundamentally different cases $\mu_{\mathbb{I}}\neq 0$ and $\mu_{\mathbb{I}}=0$.
In the case $\mu_{\mathbb{I}}\neq 0$ there is just a single BPS state, and its normalized wave function reads:\footnote{We assume that the fermion vacuum $|0\rangle$ is annihilated by all fermion annihilation operators $\psi$, and $\bar\psi$ create new states.}
\bar{e}\label{wave1}
|\Psi_{\rm BPS}\rangle=\prod\limits_{n=-\infty}^{\infty}e^{-\sqrt{\kappa_n^2+|\mu|^2}|\phi_n|^2}\frac{\left(\mu_R-\sqrt{\kappa_n^2+|\mu|^2}\right)\bar{\psi}_{1,n}+(\kappa_n+{\rm i}\mu_I)\bar{\psi}_{2,n}}{\sqrt{2\pi(\sqrt{\kappa_n^2+|\mu|^2}-\mu_R)}}|0\rangle.
 \end{equation}

In the case $\mu_{\mathbb{I}}=0$ the ground state is infinitely degenerate, and the zero mode can create a condensate.
The corresponding BPS wave-function expression admits a choice of a pair of arbitrary holomorphic functions $g_b$ and $g_f$:
\bar{e}\label{wave2}
\begin{split}
	|\Psi_{\rm BPS}\rangle=\left\{\begin{array}{ll}
		g_b(\phi_0)+g_f(\phi_0)\bar\psi_{2,0}, & {\rm if}\;\mu_{{\mathbb{R}}}>0 \\
		(g_b(\bar\phi_0)+g_f(\bar\phi_0)\bar\psi_{2,0})\bar\psi_{1,0}, & {\rm if}\;\mu_{{\mathbb{R}}}<0 \\
	\end{array} \right\}e^{-|\mu_{{\mathbb{R}}}||\phi_0|^2}\times\\
	\times\prod\limits_{n\neq 0}e^{-\sqrt{\kappa_n^2+\mu_{{\mathbb{R}}}^2}|\phi_n|^2}\frac{\left(\mu_{{\mathbb{R}}}-\sqrt{\kappa_n^2+\mu_{{\mathbb{R}}}^2}\right)\bar{\psi}_{1,n}+\kappa_n\bar{\psi}_{2,n}}{\sqrt{2\pi(\sqrt{\kappa_n^2+\mu_{{\mathbb{R}}}^2}-\mu_{{\mathbb{R}}})}}|0\rangle.
\end{split}
 \end{equation}

The theory of a complex scalar has a global $U(1)$ symmetry.
The single BPS state \eqref{wave1} at $\mu_{\mathbb{I}}\neq 0$ is preserved by $U(1)$ rotations of field modes.
On the other hand $U(1)$-rotations of $\phi_0$ in \eqref{wave2} vary functions $g_b$ and $g_f$ and are not symmetries of this state.
Therefore in this theory we distinguish two phases: the phase with \emph{unbroken} global $U(1)$ symmetry corresponding to $\mu_{\mathbb{I}}\neq 0$, the phase with \emph{broken} global $U(1)$ symmetry corresponding to $\mu_{\mathbb{I}}= 0$.

Let us consider the expectation value of the scalar field:
\bar{e}
\langle\phi\rangle:=\langle\Psi_{\rm BPS}|\phi|\Psi_{\rm BPS}\rangle.
 \end{equation}

We easily calculate this expectation value in both phases:
\bar{e}
\begin{array}{lll}
\mbox{unbroken:}& \mu_{\mathbb{I}}\neq 0, & \langle\phi\rangle=0;\\
\mbox{broken:}& \mu_{\mathbb{I}}= 0, & \langle\phi\rangle=\frac{\int d^2\phi\; \left(|g_b|^2+|g_f|^2\right)\;\phi\; e^{-|\mu_{{\mathbb{R}}}||\phi|^2}}{\int d^2\phi\; \left(|g_b|^2+|g_f|^2\right)\; e^{-|\mu_{{\mathbb{R}}}||\phi|^2}}.
\end{array}
 \end{equation}

As we see in this example operator $\phi$ can be used as a measure of symmetry breaking in this theory. Therefore we identify it with an \emph{order parameter} of this theory.

Let us consider the following two operators -- fermion number and electric charge respectively:
\bar{e}
{\cal F}=\int\limits_0^L dx^1\left[\bar\psi_+\psi_++\bar\psi_-\psi_-\right],\quad
Q=\int\limits_0^L dx^1\left[\left(\bar\phi\delta_{\bar\phi}-\phi\delta_{\phi}\right)+\bar\psi_+\psi_++\bar\psi_-\psi_-\right].\label{el_ch}
 \end{equation}
The BPS Hilbert space is stratified under the action of this operators into one-dimensional subspaces:
\bar{e}
\mathfrak{G}_{\rm BPS}=\bigoplus\limits_{{\cal F},Q}\mathbb{C}\,|{\cal F},Q\rangle.
 \end{equation}
Using this identification we can summarize the result of this section in describing the BPS Hilbert spaces of the theory of the order parameter in the following way:
\bar{e}
\begin{split}
	\mathfrak{G}_{\rm BPS}^{({\rm ord})}\left(\mu_{\mathbb{I}}\neq 0\right)=\mathbb{C}|0,0\rangle,\\
	\mathfrak{G}_{\rm BPS}^{({\rm ord})}\left(\mu_{\mathbb{I}}= 0,\mu_{{\mathbb{R}}}>0\right)=\bigoplus\limits_{n=0}^{\infty}\left(\mathbb{C}|0,-n\rangle\oplus \mathbb{C}|-1,-n-1\rangle\right),\\
	\mathfrak{G}_{\rm BPS}^{({\rm ord})}\left(\mu_{\mathbb{I}}= 0,\mu_{{\mathbb{R}}}<0\right)=\bigoplus\limits_{n=0}^{\infty}\left(\mathbb{C}|0,n\rangle\oplus \mathbb{C}|1,n+1\rangle\right).
\end{split}
 \end{equation}

\subsection{Disorder parameter}
Now we would like to turn to a dual description.
We could call an operator dual to the order operator a \emph{disorder} operator in analogy with the Ising model \cite{Polyakov:1987ez}.
Physical role of the disorder operator is to insert a vortex defect into the theory so that the phase of the order parameter winds around the vortex core, similarly we could have called it a \emph{vortex operator} (compare also to 3d monopole operator definition in \cite{Borokhov:2002ib}).

A duality in this case is a mere Fourier transform exchanging the field phase and the field winding number operator -- T-duality.

Let us extract explicitly the phase contribution into quantum fields:
\bar{e}
\phi=e^{\rho+{\rm i}\vartheta},\quad \psi_a=e^{{\rm i}\vartheta}\chi_a.
 \end{equation}
In these new variables the supercharge expression \eqref{sc} can be rewritten as:
\bar{e}\label{bar_Q_B}
\bar{\cal Q}_B=\int\limits_0^L dx^1\left[-{\rm i}\bar{\chi}_1\left(\frac{e^{-\rho}}{2}\left(\delta_{\rho}+{\rm i}\delta_{\vartheta}\right)+\mu_{{\mathbb{R}}} e^{\rho}\right)+V_3+\bar{\chi}_2e^{\rho}\left(-\partial_{1}\rho-{\rm i}\partial_{1}\vartheta+\mu_{\mathbb{I}}\right) \right],
 \end{equation}
where
\bar{e}\label{VV_3}
V_3=\frac{{\rm i}}{2} e^{-\rho}\bar{\chi}_1\bar{\chi}_2\chi_2
 \end{equation}
is a new cubic fermion interaction term produced by quadratic fermion term in $\delta_{\vartheta}$.

To pass to the disorder operator description we dualize phase operator $\vartheta$ in terms of a new field ${\mathbb{S}}$.
We will describe properties of the field ${\mathbb{S}}$ momentarily, first review the properties of phase field $\vartheta$.

First notice a large global phase shift:
\bar{e}\label{global}
\vartheta(x^1)\longrightarrow \vartheta(x^1)+2\pi n,\quad n\in{\mathbb{Z}},
 \end{equation}
is a shift symmetry of the theory.

Then notice as well that rather having purely periodic boundary conditions the phase field has a shifting twist:
\bar{e}\label{twisted}
\vartheta(x^1+L)=\vartheta(x^1)+2\pi k,\quad k\in{\mathbb{Z}}.
 \end{equation}

Function $\partial_1\vartheta(x^1)$ is a periodic function on $[0,L]$, therefore we can decompose it over normalized Fourier modes:
\bar{e}
\partial_1\vartheta(x^1)=\frac{\alpha_0}{\sqrt{L}}+\sum\limits_{n=1}^{\infty}\frac{\alpha_n}{\sqrt{2L}}\cos\kappa_n x^1+\sum\limits_{n=1}^{\infty}\frac{\beta_n}{\sqrt{2L}}\sin\kappa_n x^1.
 \end{equation}
Integrating both sides of this equality we find a decomposition for the field $\vartheta(x^1)$ over modes:
\bar{e}\label{vartheta}
\vartheta(x^1)=\vartheta_0+\frac{\alpha_0}{\sqrt{L}}x^1+\sum\limits_{n=1}^{\infty}\frac{\alpha_n}{\kappa_n}\frac{1}{\sqrt{2L}}\sin\kappa_n x^1-\sum\limits_{n=1}^{\infty}\frac{\beta_n}{\kappa_n}\frac{1}{\sqrt{2L}}\cos\kappa_n x^1.
 \end{equation}
Global symmetry \eqref{global} and twisted periodic boundary conditions \eqref{twisted} impose a periodicity constraint on physical wave functions of this theory:
\bar{e}
\Psi(\vartheta_0+2\pi)=\Psi(\vartheta_0)
 \end{equation}
As well an operator in front of the $x^1$-linear term in the expansion \eqref{vartheta} acquires only discrete eigenvalues, so we can consider an eigen basis of this operator:
\bar{e}
\frac{\alpha_0}{\sqrt{L}}\Psi_k=\frac{2\pi k}{L}\Psi_k,\quad k\in{\mathbb{Z}}.
 \end{equation}
Due to this discreteness the operator $\alpha_0$ does not contribute to variations of the field $\vartheta$, so for the variation we have the following expansion:
\bar{e}
\frac{\delta}{\delta\vartheta(x^1)}=\frac{1}{L}\frac{\partial}{\partial\vartheta_0}+\sum\limits_{n=1}^{\infty}\kappa_n\frac{1}{\sqrt{2L}}\sin\kappa_nx^1\frac{\partial}{\partial\alpha_n}-\sum\limits_{n=1}^{\infty}\kappa_n\frac{1}{\sqrt{2L}}\cos\kappa_nx^1\frac{\partial}{\partial\beta_n}.
 \end{equation}
Let us consider a Fourier transform of the wave function:
\bar{e}\label{Fourier}
\hat\Phi\left[\Psi\right]_m({\mathbb{S}}_0,\gamma_i,\eta_i):=\int\limits_0^{2\pi}e^{-{\rm i} n\vartheta_0}d\vartheta_0\sum\limits_{k=-\infty}^{+\infty}e^{2\pi i k{\mathbb{S}}_0}\prod\limits_{a=1}^{\infty}\int\limits_{-\infty}^{+\infty}d\alpha_ad\beta_a\; e^{-{\rm i}\frac{\alpha_a\eta_a+\beta_a\gamma_a}{\kappa_a}}\Psi_k(\vartheta_0,\alpha_i,\beta_i).
 \end{equation}
For operators we have:
\bar{e}\label{rel}
\hat\Phi\left[\delta_{\vartheta}\right]=-{\rm i}\,\partial_{1}{\mathbb{S}},\quad \hat\Phi\left[\partial_{1}\vartheta\right]={\rm i}\,\delta_{{\mathbb{S}}},
 \end{equation}
where field ${\mathbb{S}}$ has the following expansion:
\bar{e}
{\mathbb{S}}(x^1)={\mathbb{S}}_0+\frac{ m}{L}x^1+\sum\limits_{n=1}^{\infty}\frac{\gamma_n}{\kappa_n}\frac{1}{\sqrt{2L}}\sin\kappa_n x^1-\sum\limits_{n=1}^{\infty}\frac{\eta_n}{\kappa_n}\frac{1}{\sqrt{2L}}\cos\kappa_n x^1.
 \end{equation}
Also notice that the resulting wave-function is a periodic function of ${\mathbb{S}}_0$ with an integer period. Summarizing these observations we conclude that field ${\mathbb{S}}$ has twisted periodic boundary conditions:
\bar{e}\label{twisted_S}
{\mathbb{S}}(x^1+L)={\mathbb{S}}(x^1)+m,\quad m\in {\mathbb{Z}},
 \end{equation}
as well as an overall global shift
\bar{e}\label{global_S}
{\mathbb{S}}(x^1)\to{\mathbb{S}}(x^1)+l,\quad l\in{\mathbb{Z}}
 \end{equation}
is a symmetry of the theory.

We combine operators $\rho$ and ${\mathbb{S}}$ into a complex twisted chiral field:
\bar{e}
Y=Y_{{\mathbb{R}}}+{\rm i} Y_{\mathbb{I}}:=e^{2\rho}-{\rm i}\,{\mathbb{S}},
 \end{equation}
we will call a \emph{disorder} operator. It is simple to rewrite the supercharge in new terms:
\bar{e}\label{dual_Q_B}
\hat\Phi\left[\bar {\cal Q}_B\right]=\int\limits_0^L dx^1\left[-2{\rm i}\sqrt{Y_{{\mathbb{R}}}}\bar\chi_+\left(\delta_Y-\frac{{\rm i}}{4Y_{{\mathbb{R}}}}\partial_1\bar Y+\frac{\mu}{2}\right)-2{\rm i}\sqrt{Y_{{\mathbb{R}}}}\bar\chi_-\left(\delta_{\bar Y}+\frac{{\rm i}}{4Y_{{\mathbb{R}}}}\partial_1Y+\frac{\bar \mu}{2}\right)+V_3\right],
 \end{equation}
where the following notations for fermion fields are introduced:
$$
\chi_1=\chi_-+\chi_+,\quad \chi_2=\chi_--\chi_+.
$$

If the triple fermion term $V_3$ is ignored this supercharge is equivalent to a twisted form\footnote{Dynamical descriptions of chiral and twisted chiral field are more or less alike, one suffices to swap fermions $\psi_+$ and $\bar\psi_-$.} of Landau-Ginzburg supercharge \eqref{nlin_sup} for a 1d complex K\"ahler manifold with the following expressions for the K\"ahler potential and an induced superpotential \cite{Hori:2000kt} (for a definition of Landau-Ginzburg model with a K\"ahler target space see Appendix \ref{sec:App_LG}):
\bar{e}\label{Kahler}
\begin{split}
	&K(Y,\bar Y)=-\frac{1}{2}(Y+\bar Y){\rm log}(Y+\bar Y),\\
	&W=\mu Y.
\end{split}
 \end{equation}

Let us observe that in the proposed BPS vacuum the disorder operator $Y$ acquires non-zero vacuum expectation value:
\bar{e}\label{disorder}
\langle Y\rangle:=\Big\langle\Psi_{\rm BPS} \Big||\phi|^2-\frac{\delta}{\delta(\partial_1\vartheta)}\Big|\Psi_{\rm BPS}\Big\rangle=\sum\limits_n \frac{1}{L} \langle\Psi_{n} ||\phi_n|^2-\partial_{\kappa_n}|\Psi_{n}\rangle.
 \end{equation}

It is easy to calculate expectation value \eqref{disorder} in the limit $L\to \infty$. The summation in the r.h.s. of \eqref{disorder} is substituted by an integration\footnote{A rule for substituting a summation by an integration is simply canonical:
$$
\sum\limits_{n=-\infty}^{\infty}\frac{1}{L}\;f\left(\frac{n}{L}\right)\mathop{\longrightarrow}^{L\to \infty}\int\limits_{-\infty}^{\infty}d\nu\;f(\nu)
$$} and diverges. This is a standard divergence of QFT loop calculations and it is needed to be regularized. For regularization we introduce cutoff parameter $\Lambda$:
\bar{e}\label{Y_vev}
\begin{split}
	\langle Y\rangle =\frac{1}{4\pi}\int\limits_{-\Lambda}^{\Lambda}dk\;\frac{1}{\sqrt{k^2+|\mu|^2}}\left(1+\frac{{\rm i}\mu_{\mathbb{I}}}{\sqrt{k^2+|\mu|^2}-\mu_{{\mathbb{R}}}}\right)=\\
	=\frac{1}{2\pi}\left[{\rm log}\frac{2\Lambda}{|\mu|}+{\rm i}\left(\arctan\frac{\Lambda}{\mu_{\mathbb{I}}}+\arctan\frac{\mu_{{\mathbb{R}}}}{\mu_{\mathbb{I}}}\right)\right]+O\left(\frac{|\mu|}{\Lambda}\right)=\\
	=\frac{1}{2\pi}\left({\rm log}\frac{2\Lambda}{\mu}+\pi{\rm i}\right)+O\left(\frac{|\mu|}{\Lambda}\right).
\end{split}
 \end{equation}

The resulting expression is a multivalued function. The logarithm multivaluedness is inherited from physical symmetry \eqref{global}. In principle, field $Y$ is not a physical observable, rather $e^{-2\pi Y}$ is. So we can rewrite an equation defining theory vacua in an invariant form:
\bar{e}\label{defect_vev}
\mu=-2 \Lambda\, e^{-2\pi \langle Y\rangle}.
 \end{equation}
This vacuum equation has obvious ${\mathbb{Z}}$ symmetry $Y\to Y+2\pi {\rm i}$ shifting the sheet of the logarithm cover that is \emph{broken} by the vacuum solution \eqref{Y_vev}, this broken phase is described by a non-zero expectation value of the disorder operator.

We should notice that the operator
$$
e^{-2\pi\, Y(y)}
$$
is a vortex operator inserting a vortex defect singularity in a point $y$ on the world-sheet. It is easy to observe this if one performs a Wick rotation $x^0=-{\rm i}\, x^2$ to make the action Euclidean, then relations \eqref{rel} can be treated semi-classically as the following:
$$
\partial_i{\mathbb{S}}=-{\rm i}\epsilon_{ij}\partial_j\vartheta.
$$
Introduce a ``Dirac string" integration path $\wp(y)$ going from infinity to point $y$. Insertion of $e^{-2\pi \, Y(y)}$ in the Euclidean path integral performs a modification of the action by a new term:
$$
-2\pi {\rm i}\, {\mathbb{S}}= \int\limits_{\wp(y)}2\pi\epsilon_{ij}\partial_i\vartheta\,dx^j.
$$
Such term in the action introduces a boundary condition for phase $\vartheta$ that jumps across the Dirac string $\wp$ by $2\pi$, the vortex defect core located in $y$ is a position of singularity of $\vartheta$:
$$
\begin{array}{c}
\begin{tikzpicture}
\draw[fill=black] (0,0) circle (0.08);
\draw[thick, dashed] (-4,-0.5) to[out=300,in=240] (-3,-1);
\draw[thick] (-3,-1) to[out=60,in=180] (-2,0.5) to[out=0,in=180] (-1,-0.5) to[out=0,in=180] (0,0);
\node[left] at (-4,-0.5) {$\wp(y)$};
\node[right] at (0,0) {$y$};
\draw[->] (-0.2,0) to[out=270,in=180] (0.1,-0.3) to[out=0,in=270] (0.4,0) to[out=90,in=0] (0.1,0.3) to[out=180,in=90] (-0.2,0);
\node[right] at (0.4,0) {$\Delta\vartheta=2\pi$};
\end{tikzpicture}
\end{array}
$$

To complete dualization of the supercharge ${\cal Q}_B$ in \eqref{dual_Q_B} we need to calculate the contribution of the triple fermion operator $V_3$ \eqref{VV_3}. Again we do it up to the first loop order substituting double fermion contributions by their expectation values:
\bar{e}\label{V_3}
V_3\to \frac{{\rm i}}{2\sqrt{Y_R}}\bar\chi_-\left(\langle:\bar\psi_2\psi_2:\rangle-\langle\bar\psi_1\psi_2\rangle \right)+\frac{{\rm i}}{2\sqrt{Y_R}}\bar\chi_+\left(\langle:\bar\psi_2\psi_2:\rangle+\langle\bar\psi_1\psi_2\rangle \right).
 \end{equation}
For the expectation values we have:
\bar{e}
\begin{split}
	\langle:\bar\psi_2\psi_2:\rangle=\frac{\mu_{{\mathbb{R}}}}{4\pi}\int\limits_{-\Lambda}^{\Lambda}\frac{dk}{\sqrt{k^2+|\mu|^2}}=\frac{\mu_{{\mathbb{R}}}}{2\pi}{\rm log}\frac{2\Lambda}{|\mu|}=-Y_{{\mathbb{R}}}\;{\rm Re}\left(2 \Lambda\, e^{-2\pi  Y}\right),\\
	\langle\bar\psi_1\psi_2\rangle=-\frac{{\rm i}\mu_{\mathbb{I}}}{4\pi}\int\limits_{-\Lambda}^{\Lambda}\frac{dk}{\sqrt{k^2+|\mu|^2}}=-\frac{{\rm i}\mu_{\mathbb{I}}}{2\pi}{\rm log}\frac{2\Lambda}{|\mu|}={\rm i} Y_{{\mathbb{R}}}\;{\rm Im}\left(2\Lambda\, e^{-2\pi Y}\right).
\end{split}
 \end{equation}
Thus we see that this term reproduces a shift\footnote{Clearly, \eqref{V_3} gives only a half of that contribution that we expect to get to derive \eqref{sup1}. We assume that this discrepancy can be eliminated by redefining ambiguous cut-off parameter $\Lambda$.} of the superpotential:
\bar{e}\label{sup1}
W=\mu Y-\frac{1}{\pi}\Lambda e^{-2\pi Y}
 \end{equation}
so that its critical points define exactly the vev of the defect field \eqref{defect_vev}.

We can redefine the field $Y$ by an overall shift so that the superpotential has no additional parameters:
\bar{e}
W=\mu Y+e^{-2\pi Y}.
 \end{equation}

As well, notice that for our purpose of soliton counting an explicit K\"ahler metric \eqref{Kahler} is irrelevant since the quantum numbers of solitonic configurations include the superpotential only. Therefore we assume that our Landau-Ginzburg model has a flat target space for simplicity.

Thus we conclude that the Fourier transform of the supercharge $\bar{\cal Q}_B$ is equivalent to the following supercharge operator in the Landau-Ginzburg model:
\bar{e}\label{A-twist}
\begin{split}
	{\cal Q}_A=\int\limits_0^L dx^1\left[-{\rm i}\chi_+\left(\delta_{Y}+{\rm i} \partial_1\bar Y-\left(\mu-e^{-2\pi Y}\right)\right)-{\rm i}\bar\chi_-\left(\delta_{\bar Y}-{\rm i} \partial_1Y-\left(\bar\mu-e^{-2\pi \bar Y}\right)\right)\right].
\end{split}
 \end{equation}

${\cal Q}_A$ in a complete analogy with $\bar{\cal Q}_B$ defines a BPS Hilbert space in the corresponding Landau-Ginzburg model:
$$
{\cal Q}_A|\Psi_{\rm BPS}\rangle= \bar{\cal Q}_A|\Psi_{\rm BPS}\rangle=0,
$$
we could  call a BPS Hilbert space in the theory of disorder parameter.
We expect the following isomorphism:
\bar{e}\label{baby_mirror}
\mathfrak{G}_{\rm BPS}^{({\rm ord})}\cong\mathfrak{G}_{\rm BPS}^{({\rm disord})}.
 \end{equation}
Relation \eqref{baby_mirror} is a baby version of the \emph{Higgs-Coulomb duality} on a cylinder.
This isomorphism is primarily given by the Fourier transform \eqref{Fourier}.
However during our consideration we have made a set of modifications and simplifications to arrive from $\bar Q_B$ to $Q_A$. Therefore we would like to check explicitly if \eqref{baby_mirror} holds.

In the literature \cite{Aspinwall:2004jr} this pair of models is also referred to as A- and B-model
by the type of the supersymmetry twist.

\subsection{Soliton condensate}
To describe the states of an A-model semi-classically it suffices to consider stationary field configurations preserving A-twist \eqref{A-twist}.
Variation operators $\delta_Y$, $\delta_{\bar Y}$ correspond to momenta operators $\partial_0 \bar Y$, $\partial_0 Y$ and are zeroes on stationary field configurations.
The semi-classical BPS field configuration satisfies the following differential equation:
\bar{e}\label{sol1}
{\rm i} \partial_1\bar Y-\left(\mu-e^{-2\pi\, Y}\right)=0,
 \end{equation}
with a twisted periodic boundary conditions for field $Y$:
\bar{e}\label{bc}
Y(L)=Y(0)+{\rm i} k,\quad k\in {\mathbb{Z}}.
 \end{equation}

Using the standard techniques we derive that along the solutions the derivative of the superpotential has a stationary phase:
\bar{e}
\partial_1W={\rm i}\,\left|\mu-e^{-2\pi\, Y}\right|^2.
 \end{equation}

And for a valid solution to \eqref{sol1}, \eqref{bc}  called a \emph{soliton} there is a constraint:
\bar{e}\label{soliton}
-{\rm i}\Delta W\Big|_0^L=\mu\, k\in {\mathbb{R}}_{\geq 0}.
 \end{equation}

A soliton has a well-defined electric charge, indeed, dualizing \eqref{el_ch} we find:
\bar{e}
{\cal Y}=\Delta\, {\rm Im}\,Y\Big|_{0}^L=k.
 \end{equation}
Topological charge $k$ is called a soliton number.

Constraint \eqref{soliton} implies that for generic chiral mass $\mu_{\mathbb{I}}\neq 0$ the only possible topological solution corresponds to $k=0$.
This is a simple solution when the field $Y$ takes a constant expectation value defined by the vacuum equation \eqref{defect_vev}.
On the other hand when $\mu_{\mathbb{I}}=0$ depending on the sign of $\mu_{{\mathbb{R}}}$ equation \eqref{soliton} admits solitonic solutions with either positive or negative topological charge:
$$
\begin{array}{ll}
\mu_{{\mathbb{R}}}>0: & k\in{\mathbb{Z}}_{\geq 0};\\
\mu_{{\mathbb{R}}}<0: & k\in{\mathbb{Z}}_{\leq 0}.
\end{array}
$$
\begin{figure}
	\begin{center}
		\begin{tikzpicture}
		\node at (-7,0) {\includegraphics[scale=0.6]{figs/loop_sol_2.pdf}};
		\node at (0,0) {\includegraphics[scale=0.4]{figs/loop_sol_1.pdf}};
		\begin{scope}[shift={(4,1)}]
		\draw[fill=blue] (-0.1,-0.1) -- (-0.1,0.1) -- (0.1,0.1) -- (0.1,-0.1) -- cycle;
		\node[right] at (0.1,0) {$k=1$};
		\end{scope}
		\begin{scope}[shift={(4,0)}]
		\draw[fill=red] (-0.1,-0.1) -- (-0.1,0.1) -- (0.1,0.1) -- (0.1,-0.1) -- cycle;
		\node[right] at (0.1,0) {$k=2$};
		\end{scope}
		\begin{scope}[shift={(4,-1)}]
		\draw[fill=purple] (-0.1,-0.1) -- (-0.1,0.1) -- (0.1,0.1) -- (0.1,-0.1) -- cycle;
		\node[right] at (0.1,0) {$k=3$};
		\end{scope}
		\node at (0,-2.5) {(b)};
		\node at (-7,-2.5) {(a)};
		\end{tikzpicture}
	\caption{1-, 2- and 3-soliton solutions: a) soliton trajectories on torus ${\cal T}$ with coordinates $(e^{2\pi{\rm i} \frac{x^1}{L}},e^{-2\pi{\rm i} Y_I})$, b) soliton trajectories on complex $e^{-2\pi \, Y}$-plane.\label{fig_solitons}}
	\end{center}
\end{figure}
At this point it is clear that the order operators $\phi$ and $\bar \phi$ that shifted the topological charge of the BPS state and where perturbative in the previous picture may be identified with non-perturbative operators creating solitons and anti-solitons:
$$
\phi=\sum\limits_{k=0}^{\infty}|k+1\rangle\langle k|,\quad \bar\phi=\sum\limits_{k=0}^{\infty}|-k-1\rangle\langle -k|.
$$

The soliton equation \eqref{sol1} describes a motion of a particle in a 2d real phase space spanned by $({\rm Re}\,Y,{\rm Im}\,Y)$ induced by a Hamiltonian flow with a Hamiltonian given by ${\rm Re}\,W$.
Since the number of Hamiltonians for this system is exactly a half of the phase space dimension the soliton equation is integrable, and one can explicitly write a solution in terms of action-angle variables.
Since this procedure for solution derivation is rather standard and the resulting expression is rather involved we will not present it here.
Rather we depict the results of numerical solutions for $\mu=1.0$ and $L=10.0$ (see Fig.\ref{fig_solitons}).
We depict soliton trajectories in two ways.
On one hand we depict trajectories as result of applying a map $x^1\mapsto e^{-2\pi\, Y(x^1)}$ to the interval $x^1\in[0,L]$, notice all the solitons are represented by closed loops as expected.
On the other hand we consider a torus ${\cal T}\cong S^1\times S^1$ spanned by coordinates $(e^{2\pi{\rm i} \frac{x^1}{L}},e^{-2\pi{\rm i} Y_I})$.
On this torus $k$-solitons are again represented by closed loops winding around the torus $k$ times.

Finally, we should note that a soliton solution has a moduli space spanned by $u$ corresponding to a transition in the spatial direction, so that if $Y_*(x^1)$ is a solution then $Y_*(x^1+u)$ is also a solution.
This modulus is bounded to an interval $u\in[0,L)$ since solution $Y_*(x^1+L)$ is equivalent to $Y_*(x^1)$.
Supersymmetry relates each modulus to fermionic zero mode $\chi$ in the soliton background.
The corresponding spinor is given by $(\partial_uY_*,\partial_u\bar Y_*)$.
Therefore to each $k$-soliton solution one associates the corresponding pair of quasi-classical wave-functions:
$$
\Psi_k,\quad \chi^{\dagger}\Psi_k.
$$

Summarizing the calculations of the soliton spectra we conclude that the BPS Hilbert spaces in theories of the order and disorder parameters are isomorphic as bi-graded vector spaces in all phases of the theory, therefore we confirm \eqref{baby_mirror}.

To conclude this section let us notice that in calling $Y$ a disorder parameter we have admitted a certain abuse of notations, since a physical observable corresponding to the disorder parameter is an operator $e^{-2\pi Y}$.
Nevertheless we would like to save this terminology and keep calling non-physical operator $Y$ disorder operator since it turns out to be more suitable for further considerations.
Let us note that a $U(1)$ rotation group of $e^{-2\pi Y}$ acting by imaginary shifts on $Y$ is a symmetry of the Landau-Ginzburg Lagrangian broken explicitly by the chiral mass operator $\mu$.
In the case $\mu_{\mathbb{I}}=0$ this symmetry is restored on the BPS solitons:
$$
\left\langle e^{-2\pi Y}\right\rangle=0.
$$
Indeed the value of $\left\langle e^{-2\pi Y}\right\rangle$ is averaged over the soliton trajectory depicted in Fig.\ref{fig_solitons}(b).
After this we have to average over the position of the soliton core, or, in other words, to integrate over the soliton collective coordinate.
The latter action rotates soliton trajectories  in Fig.\ref{fig_solitons}(b) around zero, therefore the average value of $e^{-2\pi Y}$ on the soliton trajectories is zero.
We can summarize this information in the following table:
\bar{e}
\begingroup
\renewcommand*{\arraystretch}{1.4}
\begin{array}{c|c|c|c|c}
\mu_{\mathbb{I}} & U(1)_{\rm ord} & \mbox{Order op.} & U(1)_{\rm disord} & \mbox{Disorder op.}\\
\hline
\mu_{\mathbb{I}}\neq 0 & \mbox{unbroken} & \langle\phi\rangle=0 & \mbox{broken} & \left\langle e^{-2\pi Y}\right\rangle\neq 0\\
\mu_{\mathbb{I}}= 0 & \mbox{broken} & \langle\phi\rangle\neq 0 & \mbox{unbroken} & \left\langle e^{-2\pi Y}\right\rangle= 0
\end{array}
\endgroup
 \end{equation}

\section{Localization on a strip and BPS states}\label{sec:Localization}

\subsection{Localization and renormalization group}

\subsubsection{Localization in Schr\"odinger picture} \label{sec:Heis_loc}

A canonical generic approach to localization in a quantum mechanical system with a supersymmetry in the Schr\"odinger picture was proposed in a seminal paper \cite{Witten:1982im} (see also reviews in \cite{GMW,Behtash:2017rqj}).
Here let us just mention some basic steps we will apply in our consideration.

A quantum field theory can be treated as an ordinary quantum mechanical system describing the motion in the following space:
$$
{\cal T}_{\rm QM}={\rm Map}\left({\cal V}_{d-1}\longrightarrow {\cal T}\right).
$$
Supersymmetry produces supercharge operators ${\cal Q}$ and ${\cal Q}^{\dagger}$ on the Hilbert space of states satisfying the following superalgebra relation:
\bar{e}
\left\{{\cal Q},{\cal Q}^{\dagger}\right\}=2\left({\cal H}-|Q_{\rm top}|\right)\geq 0,
 \end{equation}
where ${\cal H}$ is the Hamiltonian of the system, and $Q_{\rm top}$ is a topological charge of the field configuration. The BPS bound
$$
{\cal H}\geq |Q_{\rm top}|
$$
is saturated by BPS states annihilated by both supercharges:
\bar{e}\label{BPS_wave}
{\cal Q}|\Psi_{\rm BPS}\rangle={\cal Q}^{\dagger}|\Psi_{\rm BPS}\rangle=0.
 \end{equation}

Suppose $x^i$ are coordinates on ${\cal T}_{\rm QM}$.
Supersymmetry mixes bosonic fields $x^i$ with fermionic fields $\psi_i$ and $\psi_i^{\dagger}$.
A behavior of fermionic field operators is analogous to the behavior of differential forms, so one could identify:
\bar{e}
\psi_i^{\dagger}\rightsquigarrow dx^i\wedge,\quad \psi_i\rightsquigarrow g^{ij}\iota_{\partial/\partial x^j},
 \end{equation}
where $\iota$ is an interior product operator, and $g_{ij}$ is a metric tensor for ${\cal T}_{\rm QM}$.
This allows one to identify the quantum mechanical quantities with geometric quantities.

In particular, the supercharges ${\cal Q}$ and ${\cal Q}^{\dagger}$ in the theory have a geometrical meaning of an extended differential and its Hodge dual.
Depending on the initial geometric structures on ${\cal T}_{\rm QM}$ -- Riemannian metric, complex structure, equivariant group action -- this differential will inherit its properties, so one will associate supercharge ${\cal Q}$ with a de Rham differential, a Dolbeault differential, an equivariant Cartan model differential, etc.
In addition the supercharge may acquire contributions from scalar functions characterizing a potential of the system.
Under this treatment wave functions of BPS states are identified with harmonic forms on ${\cal T}_{\rm QM}$.

Let us consider a system with concrete properties implying that a generic system will behave similarly. Suppose ${\cal T}_{\rm QM}$ admits an action of a Lie group $G$ and we could pick up a Morse height function $\mathfrak{H}$ on ${\cal T}_{\rm QM}$. In this case the supercharge reads:
\bar{e}
{\cal Q}=d+(d\mathfrak{H})+\iota_V=\psi^{\dagger}_i\frac{\partial}{\partial x^i}+\psi^{\dagger}_i(\partial_i \mathfrak{H})+\psi_ig^{ij}V_j,
 \end{equation}
where $V$ is a Killing vector field created by the $G$-action on ${\cal T}_{\rm QM}$.

A localization mechanism follows four steps:
\begin{enumerate}
	\item Using Hodge decomposition one identifies harmonic forms annihilated by both ${\cal Q}$ and ${\cal Q}^{\dagger}$ with elements of the cohomology group:
	$$
	\mathfrak{G}_{\rm BPS}\cong H^*({\cal Q}).
	$$
	\item Consider an isomorphism of cohomologies:
	$$
	\phi^{(\hbar)}:\; H^*({\cal Q})\longrightarrow H^*({\cal Q}(\hbar)),
	$$
	where the map $\phi^{(\hbar)}$ is a mere multiplication by an operator:
	$$
	\Phi_{\hbar}:={\rm exp}\left(-\left(\hbar^{-1}-1\right)\mathfrak{H}+({\rm log}\, \hbar)\sum\limits_i\psi^{\dagger}_i\psi_i\right).
	$$
	For a transformed supercharge we find:
	$$
	{\cal Q}(\hbar):=\hbar\,\Phi_{\hbar}{\cal Q} \Phi_{\hbar}^{-1}=\hbar\, d+(d\mathfrak{H})+\iota_V.
	$$
	In principle the transformed supercharge belongs to a \emph{different} quantum system with a new Hamiltonian:
	\bar{e}\label{newHamiltonian}
	\begin{split}
	{\cal H}(\hbar):=&|Q_{\rm top}(\hbar)|+\frac{1}{2}\left\{{\cal Q}(\hbar),{\cal Q}^{\dagger}(\hbar)\right\}=\\
	=&\hbar^2\Delta +\left|\vec \nabla \mathfrak{H}\right|^2+\left|\vec V\right|^2+{\rm fermions}.
	\end{split}
	 \end{equation}
	\item On one hand operator $\Phi_{\hbar}$ restores the Plank constant dependence often omitted in QFT calculations, on the other hand
	it establishes an invariance of the BPS Hilbert spaces under variations of $\hbar$:
	$$
	\mathfrak{G}_{\rm BPS}(\hbar)\cong \mathfrak{G}_{\rm BPS}(\hbar').
	$$
	\item Using this invariance it is easier to compute corresponding BPS wave-functions in the semi-classical limit:
	$$
	\hbar\longrightarrow 0.
	$$
\end{enumerate}

A calculation process in the semi-classical limit has its own known loopholes -- a necessity in certain situations to consider non-perturbative instanton corrections \cite{D-book_1}. We will not go into details widely presented in the literature, see e.g. \cite{GMW}.
In our particular situation of supersymmetric quantum system the computation of $\mathfrak{G}_{\rm BPS}$ boils down to a calculation of cohomologies of a \emph{Morse-Smale-Witten} (MSW) complex $({\bf M}^*,{\bf Q})$.

Zero locus of the potential term \eqref{newHamiltonian} corresponds to the classical vacua, therefore we call it a vacuum locus and denote it $\mathfrak{V}$.
Assume for now $\mathfrak{V}$ is a set of isolated points.
For each point $p\in\mathfrak{V}$ we change coordinates $x$ on ${\cal T}_{\rm QM}$ as
$$
x\longrightarrow p+\hbar^{\frac{1}{2}}x.
$$
The Hamiltonian can be decomposed as:
$$
{\cal H}=|Q_{\rm top}(\hbar)|+\hbar\; {\cal H}_p^{(0)}+O(\hbar^2).
$$
The Hamiltonian ${\cal H}_p^{(0)}$ is just a free particle Hamiltonian and its ground state wave function $\Psi_p$ can be easily calculated.
$\Psi_p$ is the zeroth order perturbative approximation to the actual BPS wave-function.
The MSW complex as a vector space is spanned by perturbative BPS wave-functions:\footnote{We have to warn the reader that the construction of the MSW complex has its own peculiarities depending on additional structures carried on by the quantum system. Here we review just the most basic one. For example, if ${\cal T}_{\rm QM}$ has a complex structure compatible with the supercharge Hamiltonian ${\cal H}_p^{(0)}$ would correspond to a model of a free particle on a plane put in a magnetic field perpendicular to that plane, therefore the ground states of such Hamiltonian will be described by a condensate of lowest Landau level wave functions \cite{landau1958quantum} with non-negative angular momenta corresponding to a structure sheaf of an affine complex line.}
\bar{e}\label{MSW}
{\bf M}:=\bigoplus\limits_{p\in\mathfrak{V}} \mathbb{C} \; \Psi_p.
 \end{equation}
This vector space is graded by the fermion number $f$. The differential of the complex
$$
{\bf Q}:\quad {\bf M}^{f}\longrightarrow {\bf M}^{f+1}
$$
is defined through its matrix elements $\langle\Psi_p|{\bf Q}|\Psi_{p'}\rangle$ that up to a non-zero renormalization factor coincide with matrix elements of $\langle\Psi_p|{\cal Q}^{\dagger}(\hbar)|\Psi_{p'}\rangle$:
\bar{e}\label{differential}
\langle\Psi_p|{\bf Q}|\Psi_{p'}\rangle=\delta_{f_p,f_{p'}+1}\times\sum\limits_{{\rm 1-instantons}\;(p\to p')}\Delta,
 \end{equation}
where $n$-instantons interpolating between points $p$ and $p'$ are solutions to the following differential equation boundary value problem:
\bar{e}\label{instanton}
\begin{split}
&\partial_{\tau}x^i(\tau)=-g^{ij}\partial_{x^j}\mathfrak{H}(x(\tau)),\\
&\lim\limits_{\tau\to-\infty}x^i(\tau)=p,\quad \lim\limits_{\tau\to-\infty}x^i(\tau)=p'.
\end{split}
 \end{equation}
1-instanton solutions to \eqref{instanton} have only a single translation modulus $m$ mapping a solution $x^i(\tau)$ to another solution $x^i(\tau+m)$. $\Delta$ in \eqref{differential} is a $\pm 1$-valued contribution
$$
\Delta=\frac{{\rm Det}'\hat D}{|{\rm Det}'\hat D|},
$$
where $\hat D$ is a Dirac operator in the instanton background:
$$
\hat D^i_k:=\partial_{\tau}\delta^i_k+\partial_{x^k}\left(g^{ij}\partial_{x^j}\mathfrak{H}\right),
$$
and we subtract the contribution of the zero mode corresponding to the modulus $m$ from the determinant.

The BPS Hilbert space is defined as a cohomology:
\bar{e}
\mathfrak{G}_{\rm BPS}^*\cong H^*({\bf M},{\bf Q}),
 \end{equation}
where cohomological grading coincides with the fermion number grading of the Hilbert space.

\subsubsection{Wilsonian renormalization ``exact" in one loop}

In the previous section we assumed that $\mathfrak{V}$ is a set of isolated points, now we rather assume $\mathfrak{V}$ to be a single component connected hypersurface in ${\cal T}_{\rm QM}$.
One can divide coordinates $x^i$ spanning ${\cal T}_{\rm QM}$ in two groups: perpendicular to $\mathfrak{V}$, or ``fast" variables $x_f$; and tangent to $\mathfrak{V}$, or ``slow" variables $x_s$. Slow variables $x_s$ are also referred to as \emph{vacuum moduli}.
Perturbative modes of $x_f$-fields around vacuum value $x_f=0$ have non-zero masses, therefore their temporal frequencies are rather high, at least greater or equal to corresponding masses in their absolute values, this is why we call these modes fast.
The perturbative slow modes $x_s$ are analogs of Goldstone modes and have zero masses, they can form even stationary field configurations -- condensates.

Separation of slow and fast field modes is a usual prologue to the Wilsonian renormalization group \cite{ZinnJustin:2002ru}.
We easily calculate the action of the renormalization group on the supercharges.
Let us just redefine variables:
$$
x_f\longrightarrow \hbar^{\frac{1}{2}}x_f.
$$
For the supercharge we have the following decomposition:
\bar{e}
{\cal Q}(\hbar)=\sum\limits_{n=0}^{\infty} \hbar^{\frac{n+1}{2}}{\cal Q}^{(n)},
 \end{equation}
where
\bar{e}
{\cal Q}^{(0)}=d_f+\Omega(x_s)\cdot x_f.
 \end{equation}
Here $\Omega(x_s)\neq 0$ defines a linear functional in $x_f$, and differential $d_f$ defines differentiation only in fast variables $x_f$. We search for a BPS wave function having a suitable decomposition:
\bar{e}\label{expan}
\Psi_{\rm BPS}(\hbar)=\sum\limits_{n=0}^{\infty}\hbar^{\frac{n}{2}}\Psi^{(n)}.
 \end{equation}
Substituting such $\hbar$-expansions into \eqref{BPS_wave} we have at the zeroth order:
\bar{e}\label{0th}
{\cal Q}^{(0)}\Psi^{(0)}=\bar{\cal Q}^{(0)}\Psi^{(0)}=0.
 \end{equation}
These equations are simple linear differential equations in $x_f$, $\Psi^{(0)}$ has a natural form of a wave function localized at $x_f=0$ with quantum corrections suppressed by a Gaussian exponent.
Solutions to the linear differential equations \eqref{0th} form a one-dimensional linear space $C\cdot \Psi^{(0)}$, where $C$ is usually referred to as an integration constant.
Since fields $x_s$ in \eqref{0th} play a role of mere parameters we conclude that if  $\Psi^{(0)}(x_f,x_s)$ is a solution to \eqref{0th} then any solution  to \eqref{0th} has the following form:
$$
C(x_s)\cdot \Psi^{(0)}(x_f,x_s),
$$
where $C(x_s)$ is a generic functional of $x_s$ remaining undefined so far.
At the first order of approximation we find the following equations:
\bar{e}
\begin{split}
	{\cal Q}^{(1)}\cdot C\cdot\Psi^{(0)}+{\cal Q}^{(0)}\cdot \Psi^{(1)}=0,\\
	\bar{\cal Q}^{(1)}\cdot C\cdot\Psi^{(0)}+\bar{\cal Q}^{(0)}\cdot \Psi^{(1)}=0.
\end{split}
 \end{equation}
Multiplying both equations from the left by $\Psi^{(0)\dagger}$ and integrating over $x_f$ we derive equations defining $C$:
\bar{e}
{\cal Q}_{\rm eff}C=\bar{\cal Q}_{\rm eff} C=0,
 \end{equation}
where
\bar{e}\label{Q_eff}
\begin{split}
	{\cal Q}_{\rm eff}(x_s)=\int dx_f\; \Psi^{(0)\dagger}(x_f,x_s){\cal Q}^{(1)}(x_f,x_s)\Psi^{(0)}(x_f,x_s),\\
	\bar {\cal Q}_{\rm eff}(x_s)=\int dx_f\; \Psi^{(0)\dagger}(x_f,x_s)\bar{\cal Q}^{(1)}(x_f,x_s)\Psi^{(0)}(x_f,x_s).
\end{split}
 \end{equation}
Superchargers \eqref{Q_eff} represent a one-loop Wilsonian renormalization of initial supercharges, therefore we call them \emph{effective supercharges} defining the IR theory, and $C$ is an \emph{effective wave-function}.

Since we are working in the semi-classical limit $\hbar\to 0$ all the higher orders except the zeroth one in expansion \eqref{expan} can be neglected, and we have just described a procedure to calculate it.
This procedure allows one to reformulate the problem of defining BPS wave-functions in terms of an effective theory, and since only one loop renormalization correction is taken into account we call it one-loop exact.

As well there is a non-trivial Jacobian modification to a Hilbert space norm of the  effective wave-functions:
\bar{e}
\begin{split}
\langle C'|C\rangle&=\int dx_s \;J(x_s)\;\bar C'(x_s)C(x_s),\\
\mbox{where }J(x_s)&=\int dx_f\; \Psi^{(0)\dagger}(x_f,x_s)\Psi^{(0)}(x_f,x_s).
\end{split}
 \end{equation}

\subsection{Localization in GLSM with an interface}\label{sec:loc_GLSM}
\subsubsection{Gauged linear sigma-model}
Now let us turn back to 2d ${\cal N}=(2,2)$ gauged linear sigma models (GLSM).
We consider a $U(1)$-theory with $n_f$ chiral multiplets with charges $Q_a$ and masses $\mu_a$ for $a=1,\ldots,n_f$ on an interval $[0,L]$.
GLSM with a generic gauge group can be considered in a similar fashion.
At boundaries of the interval we choose Chan-Paton boundary conditions (see Appendix  \ref{sec:App_boundary}).
Chan-Paton factors contribute twice.
The first contribution is to the supercharge, it is given by a holomorphic function $\bar {\cal Q}_{\rm bdry}$ of chiral fields $\phi_a$ and boundary anti-fermion fields $\bar \chi_i$.
Another contribution is to the boundary electric charge $\bf q$ of the brane.
An expression for the B-twist in this theory with a restored $\hbar$-dependence reads:
\bar{e}\label{supercharge}
\begin{split}
	\bar{\cal Q}_B=\int dx^1\;\Bigg[\lambda_1\left(-{\rm i}\hbar\delta_{\sigma_{\mathbb{I}}}-{\rm i}\partial_1\sigma_{{\mathbb{R}}}-\hbar\delta_{A_1}+{\rm i}\hbar \theta\right)+\\
	+\lambda_2\left(\hbar\delta_{\sigma_{{\mathbb{R}}}}-\partial_1\sigma_{\mathbb{I}}+\left(\sum\limits_a Q_a|\phi_a|^2-r\right)\right)
	-{\rm i}\sqrt{2}\sum\limits_{a}\bar\psi_{\dot 1,a}\left(\hbar\delta_{\bar\phi_a}+(Q_a\sigma_{{\mathbb{R}}}-\mu_{{\mathbb{R}},a})\phi_a\right)-\\-\sqrt{2}\sum\limits_{a}\bar\psi_{\dot 2,a}\left(\left(\partial_1\phi_a+{\rm i} Q_a A_1\phi_a\right)-(Q_a\sigma_{\mathbb{I}}-\mu_{\mathbb{I},a})\phi_a\right)\Bigg]+\bar{\cal Q}_{\rm bdry},
\end{split}
 \end{equation}

As well we will need the Gauss current operator expression:
\bar{e}\label{CJ}
{\cal J}= \sum\limits_{a}Q_a\left(\left(\bar\phi_a \delta_{\bar\phi_a}-\phi_a \delta_{\phi_a}\right)+\bar\psi_{\dot 1,a}\psi_{1,a}+\bar\psi_{\dot 2,a}\psi_{2,a}\right)+{\rm i}\partial_1 \delta_{A_1}+{\bf q}\delta({\rm bdry}),
 \end{equation}
where $\bf q$ is an electric charge produced by the boundary branes.

Supercharge satisfies the following superalgebra:
\bar{e}
{\cal Q}_B^2={\cal Z},\quad \left\{{\cal Q}_B,\bar{{\cal Q}}_B\right\}=2{\cal H}+2{\rm Re}(\zeta^{-1}\tilde {\cal Z}).
 \end{equation}
\begin{comment}
where central charges are:
\bar{e}
\begin{split}
	{\cal Z}=&-{\rm i}\lambda_1\lambda_2\Big|_{\rm bdry}\\
	\zeta^{-1}\tilde{\cal Z}=&{\rm i}\Bigg(\sum\limits_{a=1,2}\frac{(\bar\psi_{\dot 1,a}+\bar\psi_{\dot 2,a})(\psi_{1,a}-\psi_{2,a})}{2}+\\
	&+(\sigma_{{\mathbb{R}}}-{\rm i}\sigma_{\mathbb{I}})\left(\hbar^{-1}\left(r-\sum\limits_aQ_a|\phi_a|^2\right)+\left(-\delta_{A_1}+{\rm i}\theta\right)\right)\Bigg)\Big|_{\rm bdry}-\\
	&+{\rm i}\hbar^{-1}\sum\limits_{a}(\mu_{{\mathbb{R}},a}-{\rm i}\mu_{\mathbb{I},a})|\phi_a|^2\Big|_{\rm bdry}-\\
	&-\sum\limits_{a}(\mu_{{\mathbb{R}},a}-{\rm i}\mu_{\mathbb{I},a})\int dx^1\;\left(\bar\phi_a\delta_{\bar\phi_a}-\phi_a\delta_{\phi_a}+\bar\psi_{1,a}\psi_{\dot 1,a}+\bar\psi_{\dot 2,a}\psi_{2,a}\right)
\end{split}
 \end{equation}
B-brane boundary conditions are defined by non-perturbative constraints:
\bar{e}
\bar{\cal Q}_{B}^2=0,\quad \left[{\cal Q}_B,{\rm Re}(\zeta^{-1}\tilde {\cal Z})\right]=0
 \end{equation}
\end{comment}

From this algebraic relations it is clear that the Hamiltonian spectrum is bounded from below.
Wave functions of physical BPS states are annihilated by the following operators:
\bar{e}
{\cal Q}_B|\Psi_{\rm BPS}\rangle=\bar{\cal Q}_B|\Psi_{\rm BPS}\rangle={\cal J}|\Psi_{\rm BPS}\rangle=0.
 \end{equation}
On the BPS Hilbert space the Hamiltonian acquires an eigen value:
\bar{e}
{\cal H}|\Psi_{\rm BPS}\rangle=-{\rm Re}(\zeta^{-1}\tilde {\cal Z}) |\Psi_{\rm BPS}\rangle.
 \end{equation}

To introduce an interface dependence of parameters on the spatial coordinate $x^1$ it suffices to substitute constant parameters in the expression for the supercharge \eqref{supercharge} by functions:
\bar{e}
\theta\to\theta(x^1),\quad r\to r(x^1),\quad \mu_a\to \mu_a(x^1).
 \end{equation}
Using the inverse Lagrange transform on the Hamiltonian operator ${\cal H}$ one can reconstruct the original action of the theory in the presence of an interface.
We will restrict ourselves to an insertion of an interface defect only for FI parameters and $\theta$-angle.
In this case it is easy to trace back the interface modification needed for the action to be invariant under the B-twist.
We just need to modify the corresponding term \eqref{S_FI}:
\bar{e}
{\cal S}_{\rm FI,\,\theta}'=\int dx^0dx^1\,\left[-r\,{\bf D}+\theta\, F_{01}+\partial_1r\,\sigma_{\mathbb{I}}-\partial_1\theta\,\sigma_{{\mathbb{R}}}\right].
 \end{equation}

The central charge in this model reads:
\bar{e}
\begin{split}
\tilde {\cal Z}=\Delta{\rm i}\left\{\hbar\sum\limits_a \bar\psi_{-,a}\psi_{+,a}+\hbar\bar\sigma\bar  t'-\sum\limits_a\left(Q_a\bar\sigma-\bar\mu_a\right)|\phi_a|^2-\hbar \bar\sigma\delta_{A_1}\right\}+\\
+\hbar\sum\limits_{a}\bar\mu_a\int dx^1\,\left(\bar\phi_a\delta_{\bar\phi_a}-\phi_a\delta_{\phi_a}+\bar\psi_{1,a}\psi_{1,a}+\bar\psi_{2,a}\psi_{2,a}\right),
\end{split}
 \end{equation}
where $t'$ is an FI parameter complexified by the topological angle and shifted by the boundary charge:
\bar{e}
t'=\hbar^{-1}r-{\rm i} (\theta+{\bf q}_{\rm bdry})
 \end{equation}

The gauge symmetry allows one to vary the phase of the wave-function:
\bar{e}
\Psi\to {\rm exp}\left({\rm i} \int\varphi(x^1)A_1(x^1)\;dx^1\right)\, \Psi.
 \end{equation}
This variation does not change the integrability properties of the wave function, however it adds a shift to the Gauss charge operator and to the topological angle:
\bar{e}
\begin{split}
	{\cal J}(x^1)\to{\cal J}(x^1)-\partial_1\varphi(x^1),\quad \theta(x^1)\to \theta(x^1)-\varphi(x^1).
\end{split}
 \end{equation}
We will use this shift to delete the contribution form the boundary charges:
\bar{e}
\varphi={\bf q}\;\Theta({\rm bdry}),
 \end{equation}
where $\Theta$ is the Heaviside step function.
So we can move shifts due to the boundary brane electric charges to contribute as step-like shifts.
We expect that the parallel transport is a homotopy invariant of interface path $\wp$.
One can smear step-like contributions at the boundaries  using this homotopy invariance (see Fig.\ref{fig:bdry_chg}).
In what follows rather than considering the contribution of the boundary charges to the Gauss law we will imply that they perform appropriate shifts to the boundary values of the topological angle function $\theta(x^1)$ on the interface.
Surely, the parameter space ${\cal P}$ spanned by $t'$ will have singular loci where the theory is ill-defined.
The homotopy of $\wp$ has to take these singularities into account.

\begin{figure}[h!]
	\begin{center}
		\begin{tikzpicture}
			\draw[ultra thick] (0,0) to[out=0,in=180] (1.3,0.4) to[out=0,in=180] (2.5,-0.3) to[out=0,in=180] (4,0);
			\draw[<->] (0,2) -- (0,0) -- (5,0);
			\draw[dashed] (4,0) -- (4,2);
			\node[right] at (5,0) {$x^1$};
			\node[above] at (0,2) {$\theta(x^1)$};
			\node[below] at (0,0) {$0$};
			\node[below] at (4,0) {$L$};
			\draw[ultra thick, black!40!green] (0,1) to[out=0,in=180] (2,2.3) to[out=0,in=180] (4,2);
			\draw[ultra thick, blue] (0,1) -- (0.2,1) -- (0.2,0);
			\draw[ultra thick, red] (4,2) -- (3.8,2) -- (3.8,0);
			\node[left,blue] at (0,1) {${\bf q}_{\rm left}$};
			\node[right,red] at (4,2) {${\bf q}_{\rm right}$};
			\draw[->] (1.8,0.5) to[out=120,in=240] (1.8,2);
			\node[right] at (1.8,1.25) {homotopy};
			\node at (2,-0.5) {$\theta_{\rm initial}$};
			\node[black!40!green] at (2,2.7) {$\theta_{\rm final}$};
		\end{tikzpicture}
	\caption{Boundary brane charge contribution to the topological angle.}\label{fig:bdry_chg}
	\end{center}
\end{figure}

To pursue our goal to study the IR physics of BPS states in this theory we follow the steps discussed in Section \ref{sec:Heis_loc} and construct the MSW complex.
The first step in this paradigm is to define the vacuum locus $\mathfrak{V}$. $\mathfrak{V}$ is spanned by field configurations satisfying the following set of equations:
\bar{e}\label{sol_vac}
\begin{split}
	\partial_1\sigma_{{\mathbb{R}}}=O(\hbar),\\
	-\partial_1\sigma_{\mathbb{I}}+\left(\sum\limits_aQ_a|\phi_a|^2-r\right)=O(\hbar),\\
	(Q_a\sigma_{{\mathbb{R}}}-\mu_{{\mathbb{R}},a})\phi_a=O(\hbar),\\
	\left(\partial_1\phi_a+{\rm i} Q_a A_1\phi_a\right)-(Q_a\sigma_{\mathbb{I}}-\mu_{\mathbb{I},a})\phi_a=O(\hbar).
\end{split}
 \end{equation}
By choosing $O(\hbar)$ as the right hand side for those equations rather than just zeroes we would like to stress again that we allow quantum corrections to ``blur" $\mathfrak{V}$.

Constant solutions to those equations satisfy the following constraints:
\bar{e}
\sum\limits_aQ_a|\phi_a|^2-r=O(\hbar),\quad (Q_a\sigma-\mu_a)\phi_a=O(\hbar).
 \end{equation}

Based on parameter values of these equations we could distinguish three phases:
\begin{enumerate}
	\item {\bf Coulomb branch.} It appears when $r=O(\hbar)$, $\mu_a=O(1)$. The solution is dominated by expectation values of the scalar in the gauge multiplet $\sigma=O(1)$. The chiral multiplets are considered to perform quantum corrections $\phi_a=O\left(\hbar^{\frac{1}{2}}\right)$.
	\item {\bf ``Soft" Higgs branch.} It appears when $r=O(1)$, $\mu_a=O(\hbar)$. Here the situation is opposite to the Coulomb branch, the scalars from the chiral multiplet acquire expectation values $|\phi_a|=O(1)$, and the scalar from the gauge multiplet delivers corrections $\sigma=O(\hbar)$.
	\item {\bf Mixed/``rigid" Higgs branch/crystal Coulomb branch.} It appears when $r=O(1)$, $\mu_a=O(1)$. The theory is localized in one of the fixed points on the soft Higgs branch: $|\phi_a|=r^{\frac{1}{2}}+O(\hbar)$, $\sigma=Q_a^{-1}\mu_a+O(\hbar)$.
\end{enumerate}

In the literature a more preferable name for the third option is a mixed branch since scalars belonging to both gauge and chiral multiplets acquire expectation values.
We would like to save also both names ``rigid Higgs branch" and ``crystal Coulomb branch" and stress that a distinction between the rigid and the soft Higgs branches is rather contingent.
In principle the Morse height functional can be split in two gauge-invariant parts separating $r$ and $\mu_a$.
Those parts can be re-scaled also separately introducing two RG flow parameters $\hbar_1$ and $\hbar_2$.
The mass parameters in the resulting supercharge expression will scale as
$$
\mu_a\sim \hbar_1/\hbar_2.
$$
The general theory predicts that GR flows in $\hbar_1$ and $\hbar_2$ commute. The soft Higgs branch will appear naturally if we take limit $\hbar_1\to 0$ first, then limit $\hbar_2\to 0$ will perform a further flow to the rigid Higgs branch.
Mathematically this corresponds to a further localization of coherent sheaves on the soft Higgs branch moduli space due to equivariant action with $\mu_a$ parameterizing equivariant tori.
We will comment on ``crystal Coulomb branch" name in Section \ref{sec:quiver}.

So we would like to distinguish Higgs and Coulomb phases of the theory not by fields expectation values rather by orders of the parameter $r$ that we allow to vary along the interface.
So a BPS state in our theory will look like a thick ``sandwich" of various phases.
Since these phases are just dual descriptions of the same physics we do not expect an appearance of domain walls between those phases.

It is clear that $\mathfrak{V}$ may have moduli.
We would like to flow along $\hbar\to 0$ to integrate out degrees of freedom perpendicular to $\mathfrak{V}$ and derive an effective description.
In the remaining part of this section we will describe effective theories for both Higgs and Coulomb branches for constant values of parameters $\mu_a$ and $t$.
In the remaining part we will briefly remind the web formalism technique \cite{GMW} allowing one to compute the MSW complex cohomologies for supersymmetric interfaces including non-trivial dependence of parameters on the spatial coordinate $x^1$ in a universal way.

\subsubsection{Brane boundary conditions}
To proceed we need to choose boundary conditions on the ends of interval $[0,L]$ by imposing constraints that are invariant under the RG flow.
Only in this way we could guarantee that the actual Hilbert spaces of BPS states are dual to each other.

In general, the supercharge we have chosen ${\cal Q}_B$ to localize with respect to is not nilpotent (see \eqref{AB-algebra}).
For its nilpotency one has to impose constraints on the superpotential and charge ${\cal Z}$ leading to boundary conditions for field modes.
The boundary conditions for gauge multiplet naturally follow from manifestly RG invariant constraint:
$$
{\cal Q}_B^2=0.
$$

To set RG invariant boundary conditions for the chiral fields we follow approach of \cite{Dimofte:2019zzj,Cordes:1994fc} and rewrite the chiral fields in terms of superfields of the B-type supersymmetry preserved by the boundary.
A relation between the bulk and boundary supersymmetries is described in Appendix \ref{sec:brane_cond}.

For chiral fields we choose Neumann type boundary conditions allowing the branes to cover the whole vacuum variety. If some subvariety is needed we will use Chan-Paton factors carrying a complex associated to the corresponding structure sheaf. Thus for scalars we have a gauge-invariant version of \eqref{chiral_brane}:
\bar{e}
D_1\phi_a-(Q_a\sigma_{\mathbb{I}}-\mu_{\mathbb{I},a})\phi_a\big|_{\rm bdry}=0,\quad (Q_a\sigma_{{\mathbb{R}}}-\mu_{{\mathbb{R}},a})\phi_a\big|_{\rm bdry}=0.
 \end{equation}
Boundary conditions for superpartners can be produced by B-twist actions.

Surely, proposed boundary conditions admit generalizations \cite{Hori:2013ika} by putting additional operators and charges on the brane, we will not consider these instances for keeping things simple.

\subsection{Coulomb branch: Landau-Ginzburg (LG) model}\label{sec:LG_model}
\subsubsection{Wilsonian renormalization}
According to the Coulomb branch localization prescription we expect that the Higgs fields $\phi_a$ do not acquire expectation values. For this situation to represent a valid classical vacuum it has to satisfy the D-term equation implying $r=0$. However this constraint has to be satisfied on the classical level only, we are able to admit a softer quantum limit $r=O(\hbar)$. Therefore the following redefinition is suitable:
$$
r=\hbar \tilde r,\quad \tilde r=O(\hbar^0).
$$
We decompose remaining fields accordingly:
\bar{e}
\phi_a\to \hbar^{\frac{1}{2}}\phi_a,\quad \sigma\to \Sigma+\hbar^{\frac{1}{2}}\sigma,
 \end{equation}
where $\Sigma$ is a slow component of the scalar $\sigma$ satisfying
$$
|\partial_1\Sigma|= O(\hbar).
$$

After integration over the fast fields $\phi_a$ and $\sigma$ and rescaling the spatial coordinate $x^1 \to \hbar^{-1}x^1$ the supercharge and the Gauss law constraint have the following form:
\bar{e}
\begin{split}
	\bar{\cal Q}_{\rm eff}=\int dx^1\;\Bigg[\lambda_1\left(-{\rm i}\delta_{\Sigma_{\mathbb{I}}}-{\rm i}\partial_1\Sigma_{{\mathbb{R}}}-\delta_{A_1}+{\rm i}\theta\right)
	+\lambda_2\left(\delta_{\Sigma_{{\mathbb{R}}}}-\partial_1\Sigma_{\mathbb{I}}+\left(\sum\limits_a Q_a\;{\rm Re}\,Y_a-\tilde r\right)\right)\Bigg],\\
	{\cal J}={\rm i}\partial_1 \delta_{A_1}-\sum\limits_aQ_a\;\partial_1{\rm Im}\,Y_a,
\end{split}
 \end{equation}
where the following notion for the disorder operators $Y_a$ is introduced:
\bar{e}
\begin{split}
	{\rm Re}\,Y_a&=\big\langle \Psi^{(0)}\big||\phi_a|^2\big|\Psi^{(0)}\big\rangle,\\
	\partial_1{\rm Im}\,Y_a&=\big\langle \Psi^{(0)}\big|-\left(\bar\phi_a\delta_{\bar\phi_a}-\phi_a\delta_{\phi_a}+\bar\psi_{\dot 1,a}\psi_{1,a}+\bar\psi_{\dot 2, a}\psi_{2,a}\right)\big|\Psi^{(0)}\big\rangle.
\end{split}
 \end{equation}
Expectation values of $Y_a$ can be easily computed at one loop along the lines of Section \ref{sec:Preamble}, we put an explicit computation in Appendix \ref{sec:disorder}.
The result reads:
\bar{e}
Y_a=-\frac{1}{2\pi}{\rm log}\frac{Q_a\Sigma-\mu_a}{\Lambda}.
 \end{equation}
As in Section \ref{sec:Preamble} logarithm multi-valuedness corresponds to a discrete ambiguity in a choice of a Dirac string contribution.

The Gauss law constraint can be easily resolved by the phase dependence of the wave function on the gauge field:
$$
{\rm exp}\left(-{\rm i}\int dx^1A_1\sum\limits_a Q_a\;{\rm Im}\,Y_a\right)\Psi(\Sigma).
$$

The resulting effective supercharge reads:
\bar{e}\label{LG_eff_Q}
\bar{\cal Q}_{\rm LG}=\int dx^1\left[\zeta\left(\lambda^2-\lambda^1\right)\left(\delta_{\bar\Sigma}+\frac{{\rm i}}{2}\partial_1\Sigma-\zeta^{-1}\frac{\bar W'}{2}\right)+\zeta^{-1}\left(\lambda^1+\lambda^2\right)\left(\delta_{\Sigma}-\frac{{\rm i}}{2}\partial_1\bar\Sigma-\zeta\frac{W'}{2}\right)\right],
 \end{equation}
where we used an effective superpotential generated in this model:
\bar{e}
W(\Sigma)=t\,\Sigma+\sum\limits_a\frac{Q_a\Sigma-\mu_a}{2\pi}\left[{\rm log}\left(\frac{Q_a\Sigma-\mu_a}{\Lambda}\right)-1\right],
 \end{equation}
and the FI parameter is complexified with the use of a topological angle contribution:
$$
t=\tilde r-{\rm i} \theta.
$$
The effective central charge reads:
\bar{e}
\tilde Z={\rm i}\zeta^{-1}\,\int dx^1\;\partial_1\bar W.
 \end{equation}

This supercharge defines a Landau-Ginzburg model on a complex plane spanned by $\Sigma$ with (twisted) superpotential $W$ (compare to \eqref{nlin_sup}).

\subsubsection{Landau-Ginzburg model}
Let us consider a more generic Landau-Ginzburg (LG) model on an $m$-dimensional K\"ahler manifold $X_{\rm LG}$ spanned by scalars $\Sigma^I$, $I=1,\ldots, m$ with the metric tensor $g_{I\bar J}$. Despite the case we derived above corresponds to a simple flat complex plane a more generic analysis will still go through. The superpotential is given by a holomorphic function $W$. The vacuum locus $\mathfrak{V}$ is spanned by solutions to the following equation:
\bar{e}\label{LG_soliton}
{\rm i}\partial_1\Sigma^I=\zeta^{-1}g^{I\bar J}\overline{\partial_{\Sigma^J}W}.
 \end{equation}
Constant field configurations satisfying \eqref{LG_soliton} correspond to roots of an algebraic system:
$$
\partial_{\Sigma^J}W(\Sigma)=0.
$$
We call these roots LG vacua and denote by an index with an asterisk as $\Sigma_{*i}$.

To define appropriate boundary conditions apply the B-twist to ${\cal Q}_B^2+\bar{\cal Q}_B^2$. We will find that the result is equivalent to a sum of supercurrents through the boundary. Literally repeating arguments of \cite[Section 39.2.2]{D-book_1} we will arrive to a conclusion that the boundary D-brane has a form of a special Lagrangian. As a basis of such special Lagrangians we choose Lefshetz thimbles.

A Lefshetz thimble ${\cal L}_i$ is labeled by LG vacuum $(*i)$ and as a manifold is given by a union of all trajectories satisfying the following asymptotic value problem:
\bar{e}
\partial_\tau\Sigma^I=-\zeta^{-1}g^{I\bar J}\overline{\partial_{\Sigma^J}W},\quad \lim\limits_{\tau\to-\infty}\Sigma(\tau)=\Sigma_{*i}.
 \end{equation}
For equations defining $\mathfrak{V}$ and the Lefshetz thimble we easily derive:
\bar{e}
\partial_1(\zeta W)=-{\rm i} g^{I\bar J}\partial_{\Sigma^I}W\overline{\partial_{\Sigma^J}W}\in-{\rm i} {\mathbb{R}}_{\geq 0},\quad \partial_\tau(\zeta W)=-g^{I\bar J}\partial_{\Sigma^I}W\overline{\partial_{\Sigma^J}W}\in- {\mathbb{R}}_{\geq 0}.
 \end{equation}
Therefore, in the $\zeta W$-plane the thimbles are represented by rays starting in critical values $W_{*i}$ and flowing parallel to the real axis to the left, and solutions to \eqref{LG_soliton} are segments connecting those rays. In the field space $X_{LG}$ the Lefshetz thimbles are finger-looking Lagrangian fibrations of $S^{m-1}$ with the sphere shrinking at the thimble tip. Solutions to \eqref{LG_soliton} are BPS strings stretched between different thimbles or zero size strings concentrated at thimble tips (see Fig.\ref{fig:thimbles}).

\begin{figure}[h!]
	\begin{center}
		\begin{tikzpicture}
			\draw[<-] (-5,0.25) -- (1.5,0.25);
			\draw[->] (1.5,0.25) -- (3,-1.25);
			\node[right] at (3,-1.25) {$-{\rm Im}\,\zeta W$};
			\node[left] at (-5,0.25) {$-{\rm Re}\,\zeta W$};
			\draw[->] (1.5,0.25) -- (1.5,2);
			\draw[ultra thick, red,->] (-2,0) -- (-1.3,-0.7);
			\draw[ultra thick, red] (-1.3,-0.7) -- (-1,-1);
			\node[above right] at (1.5,2) {$\Sigma$-fiber};
			\node[right] at (0.1,0) {$\zeta W_{*i}$};
			\node[right] at (1.4,-1) {$\zeta W_{*j}$};

			\draw[ultra thick, ->]  (0,0) -- (-4,0);
			\draw[ultra thick] (-0.1,-0.1) -- (0.1,0.1) (0.1,-0.1) -- (-0.1,0.1);
			\begin{scope}[shift={(0,2)}]
			\begin{scope}[yscale=0.3]
			\draw[fill=white!40!gray] (-4,-1) -- (-2,-1) to[out=0,in=270] (0,0) to[out=90,in=0] (-2,1) -- (-4,1);
			\end{scope}
			\begin{scope}[shift={(-4,0)}]
			\begin{scope}[xscale=0.5]
				\draw[fill=white!40!gray] (0,0) circle (0.3);
			\end{scope}
			\end{scope}
			\end{scope}
		\draw[fill=black] (0,2) circle (0.05);
		\draw[thick, dashed] (0,0) -- (0,2);

		\draw[red,dashed] (-2,0) -- (-2,2);

		\begin{scope}[shift={(1.3,-1)}]
			\draw[ultra thick, ->]  (0,0) -- (-4,0);
			\draw[ultra thick] (-0.1,-0.1) -- (0.1,0.1) (0.1,-0.1) -- (-0.1,0.1);
			\begin{scope}[shift={(0,2)}]
				\begin{scope}[yscale=0.3]
					\draw[fill=white!40!gray] (-4,-1) -- (-2,-1) to[out=0,in=270] (0,0) to[out=90,in=0] (-2,1) -- (-4,1);
				\end{scope}
				\begin{scope}[shift={(-4,0)}]
					\begin{scope}[xscale=0.5]
						\draw[fill=white!40!gray] (0,0) circle (0.3);
					\end{scope}
				\end{scope}
			\end{scope}
			\draw[fill=black] (0,2) circle (0.05);
			\draw[thick, dashed] (0,0) -- (0,2);
		\end{scope}

		\draw[red,dashed] (-1,-1) -- (-1,1);
		\node[left] at (-4.25,2) {${\cal L}_i$};
		\node[left] at (-2.95,1) {${\cal L}_j$};
		\draw[ultra thick, red] (-2,2) to[out=350,in=150] (-1.3,2.5) to[out=330,in=60]  (-1,1);
		\draw[fill=red, red] (-2,0) circle (0.07) (-1,-1) circle (0.07) (-2,2) circle (0.07) (-1,1) circle (0.07);
		\node[right, red] at (-1.2,2.5) {BPS string};
		\end{tikzpicture}
	\caption{Lefshetz thimbles.}\label{fig:thimbles}
\end{center}
\end{figure}

Consider an auxiliary category ${\cal C}$ where objects are Lefshetz thimbles and morphisms are corresponding BPS Hilbert spaces for corresponding choices of boundary conditions:
$$
{\rm Hom}^*({\cal L}_i,{\cal L}_j):=\mathfrak{G}_{\rm BPS}({\cal L}_i,{\cal L}_j).
$$
A Fukaya-Seidel category is a category where objects are Lagrangian manifolds. Having defined superpotential $W$ as well we can define the morphisms similarly as BPS Hilbert spaces for appropriate boundary conditions.\footnote{In principle, this definition requires a modification known as a wrapped Fukaya category \cite{Auroux,aganagic2021knot} since we allow Lagrangian manifolds to approach singularities. In this case their asymptotic behavior has to be fixed by additional data.} We clearly see that morphism space for a pair of thimbles ${\cal L}_i$ and ${\cal L}_j$ is nonzero only if
\bar{e}\label{Lef_ord}
{\rm Im}\,\zeta W_{*i}\geq {\rm Im}\,\zeta W_{*j}.
 \end{equation}
Therefore ordering thimbles according to ${\rm Im}\,\zeta W$ values one acquires naturally a sequence satisfying an exceptional collection constraint automatically.
The Fukaya-Seidel category is a triangulated category, and it can be restored from its basis through shifts and exact triangles \cite{Seidel_book}.
Thus for our purposes we accept as a definition for a Fukaya-Seidel category of brane boundary conditions a derived category of the auxiliary category ${\cal C}$.
Some details on a definition of a derived category are presented in Appendix \ref{sec:categories}.

Physically, this setup implies that as boundary conditions we accept Lefshetz thimbles with possible boundary operator insertions \cite{GMW} allowing a brane to carry a non-trivial complex.
We will consider some simple examples of boundary operators when turn to the instantons in Section \ref{sec:Web}.

The very construction implies that we consider our theory in the limit of a large interval $L\to\infty$. Physical sizes of BPS strings in the spatial dimension and $\zeta W$-plane are related:
$$
L=-\int\frac{d\,{\rm Im}\,\zeta W}{\left|g^{I\bar J}\partial_{\Sigma^I}W\overline{\partial_{\sigma^J}W}\right|}.
$$
To approach the limit $L\to\infty$ we need to move the BPS string segment in the $\zeta W$-plane towards the thimble tips -- LG vacua --  where the denominator diverges.
Near LG vacuum $(*i)$ one can linearize equation \eqref{LG_soliton} for deviation $\delta\Sigma$ from the vacuum value:
\bar{e}\label{LG_sol_lin}
\partial_1\delta\Sigma^I=-{\rm i}\zeta^{-1}\left(g^{I\bar J}\overline{\partial^2_{JK}W}\right)_{*i}\overline{\delta\Sigma^K}
 \end{equation}
A solution of this linear equation decays exponentially fast, as a solution to \eqref{LG_soliton} approaches the vacuum value.
This shrinks a core of a solution to \eqref{LG_soliton} where the solution differs from either constant vacuum solution to a narrow region of size $\lambda^{-1}$, where $\lambda$ is the minimal absolute value of eigen values of the linear operator in \eqref{LG_sol_lin}.
Such a solution behaves as a quasi-particle -- a \emph{soliton} --  or a domain wall in $(x^0,x^1)$-space-time.

Depending on a relation between ${\rm Re}\,\zeta W$ for boundary vacua a soliton can be either confined at one of the branes, or freely moving in the spatial $x^1$-direction:
\bar{e}\label{quasi_par_loc}
\begin{array}{c}
\begin{tikzpicture}
	\draw[ultra thick] (-1,0) -- (1,0) (-1,-0.1) -- (-1,0.1) (1,-0.1) -- (1,0.1);
	\node[above] at (-1,0.2) {$0$};
	\node[above] at (1,0.2) {$L$};
	\shade[ball color = red] (-1,0) circle (0.15);
	\node[below] at (0,-0.2) {${\rm Re}\,\zeta W_{*i}>{\rm Re}\,\zeta W_{*j}$};
	\node[left] at (-1.2,0) {$*i$};
	\node[right] at (1.2,0) {$*j$};

	\begin{scope}[shift={(4,0)}]
		\draw[ultra thick] (-1,0) -- (1,0) (-1,-0.1) -- (-1,0.1) (1,-0.1) -- (1,0.1);
		\node[above] at (-1,0.2) {$0$};
		\node[above] at (1,0.2) {$L$};
		\shade[ball color = red] (0,0) circle (0.15);
		\node[below] at (0,-0.2) {${\rm Re}\,\zeta W_{*i}={\rm Re}\,\zeta W_{*j}$};
		\node[left] at (-1.2,0) {$*i$};
		\node[right] at (1.2,0) {$*j$};
		\node[above] at (0,0.15) {$x_c$};
	\end{scope}
	\begin{scope}[shift={(8,0)}]
		\draw[ultra thick] (-1,0) -- (1,0) (-1,-0.1) -- (-1,0.1) (1,-0.1) -- (1,0.1);
		\node[above] at (-1,0.2) {$0$};
		\node[above] at (1,0.2) {$L$};
		\shade[ball color = red] (1,0) circle (0.15);
		\node[below] at (0,-0.2) {${\rm Re}\,\zeta W_{*i}<{\rm Re}\,\zeta W_{*j}$};
		\node[left] at (-1.2,0) {$*i$};
		\node[right] at (1.2,0) {$*j$};
	\end{scope}
\end{tikzpicture}
\end{array}
 \end{equation}
In the latter case the core position $x_c$ is a translation modulus of a soliton solution at $L\to\infty$. Depending on situation we name corresponding solutions left or right confined soliton or a free $ij$-soliton. We should stress that the modulus $x_c$ is fictitious and does not appear at finite $L$,  at finite $L$ there is still a small interaction between the quasi-particle and boundaries forcing $x_c$ to have a fixed equilibrium value.
\subsection{``Soft" Higgs branch: coherent sheaves}

Let us start with the strict case $\mu_a=0$.

In this case vacuum equations \eqref{sol_vac} have a trivial almost stationary solution:
\bar{e}
\begin{split}
	\sigma_{{\mathbb{R}}}=\sigma_{\mathbb{I}}=0,\\
	\phi_a=e^{{\rm i} Q_a \vartheta_0}\Phi_a,\\
	A_1=-\partial_1\vartheta_0,
\end{split}
 \end{equation}
where $\Phi_a$ are Higgs branch moduli satisfying
\bar{e}\label{Hb}
\sum\limits_{a=1}^n Q_a|\Phi_a|^2=r,
 \end{equation}
and $\vartheta_0$ is an arbitrary function.
Let us argue there are no other soliton solutions.
We can deduce a secondary equation:
\bar{e}\label{no_soli}
\left(-\partial_1^2+\sum\limits_a Q_a^2|\phi_a|^2\right)\sigma_{\mathbb{I}}=0.
 \end{equation}
Notice the operator in brackets in \eqref{no_soli} has only positive eigenvalues for constant vacuum boundary conditions, therefore this equation has no soliton solutions except the trivial one.

The equivariant term in the supercharge localizes filed configurations to constant modes up to a gauge transformation.
After integration over quickly oscillating modes we derive an effective supercharge depending only on constant modes of $\phi_a$ and $\sigma_{{\mathbb{R}}}$.
We put details on the one-loop renormalization in this system in Appendix \ref{sec:Ren_soft}.
The effective supercharge reads:
\bar{e}\label{eff_supercharge}
\begin{split}
	\bar{\cal Q}_B^{(\rm eff)}=-{\rm i}\sqrt{2}\sum\limits_a\bar\eta_{a}\left(\hbar \partial_{\bar \Phi_{a}}+Q_a\Sigma_{{\mathbb{R}}}\Phi_{a}\right)+\\
	+\nu\left(\partial_{\Sigma_{{\mathbb{R}}}}+\left(\sum\limits_aQ_a|\Phi_{a}|^2-r\right)\right)+\bar{{\cal Q}}_{\rm bdry}.
\end{split}
 \end{equation}
Boundary electric charge constraint also reduces to a contribution of solely constant modes:
\bar{e}\label{chrg}
\sum\limits_a Q_a\left(\bar \Phi_{a}\frac{\partial}{\partial\bar \Phi_{a}}- \Phi_{a}\frac{\partial}{\partial \Phi_{a}}+\bar\eta_{a}\eta_{a }\right)=\Delta{\bf q}.
 \end{equation}
Consider a projective variety $X$ given by tuples of complex numbers $z_a$ $a=1,\ldots,n$ modulo a transform:
$$
z_a\sim \lambda^{Q_a} z_a,\quad \lambda\in \mathbb{C}^{\times}.
$$
Now redefining fields as:
$$
\Phi_{a}=e^{Q_a(\rho+{\rm i}\vartheta)}z_a, \quad\rho,\vartheta\in {\mathbb{R}}.
$$
we will find that both fields $\rho$ and $\vartheta$ acquire expectation values in the IR.
The expectation value of field $\rho$ is fixed by the D-term constraint, and the value of field $\vartheta$ is fixed by eigenvalue \eqref{chrg}.
We derive that the effective theory describes a particle moving in $X$ so that the effective supercharge can be decomposed as:
\bar{e}
\bar {\cal Q}_{\rm eff}=\bar\nabla+\bar{\cal Q}_{\rm bdry},
 \end{equation}
where $\bar\nabla$ is a $\bar\partial$-component of a Dolbeault differential on $X$ extended by the corresponding Levi-Civita connection and the $U(1)$ connection with charge $\Delta{\bf q}$.
We will give a derivation of this fact in Appendix \ref{s:sheaf_coho_CP_n} for the case $X=\mathbb{C}\mathbb{P}^{n-1}$. $\bar{\cal Q}_{\rm bdry}$ has contributions from two branes located at interval ends $x^1=0$ and $x^1=L$.
We call these contributions $\bar{\cal Q}_{\cal A}$ and $\bar{\cal Q}_{\cal B}$ correspondingly:
\bar{e}\label{superch}
\bar {\cal Q}_{\rm eff}=\bar\nabla+\bar{\cal Q}_{{\cal A}}-\bar{\cal Q}_{{\cal B}}.
 \end{equation}
Both $\bar{\cal Q}_{\cal A}$ and $\bar{\cal Q}_{\cal B}$ define complexes of vector bundles, or more generally, coherent sheaves we call ${\cal A}$ and ${\cal B}$.
Details on coherent sheaf properties can be found in Appendix \ref{s:DCoh}. Supercharge \eqref{superch} defines a differential on a complex with the following cochains:
\bar{e}
{\cal C}^n:=\bigoplus\limits_{k=0}^n\Omega^{(0,k)}(X)\otimes {\rm Hom}_{{\cal O}_X}({\cal A},{\cal B}[n-k]).
 \end{equation}
This complex gives an injective resolution of a Hom-sheaf of ${\cal A}$ and ${\cal B}$. Since we have identified the BPS Hilbert space with the cohomologies of $\bar {\cal Q}_{\rm eff}$ we can rewrite our Hilbert space as cohomologies of a derived functor:
\bar{e}
\mathfrak{G}_{\rm BPS,\; soft\; Higgs}^{{\cal F}}\cong {\bf R}^{{\cal F}}{\rm Hom}_{D{\rm Coh}(X)}\left({\cal A},\underline{{\cal B}}\right),
 \end{equation}
where derivations are taken in the underlined argument, and homological degree ${\cal F}$ corresponds to the fermion number eigenvalue.

A simple application of this setup to a textbook example of sheaf valued cohomologies of $\mathbb{C}\mathbb{P}^n$ can be found in Appendix \ref{s:sheaf_coho_CP_n}.

\subsection{``Rigid" Higgs branch: equivariant action}\label{sec:RigidHiggs}

The roles of real and imaginary parts of the chiral masses $\mu_i$ appear to be different.
Real parts $\mu_{{\mathbb{R}},a}$ contribute to the Morse height functional, imaginary ones $\mu_{\mathbb{I},a}$ contribute to the equivariant action on the filed space.
Therefore we can refer to the rigid Higgs branch as an equivariant derived category as opposed to the soft Higgs branch.
Unfortunately, a definition of derived equivariant categories of coherent sheaves is rather involved \cite{Equivariant_sheves} and lacks a simple operational definition in the literature like a Cartan model or a BRST model for equivariant cohomologies \cite{Cordes:1994fc}.
So one could consider this physical model as a Cartan model-like version of equivariant cohomologies valued in sheaves.

Constant vacua of the theory correspond to the points on the vacuum manifold fixed with respect to the complexified gauge group:
\bar{e}
b^{\rm th}\;{\rm fixed \;point}:\quad \sigma=Q_b^{-1}\mu_b,\quad |\phi_{a=b}|=r^{\frac{1}{2}},\quad \phi_{a\neq b}=0,\quad\mbox{if}\;Q_b^{-1}r>0.
 \end{equation}

We call these fixed points $\sigma$-model vacua for simplicity.
We would like to study the BPS wave function behavior in a presence of such a $\sigma$-model vacuum.
The following decomposition allows one to fix automatically the gauge by extracting overall phase $\vartheta$:
\bar{e}
\begin{split}
	\phi_{a=b}=e^{{\rm i} Q_b\vartheta}\left(r^{\frac{1}{2}}+\hbar^{\frac{1}{2}}\rho\right),\quad \phi_{a\neq b}=e^{{\rm i} Q_a\vartheta}\hbar^{\frac{1}{2}}f_a,\quad \psi_{\alpha,a}=e^{-{\rm i} Q_a\vartheta}\eta_{\alpha,a},\quad  A_1=-\partial_1\vartheta.
\end{split}
 \end{equation}

The supercharge contribution from $f_a$-modes reads:
\bar{e}
\begin{split}
\bar Q_B=\sum\limits_{a\neq b}\int dx^1\;\bigg(-{\rm i}\sqrt{2}\bar\eta_{\dot 1,a}\left(\delta_{\bar f_a}+\left[Q_aQ_b^{-1}\mu_{{\mathbb{R}},b}-\mu_{{\mathbb{R}},a}\right]f_a\right)-\\
-\sqrt{2}\bar\eta_{\dot 2,a}\left(\partial_1 f_a-\left[Q_aQ_b^{-1}\mu_{\mathbb{I},b}-\mu_{\mathbb{I},a}\right]f_a\right)\bigg).
\end{split}
 \end{equation}
The peculiarity of this supercharge is that the equivariant vector field term has a fixed point in maps from $[0,L]$:
$$
f_{a,0}\sqrt{\frac{2\mu'_{\mathbb{I}}}{e^{2\mu_{\mathbb{I}}'L}-1}}e^{\mu_{\mathbb{I}}'x^1},\quad \mu_{\mathbb{I}}'=Q_aQ_b^{-1}\mu_{\mathbb{I},b}-\mu_{\mathbb{I},a}.
$$
The supercharge for this mode is purely chiral and corresponds to a supercharge of a particle in a magnetic field perpendicular to the complex plane spanned by complex coordinate $f_{a,0}$:
$$
\bar Q_B=-{\rm i}\sqrt{2}\bar\eta_{\dot 1, a ,0}\left(\frac{\partial}{\partial\bar f_{a,0}}+\mu_{{\mathbb{R}}}'f_{a,0}\right),\quad \mu_{{\mathbb{R}}}'=Q_aQ_b^{-1}\mu_{{\mathbb{R}},b}-\mu_{{\mathbb{R}},a}.
$$
Therefore rather having just a single BPS state we have the whole lowest Landau level Hilbert space with states labeled by angular momentum number $\ell$:
\bar{e}\label{cond_wf}
\Psi=\left\{\begin{array}{lll}
f_{a,0}^{\ell}e^{-|\mu_{{\mathbb{R}}}'||f_{a,0}|^2}|0\rangle, & \mu_{{\mathbb{R}}}'>0, & \ell\geq 0;\\
\bar f_{a,0}^{\ell}e^{-|\mu_{{\mathbb{R}}}'||f_{a,0}|^2}\bar\eta_{\dot 1,a,0}|0\rangle, & \mu_{{\mathbb{R}}}'<0, & \ell\geq 0.
\end{array}\right.
 \end{equation}
This Landau Hilbert space is a condensate of gauge invariant mesons
\bar{e}\label{meson}
f_a=\phi_a\phi_b^{-Q_a/Q_b},
 \end{equation}
analogous to one observed in the cylindrical case in Section \ref{sec:Preamble}, the wave functions form a sheaf of holomorphic or anti-holomorphic (depending on the sign of $\mu_{{\mathbb{R}}}'$) functions in a local chart containing the fixed point.

The angular momentum number $\ell$ is captured by the equivariant degree, or the central charge shift:
\bar{e}
\Delta\tilde Z=-{\rm i}\left(Q_aQ_b^{-1}\mu_b-\mu_a\right)\ell.
 \end{equation}

When the interval $L$ between branes becomes large zero mode condensate may be attracted to either left or right brane depending on the sign of $\mu_{\mathbb{I}}'$:
\bar{e}\label{condensate}
\begin{array}{c}
\begin{tikzpicture}
	\node[left] at (0,0) {$\lim\limits_{L\to\infty}\sqrt{\dfrac{2\mu'_{\mathbb{I}}}{e^{2\mu_{\mathbb{I}}'L}-1}}e^{\mu_{\mathbb{I}}'x^1}$};
	\node[right] at (6,1) {$\delta(L-x)$};
	\node[right] at (6,0) {const};
	\node[right] at (6,-1) {$\delta(x)$};
	\draw[->] (0.1,0.2) to[out=45,in=180] (1,1) -- (6,1);
	\draw[->] (0.1,0) -- (6,0);
	\draw[->] (0.1,-0.2) to[out=315,in=180] (1,-1) -- (6,-1);
	\node[above right] at (1.2,1) {$\mu_{\mathbb{I}}'>0,\;{\rm Re}\Delta\tilde Z>0$};
	\node[above right] at (1.2,0) {$\mu_{\mathbb{I}}'=0,\;{\rm Re}\Delta\tilde Z=0$};
	\node[above right] at (1.2,-1) {$\mu_{\mathbb{I}}'<0,\;{\rm Re}\Delta\tilde Z<0$};
\end{tikzpicture}
\end{array}
 \end{equation}

This behavior is a duality counterpart of localization \eqref{quasi_par_loc}.

For generic chiral masses $\mu_a$ vacuum equations \eqref{sol_vac} also admit solitonic solutions interpolating between two constant solutions in asymptotic similarly to the Landau-Ginzburg model.
Linearized equations in the neighborhood of a $\sigma$-model vacuum predict that a solitonic particle has a finite size:
\bar{e}\label{sigma_sol_size}
{\rm min}\left(r^{\frac{1}{2}},|\mu_a|\right)^{-1}.
 \end{equation}
Unfortunately, unlike the LG model case we are not aware about techniques that will allow one to construct such a solution in a general setting.
We will present an example of explicit calculation of a soliton solution using purely $\sigma$-model means without reference to the dual LG theory in Section \ref{sec:sigma-model-sol} for the case of equivariant $\mathbb{C}\mathbb{P}^1$.

\subsection{Parallel transport and web formalism}\label{sec:Web}
\subsubsection{Interfaces}
Parameter space ${\cal P}$ is spanned by FI parameters and topological angles for all gauge groups as well as chiral mass parameters.
In our particular case of a single $U(1)$ gauge group we allow only corresponding FI parameter and topological angle to vary along the interface, and it is convenient to combine them in a single complex parameter:
$$
t=r-{\rm i} \theta.
$$
So in our particular case ${\cal P}$ is just a complex plane of $t$.
However we should note that the technology of web formalism of \cite{GMW} works for generic ${\cal N}=(2,2)$ 2d theories with massive vacua.

An interface path $\wp$ is a map:
$$
\wp:\quad [0,L]\longrightarrow {\cal P}.
$$

The vacuum locus $\mathfrak{V}$ corresponds to a solution of a the vacuum equations \eqref{sol_vac} for the $\sigma$-model and \eqref{LG_soliton} correspondingly with constant parameter $t$ substituted with the interface path:
$$
t=\wp\left(x^1\right).
$$
This problem can be easily solved in the limit of \emph{adiabatic} approximation.
We assume that the variation of parameter $t$ along $\wp$ is \emph{adiabatic} if a derivative $|\partial_1t|$ is much smaller than all the soliton masses.
In a theory with generically massive vacua we always can choose such a path since we consider a limit $L\to \infty$, so that quantity $|\partial_1t|\geq|\Delta t|/L$ is allowed to be as small as necessary.
In this limit $t$ may be assumed to be almost a constant, therefore a solution for generic adiabatic function $t(x^1)$ can be ``glued" from solutions for constant $t$ we considered in the previous part of this section using a diagrammatic technique.

All our diagrams are constructed in a line segment $[0,L]$ and consist of vertices and links. Vertices are divided into bulk ones and boundary ones.

Bulk vertices correspond to $ij$-solitons having a transition modulus, after inserting such a vertex the soliton modulus is no more a free parameter, rather it is fixed to satisfy a stability constraint:
\bar{e}\label{stability}
\begin{array}{c}
\begin{tikzpicture}
	\draw[ultra thick] (-0.5,0) -- (0.5,0);
	\shade[ball color = red] (0,0) circle (0.15);
	\node[left] at (-0.5,0) {$*i$};
	\node[right] at (0.5,0) {$*j$};
	\node[above] at (0,0.15) {$x_c$};
\end{tikzpicture}
\end{array},\quad \zeta\tilde{\cal Z}^{(i\to j)}\left(t(x_c)\right)\in -{\rm i} {\mathbb{R}}_{\geq 0}.
 \end{equation}

Correspondingly boundary vertices are represented by solitons confined at boundaries accompanied by stability constraints or just empty boundary with a choice of a constant vacuum $*i$:
\bar{e}
\begin{split}
\begin{array}{c}
\begin{tikzpicture}
	\draw[ultra thick] (0,0) -- (0.5,0) (0,-0.1) -- (0,0.1);
	\node[left] at (0,0) {$*i$};
	\node[right] at (0.5,0) {$*i$};
	\node[above] at (0,0.1) {$0$};
\end{tikzpicture}
\end{array}\quad \begin{array}{c}
\begin{tikzpicture}
	\draw[ultra thick] (0,0) -- (0.5,0) (0.5,-0.1) -- (0.5,0.1);
	\node[left] at (0,0) {$*i$};
	\node[right] at (0.5,0) {$*i$};
	\node[above] at (0.5,0.1) {$L$};
\end{tikzpicture}
\end{array}\\
\begin{array}{c}
	\begin{tikzpicture}
		\draw[ultra thick] (0,0) -- (0.5,0);
		\shade[ball color = red] (0,0) circle (0.15);
		\node[left] at (-0.15,0) {$*i$};
		\node[right] at (0.5,0) {$*j$};
		\node[above] at (0,0.15) {$0$};
	\end{tikzpicture}
\end{array},\quad {\rm Re}\,\zeta{\cal Z}^{(i\to j)}\left(t(0)\right)>0 \quad {\rm Im}\,\zeta{\cal Z}^{(i\to j)}\left(t(0)\right)<0\\
\begin{array}{c}
	\begin{tikzpicture}
		\draw[ultra thick] (0,0) -- (0.5,0);
		\shade[ball color = red] (0.5,0) circle (0.15);
		\node[left] at (0,0) {$*i$};
		\node[right] at (0.65,0) {$*j$};
		\node[above] at (0.5,0.15) {$L$};
	\end{tikzpicture}
\end{array},\quad {\rm Re}\,\zeta{\cal Z}^{(i\to j)}\left(t(L)\right)<0 \quad {\rm Im}\,\zeta{\cal Z}^{(i\to j)}\left(t(L)\right)<0
\end{split}
 \end{equation}

Links connect vertices in such a way that linked vacua match.
From a diagram we can produce a solution function on a segment $[0,L]$ to the soliton equation in the following way.
One needs just to scan a diagram along $x^1\in[0,L]$ from the left to the right:
\begin{enumerate}
	\item For a generic point of a link the solution is the corresponding solution in vacuum $*i$ for parameter values $t(x^1)$.
	\item If $x^1$ hits a diagram vertex at some $x_c$ we implant corresponding soliton solution in a spatial interval $(x_c-m^{-1},x_c+m^{-1})$, where $m$ is the corresponding soliton mass.
	This solution approaches corresponding vacua at the interval ends exponentially fast $O(e^{-m|x|})$.
\end{enumerate}

All such diagrams are points in the vacuum locus $\mathfrak{V}$.
The MSW complex \eqref{MSW} as a vector space is spanned by wave functions corresponding to Gaussian fluctuations around $p\in\mathfrak{V}$.
In our case it is bi-graded by fermion number ${\cal F}$ and twisted central charge $\tilde{\cal Z}$.
A universal way to produce the diagrams and calculate corresponding quantum numbers could be given for the LG model in terms of spectral networks techniques.
The duality allows one to extend this technique to $\sigma$-models.

The resemblance between soliton behavior and that of quasi-particles on a 2d world-volume will become more transparent when we consider instantons. We will incorporate a specific notations for 2d quasi-particles:
$$
\mathfrak{X}_{ij},
$$
implying that the world-line of such a particle represents a domain wall between $*i$ and $*j$ constant vacua. Suppose on a diagram we have a sequence of bound quasi-particles:
$$
\mathfrak{X}_{i_1i_2},\mathfrak{X}_{i_2i_3},\ldots, \mathfrak{X}_{i_{n-1}i_n}.
$$
Using technique described in this section one extracts a vacuum field configuration, then it is a simple task to derive a  perturbative wave function as perturbative quantum fluctuations around vacuum filed configuration, and then a state vector associated to this critical point. This state has a fixed fermion number $f$, and we will denote this wave function in the following way:
\bar{e}\label{bpsi}
{\boldsymbol{\psi}}_{i_1i_n}^{(f)}\left[\mathfrak{X}_{i_1i_2},\mathfrak{X}_{i_2i_3},\ldots, \mathfrak{X}_{i_{n-1}i_n}|\wp\right].
 \end{equation}
We do not choose normalization for these vectors, so ${\boldsymbol{\psi}}$ is a whole 1d state subspace in the Hilbert space.
To denote a wave function associated to quantum fluctuations around a constant vacuum $*i$ we will use an argument $1$, and what constant vacuum is in consideration will be seen from subscripts of ${\boldsymbol{\psi}}$. Explicit path notation $\wp$ will be omitted in ${\boldsymbol{\psi}}$ when it is obvious.

A concatenation rule for diagrams is reflected in a relation for wave functions (fermion numbers add due to its extensive property, see Appendix \ref{sec:App_FS_cat}):
\bar{e}
\begin{split}
{\boldsymbol{\psi}}_{i_1i_k}^{(f_1)}\left[\mathfrak{X}_{i_1i_2},\ldots, \mathfrak{X}_{i_{k-1}i_k}|\wp_1\right]\otimes {\boldsymbol{\psi}}_{i_ki_n}^{(f_2)}\left[\mathfrak{X}_{i_ki_{k+1}},\ldots, \mathfrak{X}_{i_{n-1}i_n}|\wp_2\right]
={\boldsymbol{\psi}}_{i_1i_n}^{(f_1+f_2)}\left[\mathfrak{X}_{i_1i_2},\ldots, \mathfrak{X}_{i_{n-1}i_n}|\wp_1\circ \wp_2\right].
\end{split}
 \end{equation}

\subsubsection{Instantons}
To construct the differential in the MSW complex one needs to consider instantons \eqref{instanton}. The simplest way to reproduce an instanton equation is to consider the corresponding supersymmetry variation of fermionic fields and apply the Wick rotation to the temporal direction:
$$
x^0\longrightarrow -{\rm i} x^2.
$$
After this rotation it is convenient to parameterize the world-sheet by a complex coordinate:
$$
z=x^1+{\rm i} x^2.
$$
In this way we derive instanton equations for the GLSM:
\bar{e}\label{sigma-instanton}
\begin{split}
	F_{z\bar z}=0,\quad
	2\partial_{\bar z}\sigma=-{\rm i}\zeta^{-1}\left(r-\sum\limits_{a}Q_a|\phi_a|^2\right),\\
	2D_{\bar z}\phi_a={\rm i}\zeta^{-1}(Q_a\bar\sigma-\bar\mu_a)\phi_a,\quad
	2D_{\bar z}\bar\phi_a={\rm i}\zeta^{-1}(Q_a\bar\sigma-\bar\mu_a)\bar\phi_a,
\end{split}
 \end{equation}
and the Landau-Ginzburg model:
\bar{e}\label{LG-instanton}
2\partial_{\bar z}\Sigma=-{\rm i}\zeta^{-1}\bar W'.
 \end{equation}

In field theories we can impose a BPS bound on the Euclidean action to be bound below by a topological term. Instantons as BPS solutions saturate this bound, so that the  instanton field configurations satisfy integral equations:
\bar{e}\label{BPS_inst}
\begin{split}
\int\limits_A d^2x\left[\sum\limits_a \left(|D_1\phi_a|^2+|D_2\phi_a|^2+2|Q_a\sigma-\mu_a|^2|\phi_a|^2\right)+4|\partial_{\bar z}\sigma|^2+\left(r-\sum\limits_a Q_a|\phi_a|^2\right)^2\right]=\\
=-2{\rm Re}\;\zeta^{-1}\ointctrclockwise\limits_{\partial A}d\bar z\;\left(\bar\sigma\left(r-\sum\limits_aQ_a|\phi_a|^2\right)+\sum\limits_a\mu_a|\phi_a|^2\right);\\
\int\limits_A d^2 x\left(4|\partial_{\bar z}\Sigma|^2+|W'|^2\right)=-2{\rm Re}\ointctrclockwise\limits_{\partial A}\zeta W\;dz.
\end{split}
 \end{equation}
Then we come to a generic argument of \cite{GMW} that instanton equations \eqref{sigma-instanton} and \eqref{LG-instanton} do not have point-like solutions approaching a single LG or $\sigma$-model vacuum at infinity.
For a single vacuum at infinity the right hand side of both relations in \eqref{BPS_inst} is zero, then constraint \eqref{BPS_inst} admits only trivial constant (up to gauge transformations) field configurations, unlike the choice of an A-twist \cite{Hori:2000kt} in the $\sigma$-model, where point-like BPS vortex solutions appear.
Instead in our situation solutions may localize to 1d soliton quasi-particle world-lines.

Any steady soliton particle gives an Euclidean $x^2$-time independent solution to \eqref{sigma-instanton} or \eqref{LG-instanton}. Moving soliton quasi-particles are also solutions to the instanton equations.
Notice that \eqref{sigma-instanton} and \eqref{LG-instanton} are invariant under the following change of variables:
\bar{e}
z\longrightarrow e^{{\rm i}\varphi}z,\quad \zeta \longrightarrow e^{-{\rm i}\varphi}\zeta.
 \end{equation}
This transformation acts as an Euclidean boost on $(x^1,x^2)$-space-time with rapidity $\varphi$.
The soliton with a free translation modulus can be put in the bulk and form a core of such an instanton solution.
For such a soliton to satisfy the stability condition \eqref{stability} and to preserve B-twist with chosen parameter $\zeta$ its boost rapidity should satisfy:
\bar{e}\label{rapidity}
e^{{\rm i}\varphi}=-\zeta^{-1}\frac{\tilde {\cal Z}}{|\tilde {\cal Z}|}.
 \end{equation}
Due to the presence of the interface background the free soliton central charge depends adiabatically on $x^1$, its world-line forms a curved trajectory $(x^1(\tau),x^2(\tau))$ where $\tau$ is a trajectory proper time:
\bar{e}
\dot x^1(\tau)\cdot {\rm Re}\left[\zeta^{-1}\tilde Z\left(x^1(\tau)\right)\right]+\dot x^2(\tau)\cdot{\rm Im}\left[\zeta^{-1}\tilde Z\left(x^1(\tau)\right)\right]=0.
 \end{equation}

Allowing solitons to move around we have to allow them to interact through scattering, both mutual one and one with the boundary branes.
The scattering processes are localized in the Euclidean space-time in point-like defects we call scattering vertices.
The corresponding scattering amplitude is given by a QFT path integral for fixed parameter value $t\in{\cal P}$ and a fan of asymptotic constant vacua:
\bar{e}
\begin{array}{ll}
\mbox{bulk:} & \mathfrak{a}\left[\{i_1,i_2,\ldots,i_n\}|t\right]\sim \begin{array}{c}
\begin{tikzpicture}
	\draw[dashed, fill=blue, opacity=0.4] (0,0) circle (1);
	\draw[thick] (0,-1) -- (0,0) (-0.7071,-0.7071) -- (0,0)
	(-0.7071,0.7071) -- (0,0) (0,0) -- (1,0);
	\draw[fill=red] (0,0) circle (0.1);
	\node[below] at (-0.382683, -0.92388) {$\scriptstyle *i_1$};
	\node[left] at (-1,0) {$\scriptstyle *i_n$};
	\node[below right] at (0.7071,-0.7071) {$\scriptstyle *i_2$};
	\draw[fill=black]

	(1.20104, 0.497488) circle (0.02)
	(0.919239, 0.919239) circle (0.02)
	(0.497488, 1.20104) circle (0.02)
	(0,1.3) circle (0.02)
	(-0.497488, 1.20104) circle (0.02);

\end{tikzpicture}
\end{array};\\
\mbox{boundary:} & \mathfrak{b}_l\left[\{i_0,i_2,\ldots,i_n\}|t\right]\sim\begin{array}{c}
\begin{tikzpicture}
	\draw[ultra thick] (0,-1) -- (0,1);
	\draw[dashed,  fill=blue, opacity=0.4] ([shift=(-90:1)]0,0) arc (-90:90:1);
	\draw[thick] (0,0) -- (0.7071,-0.7071) (0,0) -- (0.7071,0.7071);
	\draw[fill=black] (1.20104, 0.497488) circle (0.02) (1.20104, -0.497488) circle (0.02) (1.3,0) circle (0.02);
	\node[left] at (0,-0.7) {$\scriptstyle *i_0$};
	\node[left] at (0,0.7) {$\scriptstyle *i_n$};
	\node[below right] at (0.382683, -0.92388) {$\scriptstyle *i_1$};
	\node[above right] at (0.382683, 0.92388) {$\scriptstyle *i_{n-1}$};
	\draw[fill=red] (0,0) circle(0.1);
\end{tikzpicture}
\end{array} ,\quad \mathfrak{b}_r\left[\{i_0,i_2,\ldots,i_n\}|t\right]\sim \begin{array}{c}
\begin{tikzpicture}[xscale=-1]
	\draw[ultra thick] (0,-1) -- (0,1);
	\draw[dashed,  fill=blue, opacity=0.4] ([shift=(-90:1)]0,0) arc (-90:90:1);
	\draw[thick] (0,0) -- (0.7071,-0.7071) (0,0) -- (0.7071,0.7071);
	\draw[fill=black] (1.20104, 0.497488) circle (0.02) (1.20104, -0.497488) circle (0.02) (1.3,0) circle (0.02);
	\node[right] at (0,-0.7) {$\scriptstyle *i_n$};
	\node[right] at (0,0.7) {$\scriptstyle *i_0$};
	\node[below left] at (0.382683, -0.92388) {$\scriptstyle *i_{n-1}$};
	\node[above left] at (0.382683, 0.92388) {$\scriptstyle *i_{1}$};
	\draw[fill=red] (0,0) circle(0.1);
\end{tikzpicture}
\end{array}.
\end{array}
 \end{equation}
Asymptotic scattering soliton states are uniquely defined by this data as boosted solitons with rapidity defined by \eqref{rapidity}.
Vacuum fans for the bulk vertices with cyclically permuted vacua are equivalent.
On the boundaries we allow confined soliton states (see Sections \ref{sec:LG_model} and \ref{sec:RigidHiggs}) that can be also defined using a pair of vacua.
If we would like to specify an empty boundary brane as an asymptotic state we just set $*i_0=*i_1$ or $*i_{n-1}=*i_n$.

Solely bulk vertices satisfy a set of $A_{\infty}$-relations as Lagrangian disks in the Fukaya-Seidel category \cite{Auroux}, and being mixed up with boundary vertices they satisfy $L_{\infty}$-relations \cite{GMW}.
We will not consider and use these relations in the paper, rather we calculate necessary vertices using other methods.

Eventually, the supercharge matrix element is given by a sum over instantons -- web diagrams consisting of disks containing vertices and glued together along the boosted soliton trajectories.
Web diagrams give a name to this formalism \cite{Gaiotto:2015zna}.

\subsubsection{Interface homotopy and bootstrapping scattering vertices.}\label{sec:bootstrap}

The role of instantons becomes extra crucial when invariance of the interface MSW complex  under interface path homotopy is in consideration.
Let us introduce a homotopy parameter $h$ and a homotopy family of paths $\wp(h)$ interpolating between $\wp_0$ and $\wp_1$:
$$
\begin{array}{c}
\begin{tikzpicture}
	\draw[->] (-0.5,0) -- (3.5,0);
	\draw[ultra thick] (0,-0.08) -- (0,0.08) (3,-0.08) -- (3,0.08);
	\node(A)[above] at (0,0) {$\wp_0$};
	\node(B)[above] at (3,0) {$\wp_1$};
	\path (A) edge[<->] node[above] {\scriptsize homotopy} (B);
	\node[right] at (3.5,0) {$h$};
	\node[below] at (0,0) {$0$};
	\node[below] at (3,0) {$1$};
\end{tikzpicture}
\end{array}
$$
The  homotopy morphism produces therefore a two-parametric family of theory couplings and effective superpotentials:
$$
t=\wp(x^1,h),\quad W(\phi, x^1,h).
$$

Isomorphism of MSW complex cohomologies, or quasi-isomorphism of MSW complexes is guaranteed (see Section 10.7 of \cite{GMW}) by an existence of a pair of chain maps $U$ and $\tilde U$ between corresponding complexes satisfying:
\bar{e}\nonumber
\begin{split}
\begin{array}{c}
\begin{tikzpicture}
	\node[left] at (0,0) {$({\bf M}^*_0,{\bf Q}_0)$};
	\node[right] at (1.5,0) {$({\bf M}^*_1,{\bf Q}_1)$};
	\draw[->] (0,0.07) -- (1.5,0.07);
	\draw[<-] (0,-0.07) -- (1.5,-0.07);
	\node[above] at (0.75,0.07) {$U$};
	\node[below] at (0.75,-0.07) {$\tilde U$};
\end{tikzpicture}
\end{array}\!\!\!\!,\\
U\cdot\tilde U={\bf Id}+T_1\cdot {\bf Q}_0+{\bf Q}_0\cdot T_2,\\
\tilde U\cdot U={\bf Id}+T_3\cdot {\bf Q}_1+{\bf Q}_1\cdot T_4,
\end{split}
 \end{equation}
where $T_i$ are some maps.
Analogously to the non-perturbative corrections to the supercharge the map $U$ is saturated by contributions from a ``forced" instanton equation with appropriate boundary conditions:
\bar{e}\label{forced_instanton}
(\partial_{1}+{\rm i}\partial_h)\Sigma=-{\rm i}\zeta^{-1}\overline{\partial_{\Sigma}W(\Sigma,x^1,h)}.
 \end{equation}
There is a similar analog for \eqref{sigma-instanton}.

Another map $\tilde U$ is constructed using the inverse homotopy map. One constructs solutions for this equation in the same fashion as we did for the forced soliton equation by gluing those from solution pieces for constant parameters.
Thus one ends up again with summing over web diagrams where instantons contribute as vertices.

Homotopic invariance of the BPS Hilbert space leads to a categorification of a 2d wall-crossing formula \cite{GMW,Khan:2020hir}.
Due to wall-crossing solitons form new stable BPS bound-states from old ones, or some BPS states become unstable and decay.
It is natural to describe these decay-recombination processes by scattering amplitudes. We also can mimic the decay/recombination processes by a homotopic moves connecting different regions of the parameter space where different BPS spectra are established.
The decay/recombination amplitudes will be saturated by soliton scattering vertices $\mathfrak{a}$ contributing to \eqref{forced_instanton}.

We conclude that one way to bypass the necessity to calculate a solution to instanton equations \eqref{sigma-instanton} and \eqref{LG-instanton} is to bootstrap the amplitudes using the wall-crossing. In quite many cases the soliton states in the model are not degenerate: for a given central charge eigen value the dimension of the graded component of the BPS Hilbert space is 0 or 1.
To achieve such level of granulation of the graded BPS Hilbert space one might need to include extra gradings using other charges commuting with the supercharge. In this case the wall-crossing becomes simple and is described by primitive-like wall-crossing formulas \cite{Denef:2007vg}, from those we can restore certain scattering amplitude values.
To illustrate this approach with an example let us assume that we have a theory with at least three constant vacua $*i$, $*j$ and $*k$. Consider a homotopy between two phases where the spectra are inhabited by two stable solitons $\mathfrak{X}_{ij}$ and $\mathfrak{X}_{jk}$, and in one of the phases there is a new stable bound soliton state $\mathfrak{X}_{ik}$ that is a recombination of $\mathfrak{X}_{ij}$ and $\mathfrak{X}_{jk}$ due to wall-crossing.
This is the only scenario for $\mathfrak{X}_{ik}$ to appear in the spectrum therefore \eqref{forced_instanton} has a solution interpolating between interface asymptotic states containing a pair of bound $\mathfrak{X}_{ij}$ and $\mathfrak{X}_{jk}$ in the far past and $\mathfrak{X}_{ik}$ in the far future:
\bar{e}
\begin{array}{c}
\begin{tikzpicture}
	\draw[->] (0,-0.5) -- (0,2.5);
	\draw[fill=blue, opacity=0.4] (0,0) -- (3,0) -- (3,2) -- (0,2) -- cycle;
	\draw[ultra thick] (1,0) to[out=90,in=225] (1.5,1) (2,0) to[out=90,in=315] (1.5,1) (1.5,1) -- (1.5,2);
	\draw[dashed, thick] (1.5,1) to[out=135,in=270] (0.5,2) (1.5,1) to[out=45,in=270] (2.5,2);
	\draw[dashed] (0,1) -- (1.5,1);
	\draw[ultra thick] (-0.05,0) -- (0.05,0) (-0.05,1) -- (0.05,1) (-0.05,2) -- (0.05,2);
	\node[left] at (0,2.5) {$h$};
	\node[left] at (0,2) {$1$};
	\node[left] at (0,1) {$h_{\rm crit}$};
	\node[left] at (0,0) {$0$};
	\shade[ball color = red] (1,0) circle (0.1);
	\shade[ball color = blue] (2,0) circle (0.1);
	\shade[ball color = orange] (1.5,2) circle (0.1);
	\draw[fill=red] (1.5,1) circle (0.13);
	\node[right, fill=white] at (2.1,1) {$\mathfrak{a}_{\rm crit}\left[\{i,j,k\}\right]=\pm 1$};
	\node[below] at (1,-0.1) {$\mathfrak{X}_{ij}$};
	\node[below] at (2,-0.1) {$\mathfrak{X}_{jk}$};
	\node[above] at (1.5,2.1) {$\mathfrak{X}_{ik}$};
\end{tikzpicture}
\end{array},\quad \mathfrak{a}\left[\{i,j,k\}|h\right]=\left\{\begin{array}{ll}
\mathfrak{a}_{\rm crit}, & h\geq h_{\rm crit};\\
0, & h<h_{\rm crit}.
\end{array}\right.
 \end{equation}
The same solution contributes to scattering amplitude $\mathfrak{a}$ for vacuum fan $\{i,j,k\}$.
In principle, for calculating the supercharge non-perturbative corrections in practice we need only a sign value of the amplitude.

Having some a priori information about the behavior of the soliton states induced by wall-crossing one is able to bootstrap some soliton scattering amplitude values without solving instanton equations.

\subsubsection{Categorical parallel transport}

As we discussed in Section \ref{sec:LG_model} the brane category of the IR Landau-Ginzburg theory corresponds to the Fukaya-Seidel category.
The data of this category are defined by the K\"ahler manifold and a  holomorphic superpotential function $W$ on it.
Transition between Fukaya-Seidel categories with close superpotentials $W$ and $W'$ is a transition functor described well in the literature \cite{Seidel_book}.
The slow,  ``adiabatic" variation of parameters along  an interface path $\wp$ is lifted to a path in a family of superpotentials.
We used the adiabaticity property to extrapolate the behavior of the physical system on the interface by a physical system at constant parameter values.
The adiabaticity allows one to parallel transport a physical system along a path in a parameter space identifying states between close systems.
It is natural to expect that a similar phenomenon is established by the brane categories, so that brane boundary conditions ${\cal B}$ and an interface defect $\mathfrak{J}_{\wp}$ could be substituted by a brane hologram $\beta_{\wp}({\cal B})$, so that:
\bar{e}
\begin{array}{c}
	\begin{tikzpicture}
		\draw[dashed,fill=blue, opacity=0.2] (0,0) -- (3,0) -- (3,2) -- (0,2) -- cycle;
		\draw[dashed,fill=blue, opacity=0.4] (3,0) -- (4.5,0) -- (4.75,0.25) -- (4.5,0.5) -- (4.75,0.75) -- (4.5,1) -- (4.75,1.25) -- (4.5,1.5)  -- (4.75,1.75) -- (4.5,2) -- (3,2) -- cycle;
		\draw[ultra thick] (0,0) -- (0,2) (3,0) -- (3,2);
		\node[left] at (0,1) {${\cal B}$};
		\node[right] at (3,1) {$\beta_{\wp}\left({\cal B}\right)$};
		\draw[<->] (-1,2) -- (-1,-0.5) -- (4.5,-0.5);
		\node[right] at (4.5,-0.5) {$x^1$};
		\node[above] at (-1,2) {$x^0$};
		\draw[ultra thick] (0,-0.4) -- (0,-0.6) (3,-0.4) -- (3,-0.6);
		\node[below] at (0,-0.6) {$0$};
		\node[below] at (3,-0.6) {$0'$};
		\node at (1.5,1) {$\mathfrak{J}_{\wp}$};
		\draw[->] (0.25,0.5) -- (2.75,0.5);
	\end{tikzpicture}
\end{array},\quad \mathfrak{G}_{\rm BPS}(\mathfrak{J}_{\wp}|{\cal B},X)\cong \mathfrak{G}_{\rm BPS}(\mathfrak{J}_{\rm triv}|\beta_{\wp}({\cal B}),X),\;\forall X.
 \end{equation}

Here $\beta_{\wp}$ is a parallel transport functor, and $\mathfrak{J}_{\rm triv}$ is a simple interface associated with a trivial path $[0,L]\to\mbox{point}$. Moreover, applying adiabaticity we can claim that $\beta_{\wp}$ is a functor coinciding with the transition functor of the Fukaya-Seidel category through a family of superpotentials along $\wp$, it can be calculated explicitly using various techniques.

An actual proof of the proposal above is extremely involved, a large portion of \cite{GMW} is devoted to this issue, and we have no means to repeat the proof here.

The action of the transition functor of the Fukaya-Seidel category has the most transparent form for an exceptional collection basis of Lefshetz thimbles.
We will discuss explicit relations in Appendix \ref{sec:App_FS_cat}.
The derived category of coherent sheaves is also a triangulated category and admits a choice of a basis among the objects in the form of an exceptional collection, say, a basis of ${\cal O}(k)$ sheaves for $\mathbb{C}\mathbb{P}^n$.
The same statement follows naturally from the mirror symmetry \cite{D-book_1}, however there is no naive choice of basis elements nor ordering rule like \eqref{Lef_ord}.

Having constructed equivalences of categories of brane boundary conditions for various phases one extends the action of the brane parallel transport functor to an abstract category of brane boundary conditions:
\bar{e}\label{parallel_transport}
\beta_{\wp}:\quad \mathfrak{D}_{p_1}\longrightarrow \mathfrak{D}_{p_2}.
 \end{equation}
One can substitute this abstract category with any category suitable for each phase in corresponding chambers of the parameter space $p_{1,2}\in {\cal P}$.

The parallel transport functor in the Fukaya-Seidel category satisfies a composition law
\bar{e}
\beta_{\wp_2}\circ\beta_{\wp_1}=\beta_{\wp_1\circ\wp_2}.
 \end{equation}
This statement is also extendable to the abstract brane category.

\section{Equivariant \texorpdfstring{$\mathbb{C}\mathbb{P}^1$}{CP1}: soliton spectrum}\label{sec:equiv_CP_1}

\subsection{Model description}

Field defects (kinks, quasi-particles, domain walls) in sigma-models with a target space given by $\mathbb{C}\mathbb{P}^{\kappa}$ in 1+1 and higher dimensions attract interests of many researchers \cite{Eto:2006pg,Eto:2007aw,Eto:2007uc,Shifman:2010id,Fujimori:2016ljw,Harland:2009mf} as a toy model to test ideas about instanton behavior of Yang-Mills theories in 3+1 dimensions \cite{Polyakov:1987ez}.
In this section we will construct spectra of BPS domain wall solutions in a 2d ${\cal N}=(2,2)$ $\mathbb{C}\mathbb{P}^1$-model in both Higgs and Coulomb branch descriptions of the IR physics and will find a complete agreement between these spectra.
Depending on a concrete model these defects have different names, we will use a unifying name ``soliton"  for these quasi-particles to follow notions of \cite{GMW} as these defects are supported on 1d world-lines.
Eventually we will discuss scattering of these soliton solutions contributing to instantons discussed in Section \ref{sec:Web}.

\bigskip

$\mathbb{C}\mathbb{P}^1$ turns out to be the first rather simple, yet non-trivial, model establishing physical effects we are after.
Moreover, it turns out the $\mathbb{C}\mathbb{P}^1$ model could be considered as a primitive building block for other more involved models.
Therefore we would like to consider this setup in great details.

The key observation we are chasing in this section is a support of the duality relation we described in the previous section.
We will calculate spectra of BPS states on both branches explicitly and from the first principles.
A comparison of the spectra leads to a conclusion that the Hilbert spaces for two branches are isomorphic:
\bar{e}\label{CP_1_mirror}
\mathfrak{G}_{\rm BPS,\, Higgs}\left(\mathbb{C}\mathbb{P}^1\right)\cong \mathfrak{G}_{\rm BPS,\, Coulomb}\left(\mathbb{C}\mathbb{P}^1\right).
 \end{equation}

The $\mathbb{C}\mathbb{P}^1$ model is described by a $U(1)$ gauged linear sigma-model with 2 chiral multiplets of the same charge $Q_1=Q_2=1$ and complex masses $\mu_1$ and $\mu_2$. The matter content of this theory may be encoded in a simple quiver:
$$
\begin{array}{c}
	\begin{tikzpicture}
		\node at (-2,0) {$1$};
		\node at (0,0) {$1$};
		\node at (2,0) {$1$};
		\draw (0,0) circle (0.3);
		\begin{scope}[shift={(2,0)}]
			\draw (-0.3,-0.3) -- (-0.3,0.3) -- (0.3,0.3) -- (0.3,-0.3) -- cycle;
		\end{scope}
	\begin{scope}[shift={(-2,0)}]
		\draw (-0.3,-0.3) -- (-0.3,0.3) -- (0.3,0.3) -- (0.3,-0.3) -- cycle;
	\end{scope}
		\draw[->] (1.7,0) -- (0.3,0);
		\draw[->] (-1.7,0) -- (-0.3,0);
	\end{tikzpicture}
\end{array}
$$
We immediately write down an expression for the effective superpotential in the Landau-Ginzburg phase of the theory following a recipe of Section \ref{sec:LG_model}:
\bar{e}\label{CP_1_sup}
W(\Sigma)=t\Sigma+\frac{1}{2\pi}\sum\limits_{a=1}^2(\Sigma-\mu_a)\left({\rm log}\frac{\Sigma-\mu_a}{\Lambda}-1\right),
 \end{equation}
and corresponding spectral curve (see Appendix \ref{sec:App_SN} for details):
\bar{e}
(\lambda-\mu_1)(\lambda-\mu_2)=z,
 \end{equation}
where we identify the spectral parameters with the complexified FI parameter and superpotential expectation value derivative:
\bar{e}
z=e^{-2\pi t},\quad \lambda=-2\pi \,z\frac{d}{dz}W.
 \end{equation}

Properties of this spectral curve are simpler to describe using a relative complex mass parameter:
$$
\mu:=\mu_1-\mu_2.
$$

There is a single ramification point:
$$
z_{r}=-\mu^2/4,
$$
and the topology of the spectral network depends on the argument of $\mu$. The topology experiences jumps at half-integer portions of $\pi$ (see Fig.\ref{fig:SN_CP_1}).

\begin{figure}[h!]
	\begin{center}
		\begin{tikzpicture}
			\begin{scope}
				\node at (0,0) {\includegraphics[scale=0.4]{figs/CP_1_sol_1.pdf}};
				\draw[thick, blue] (-1.5,-1) to[out=0,in=210] (1.2,-0.05);
				\node[left,blue] at (-1.5,-1) {$\wp$};
				\draw[fill=orange] (-0.57,-0.9) circle (0.05) (-0.1,-0.73) circle (0.05) (0.2,-0.6) circle (0.05) (0.43,-0.5) circle (0.05) (0.6,-0.4) circle (0.05);
				\node[below,orange] at (-0.57,-0.9) {1};
				\node[below,orange] at (-0.1,-0.73) { \small 2};
				\node[below,orange] at (0.2,-0.6) { \footnotesize 3};
				\node[below,orange] at (0.43,-0.5) { \scriptsize 4};
				\node[below,orange] at (0.6,-0.4) { \tiny 5};
				\begin{scope}[shift={(-1.2,-0.05)}]
					\draw[ultra thick, orange] (-0.1,-0.1) -- (0.1,0.1) (0.1,-0.1) -- (-0.1,0.1);
				\end{scope}
				\draw[fill=red] (1.2,-0.05) circle (0.08);
				\node[below] at (0,-2) {a) $\begin{array}{c}0<{\rm Arg}\;\mu<\pi/2\\
						\pi<{\rm Arg}\;\mu<3\pi/2
					\end{array}$};
			\end{scope}
			%
			\begin{scope}[shift={(8,0)}]
				\node at (0,0) {\includegraphics[scale=0.4]{figs/CP_1_sol_2.pdf}};
				\begin{scope}[shift={(-1.2,0)}]
					\draw[ultra thick, orange] (-0.1,-0.1) -- (0.1,0.1) (0.1,-0.1) -- (-0.1,0.1);
				\end{scope}
				\draw[fill=red] (1.2,0) circle (0.08);
				\node[below] at (0,-2) {b) $\begin{array}{c}\pi/2<{\rm Arg}\;\mu<\pi\\
				3\pi/2<{\rm Arg}\;\mu<2\pi
				\end{array}$};
			\end{scope}
			%
			\begin{scope}[shift={(0,-5)}]
				\node at (0,0) {\includegraphics[scale=0.4]{figs/CP_1_sol_3.pdf}};
				\begin{scope}[shift={(-1.2,0)}]
					\draw[ultra thick, orange] (-0.1,-0.1) -- (0.1,0.1) (0.1,-0.1) -- (-0.1,0.1);
				\end{scope}
				\draw[fill=red] (1.2,0) circle (0.08);
				\node[below] at (0,-2) {c) ${\rm Arg}\;\mu=0,\;\pi$};
			\end{scope}
			%
			\begin{scope}[shift={(8,-5)}]
				\node at (0,0) {\includegraphics[scale=0.4]{figs/CP_1_sol_4.pdf}};
				\begin{scope}[shift={(-1.2,-0.05)}]
					\draw[ultra thick, orange] (-0.1,-0.1) -- (0.1,0.1) (0.1,-0.1) -- (-0.1,0.1);
				\end{scope}
				\draw[fill=red] (1.2,-0.05) circle (0.08);
				\node[below] at (0,-2) {d) ${\rm Arg}\;\mu=\pi/2,\;3\pi/2$};
			\end{scope}
		\end{tikzpicture}
	\caption{Spectral network topologies in $z$-plane for equivariant $\mathbb{C}\mathbb{P}^1$.}\label{fig:SN_CP_1}
	\end{center}
\end{figure}

\subsection{Coulomb branch soliton spectrum}

The vacua on the Coulomb branch of the theory are defined by solutions of the vacuum equation:
\bar{e}\label{CP_1_vacuum}
(\Sigma-\mu_1)(\Sigma-\mu_2)=\Lambda^2e^{-2\pi t}.
 \end{equation}
We are interested in solving it in a regime when Coulomb and Higgs branch overlap:
$$
{\rm Re}\;t\gg 0.
$$
In this case vacuum expectation value of field $\Sigma$ is given by a small deviation from either of complex mass values:
\bar{e}
\Sigma_{*1,2}=\mu_{1,2}\pm \frac{\Lambda^2}{\mu_1-\mu_2}e^{-2\pi t}+O\left(e^{-4\pi t}\right).
 \end{equation}
Despite the complex scalar expectation value is fixed by the vacuum choice the log function in the superpotential contributes with a multi-valuedness ambiguity. Therefore for the superpotential we acquire two infinite series of vacuum values labeled by integers:
\bar{e}
\begin{split}
	W_{*1,k}=t\mu_1+\frac{\mu_1-\mu_2}{2\pi}\left({\rm log}\frac{\mu_1-\mu_2}{\Lambda}-1\right)+{\rm i} k+O\left(e^{-2\pi t}\right),\quad k\in{\mathbb{Z}};\\
	W_{*2,k}=t\mu_2+\frac{\mu_2-\mu_1}{2\pi}\left({\rm log}\frac{\mu_2-\mu_1}{\Lambda}-1\right)+{\rm i} k+O\left(e^{-2\pi t}\right),\quad k\in{\mathbb{Z}}.
\end{split}
 \end{equation}
Those integers correspond to a choice of a line bundle ${\cal O}(k)$ as a Chan-Paton factor for the brane boundary conditions. To describe the soliton spectrum on the Coulomb branch we will depict vacua positions in the $W$-plane in Fig.\ref{fig:CP_1_vacua}.\footnote{A measurable value of $t$ gets renormalized in this picture: $$t_{\rm mes}:=\frac{W_{*1,0}-W_{*2,0}}{\mu_1-\mu_2}=t\pm\frac{{\rm i}}{2}-\frac{1}{\pi}+O\left((\Lambda/\mu) e^{-2\pi t}\right).$$ We neglect this shift in our consideration.}

\begin{figure}
	\begin{center}
		\begin{tikzpicture}[xscale=1.5, yscale=0.8]
			\begin{scope}
			\foreach \i in {1,...,19}
			{
				\draw[gray] (0.05 * \i - 1,0) -- (0.05 * \i - 1.5,2);
				\draw[gray] (-1 - 0.025 * \i,0.1 * \i) -- (- 0.025 * \i ,0.1 * \i);
			}
			\draw[ultra thick] (-1,0) -- (0,0) -- (-0.5,2) -- (-1.5,2) -- cycle (-1,0) -- (-0.5,2);
			\draw[fill=blue] (-1,0) ellipse (0.05333 and 0.1) (1,0) ellipse (0.05333 and 0.1)  (0,0) ellipse (0.05333 and 0.1)  (1,0) ellipse (0.05333 and 0.1)  (2,0) ellipse (0.05333 and 0.1)  (3,0) ellipse (0.05333 and 0.1)  (-1.5,2) ellipse (0.05333 and 0.1)  (-0.5,2) ellipse (0.05333 and 0.1)  (0.5,2) ellipse (0.05333 and 0.1)  (1.5,2) ellipse (0.05333 and 0.1)  (2.5,2) ellipse (0.05333 and 0.1)  (3.5,2) ellipse (0.05333 and 0.1) ;
			\draw[->] (1,0) to node[above] {$a$} (1.92,0);
			\begin{scope}[shift={(0.5,2)}]
				\draw[->] (1,0) to node[below] {$a$} (1.92,0);
			\end{scope}
			\draw[->] (1,0) to node[right]{$b$} (1.4806, 1.92239);
			\node[right] at (5,1) {$\begin{array}{l}
					a={\rm i}(\mu_1-\mu_2)\\
					b=t(\mu_1-\mu_2)
				\end{array}$};
			\node[right] at (3.2,0) {$W_{*1,{\mathbb{Z}}}$};
			\node[right] at (3.7,2) {$W_{*2,{\mathbb{Z}}}$};
			%
			\node[below left] at (-1,-0.08) {$-1$};
			\node[below right] at (0,-0.08) {$0$};
			\node[below] at (1,-0.08) {$+1$};
			\node[below] at (2,-0.08) {$+2$};
			\node[below] at (3,-0.08) {$+3$};
			%
			\node[above left] at (-1.5,2.08) {$-2$};
			\node[above right] at (-0.5,2.08) {$-1$};
			\node[above] at (0.5,2.08) {$0$};
			\node[above] at (1.5,2.08) {$+1$};
			\node[above] at (2.5,2.08) {$+2$};
			\node[above] at (3.5,2.08) {$+3$};
			%
			\node[below] at (-0.5,0) {$\mathfrak{X}_{A,-1}$};
			\node[above] at (-1,2) {$\mathfrak{X}_{B,-1}$};
			\node[left] at (-1.25,1) {$\mathfrak{X}_{C,-1}$};
			\node[right] at (-0.25,1) {$\mathfrak{X}_{C,-1}$};
			\draw[thick,->] (-2.5,0.2) to[out=0,in=180] (-1.5,0.2) to[out=0,in=150] (-0.75,1);
			\node[left] at (-2.5,0.2) {$\mathfrak{X}_{C,0}$};
			\end{scope}
		\end{tikzpicture}
\caption{Equivariant $\mathbb{C}\mathbb{P}^1$ vacua positions in $W$-plane}\label{fig:CP_1_vacua}
\end{center}
\end{figure}

The soliton equation \eqref{LG_soliton} has a solution interpolating from vacuum $(*i)$ to vacuum $(*j)$ if and only if the separation between vacua satisfies constraint $W_{*j}-W_{*i}\in {\mathbb{R}}_{\geq 0}$.

Therefore we calculate the following soliton spectrum on the Coulomb branch becoming a stable BPS state if the corresponding constraint is satisfied:\footnote{By the superscript we will denote the soliton flow direction between vacua.}
\begingroup
\renewcommand*{\arraystretch}{1.7}
\bar{e}\label{CP_1_solitons}
\begin{array}{c|c|c}
	\mbox{Name} & \mbox{Constraint} & \mbox{Central charge}\\
	\hline
	\mathfrak{X}_{A,k}& \begin{array}{c}
		\mu_{\mathbb{I},1}=\mu_{\mathbb{I},2}\\
	\end{array}& {\cal Z}_{\rm Clmb}^{(1\to 1)}=-|\mu_{{\mathbb{R}},1}-\mu_{{\mathbb{R}},2}|k,\;k\in{\mathbb{Z}}_{\geq 0}\\
\hline
\mathfrak{X}_{B,k}& \begin{array}{c}
	\mu_{\mathbb{I},1}=\mu_{\mathbb{I},2}\\
\end{array}& {\cal Z}_{\rm Clmb}^{(2\to 2)}=-|\mu_{{\mathbb{R}},1}-\mu_{{\mathbb{R}},2}|k,\;k\in{\mathbb{Z}}_{\geq 0}\\
\hline
\mathfrak{X}_{C,k}& \begin{array}{c}{\rm Re}\, t\,\frac{\mu_{{\mathbb{R}},1}-\mu_{{\mathbb{R}},2}}{\mu_{\mathbb{I},1}-\mu_{\mathbb{I},2}}-{\rm Im}\,t=:k\in{\mathbb{Z}}\\
	\mu_{\mathbb{I},1}>\mu_{\mathbb{I},2}
\end{array} & {\cal Z}^{(1\to 2)}_{\rm Clmb}=-\frac{{\rm Re}\,t}{\mu_{\mathbb{I},1}-\mu_{\mathbb{I},2}}|\mu_1-\mu_2|^2\\
\hline
\mathfrak{X}_{D,k}& \begin{array}{c}{\rm Re}\, t\,\frac{\mu_{{\mathbb{R}},1}-\mu_{{\mathbb{R}},2}}{\mu_{\mathbb{I},1}-\mu_{\mathbb{I},2}}-{\rm Im}\,t=:k\in{\mathbb{Z}}\\
	\mu_{\mathbb{I},1}<\mu_{\mathbb{I},2}
\end{array} & {\cal Z}^{(2\to 1)}_{\rm Clmb}=-\frac{{\rm Re}\,t}{\mu_{\mathbb{I},1}-\mu_{\mathbb{I},2}}|\mu_1-\mu_2|^2
\end{array}
 \end{equation}
\endgroup

Consider a path $\wp$ depicted in Fig.\ref{fig:SN_CP_1} (a).
This path flows in the direction  ${\rm Re}\,t\to+\infty$ in the $t$-plane and intersects the spectral network in a sequence of points.
Clearly this sequence is associated to either $\mathfrak{X}_{C,k}$ or $\mathfrak{X}_{D,k}$ soliton family depending on a sign of $\mu_{\mathbb{I},1}- \mu_{\mathbb{I},2}$.
So that as soon $\wp$ intersects the spectral network in a point the expression
$$
{\rm Re}\, t\,\frac{\mu_{{\mathbb{R}},1}-\mu_{{\mathbb{R}},2}}{\mu_{\mathbb{I},1}-\mu_{\mathbb{I},2}}-{\rm Im}\,t
$$
hits an integer value.

Solitons from $\mathfrak{X}_{A,k}$ and $\mathfrak{X}_{B,k}$ series have fixed masses independent of the $t$-parameter, therefore they do not produce in the $z$-plane a branching point where masses of a corresponding effective particle will flow to zero.
The only way they produce a contribution in this setting is a critical behavior at ${\rm Arg}\,\mu=0,\,\pi$ (Fig.\ref{fig:SN_CP_1} (c)).
At this critical value solitons from all families have co-directed central charges as vectors in a complex plane, therefore the solitons experience the wall-crossing phenomenon when some composite solitons decay to or recombine from more elementary ones.
The cluster coordinates corresponding to soliton partition function  experience a cluster transformation when the partition function associated with a soliton from $\mathfrak{X}_{C,k}$ or $\mathfrak{X}_{D,k}$ series is conjugated by a partition function associated to a gas of solitons from $\mathfrak{X}_{A,k}$ or $\mathfrak{X}_{B,k}$ series \cite{Manschot:2010qz,Gaiotto:2012rg,Galakhov:2014xba}.
We will consider some details of this wall-crossing process when discuss instantons in this model.

\subsection{Higgs branch soliton spectrum} \label{sec:sigma-model-sol}

On the Higgs branch the theory has also two classical vacua:
\bar{e}
\begin{array}{lll}
\mbox{1)}&\sigma_{*1}=\mu_1,& \vec \phi_{*1}=\left(r^{\frac{1}{2}},0\right);\\
\mbox{2)}&\sigma_{*2}=\mu_2,& \vec \phi_{*2}=\left(0,r^{\frac{1}{2}}\right).
\end{array}
 \end{equation}

As it was discussed in Section \ref{sec:RigidHiggs} in the case $\mu_{\mathbb{I},1}=\mu_{\mathbb{I},2}$ BPS wave functions in the corresponding vacua are given by the following expressions (see \eqref{cond_wf}):
\bar{e}\label{CP_1_const_wave}
\Psi_1\sim\left\{\begin{array}{ll}
	(\phi_2/\phi_1)^k|0\rangle,& \mu_{{\mathbb{R}},1}>\mu_{{\mathbb{R}},2}\\
	(\bar\phi_2/\bar\phi_1)^k\bar\psi_{2,\dot 1}|0\rangle,& \mu_{{\mathbb{R}},1}<\mu_{{\mathbb{R}},2}
\end{array}\right.,\quad \Psi_2\sim\left\{\begin{array}{ll}
(\phi_1/\phi_2)^k|0\rangle,& \mu_{{\mathbb{R}},1}<\mu_{{\mathbb{R}},2}\\
(\bar\phi_1/\bar\phi_2)^k\bar\psi_{1,\dot 1}|0\rangle,& \mu_{{\mathbb{R}},1}>\mu_{{\mathbb{R}},2}
\end{array}\right.,
 \end{equation}
where $\phi$ and $\psi$ are constant field modes.

Corresponding supercharges are:
\bar{e}
\begin{split}
{\cal Z}_{\rm Higgs}^{(1\to 1)}=-|\mu_{{\mathbb{R}},1}-\mu_{{\mathbb{R}},2}|k,\;k\in{\mathbb{Z}}_{\geq 0},\\
{\cal Z}_{\rm Higgs}^{(2\to 2)}=-|\mu_{{\mathbb{R}},1}-\mu_{{\mathbb{R}},2}|k,\;k\in{\mathbb{Z}}_{\geq 0}.
\end{split}
 \end{equation}
So these states resemble solitonic states $\mathfrak{X}_{A,k}$ and $\mathfrak{X}_{B,k}$ on the Coulomb branch.

In the case $\mu_{\mathbb{I},1}\neq \mu_{\mathbb{I},2}$ we have to solve a system of equations describing the critical field configuration:
\bar{e}\label{complex_vortex}
\begin{split}
	-\partial_1\sigma_{\mathbb{I}}+\left(\sum\limits_{a=1}^2|\phi_a|^2-r\right)=0,\\
	D_1\phi_a-(\sigma_{\mathbb{I}}-\mu_{\mathbb{I},a})\phi_a=0,\; a=1,2.
\end{split}
 \end{equation}
Remaining equations of system \eqref{sol_vac} can have a non-trivial solution only if
\bar{e}\label{mass_const}
\sigma_{{\mathbb{R}}}=\mu_{{\mathbb{R}},1}=\mu_{{\mathbb{R}},2}.
 \end{equation}

Let us perform the following change of variables:
\bar{e}
\sigma_{\mathbb{I}}=\frac{\mu_{\mathbb{I},1}+\mu_{\mathbb{I},2}}{2}+s,\quad \Delta\mu=\frac{\mu_{\mathbb{I},1}-\mu_{\mathbb{I},2}}{2},\quad A_1=\partial_1\vartheta,\quad \phi_1=e^{-{\rm i}\theta}r^{\frac{1}{2}}e^{\Phi}\cos\tau,\quad \phi_2=e^{-{\rm i}\theta} r^{\frac{1}{2}}e^{\Phi}\sin\tau.
 \end{equation}
In terms of the new variables equations \eqref{complex_vortex} are simplified:
\bar{e}\label{compl_v_simp}
\begin{split}
	\partial_1\tau&=\Delta \mu \,\sin 2\tau,\\
	s&=\partial_1\Phi+\Delta \mu\,\cos2\tau,\\
	\partial_1s&=r\left(e^{2\Phi}-1\right).
\end{split}
 \end{equation}
The first equation in the column \eqref{compl_v_simp} can be solved analytically:
\bar{e}\label{Higgs_sol}
\cos\tau=\frac{1}{\sqrt{1+e^{4\Delta \mu \left(x^1+c\right)}}},\quad \sin\tau=\frac{e^{2\Delta \mu \left(x^1+c\right)}}{\sqrt{1+e^{4\Delta \mu \left(x^1+c\right)}}},
 \end{equation}
This solution represents a domain wall of thickness\footnote{Solutions to the Liouville equation introduce another suppressing exponent $r^{-\frac{1}{2}}$.
So the actual width of the soliton core is given by \eqref{sigma_sol_size}.}
 $|\Delta \mu|^{-1}$ and located at $x^1=c$.
We call this solution a \emph{polar soliton} since it interpolates between poles of $\mathbb{C}\mathbb{P}^1$.
Depending on the sign of $\Delta \mu$ the soliton flows from vacuum 1 to vacuum 2 or in the inverse direction:
\bar{e}
\begin{array}{c}
\begin{tikzpicture}
\node[left] at (0,0) {$	{\vec \phi}_{*1}=\left(r^{\frac{1}{2}},0\right)$};
\node[right] at (3,0) {${\vec \phi}_{*2}=\left(0,r^{\frac{1}{2}}\right)$};
\draw[-stealth] (0,0.05) -- (3,0.05);
\draw[stealth-] (0,-0.05) -- (3,-0.05);
\node[above] at (1.5,0.05) {\footnotesize$\Delta\mu>0$};
\node[below] at (1.5,-0.05) {\footnotesize$\Delta\mu<0$};
\end{tikzpicture}
\end{array}
 \end{equation}
Unfortunately, we are unable to solve remaining equations in system \eqref{compl_v_simp} analytically since the resulting equation for $\Phi$ is a Liouville equation with a non-trivial external source.
Rather we could perform perturbative expansion in a regime $\Delta\mu/\sqrt{r}\ll 1$ confirming each soliton solution \eqref{Higgs_sol} is accompanied by a single solution to the complete system.
The resulting expressions turn out to be rather bulky and we will never use their explicit form, therefore we omit them in our consideration.

As usual the low energy dynamics of this solutions is governed by zero modes  -- variations of fields $\delta \phi_a$, $\delta\sigma_{\mathbb{I}}$, $\delta A_1$ along the soliton moduli we will calculate momentarily.
We would like to accompany the usual zero mode equations by an additional constraint for field variation following from the Gauss law \eqref{Gauss}:
$$
\partial_1\delta A_1={\rm i}\sum\limits_{a=1}^2\left(\bar\phi_a\delta\phi_a-\phi_a\delta\bar\phi_a\right).
$$

Chosen B-twist \eqref{AB-twists} relates fermionic and bosonic zero modes in the usual way:
\bar{e}
\delta\phi_a=\sqrt{2}\epsilon\,\psi_{1,a},\quad \delta\sigma_{\mathbb{I}}-{\rm i}\delta A_1=2\epsilon\,\bar\lambda_1,
 \end{equation}
allowing one to re-assemble the operators annihilating zero modes in a Dirac operator:
\bar{e}
\nabla\left(\begin{array}{c}
\psi_{1,1}\\
\psi_{1,2}\\
\bar\lambda_1
\end{array}\right)=0,\quad \nabla=\left(\begin{array}{ccc}
D_1-(\sigma_{\mathbb{I}}-\mu_{\mathbb{I},1}) & 0 & -\sqrt{2}\phi_1\\
0& D_1-(\sigma_{\mathbb{I}}-\mu_{\mathbb{I},2}) & -\sqrt{2}\phi_2\\
-\sqrt{2}\bar\phi_1 & -\sqrt{2}\bar\phi_2 & \partial_1
\end{array}\right).
 \end{equation}
Again we would like to redefine variables in the following way:
\bar{e}
\psi_{1,a}:=\phi_a \alpha_a,\quad \bar\lambda_1:=\beta.
 \end{equation}
In terms of these new variables the zero mode equations read (we suppress the superscript $1$ for the spatial coordinate $x^1$):
\bar{e}\label{system}
\begin{split}
	\partial_x\alpha_1-\sqrt{2}\beta=0,\\
	\partial_x\alpha_2-\sqrt{2}\beta=0,\\
	\partial_x\beta-\sqrt{2}|\phi_1|^2\alpha_1-\sqrt{2}|\phi_2|^2\alpha_2=0.
\end{split}
 \end{equation}
The first two equations of this system can be solved easily:
\bar{e}
\alpha_1(x)=\alpha_0(x)+c_0,\quad\alpha_2(x)=\alpha_0(x)-c_0,\quad \beta=\frac{1}{\sqrt{2}}\partial_x\alpha_0(x).
 \end{equation}
where $c_0$ is an integration constant.
The system \eqref{system} reduces to the following equation for $\alpha_0$:
\bar{e}
-\partial_x^2\alpha_0+2(|\phi_1|^2+|\phi_2|^2)\alpha_0+2c_0\left(|\phi_1|^2-|\phi_2|^2\right)=0.
 \end{equation}
It is simpler to solve this equation in an approximation regime $\Delta\mu/\sqrt{r}\ll 1$ when we have:
\bar{e}\label{approx}
|\phi_1|^2\approx r\cos\tau^2,\quad|\phi_2|^2\approx r\sin\tau^2.
 \end{equation}
In this case a generic solution reads:\footnote{Green function for a massive 1d Laplace operator reads:
	$$
	\left(-\partial_x^2+m^2\right)G_m(x)=\delta(x),\quad G_m(x)=\frac{1}{2m}e^{-m|x|},\quad m>0.
	$$}
\bar{e}
\alpha_0=c_1e^{-\sqrt{2r}x}+c_2e^{\sqrt{2r}x}-\frac{c_0 r}{\sqrt{2r}}\int\limits_{-\infty}^{+\infty}dy\,e^{-\sqrt{2r}|x-y|}\frac{1-e^{2(\mu_{\mathbb{I},1}-\mu_{\mathbb{I},2}) y}}{1+e^{2(\mu_{\mathbb{I},1}-\mu_{\mathbb{I},2}) y}}.
 \end{equation}
The latter integral can be rewritten in terms of hypergeometric series, however the resulting expression is rather bulky, moreover applying assumption $\Delta\mu/\sqrt{r}\ll 1$ we could substitute the integration kernel by a kernel supported in a single point:
$$
e^{-\sqrt{2r}|x|}\to \frac{2}{\sqrt{2r}}\delta(x).
$$
Thus one derives:
\bar{e}\label{alpha_0}
\alpha_0\approx c_1e^{-\sqrt{2r}x}+c_2e^{\sqrt{2r}x}-c_0\frac{1-e^{2(\mu_{\mathbb{I},1}-\mu_{\mathbb{I},2}) x}}{1+e^{2(\mu_{\mathbb{I},1}-\mu_{\mathbb{I},2}) x}}.
 \end{equation}
Zero modes governed by integration constants $c_1$ and $c_2$ are concentrated near interval boundaries and shift the boundary brane electric charges (see \eqref{condensate}).
The zero mode governed by $c_0$ is a mode bound to the complex vortex soliton core.

The mass constraint \eqref{mass_const} for the vacuum field configuration needs to be satisfied up to the quantum $\hbar$ order. Therefore we could put:
$$
\mu_{{\mathbb{R}},1}=\hbar \tilde\mu_{{\mathbb{R}},1},\quad \mu_{{\mathbb{R}},2}=\hbar \tilde\mu_{{\mathbb{R}},2},
$$
where $\tilde\mu_{{\mathbb{R}},a}$ are of order 1.
The central charge acquires a constant correction to the equilibrium zero value due to this core zero mode -- ``vacuum supercharge":
\bar{e}
\bar Q_0=\int\left(\bar\lambda_1\theta+\sqrt{2}\tilde\mu_{{\mathbb{R}},1}\bar\phi_1\psi_{1,1}+\sqrt{2}\tilde\mu_{{\mathbb{R}},2}\bar\phi_2\psi_{1,2}\right)\,dx
 \end{equation}
The energy of the state acquires additional  gap:
$$
E_0\sim|\tilde{\cal Z}|+|\bar Q_0|^2.
$$
\begin{comment}
Therefore for the core zero mode we derive
After massaging this expression a little we derive:
\bar{e}
\bar Q_0=\int\left(\beta\theta+\frac{\mu_{{\mathbb{R}},1}+\mu_{{\mathbb{R}},2}}{2}\partial_x\beta+\frac{\mu_{{\mathbb{R}},1}-\mu_{{\mathbb{R}},2}}{2}\frac{|\phi_1|^2-|\phi_2|^2}{|\phi_1|^2+|\phi_2|^2}\partial_x\beta+\frac{\mu_{{\mathbb{R}},1}-\mu_{{\mathbb{R}},2}}{\sqrt{2}}\frac{4|\phi_1|^2|\phi_2|^2}{|\phi_1|^2+|\phi_2|^2}c_0\right)\,dx
 \end{equation}
We expect $\beta(x\to\pm \infty)=0$, therefore
\end{comment}
For the state to be BPS it should satisfy  constraint $|Q_0|^2=0$.
After massaging the vacuum supercharge expression a little and applying $\beta(x\to\pm \infty)=0$ we derive:
\bar{e}
\bar Q_0=\int\left(\beta\theta-\frac{\tilde\mu_{{\mathbb{R}},1}-\tilde\mu_{{\mathbb{R}},2}}{2}\left(\partial_x\frac{|\phi_1|^2-|\phi_2|^2}{|\phi_1|^2+|\phi_2|^2}\right)\beta+\frac{\tilde\mu_{{\mathbb{R}},1}-\tilde\mu_{{\mathbb{R}},2}}{\sqrt{2}}\frac{4|\phi_1|^2|\phi_2|^2}{|\phi_1|^2+|\phi_2|^2}c_0\right)\,dx.
 \end{equation}
For the approximate expression \eqref{alpha_0} for the core zero mode we derive another BPS constraint:
\bar{e}
\bar Q_0\approx\frac{2c_0}{\sqrt{2}|\mu_{\mathbb{I},1}-\mu_{\mathbb{I},2}|}\left[\theta(\mu_{\mathbb{I},1}-\mu_{\mathbb{I},2})+r(\tilde\mu_{{\mathbb{R}},1}-\tilde\mu_{{\mathbb{R}},2})\right]=0.
 \end{equation}

In principle the zero mode $\phi$ has its own monomial contribution $\phi^k$ to the wave-function (as in \eqref{CP_1_const_wave}), this contribution shifts the effective topological angle $\theta$ by $k$, therefore the BPS constraint reads (compare to \eqref{CP_1_solitons}):
\bar{e}
k={\rm Re}\, t\,\frac{\mu_{{\mathbb{R}},1}-\mu_{{\mathbb{R}},2}}{\mu_{\mathbb{I},1}-\mu_{\mathbb{I},2}}-{\rm Im}\,t\in{\mathbb{Z}}.
 \end{equation}
If this constraint is satisfied the complex vortex soliton flows in a direction defined by the difference between masses $\mu_{\mathbb{I},1}$ and $\mu_{\mathbb{I},2}$:
\begingroup
\renewcommand*{\arraystretch}{1.7}
\bar{e}
\begin{array}{ll}
	{\cal Z}^{(1\to 2)}_{\rm Higgs}=-\dfrac{{\rm Re}\,t}{\mu_{\mathbb{I},1}-\mu_{\mathbb{I},2}}|\mu_1-\mu_2|^2,& \mu_{\mathbb{I},1}>\mu_{\mathbb{I},2};\\
	{\cal Z}^{(2\to 1)}_{\rm Higgs}=-\dfrac{{\rm Re}\,t}{\mu_{\mathbb{I},1}-\mu_{\mathbb{I},2}}|\mu_1-\mu_2|^2,& \mu_{\mathbb{I},1}<\mu_{\mathbb{I},2}.
\end{array}
 \end{equation}
\endgroup
Comparing the eventual Higgs branch BPS soliton spectrum with the Coulomb branch soliton spectrum \eqref{CP_1_solitons} we find them to be {\bf identical}. This identity is a manifestation of the \emph{cHC duality} \eqref{HC_du} in this simple case of equivariant $\mathbb{C}\mathbb{P}^1$ \eqref{CP_1_mirror}.

\subsection{Instantons and boundary operators}
As we discussed in the previous section instanton contributions become crucial when homotopy relations between interface paths $\wp$ are in consideration.
One type of instantons appearing in this model is a standard instanton contributing to the homotopy of the interface path around the branching point \cite{Galakhov:2017pod}.
We will not consider it here.

Rather we concentrate on another type of instanton vertices delivering a homotopy invariance depicted in Fig.\ref{fig:bdry_chg} through the corresponding boundary operator.
The same instanton vertex governs soliton recombination during critical wall-crossing between soliton spectra (a) and (b) depicted in Fig.\ref{fig:SN_CP_1}.
This process could be described as a cluster transformation of soliton partition functions in coupled  2d-4d systems due to ``flavor" solitons $\mathfrak{X}_{A,k}$ and $\mathfrak{X}_{B,k}$ that are effective vector multiplet 4d quasi-particles coupled to a 2d defect \cite{Gaiotto:2009hg,Klemm:1996bj}.
These solitons interpolate between the same type LG vacua \eqref{CP_1_vacuum} on different sheets of $W$-cover (see Fig.\ref{fig:CP_1_vacua}).
The instanton vertices represent the following reactions:
\bar{e}\label{sol_scatt}
\begin{split}
\mathfrak{X}_{A,-1}+\mathfrak{X}_{C,-1} \longleftrightarrow \mathfrak{X}_{C,0},\\
\mathfrak{X}_{B,-1}+\mathfrak{X}_{C,-1} \longleftrightarrow \mathfrak{X}_{C,0}.
\end{split}
 \end{equation}
To prove that corresponding scattering vertices are non-zero we apply the bootstrap method of Section \ref{sec:bootstrap}.

It will be more spectacular if we consider a dual model.
First we restore disorder operator $Y$ in the superpotential expression \eqref{CP_1_sup} then integrate over field $\Sigma$.
The resulting class of models can be described by the following superpotential:
$$
W=\mu Y-R \,e^{Y}-e^{-Y},\quad R\ll 1,
$$
where $\mu$ is some combination of mass parameters and $R\sim e^{-2\pi t}\ll 1$.
This model has two LG vacuum series corresponding to two vacua series in the $W$-plane (Fig.\ref{fig:CP_1_vacua}):
$$
Y_{*1}={\rm log}\frac{\mu+\sqrt{\mu^2+4R}}{2R}+2\pi{\rm i}{\mathbb{Z}},\quad Y_{*2}={\rm log}\frac{\mu-\sqrt{\mu^2+4R}}{2R}+2\pi{\rm i}{\mathbb{Z}}.
$$
The wall-crossing decay/recombination occur when the central charges of solitons are co-directed in the complex plane. For flavor solitons $\Delta W$ is purely imaginary for real parameter $\mu$.
The marginal stability wall separating chambers with different BPS spectra is located in the point where ${\rm Re}\,(\Delta W)$ for $\mathfrak{X}_{C,k}$ changes sign:
$$
{\rm Re}\,(\Delta W)=\mu\;{\rm log}\frac{\sqrt{\mu^2+4R}+\mu}{\sqrt{\mu^2+4R}-\mu}-2\sqrt{\mu^2+4R}.
$$
The behavior of this function is depicted in a plot in Fig.\ref{fig:plot}.
\begin{figure}[h!]
	\begin{center}
		\begin{tikzpicture}
			\node {\includegraphics[scale=0.4]{figs/CP_1_muDeltaW.pdf}};
			\node[black!10!orange] at (-2,-1) {$\mu_{\rm crit}=0.3018$};
			\node at (4.5,-1.7) {$\scriptstyle\mu$};
			\node at (-5,1.8) {$\scriptstyle{\rm Re}\,\Delta W$};
		\end{tikzpicture}
	\caption{Plot ${\rm Re}\,\Delta W(\mu).$}\label{fig:plot}
\end{center}
\end{figure}

The soliton equation in this model is integrable, therefore $\zeta$-solitons in Fig.\ref{fig:CP_1_vacua} are stable as long as the inverse map $W^{-1}$ to $Y$-plane is holomorphic on a disk inside the rectangle generated by soliton paths $\mathfrak{X}_{C,-1}$, $\mathfrak{X}_{A,-1}$, $\mathfrak{X}_{B,-1}$. We can construct a family of conformal $W^{-1}$-maps to $Y$-plane for this disk as a function of parameter $\mu$. In Fig.\ref{fig:holo_disk} we present this family for a numerical value $R=0.01$, clearly the holomorphic disk shrinks as one approaches to and cease to exist after
$$
\mu_{\rm crit}=0.3018\ldots
$$
\begin{figure}[h!]
	\begin{center}
		\begin{tikzpicture}
			\begin{scope}
			\node {\includegraphics[scale=0.18]{figs/CP_1_disk_04.pdf}};
			\node at (0,2) {$\scriptstyle{\rm Re}\,Y$};
			\node at (-2,0) {$\scriptstyle{\rm Im}\,Y$};
			\node[black!10!blue] at (1.8,1.4) {$\scriptstyle Y_{\rm I}$};
			\node[black!10!blue] at (1.8,-0.8) {$\scriptstyle Y_{\rm II}$};
			\node[black!10!blue] at (-2,0.8) {$\scriptstyle Y_{\rm III}$};
			\node[black!10!blue] at (-2,-1.8) {$\scriptstyle Y_{\rm IV}$};
			\end{scope}
		\begin{scope}[shift={(5,0)}]
			\node {\includegraphics[scale=0.18]{figs/CP_1_disk_07.pdf}};
			\node at (0,2) {$\scriptstyle{\rm Re}\,Y$};
			\node at (-2,0) {$\scriptstyle{\rm Im}\,Y$};
			\node[black!10!blue] at (1.8,1.4) {$\scriptstyle Y_{\rm I}$};
			\node[black!10!blue] at (1.8,-0.8) {$\scriptstyle Y_{\rm II}$};
			\node[black!10!blue] at (-2,0.8) {$\scriptstyle Y_{\rm III}$};
			\node[black!10!blue] at (-2,-1.8) {$\scriptstyle Y_{\rm IV}$};
		\end{scope}
	\begin{scope}[shift={(10,0)}]
		\node {\includegraphics[scale=0.18]{figs/CP_1_disk_10.pdf}};
		\node at (0,2) {$\scriptstyle{\rm Re}\,Y$};
		\node at (-2,0) {$\scriptstyle{\rm Im}\,Y$};
		\node[black!10!blue] at (1.8,1.4) {$\scriptstyle Y_{\rm I}$};
		\node[black!10!blue] at (1.8,-0.8) {$\scriptstyle Y_{\rm II}$};
		\node[black!10!blue] at (-2,0.8) {$\scriptstyle Y_{\rm III}$};
		\node[black!10!blue] at (-2,-1.8) {$\scriptstyle Y_{\rm IV}$};
	\end{scope}
\begin{scope}[shift={(0,-2.5)}]
	\draw[->] (-2,0) -- (11,0);
	\node[right] at (11,0) {$\mu$};
	\draw[ultra thick] (-1.8,-0.1) -- (-1.8,0.1) (0,-0.1) -- (0,0.1) (5,-0.1) -- (5,0.1) (10,-0.1) -- (10,0.1);
	\node[below] at (-1.8,-0.1) {$\mu_{\rm crit}$};
	\node[below] at (0,-0.1) {$0.4$};
	\node[below] at (5,-0.1) {$0.7$};
	\node[below] at (10,-0.1) {$1.0$};
	\draw (-1.5,0.4) to[out=330,in=120] (0,0) to[out=60,in=210] (1.5,0.4);
	\begin{scope}[shift={(5,0)}]
		\draw (-1.5,0.4) to[out=330,in=120] (0,0) to[out=60,in=210] (1.5,0.4);
	\end{scope}
\begin{scope}[shift={(10,0)}]
	\draw (-1.5,0.4) to[out=330,in=120] (0,0) to[out=60,in=210] (1.5,0.4);
\end{scope}
\end{scope}
		\end{tikzpicture}
	\caption{Shrinking holomorphic disk movie in $\mathbb{C}\mathbb{P}^1$ model.}\label{fig:holo_disk}
\end{center}
\end{figure}

Thus we conclude that there is a \emph{non-zero} soliton scattering amplitude for processes like \eqref{sol_scatt}. In principle, we conclude that any amplitude for a triplet of solitons forming a triangle with vertices positioned in local vacua superpotential values are non-zero if all three are stable:
\bar{e}\label{amplitudes}
\begin{split}
	\mathfrak{a}\left[W_{*1,k_1},W_{*1,k_2},W_{*2,k_3}||{\rm Re},t|\gg 1\right]=1,\quad \forall\;k_1,k_2,k_3;\\
	\mathfrak{a}\left[W_{*1,k_1},W_{*2,k_2},W_{*2,k_3}||{\rm Re},t|\gg 1\right]=1,\quad \forall\;k_1,k_2,k_3.
\end{split}
 \end{equation}

In such a simple model of a single field $Y$ we can perform even some simple numerical estimates, in particular, the instanton equation \eqref{LG-instanton} can be solved numerically.

To do so let us make few preparations. First of all, let us choose notations for vacua:
\bar{e}\nonumber
\begin{split}
	Y_{\rm I}=Y_{*1},\quad Y_{\rm II}=Y_{*1}-2\pi {\rm i},\quad Y_{\rm III}=Y_{*2}-2\pi{\rm i},\quad Y_{\rm IV}=Y_{*2}-4\pi {\rm i}
\end{split}
 \end{equation}
Further let us choose the phase $\zeta={\rm i}$ and coordinates in $z$-plane:
\bar{e}\nonumber
z=\alpha\tilde x^1+\beta\tilde x^2,\quad \alpha=\frac{\bar W_2-\bar W_1}{|W_2-W_1|},\; \beta=\frac{\bar W_3-\bar W_1}{|W_3-W_1|}.
 \end{equation}
So that the instanton equation reads:
\bar{e}
(\alpha\partial_{\tilde 1}+\beta\partial_{\tilde 2})Y=h\,\overline{\partial_Y W},
 \end{equation}
where $h$ is an overall scaling factor for the world-sheet plane to control a fast convergence of the soliton solution to a LG vacuum.
We take $h=20.0$.
If such a coordinate transformation is taken trajectories of asymptotic soliton quasi-particles will be parallel to axes $\tilde x^1$ and $\tilde x^2$.
We depict the result of such numerical simulation in Fig.\ref{fig:inst}(a) in terms of a density function:
$$
\rho\left(\tilde x^1,\tilde x^2\right):=|Y\left(\tilde x^1,\tilde x^2\right)-Y_{\rm I}|\cdot |Y\left(\tilde x^1,\tilde x^2\right)-Y_{\rm II}|\cdot |Y\left(\tilde x^1,\tilde x^2\right)-Y_{\rm III}|\cdot |Y\left(\tilde x^1,\tilde x^2\right)-Y_{\rm IV}|.
$$
When density $\rho$ is zero it means that the field $Y$ is in one of the LG vacuum states.
Clearly, the simulation depicts as two asymptotic soliton states are scattered through a s-channel process to a new pair of asymptotic soliton states.
The intermediate state in s-channel is also a soliton, so we can depict this process diagrammatically as in Fig.\ref{fig:inst} (b).
This process incorporates two scattering vertices \eqref{sol_scatt}.

\begin{figure}[h!]
	\begin{center}
		\begin{tikzpicture}
			\node at (0,0) {\includegraphics[scale=0.4]{figs/CP_1_inst_3d.pdf}};
			\node at (4.5,0) {\includegraphics[scale=0.2]{figs/CP_1_inst_scatt.pdf}};
			\node at (7,0) {\includegraphics[scale=0.35]{figs/CP_1_inst_scatt_scale.pdf}};
			\node[below] at (3.5,-2.3) {(a)};
			\begin{scope}[shift={(10.5,0)}]
				\draw[dashed,fill=blue,opacity=0.2] (-1.5,-1.5) -- (-1.5,1.5) -- (1.5,1.5) -- (1.5,-1.5) -- cycle;
				\draw[ultra thick] (-1.5,-0.5) -- (-0.5,-0.5) -- (-0.5,-1.5) (1.5,0.5) -- (0.5,0.5) -- (0.5,1.5) (-0.5,-0.5) -- (0.5,0.5);
				\draw[fill=black] (-0.5,-0.5) circle (0.08) (0.5,0.5) circle (0.08);
				\node[below] at (-0.5,-1.5) {$\mathfrak{X}_{A,-1}$};
				\node[left] at (-1.5,-0.5) {$\mathfrak{X}_{C,-1}$};
				\node[above] at (0.5,1.5) {$\mathfrak{X}_{B,-1}$};
				\node[right] at (1.5,0.5) {$\mathfrak{X}_{C,-1}$};
				\node[above left] at (0,0) {$\mathfrak{X}_{C,0}$};
				\node[below left] at (-1.5,-1.5) {$W_{\rm I}$};
				\node[below right] at (1.5,-1.5) {$W_{\rm II}$};
				\node[above left] at (-1.5,1.5) {$W_{\rm III}$};
				\node[above right] at (1.5,1.5) {$W_{\rm IV}$};
			\end{scope}
		\node[below] at (10.5,-2.3) {(b)};
		\end{tikzpicture}
	\caption{ (a) Density $\rho(t,s)$ for numerically simulated soliton scattering process in the model $W(Y)=1.0\times Y-0.01\times e^Y-e^{-Y}$;  and (b) diagrammatic depiction of the soliton scattering process.}\label{fig:inst}
\end{center}
\end{figure}

\section{Categorified analytic continuation of hypergeometric series}\label{sec:cat_contin_hyper}

\subsection{Model description}

We would like to discuss the construction of the BPS Hilbert space as a route for categorifying the analytic continuation as proposed in Section \ref{sec:Atiyah_intro}.
First we review the disk partition function construction associated to the Grothendieck group of the category.

As a model for the conifold transition we pick a $U(1)$ B-twisted $\sigma$-model with $N_f=4$ chiral multiplets with charges and masses.
This matter content can be formulated in the following quiver diagram:
\bar{e}
\begin{array}{c|c|c|c|c}
	{\rm field} & \phi_1 & \phi_2 & \phi_3 & \phi_4\\
	\hline
	Q_a & +1 & +1 & -1 & -1\\
	\hline
	\mu_a & \mu_1 & \mu_2 & -\mu_3 & -\mu_4\\
\end{array},\quad \begin{array}{c}
\begin{tikzpicture}
	\node at (-2,0) {$1$};
	\node at (0,0) {$1$};
	\node at (2,0) {$1$};
	\draw (0,0) circle (0.3);
	\begin{scope}[shift={(2,0)}]
		\draw (-0.3,-0.3) -- (-0.3,0.3) -- (0.3,0.3) -- (0.3,-0.3) -- cycle;
	\end{scope}
	\begin{scope}[shift={(-2,0)}]
		\draw (-0.3,-0.3) -- (-0.3,0.3) -- (0.3,0.3) -- (0.3,-0.3) -- cycle;
	\end{scope}
	\draw[<-] (1.7,0.1) -- (0.3,0.1);
	\draw[->] (1.7,-0.1) -- (0.3,-0.1);
	\draw[->] (-1.7,0.1) -- (-0.3,0.1);
	\draw[<-] (-1.7,-0.1) -- (-0.3,-0.1);
	\node[above] at (-1,0.1) {$\mu_1$};
	\node[below] at (-1,-0.1) {$-\mu_3$};
	\node[above] at (1,0.1) {$-\mu_4$};
	\node[below] at (1,-0.1) {$\mu_2$};
\end{tikzpicture}
\end{array}.
 \end{equation}
Depending on the sign of FI parameter $r$ the Higgs branch is a conifold resolution:
\bar{e}
\begin{split}
X_+:=X_{\rm Higgs}(r>0)=\left[{\cal O}(-1)_{(\phi_3,\phi_4)}^{\oplus 2}\longrightarrow \mathbb{C}\mathbb{P}^1_{(\phi_1:\phi_2)}\right];\\
X_-:=X_{\rm Higgs}(r<0)=\left[{\cal O}(-1)_{(\phi_1,\phi_2)}^{\oplus 2}\longrightarrow \mathbb{C}\mathbb{P}^1_{(\phi_3:\phi_4)}\right].
\end{split}
 \end{equation}
where pairs of field $(\phi_1,\phi_2)$ and $(\phi_3,\phi_4)$ play roles of a base and a fiber.

The disk partition function for a boundary condition given by the structure sheaf reads \cite{Hori:2013ika}:
\bar{e}\label{conifold_pf}
Z(t)=\int\limits_{-{\rm i}\infty}^{{\rm i}\infty}d\sigma\; e^{2\pi\, t\, \sigma}\,\Gamma(\sigma-{\rm i}\mu_1)\Gamma(\sigma-{\rm i}\mu_2)\Gamma(-\sigma+{\rm i}\mu_3)\Gamma(-\sigma+{\rm i}\mu_4)
 \end{equation}

We can calculate this integral by the standard Cauchy formula method after closing the integration contour at infinity.
The way we close the contour depends on the sign of ${\rm Re}\;t$.
Rather than considering different operator insertions for the bundle on the boundary brane we can consider various integration cycles $\gamma_a$, $a=1,\ldots,4$ encircling different series of poles from the gamma-functions.
Clearly, these four choices correspond to constant vacua choices $*a$ when the corresponding field acquires expectation value $|\phi_a|=r^{\frac{1}{2}}$:
\bar{e}
\begin{array}{c}
	\begin{tikzpicture}
		\begin{scope}
			\draw[fill=black] (0,0) circle (0.05) (-0.5,0) circle (0.05) (-1,0) circle (0.05) (-1.5,0) circle (0.05) (-2,0) circle (0.05);
			\draw[->,thick,red] (-2.3,-0.4) to[out=0,in=180] (0,-0.4) to[out=0,in=270] (0.2,0) to[out=90,in=0] (0,0.4) to[out=180,in=0] (-1,0.4);
			\draw[thick,red] (-2.3,0.4) -- (-1,0.4);
			\node[left,red] at (-2.3,0.4) {$\gamma_1$};
			\node[right] at (0.2,0) {${\rm i}\mu_1$};
		\end{scope}
		\begin{scope}[shift={(0,-1.5)}]
			\draw[fill=black] (0,0) circle (0.05) (-0.5,0) circle (0.05) (-1,0) circle (0.05) (-1.5,0) circle (0.05) (-2,0) circle (0.05);
			\draw[->,thick,blue] (-2.3,-0.4) to[out=0,in=180] (0,-0.4) to[out=0,in=270] (0.2,0) to[out=90,in=0] (0,0.4) to[out=180,in=0] (-1,0.4);
			\draw[thick,blue] (-2.3,0.4) -- (-1,0.4);
			\node[left,blue] at (-2.3,0.4) {$\gamma_2$};
			\node[right] at (0.2,0) {${\rm i}\mu_2$};
		\end{scope}
		\begin{scope}[shift={(3,0)}]
			\begin{scope}[scale=-1]
				\draw[fill=black] (0,0) circle (0.05) (-0.5,0) circle (0.05) (-1,0) circle (0.05) (-1.5,0) circle (0.05) (-2,0) circle (0.05);
				\draw[->,thick,black!40!green] (-2.3,-0.4) to[out=0,in=180] (0,-0.4) to[out=0,in=270] (0.2,0) to[out=90,in=0] (0,0.4) to[out=180,in=0] (-1,0.4);
				\draw[thick,black!40!green] (-2.3,0.4) -- (-1,0.4);
				\node[right,black!40!green] at (-2.3,0.4) {$\gamma_3$};
				\node[left] at (0.2,0) {${\rm i}\mu_3$};
			\end{scope}
		\end{scope}
		\begin{scope}[shift={(3,-1.5)}]
			\begin{scope}[scale=-1]
				\draw[fill=black] (0,0) circle (0.05) (-0.5,0) circle (0.05) (-1,0) circle (0.05) (-1.5,0) circle (0.05) (-2,0) circle (0.05);
				\draw[->,thick,purple] (-2.3,-0.4) to[out=0,in=180] (0,-0.4) to[out=0,in=270] (0.2,0) to[out=90,in=0] (0,0.4) to[out=180,in=0] (-1,0.4);
				\draw[thick,purple] (-2.3,0.4) -- (-1,0.4);
				\node[right,purple] at (-2.3,0.4) {$\gamma_4$};
				\node[left] at (0.2,0) {${\rm i}\mu_4$};
			\end{scope}
		\end{scope}
	\end{tikzpicture}
\end{array}
 \end{equation}

For each of the basis integration cycle we can easily calculate the integral as a sum over pole contributions in terms of hypergeometric series (see Appendix \ref{sec:hypergeom} for reference):
\bar{e}
\begin{split}
	&Z_{\gamma_1}=-2\pi{\rm i} \,e^{2\pi{\rm i} t \mu_1}\,\left[\Gamma({\rm i}\mu_{12})\Gamma(-{\rm i}\mu_{13})\Gamma(-{\rm i}\mu_{14})\right]\;{}_2F_1\left[\begin{array}{c}
		-{\rm i}\mu_{13};\;-{\rm i}\mu_{14}\\
		1-{\rm i}\mu_{12}
	\end{array}\right]\left(e^{-2\pi t}\right),\\
	&Z_{\gamma_2}=-2\pi{\rm i} \,e^{2\pi{\rm i} t \mu_2}\,\left[\Gamma({\rm i}\mu_{21})\Gamma(-{\rm i}\mu_{23})\Gamma(-{\rm i}\mu_{24})\right]\;{}_2F_1\left[\begin{array}{c}
		-{\rm i}\mu_{23};\;-{\rm i}\mu_{24}\\
		1-{\rm i}\mu_{21}
	\end{array}\right]\left(e^{-2\pi t}\right),\\
	&Z_{\gamma_3}=2\pi{\rm i}\,e^{2\pi{\rm i} t \mu_3}\,\left[\Gamma({\rm i}\mu_{43})\Gamma(-{\rm i}\mu_{13})\Gamma(-{\rm i}\mu_{23})\right]\;{}_2F_1\left[\begin{array}{c}
		-{\rm i}\mu_{13};\;-{\rm i}\mu_{23}\\
		1-{\rm i}\mu_{43}
	\end{array}\right]\left(e^{2\pi t}\right),\\
	&Z_{\gamma_4}=2\pi{\rm i}\,e^{2\pi{\rm i} t \mu_4}\,\left[\Gamma({\rm i}\mu_{34})\Gamma(-{\rm i}\mu_{14})\Gamma(-{\rm i}\mu_{24})\right]\;{}_2F_1\left[\begin{array}{c}
		-{\rm i}\mu_{14};\;-{\rm i}\mu_{24}\\
		1-{\rm i}\mu_{34}
	\end{array}\right]\left(e^{2\pi t}\right),
\end{split}
 \end{equation}
where $\mu_{ab}=\mu_a-\mu_b$.
A unit circle $|e^{2\pi t}|=1$ corresponding to value $r=0$ divides the Riemann sphere spanned by $e^{2\pi t}$ in two hemispheres.
Correspondingly, series for $\gamma_{1,2}$ converge absolutely in the hemisphere containing point $e^{2\pi t}=\infty$, and series for $\gamma_{3,4}$ converge in the opposite hemisphere containing point $e^{2\pi t}=0$ (see Fig.\ref{fig:wp}(c)).

The calculated series may be continued from one hemisphere to the other using the fact that the hypergeometric function is a solution to the hypergeometric equation,  and a pair of solutions are flat sections of a holomorphic $SL(2,\mathbb{C})$ connection.
This connection allows one to parallel transport a hypergeometric solution across the unit circle boundary and re-decompose it on the other side over basis convergent solutions. We can summarize this linear  map between hypergeometric series bases in the following form:
\bar{e}
\begingroup
\renewcommand*{\arraystretch}{1.7}
\left(\begin{array}{c}
	Z_{\gamma_1}\\
	Z_{\gamma_2}
\end{array}\right)={\bf A}
\left(\begin{array}{c}
	Z_{\gamma_3}\\
	Z_{\gamma_4}
\end{array}\right),\quad {\bf A}=\left(\begin{array}{cc}
	\dfrac{\chi_3}{\chi_1}\cdot\dfrac{\lambda_{23}}{\lambda_{12}} & \dfrac{\chi_4}{\chi_1}\cdot\dfrac{\lambda_{24}}{\lambda_{12}}\\
	\dfrac{\chi_3}{\chi_2}\cdot\dfrac{\lambda_{13}}{\lambda_{21}} & \dfrac{\chi_4}{\chi_2}\cdot\dfrac{\lambda_{14}}{\lambda_{21}}
\end{array}\right),
\endgroup
 \end{equation}
where
\bar{e}
\chi_a:=e^{\pi\mu_a},\quad \lambda_{ab}:=\frac{\chi_a}{\chi_b}-\frac{\chi_b}{\chi_a}.
 \end{equation}
Matrix $\bf A$ follows directly from \eqref{analytic}, it is induced by a parallel transport along a path $\wp$ on Fig.\ref{fig:wp}(b), we will call it an analytic continuation matrix.

\subsection{Analytic continuation in the language of the Grothendieck group}

As we mentioned in Section \ref{sec:dec_cat_par} the disk partition function represents a map from  the category to its Grothendieck group given by a vector space.
In this section we will discuss a simpler behavior of the Grothendieck groups under the parallel transport.
This will allow us to capture static behavior of solitons confined to the defect.
For this we use the methods of spectral networks (see Appendix~\ref{sec:App_SN}).

We consider the superpotential of this theory having restored the contributions of the disorder operator fields $Y_a$:
\bar{e}\label{Conif_sp}
W(\Sigma,Y_a)=t\Sigma+\frac{1}{2\pi}\sum\limits_{a=1}^2\left[(\Sigma-\mu_a)Y_a-\Lambda e^{Y_a}\right]+\frac{1}{2\pi}\sum\limits_{b=3}^4\left[(\mu_b-\Sigma)Y_b-\Lambda e^{Y_b}\right].
 \end{equation}
Clearly, substituting this expression in the formula for the disk partition function \eqref{Clmb_br_pf} and integrating over disorder operator fields $Y_a$ we will get an expression of type \eqref{conifold_pf}.
Unfortunately, the topology of Lefschetz thimbles in terms of original twisted field $\Sigma$ is rather messy, we would rather use a dual model where it will be more transparent. For that we first integrate over $\Sigma$-field in \eqref{Conif_sp}.
It plays a role of a Lagrange multiplier imposing a constraint:
$$
2\pi t +Y_1+Y_2-Y_3-Y_4=0,
$$
realized in terms of the partition function as a delta-function in the integrand.
We easily resolve this constraint\footnote{The author would like to thank Mina Aganagic for a suggestion to apply here this variable redefinition.} in terms of new variables $y_1$, $y_2$ and $x$:
\bar{e}
\begin{split}
	Y_1&=y_2-{\rm log}\left(1-x^{-1}e^{-2\pi t}\right)-2\pi t-{\rm log}(-x)+w,\\
	Y_2&=y_1-{\rm log}(1-x)+{\rm log}(-x),\\
	Y_3&=y_1-{\rm log}(1-x),\\
	Y_4&=y_2-{\rm log}\left(1-x^{-1}e^{-2\pi t}\right).
\end{split}
 \end{equation}
Then fields $y_1$ and $y_2$ decouple: corresponding effective theories describe two free neutral massive scalars of masses $\mu_{32}$ and $\mu_{41}$ with a single constant vacuum field configuration.
In the partition function integral these variables are also separated giving simple contributions:
\bar{e}\label{extra_PF}
\begin{split}
	\int dy_1\,e^{y_1({\rm i} \mu_{32})-\Lambda e^{y_1}}\sim\Gamma({\rm i}\mu_{32}),\quad 	\int dy_2\,e^{y_2({\rm i} \mu_{41})-\Lambda e^{y_2}}\sim\Gamma({\rm i}\mu_{41}).
\end{split}
 \end{equation}

We denote the resulting partition function as $\Xi$ to stress it is a LG partition function as opposed to the $\sigma$-model partition function denoted by $Z$.
The resulting expression depends on a choice of Lagrangian integration cycle ${\cal L}$:
\bar{e}
\Xi_{{\cal L}}(t)=e^{2\pi{\rm i} t\,\mu_4}\;\Gamma({\rm i}\mu_{32})\Gamma({\rm i}\mu_{41})\;\int\limits_{{\cal L}} \frac{dx}{x}\;x^{{\rm i}\mu_{42}}(1-x)^{{\rm i}\mu_{23}}\left(1-x\,e^{2\pi t}\right)^{{\rm i}\mu_{14}}.
 \end{equation}
This partition function corresponds (up to an overall multiplier) to an LG model of field $x$ with a superpotential:
\bar{e}\label{sup_0}
W(x)=\frac{1}{2\pi}\left[\mu_{42}\,{\rm log} \,x+\mu_{23}\,{\rm log}(1-x)+\mu_{14}{\rm log}\left(1-x \,e^{2\pi t}\right)\right].
 \end{equation}

Now it is easy to describe the topology of the Lefschetz thimbles.
Those are curves in the $x$-plane that spiral towards singularities at $0$, $1$, $\infty$ and $e^{2\pi t}$ similarly to the situation depicted in Fig.\ref{fig:SN_CP_1} for $\mathbb{C}\mathbb{P}^1$.
The topology of Lefshetz thimbles jumps across S-walls.
Using this superpotential it is easy to derive a spectral cover equation:
\bar{e}
	z(z\lambda-\mu_1)(z\lambda-\mu_2)=(z\lambda-\mu_3)(z\lambda-\mu_4),
 \end{equation}
where we introduced a parameter:
$$
z:=e^{2\pi t},
$$
and the meromorphic SW differential reads (see Appendix \ref{sec:App_SN} for definitions):
$$
\Omega =2\pi {\rm i}\; \lambda\; dz.
$$
\begin{figure}
	\begin{center}
	\begin{tikzpicture}
		\node {\includegraphics[scale=0.7]{figs/Conifold_SN.pdf}};
		\draw[fill=blue] (-4.1,0) circle (0.07);
		\node[blue] at (-4,-0.7) {$-1$};
		\draw[fill=blue] (4.1,0) circle (0.07);
		\node[blue] at (4,0.7) {$+1$};
	\end{tikzpicture}
	\caption{Spectral networks for a conifold with the following choice of parameters:
	$\mu_1=1.2\times e^{0.07\pi {\rm i}}$, $\mu_2=2.2\times e^{0.07\pi {\rm i}}$, $\mu_3=3.248\times e^{0.07\pi {\rm i}}$, $\mu_4=2.248\times e^{0.07\pi {\rm i}}$. The spectral networks is drawn in the plane $z=e^{2\pi t}$ under a conformal map $z\mapsto \frac{1+z}{1-z}$, so that $0\mapsto 1$, $\infty\mapsto -1$.}\label{fig:Conif_SN}
	\end{center}
\end{figure}

Using this differential we construct a collection of S-walls on the phase diagram spanned by $z=e^{2\pi t}$ (see Fig.\ref{fig:Conif_SN}) and schematize an interface path and corresponding Lefshetz thimble topologies in different chambers (see Fig.\ref{fig:phase}).

\begin{figure}
	\begin{center}
		$\begin{array}{c}
			\begin{tikzpicture}[scale=0.7]
				\begin{scope}[yscale=0.5]
					\draw[ultra thick] (0,0) circle (5);
				\end{scope}
				\begin{scope}[xscale=0.2]
					\draw[dashed, blue] (0,2.5) arc (90:270:2.5);
					\begin{scope}[xscale=-1]
						\draw[thick, blue] (0,2.5) arc (90:270:2.5);
					\end{scope}
				\end{scope}
				\draw[ultra thick, color=purple] (-5,0) to[out=30,in=180] (0,1.5) to[out=0,in=150] (5,0);
				\draw[thick] (-2,0) to[out=90,in=210] (0.3,2) (2,0) to[out=120,in=315] (0.3,2);
				\draw[thick] (-2,0) to[out=310,in=171.064] (-1.5,-2.38485) (-1,2.44949) to[out=5.82755,in=135] (0.3,2);
				\draw[thick, dashed] (-1.5,-2.38485) to[out=351.064,in=185.828] (-1,2.44949);
				\begin{scope}[xscale=-1]
					\draw[thick] (-2,0) to[out=270,in=171.064] (-1.5,-2.38485) (-1,2.44949) to[out=5.82755,in=120] (-0.3,2);
					\draw[thick, dashed] (-1.5,-2.38485) to[out=351.064,in=185.828] (-1,2.44949);
				\end{scope}
			\begin{scope}[yscale=-1]
				\draw[thick] (-2,0) .. controls (-2.62547, 2.14045) .. (-3,2) (-3,-2) .. controls (-3.37453, -1.85955) and (-3.66718, 1.72188) .. (-4, 1.5) (-4,-1.5) .. controls (-4.33282, -1.27812) and (-4.7,0) .. (-5,0);
				\draw[thick, dashed] (-3,2) .. controls (-3.37453, 1.85955) and (-2.62547, -2.14045) .. (-3,-2) (-4,1.5) .. controls (-4.33282, 1.27812) and (-3.66718, -1.72188) ..  (-4,-1.5);
			\end{scope}
				\begin{scope}[xscale=-1]
					\draw[thick] (-2,0) .. controls (-2.62547, 2.14045) .. (-3,2) (-3,-2) .. controls (-3.37453, -1.85955) and (-3.66718, 1.72188) .. (-4, 1.5) (-4,-1.5) .. controls (-4.33282, -1.27812) and (-4.7,0) .. (-5,0);
					\draw[thick, dashed] (-3,2) .. controls (-3.37453, 1.85955) and (-2.62547, -2.14045) .. (-3,-2) (-4,1.5) .. controls (-4.33282, 1.27812) and (-3.66718, -1.72188) ..  (-4,-1.5);
				\end{scope}
				\draw[fill=blue] (-5,0) circle (0.08) (0.3,2) circle (0.08) (5,0) circle (0.08);
				\node[above left] at (-5,0) {$\infty$};
				\node[above right] at (5,0) {$0$};
				\node[right] at (0.3,2) {$1$};
				\foreach \i in {0,1,...,19}
				{
					\draw[thick, purple] (\i / 5 - 2, 0) to[out=90,in=180] (\i / 5 + 0.1 - 2, 0.1) to[out=0,in=90] (\i / 5 + 0.2 - 2, 0);
				}
				\begin{scope}[shift={(-2,0)}]
					\draw[ultra thick, purple] (-0.1,-0.1) -- (0.1,0.1) (-0.1,0.1) -- (0.1,-0.1);
				\end{scope}
				\begin{scope}[shift={(2,0)}]
					\draw[ultra thick, purple] (-0.1,-0.1) -- (0.1,0.1) (-0.1,0.1) -- (0.1,-0.1);
				\end{scope}
				\draw[->] (5.2,0) -- (5.8,0);
				\draw[->] (-5.2,0) -- (-5.8,0);
				\node[left] at (-6,0) {$\begin{array}{c} \mbox{Higgs} \\ \mbox{branch I}\\ (r>0) \end{array}$};
				\node[right] at (6,0) {$\begin{array}{c} \mbox{Higgs} \\ \mbox{branch II}\\ (r<0)\end{array} $};
				\draw[fill=green!60!black] (-0.8,1.47) circle (0.1) (0.8,1.47) circle (0.1);
				\draw[thick, green!60!black, dashed] (-1,1.47) to[out=90,in=180] (-0.8,1.67) -- (0.8,1.67) to[out=0,in=90] (1,1.47) to[out=270,in=0] (0.8,1.27) -- (-0.8,1.27) to[out=180,in=270] (-1,1.47);
				\node[above] at (0,2.6) {\color{green!60!black} Group B};
				\draw[fill=orange] (-3.4,0.8) circle (0.1);
				\begin{scope}[shift={(-5,0)}]
					\begin{scope}[rotate=26]
						\draw[dashed, thick, orange] (-0.2,0) to[out=90,in=180] (0,0.2) -- (1.8,0.2) to[out=0,in=90] (2,0) to[out=270,in=0] (1.8,-0.2) -- (0,-0.2) to[out=0,in=270] (-0.2,0);
					\end{scope}
				\end{scope}
				\node[above left] at (-4,1.5) {\color{orange} Group A};
				\begin{scope}[xscale=-1]
					\draw[fill=red] (-2.35,1.2) circle (0.1) (-3.65,0.7) circle (0.1);
					\begin{scope}[shift={(-5,0)}]
						\begin{scope}[rotate=24.36]
							\draw[dashed, thick, red] (-0.2,0) to[out=90,in=180] (0,0.2) -- (2.91,0.2) to[out=0,in=90] (3.11,0) to[out=270,in=0] (2.91,-0.2) -- (0,-0.2) to[out=0,in=270] (-0.2,0);
						\end{scope}
					\end{scope}
					\node[above right] at (-4,1.5) {\color{red} Group C};
				\end{scope}
				%
				\draw[->] (-3,3) to[out=270,in=90] (-1.5,1.4);
				%
				\draw[->] (3,3) to[out=270,in=90] (1.5,1.4);
				%
				\draw[->] (-3.5,-2.5) to[out=90,in=270] (0,1.5);
				%
				\node[above] at (-3,3) {${\bf P}_{I}$};
				\node[above] at (3,3) {${\bf P}_{III}$};
				\node[below] at (-3.5,-2.5) {${\bf P}_{II}$};
				%
				\node[below] at (0,-2.5) {\color{blue} $r=0$};
				\node[below right] at (3,-2) {Coulomb branch};
			\end{tikzpicture}\\
			\begin{array}{|c|c|c|}
				\hline
				{\bf P}_{I} & {\bf P}_{II} & {\bf P}_{III}\\
				\hline
				\begin{array}{c}
					\begin{tikzpicture}
						\draw[thick] (1,0) to[out=180,in=0] (0,0.3) to[out=180,in=90] (-0.3,0) to[out=270,in=180] (0,-0.3) to[out=0,in=180] (1,0) (1,0) -- (2,0);
						\draw[fill=red] (0,0) circle (0.08) (1,0) circle (0.08) (2,0) circle (0.08) (2,-0.7) circle (0.08);
						\node[left] at (-0.3,0) {$0$};
						\node[right] at (2,0) {$1$};
						\node[below] at (1,0) {$e^{-2\pi t}$};
						\node[right] at (2,-0.7) {$\infty$};
						\node[above] at (0,0.3) {${\cal L}_2$};
						\node[above] at (1.5,0) {${\cal L}_1$};
					\end{tikzpicture}
				\end{array}&\begin{array}{c}\begin{tikzpicture}
						\draw[thick] (0,0) to[out=30,in=150] (2,0) (0,0) to[out=330,in=210] (2,0);
						\draw[fill=red] (0,0) circle (0.08) (1,0) circle (0.08) (2,0) circle (0.08) (2,-0.7) circle (0.08);
						\node[left] at (0,0) {$e^{-2\pi t}$};
						\node[right] at (2,0) {$1$};
						\node[right] at (1,0) {$0$};
						\node[right] at (2,-0.7) {$\infty$};
						\node[above] at (1,0.3) {${\cal L}_1$};
						\node[below] at (1,-0.3) {${\cal L}_4$};
				\end{tikzpicture}\end{array}&\begin{array}{c}
					\begin{tikzpicture}
						\draw[thick] (1,0) to[out=180,in=0] (0,0.3) to[out=180,in=90] (-0.3,0) to[out=270,in=180] (0,-0.3) to[out=0,in=180] (1,0) (1,0) -- (2,0);
						\draw[fill=red] (0,0) circle (0.08) (1,0) circle (0.08) (2,0) circle (0.08) (2,-0.7) circle (0.08);
						\node[left] at (-0.3,0) {$0$};
						\node[below] at (1,0) {$1$};
						\node[right] at (2,0) {$e^{-2\pi t}$};
						\node[right] at (2,-0.7) {$\infty$};
						\node[above] at (0,0.3) {${\cal L}_3$};
						\node[above] at (1.5,0) {${\cal L}_1$};
					\end{tikzpicture}
				\end{array}\\
				\hline
			\end{array}
		\end{array}$
	\end{center}
	\caption{Conifold Coulomb branch parameterized by $z=e^{2\pi t}$.}\label{fig:phase}
\end{figure}

Comparing S-wall patterns of Fig.\ref{fig:SN_CP_1} and Fig.\ref{fig:phase} we observe that the conifold Coulomb branch is actually glued out of two $\mathbb{C}\mathbb{P}^1$ Coulomb branches. The cut line goes along the unit circle $r=0$. At singularities $0$ and $\infty$ the Coulomb branch is sewed with Higgs branches as in the $\mathbb{C}\mathbb{P}^1$ case.

S-walls divide the Coulomb branch in three chambers ${\bf P}_I$, ${\bf P}_{II}$ and ${\bf P}_{III}$ with corresponding topologies of Lefschetz thimbles depicted schematically in a table in Fig.\ref{fig:phase}. Corresponding expressions for the partition functions are:
\bar{e}\nonumber
\begin{split}
	\Xi_{{\cal L}_1}=&\frac{\Gamma({\rm i}\mu_{32})\Gamma(1+{\rm i}\mu_{23})\Gamma({\rm i}\mu_{41})\Gamma(1+{\rm i}\mu_{14})}{\Gamma(2+{\rm i}\mu_{14}+{\rm i}\mu_{23})}e^{2\pi{\rm i} t\,\mu_3}\left(1-e^{2\pi t}\right)^{1+{\rm i}\mu_{23}+{\rm i}\mu_{14}}\times\\
	&\times {}_2F_1\left[\begin{array}{c}
		1+{\rm i}\mu_{14};\;1+{\rm i}\mu_{24}\\
		2+{\rm i}\mu_{14}+{\rm i}\mu_{23}
	\end{array}\right]\left(1-e^{2\pi t}\right),\\
	\Xi_{{\cal L}_2}=&\frac{\sinh \pi\mu_{42}\sinh\pi\mu_{21}}{\sinh\pi\mu_{41}}e^{2\pi{\rm i} t\,\mu_2}\left[\Gamma({\rm i}\mu_{32})\Gamma({\rm i}\mu_{42})\Gamma({\rm i}\mu_{21})\right]\;{}_2F_1\left[\begin{array}{c}
		{\rm i}\mu_{42};\;{\rm i}\mu_{32}\\
		1+{\rm i}\mu_{12}
	\end{array}\right]\left(e^{-2\pi t}\right),\\
	\Xi_{{\cal L}_3}=&\frac{\sinh \pi\mu_{42}\sinh\pi\mu_{34}}{\sinh\pi\mu_{32}}e^{2\pi{\rm i} t\,\mu_4}\left[\Gamma({\rm i}\mu_{41})\Gamma({\rm i}\mu_{42})\Gamma({\rm i}\mu_{34})\right]\;{}_2F_1\left[\begin{array}{c}
		{\rm i}\mu_{42};\;{\rm i}\mu_{41}\\
		1+{\rm i}\mu_{43}
	\end{array}\right]\left(e^{2\pi t}\right),\\
	\Xi_{{\cal L}_4}=&e^{\pi(\mu_{32}+\mu_{42})}\;\Xi_1+\Xi_2=e^{\pi \mu_{34}}\;\Xi_{{\cal L}_1}-\Xi_{{\cal L}_3}.
\end{split}
 \end{equation}

We perform a parallel transport along path $\wp$ connecting Higgs branches at $z=\infty$ and $0$.
This path intersect S-walls in a selection of points we divide in three groups (see Fig.\ref{fig:phase}).
In groups A and C there are infinitely many intersection points from the spirals, these intersection points are in one-to-one correspondence with a tower of solitons obtained from intersection points at Fig.\ref{fig:SN_CP_1}(a).
There are only two intersection points in group B.

A pair of hypergeometric functions form a basis in the space of solutions to a hypergeometric equation.
Transition between bases $Z_{\gamma}$ and $\Xi_{{\cal L}}$ is given by Stokes matrices ${\bf S}_A$, ${\bf S}_B$, ${\bf S}_C$ that are given by supersymmetric indices of solitons contributing to intersection points of $\wp$ with the spectral network.
So we can summarize transitions between different bases in the following form:
\bar{e}
\begin{array}{c}
	\begin{tikzpicture}
		\node (A) at (0,0) {$
			\left(\begin{array}{c}
				Z_{\gamma_1}\\ Z_{\gamma_2}
			\end{array}\right)
			$};
		\node (B) at (3,0) {$
			\left(\begin{array}{c}
				\Xi_{{\cal L}_1}\\ \Xi_{{\cal L}_2}
			\end{array}\right)
			$};
		\node (C) at (6,0) {$
			\left(\begin{array}{c}
				\Xi_{{\cal L}_1}\\ \Xi_{{\cal L}_3}
			\end{array}\right)
			$};
		\node (D) at (9,0) {$
			\left(\begin{array}{c}
				Z_{\gamma_3}\\ Z_{\gamma_4}
			\end{array}\right)
			$};
		\path (A) edge[->] node[above] {$G_I{\bf \color{orange} S}_{\color{orange} A}$} (B)  (B) edge[->] node[above] {${\bf \color{green!60!black} S}_{\color{green!60!black} B}$} (C) (C) edge[->] node[above] {${\bf \color{red} S}_{\color{red} C}G_{II}$} (D);
	\end{tikzpicture}
\end{array}
 \end{equation}
so that
$$
{\bf A}=G_I{\bf \color{orange} S}_{\color{orange} A} {\bf \color{green!60!black} S}_{\color{green!60!black} B} {\bf \color{red} S}_{\color{red} C}G_{II}.
$$
Here $G_{I}$ and $G_{II}$ are diagonal matrices of normalization coefficients:
$$
G_I={\rm diag}\left(\begin{array}{cc}
\frac{\chi_3\chi_4}{\chi_1\chi_2}\frac{\lambda_{23}}{\lambda_{13}},&\frac{\lambda_{14}}{\lambda_{21}\lambda_{24}}
\end{array}\right),\quad G_{II}={\rm diag}\left(\begin{array}{cc}
\frac{\lambda_{13}}{\lambda_{14}},&\frac{\lambda_{24}\lambda_{34}}{\lambda_{23}}
\end{array}\right);
$$
and the Stokes matrices are:
\begingroup
\renewcommand*{\arraystretch}{1.7}
\bar{e}
{\bf S}_A=\left(\begin{array}{cc}
	1 & \frac{\chi_1\chi_2}{\chi_3\chi_4}\cdot\frac{1}{\lambda_{12}}\\
	0 & 1 \\
\end{array}\right),\quad {\bf S}_B=\left(\begin{array}{cc}
	1 & 0 \\
	\frac{\chi_3}{\chi_4}-\frac{\chi_3\chi_4}{\chi_2^2} & -1
\end{array}\right),\quad {\bf S}_C=\left(\begin{array}{cc}
	1 & \frac{1}{\lambda_{34}}\\
	0 & 1 \\
\end{array}\right).
 \end{equation}
\endgroup

\subsection{Herbst-Hori-Page invariant branes and parallel transport}
Rather than considering various Lagrangian cycles as integration contours we can produce various solutions to the hypergeometric equations by modifying brane boundary conditions.
There is a single brane on the disk boundary and we can insert a supersymmetric Wilson line of charge $k$.
Clearly \cite{Hori:2013ika} it produces a shift of the complexified FI parameter in the partition function argument:
$$
t\longrightarrow t-{\rm i} k.
$$
It will be simpler to calculate the result of this shift in the basis $Z_{\gamma}$. Let us denote the corresponding shift operator by ${\cal W}_k$:
\bar{e}\label{Wilson_line}
{\cal W}_k\,Z_{\gamma_a}=\chi_a^{2}Z_{\gamma_a}.
 \end{equation}

The space of hypergeometric solutions is two-dimensional, therefore the basis in the Wilson loop operators is also two-dimensional.
As basis elements one could choose, say, ${\cal W}_0$ and ${\cal W}_1$ that correspond to pullbacks of an exceptional set of bundles ${\cal O}$ and ${\cal O}(1)$ from base $\mathbb{C}\mathbb{P}^1$ to the resolved conifold.
To be precise let us note that since the fields spanning the resolution $X_{\pm}$ base $\mathbb{C}\mathbb{P}^1$ are $\phi_{1,2}$ or $\phi_{3,4}$ depending on the sign of $r$.
Therefore since $\phi_{1,2}$ and $\phi_{3,4}$ are charged oppositely with respect to the gauge $U(1)$ Wilson line insertions produce bundles with opposite Chern classes (for definition of bundles on conifold resolutions see Appendix \ref{s:BKS}):
$$
\begin{array}{c}
\begin{tikzpicture}
	\node(A) {${\cal W}_k$};
	\node(B) at (3,0) {${\cal Y}_+(k)$};
	\node(C) at (-3,0) {${\cal Y}_-(-k)$};
	\path (A) edge[->] node[above] {$r>0$} (B) (A) edge[->] node[above] {$r<0$} (C);
\end{tikzpicture}
\end{array}
$$

The result of the brane parallel transport can be calculated constructing invariant branes using a brane grade restriction rule as in \cite{HHP}.
We will call these branes Herbst-Hori-Page branes, or HHP branes for short.
Without going into details we could stress that the result of the parallel transport along $\wp$ coincides with the action of the Fourier-Mukai transform with a certain kernel we summarized in Appendix \ref{s:BKS}.
HHR invariant branes correspond to basis branes ${\cal Y}_{\pm}(k)$.
So that for corresponding bundle K-theory classes we have (compare with Appendix \ref{s:BKS}):
\bar{e}
\begin{split}
	&[{\cal Y}_+(0)]\mathop{\longrightarrow}^{\wp}[{\cal Y}_-(0)],\\
	&[{\cal Y}_+(1)]\mathop{\longrightarrow}^{\wp}[{\cal Y}_-(-1)]=-[{\cal Y}_-(0)]+2[{\cal Y}_-(1)].
\end{split}
 \end{equation}

However in this consideration a natural question arises if we can follow the HHP invariant branes transition through the LG phase and if they have a natural representation in the LG phase.
Our proposal is to identify HHP branes with two basis Lagrangian cycles ${\cal L}_1$ and ${\cal L}_4$ in chamber ${\bf P}_{II}$ in Fig.\ref{fig:phase}.
Corresponding partition functions satisfy the following relations:
\bar{e}\label{K-thy}
\begin{split}
	\Xi_{{\cal L}_1}=&\tilde Z_{\gamma_1}+\tilde Z_{\gamma_2}=\tilde Z_{\gamma_3}+\tilde Z_{\gamma_4},\\
	\chi_3\chi_4\Xi_{{\cal L}_4}=&{\color{black!40!green}\left(\frac{\chi_3\chi_4}{\chi_1\chi_2}\right)^2}\left[{\color{blue}\chi_1^2}\tilde Z_{\gamma_1}+{\color{blue}\chi_2^2}\tilde Z_{\gamma_2}\right]={\color{blue}\chi_3^2}\tilde Z_{\gamma_3}+{\color{blue}\chi_4^2}\tilde Z_{\gamma_4},
\end{split}
 \end{equation}
where\footnote{
A discrepancy between normalizations of $Z_k$ and $\tilde Z_k$ is induced by our rather inaccurate treatment of gamma function integrals like \eqref{extra_PF}.
In practice, these relations depend on a particular choice of the integration contour dictated by the QFT, for instance, relation
$$
\int\limits_{-\infty}^{\infty} dy e^{\mu \;y-e^{y}} =\Gamma(\mu)
$$
holds only for ${\rm Re}\;\mu>0$.
If the situation is different we have to choose a different integration cycle.
Similarly, in the definition of the disk partition function \eqref{conifold_pf} we have chosen simple brane boundary conditions.
As we will see in what follows this produces condensates of neutral mesons \eqref{cond_wf} that have to be dualized to the LG phase.
To cancel contributions of these condensates one may choose a zero section sheaf following recipe of \eqref{Weil}.
Such a sheaf produces a non-trivial D-brane operator \eqref{Z_B} $f(\sigma)\sim\sin\pi\sigma$ shifting normalizations of disk partition functions.
See also \cite[Section 2.1.1]{Zenkevich:2017ylb}.
}
$$
\tilde Z_{\gamma_1}=\frac{\chi_1\chi_2}{\chi_3\chi_4}\frac{\lambda_{13}\lambda_{14}}{\lambda_{32}\lambda_{41}}Z_{\gamma_1},\quad \tilde Z_{\gamma_2}=\frac{\chi_1\chi_2}{\chi_3\chi_4}\frac{\lambda_{23}\lambda_{24}}{\lambda_{32}\lambda_{41}}Z_{\gamma_2},\quad \tilde Z_{\gamma_3}=\frac{\lambda_{31}\lambda_{32}}{\lambda_{32}\lambda_{41}}Z_{\gamma_3},\quad \tilde Z_{\gamma_4}=\frac{\lambda_{41}\lambda_{42}}{\lambda_{32}\lambda_{41}}Z_{\gamma_4}.
$$
Clearly, Lefshetz thimble ${\cal L}_1$ produces a Wilson line boundary condition ${\cal W}_0$ on both Higgs branches for $r>0$ and $r<0$ decomposing to ${\cal Y}_\pm(0)$ respectively.
Lefshetz thimble ${\cal L}_4$ produces a Wilson line boundary condition ${\cal W}_1$ according to \eqref{Wilson_line} that decomposes to ${\cal Y}_+(1)$ and ${\cal Y}_-(-1)$ for $r>0$ and $r<0$ correspondingly.
An overall extra factor in the expression for ${\cal L}_4$ partition function on one side of the correspondence represents an equivariant morphism of multiplication by:
$$
\left(\phi_1\phi_2\phi_3\phi_4\right)\otimes
$$
This element is contained in the ideal sheaf ${\cal J}$ of zero sections of the conifold resolution $X_{\pm}$.
Following this logic we observe that the result of the parallel transport repeats the algebraic geometry calculation \cite{Kapranov:2014uwa} (see also Appendix \ref{s:BKS}):
\bar{e}\label{PT_cat}
\begin{split}
	&{\cal Y}_+(0)\mathop{\longrightarrow}^{\wp} {\cal Y}_-(0),\\
	&{\cal Y}_+(1)\mathop{\longrightarrow}^{\wp} {\cal Y}_-(-1)\otimes {\cal J}.
\end{split}
 \end{equation}
Unfortunately, the calculation presented in this section is performed in the K-theory of bundles rather than a derived category of coherent sheaves. Nevertheless, it schematizes how brane categories transform when we transit from one phase of the theory to another. To lift this calculation to the category level we need to check that non-perturbative instanton corrections to the MSW complex differential produce correct BPS Hilbert spaces. We will address this question in the following subsection.

\subsection{Fourier-Mukai kernel}

Manifolds $X_+$ and $X_-$ are isomorphic.
Rather than checking categorical parallel transport formula for sheaves \eqref{PT_cat} based on the Grothendieck group behavior we would check that HHP branes are indeed invariant on the category level.
To do so we need only a half of path $\wp$ flowing, say from phase ${\bf P}_{II}$ to $0$ denoting this path as $\wp_-$, for another part $\wp_+$ flowing from $\infty$ to ${\bf P}_{II}$ the reasoning will be analogous.

For the partition functions we have:
\bar{e}
\begin{array}{c}
	\begin{tikzpicture}
		\node (B) at (3,0) {$
			\left(\begin{array}{c}
				\Xi_{{\cal L}_1}\\ \Xi_{{\cal L}_4}
			\end{array}\right)
			$};
		\node (C) at (6,0) {$
			\left(\begin{array}{c}
				\Xi_{{\cal L}_1}\\ \Xi_{{\cal L}_3}
			\end{array}\right)
			$};
		\node (D) at (9,0) {$
			\left(\begin{array}{c}
				Z_{\gamma_3}\\ Z_{\gamma_4}
			\end{array}\right)
			$};
		\path (B) edge[->] node[above] {${\bf  S}_{0}$} (C) (C) edge[->] node[above] {${\bf S}_{C}G_{II}$} (D);
	\end{tikzpicture}
\end{array},
 \end{equation}
where
\bar{e}
{\bf S}_0=\left(\begin{array}{cc}
	1 & 0 \\
	\frac{\chi_3}{\chi_4} & -1
\end{array}\right)
 \end{equation}
is a Stokes matrix corresponding to the right intersection point in group B, so that
$$
{\bf S}_0{\bf S}_C=\left(\begin{array}{cc}
	1 & 0 \\
	\frac{\chi_3}{\chi_4} & -1
\end{array}\right)\cdot\left(\begin{array}{cc}
1 & \frac{1}{\lambda_{34}}\\
0 & 1 \\
\end{array}\right)=\left(\begin{array}{cc}
	1 & \lambda_{34}^{-1} \\
	\frac{\chi_3}{\chi_4} & \frac{\chi_4}{\chi_3}\lambda_{34}^{-1}
\end{array}\right).
$$

Stokes coefficient $\lambda_{34}^{-1}$ in matrix ${\bf S}_C$ is a partition function contribution from an infinite series of poles in group C, in particular, the very coefficient is a trivial consequence of series summation:
$$
\lambda_{34}^{-1}=\sum\limits_{k=0}^{\infty}\left(\chi_3^{-1}\chi_4\right)^{2k+1}.
$$
Each such solitonic contribution to the partition function is reflected in wave functions spanning the BPS Hilbert space.
So we write easily an expression for the MSW complex vector space associated to an interface parameterized by path $\wp_-$:\footnote{Here to denote component vectors of the MSW complex we use notations \eqref{bpsi}. Homological degrees -- fermion numbers -- of distinct vectors are defined by homological degrees of complex \eqref{subcomplex} and can be calculated alternatively using Maslov index \eqref{Maslov} (see also \cite[Appendix A]{Galakhov:2016cji}).}
\bar{e}
\begin{split}
{\bf M}^*=\Psi_L\otimes\left(\begin{array}{cc}
{\boldsymbol{\psi}}_{33}^{(0)}[1] & 0\\
{\boldsymbol{\psi}}_{43}^{(0)}[\mathfrak{X}_{43;0}] & {\boldsymbol{\psi}}_{44}^{(-1)}[1]\\
\end{array}\right)\otimes \left(\begin{array}{cc}
{\boldsymbol{\psi}}_{33}^{(0)}[1] & \bigoplus\limits_{k\geq 0}{\boldsymbol{\psi}}_{34}^{(0)}[\mathfrak{X}_{34;k}]\\
0 & {\boldsymbol{\psi}}_{44}^{(0)}[1]\\
\end{array}\right)\otimes \Psi_R=\\
=\Psi_L\otimes\left(\begin{array}{cc}
{\boldsymbol{\psi}}_{33}^{(0)}[1] & \bigoplus\limits_{k\geq 0}{\boldsymbol{\psi}}_{34}^{(0)}[\mathfrak{X}_{34;k}]\\
{\boldsymbol{\psi}}_{43}^{(0)}[\mathfrak{X}_{43;0}] & \bigoplus\limits_{k\geq 0}{\boldsymbol{\psi}}_{44}^{(0)}[\mathfrak{X}_{43;0},\mathfrak{X}_{34;k}]\oplus{\boldsymbol{\psi}}_{44}^{(-1)}[1] \\
\end{array}\right)\otimes \Psi_R.
\end{split}
 \end{equation}
So far we have not specified boundary conditions yet, therefore we use wave functions $\Psi_L$ and $\Psi_R$ to encode these data.
$\Psi_L$ corresponds to a brane boundary condition on the left.
We consider it to be given by either Lagrangian brane ${\cal L}_1$ or ${\cal L}_4$.
Solutions for ${\cal L}_1$ and ${\cal L}_4$ approach asymptotically constant vacua $*3$ and $*4$ correspondingly.
We can represent these two choices in a form of vectors of complexes:
$$
\left(\begin{array}{cc}
\psi_{33}^{(0)}[{\cal L}_1]& 0\\
\end{array}\right),\quad  \left(\begin{array}{cc}
0& \psi_{44}^{(0)}[{\cal L}_4]\\
\end{array}\right).
$$
$\Psi_R$ and boundary conditions at the right end of the interval will remain unspecified.

Let us reshuffle some terms of an expression for this complex and start with the bottom right element in the matrix:
$$
\bigoplus\limits_{k\geq 0}{\boldsymbol{\psi}}_{44}^{(0)}[\mathfrak{X}_{43;0},\mathfrak{X}_{34;k}]\oplus{\boldsymbol{\psi}}_{44}^{(1)}[1] =\underline{\left({\boldsymbol{\psi}}_{44}^{(0)}[\mathfrak{X}_{43;0},\mathfrak{X}_{34;0}]\oplus{\boldsymbol{\psi}}_{44}^{(-1)}[1]\right)}\oplus \bigoplus\limits_{k\geq 1}{\boldsymbol{\psi}}_{44}^{(0)}[\mathfrak{X}_{43;0},\mathfrak{X}_{34;k}].
$$
The underlined term in this expression is a subcomplex:
\bar{e}\label{subcomplex}
\begin{array}{c}
	\begin{tikzcd}
		0 \ar[r]& {\boldsymbol{\psi}}_{44}^{(-1)}[1] \ar[r,"{\bf Q}"]& {\boldsymbol{\psi}}_{44}^{(0)}[\mathfrak{X}_{43;0},\mathfrak{X}_{34;0}] \ar[r]& 0
	\end{tikzcd}
\end{array},
 \end{equation}
that is quasi-isomorphic to zero. This is a simple conclusion following from the fact that a homotopy move of an interface path forcing it to go through the ramification point of the spectral network gives a family of quasi-isomorphic MSW complexes (see \cite{Galakhov:2017pod}).
In particular, all the states in MSW complex will contain either both cochains of this two-term complex or none.
The instanton map saturating the supercharge differential $\bf Q$ in this complex is given by a soliton web diagram depicted in Fig.\ref{fig:web}(a).

In a similar way we treat a tower of solitons contribution $\mathfrak{X}_{34;k}$ as condensate \eqref{condensate}:
\bar{e}
\begin{array}{c}
\begin{tikzpicture}
	\node (A) at (0,0) {${\boldsymbol{\psi}}_{34}^{(0)}[\mathfrak{X}_{34;k}]$};
	\node (B) at (5,0) {${\boldsymbol{\psi}}_{34}^{(0)}[\mathfrak{X}_{34;0}]\otimes \mathbb{C} \left(\phi_3\phi_4^{-1}\right)^k$};
	\path (A) edge[<->] node[above] {$\mathfrak{a}_k$} (B);
\end{tikzpicture}
\end{array}.
 \end{equation}
This transition is governed by a soliton scattering amplitude $\mathfrak{a}_k$ \eqref{amplitudes}.
Indeed both wave functions set up equivalent boundary conditions for soliton webs (see Fig.\ref{fig:web}(b)).
Condensed solitons $\mathfrak{X}_{44;k}$ are equivalent to condensate \eqref{condensate} as we have shown in a comparison of Higgs and Coulomb branch soliton spectra in $\mathbb{C}\mathbb{P}^1$ model in Section \ref{sec:equiv_CP_1}.

\begin{figure}[h!]
	\begin{center}
		\begin{tikzpicture}
			\node (A) at (0,0) {$\begin{array}{c}
					\begin{tikzpicture}[scale=0.8]
						\draw[dashed,fill=blue, opacity=0.4] (0,0) -- (3,0) -- (3.25,0.25) -- (3,0.5) -- (3.25,0.75) -- (3,1) -- (3.25,1.25) -- (3,1.5) -- (3.25,1.75) -- (3,2) -- (0,2) -- (-0.25,1.75) -- (0,1.5) -- (-0.25,1.25) -- (0,1) -- (-0.25,0.75) -- (0,0.5)  -- (-0.25,0.25) -- (0,0);
						\draw[<->] (-0.5,2) -- (-0.5,-0.25) -- (3.5,-0.25);
						\node[above left] at (-0.5,2) {$x^2$};
						\node[right] at (3.5,-0.25) {$x^1$};
						\draw[thick] (1,2) to[out=270,in=180] (1.5,1) to[out=0,in=270] (2,2);
						\shade[ball color = red] (1,2) circle (0.08);
						\shade[ball color = red] (2,2) circle (0.08);
						\node[above left] at  (1,1.8) {$\mathfrak{X}_{43;0}$};
						\node[above right] at  (2,1.8) {$\mathfrak{X}_{34;0}$};
						\node at (1.5,1.5) {$*3$};
						\node at (1.5,0.5) {$*4$};
					\end{tikzpicture}
				\end{array}$};
		\node (B) at (7,0) {$\begin{array}{c}
				\begin{tikzpicture}[scale=0.8]
					\draw[dashed,fill=blue, opacity=0.4] (0,0) -- (3,0) -- (3.25,0.25) -- (3,0.5) -- (3.25,0.75) -- (3,1) -- (3.25,1.25) -- (3,1.5) -- (3.25,1.75) -- (3,2) -- (0,2) -- (-0.25,1.75) -- (0,1.5) -- (-0.25,1.25) -- (0,1) -- (-0.25,0.75) -- (0,0.5)  -- (-0.25,0.25) -- (0,0);
					\draw[<->] (-0.5,2) -- (-0.5,-0.25) -- (3.5,-0.25);
					\node[above] at (-0.5,2) {$x^2$};
					\node[right] at (3.5,-0.25) {$x^1$};
					\draw[thick] (1,0) -- (1,1) to[out=135,in=270] (0.5,2) (1,1) -- (2.5,1.375);
					\draw[thick, dashed] (2.5,1.375) -- (3,1.5);
					\shade[ball color = red] (1,0) circle (0.08);
					\shade[ball color = blue] (0.5,2) circle (0.08);
					\draw[fill=red] (1,1) circle (0.1);
					\node[left] at (1,1) {$\mathfrak{a}_k$};
					\node[above right] at (1,-0.2) {$\mathfrak{X}_{34;k}$};
					\node[above right] at (0.5,1.8) {$\mathfrak{X}_{34;0}$};
					\node[right] at (2,1) {$\mathfrak{X}_{44;k}$};
				\end{tikzpicture}
			\end{array}\!\!\longleftrightarrow\!\! \begin{array}{c}
				\begin{tikzpicture}[scale=0.8]
					\draw[dashed,fill=blue, opacity=0.4] (2,0) -- (3,0) -- (3.25,0.25) -- (3,0.5) -- (3.25,0.75) -- (3,1) -- (3.25,1.25) -- (3,1.5) -- (3.25,1.75) -- (3,2) -- (2,2) -- cycle;
					\draw[<->] (1.5,2) -- (1.5,-0.25) -- (3.5,-0.25);
					\node[above] at (1.5,2) {$x^2$};
					\node[right] at (3.5,-0.25) {$x^1$};
					\draw[thick] (2,1.25) -- (2.5,1.375);
					\draw[thick, dashed] (2.5,1.375) -- (3,1.5);
					\draw[ultra thick] (2,0) -- (2,2);
					\node[right] at (2,1) {$\mathfrak{X}_{44;k}$};
					\draw[fill=red] (2,1.25) circle (0.1);
					\shade[ball color = orange] (2,0) circle (0.08);
				\end{tikzpicture}
			\end{array}$};
		\node at (0,-1.5) {(a)};
		\node at (7,-1.5) {(b)};
		\end{tikzpicture}
	\caption{Web diagrams}\label{fig:web}
	\end{center}
\end{figure}

If ${\cal L}_1$ is chosen as a boundary condition the resulting MSW complex reads:
\bar{e}\label{complex}
\left({\boldsymbol{\psi}}^{(0)}_{33}\left[{\cal L}_1\right]\otimes \Psi_R\right)\oplus \left({\boldsymbol{\psi}}^{(0)}_{34}\left[{\cal L}_1;\mathfrak{X}_{34;0}\right]\otimes \bigoplus\limits_{k\geq 0}\mathbb{C} \left(\phi_3\phi_4^{-1}\right)^k\otimes \Psi_R\right)\leftrightarrow \Psi_L[{\cal Y}(0)]\otimes \Psi_R.
 \end{equation}
This complex is acted upon by ${\bf Q}$ and is equivalent to a choice of sheaf ${\cal Y}(0)$ as a brane boundary condition in the sigma model.
Indeed if such a sheaf is chosen due to equivariant localization it reduces to either constant vacuum $*3$ or $*4$.
In the case of vacuum $*4$ there is a tower of condensed solitons \eqref{condensate} for our choice of numerical values of complex masses $\mu_a$.
These are exactly two terms in complex \eqref{complex}.

Similarly, for ${\cal L}_4$ choice as a left boundary brane we have:
\bar{e}
\left({\boldsymbol{\psi}}^{(0)}_{43}\left[{\cal L}_4;\mathfrak{X}_{43;0}\right]\otimes \Psi_R\right)\oplus \left({\boldsymbol{\psi}}^{(0)}_{44}\left[{\cal L}_4;\mathfrak{X}_{43;0};\mathfrak{X}_{34;0}\right]\otimes \bigoplus\limits_{k\geq 1}\mathbb{C} \left(\phi_3\phi_4^{-1}\right)^k\otimes \Psi_R\right)\leftrightarrow \Psi_L[{\cal Y}(-1)]\otimes \Psi_R.
 \end{equation}
Due to a shift in the electric charges $k$ this boundary condition is equivalent to a sheaf ${\cal Y}(-1)$.

We conclude that the parallel transport formula \eqref{PT_cat} we conjectured based on the K-theory results is valid for the categories as well.

The theory at the end of the interval $[0,L]$ is in its sigma-model phase, so the brane boundary conditions are represented by derived categories of coherent sheaves and the parallel transport is a functor:
\bar{e}
\beta_{\wp}:\quad D^{(b)}{\rm Coh}(X_+)\longrightarrow D^{(b)}{\rm Coh}(X_-).
 \end{equation}
According to Orlov's theorem \cite{Huybrechts,orlov2003derived} this functor can be represented as a Fourier-Mukai (FM) transform with some kernel ${\cal K}$ (see Appendix \ref{sec:App_FM} for our conventions):
\bar{e}
\beta_{\wp}={\boldsymbol{\Phi}}_{{\cal K}}.
 \end{equation}

Fortunately, sheaves ${\cal Y}(k)$ and ${\cal J}$ are equivariant, therefore the action of the parallel transport functor on those sheaves will be preserved even if the equivariant action is lifted (see a definition of equivariant derived category in \cite{Equivariant_sheves}).
It can be explicitly shown that for the following kernel:
\bar{e}\label{kernel}
{\cal K}={\cal O}_{\{S_{(-)}=S_{(+)}\}},
 \end{equation}
the FM transform action on branes ${\cal Y}(0)$ and ${\cal Y}(1)$ will coincide with the action of parallel transport \eqref{PT_cat}.
This check is performed in \cite{Kapranov:2014uwa} and in Appendix \ref{s:BKS} using slightly different tools in algebraic geometry.

To conclude this section let us comment on how the kernel in the form \eqref{kernel} could have been guessed a priori without performing the algebraic geometry calculation.

Let us consider the theory in the neighborhood of the left brane in either constant vacuum $*1$ or $*2$ when either field $\phi_1$ or field $\phi_2$ acquires an expectation value.
According to our calculations in Section \ref{sec:RigidHiggs} the IR description will contain electrically neutral effective meson scalars \eqref{meson}.
We can combine chiral fields in meson operators in different ways, however the only holomorphic combinations are the following four we arrange in a matrix:
\bar{e}\label{S}
S=\left(\begin{array}{cc}
	M_{11} & M_{12}\\
	M_{21} & M_{22}
\end{array}\right)=\left(\begin{array}{cc}
\phi_2\phi_4& -\phi_1\phi_4\\
-\phi_2\phi_3 & \phi_1\phi_3\\
\end{array}\right).
 \end{equation}
Effective theory for the right brane with other two choices -- $*3$ and $*4$ --  as constant vacua will have the same set of neutral holomorphic meson fields.
According to Section \ref{sec:RigidHiggs} the localization position of meson degrees of freedom depends on effective masses:
\bar{e}
\mu_{\rm eff}(M_{ij})=\left(\begin{array}{cc}
\mu_4-\mu_2 & \mu_4-\mu_1\\
\mu_3-\mu_2 & \mu_3-\mu_1\\
\end{array}\right).
 \end{equation}
A meson condensate wave function is a generic holomorphic or anti-holomorphic function of the meson field -- structure sheaf of an affine plane spanned by this field.
A condensate of neutral meson chiral field $\mathfrak{m}$ corresponds to a ring of lowest Landau level wave functions $\mathbb{C}[\mathfrak{m}]$ and reproduces a condensate variety as points in
$$
{\rm Spec}\; \mathbb{C}[\mathfrak{m}].
$$
In our case matrix the whole meson matrix $S$ condenses and parameterizes a fiber of $X_{\pm}$ (see such parameterization in Appendix \ref{s:BKS}).
We could even distinguish values of $S$ on two ends of spatial interval $[0,L]$:
$$
S_{(\pm)}\subset X_{\pm}.
$$
$M_{ij}$ localized at the left or right brane depending on the sign of ${\rm Im}\,\mu(M_{ij})$. In the case ${\rm Im}\,\mu(M_{ij})=0$ meson $M_{ij}$ is delocalized and is represented by a constant mode for the whole interface. We can even decouple a half of them from the IR theory explicitly, see decoupling of partition functions in \eqref{extra_PF}.
These observations indicate that even if we have distinguished subvarieties $S_{(+)}$ and $S_{(-)}$ for the left and the right branes the BPS state wave function depends on common meson degrees of freedom $M_{ij}(x^1)$ so that:\footnote{We could probe this constraint by applying such a morphism with a constant chiral mode to skyscraper sheaves.
Suppose on the left and on the right branes one puts stalks supported in points $z_L$ and $z_R$ correspondingly.
The resulting boundary supercharge contribution reads (see \eqref{stalk}):
$$
\bar{\cal Q}_{\rm bdry}=\chi_L\left(\phi(0)-z_L\right)+\chi_R\left(\phi(0)-z_R\right).
$$
The effective supercharge for the constant mode $\phi_0$ has the following form:
$$
\bar{\cal Q}\sim \bar\psi_{\dot 1}\partial_{\bar\phi_0}+\left(\chi_L+\chi_R\right)\left(\phi_0-\frac{z_L+z_R}{2}\right)-\frac{\chi_L-\chi_R}{2}(z_L-z_R).
$$
As a result the Laplacian is bounded from below $\left\{\bar{\cal Q},{\cal Q}\right\}\geq \frac{1}{2}|z_L-z_R|^2$, and there is no BPS state in the spectrum unless $z_L=z_R$.
}
$$
S_{(+)}=M_{ij}(0)=M_{ij}(L)=S_{(-)}.
$$
An alternative way to impose this constraint is to include the corresponding $\delta$-function in the IR BPS wave-function:
\bar{e}\label{Psi}
\Psi_L(S_{(+)})\otimes \Psi_{\rm bulk}\otimes \Psi_R(S_{(+)})\;\cdot\;\delta\left(\{S_{(+)}=S_{(-)}\}\subset X_+\times X_-\right).
 \end{equation}
Geometrically the $\delta$-function contribution implies that kernel ${\cal K}$ is supported on the corresponding locus, and we checked that ${\cal K}$ itself is just a structure sheaf of this locus \eqref{kernel}.

\section{Categorification of braid group action}\label{sec:quiver}

\subsection{Braid group in Wess-Zumino-Witten model}

\subsubsection{Braid group}

Consider a generic configuration of $N$ points in a complex plain $z_i\in\mathbb{C}$, $i=1,\ldots,N$.
Suppose this set is ordered by any ordering function, say, a real part, an imaginary part or an absolute value.
So if we choose the latter variant we say that inequality $i<j$ holds for indices when
$$
|z_i|<|z_j|.
$$
Consider a counterclockwise permutation move of two neighbor points with indices $i$ and $i+1$.
Let us parameterize this move by ``time" interval $I_t$.
World-lines of  these points in ``space-time" $\mathbb{C}\times I_t$ form a simple braid of $N$ strands where only two neighbor strands are braided:
\bar{e}\label{bii+1_+}
b_{i,i+1}=\begin{array}{c}
\begin{tikzpicture}
	\draw (-2.1,-0.3) -- (-2.5,-0.7) -- (1.6,-0.7) -- (2,-0.3) -- cycle;
	\draw[thick] (-0.5,-0.5) to[out=90,in=210] (0,0) to[out=30,in=270] (0.5,0.5) (0.5,-0.5) to[out=90,in=330] (0.1,-0.07) (-0.1,0.07) to[out=150,in=270] (-0.5,0.5);
	\draw[thick] (-1.5,-0.5) -- (-1.5,0.5) (-2,-0.5) -- (-2,0.5) (1.5,-0.5) -- (1.5,0.5);
	\draw[fill=black] (-2,-0.5) circle (0.07) (-1.5,-0.5) circle (0.07) (-0.5,-0.5) circle (0.07) (0.5,-0.5) circle (0.07) (1.5,-0.5) circle (0.07)
	%
	(-2,0.5) circle (0.07) (-1.5,0.5) circle (0.07) (-0.5,0.5) circle (0.07) (0.5,0.5) circle (0.07) (1.5,0.5) circle (0.07)
	%
	(-1.25,-0.5) circle (0.03) (-1,-0.5) circle (0.03) (-0.75,-0.5) circle (0.03)
	%
	(1.25,-0.5) circle (0.03) (1,-0.5) circle (0.03) (0.75,-0.5) circle (0.03)
	%
	(-1.25,0.5) circle (0.03) (-1,0.5) circle (0.03) (-0.75,0.5) circle (0.03)
	%
	(1.25,0.5) circle (0.03) (1,0.5) circle (0.03) (0.75,0.5) circle (0.03);
	\begin{scope}[shift={(0,1)}]
		\draw (-2.1,-0.3) -- (-2.5,-0.7) -- (1.6,-0.7) -- (2,-0.3) -- cycle;
	\end{scope}
	\node[right] at (2,-0.5) {$\mathbb{C}$};
	\node[right] at (2,0.5) {$\mathbb{C}$};
	\node[below] at (-2,-0.7) {$1$};
	\node[below] at (-1.5,-0.7) {$2$};
	\node[below] at (-0.5,-0.7) {$i$};
	\node[below] at (0.5,-0.7) {$i+1$};
	\node[below] at (1.5,-0.7) {$N$};
	\node[above] at (-2,0.7) {$1$};
	\node[above] at (-1.5,0.7) {$2$};
	\node[above] at (-0.5,0.7) {$i$};
	\node[above] at (0.5,0.7) {$i+1$};
	\node[above] at (1.5,0.7) {$N$};
\end{tikzpicture}
\end{array}
 \end{equation}
The inverse element is generated by a clockwise permutation move:
\bar{e}\label{bii+1_-}
b_{i,i+1}^{-1}=\begin{array}{c}
	\begin{tikzpicture}
		\draw (-2.1,-0.3) -- (-2.5,-0.7) -- (1.6,-0.7) -- (2,-0.3) -- cycle;
		\begin{scope}[xscale=-1]
		\draw[thick] (-0.5,-0.5) to[out=90,in=210] (0,0) to[out=30,in=270] (0.5,0.5) (0.5,-0.5) to[out=90,in=330] (0.1,-0.07) (-0.1,0.07) to[out=150,in=270] (-0.5,0.5);
		\end{scope}
		\draw[thick] (-1.5,-0.5) -- (-1.5,0.5) (-2,-0.5) -- (-2,0.5) (1.5,-0.5) -- (1.5,0.5);
		\draw[fill=black] (-2,-0.5) circle (0.07) (-1.5,-0.5) circle (0.07) (-0.5,-0.5) circle (0.07) (0.5,-0.5) circle (0.07) (1.5,-0.5) circle (0.07)
		%
		(-2,0.5) circle (0.07) (-1.5,0.5) circle (0.07) (-0.5,0.5) circle (0.07) (0.5,0.5) circle (0.07) (1.5,0.5) circle (0.07)
		%
		(-1.25,-0.5) circle (0.03) (-1,-0.5) circle (0.03) (-0.75,-0.5) circle (0.03)
		%
		(1.25,-0.5) circle (0.03) (1,-0.5) circle (0.03) (0.75,-0.5) circle (0.03)
		%
		(-1.25,0.5) circle (0.03) (-1,0.5) circle (0.03) (-0.75,0.5) circle (0.03)
		%
		(1.25,0.5) circle (0.03) (1,0.5) circle (0.03) (0.75,0.5) circle (0.03);
		\begin{scope}[shift={(0,1)}]
			\draw (-2.1,-0.3) -- (-2.5,-0.7) -- (1.6,-0.7) -- (2,-0.3) -- cycle;
		\end{scope}
		\node[right] at (2,-0.5) {$\mathbb{C}$};
		\node[right] at (2,0.5) {$\mathbb{C}$};
		\node[below] at (-2,-0.7) {$1$};
		\node[below] at (-1.5,-0.7) {$2$};
		\node[below] at (-0.5,-0.7) {$i$};
		\node[below] at (0.5,-0.7) {$i+1$};
		\node[below] at (1.5,-0.7) {$N$};
		\node[above] at (-2,0.7) {$1$};
		\node[above] at (-1.5,0.7) {$2$};
		\node[above] at (-0.5,0.7) {$i$};
		\node[above] at (0.5,0.7) {$i+1$};
		\node[above] at (1.5,0.7) {$N$};
	\end{tikzpicture}
\end{array}
 \end{equation}
We will treat such braids being equivalent up to ambient isotopy.
Also we endow them with a group composition law reflected in a natural braid concatenation.
Under these circumstances braids $b_{i,i+1}$ and $b^{-1}_{i,i+1}$ for different indices $i$ generate braid group $\mathfrak{B}\mathfrak{r}_N$ of $N$ strands and satisfy the following set of relations:
\bar{e}\label{braid_group}
\begin{split}
	& b_{i,i+1}b_{i,i+1}^{-1}=b_{i,i+1}^{-1}b_{i,i+1}={\bf Id};\\
	& b_{i,i+1}b_{j,j+1}=b_{j,j+1}b_{i,i+1},\quad {\rm if}\; |i-j|>1;\\
	& b_{i,i+1}b_{i+1,i+2}b_{i,i+1}=b_{i+1,i+2}b_{i,i+1}b_{i+1,i+2};
\end{split}
 \end{equation}
where by $\bf Id$ we imply a collection of $N$ unbraided strands.

\subsubsection{Wess-Zumino-Witten model}\label{sec:WZW_model}

Wess-Zumino-Witten(WZW) 2d conformal model is characterized by a level $k$ and a Lie group $G$.
A holomorphic primary field $g_{\lambda}(z)$ depends on a complex coordinate $z$ on a 2d surface and transforms as a representation of Lie algebra $\mathfrak{g}$ of the highest weight $\lambda$ \cite{DiFrancesco:639405}.
$N$-point conformal blocks are holomorphic functions of positions of punctures:
\bar{e}
{\cal B}_{\vec\lambda}^{\vec\alpha}(\vec z)=\left\langle g_{\lambda_1}^{\alpha_1}(z_1)g_{\lambda_2}^{\alpha_2}(z_2)\ldots g_{\lambda_N}^{\alpha_N}(z_N)\right\rangle,
 \end{equation}
where $\alpha_i$ parameterize vectors of corresponding representations of weights $\lambda_i$.
Conformal blocks for various choices of $\vec \lambda$ and $\vec\alpha$ form a vector space of $N$-point conformal blocks $\mathfrak{B}_N$.
$\mathfrak{B}_N$ is a space of solutions to Knizhnik-Zamolodchikov equations.
In other words, a vector
\bar{e}
\Psi(\vec z):=\sum\limits_{\vec \lambda,\vec \alpha}c_{\vec\alpha}^{\vec\lambda}{\cal B}_{\vec\lambda}^{\vec\alpha}(\vec z)
 \end{equation}
is annihilated by a Knizhnik-Zamolodchikov (KZ) connection \cite{Knizhnik:1984nr}:
\bar{e}\label{KZ}
\nabla_a^{\rm KZ}\Psi=\left(\partial_{z_a}-A_a\right)\Psi=\left(\partial_{z_a}+\epsilon\sum\limits_{b\neq a}\frac{\eta_{ij}T^i_aT_b^j}{z_a-z_b}\right)\Psi=0,\quad a=1,\ldots,N,
 \end{equation}
where $\eta$ is the Killing form, and $\epsilon=(k+c_2(\mathfrak{g}))^{-1}$, where $c_2(\mathfrak{g})$ is the dual Coxeter number.
Generators $T$ act as generators $t^i$ of corresponding Lie algebra $\mathfrak{g}$ on fields:
\bar{e}
T_a^i\left\langle g_{\lambda_1}^{\alpha_1}(z_1)\ldots g_{\lambda_N}^{\alpha_N}(z_N)\right\rangle=\left\langle g_{\lambda_1}^{\alpha_1}(z_1)\ldots \left(\left(t^i\right)_{\alpha_a'}^{\alpha_a}g_{\lambda_a}^{\alpha_a'}(z_a)\right)\ldots g_{\lambda_N}^{\alpha_N}(z_N)\right\rangle.
 \end{equation}

The KZ connection is flat on the configuration space of $N$ points outside the singular locus where points $z_a$ collide.
Corresponding parallel transport map will be an invariant of the parallel transport path homotopy class. Consider paths associated to simple tangles $b_{i,i+1}^{\pm}$ in \eqref{bii+1_+}, \eqref{bii+1_-} and denote corresponding paths projected to $\mathbb{C}$ as $\wp_{i,i+1}^{\pm}$.
Parallel transport operators:
\bar{e}
\begin{split}
&U_{i,i+1}:\quad \mathfrak{B}_N\longrightarrow\mathfrak{B}_N,\\
&U_{i,i+1}={\rm P}{\rm exp}\int\limits_{\wp_{i,i+1}}A_adz^a.
\end{split}
 \end{equation}
satisfy braid group relations \eqref{braid_group} due to parallel transport flatness and form a representation of the braid group $\mathfrak{B}\mathfrak{r}_N$ on conformal blocks $\mathfrak{B}_N$.

A categorification of the braid group representation for generic Lie groups and representations is a hard problem \cite{Aganagic:2020olg,aganagic2021knot}, therefore we restrict ourselves to the simplest case of $G=SU(2)$ and its spin-1/2 representations.
We put the WZW model on a cylinder, for this purpose two primary fields are located in $0$ and $\infty$, corresponding (not necessarily integral) weights are $\lambda_0$ and $\lambda_{\infty}$, remaining fields are in a spin-1/2 representation of $SU(2)$:
\bar{e}\label{cylinder}
\begin{array}{c}
\begin{tikzpicture}
	\draw[thick] (-2,0.3) -- (2,0.3) (-2,-0.3) -- (2,-0.3);
	\begin{scope}[shift={(-2,0)}]
		\begin{scope}[xscale=0.5]
			\draw[thick] (0,0) circle (0.3);
		\end{scope}
	\end{scope}
	\begin{scope}[shift={(2,0)}]
		\begin{scope}[xscale=0.5]
			\draw[thick] (0,0) circle (0.3);
		\end{scope}
	\end{scope}
\node[left] at (-2.2,0) {$\langle \lambda_0|$};
\node[right] at (2.2,0) {$|\lambda_{\infty}\rangle$};
\draw[fill=black] (-1.5,0) circle (0.06) (-1,0) circle (0.06) (1.5,0) circle (0.06) (0.25,0) circle (0.03) (0.5,0) circle (0.03) (0,0) circle (0.03);
\node[below] at (-1.5,-0.3) {$z_0$};
\node[below] at (-1,-0.3) {$z_1$};
\node[below] at (1.5,-0.3) {$z_{m-1}$};
\end{tikzpicture}
\end{array}
 \end{equation}

Index $\alpha$ for primary fields $g$ in spin-1/2 representation takes two values corresponding to +1/2 and -1/2 spin projections. We will denote these indices as ``$+$" and ``$-$" correspondingly for the sake of brevity.
The action of the parallel transport maps splits the space of conformal blocks in invariant subspaces of fixed weights.
Therefore the relative numbers of positive and negative spins are invariants of the braid group action. Suppose among our $m$ spins $n$ spins are positive and $m-n$ are negative.
We denote the set of all such spin configurations as
$$
{\cal Y}_{n,m-n}.
$$

In what follows we will exploit an isomorphism between the set of spin configurations ${\cal Y}_{n,m-n}$ and all Young diagrams that can be embedded in a $(m-n)\times n$ rectangular field.
This isomorphism is constructed as follows.
Consider a rectangular $(m-n)\times n$ field of unit cells.
We construct a path going from the top left corner of this field towards the bottom right corner.
This path goes along the borders of cells. At each crossroad one is able to turn either downwards or to the right, the downward flow corresponds to spin ``$+$" and the right flow corresponds to spin ``$-$".
A road-map describing the route and written as a sequence of ``$+$"'s and ``$-$"'s is in a one-to-one correspondence with a spin configuration in ${\cal Y}_{n,m-n}$.
The corresponding Young diagram is given by a box profile under the path.
For example, there are two following representations of partition $\{3,2,1,1\}$ in ${\cal Y}_{3,5}$
\bar{e}\label{Xi_iso}
(-,+,-,+,-,-,+,-)\quad\leftrightarrow \quad
\begin{array}{c}
\begin{tikzpicture}[scale=0.5]
	\begin{scope}[shift={(0,0)}]
		\draw[fill=gray] (0,0) -- (0,1) -- (1,1) -- (1,0) -- cycle;
	\end{scope}
\begin{scope}[shift={(1,0)}]
	\draw[fill=gray] (0,0) -- (0,1) -- (1,1) -- (1,0) -- cycle;
\end{scope}
\begin{scope}[shift={(2,0)}]
	\draw[fill=gray] (0,0) -- (0,1) -- (1,1) -- (1,0) -- cycle;
\end{scope}
\begin{scope}[shift={(3,0)}]
	\draw[fill=gray] (0,0) -- (0,1) -- (1,1) -- (1,0) -- cycle;
\end{scope}
\begin{scope}[shift={(4,0)}]
	\draw[fill=white!40!blue] (0,0) -- (0,1) -- (1,1) -- (1,0) -- cycle;
\end{scope}
\begin{scope}[shift={(0,1)}]
	\begin{scope}[shift={(0,0)}]
		\draw[fill=gray] (0,0) -- (0,1) -- (1,1) -- (1,0) -- cycle;
	\end{scope}
	\begin{scope}[shift={(1,0)}]
		\draw[fill=gray] (0,0) -- (0,1) -- (1,1) -- (1,0) -- cycle;
	\end{scope}
	\begin{scope}[shift={(2,0)}]
		\draw[fill=white!40!blue] (0,0) -- (0,1) -- (1,1) -- (1,0) -- cycle;
	\end{scope}
	\begin{scope}[shift={(3,0)}]
		\draw[fill=white!40!blue] (0,0) -- (0,1) -- (1,1) -- (1,0) -- cycle;
	\end{scope}
	\begin{scope}[shift={(4,0)}]
		\draw[fill=white!40!blue] (0,0) -- (0,1) -- (1,1) -- (1,0) -- cycle;
	\end{scope}
\end{scope}
\begin{scope}[shift={(0,2)}]
	\begin{scope}[shift={(0,0)}]
		\draw[fill=gray] (0,0) -- (0,1) -- (1,1) -- (1,0) -- cycle;
	\end{scope}
	\begin{scope}[shift={(1,0)}]
		\draw[fill=white!40!blue] (0,0) -- (0,1) -- (1,1) -- (1,0) -- cycle;
	\end{scope}
	\begin{scope}[shift={(2,0)}]
		\draw[fill=white!40!blue] (0,0) -- (0,1) -- (1,1) -- (1,0) -- cycle;
	\end{scope}
	\begin{scope}[shift={(3,0)}]
		\draw[fill=white!40!blue] (0,0) -- (0,1) -- (1,1) -- (1,0) -- cycle;
	\end{scope}
	\begin{scope}[shift={(4,0)}]
		\draw[fill=white!40!blue] (0,0) -- (0,1) -- (1,1) -- (1,0) -- cycle;
	\end{scope}
\end{scope}
\path[thick,>=stealth'] (0,3) edge[->] (1,3) (1,3) edge[->] (1,2) (1,2) edge[->] (2,2) (2,2) edge[->] (2,1) (2,1) edge[->] (3,1) (3,1) edge[->] (4,1) (4,1) edge[->] (4,0) (4,0) edge[->] (5,0);
\draw[fill=orange] (0,3) circle (0.08) (1,3) circle (0.08) (1,2) circle (0.08) (2,2) circle (0.08) (2,1) circle (0.08) (3,1) circle (0.08) (4,1) circle (0.08) (4,0) circle (0.08) (5,0) circle (0.08);
\end{tikzpicture}
\end{array}
 \end{equation}
Compare to homological projective duality of \cite{Chen:2020iyo}.

\subsubsection{Picard-Lefschetz parallel transport}

Conformal block expressions can be constructed using so called free field representation \cite{Dotsenko:1984nm,1990IJMPA...5.2495G}.
Resulting expressions take the form of integral representation of solution to the KZ equations \eqref{KZ}.
In our particular case, this solution reads \cite{schechtman1991arrangements,etingof1998lectures}:
\bar{e}\label{free_field}
\begin{split}
&\Psi_{{\cal L}}(z_1,\ldots,z_m)=\prod\limits_{a=0}^{m-1}z_a^{-\epsilon\lambda_0/2}\prod\limits_{0\leq a<b\leq m-1}(z_a-z_b)^{-\epsilon/2}\times\\
&\times\int\limits_{{\cal L}}\prod\limits_{i=1}^n dx_i \;\prod\limits_{i=1}^{n}x_i^{\epsilon\lambda_0}\;\prod\limits_{i=1}^n\prod\limits_{a=0}^{m-1} (x_i-z_a)^{\epsilon}\;\prod\limits_{1\leq i<j\leq n} (x_i-x_j)^{-2\epsilon}.
\end{split}
 \end{equation}

The integration contour ${\cal L}$ is chosen in such a way that the integral is absolutely convergent.
The basis of integration cycles is isomorphic to the basis of conformal blocks.
However a direct identification of a cycle with fixed values of $\vec\lambda$ and $\vec\alpha$ is rather involved.
In the limit $|\lambda_0|\gg 1$ it is easy to show \cite{Galakhov:2017pod} that the basis of conformal blocks is identical to the basis of Lefschetz thimbles.

The form of Knizhnik-Zamolodchikov equations suggests a natural WKB approximation \cite{Reshetikhin:1994qw} to the parallel transport problem of calculating $U_{i,i+1}$ \cite{Galakhov:2014aha}.
That is analogous to Picard-Lefschetz parallel transport discussed in Appendix \ref{sec:App_FS_cat}.

Eventually we can categorify the Picard-Lefschetz parallel transport as a Fukaya-Seidel category of a Landau-Ginzburg model with a superpotential:
\bar{e}\label{check_sup}
\check{W}_{\rm LG}(\vec x)=\epsilon\lambda_0\sum\limits_{i=1}^{n}{\rm log}\;x_i+\epsilon\sum\limits_{i=1}^n\sum\limits_{a=0}^{m-1} {\rm log}\;(x_i-z_a)-2\epsilon\sum\limits_{1\leq i<j\leq n} {\rm log}\;(x_i-x_j).
 \end{equation}
This superpotential is a generalization of the superpotential we used to construct the categorification of the hypergeometric series analytic continuation in \eqref{sup_0}.
Analogous  LG model can be derived as an effective model on collective coordinates of monopole-like solutions in a 5d theory compactified on a cigar \cite{Gaiotto:2011nm,GMW}.

\subsection{Assets of quiver varieties}

\subsubsection{Nakajima quiver varieties}

Nakajima quiver varieties parameterize classical Higgs branches of super Yang-Mills-Higgs theories.
The gauge-matter content of these theories can be also summarized in the form of a quiver according to the rules listed in Section \ref{sec:Intro_mirror}.
We will consider quiver varieties of so called ${\rm A}_k$-type:
\bar{e}
\begin{array}{c}
\begin{tikzpicture}
	\node at (0,0) {$v_1$};
	\draw (0,0) circle (0.3);
	\begin{scope}[shift={(0,-2)}]
		\node at (0,0) {$w_1$};
		\draw (-0.3,-0.3) -- (-0.3,0.3) -- (0.3,0.3) -- (0.3,-0.3) -- cycle;
	\end{scope}
	\draw[->] (0.15, -0.259808) -- (0.15, -1.7);
	\draw[<-] (-0.15, -0.259808) -- (-0.15, -1.7);
	\draw[->] (0.15, 0.259808) to[out=60,in=0] (0,1) to[out=180,in=120] (-0.15, 0.259808);
	\node[above] at (0,1) {$P_1$};
	\node[left] at (-0.15,-1) {$\Gamma_1$};
	\node[right] at (0.15,-1) {$\Delta_1$};
	%
	\draw[->] (0.259808, 0.15) -- (1.74019, 0.15);
	\draw[<-] (0.259808, -0.15) -- (1.74019, -0.15);
	\node[above] at (1,0.15) {$A_1$};
	\node[below] at (1,-0.15) {$B_1$};
	%
	\begin{scope}[shift={(2,0)}]
	\node at (0,0) {$v_2$};
	\draw (0,0) circle (0.3);
	\begin{scope}[shift={(0,-2)}]
		\node at (0,0) {$w_2$};
		\draw (-0.3,-0.3) -- (-0.3,0.3) -- (0.3,0.3) -- (0.3,-0.3) -- cycle;
	\end{scope}
	\draw[->] (0.15, -0.259808) -- (0.15, -1.7);
	\draw[<-] (-0.15, -0.259808) -- (-0.15, -1.7);
	\draw[->] (0.15, 0.259808) to[out=60,in=0] (0,1) to[out=180,in=120] (-0.15, 0.259808);
	\node[above] at (0,1) {$P_2$};
	\node[left] at (-0.15,-1) {$\Gamma_2$};
	\node[right] at (0.15,-1) {$\Delta_2$};
	%
	\draw[->] (0.259808, 0.15) -- (1.74019, 0.15);
	\draw[<-] (0.259808, -0.15) -- (1.74019, -0.15);
	\node[above] at (1,0.15) {$A_2$};
	\node[below] at (1,-0.15) {$B_2$};
	\end{scope}
	\begin{scope}[shift={(8,0)}]
		\node at (0,0) {$v_k$};
		\draw (0,0) circle (0.3);
		\begin{scope}[shift={(0,-2)}]
			\node at (0,0) {$w_k$};
			\draw (-0.3,-0.3) -- (-0.3,0.3) -- (0.3,0.3) -- (0.3,-0.3) -- cycle;
		\end{scope}
		\draw[->] (0.15, -0.259808) -- (0.15, -1.7);
		\draw[<-] (-0.15, -0.259808) -- (-0.15, -1.7);
		\draw[->] (0.15, 0.259808) to[out=60,in=0] (0,1) to[out=180,in=120] (-0.15, 0.259808);
		\node[above] at (0,1) {$P_k$};
		\node[left] at (-0.15,-1) {$\Gamma_k$};
		\node[right] at (0.15,-1) {$\Delta_k$};
	\end{scope}
\begin{scope}[shift={(6,0)}]
	\draw[->] (0.259808, 0.15) -- (1.74019, 0.15);
	\draw[<-] (0.259808, -0.15) -- (1.74019, -0.15);
	\node[above] at (1,0.15) {$A_{k-1}$};
	\node[below] at (1,-0.15) {$B_{k-1}$};
\end{scope}
\draw[fill=black] (4.5,0) circle (0.05) (5,0) circle (0.05) (5.5,0) circle (0.05);
\end{tikzpicture}
\end{array}
 \end{equation}

The theory is accompanied by a canonical superpotential:
\bar{e}\label{Naka_sup}
W=\sum\limits_{i=1}^{k}{\rm Tr}\; P_i\left(B_iA_i-A_{i+1}B_{i+1}-\Gamma_i\Delta_i\right).
 \end{equation}
Here and in what follows we imply that fields with indices $i$ outside the interval $1,\ldots,k$ are zero.

Masses of chiral fields -- equivariant weights --  are compatible with the superpotential, so that the overall equivariant weight of the superpotential is 0.
We denote mass parameters $\epsilon_1$, $\epsilon_2$ as a reference to the canonical parameterization of the $\Omega$-background deformation \cite{Nekrasov:2002qd,Nekrasov:2003rj,Moore:1998et}:
\bar{e}
\mu(A_i)=\epsilon_1,\;\;\;\mu(B_i)=\epsilon_2,\;\;\;\mu(\Gamma_i)=0,\;\;\; \mu(\Delta_i)=\epsilon_1+\epsilon_2,\;\;\;\mu(P_i)=-\epsilon_1-\epsilon_2.
 \end{equation}
These masses are introduced as additional twisted masses --  flavor $U(1)$ framing multiplets associated with quiver arrows \cite{Galakhov:2020vyb}.

As before we denote FI parameters and topological angles associated to  gauge groups of quiver nodes correspondingly:
$$
r_i,\quad \theta_i,\quad i=1,\ldots,k.
$$

The classical Higgs branch is described by a locus of chiral fields satisfying D-term and F-term constraints:
\bar{e}\label{Nakajima}
\begin{split}
	{\bf D}\mbox{-term:}\quad& A_{i-1}
A_{i-1}^{\dagger}+B_iB_i^{\dagger}+\Gamma_i\Gamma_i^{\dagger}-A_i^{\dagger}A_i-B_{i-1}^{\dagger}B_{i-1} -\Delta_{i}^{\dagger}\Delta_i=r_i,\quad i=1,\ldots,k;\\
{\bf F}\mbox{-term:}\quad& A_{i}
B_{i}=A_{i+1}B_{i+1}-\Gamma_i\Delta_i,\quad i=1,\ldots,k;\\
&P_i=0,\quad i=1,\ldots,k.
\end{split}
 \end{equation}
The latter constraint $P_i=0$ is very simple and maps $P_i$ do not appear anywhere else, so in a canonical description of Nakajima quiver varieties these constraints and fields $P_i$ are usually omitted, we will proceed in a similar fashion.
Also we admit the following usual simplification of notations.
Since all the nodes are connected by a doublet of arrows we substitute this doublet by a single link:
$$
\begin{array}{c}
	\begin{tikzpicture}[scale=0.6]
		\draw (0,0) circle (0.3) (2,0) circle (0.3);
		\draw[->] (0.259808, 0.15) -- (1.74019, 0.15);
		\draw[<-] (0.259808, -0.15) -- (1.74019, -0.15);
	\end{tikzpicture}
\end{array}\leftrightsquigarrow \begin{array}{c}
\begin{tikzpicture}[scale=0.6]
	\draw (0,0) circle (0.3) (2,0) circle (0.3);
	\draw (0.3,0) -- (1.7,0);
\end{tikzpicture}
\end{array}
$$

Classical vacua satisfying \eqref{Nakajima} connected by the action of the gauge group are equivalent.
Therefore the vacuum manifold -- the classical Higgs branch -- in this case is described by the following quotient called quiver representation moduli space:
\bar{e}
{\cal R}(\vec v,\vec w):=\{{\bf D}\mbox{-term},{\bf F}\mbox{-term}\}\Big/\prod\limits_{i=1}^kU(v_i).
 \end{equation}
This manifold is smooth for values of $r_i$ belonging to certain stability chambers.
In what follows we will exploit mostly a stability chamber called cyclic chamber:
$$
r_i>0,\quad i=1,\ldots,k.
$$
As we mentioned in Section \ref{sec:Intro_mirror} the $\bf D$-term constraint can be traded for a stability condition and enlargement of the gauge group to its complexified analog, so the quiver representation moduli space admits an isomorphic description:
\bar{e}
{\cal R}(\vec v,\vec w)=\{\mbox{stability},{\bf F}\mbox{-term}\}\Big/\prod\limits_{i=1}^kGL(v_i,\mathbb{C}).
 \end{equation}
For Nakajima quiver varieties this theorem is proven in \cite{nakajima1994instantons}.
We will denote a stable Nakajima quiver variety representation corresponding to the cyclic chamber as ${\cal R}_+(\vec v,\vec w)$.
It is a smooth algebraic variety of complex dimension:
\bar{e}\label{Nakajima_dim}
{\rm dim}_{\mathbb{C}}\;{\cal R}(\vec v,\vec w)=2(\vec w,\vec v)-2(\vec v,\vec v)+(\vec v,A\vec v),
 \end{equation}
where $A$ is the quiver adjacency matrix:
$$
A_{i,j}=\delta_{i,j+1}+\delta_{i+1,j}.
$$

\subsubsection{Spectral duality}
The spectral duality establishes isomorphism of vacuum moduli spaces for certain theories (see e.g. \cite{Mironov:2012uh}).
We will be interested in the case of 3d theories \cite{Gaiotto:2013bwa,Bullimore:2014awa}.
This duality is also tightly related to a 3d mirror symmetry (see e.g.  \cite{Gaiotto:2013bwa,Bullimore:2016nji,Bullimore:2014awa,Intriligator:1996ex,deBoer:1996ck}) and $q$-Langlands correspondence (see e.g. \cite{Kapustin:2006pk,Aganagic:2016jmx,Aganagic:2017smx,Okounkov:2016sya,Rimanyi:2019zyi,2020arXiv201108603D}).

One could extract an equivalence of solutions to Bethe ansatz equations describing vacuum moduli spaces for theories with target spaces described by a dual pair of quivers $\mathfrak{Q}$ and $\mathfrak{Q}^!$. This duality is well established when both $\mathfrak{Q}$ and $\mathfrak{Q}^!$ describe so called $T[SU(k)]$ theories \cite{Bullimore:2014awa,Aprile:2018oau}:
\bar{e}\label{T(SU(k))}
\begin{array}{c}
	\begin{tikzpicture}[scale=0.4]
		\draw[thick] (0,0) -- (7,0) (11,0) -- (15,0);
		\draw[thick,dashed] (7,0) -- (11,0);
		\draw[fill=white] (3,0) circle (0.3) (6,0) circle (0.3) (12,0) circle (0.3) (15,0) circle (0.3);
		\draw[fill=white] (-0.3,-0.3) -- (-0.3,0.3) -- (0.3,0.3) -- (0.3,-0.3) -- cycle;
		\node[above] at (0,0.3) {$k$};
		\node[above] at (3,0.3) {$k-1$};
		\node[above] at (6,0.3) {$k-2$};
		\node[above] at (12,0.3) {$2$};
		\node[above] at (15,0.3) {$1$};
		\node[left] at (-0.5,0) {$T[SU(k)]=$};
	\end{tikzpicture}
\end{array}
 \end{equation}

The duality exchanges the roles of complex mass parameters $\mu$ associated with the framing nodes and complexified FI parameters $r_i-{\rm i}\theta_i$ associated with the gauge nodes.

Similarly, the spectral duality \cite{Gaiotto:2013bwa} connects the theory with the target space given by the cotangent bundle to a Grassmannian:
\bar{e}
\begin{array}{c}
	\begin{tikzpicture}[scale=0.4]
		\draw[thick] (0,0) -- (3,0);
		\draw[fill=white] (3,0) circle (0.3);
		\draw[fill=white] (-0.3,-0.3) -- (-0.3,0.3) -- (0.3,0.3) -- (0.3,-0.3) -- cycle;
		\node[above] at (0,0.3) {$m$};
		\node[above] at (3,0.3) {$n$};
		\node[left] at (-0.5,0) {$T^*{\rm Gr}(n,m)=$};
	\end{tikzpicture}
\end{array}
 \end{equation}
and a theory of type we denote as ${\cal S}_{n,m-n}$:
\bar{e}\label{S_{n,m-n}}
\begin{array}{c}
\begin{tikzpicture}[scale=0.4]
	\draw[thick] (0,0) -- (3,0) (6,0) -- (12,0) (15,0) -- (21,0) (24,0) -- (27,0) (9,0) -- (9,-1.3) (18,0) -- (18,-1.3);
	\draw[thick,dashed] (3,0) -- (6,0) (12,0) -- (15,0) (21,0) -- (24,0);
	\draw[fill=white] (0,0) circle (0.3) (3,0) circle (0.3) (6,0) circle (0.3) (9,0) circle (0.3) (12,0) circle (0.3) (15,0) circle (0.3) (18,0) circle (0.3) (21,0) circle (0.3) (24,0) circle (0.3) (27,0) circle (0.3);
	\node[above] at (0,0.3) {$1$};
	\node[above] at (3,0.3) {$2$};
	\node[above] at (6,0.3) {$n-1$};
	\node[above] at (9,0.3) {$n$};
	\node[above] at (12,0.3) {$n$};
	\node[above] at (15,0.3) {$n$};
	\node[above] at (18,0.3) {$n$};
	\node[above] at (21,0.3) {$n-1$};
	\node[above] at (24,0.3) {$2$};
	\node[above] at (27,0.3) {$1$};
	\begin{scope}[shift={(9,-1.3)}]
			\draw[fill=white] (-0.3,-0.3) -- (-0.3,0.3) -- (0.3,0.3) -- (0.3,-0.3) -- cycle;
	\end{scope}
	\begin{scope}[shift={(18,-1.3)}]
			\draw[fill=white] (-0.3,-0.3) -- (-0.3,0.3) -- (0.3,0.3) -- (0.3,-0.3) -- cycle;
		\end{scope}
	\node[below] at (9,-1.6) {$1$};
	\node[below] at (18,-1.6) {$1$};
	\node[left] at (-0.5,0) {${\cal S}_{n,m-n}=$};
	\begin{scope}[shift={(0,1.5)}]
		\draw (-0.3,0) to[out=90,in=180] (0,0.3) -- (13.2,0.3) to[out=0,in=270] (13.5,0.6) to[out=270,in=180] (13.8,0.3) -- (27,0.3) to[out=0,in=90] (27.3,0);
		\node[above] at (13.5,0.6) {\small $m-1$ nodes};
	\end{scope}
\end{tikzpicture}
\end{array}
 \end{equation}

In the IR these theories experience a symmetry breaking phenomenon.
Effective fields remaining present in the IR description on the Coulomb branch are eigen values of the scalars $\sigma$ in the gauge multiplet.
We can denote those fields by two indices $\Sigma^{(a)}_{\alpha}$ where index $a$ runs over nodes $a=1,\ldots, m-1$ and $\alpha=1,\ldots, v_a$.

A procedure to generate an effective superpotential is analogous to discussed in Section \ref{sec:Preamble} (see also \cite{Aganagic:2001uw}).
Analogously we could reduce dimensionally the 3d superpotential \cite{Gaiotto:2013bwa} or take a logarithm of disk partition function integrand \cite[Section 10]{Hori:2013ika} substituting gamma-functions by their Stirling's approximations.
The result for a Nakajima quiver variety reads:
\bar{e}\label{uncheck_sup}
\begin{split}
&W_{\rm LG}=\sum\limits_{a=1}^Nt_a \Sigma_{\alpha}^{(i)}+\sum\limits_{a=1}^N\sum\limits_{\substack{\alpha,\beta=1\\ \alpha\neq \beta}}^{N}w\left(\Sigma_{\alpha}^{(a)}-\Sigma_{\beta}^{(a)}-\epsilon_1-\epsilon_2\right)+\\
&+\sum\limits_{a=1}^{N-1}\sum\limits_{\alpha=1}^{v_a}\sum\limits_{\beta=1}^{v_{a+1}}w\left(\Sigma_{\beta}^{(a+1)}-\Sigma_{\alpha}^{(a)}-\epsilon_1\right)+\sum\limits_{a=1}^{N-1}\sum\limits_{\alpha=1}^{v_a}\sum\limits_{\beta=1}^{v_{a+1}}w\left(\Sigma_{\alpha}^{(a)}-\Sigma_{\beta}^{(a+1)}-\epsilon_2\right)+\\
&+\sum\limits_{a=1}^{N}\sum\limits_{p=1}^{w_a}w\left(\Sigma_{\alpha}^{(a)}-\mu_p^{(a)}\right)+\sum\limits_{a=1}^{N}\sum\limits_{p=1}^{w_a}w\left(\mu_p^{(a)}-\Sigma_{\alpha}^{(a)}-\epsilon_1-\epsilon_2\right).
\end{split}
 \end{equation}
where $\mu_p^{(a)}$, $p=1,\ldots,w_a$ are masses associated with the framing nodes, $t_a$ are complexified FI parameters:
$$
t_a=r_a-{\rm i}\theta_a,
$$
and $w$ is an elementary single chiral superpotential \eqref{small_superpotential}.

Analogous dimensional reduction of $T^*{\rm Gr}(n,m)$ will produce an effective theory with superpotential \eqref{check_sup}.
A spectral duality between $T^*{\rm Gr}(n,m)$ and ${\cal S}_{n,m-n}$ indicates that corresponding vacuum moduli spaces are isomorphic, in particular, in the Landau-Ginzburg phase critical point sets of $\check W_{\rm LG}$ \eqref{check_sup} and $W_{\rm LG}$ \eqref{uncheck_sup} are isomorphic under the following identification of parameters:
\bar{e}
z_a=:e^{2\pi \tau_a},\quad t_a=\tau_a-\tau_{a-1}.
 \end{equation}

Here let us review a quick two line proof of this fact for the simplest case $n=1$.
A complete analytic proof of this duality relation for generic $m$ and $n$ is rather involved  technically, it is described in \cite{Koroteev:2021lvp}.

In the case $n=1$ theory ${\cal S}_{1,m-1}$ has the following quiver depiction:
\bar{e}
\begin{array}{c}
	\begin{tikzpicture}[scale=0.4]
		\draw[thick] (-3,0) -- (3,0) (6,0) -- (12,0);
		\draw[thick,dashed] (3,0) -- (6,0);
		\draw[fill=white] (0,0) circle (0.3) (3,0) circle (0.3) (6,0) circle (0.3) (9,0) circle (0.3);
		\node[above] at (-3,0.3) {$1$};
		\node[above] at (0,0.3) {$1$};
		\node[above] at (3,0.3) {$1$};
		\node[above] at (6,0.3) {$1$};
		\node[above] at (9,0.3) {$1$};
		\node[above] at (12,0.3) {$1$};
		\begin{scope}[shift={(12,0)}]
			\draw[fill=white] (-0.3,-0.3) -- (-0.3,0.3) -- (0.3,0.3) -- (0.3,-0.3) -- cycle;
		\end{scope}
		\begin{scope}[shift={(-3,0)}]
			\draw[fill=white] (-0.3,-0.3) -- (-0.3,0.3) -- (0.3,0.3) -- (0.3,-0.3) -- cycle;
		\end{scope}
		\node[left] at (-3.5,0) {${\cal S}_{1,m-1}=$};
	\end{tikzpicture}
\end{array}\;.
 \end{equation}

Corresponding superpotential reads:
\bar{e}
W_{\rm LG}=\sum\limits_{a=0}^{m-1}w\left(\Sigma^{(a)}-\Sigma^{(a+1)}\right)+\sum\limits_{a=0}^{m-1}w\left(\Sigma^{(a+1)}-\Sigma^{(a)}-\epsilon\right)+\sum\limits_{a=1}^{m-1}t_a\Sigma^{(a)},
 \end{equation}
where we shifted fields $\Sigma^{(a)}\to\Sigma^{(a)}+a\epsilon_1$,
\bar{e}
\epsilon=\epsilon_1+\epsilon_2,
 \end{equation}
and we assume boundary conditions for framing nodes:
$$
\Sigma^{(0)}=\mu_+,\quad \Sigma^{(m)}=\mu_-,
$$
where complex masses $\mu_+$ and $\mu_-$ are associated with framing nodes $w_1$ and $w_{m-1}$ correspondingly.

In our notations constant vacuum equations read:
\bar{e}
e^{2\pi\tau_{a-1}}\left(1-\frac{\epsilon}{\Sigma^{(a-1)}-\Sigma^{(a)}}\right)=e^{2\pi\tau_{a}}\left(1-\frac{\epsilon}{\Sigma^{(a)}-\Sigma^{(a+1)}}\right).
 \end{equation}
Clearly the right hand side of this equation is equivalent to the left hand side with indices shifted by one.
We conclude that both expressions for the l.h.s. and the r.h.s. are independent of index $a$.
Introduce a new variable:
\bar{e}\label{xx}
x:=e^{2\pi\tau_{a}}\left(1-\frac{\epsilon}{\Sigma^{(a)}-\Sigma^{(a+1)}}\right).
 \end{equation}

It is simple to solve \eqref{xx} for $\Sigma^{(a)}$ variables, then one arrives to a single constraint for $\Sigma^{(a)}$ to match the boundary conditions:
\bar{e}
\frac{\mu_+-\mu_--m\epsilon}{x}+\sum\limits_{a=0}^{m-1}\frac{\epsilon}{x-z_a}=0.
 \end{equation}

The latter equation describes critical points of the dual superpotential of $T^*{\rm Gr(1,m)}$ theory (compare to \eqref{check_sup}):
\bar{e}
{\check W}_{\rm LG}=\epsilon\left(\frac{\mu_+-\mu_-}{\epsilon}-m\right){\rm log}\; x+\epsilon \sum\limits_{a=0}^{m-1}{\rm log}(x-z_a).
 \end{equation}

\subsubsection{Maffei's isomorphism}

Consider an $N$-step flag variety of hyperplanes $F_a$ in a $d$-dimensional complex space (see \cite{Affleck:2021ypq}):
\bar{e}
{\cal F}^d_{\vec q}:=\left\{\begin{array}{l}
\mathbb{C}^d=F_0\supseteq F_1 \supseteq F_2\supseteq \ldots \supseteq F_N \supseteq F_{N+1}=\{0\}\\
{\rm dim}F_a-{\rm dim}F_{a+1}=q_a\geq 0,\quad a=0,\ldots,N
\end{array}\right\}.
 \end{equation}

A cotangent bundle to a flag variety is produced by adding a nilpotent element $z$:
\bar{e}\label{cotang}
{\cal N}^{d}_{\vec q}:=\left\{(z,{\cal F}_{\vec q}^d)\in {\rm End}(\mathbb{C}^d)\times {\cal F}_{\vec q}^d\;\big|\; z(F_a)\subseteq F_{a+1}\right\}.
 \end{equation}

We say that $z$ is an element of type $\lambda$, where $\lambda=\{\lambda_1\geq \lambda_2\geq \ldots\geq \lambda_p\}$ is a partition of $d$, if nilpotent matrix  $z$ has blocks of sizes $\lambda_1,\ldots, \lambda_p$ in its Jordan decomposition. A slice in ${\cal N}^{d}_{\vec q}$ where $z$ is an element of type $\lambda$ we denote as
$$
{\cal N}^{d}_{\vec q,\lambda}
$$

Cotangent bundle ${\cal N}^{d}_{\vec q}$ is isomorphic to a Nakajima quiver variety with dimensional vectors
$$
\vec v=\left({\rm dim}\; F_1,\;{\rm dim}\; F_2,\ldots, {\rm dim} \;F_N\right),\quad \vec w=(d,0,0,\ldots).
$$
in the cyclic chamber \cite{Rimanyi:2019zyi,nakajima1994instantons}. To establish this isomorphism it is useful to identify $W_1$ with $V_0$ and maps:
$$
A_0:=\Gamma_1,\quad B_0:=\Delta_1.
$$
If the vector space $\mathbb{C}^d$ is identified with $V_0$ then hyperplanes $F_a$ are derived as
\bar{e}
F_a={\rm Im}\;H_a,
 \end{equation}
where maps $H_a$ read:
\bar{e}\label{embedding}
H_a=A_0^\dagger\cdot\ldots\cdot A_{a-1}^\dagger:\quad V_a\longrightarrow V_0.
 \end{equation}
Nilpotent element $z$ is defined in this framework as:
\bar{e}
z=(B_0A_0)^\dagger.
 \end{equation}

We should note that according to our prescriptions element $z$ has equivariant weight $\epsilon=\epsilon_1+\epsilon_2$. Vectors in $V_0$ belonging to the same Jordan subspace form a module over a vector $\vec u$:
$$
\vec u,\; z {\vec u},\; z^2 {\vec u},\; z^3 {\vec u},\ldots.
$$
This imposes a constraint on non-dynamical expectation values of field $\sigma$ associated with the framing node $W_1$. To produce a nilpotent element of type $\lambda$ it has to have the following form:
\bar{e}
\sigma={\rm diag}\left(\mu_1,\mu_1-\epsilon,\ldots,\mu_1-\epsilon \lambda_1,\;\;\mu_2,\ldots,\mu_2-\epsilon \lambda_2,\;\;\ldots,\mu_p-\epsilon\lambda_p\right).
 \end{equation}

Maffei's isomorphism  \cite{maffei2005quiver} relates Nakajima quiver varieties and slices in the cotangent bundle to a flag variety:
\bar{e}
{\cal R}_+(\vec v, \vec w)\cong {\cal N}_{\vec q, \lambda}^d,
 \end{equation}
where
$$
q_a=-v_a+v_{a+1}+\sum\limits_{j=1}^aw_j,
$$
and $\lambda$ is a partition of type $1^{w_N}2^{w_N-1}\ldots N^{w_1}$, $d=\sum_a q_a$.
In general Maffei's isomorphism is rather involved, nevertheless it can be simplified a lot for equivariant fixed points.

Variety ${\cal N}_{\vec q, \lambda}^d$ describes a transverse slice in a point defined by nilpotent element $z$ in a convolution space representation of orbit closure resolution $\tilde{\rm Gr}_{\vec \chi}$ in an affine Grassmannian of $PGL_2$ \cite{CautisKamnitzerI,CautisKamnitzerII,Mirkovic_Vybornov,Aganagic:2020olg,aganagic2021knot}.

\subsubsection{Fixed points}
For the GLSM with a quiver variety target space we have introduced a non-trivial superpotential \eqref{Naka_sup}.
On one hand superpotential adds a twist to the supercharge differential operator \cite{Galakhov:2020vyb}, nevertheless localization procedure is still applicable.
The theory localizes to the critical locus of the height function and superpotential fixed with respect to gauge symmetry.
On the other hand superpotential $W$ complicates consideration of the GLSM model, it introduces a non-trivial boundary condition for the boundary supercharge leading to an effect of matrix factorization \cite{Khovanov:2004bc,Hori:2013ika}:
$$
\bar{\cal Q}_{\rm bdry}^2=W\cdot{\bf Id}\;.
$$
We will try to avoid this difficulty using an equivariant localization trick.
We split localization flows as we proposed in Section~\ref{sec:loc_GLSM} so that first we localize to the locus \eqref{Nakajima} so that the IR target space is a cotangent bundle to \eqref{Nakajima} (similarly to an example in Appendix \ref{s:sheaf_coho_CP_n}), then we continue localization to the equivariant fixed points.
Fortunately, on the critical locus \eqref{Nakajima} $W\equiv0$ and there is no need to consider matrix factorization for coherent sheaves on the IR target space.
For other types of quiver varieties with singular moduli spaces this trick will not work as higher quantum corrections will contribute \cite{Galakhov:2020vyb}.

\bigskip

Equivariant fixed points on a Nakajima quiver variety in the cyclic chamber are enumerated by $\sum\limits_i w_i$-tuples of $N$-colored Young diagrams.
In the case of theory ${\cal S}_{n,m-n}$ this tuple is just a doublet of Young diagrams $\vec Y=\{Y_1,Y_2\}$.
Moreover, for specified quiver dimensions two Young diagrams complement each other, so that an actual fixed point is labeled by a single Young diagram that can be embedded in a $(m-n)\times n$ cell field.
If $Y_1$ fills the gray area in diagram depicted in Fig.\ref{fig:S_fixed}(a), then $Y_2$ transposed and reflected afterwards along both axes with respect to the origin is filling the complementary blue area.
There is an additional coloring of the diagrams associated with quiver nodes.
We enumerate/color nodes of diagram \eqref{S_{n,m-n}} from left to right starting with color 1, so that nodes linked to the framing nodes have colors $n$ and $m-n$ respectively.
Coloring of field cells goes in diagonal rows as it is depicted in Fig.\ref{fig:S_fixed}(a) starting with color 1 in the top left corner and ending with color $m-1$ in the right bottom corner.

We denote the left bottom corner as ``$-$" and along the gray area use coordinates $x_1$ and $y_1$ so that the most bottom left cell has coordinates $(0,0)$.
Also the top right corner is denoted as ``$+$", along the blue area coordinates $x_2$ and $y_2$ are used, so that the most top right cell has coordinates $(0,0)$.

\begin{figure}
	\begin{center}
	\begin{tikzpicture}
	\node {
		$\begin{array}{c}
			\begin{tikzpicture}
				\begin{scope}[scale=0.7]
					\draw[fill=gray,thick] (0,0) -- (0,5) -- (1,5) -- (1,4) -- (2,4) -- (2,3) -- (5,3) -- (5,2) -- (7,2) -- (7,0) -- cycle;
					\draw[fill=white!40!blue,thick] (8,5) --  (1,5) -- (1,4) -- (2,4) -- (2,3) -- (5,3) -- (5,2) -- (7,2) -- (7,0) -- (8,0) -- cycle;
					\foreach \i in {1,...,7}
					{
						\draw (\i,0) -- (\i,5);
					}
					\foreach \i in {1,...,4}
					{
						\draw (0,\i) -- (8,\i);
					}
					\node at (0.5,4.5) {$1$};
					\node at (0.5,3.5) {$2$};
					\node at (1.5,4.5) {$2$};
					\foreach \i in {0,...,4}
					{
						\node at (\i+0.5,\i+0.5) {$n$};
					}
					\foreach \i in {0,...,4}
					{
						\node at (-\i+7.5,-\i+4.5) {\tiny $m-n$};
					}
					\node at (7.5,0.5) {\tiny $m-1$};
					\node at (6.5,0.5) {\tiny $m-2$};
					\node at (7.5,1.5) {\tiny $m-2$};
				\end{scope}
				\node at (-0.5,-0.5) {$-$};
				\node at (6.1, 4.) {$+$};
				\draw (-0.5,-0.5) circle (0.3) (6.1, 4.) circle (0.3);
				\draw[->] (-0.2,-0.5) -- (2,-0.5);
				\draw[->] (-0.5,-0.2) -- (-0.5,2);
				\node[right] at (2,-0.5) {$x_1$};
				\node[above] at (-0.5,2) {$y_1$};
				\draw[->] (6.1, 3.7) -- (6.1, 1.5);
				\draw[->] (5.8, 4.) -- (3.6, 4.);
				\node[below] at (6.1, 1.5) {$x_2$};
				\node[left] at (3.6, 4.) {$y_2$};
			\end{tikzpicture}
		\end{array}$
	};
	\node at (0,-3) {(a)};
	%
	%
	\begin{scope}[shift={(8,0)}]
	\node {
		$\begin{array}{c}
			\begin{tikzpicture}
				\begin{scope}[scale=0.7]
					\draw[fill=gray,thick] (0,0) -- (0,5) -- (1,5) -- (1,4) -- (2,4) -- (2,3) -- (5,3) -- (5,2) -- (7,2) -- (7,0) -- cycle;
					\draw[fill=white!40!blue,thick] (8,5) --  (1,5) -- (1,4) -- (2,4) -- (2,3) -- (5,3) -- (5,2) -- (7,2) -- (7,0) -- (8,0) -- cycle;
					\foreach \i in {1,...,7}
					{
						\draw (\i,0) -- (\i,5);
					}
					\foreach \i in {1,...,4}
					{
						\draw (0,\i) -- (8,\i);
					}
					%
					\draw[thick,fill=orange] (4,2) -- (4,3) -- (5,3) -- (5,2) -- cycle;
					%
					\foreach \i in {1,...,5}
					\foreach \j in {1,...,5}
					{
						\pgfmathparse{int(\j-\i+1)}
						\let\r\pgfmathresult
						\ifnum \r > 0
						\breakforeach
						\mathfrak{i}

						\node at (\j-0.5,-\i+4.5+\j) {\i};

					}
					%
					\foreach \i in {6,...,7}
					\foreach \j in {1,...,5}
					{
						\node at (\j+\i-5.5,-0.5+\j) {\i};

					}
				%
				\foreach \i in {1,...,5}
				\foreach \j in {1,...,5}
				{
					\pgfmathparse{int(\j-\i+1)}
					\let\r\pgfmathresult
					\ifnum \r > 0
					\breakforeach
					\mathfrak{i}

					\pgfmathparse{int(\i-\j+8)}
					\let\s\pgfmathresult
					\node at (8.5-\j,-\i+5.5) {\s};

				}
				\draw[->,thick,>=stealth'] (0.5,0.5) -- (4.2,2.2);
				\draw[->,thick,>=stealth'] (7.5,4.5) -- (4.8,2.8);
				\node[left] at (0,0.5) {${\vec \rho}_-$};
				\node[right] at (8,4.5) {${\vec \rho}_+$};
				\end{scope}
			\end{tikzpicture}
		\end{array}$
	};
	\node at (0,-3) {(b)};
	\end{scope}
	\end{tikzpicture}
		\caption{Diagrammatic depiction of equivariant fixed points on ${\cal S}_{n,m-n}$.}\label{fig:S_fixed}
	\end{center}
\end{figure}

In Section \ref{sec:WZW_model} we have already identified a set of such Young diagrams with spin arrangements ${\cal Y}_{n,m-n}$.

On the 3d mirror dual side diagrams ${\cal Y}_{n,m-n}$ define Schubert cells of Grassmannian ${\rm Gr}(n,m)$ (see, for example,  \cite{2015arXiv150803010S}).

There is a simple procedure to present field vevs in the corresponding constant vacuum having a diagrammatic labeling of the equivariant fixed point.
In the IR vacuum the gauge group is broken to its Weyl subgroup and scalar field $\sigma$ in the gauge multiplets acquire diagonal expectation values of order of complex masses \cite{Galakhov:2020vyb}:
$$
\left\langle\sigma^{(a)}\right\rangle={\rm diag}\left(\sigma^{(a)}_{1},\sigma^{(a)}_{2},\ldots,\sigma^{(a)}_{v_a}\right),\quad a=1,\ldots,m-1.
$$
The remnant Weyl subgroup of the gauge group acts on these expectation values by permutations:
$$
\prod\limits_{i=1}^{m-1}S_{v_i}.
$$
Therefore vacuum values $\left\langle\sigma^{(a)}\right\rangle$ different by reordering $\sigma^{(a)}_{\alpha}$ inside each $a$-group are not distinguished.
These expectation values can be extracted directly from diagrams like in Fig.\ref{fig:S_fixed}(a).
Each cell in the diagram corresponds to a particular expectation value.
Denote the complex mass associated with the framing node $w_n$ as $\mu_+$ and the mass associated with node $w_{m-n}$ as $\mu_-$.
An association between cells and $\sigma^{(a)}$ vevs depends on the area type the cell belongs to and also its $x$, $y$ and color $c$ coordinates:
\bar{e}\label{eigen}
\begin{array}{rcl}
\mbox{{\color{gray} Gray} cell with }(x_1,y_1,c) & \longleftrightarrow& \sigma^{(c)}=\mu_-+x_1\epsilon_1+y_1\epsilon_2;\\
\mbox{{\color{white!40!blue} Blue} cell with }(x_2,y_2,c') & \longleftrightarrow& \sigma^{(c')}=\mu_++x_2\epsilon_1+y_2\epsilon_2.
\end{array}
 \end{equation}

After symmetry breaking components of chiral fields behave as independent effective IR fields. We could introduce the following notations for their indices:
$$
\phi_{(b,\beta)(a,\alpha)}.
$$
This field is a component of the chiral field represented by an arrow $a\to b$ linking nodes $a$ and $b$.
Indices $\alpha$ and $\beta$ are just matrix indices of the corresponding linear map associated with arrow $a\to b$.
Effective IR mass of such field reads:
\bar{e}\label{Nakajima_meson}
\mu\left(\phi_{(b,\beta)(a,\alpha)}\right)=\sigma_\beta^{(b)}-\sigma_\alpha^{(a)}-\mu(\mbox{arrow }a\to b).
 \end{equation}

Only chiral fields that turn out to be effectively massless acquire IR expectation values of order $\sim\sqrt{r_i}$.
We provide an explicit example of effective theory calculation in Appendix \ref{sec:app_quiver}.
A nice geometric way to enumerate chiral fields acquiring expectation values is terms of composite meson operators.
As in the the case of fields $\sigma$ such meson operators acquiring an expectation value in the IR are defined by cells of Young diagrams.
So a cell in the gray area with coordinates and color $(x_1,y_1,c)$ corresponds to the following operator:
\bar{e}\label{quiver_meson}
\left( e_{(x_1,y_1,c)},\left( B_{n+y_1-x_1}\cdot\ldots \cdot B_{n+x_1-1}\cdot A_{n+x_1}\cdot\ldots\cdot A_{n+1} \right)\Gamma_n\right),
 \end{equation}
where $e_{(x_1,y_1,c)}$ is a unit eigenvector of matrix $\langle\sigma^{c}\rangle$ with eigenvalue $\sigma^{(c)}$ corresponding to cell $(x_1,y_1,c)$ \eqref{eigen} and $( *,*)$ is a simple scalar product.
An expression for the meson operator associated with a cell in the blue area is given by an expression similar to \eqref{quiver_meson} starting with $\Gamma_{m-n}$.
Clearly operators $A_a$ and $B_a$ in brackets in \eqref{quiver_meson} can be re-arranged freely due to $\bf F$-term condition \eqref{Nakajima}.

\bigskip

Young diagram depiction represents a part of a ${\mathbb{Z}}^2$-lattice.
Expectation values $\langle\sigma^{(c)}\rangle$ acquire positions in nodes of this lattice embedded in $\mathbb{C}$.
In a complete analogy with a 3d crystal description of toric fixed points on Calabi-Yau threefolds \cite{Ooguri:2008yb,Okounkov:2003sp,Yamazaki:2010fz,Li:2020rij,Aganagic:2010qr} Young diagram labeling for fixed points on a Nakajima variety can be associated with 2d crystals.
A finite 2d crystal is formed by vevs of the scalar in the gauge multiplet so it is also natural to call this phase we called ``rigid Higgs branch" in Section \ref{sec:RigidHiggs} a crystal Coulomb branch.

\bigskip

Now we are in a position to describe an inverse isomorphism map $\Xi$ mapping a fixed point on ${\cal S}_{n,m-n}$ to a spin configuration. This construction goes as follows.

Maffei's isomorphism maps ${\cal S}_{n,m-n}$ to a flag variety with $a_i=1$ in other words to a slice in a complete flag $T[SU(m)]$ moduli space.
Let us denote the corresponding flag cotangent bundle as ${\cal N}_{n,m-n}$  and vector spaces associated with gauge and framing nodes for original ${\cal S}_{n,m-n}$ and its Maffei's image as $V$,  $W$ and $\tilde V$, $\tilde W$ correspondingly. For the largest space we have:
\bar{e}
\tilde V_0=\tilde W_1=\mathbb{C}^m=W_n^{\oplus n}\oplus W_{m-n}^{\oplus (m-n)},
 \end{equation}
where $W_n^{\oplus n}$ is just $n$ copies of 1d space $W_n$.
Operator $z$ in this basis has the following form:
$$
z=\left(\begin{array}{cc}
	J_{n} & 0\\
	0 & J_{m-n}\\
\end{array}\right)
$$
where $J_k$ are Jordan blocks of appropriate sizes.

We assign to $W_n$ and $W_{m-n}$ 1d spaces spanned by 1/2-spin projection vectors:
\bar{e}
W_n={\rm Span}\left\{ \left|+\frac{1}{2},\frac{1}{2}\right\rangle\right\},\quad W_{m-n}={\rm Span}\left\{ \left|-\frac{1}{2},\frac{1}{2}\right\rangle\right\},
 \end{equation}
or simply $|+\rangle$ and $|-\rangle$ for brevity.
Then vectors of $W_n^{\oplus n}$ and  $W_{m-n}^{\oplus (m-n)}$ can be denoted as $|+,a\rangle$, $a=1,\ldots,n$ and $|-,b\rangle$, $b=1,\ldots,m-n$ correspondingly.
Also introduce a forgetful morphism:
\bar{e}
\xi:\quad |+,i\rangle\mapsto |+\rangle,\quad |-,i\rangle\mapsto |-\rangle.
 \end{equation}

Flag cotangent bundle ${\cal N}_{n,m-n}$ admits an orthogonal decomposition in lines with a natural norm on $\tilde V_0$:
\bar{e}\label{ortho}
\begin{split}
&\tilde V_0=\mathbb{C}^m=\ell_0\oplus\ell_1\oplus\ldots\oplus\ell_{m-1},\quad {\rm dim}\;\ell_a=1,\quad \ell_a \perp \ell_b \;\mbox{for}\;a\neq b;\\
&F_a=\bigoplus\limits_{k=a}^{m-1}\ell_k.
\end{split}
 \end{equation}

It turns out that for fixed points $\xi(\ell_i)$ is parallel to either $|+\rangle$ and $|-\rangle$.
Thus we construct the following map on Young diagrams $Y\in{\cal Y}_{n,m-n}$.
First we construct a quiver representation of ${\cal S}_{n,m-n}$ based on fixed point labeling, then we apply to it Maffei's isomorphism to derive a point on flag cotangent bundle ${\cal N}_{n,m-n}$, and, finally we apply the forgetful morphism to flag's orthogonal decomposition:
\bar{e}
\Xi:\quad Y\mapsto \left(\xi(\ell_0),\xi(\ell_1),\ldots,\xi(\ell_{m-1})\right).
 \end{equation}
The letter word is an arrangement of spins with $n$ ``$+$"-spins and $m-n$ ``$-$"-spins, and $\Xi$ is an isomorphism. As might be expected $\Xi$ is equivalent to isomorphism \eqref{Xi_iso} constructed in Section \ref{sec:WZW_model}.

\subsection{Categorified tangles as interfaces}

\subsubsection{Braid group action on quiver varieties}
Let us assume that parameters $z_a$ of the WZW conformal block are arranged on the cylinder as depicted in diagram \eqref{cylinder}:
\bar{e}
|z_0|<|z_1|<|z_2|<\ldots<|z_{m-1}|.
 \end{equation}
In this case parameters $\tau_a$ are ordered according to their real parts:
\bar{e}
{\rm Re}\;\tau_0<{\rm Re}\;\tau_1<{\rm Re}\;\tau_2<\ldots<{\rm Re}\;\tau_{m-1}.
 \end{equation}
Then the dual ${\cal S}_{n,m-n}$ model is in a cyclic chamber of the parameter space:
\bar{e}
r_1>0,\;r_2>0,\;\ldots,\;r_{m-1}>0.
 \end{equation}
Path $\wp_{a,a+1}$ defined by braid element $b_{a,a+1}$ permutes $\tau_a$ and $\tau_{a+1}$ and flows outside the cyclic chamber of the stability parameters to a new chamber:
\bar{e}\label{new_chamb}
r_1>0,\;\ldots,\;r_a>0,\;r_{a+1}<0,\;r_{a+2}>0,\;\ldots,\;r_{m-1}>0.
 \end{equation}
On a wall in the parameter space separating two chambers the quiver variety may become \emph{singular}.
Physically, we may observe a phenomenon similar to one discussed in Section \ref{sec:cat_contin_hyper} when Higgs branch description fails to mimic the effective behavior of the theory, and one has to switch to the Coulomb branch description via a LG model.
We would like to calculate categorified parallel transport induced by $\wp_{a,a+1}$, however it is inconvenient to work with chamber \eqref{new_chamb}.
We are not aware if there is a nice combinatorial way to count fixed points in that chamber as we did with the cyclic one.
Fortunately, there is an isomorphism $\varphi$ of the original quiver variety ${\cal S}_{n,m-n}$ moduli space in chamber \eqref{new_chamb} to another variety ${\cal S}'_{n,m-n}$ moduli space in a cyclic chamber:
\bar{e}\label{Phi_{i,i+1}}
\begin{array}{c}
\begin{tikzpicture}
	\node[shape
	= rectangle,
	rounded corners=10pt,
	draw,
	fill           = black!20!white,
	align          = center,
	thick	](A) at (0,0) {$\begin{array}{c}
			\begin{tikzpicture}[scale=0.4]
				\draw[thick] (-1,0) -- (11,0);
				\draw[fill=black!20!white] (0,0) circle (0.3) (5,0) circle (0.3) (10,0) circle (0.3);
				\node[above] at (0,0.3) {$r_{a}>0$};
				\node[above] at (5,0.3) {$r_{a+1}>0$};
				\node[above] at (10,0.3) {$r_{a+2}>0$};
			\end{tikzpicture}
		\end{array}$};
	%
	\node[shape
	= rectangle,
	rounded corners=5pt,
	draw,
	fill           = black!20!white,
	align          = center,
	thick	](B) at (9,0) {$\begin{array}{c}
			\begin{tikzpicture}[scale=0.4]
				\draw[thick] (-1,0) -- (11,0);
				\draw[fill=black!20!white] (0,0) circle (0.3) (5,0) circle (0.3) (10,0) circle (0.3);
				\node[above] at (0,0.3) {$r_{a}>0$};
				\node[above] at (5,0.3) {$r_{a+1}<0$};
				\node[above] at (10,0.3) {$r_{a+2}>0$};
			\end{tikzpicture}
		\end{array}$};
	%
	\node[shape
	= rectangle,
	minimum height=1cm,
	rounded corners=10pt,
	draw,
	fill           = black!20!white,
	align          = center,
	thick	](C) at (0,-2.5) {LG model};
	%
	\node[shape
	= rectangle,
	rounded corners=10pt,
	draw,
	fill           = black!20!white,
	align          = center,
	thick	](D) at (9,-2.5) {$\begin{array}{c}
			\begin{tikzpicture}[scale=0.4]
				\draw[thick] (-1,0) -- (11,0);
				\draw[fill=black!20!white] (0,0) circle (0.3) (5,0) circle (0.3) (10,0) circle (0.3);
				\node[above] at (0,0.3) {$r'_{a}>0$};
				\node[above] at (5,0.3) {$r'_{a+1}>0$};
				\node[above] at (10,0.3) {$r'_{a+2}>0$};
			\end{tikzpicture}
		\end{array}$};
	%
	\path (A) edge[->,thick] node[above] {$\wp_{a,a+1}$} (B) (A) edge[->,thick] node[left] {\tiny melting}(C) (C) edge[->,thick] node[below] {\tiny solidifying}(D) ([shift={(-2.92893pt,2.92893pt)}]A.south east) edge[->,thick] node[below left] {${\boldsymbol{\Phi}}_{a,a+1}$} ([shift={(2.92893pt,-2.92893pt)}]D.north west) (B) edge[draw=none] node[right] {\rotatebox[origin=c]{-90}{$\cong$}} node [right=0.5cm] {$\varphi$} (D);
\end{tikzpicture}
\end{array}
 \end{equation}
Indeed both theories are dual to the same LG model with only reshuffled indices of punctures $z_a$.
In the construction of the parallel transport if it is along a path going through a singular variety one is unable to proceed directly and has to dualize theory observables to the LG model and then map them back.
Along this route rather than dualizing directly back the LG model to a new chamber of ${\cal S}_{n,m-n}$ we simply reshuffle subscripts of punctures $z_a$ and map theory to $S'_{n,m-n}$ as it it is depicted in diagram \eqref{Phi_{i,i+1}}.
This identification gives a precise map between parameters of ${\cal S}_{n,m-n}$ and ${\cal S}'_{n,m-n}$:
\bar{e}
\begin{split}
	&r'_a=r_a+r_{a+1},\quad r'_{a+1}=-r_{a+1},\quad r'_{a+2}=r_{a+2}+r_{a+1},\\
	&\mbox{if }b<a\mbox{ or }b>a+2\quad r'_b=r_b.
\end{split}
 \end{equation}

For a LG model, in general, a deviation of effective IR fields $\Sigma$ from vacuum values are suppressed by the effective superpotential behavior, this suppression is much ``softer" than quantum suppression of deviations of fields $\sigma$ from crystal nodes in GLSMs.
Borrowing an analogy from condensed matter physics we could call the LG phase ``\emph{liquid}"\footnote{Similarly one could have borrowed as a physical analogy a resemblance between conformal blocks given by disk amplitudes in a theory with superpotential ${\check W}_{LG}$ and averages of Penner type matrix models (or $\beta$-ensembles) \cite{Dijkgraaf:2009pc,Itoyama:2009sc,Eguchi:2009gf,Schiappa:2009cc,Mironov:2010ym,Mironov:2011jn}.
In the canonical large $N$ limit effective particles -- matrix eigen values -- form Wigner-like \emph{droplets} confined in potential extrema.
}, the transition from the GLSM phase to the LG phase we could call ``melting" and the inverse process ``solidifying".
Using this terminology one arrives to a picturesque image of the parallel transition process: as one varies parameters along a path through a liquid phase a crystal state first melts then solidifies to, in principle, a new crystal.

\bigskip

To get a slight glimpse of how this isomorphism could work on original varieties let us consider the simple case of ${\cal S}_{1,m-1}$.
If we are searching for constant vacua the dominant part of the height function defining these vacua is one without derivatives:
\bar{e}\label{height_red}
\mathfrak{H}_{\rm red}=-\int dx^1\;\sum\limits_{b}{\rm Tr}\;\sigma^{(b)}_{\mathbb{I}}\left(|A_{b-1}|^2+|B_b|^2-|A_b|^2-|B_{b-1}|^2-{\rm Re}\;t_b\right).
 \end{equation}

Clearly, \eqref{height_red} is invariant with respect to the following change of coordinates:
\bar{e}
\begin{array}{c}
	\begin{array}{ccc}
		t_{a-1}\to t_{a-1}+t_a, & t_a\to -t_a, & t_{a+1}\to t_{a+1}+t_a;\\
		\sigma^{(a-1)} \to \sigma^{(a-1)}, & \sigma^{(a)}\to \sigma^{(a-1)}+\sigma^{(a+1)}-\sigma^{(a)}, &\sigma^{(a+1)}\to \sigma^{(a+1)};
	\end{array}\\
	\begin{array}{cccc}
		A_a\to A_{a+1}, & B_a\to B_{a+1}, & A_{a+1}\to A_{a}, & B_{a+1}\to B_{a}.
	\end{array}
\end{array}
 \end{equation}

We denote a complete move from ${\cal S}_{n,m-n}$ to ${\cal S}'_{n,m-n}$ as ${\boldsymbol{\Phi}}_{a,a+1}$ as depicted in diagram \eqref{Phi_{i,i+1}}:
\bar{e}
{\boldsymbol{\Phi}}_{a,a+1}=\varphi\circ\beta_{\wp_{a,a,+1}}.
 \end{equation}

${\boldsymbol{\Phi}}_{a,a+1}$ will be also represented by a Fourier-Mukai transform with a specific kernel. In what follows we will calculate this kernel.

\subsubsection{Fourier-Mukai transform}

In this subsection we will study properties of the parallel transport along the path $\wp_{a,a+1}$ and calculate morphism ${\boldsymbol{\Phi}}_{a,a+1}$ in a form of a Fourier-Mukai transform on the flag cotangent bundle ${\cal N}_{\vec q,\lambda}^d$ associated with theory ${\cal S}_{n,m-n}$.
Along the path $\wp_{a,a+1}$ taking $r_{a+1}>0$ to $r_{a+1}<0$ the variety becomes singular at $r_{a+1}=0$, however  this may not affect equivariant fixed points.
Equivariant localization may prevent field values from hitting the singularity and the effective description from collapsing.
As a matter of fact the situation depends on the spin arrangement in the fixed point and precisely on two spins located at positions $a$ and $a+1$.

Let us introduce the following space (were we use orthogonal decomposition \eqref{ortho}):
\bar{e}
E_{a,a+1}:=F_{a+1}/F_{a-1}=\ell_a\oplus\ell_{a+1}.
 \end{equation}

There are 4 possible spin configurations divided in two groups when spins are parallel and anti-parallel.
Jordan decomposition of operator $z$ restricted to $E_{a,a+1}$ is different for these two groups:
\bar{e}
\begin{array}{c|c}
	\begin{array}{ccc}
		(\underset{a}{-},\underset{a+1}{-}) & \quad &	(\underset{a}{+},\underset{a+1}{+})
	\end{array} &
	\begin{array}{ccc}
		(\underset{a}{-},\underset{a+1}{+}) & \quad &	(\underset{a}{+},\underset{a+1}{-})
	\end{array}\\
	z\big|_{E_{a,a+1}}=J_2=\left(\begin{array}{cc}
		0 & 1\\
		0 & 0\\
	\end{array}\right) & z\big|_{E_{a,a+1}}=J_1\oplus J_1=\left(\begin{array}{cc}
	0 & 0\\
	0 & 0\\
\end{array}\right)
\end{array}
 \end{equation}

Notice that in the former case when spins are co-aligned the information about the very flag can be restored from the information about $E_{a,a+1}$ and operator $z$, one can construct both lines $\ell_a$ and $\ell_{a+1}$ as a kernel of $z$ and its orthogonal complement:
$$
\ell_a={\rm Ker}\; z\big|_{E_{a,a+1}},\quad \ell_{a+1}=\left({\rm Ker}\; z\big|_{E_{a,a+1}}\right)^{\perp}.
$$
In the latter case when spins are opposite the information about embedding $\ell_a\hookrightarrow E_{a,a+1}$ is stored neither in $E_{a,a+1}$ nor in $z$.

To observe that in one case the variety becomes singular notice that  all $\ell_a$ are lines, so they can be considered as elements of $\mathbb{C}\mathbb{P}^1$.
In the physical theory all these $\mathbb{C}\mathbb{P}^1$'s have finite volumes controlled by expectation values of chiral condensates defining embedding of the lines in $\mathbb{C}^m$ \eqref{embedding}.
Denote corresponding volumes $U_a$, eventually they are functions of FI stability parameters $r_a$:
$$
\vec U({\vec r}).
$$

As $r_{a+1}$ goes to zero along $\wp_{a,a+1}$ volume vectors in the cases of co-aligned and opposite spins behave in a different way:
\bar{e}
\begin{split}
&{\vec U}_{J_2}=(O(1),\ldots,O(1),\ldots,O(1)),\\
&{\vec U}_{J_1\oplus J_1}=(O(1),\ldots,\underset{a}{O(1)},\underset{a+1}{\sim{\color{red}\sqrt{r_{a+1}}}},\underset{a+2}{O(1)},\ldots,O(1)).
\end{split}
 \end{equation}

In the first case the classical vacuum stays away from the singularity along the whole path $\wp_{a,a+1}$, so the transition is smooth and does not require gluing in a Coulomb branch resolution.
In the case when spins are opposite some classical expectation values of fields disappear in the point $r_{a+1}=0$, as in the the case of the conifold discussed in Section \ref{sec:cat_contin_hyper} the Higgs branch description fails to reflect the IR behavior and we have to switch to the Coulomb branch description.
Let us consider the situation of opposite spins in more details.

If two spin arrangements differ by a permutation:
$$
(\underset{a}{-},\underset{a+1}{+}) \longrightarrow	(\underset{a}{+},\underset{a+1}{-}),
$$
corresponding Young diagrams differ by a single cell of color $a+1$ at the boundary of the Young diagram \emph{migrating} from the gray area to the blue area, see an example in Fig.\ref{fig:S_fixed}(b) for a pair of the following fixed points where the \emph{migrating cell} is denoted by the orange color marker:
\bar{e}
\begin{array}{c}
\begin{tikzpicture}
	\node(0) at (0,0) {$-$};
	\node[left] at (0) {$($};
	\node[below right] at (0) {$,$};
	\node(1) at (0.5,0) {$+$};
	\node[below right] at (1) {$,$};
	\node(2) at (1,0) {$-$};
	\node[below right] at (2) {$,$};
	\node(3) at (1.5,0) {$+$};
	\node[below right] at (3) {$,$};
	\node(4) at (2,0) {$-$};
	\node[below right] at (4) {$,$};
	\node(5) at (2.5,0) {$-$};
	\node[below right] at (5) {$,$};
	\node(6) at (3,0) {$-$};
	\node[below right] at (6) {$,$};
	\node(7) at (3.5,0) {$+$};
	\node[below right] at (7) {$,$};
	\node(8) at (4,0) {$-$};
	\node[below right] at (8) {$,$};
	\node(9) at (4.5,0) {$-$};
	\node[below right] at (9) {$,$};
	\node(10) at (5,0) {$+$};
	\node[below right] at (10) {$,$};
	\node(11) at (5.5,0) {$+$};
	\node[below right] at (11) {$,$};
	\node(12) at (6,0) {$-$};
	\node[right] at (12) {$)$};
	%
	\begin{scope}[shift={(0,-1.5)}]
	\node(0) at (0,0) {$-$};
	\node[left] at (0) {$($};
	\node[below right] at (0) {$,$};
	\node(1) at (0.5,0) {$+$};
	\node[below right] at (1) {$,$};
	\node(2) at (1,0) {$-$};
	\node[below right] at (2) {$,$};
	\node(3) at (1.5,0) {$+$};
	\node[below right] at (3) {$,$};
	\node(4) at (2,0) {$-$};
	\node[below right] at (4) {$,$};
	\node(5) at (2.5,0) {$-$};
	\node[below right] at (5) {$,$};
	\node(6) at (3,0) {$+$};
	\node[below right] at (6) {$,$};
	\node(7) at (3.5,0) {$-$};
	\node[below right] at (7) {$,$};
	\node(8) at (4,0) {$-$};
	\node[below right] at (8) {$,$};
	\node(9) at (4.5,0) {$-$};
	\node[below right] at (9) {$,$};
	\node(10) at (5,0) {$+$};
	\node[below right] at (10) {$,$};
	\node(11) at (5.5,0) {$+$};
	\node[below right] at (11) {$,$};
	\node(12) at (6,0) {$-$};
	\node[right] at (12) {$)$};
	\end{scope}
	%
	\begin{scope}[shift={(0,-2)}]
		\node(0) at (0,0) {\tiny 0};
		\node(1) at (0.5,0) {\tiny 1};
		\node(2) at (1,0) {\tiny 2};
		\node(3) at (1.5,0) {\tiny 3};
		\node(4) at (2,0) {\tiny 4};
		\node(5) at (2.5,0) {\tiny 5};
		\node(6) at (3,0) {\tiny 6};
		\node(7) at (3.5,0) {\tiny\color{orange} 7};
		\node(8) at (4,0) {\tiny 8};
		\node(9) at (4.5,0) {\tiny 9};
		\node(10) at (5,0) {\tiny 10};
		\node(11) at (5.5,0) {\tiny 11};
		\node(12) at (6,0) {\tiny 12};
	\end{scope}
	\draw[thick] (3,-0.3) -- (3,-0.5) -- (3.5,-0.5) -- (3.5,-0.3) (3,-1.2) -- (3,-1) -- (3.5,-1) -- (3.5,-1.2) (3.25,-0.5) -- (3.25,-1);
\end{tikzpicture}
\end{array}
 \end{equation}

Let us denote vectors pointing to the migrating cell from the $+$ and $-$ corners as ${\vec\rho}_+$ and ${\vec\rho}_-$ correspondingly.
According to association \eqref{Nakajima_meson} these vertices correspond to meson operators, we denote these operators as $y_+$ and $y_-$ respectively.
These are exactly the operators whose expectation values go to zero as one approaches singularity at $r_{a+1}=0$:
\bar{e}
\left\langle|y_+|^2\right\rangle\sim r_{a+1},\quad \left\langle|y_-|^2\right\rangle\sim r_{a+1}.
 \end{equation}
As expectation values of these fields approach $\hbar$ in orders of magnitude the fields are no longer considered as classical.

Deviations of gauge multiplets from classical values $\langle\sigma\rangle$ located in cell positions of the Young diagrams are suppressed by masses of the fields corresponding to the tangent bundle and generated through the Higgs mechanism by chiral fields expectation values.
As fields $y_{\pm}$ become quantum the Higgs mass of the gauge field corresponding to the migrating cell becomes null.
We denote corresponding scalar in this gauge multiplet as $\sigma_0$.

\bigskip

Effective masses, or equivariant weights, of fields $y_{\pm}$ read:
\bar{e}\label{y_masses}
\mu(y_{\pm})=\sigma_0-\tilde \mu_{\pm},
 \end{equation}
where
$$
\tilde\mu_{\pm}:=\mu_{\pm}+{\vec \rho}_{\pm}\cdot\vec\epsilon,
$$
where we introduced the following notations:
$$
{\vec \rho}_{-}\cdot\vec\epsilon=x_1\epsilon_1+y_1\epsilon_2,\quad {\vec \rho}_{+}\cdot\vec\epsilon=x_2\epsilon_1+y_2\epsilon_2.
$$

When the migrating cell is in the gray area of diagram in Fig\ref{fig:S_fixed}(b) $\sigma_0=\tilde\mu_-$, so that mass of field $y_-$ is zero according to \eqref{y_masses}, so $y_-$ condensates and produces the $(-,+)$ equivariant point.
Similarly, if the migrating cell is in the blue area $\sigma_0=\tilde\mu_+$, $y_+$ condensates and produces the $(+,-)$ point.

A tangent space to the quiver variety produces other quantum fields we divide in two groups.
The first group of two fields has masses dependent on $\sigma_0$, we call them $z_+$ and $z_-$:
\bar{e}
\mu(z_{\pm})=\epsilon+\tilde\mu_{\pm}-\sigma_0.
 \end{equation}

Also there is a group of tangent fields with masses independent of $\sigma_0$, there are $2(n-1)$ of them:
\bar{e}
u_1,\ldots,u_{2(n-1)}.
 \end{equation}

$\bf D$-term and $\bf F$-term constraints for these fields read:
\bar{e}
\begin{split}
	& |y_+|^2+|y_-|^2-|z_+|^2-|z_-|^2=r_{a+1},\\
	& y_+z_+-y_-z_-=0.
\end{split}
 \end{equation}

Analyzing the field spectra and constraints we conclude that near singularity $r_{a+1}\to 0$ our ${\cal S}_{n,m-n}$ theory flows to simply ${\cal S}_{1,1}$ theory and a collection of $2(n-1)$ free neutral chiral scalars:
\bar{e}
\begin{array}{c}
\begin{tikzpicture}
	\node (A) {$\begin{array}{c}
			\begin{array}{c}
				\begin{tikzpicture}
					\node at (-2,0) {$1$};
					\node at (0,0) {$1$};
					\node at (2,0) {$1$};
					\draw (0,0) circle (0.3);
					\begin{scope}[shift={(2,0)}]
						\draw (-0.3,-0.3) -- (-0.3,0.3) -- (0.3,0.3) -- (0.3,-0.3) -- cycle;
					\end{scope}
					\begin{scope}[shift={(-2,0)}]
						\draw (-0.3,-0.3) -- (-0.3,0.3) -- (0.3,0.3) -- (0.3,-0.3) -- cycle;
					\end{scope}
					\draw[<-] (1.7,0.1) -- (0.3,0.1);
					\draw[->] (1.7,-0.1) -- (0.3,-0.1);
					\draw[->] (-1.7,0.1) -- (-0.3,0.1);
					\draw[<-] (-1.7,-0.1) -- (-0.3,-0.1);
					\node[above] at (-1,0.1) {$y_+$};
					\node[below] at (-1,-0.1) {$z_+$};
					\node[above] at (1,0.1) {$z_-$};
					\node[below] at (1,-0.1) {$y_-$};
				\end{tikzpicture}
			\end{array}\oplus\mbox{Free } \left(u_1,\ldots,u_{2(n-1)}\right)
		\end{array}$};
	\node(B) at ([shift={(-3,0)}]A.west) {${\cal S}_{n,m-n}$};
	\path (B) edge[->] node[above] {$r_{a+1}\to 0$} (A);
	\node at ([shift={(-1.95,-0.3)}]A.north) {$\sigma_0$};
\end{tikzpicture}
\end{array}
 \end{equation}

Spontaneously broken initial gauge symmetry is restored in the IR to $U(1)$, effective phonons with Higgs masses
$$
m_{\rm Higgs}\sim\langle|y_+|^2+|y_-|^2\rangle
$$
become massless and destroy crystal links fixing position of the migrating cell $\sigma_0$ in the lattice.
However as we see there is no need to melt the whole crystal, rather just to dislocate a single migrating cell is sufficient.

\bigskip

We should note that the number of actual tangent fields matches the quiver variety dimension:
\bar{e}
2\mbox{ of }y_{\pm}\;+\;2\mbox{ of }z_{\pm}\;+\;2(n-1)\mbox{ of }u_a\;-\;\mbox{({\bf D}-term)}\;-\;\mbox{({\bf F}-term)}\;=\;2n\;=\;{\rm dim}\;S_{n,m-n},
 \end{equation}
according to \eqref{Nakajima_dim}.

Effective theory ${\cal S}_{1,1}$ describes a conifold transition we discussed in Section \ref{sec:cat_contin_hyper}.
Path $\wp_{a,a+1}$ bringing $r_{a+1}$ from the positive values to the negative values coincides with the analytic continuation of hypergeometric functions we have constructed categorification of in terms of a Fourier-Mukai transform.
The kernel of the Fourier-Mukai transform was defined by a support of the delta-function contribution to the BPS state wave-function \eqref{Psi}.
Briefly speaking we had to identify all the neutral mesons on the ends of the interface segment $[0,L]$.
In this case in addition to the usual conifold neutral mesons there are new mesons $u_1,\ldots,u_{2(n-1)}$ that are neutral with respect to $U(1)$ of effective ${\cal S}_{1,1}$.
Therefore, effective wave functions of the BPS states in the presence of an interface generated by a parallel transport along $\wp_{a,a+1}$ have the following multiplier (compare to \eqref{Psi}):
\bar{e}\label{Psi_Naka}
\begin{split}
&\Psi_{\rm BPS}\sim \delta\left(y_+z_+-(y_+z_+)'\right) \delta\left(y_+z_--(y_+z_-)'\right)\delta\left(y_-z_+-(y_-z_+)'\right)\delta\left(y_-z_--(y_-z_-)'\right)\times\\
&\times\delta(u_1-u_1')\ldots\delta(u_{2(n-1)}-u_{2(n-1)}'),
\end{split}
 \end{equation}
where unprimed variables correspond to coordinates on initial ${\cal S}_{n,m-m}$ at $x^1=0$ and primed variables correspond to coordinates on final ${\cal S}_{n,m-n}'$ at $x^1=L$ in diagram \eqref{Phi_{i,i+1}}.

To give a geometric interpretation of this transform we would like to note that the following orthogonal decomposition over subspaces:
\bar{e}
\begin{split}
	&\tilde V_0=\ell_0\oplus\ldots\oplus \ell_{a-1}\oplus E_{a,a+1}\oplus \ell_{a+2}\oplus \ldots\oplus \ell_{m-1},
\end{split}
 \end{equation}
and operator $z$ are functions of neutral mesons only.
Moreover, knowing this decomposition is sufficient to define values of all the neutral mesons.
Information about values of $y_\pm$ remains unknown since they control embedding $\ell_a\hookrightarrow E_{a,a+1}$ that can not be reconstructed from presented data.
From this data we can reconstruct all hyperplanes $F_b$ except $F_{a+1}$:
\bar{e}
\begin{split}
	&F_b=\bigoplus\limits_{k=b}^{m-1}\ell_k,\quad \mbox{if }b\geq a+2,\\
	&F_b=\left(\bigoplus\limits_{k=b}^{a-1}\ell_k\right)\oplus E_{a,a+1}\oplus \left(\bigoplus\limits_{j=a+2}^{m-1}\ell_j\right),\quad \mbox{if }b\leq a.
\end{split}
 \end{equation}

Summarizing, we conclude that the delta-function in \eqref{Psi_Naka} is supported on the following variety:
\bar{e}
\Upsilon_{a,a+1}:=\left\{(F_*,F_*'):\;F_b=F_b',\;\mbox{if }b\neq a+1 \right\}.
 \end{equation}

The resulting parallel transport functor ${\boldsymbol{\Phi}}_{a,a+1}$ can be constructed as Fourier-Mukai transform \eqref{FM_transform} with respect to the following kernel:
\bar{e}
{\cal O}_{\Upsilon_{a,a+1}}.
 \end{equation}

Functors ${\boldsymbol{\Phi}}_{a,a+1}$ coincide with braiding functors constructed in \cite{CautisKamnitzerI} for $SL(2)$ and generalized in \cite{CautisKamnitzerII,CautisKamnitzerIV}.
They are proven to satisfy braid relations \eqref{braid_group}.
In our interpretation braid relations \eqref{braid_group} for ${\boldsymbol{\Phi}}_{a,a+1}$ follow naturally from accompanying homotopy relations for parallel transport paths $\wp_{a,a+1}$.

We present an explicit example of cotangent bundle construction ${\cal N}_{n,m-n}$ near a singular  point and counting meson degrees of freedom in Appendix \ref{sec:app_quiver}.

\subsubsection{Remarks on decategorification and quantum groups}

To conclude this section let us give few comments on decategorification of functors ${\boldsymbol{\Phi}}_{a,a+1}$.
In \cite{CautisKamnitzerI} it is shown that the K-theoretic reduction of functor ${\boldsymbol{\Phi}}_{a,a+1}$ gives the usual braid group element in a modular tensor category of ${\cal U}_{\mathfrak{q}}(sl_2)$ constructed from a permutation operator and $U_{\mathfrak{q}}(sl_2)$ R-matrix in representation $\Box\otimes\Box$:
$$
R=\mathfrak{q}^{\frac{h\otimes h}{4}}\left(1+(\mathfrak{q}-\mathfrak{q}^{-1})\;e\otimes f\right)\mathfrak{q}^{\frac{h\otimes h}{4}},
$$
where $e$, $f$ and $h$ are the standard Chevalley generators of $U_{\mathfrak{q}}(sl_2)$ \cite{MR1492989}. The parameter of the quantum group is defined by the equivariant weight of operator $z$:
\bar{e}\label{quant_q}
\mathfrak{q}=e^{\pi{\rm i}\left(\mu(A_i)+\mu(B_i)\right)}=e^{\pi {\rm i}\epsilon}.
 \end{equation}

On the other hand, a flat section of KZ connections \eqref{free_field} is a free field representation of conformal blocks in a 2d Liouville CFT with a coupling constant $b=\sqrt{\epsilon}$ \cite{Mironov:2010su}.
Corresponding parallel transport operator $U_{i,i+1}$ also produces an R-matrix in ${\cal U}_{\mathfrak{q}}(sl_2)$ \cite{Galakhov:2015fna} with a parameter defined by \eqref{quant_q}.

We conclude that indeed functor ${\boldsymbol{\Phi}}_{i,i+1}$ categorifies Berry parallel transport operator $U_{i,i+1}$:
$$
\begin{array}{c}
	\begin{tikzpicture}
		\node(A)[left] at (0,0) {${\boldsymbol{\Phi}}_{i,i+1}$};
		\node(B)[right] at (4,0) {$U_{i,i+1}$};
		\draw[-stealth] (0.2,0.06) -- (3.8,0.06);
		\draw[stealth-] (0.2,-0.06) -- (3.8,-0.06);
		\node[above] at (2,0.06) {\tiny decategorification};
		\node[below] at (2,-0.06) {\tiny categorification};
	\end{tikzpicture}
\end{array}.
$$

\section{Open problems and future directions} \label{sec:future}

In conclusion we would like to mention some open problems and possible directions worth further investigation:
\begin{itemize}
	\item To construct brane parallel transport we used as a tool the algebra of the infrared of \cite{GMW}.
	Despite the soliton scattering vertices deliver algebraic structures and satisfy $L_{\infty}$-relations an effective general mechanism to calculate or bootstrap scattering amplitudes from the first principles is still lacking in the literature.
	This is not surprising since in the heart of this calculation lies a solution to a non-linear differential instanton boundary value problem.
	On the other hand the Higgs-Coulomb duality makes this problem dual to a problem in algebraic geometry where one expects to use algebraic means to solve it.
	Therefore it is natural to guess that the problem could be reversed, and computations of brane parallel transport in GLSM using alternative methods like \cite{HHP,Clingempeel:2018iub,Chen:2020iyo} can be used to benefit solving the instanton equation in LG models, or, at least counting such solutions with signs.
	\item Another possible direction follows from the previously discussed one.
	$L_{\infty}$-relations for amplitudes appear in \cite{GMW} as structures on 2d polygons dual to web diagrams and may be extended to higher dimensional polytopes \cite{Kapranov:2014uwa}.
	A physical theory representing structures of \cite{Kapranov:2014uwa} has not been presented in the literature yet.
	Structures of \cite{GMW} are universal for 2d ${\cal N}=(2,2)$ theories with massive vacua, and equivariant GLSMs in particular.
	So a good candidate for such a physical theory may be a higher dimensional Yang-Mills theory from what a 2d GLSM is derived by dimensional reduction.
	Some indications that this is the case are given in \cite{Costello:2020ndc}.
	\item As we have seen the models discussed in the paper have a direct application to a categorification of a braid group action and, therefore, the knot theory.
	As it is explained in \cite{Aganagic:2020olg,aganagic2021knot} the string theory predicts a categorification scheme for links in generic representation and an arbitrary simple Lie group, however an algorithmic construction of link homologies for this general parameterization is absent in the literature as far as we can tell.
	Ideally, using physical approaches and intuition one expects to have an algorithmic program translatable to any computer language to calculate a link homology for arbitrary group and representations along the lines of \cite{Carqueville:2011zea}.
	Moreover, a closer work with representations higher than minuscule ones is expected to be related to a more detailed investigation of a monopole bubbling phenomenon \cite{Kapustin:2006pk,Dedushenko:2018icp}.
	\item Knot invariants, in particular, HOMFLY polynomials are used in Witten-Reshetikhin-Turaev-Viro construction \cite{reshetikhin1991invariants} of Chern-Simons invariants of 3d manifolds.
	So it is a natural tendency to try to extend categorified knot invariants to categorified invariants of 3d manifolds.
	Moreover physical theories (see e.g. \cite{Haydys:2010dv,Gukov:2017kmk,Wang:2020lqp}) suggest that a geometrical self-consistent definition for such invariants exists.
	\item Another possible application of a categorified representation of conformal blocks in WZW models is a categorification of the mapping class group that will probably deliver a new refinement to the standard representation of the mapping class group similarly to \cite{Arthamonov:2017oxw}.
	\item Appearance of Young diagram crystals in the problem about categorified braid group action colored with $m-1$ colors is not surprising and accounts to the skew Howe duality (see e.g. \cite{2015arXiv150206011M,Chun:2015gda}) where braid functors $\Phi_{i,i+1}$ are associated with a categorified action of raising/lowering operators in corresponding $\mathfrak{s}\mathfrak{l}(m)$.
	A similarity of the braid action with the crystal melting problem suggests that the action of a BPS algebra on molten crystals of Calabi-Yau 3-folds (see e.g. \cite{Li:2020rij,Galakhov:2020vyb,Rapcak:2020ueh}) and, hopefully, 4-folds (see e.g. \cite{Nekrasov:2018xsb,Kanno:2020ybd,Cao:2019tvv,Oh:2020rnj}) will acquire a physical categorification as well.

\end{itemize}

\section*{Acknowledgements}
The author is indebted to Mina Aganagic, Aleksandra Anokhina, Semeon Artamonov, Alexei Bondal, Tudor Dimofte, Eugene Gorsky, Kentaro Hori, Mikhail Kapranov, Petr Koroteev, Andrey Mironov,  Gregory W. Moore, Alexei Morozov, Hiraku Nakajima, Vivek Schende, Alexei Sleptsov, Andrey Smirnov, Lev Soukhanov, Masahito Yamazaki, Taizan Watari, Yegor Zenkevich for stimulating and illuminating discussions at various stages of this project.  This work is supported in part by WPI Research Center Initiative, MEXT, Japan.