text:
\section{Introduction}

Among all 4-dimensional supersymmetric gauge theories, $\cN=4$ super Yang-Mills
(SYM) is the
most special and rigid model. Due to this rigidity, $\cN=4$ SYM has
remarkable properties as a quantum field theory, such as conformal invariance
and UV-finiteness \cite{Mandelstam:1982cb,Brink:1982wv}. It arises in various contexts including
compactification of 10-dimensional SYM on $T^6$,
IIB string theory on D3 branes, and the AdS/CFT correspondence
\cite{Maldacena:1997re}.
Its amplitudes exhibit a rich  structure \cite{Brandhuber:2007yx,ArkaniHamed:2010kv},
and powerful tools such as integrability provided many remarkable insights \cite{Beisert:2010jr}.

On the other hand, the special structure of $\cN=4$ SYM  limits its
applicability to real-world physics.
The SUSY vacua of $\cN=4$ SYM with $SU(N)$ gauge group have a simple structure, parametrized by the
three commuting vacuum expectation values (VEVs) of the $\cN=1$ chiral
superfields.
However, deformed or softly broken versions of $\cN=4$ SYM, for example via
extra terms in the potential, may lead to more interesting vacua and low-energy
physics.
For instance, it is well-known that certain mass-deformations to the
superpotential, which preserve $\cN=1$, have a much richer vacuum
structure \cite{Vafa:1994tf,Donagi:1995cf,Dorey:2000fc,Polchinski:2000uf,Berenstein:2000ux,Berenstein:2014isa},
including stacks of fuzzy spheres which partially or completely break the gauge symmetry.
The resulting low-energy effective theory around
such vacua behaves like a 6-dimensional theory with two fuzzy
extra dimensions and a finite tower of KK modes, cf.\
\cite{Aschieri:2006uw,Andrews:2006aw,Harland:2009kg}.
Adding suitable soft SUSY breaking terms to the potential can lead to vacua with even richer geometries,
including $S^2 \times S^2$
 \cite{Behr:2005wp,CastroVillarreal:2005uu,Azuma:2005pm,Kurkcuoglu:2015eta}, $S^4$ \cite{Steinacker:2015dra}, and others.
 This generation of fuzzy extra dimensions is particularly interesting from the
matrix model point
 of view. In fact, the IKKT matrix model on noncommutative D3-brane solutions
 reduces to the $U(N)$ $\cN=4$ SYM \cite{Aoki:1999vr}, where space-time and
fuzzy extra dimensions are treated on the samen footing. Here, we focus,
however, on the familiar gauge theory setting on $\mathds{R}^{3,1}$.

The above examples lead to two obvious questions: (i) Which other, more
interesting fuzzy geometries can be
realized as vacua of deformed or softly broken $\cN=4$ SYM? (ii) What is the
low-energy physics around these vacua? In particular, is it possible to obtain a
gauge theory with chiral low-energy physics in some sense?
More specifically, we have in mind
a  gauge theory where fermions with different chirality couple
differently to the gauge fields, such as the Standard Model (SM) of particle
physics.

The issue of chirality is particularly challenging, and one might be led to quickly dismiss this possibility.
In the aforementioned examples,
the low-energy effective theory exhibits the corresponding KK tower, but it
typically results in a non-chiral theory \cite{Chatzistavrakidis:2009ix}.
On the other hand, according to \cite{Chatzistavrakidis:2011gs}, chiral fermions
do appear on intersecting
non-commutative brane solutions in the IKKT matrix model, if the
intersecting branes span all six extra dimensions.
This works nicely for flat branes, which cannot be realized in $SU(N)$
SYM, though. In contrast, intersecting configurations of curved fuzzy branes
tend to be unstable, cf.\ \cite{Azuma:2007jr}.

Remarkably, intersecting compact brane solutions of softly broken $\cN=4$ SYM without instabilities
were found in \cite{Steinacker:2014lma,Steinacker:2015mia}, by projecting
coadjoint orbits $\cO [\mu]$ of $SU(3)$ along the Cartan directions. This results
in squashed orbits $\cC[\mu]$ of dimension 4 and 6, but both types locally span all
six internal directions at the intersections.
Moreover, chiral fermionic zero modes arise at the intersections of (the sheets
of) the branes.
While it was shown that there are no negative modes around the vacua, there
do exist numerous zero modes, which lead to Yukawa couplings between the
fermionic zero modes.

These zero modes arising on such intersecting fuzzy branes $\cO [\mu]$ are particularly
interesting. The bosonic ones are denoted as \emph{Higgs sector} here, and will
be the main focus of the present paper.
It had been argued in \cite{Steinacker:2015mia} that some of these Higgs modes
may acquire a VEV, which in turn may lead to a chiral low-energy theory on
$\mathds{R}^{1,3}\times \cC[\mu]$ in suitable configurations. More specifically,
some of the fermionic zero modes (a \emph{mirror sector}) should become
massive due to the Yukawa couplings, leaving a low-energy sector of fermions
with index zero, but chiral coupling to the low-energy gauge fields.
However, the existence of such non-trivial \emph{Higgs vacua} had not been
established up to now.

In the present paper, we study this Higgs sector in detail, and show that there
exist exact solutions with non-vanishing Higgs VEV linking the $\cC[\mu]$
branes at their intersections,
leading to appropriate Yukawa couplings. We find a remarkably rich structure of
such solutions, with many similarities, but also some differences to the case
of supersymmetric gauge theories. In particular, the zero modes form a
\emph{graded ring}.
We establish a class of exact solutions with vanishing backreaction to the
branes, which is discussed and illustrated in a number of examples,
including a brane with $G_2$ structure.
For the simplest 6-dimensional (locally space-filling) branes,
we find suitable Higgs solutions which link the 3+3 left and right-handed
sheets, as required for a chiral gauge theory with 3 generations.
This suggests that a low-energy behavior reminiscent of the Standard Model might
be achieved for larger branes; however for generic branes we
can only provide evidence for the existence of such Higgs solutions.

\paragraph{Summary.}
Due to the length of the article, we provide a brief summary of the key points
already at this stage.
Our starting point is the $\cN=4$ SYM Lagrangian for gauge group $SU(N)$ with
the following cubic and quadratic soft SUSY breaking terms
\begin{equation}
V_{\mathrm{soft}}[X]= -4\, \mbox{tr} \left([X_1^+,X_2^+] X_3^+ +\mathrm{h.c.}
\right)+ 4 \sum_{i=1}^3 M_{i}^2 \, \mbox{tr} \left( X_i^- X_i^+ \right)\,
\end{equation}
with $X_i^- \coloneqq (X_i^+)^\dagger , \, i=1,2,3$.
We will focus on vacuum solutions $X_i^\pm   \in \mathrm{Mat}(N,\mathds{C})$.
Our first new insight is the following observation: The
second order, i.e.\ double commutator, equations of motion (eom) for $X$ admit
\emph{first integral equations}
\begin{align}
 \left[X_i^+,X_j^+\right] = \sum_{k=1}^3 \varepsilon_{ijk}X_k^-
 \, ,\; \forall i,j
 \qquad \text{and} \qquad
 \sum_{i=1}^3 \left[X_i^+,X_i^-\right] =0 \; .
 \label{eq:1st-integral}
\end{align}
The full equations of motion with arbitrary mass term
$M\equiv M_i \in[0,\frac{1}{2}]$ are solved by a rescaling
\begin{align}
 \widetilde{X}_i^+ = R(M) \cdot X_i^+ \, ,
 \label{eq:rescale-X}
\end{align}
with $R(M) = \frac{1}{2} (1+\sqrt{1-4M^2})$
and any $X_i^+$ that satisfy \eqref{eq:1st-integral}.
As demonstrated in
\cite{Steinacker:2014lma}, this allows for vacua describing squashed
fuzzy $SU(3)$ coadjoint orbits $\cC[\mu]$. The (bosonic and fermionic) fluctuations around such a background $X$
are governed by a matrix Laplacian $\cO_V^X$ and a Dirac operator ${\slashed{D}}^X$. For
vanishing masses $M$, the operator $\cO_V^X$ on $\cC[\mu]$ is positive
semi-definite, and the zero modes split into \emph{regular} and
\emph{exceptional} modes. The regular modes are fully classified by the
representation theory origins of $\cC[\mu]$. Moreover, regular zero modes are
characterized by the \emph{decoupling conditions}
\begin{equation}
 \left[X_i^+ , \phi_j^+\right]=0 \, , \; \forall i\neq j \,,
 \qquad \text{and} \qquad
 \left[X_i^-, \phi_i^+\right]=0 \, \; \forall i \,,
 \label{eq:decoupling}
\end{equation}
between background $X$ and fluctuations $\phi$.
Our second new insight concerns properties of the regular zero modes: first, the set
of regular zero modes, in the sense of \eqref{eq:decoupling}, form a
ring which is graded by the integral weights of $\mathfrak{s}\mathfrak{u}(3)$.
Second, the decoupling conditions between $X$ and $\phi$ are sufficient for a
decoupling of the  potential
\begin{subequations}
\begin{equation}
 V[X+\phi] = V[X]+V[\phi]
\end{equation}
and the full equations of motion also decouple,
\begin{equation}
 \mathrm{eom}(X+\phi) = \mathrm{eom}(X) + \mathrm{eom}(\phi)  \,,
 \label{eq:additivity_eom}
\end{equation}
\end{subequations}
for arbitrary mass values.

Our third new insight lies in the following construction: starting from a
background $X$, we add regular zero modes $\phi$ which satisfy
the equations of motion. Then $X+\phi$ is
an exact solution due to  \eqref{eq:additivity_eom}.
Examples include (i) zero modes $\phi$ with maximal rank that
reduce $X+ \phi$ to the fuzzy 2-sphere, and (ii)
zero modes $\phi$ with rank one such that $X+\phi$ is interpreted as
$\cC[\mu]$ brane plus string modes. The latter provide Yukawa couplings for
the analogous fermionic zero modes at the brane intersections.

While the spectra of $\cO_V^{X+\phi}$ and ${\slashed{D}}^{X+\phi}$
are not understood analytically,  numerical studies show the existence of
instabilities in the massless case. However, a surprising but simple observation shows
that the full potential can be expressed in complete squares, such that it
becomes positive semi-definite for $M\geq \frac{\sqrt{2}}{3}$. While for
$M>\frac{\sqrt{2}}{3}$ the solution with $V=0$ are trivial,
for the particular mass $M^*=\frac{\sqrt{2}}{3}$ the solutions with $V=0$ are
given precisely by \eqref{eq:1st-integral}, upon
rescaling \eqref{eq:rescale-X}.
Thus $\cO_V^{X+\phi}$ has \emph{no instabilities},
and moreover numerical studies indicate that the number of zero modes is
drastically reduced in comparison to $\cO_V^{X}$, and independent of $N$.
This implies that our non-trivial (brane+Higgs) solutions are locally stable up to a compact moduli space,
which is understood in several interesting cases.

These results are remarkable, because they point to the possibility to obtain
interesting chiral low energy gauge theories from softly broken $\cN=4$ SYM.
The $\cC[\mu]$ solutions behave as self-intersecting branes in extra dimensions,
with chiral fermionic zero modes located at the intersections.
In \cite{Steinacker:2014lma,Steinacker:2015mia} it has been argued that
some Higgs moduli on the $\cC[\mu]$ branes need to acquire VEVs
in order to stabilize the system and to give masses to undesired
fermions via Yukawa couplings. Here, we established the existence of such
Higgs VEVs, which give masses not only to fermions, but also to most of
the bosonic moduli, as shown by the spectrum of $\cO_V^{X+\phi}$.

There are two types of branes $\cC[(N,0)]$ and $\cC[(N_1,N_2)]$ with different chirality properties.
The space-filling 6-dimensional $\cC[(N_1,N_2)]$ have a built-in separation of chiral modes,
as seen through a suitable gauge boson mode of $\chi$ \eqref{chi-def}.
We focus on the simplest $\cC[(1,1)]$ brane, for which we find an exact Higgs
solution,
and the more general \emph{rigged} $\cC[(N,1)]$ branes. These are the basis
for
our approach to the Standard Model.
On the 4-dimensional counterparts $\cC[(N,0)]$, a chiral gauge theory would require a
configuration of Higgs modes which is not supported by our results.

There is another interesting general message: the underlying models are
gauge theories with a gauge group of large rank $N \gg 1$. In the trivial
vacuum, such large-$N$ gauge theories are governed by the t'Hooft coupling $\lambda} \def\L{\Lambda} \def\la{\lambda =
g^2 N$.
In contrast, the $\cC[(N_1,N_2)]$ vacua with large $N_i$ behave as
semi-classical, large fuzzy branes. The fluctuation spectrum on these vacua
consist of a tower of KK states with a finite mass gap (independent of $N$), as
well as the Higgs sector of zero modes.
As discussed in section \ref{sec:regular-zm}, this Higgs sector consists of
\emph{string-like modes}, which link some sheets of these branes with coupling
strength $g$,
as well as \emph{semi-classical  modes}, which are almost free.
Moreover, most of the latter disappear as the string-like modes acquire some
VEV.
Hence, the large number of massless fluctuations in the original large-$N$
theory is reduced to a small sector of string-like modes, which acquire VEVs, and leave
few remaining zero modes.
This mean that the original large-$N$ gauge theory reduces to an
effective low-energy theory with few modes and an interesting geometric structure.
This certainly provides strong motivation to study such scenarios in more detail.

\paragraph{Outline.}
The paper is organized as follows:
We start  by recalling $\cN=4$ SYM in section \ref{sec:background}, which
is modified by cubic and quadratic soft breaking terms. The
properties of the squashed coadjoint $SU(3)$ orbits and the classification of
the bosonic zero modes are reviewed, and the potential is organized
accordingly.
In section \ref{sec:4D-branes},
we focus on the exact (classical) $\cC[\mu]$ solutions of 4-dimensional type,
and study their zero mode sector.
We find various exact solutions in the massless case, and show that all of these are
free of instabilities in the presence of certain mass parameters.
Moving on to 6-dimensional squashed orbits and their zero modes, we show in
section \ref{sec:6D-branes} that chiral settings are simpler and more naturally
obtained here, and identify the chirality observable $\chi$, see
\eqref{chi-def}.
We find again exact solutions for the simplest 6-dimensional case $\cC[(1,1)]$,
and comment on generalizations for larger branes and the issue of three
generations.
The fermionic zero modes and their Yukawa couplings are discussed in section \ref{sec:fermions}.
In section \ref{sec:standardmodel}, we give a qualitative discussion how
low-energy theories resembling the Standard Model might be obtained using the
present framework. Finally, we conclude in section \ref{sec:conclusion}.

Two appendices complement this article, exemplifying solutions to the
equations of motion in appendix \ref{app:explicit_sol}, and  spelling out
details of
the novel combined solutions and the spectra of the vector Laplacian and the
Dirac operator  in appendix \ref{app:solutions_spectrum}.
\section{Background, zero modes and Higgs potential}
\label{sec:background}
First we recall the setting from
\cite{Steinacker:2014lma,Steinacker:2014eua,Steinacker:2015mia}.
We start with the action of $\cN=4$ $SU(N)$ SYM, which is organized most transparently
in terms of 10-dimensional SYM reduced to 4 dimensions:
\begin{equation}
\begin{aligned}
 S_{\rm YM}
=  \int d^4 y \ \frac 1{4g^2} &\mbox{tr}\Big(-F^{\mu\nu} F_{\mu\nu}
 - 2  D^\mu \phi^a D_\mu \phi_a +  [\phi^a,\phi^b][\phi_a,\phi_b] \Big)\\
&+  \mbox{tr}\Big(\bar\Psi\g^{\mu} \mathrm{i} D_\mu \Psi + \bar\Psi\Gamma^a
[\phi_a,\Psi]\Big) \, .
\end{aligned}
\label{N=4SYM}
\end{equation}
Here $F_{\mu\nu}$ is the field strength, $D_\mu = \partial_\mu - \mathrm{i}
[A_\mu,\cdot]$ the gauge covariant derivative,
$\phi^a,\ a \in \{1,2,4,5,6,7\}$ are 6 scalar fields,
$\Psi$ is a matrix-valued Majorana-Weyl (MW) spinor of $SO(9,1)$ dimensionally
reduced to 4-dimensions, and $\Gamma^a$ arise from the 10-dimensional gamma matrices.
 All fields transform take values in $\mathfrak{s}\mathfrak{u}(N)$ and transform
 in the adjoint of the $SU(N)$ gauge symmetry. The global $SO(6)_R$ symmetry is manifest.

It will be useful to work with the following complex linear combinations of
dimensionless scalar fields
\begin{equation}
\label{X-T-definition}
\begin{aligned}
m X_1^\pm &= \frac{1}{\sqrt{2}}\left(\phi_4\pm \mathrm{i} \phi_5 \right)
, \\
m X_2^\pm &= \frac{1}{\sqrt{2}}\left(\phi_6\mp \mathrm{i} \phi_7 \right)
,\\
m X_3^\pm &= \frac{1}{\sqrt{2}}\left(\phi_1\mp \mathrm{i} \phi_2 \right)
\end{aligned}
\end{equation}
with $m$ having the dimension of a mass.
For later, we also introduce the notation $X_j^\pm \equiv X_{\pm \alpha}          \def\da{{\dot\alpha}_j}$ with a
normalization such that
\begin{align}
 X^\alpha}          \def\da{{\dot\alpha} =  X_{-\alpha}          \def\da{{\dot\alpha}} \quad \forall  \alpha}          \def\da{{\dot\alpha} \in \cI = \left\{ \pm \alpha}          \def\da{{\dot\alpha}_j,\ j=1,2,3
\right\} \,.
\end{align}
To introduce a scale and to allow non-trivial \emph{brane} solutions, we add
soft terms to the potential,
\begin{align}
 {\cal V}} \def\cW{{\cal W}} \def\cX{{\cal X}[\phi] &=  \frac {m^4}{g^2}  \big(V_4[X] + V_{\mathrm{soft}}[X]\big)
 \equiv  \frac {m^4}{g^2} V[X]
 \label{eq:V_total}
 \end{align}
 where
 \begin{align}
 V_4[X] &= -\frac 14 \mbox{tr} \sum_{\alpha}          \def\da{{\dot\alpha},\beta}           \def\db{{\dot\beta}\in\cI}[X_\alpha}          \def\da{{\dot\alpha},X_\beta}           \def\db{{\dot\beta}][X^\alpha}          \def\da{{\dot\alpha},X^\beta}           \def\db{{\dot\beta}] \\
 &=- \frac{1}{2} \mbox{tr} \sum_{i,j} [X_i^+,X_j^+][X_i^-,X_j^-]
 - \frac{1}{2} \mbox{tr}   \sum_{i,j} [X_i^+,X_j^-][X_i^-,X_j^+]
 , \nonumber\\
 V_{\rm soft}[X] &= 4\, \mbox{tr} \big(-[X_1^+,X_2^+] X_3^+ - [X_2^-,X_1^-] X_3^- +
M_{i}^2 X_i^- X_i^+ \big) \equiv V_3[X]+V_2[X] \;,
 \label{V-soft}
\end{align}
thereby fixing the scale $m$.
The cubic potential can be written as
 \begin{align}
 V_3[X] = - \frac {4}3\mbox{tr} \left(\sum_{i,j,k}\varepsilon_{ijk} X_i^+ X_j^+ X_k^+
 + \text{h.c.}\right)\  \equiv \frac{\mathrm{i}}{3}  \mbox{tr} \sum_{\alpha}          \def\da{{\dot\alpha},\beta}           \def\db{{\dot\beta},\g \in \cI}
(g_{\alpha}          \def\da{{\dot\alpha}\beta}           \def\db{{\dot\beta}\g} X^\alpha}          \def\da{{\dot\alpha} X^\beta}           \def\db{{\dot\beta} X^\g)  ,
\label{cubic-flux}
\end{align}
corresponding to a holomorphic 3-form on $\mathds{C}^3$.
Rewritten in a  real basis, this is recognized as the structure constants of
$\mathfrak{s}\mathfrak{u}(3)$ projected to the root generators \cite{Steinacker:2014lma}.

The cubic term breaks the global $SU(4)_R$ symmetry to $SU(3)_R$, which is in a sense the minimal breaking possible.
The mass terms $M_i$ may break this further: for
equal  $M_i \equiv M \geq0$, the $SU(3)_R$ is maintained. For $M_1=M_2
\neq M_3 $ (or permutations thereof) one has a global $(SU(2)\times U(1))_R$,
and if all masses are distinct there is only a $(U(1)\times
U(1))_R$ left.
\subsection{Preliminaries}
\label{sec:massive_potential}
In this section we perform some algebraic manipulations of the full potential,
which allow to derive first integral equations. Also, we introduce notation for
the treatment of  perturbations.
\paragraph{Rewriting the potential.}
We reconsider the full potential $V[X]= V_4[X]+V_3[X]+V_2[X]$. To begin with,
we rewrite the quartic
potential by using the Jacobi identity of the  commutator and cyclicity
of the trace. This results in
\begin{subequations}
\begin{align}
 V_4 &= - \mbox{tr}\sum_{i,j} [X_i^+,X_j^+][X_i^-,X_j^-]
 + \frac{1}{2} \mbox{tr} \sum_{i} [X_i^+,X_i^-] \sum_{j} [X_j^+,X_j^-]  \,.
\end{align}
The cubic potential \eqref{cubic-flux} can be expressed as
\begin{align}
 V_3 = -\frac{2}{3} \mbox{tr} \left( \sum_{i,j,k} \varepsilon_{ijk} [ X_i^+, X_j^+]
X_k^+
 - \sum_{i,j,k} \varepsilon_{ijk} [X_i^-, X_j^-] X_k^-  \right) \,.
\end{align}
\end{subequations}
By completing the square, we arrive at the following expression for
the total potential
\begin{equation}
\begin{aligned}
 V[X] =\mbox{tr} \bigg( &\sum_{i,j}
 \Big( [X_i^+,X_j^+] -\frac{2}{3} \sum_k \varepsilon_{ijk} X_k^-\Big)
 \Big(-[X_i^-, X_j^-] - \frac{2}{3} \sum_{k} \varepsilon_{ijk} X_k^+ \Big) \\
&+ \frac{1}{2}  \sum_{i} [X_i^+,X_i^-] \sum_{j} [X_j^+,X_j^-]
+4 \sum_i \Big(M_i^2 -\frac{2}{9}\Big)  X_i^- X_i^+ \bigg) \,.
\end{aligned}
\end{equation}
Next, we define the suggestive notation
\begin{subequations}
\label{eq:notation_FandD}
\begin{align}
F_l &=\frac{1}{2} \sum_{i,j} \varepsilon_{lij} B_{ij} \, , \quad  B_{ij}=
[X_i^+,X_j^+] -\frac{2}{3} \sum_k \varepsilon_{ijk} X_k^-  \, , \\
D&= \sum_{i} [X_i^+,X_i^-] \,,
\end{align}
\end{subequations}
and arrive at
\begin{equation}
\label{eq:potential_rewritten}
 V[X]
= 2\mbox{tr}( F_{l} F_{l}^\dagger) + \frac{1}{2} \mbox{tr} (D D^\dagger)   + 4
\sum_i(M_i^2 - \tfrac{2}{9}) \mbox{tr}(X_i^- X_i^+)
\end{equation}
It is apparent that $\mbox{tr}( F_{l} F_{l}^\dagger),\mbox{tr} (D D^\dagger), \mbox{tr}(X_i^-
X_i^+)  \geq0$. Therefore, we conclude the following:
\begin{compactenum}[(i)]

  \item Solutions with $V<0$ can only exist if $0\leq
M_i^2 < \frac{2}{9}$ holds at least for one index $i$.
 \item  If $M_i^2 > \frac{2}{9}$ for all $i$, then the only solution with $V=0$
 is the trivial one $X_i^\pm =0$ for all $i$.
\item For the special case $M_i = \frac{\sqrt{2}}{3}$ for all $i$, the vacuum
configurations with $V=0$ are characterized by the two simultaneous conditions
\begin{equation}
\begin{aligned}
 F_l &=0\,  \quad \forall l  & &\Longleftrightarrow  \quad &
 [X_i^+,X_j^+] -\frac{2}{3} \sum_k \varepsilon_{ijk} X_k^- &=0\,  \quad
\forall
i,j \\
 D&=0  & &\Longleftrightarrow & \sum_{i} [X_i^+,X_i^-] &=0 \; .
\end{aligned}
\label{firstorder-eom}
\end{equation}
Note that although the potential simplifies to $V= 2\mbox{tr}( F_{l} F_{l}^\dagger) +
\frac{1}{2} \mbox{tr} (D D^\dagger) $, the $F_l$ are not F-terms in the usual sense,
because they cannot be integrated to a superpotential. In other words, the
would-be F-terms $F_l$ contain both the holomorphic as well as anti-holomorphic
multiplets;
therefore, they cannot originate from derivatives of a holomorphic
superpotential. Nonetheless, $D$ behaves like the D-term for $\mathcal{N}=4$
SYM.
\end{compactenum}
\paragraph{Relation to $\mathcal{N}=1^*$ SUSY.}
Although SUSY is generically broken by the addition of soft terms, there is a
special case $M_1=M_2=0, M_3 = \sqrt{2}$ where the soft potential
can be expressed as a mass deformation of an $\cN=1$ superpotential.
Suppose the superpotential is of the form
\begin{align}
 W = \mbox{tr} ([X_1^+,X_2^+]X_3^-
 - m_1 X_1^+ X_1^+
 - m_2 X_2^+ X_2^+
 - m_3 X_3^- X_3^- ) \; .
\end{align}
Then the F-term contributions to the scalar potential are as follows:
\begin{align}
 V_F = - \mbox{tr} \bigg( &[X_1^+,X_2^+] [X_1^-,X_2^-]  +  [X_1^+,X_3^-]
[X_1^-,X_3^+] +  [X_2^+,X_3^-] [X_2^-,X_3^+] \nonumber \\
&+ m_1 ([X_2^+,X_3^-]X_1^- - [X_2^-,X_3^+]X_1^+) \nonumber \\
&+ m_2 ([X_3^-,X_1^+]X_2^- - [X_3^+,X_1^-]X_2^-)\nonumber \\
&+ m_3 ([X_1^+,X_2^+]X_3^+ - [X_1^-,X_2^-]X_3^-)\nonumber \\
&+m_1^2 X_1^+ X_1^- +m_2^2 X_2^+ X_2^- +m_3^2 X_3^+ X_3^-\bigg)
\end{align}
to match the cubic potential \eqref{cubic-flux}, one is forced to set
$m_1=m_2=0$ and
$m_3=\sqrt{2}$. Hence, this has an unbroken $\cN=1$ SUSY and is known as
$\cN=1^*$ SUSY, see for
instance \cite{Vafa:1994tf,Donagi:1995cf,Dorey:2000fc,Polchinski:2000uf}.
\paragraph{Perturbation of the background.}
Let us add a  perturbation $\phi^\alpha}          \def\da{{\dot\alpha}$ to a background $X^\alpha}          \def\da{{\dot\alpha}$,
\begin{align}
Y^\alpha}          \def\da{{\dot\alpha}  = X^\alpha}          \def\da{{\dot\alpha} + \phi^\alpha}          \def\da{{\dot\alpha} \, .
\label{eq:perturbe_background}
\end{align}
As discussed in \cite{Steinacker:2015mia}, the perturbations imply further
symmetry breaking and lead to interesting low-energy physics
in the zero-mode sector of the background $X$ .
The complete potential can be worked out,
\begin{align}
  V[X+\phi]
&= V[X] + V[\phi] +
\mbox{tr}\Big(\phi^\alpha}          \def\da{{\dot\alpha} \Box_X X_\alpha}          \def\da{{\dot\alpha} + X^\alpha}          \def\da{{\dot\alpha} \Box_\phi \phi_\alpha}          \def\da{{\dot\alpha}
        +\frac 12\phi^\alpha}          \def\da{{\dot\alpha}  \big(\Box_X + 2 {\slashed{D}}_{ad} \big) \phi_\alpha}          \def\da{{\dot\alpha} - \frac 12 f^2 \Big) \nonumber\\
     &\ \ + 4\mbox{tr} \big(- \varepsilon_{ijk} \phi_i^+ X_j^+ X_k^+
      - \varepsilon_{ijk} \phi_i^+ \phi_j^+ X_k^+ + M_{i}^2 \phi_i^- X_i^+
      + \text{h.c.} \big)  \; .
 \label{V-full-expand}
\end{align}
Here
\begin{align}
 f &= i[\phi_\alpha}          \def\da{{\dot\alpha},X^\alpha}          \def\da{{\dot\alpha}]
\end{align}
can be viewed as gauge-fixing function in extra dimensions, and, following
\cite{Steinacker:2014eua}, we define following operators
\begin{align}
 \Box_X &= \sum_{a\in\cI} [X_\alpha}          \def\da{{\dot\alpha},[X^\alpha}          \def\da{{\dot\alpha},\cdot]]
 = \sum_{j=1}^3 [X_j^+, [X_j^-,\cdot ]] + [X_j^-, [X_j^+,\cdot]] ,  \\
({\slashed{D}}_{ad} \phi)_\alpha}          \def\da{{\dot\alpha} &= \sum_\beta}           \def\db{{\dot\beta} [[X_\alpha}          \def\da{{\dot\alpha},X^{\beta}           \def\db{{\dot\beta}}],\phi_{\beta}           \def\db{{\dot\beta}}]
=  ((\slashed{D}_{\rm mix}  + \slashed{D}_{\rm diag}) \phi)_\alpha}          \def\da{{\dot\alpha} \nonumber\\
(\slashed{D}_{\rm mix} \phi)_\alpha}          \def\da{{\dot\alpha} &= \sum_{\beta}           \def\db{{\dot\beta}\neq \alpha}          \def\da{{\dot\alpha}} [[X_\alpha}          \def\da{{\dot\alpha},X^{\beta}           \def\db{{\dot\beta}}],\phi_{\beta}           \def\db{{\dot\beta}}]  \nonumber\\
(\slashed{D}_{\rm diag} \phi)_\alpha}          \def\da{{\dot\alpha} &= [[X_\alpha}          \def\da{{\dot\alpha},X_{-\alpha}          \def\da{{\dot\alpha}}],\phi_{\alpha}          \def\da{{\dot\alpha}}]  \qquad\mbox{(no sum)} \ .
 \label{higgs-decouple-1}
\end{align}
\paragraph{Equations of motion.}
The equations of motion (eom) for the background $X$ can be written as
\begin{align}
 0=\big(\Box_4  + m^2(\Box_X + 4 M_{i}^2)\big)X_i^+ + 4 m^2 \varepsilon_{ijk} X_j^- X_k^-
 \label{eom-roots}
\end{align}
where $\Box_4 = -D_\mu D^\mu$ is the 4-dimensional covariant d'Alembertian.
For classical vacua, i.e.\ space-time independent $X$, the eom reduce and can
be cast in the following form:
\begin{align}
 0 = 2\sum_j \left[ \widetilde{B}_{ij} ,X_j^- \right]
 +\left[ D,X_i^+ \right] + 4M_i^2 X_i^+  \quad
 \text{with } \quad \widetilde{B}_{ij} =[X_i^+,X_j^+] - \sum_k
\varepsilon_{ijk} X_k^-
\label{eq:eom_new}
\end{align}
with $D$ as  in \eqref{eq:notation_FandD}. We observe that $B_{ij}$ of
\eqref{eq:notation_FandD} and
$\widetilde{B}_{ij}$ are equivalent upon rescaling $X$.
\paragraph{Homogeneity of potential.}
The full potential exhibits a certain homogeneity pattern, which implies
the relation $4V_4 +3 V_3+2V_2=0 $ for solutions. Hence the potential value at a
solution $X$ of the eom can be computed via
\begin{align}
 V[X]\big|_{\text{sol. eom}}= \frac{1}{4} V_3[X] +\frac{1}{2}V_2[X] \; .
 \label{energy-homogeneous}
\end{align}
\subsection{Squashed \texorpdfstring{$SU(3)$}{SU(3)} brane solutions}
\label{sec:solutions}
It is well-known that the potential \eqref{eq:V_total} with
\eqref{V-soft} has fuzzy sphere solutions $X_i^\pm \sim R_i^\pm J_i$
where $J_i$ are generators of $\mathfrak{s}\mathfrak{u}(2)$
\cite{DuboisViolette:1989at,Vafa:1994tf,Myers:1999ps,Polchinski:2000uf,Iso:2001mg,Berenstein:2002jq,Andrews:2006aw,Aschieri:2006uw}.
However as shown in \cite{Steinacker:2014lma}, there are also solutions with much richer structure
corresponding to (stacks of) squashed fuzzy coadjoint $SU(3)$ orbits $\cC_N[\mu]$, obtained by
the following ansatz
\begin{align} \
 X_i^\pm = R_i \pi(T_i^\pm)  \ .
  \label{basic-branes}
\end{align}
We denote these as \emph{(squashed) $SU(3)$ branes}, and they are the focus of
this paper. Here
\begin{align}
 T_1^\pm &\equiv T_{\pm \alpha}          \def\da{{\dot\alpha}_1}, \qquad T_2^\pm \equiv T_{\pm \alpha}          \def\da{{\dot\alpha}_2},  \qquad
 T_3^\pm \equiv T_{\pm \alpha}          \def\da{{\dot\alpha}_3}
 \label{X-T-definition-roots}
\end{align}
 are \emph{root} generators of $\mathfrak{s}\mathfrak{u}(3)_X \subset \mathfrak{s}\mathfrak{u}(N)$,
$\pi$ is any representation on $\cH \cong \mathds{C}^N$, and
$\alpha}          \def\da{{\dot\alpha}_{1}, \alpha}          \def\da{{\dot\alpha}_2$ are the simple roots with $\alpha}          \def\da{{\dot\alpha}_3 = -(\alpha}          \def\da{{\dot\alpha}_1+\alpha}          \def\da{{\dot\alpha}_2)$.
In these conventions\footnote{We use field theory conventions, while in
\cite{Steinacker:2014lma} group-theory friendly conventions were used. In
particular, the $\alpha}          \def\da{{\dot\alpha}_i$ are related to  the standard
basis $\tilde\alpha}          \def\da{{\dot\alpha}_i$ of positive roots of $\mathfrak{s}\mathfrak{u}(3)$ used in group theory via
$\alpha}          \def\da{{\dot\alpha}_1=\tilde\alpha}          \def\da{{\dot\alpha}_1,\alpha}          \def\da{{\dot\alpha}_2=\tilde\alpha}          \def\da{{\dot\alpha}_2,\alpha}          \def\da{{\dot\alpha}_3=-\tilde\alpha}          \def\da{{\dot\alpha}_3$, such that
$\alpha}          \def\da{{\dot\alpha}_1+\alpha}          \def\da{{\dot\alpha}_2+\alpha}          \def\da{{\dot\alpha}_3=0$; this is more natural here.}, the Lie algebra relations
are
\begin{align}
 [T_\alpha}          \def\da{{\dot\alpha},T_\beta}           \def\db{{\dot\beta}] &= \pm T_{\alpha}          \def\da{{\dot\alpha}+\beta}           \def\db{{\dot\beta}}, \qquad 0 \neq \alpha}          \def\da{{\dot\alpha}+\beta}           \def\db{{\dot\beta}\in \cI  \nonumber\\
 [T_{\alpha}          \def\da{{\dot\alpha}_i},T_{-\alpha}          \def\da{{\dot\alpha}_i}] &= H_{i} \equiv H_{\alpha}          \def\da{{\dot\alpha}_i} \nonumber\\
 [H,T_\alpha}          \def\da{{\dot\alpha}] &= \alpha}          \def\da{{\dot\alpha}(H) T_\alpha}          \def\da{{\dot\alpha} \ ,
 \label{Cartan-Weyl}
\end{align}
and in particular
\begin{subequations}
\label{eq:Lie_algebra_eom}
\begin{align}
 [T_i^+,T_j^+] = \varepsilon_{ijk} T_k^-
 \label{root-algebra}
\end{align}
and $\alpha}          \def\da{{\dot\alpha}_i(H_i) = (\alpha}          \def\da{{\dot\alpha}_i,\alpha}          \def\da{{\dot\alpha}_i) =2$ where
$(\cdot,\cdot)$ denotes the Killing form of $\mathfrak{s}\mathfrak{u}(3)$.
Note also that the choice of labeling of $\alpha}          \def\da{{\dot\alpha}_3 = -\alpha}          \def\da{{\dot\alpha}_1 -\alpha}          \def\da{{\dot\alpha}_2$ implies that
\begin{align}
 \sum_j [T_j^+,T_j^-] = \sum_j H_j =0 \, .
\end{align}
\end{subequations}
Using the Lie algebra relations, the equations of motion \eqref{eom-roots}
become
\begin{align}
0   &=  R_1\big(2 R_1^2 +R_2^2 + R_3^2 - 4 \frac{R_2 R_3}{R_1} + 4M_{1}^2 \big)
T_1^+ \nonumber\\
 0  &= R_2\big(R_1^2 + 2 R_2^2 + R_3^2 - 4 \frac{R_1 R_3}{R_2}  + 4M_{2}^2\big)
T_2^+  \nonumber\\
 0 &= R_3\big(R_1^2 + R_2^2 + 2R_3^2 - 4 \frac{R_1 R_2}{R_3}  + 4M_{3}^2\big)
T_3^+ .
  \label{eom-general}
\end{align}
Assuming  $M_i=0$ for simplicity, these equations have
the non-trivial solution
\begin{align}
 R_i = 1 \equiv R \ .
 \label{R-solution-equal}
\end{align}
This can be seen from \eqref{eq:eom_new}, as for vanishing masses the eom are
necessarily satisfied for any configuration with $\widetilde{B}_{ij}=0=D$.
Inspecting the relations \eqref{eq:Lie_algebra_eom} reveals that the ansatz
\eqref{basic-branes} solves the eom for \eqref{R-solution-equal}.
Based on investigations with \texttt{Mathematica} presented in
appendix \ref{app:explicit_sol}, we conclude that all $R_i$ must be equal (up to
an irrelevant sign) if all $M_i^2$ are equal.
In addition, we observe that there are no solution if one of the mass
parameters satisfies $M_i^2 \geq \frac 43$, in agreement  with earlier findings
\cite{Steinacker:2015mia}.

Let $\mu$ be the highest weight vector of an $SU(3)$ irrep $\pi_\mu$, which
enters the
definition \eqref{basic-branes} of the squashed orbit $\cC[\mu]$.
The highest weight $\mu$ has associated Dynkin
labels $n_i =2\frac{( \mu ,\alpha_i )}{( \alpha_i ,\alpha_i
)}\in \mathbb{N}$ ($\alpha_i$,  $i=1,2$ are the simple roots), and we
simply write $\mu=(n_1,n_2)$ instead of $\mu = n_1 \Lambda_1 +n_2 \Lambda_2$,
with $\Lambda_i$ the fundamental weights.
Generically the $\cC_N[\mu]$ are 6-dimensional (fuzzy) varieties, while for
$\mu =(n,0)$ and $\mu = (0,n)$ they are
4-dimensional projections of (fuzzy) $\mathds{C} P^2$.
Such a \emph{squashed} $\mathds{C} P^2$ has a triple self-intersection at the origin,
as visualized in figure
\ref{fig:squashedCP2}.
\begin{figure}[t!]
\centering
\begin{subfigure}{0.48\textwidth}
\begin{center}
 \includegraphics[width=0.45\textwidth]{cpplot.png}
 \end{center}
 \caption{}
 \label{fig:squashedCP2}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
 \includegraphics[width=0.40\textwidth]{cpplot-higgs.jpg}
 \end{center}
 \caption{}
 \label{fig:squashedCP2-Higgs}
\end{subfigure}
\caption{(\subref{fig:squashedCP2}) 3-dimensional section of squashed $\mathds{C} P^2$.
(\subref{fig:squashedCP2-Higgs}) 3-dimensional section of squashed $\mathds{C} P^2$,
with zero mode connecting 2
sheets. }
\end{figure}

The $\cC_N[\mu]$ backgrounds $X$ break $SU(3)_R$ to $(U(1) \times U(1))_K$
generated by
\begin{align}
 K_i = 2\tau_i-[H_{\alpha}          \def\da{{\dot\alpha}_i},\cdot], \quad i=1,2,3\qquad  \text{with} \quad
K_1+K_2+K_3 = 0 \; .
 \label{K-i-def}
\end{align}
where $\tau_i$ denotes the Cartan generators of $SU(3)_R$.
This is a combination of the global
$(U(1)\times U(1))_R \subset SU(3)_R$ symmetry and the $SU(3)_X\subset
SU(N)$ gauge symmetry.
We denote these $U(1)_{K_i}$-charges by
\begin{align}
 K(\phi) \quad \in \ \mathfrak{s}\mathfrak{u}(3)^*
\end{align}
for any\footnote{In particular, $K(X_\alpha}          \def\da{{\dot\alpha}) = 0$ expresses the invariance of the background.} scalar field $\phi$,
and similarly also for fermionic fields.
They define a $\mathfrak{s}\mathfrak{u}(3)$ weight lattice.
The $\mathfrak{s}\mathfrak{u}(3)_X\subset \mathfrak{s}\mathfrak{u}(N)$ gauge charges
will be denoted as usual by $\lambda} \def\L{\Lambda} \def\la{\lambda(\phi) \in \mathfrak{s}\mathfrak{u}(3)^*$, so that $K=2\tau-\lambda} \def\L{\Lambda} \def\la{\lambda$. They all live in the same $\mathfrak{s}\mathfrak{u}(3)$ weight lattice.
Furthermore, we will need the \emph{$\tau$-parity} generator $\tau$ in $U(3)_R$
defined by
\begin{align}
 \tau \phi_i^\pm = \pm \phi_i^\pm \ ,
 \label{tau-parity}
\end{align}
which is broken by the cubic potential \eqref{cubic-flux}.
\paragraph{Minimal branes.}
The simplest solutions are the minimal branes, which arise for $\mu=(1,0)$ and $\mu=(0,1)$.
 To see e.g. $\cC[(1,0)]$ explicitly, denote the 3 extremal weight vectors of $(1,0)$ with
$\{|1,\mu\rangle,|2,\mu\rangle,|3,\mu\rangle\}$.
Then the minimal squashed $\mathds{C} P^2$ brane is given by
\begin{align}
 X_i^+ &\coloneqq  T_i^+ = |i, \mu\rangle \langle i-1,\mu|
\end{align}
where $T$ is the fundamental representation of $\mathfrak{s}\mathfrak{u}(3)$, and we dropped $r$ for simplicity.
The  energy for this minimal brane for $M_i=0$ is obtained using
\eqref{energy-homogeneous} as
\begin{align}
 V_4 + V_{3} = \frac 14 V_{3} = - 2 \mbox{tr}(T_1^+[T_2^+,T_3^+] + \text{h.c.}) = -8 \ .
 \label{V-minimalCP2}
\end{align}
\subsection{Geometric significance: self-intersecting branes}
The qualitative features of the above solutions, and in particular the chirality
properties of the
fermionic zero modes discussed below, can be understood
in terms of the semi-classical geometry of the solutions, interpreted as self-intersecting branes in $\mathds{R}^6$.
As for all quantized coadjoint orbits, the semi-classical geometry of
$\cC_N[\mu]$ can be extracted using coherent states.
These are precisely the $SU(3)$ orbits $\{g \cdot |x\rangle   \, ,\; g\in SU(3)
\}$  of the
extremal weight states $|x\rangle$ of the irreps $\cH_\L\in \mathrm{End}(\cH_\mu)$,
where $\cH_\mu$ is the representation space for the $\mathfrak{s}\mathfrak{u}(3)$ representation
$\pi_\mu$.
The set of coherent states $|g x\rangle \coloneqq g \cdot |x\rangle$
forms a $U(1)$ bundle over the coadjoint orbit
$\cO[\mu]$, and the semi-classical
base manifold embedded in $\mathds{R}^6$ (the \emph{brane}) is recovered from the
expectation values of these coherent states
\begin{align}
 \cC &= \left\{\langle g x| X^\alpha}          \def\da{{\dot\alpha} | g x\rangle, \  g \in SU(3) \right\}
\subset \mathds{R}^6 .
\end{align}
This would be the full coadjoint orbit $\cO[\mu]$ if the $X^\alpha}          \def\da{{\dot\alpha}$ were
supplemented by the Cartan generators $H_3, H_8$, but
here we obtain the projection $\cC[\mu]$ of $\cO[\mu]$ along the Cartan
generators, cf.\
\cite{Steinacker:2014lma,Steinacker:2014eua,Steinacker:2015mia}.

The extremal weight states $\cW |x\rangle$, which lie on the discrete orbit of
the Weyl group $\cW$ through $|x\rangle$, are projected to the origin of weight
space. To see this, it is sufficient to note that $\langle x|X_\alpha}          \def\da{{\dot\alpha}|x\rangle = 0$
as any $X_j^\pm$ annihilates either $|x\rangle$ or $\langle x|$.
The tangent space of the sub-variety $\cC_N[\mu]$ is obtained by acting with
the three $SU(2)_\alpha}          \def\da{{\dot\alpha}$ subgroups, which correspond to the roots $X_{\alpha}          \def\da{{\dot\alpha}}$ of
$\mathfrak{s}\mathfrak{u}(3)$, on the set of extremal weight states $\cW|x\rangle$.
For $\mu=(N,0)$ and $\mu=(0,N)$, one of these actions vanishes, leading to the
4-dimensional self-intersecting
$\mathds{C} P^2$ branes depicted in figure \ref{fig:squashedCP2-Higgs}. For generic
$\mu=(N_1,N_2)$, the $\cC_N[\mu]$ are 6-dimensional sub-varieties in $\mathds{R}^6$,
i.e.\ they are locally space-filling branes. Moreover, the non-degenerate
Poisson structure on $\cC_N[\mu]$ is recovered from the
commutators $[X^\alpha}          \def\da{{\dot\alpha},X^\beta}           \def\db{{\dot\beta}] \sim i \{x^\alpha}          \def\da{{\dot\alpha},x^\beta}           \def\db{{\dot\beta}\}$, whose Pfaffian is measured by
the operator $\chi$, see \eqref{chi-def}.
Therefore the generic 6-dimensional branes, which carry a rank 6 flux due to
the symplectic form, consist of 3+3 locally space-filling sheets that cover
the origin.
\subsection{Fluctuations and zero modes}
Consider the fluctuations of the scalar fields on a background brane $\cC[\mu]$
with representation space $\cH$.
To organize the degrees of freedom $\mathrm{End}(\cH)$, which denotes the algebra of
all possible functions, we note that
the solutions \eqref{basic-branes} define an embedding $SU(3)_X \subset SU(N)$,
which acts via the adjoint on all the fields.
Consequently, we decompose the $\mathfrak{s}\mathfrak{u}(N)$-valued fields into harmonics, i.e.\
irreps of this $SU(3)_X$
\begin{align}
\mathfrak{s}\mathfrak{u}(N) \cong \mathrm{End}(\cH)\big|_{SU(3)_X} = \oplus_\L  n_\L \cH_\L
\label{eq:decomposition_End}
\end{align}
where $\cH_\L$ denotes the  irreps with highest weight $\L$ appearing with
multiplicity $n_\L$.
For the case of $\mu=(N,0)$ or $\mu=(0,N)$, this decomposition is given by
\begin{align}
 \mathrm{End}(\cH)\big|_{SU(3)_X} = \oplus_{n=0}^N  \cH_{(n,n)} \ .
\label{eq:decomposition_End-CP2}
\end{align}
While the $SU(3)_X$ gauge transformations do \emph{not}
act on the indices $\alpha}          \def\da{{\dot\alpha}$ of the scalar fields $\phi_\alpha}          \def\da{{\dot\alpha}$, the $(U(1)\times U(1))_{K_i}$ symmetry
\eqref{K-i-def} does act, and the $\alpha}          \def\da{{\dot\alpha}$ realize 3+3 of the 8 states in
$(1,1)$ of the $\mathfrak{s}\mathfrak{u}(3)$ weight lattice due to the origin of $\cC[\mu]$ as a $SU(3)$ coadjoint orbit.
This allows to organize the various harmonics, which will be very useful.

Assume first $M_i=0$. Then the squashed brane backgrounds $X_\alpha}          \def\da{{\dot\alpha}$ admit a number
of bosonic zero modes $\phi^{(0)}_\alpha}          \def\da{{\dot\alpha}$, as shown in \cite{Steinacker:2014lma}.
To see this, we note that the bilinear form defined by $\slashed{D}_{\rm mix}$
on a background \eqref{basic-branes} can be simplified, for example, as follows:
\begin{align}
 \mbox{tr} \big(\phi_i^-(\slashed{D}_{\rm mix} \phi)^+_i\big)
  &= \sum_{j\neq i} \mbox{tr} (\phi_i^-[[X_i^+,X_{j}^+],\phi_{j}^-])
  =  - \varepsilon_{ikj}\frac{R_i R_j}{R_k} \mbox{tr}(\phi_i^-[X_{k}^-,\phi_{j}^-])
  \label{Dmix-explicit}
\end{align}
using \eqref{root-algebra}, where the $\phi_j^\pm$ are
perturbations as in \eqref{eq:perturbe_background}. For $R_i\equiv R$,
this has the form of the quadratic contribution from the cubic potential
\eqref{V-full-expand}, and the quadratic terms in the potential can be combined
as follows:
\begin{align}
 V_2[\phi] = \frac 12\mbox{tr} \phi^\alpha}          \def\da{{\dot\alpha} \cO_V^X \phi_\alpha}          \def\da{{\dot\alpha}  \ ,
  \qquad \cO_V^X &\stackrel{R_i=R}{=} \Box_X + 2\slashed{D}_{\rm diag} + 2(1-
\frac{2}{R})\slashed{D}_{\rm mix} \ \nonumber\\
              &\stackrel{R_i=1}{=} \Box_X + 2\slashed{D}_{\rm diag}  -2
\slashed{D}_{\rm mix} \ .
 \label{fluct-equation}
\end{align}
It has been proven in \cite{Steinacker:2014lma} that, under these conditions,
$\cO_V^X$ is positive semi-definite for all representations $\pi$.
The zero modes of $\cO_V^X$ fall into two classes, denoted as \emph{regular}
and \emph{exceptional} zero modes. We will focus on the regular zero modes, and
show that their classification is based on a \emph{decoupling condition}, as discussed in
the next section.
\subsection{Regular zero modes and decoupling condition}
\label{sec:regular-zm}
Let $\phi_\alpha}          \def\da{{\dot\alpha}$ and $\psi_\alpha}          \def\da{{\dot\alpha}, \ \alpha}          \def\da{{\dot\alpha}\in \cI$ be two arbitrary matrix
configurations, each consisting of three complex matrices or equivalently six
hermitian matrices.
We will say that $\phi_\alpha}          \def\da{{\dot\alpha}$ and $\psi_\alpha}          \def\da{{\dot\alpha}$ satisfy the \emph{decoupling
condition} if
\begin{align}
 [\psi_\alpha}          \def\da{{\dot\alpha},\phi_\beta}           \def\db{{\dot\beta}]=0\qquad \mbox{whenever}\quad  \alpha}          \def\da{{\dot\alpha}+\beta}           \def\db{{\dot\beta} \in \cI \quad \mbox{or}\
\  \alpha}          \def\da{{\dot\alpha}+\beta}           \def\db{{\dot\beta}=0
  \label{extremal-char-1}
 \end{align}
This condition is symmetric under $\psi \leftrightarrow \phi$, and
equivalent to
 \begin{align}
 [\psi_{\alpha}          \def\da{{\dot\alpha}},\phi_{-\beta}           \def\db{{\dot\beta}}]=0\qquad \mbox{whenever}\quad  \alpha}          \def\da{{\dot\alpha}-\beta}           \def\db{{\dot\beta} \in \cI \quad \mbox{or}\ \  \alpha}          \def\da{{\dot\alpha}-\beta}           \def\db{{\dot\beta}=0 \ .
   \label{extremal-char-2}
\end{align}
We define \emph{regular zero modes} $\phi_\alpha}          \def\da{{\dot\alpha}$ of $\cO_V^X$ to be modes that
satisfy
the decoupling condition w.r.t.\ $X_\alpha}          \def\da{{\dot\alpha}$.
As we will see momentarily, these modes are in fact zero modes of $\cO_V^X$
for $R_i=R$ and $M_i =0$, and this definition is then equivalent to
the one given in \cite{Steinacker:2014lma,Steinacker:2014eua}.
The conditions \eqref{extremal-char-1} or \eqref{extremal-char-2} amount to the
requirement that $\phi_\alpha}          \def\da{{\dot\alpha}$
is annihilated by three ladder operators out of the six $X_j^\pm$. For example,
the condition for $\phi_i^+$ is
\begin{align}
 [X_j^+,\phi_i^+] &= 0 \quad \text{for}\ \ j\neq i, \quad \text{and} \quad
[X_i^-,\phi_i^+] = 0 \quad \text{(no sum)}.
 \label{extremal-3}
\end{align}
Before classifying them more explicitly, we state some important consequences.
For $M_i=0$, the regular zero modes $\phi_{\alpha}          \def\da{{\dot\alpha}}$ satisfy
\begin{equation}
\begin{aligned}
 (\slashed{D}_{\rm mix}^X \phi)_{i}^+ &= 0 \label{D-mix-vanish} \\
 \cO_{V,M_i\equiv0}^X \phi_{i}^+ &= [H_R,\phi_{i}^+] \ \stackrel{R_i=R}{=} \  0
\end{aligned}
\end{equation}
where
\begin{align}
  H_R &\coloneqq R_1^2 H_1+R_2^2 H_2 + R_3^2 H_3  \ \stackrel{R_i=R}{=} \  0 \ .
  \label{HR-def}
\end{align}
To prove the first statement of \eqref{D-mix-vanish}, consider
 \begin{align}
(\slashed{D}_{\rm mix}^X \phi)_{i}^+ &= \sum_j [[X_i^+,X_j^+],\phi_j^-]
+ \sum_{j\neq i} [[X_i^+,X_j^+],\phi_j^+] \nonumber \\
&=\sum_{j,k} \frac{R_i R_j}{R_k}
\varepsilon_{ijk}\underbrace{[X_k^-,\phi_j^-]}_{=0}
-\sum_{j\neq i} [\underbrace{[X_j^-,\phi_j^+]}_{=0},X_i^+]
-\sum_{j\neq i} [\underbrace{[\phi_j^+,X_i^+]}_{=0},X_j^-]
=0
 \label{Dmix-zero-explicit}
\end{align}
where we have used the algebra relation \eqref{root-algebra} and the Jacobi
identity. All indicated commutators vanish due to decoupling condition
\eqref{extremal-3}.
For the second statement of \eqref{D-mix-vanish}, consider
\begin{align}
 \Box_X \phi_{i}^+ &=
 \sum_j [X_j^+,[X_j^-,\phi_i^+]]
 +\sum_j [X_j^-,[X_j^+,\phi_i^+]]
 =\sum_{j\neq i} [X_j^+,[X_j^-,\phi_i^+]]
 + [X_i^-,[X_i^+,\phi_i^+]] \nonumber \\
 &= \sum_{j\neq i} [[X_j^+,X_j^-],\phi_i^+] -[[X_i^+,X_i^-],\phi_i^+]
 =\sum_{j\neq i}R_j^2 [H_j,\phi_i^+] - R_i^2 [H_i,\phi_i^+] \\
  \slashed{D}_{\rm diag}\phi_{i}^+ &= R_i^2 [[T_{i},T_{-i}],\phi_{i}^+]
  = R_i^2 [H_i,\phi_{i}^+] \
\end{align}
using \eqref{extremal-3}. Therefore
\begin{align}
 \cO_{V,M_i\equiv0}^X \phi_{i}^+ =  (\Box_X + 2\slashed{D}_{\rm diag})\phi_{i}^+
  = [H_R,\phi_{i}^+]
  \stackrel{R_i=R}{=} 0
 \label{bosonic-zeromodes-ri}
\end{align}
Consequently, the regular zero modes, as defined by the decoupling condition,
are indeed zero modes of the $SU(3)$ branes if all $R_i$ are equal \emph{and}
all mass parameters $M_i$ vanish. We shall keep this name also in the general
case of non-vanishing masses and distinct $R_i$, which is discussed in section
\ref{sec:masses}.

Now we relate this to the group-theoretical classification of
the regular zero modes on a squashed $SU(3)$ brane $\cC[\mu]$ given in
\cite{Steinacker:2014lma,Steinacker:2015mia}.
The decoupling conditions imply that any regular zero mode $\phi_\alpha}          \def\da{{\dot\alpha}$ for any fixed $\alpha}          \def\da{{\dot\alpha}\in\cI$  is an extremal weight vector
with $\mathfrak{s}\mathfrak{u}(3)_X$ weight $\lambda} \def\L{\Lambda} \def\la{\lambda$ in some irrep
$\cH_{\L} \subset \mathrm{End}(\cH)$ of the decomposition \eqref{eq:decomposition_End}.
In view of \eqref{extremal-3}, the \emph{arrow}
$\lambda} \def\L{\Lambda} \def\la{\lambda$ must be the extremal weight vector in the  Weyl chamber opposite to the
\emph{polarization} $\alpha}          \def\da{{\dot\alpha}$ (or possibly on its wall).
Recalling the unbroken  $U(1)_{K_i}$ of the background, this means that it is
one of the six extremal  $U(1)_{K_i}$ weights\footnote{These are the corners
of the convex set of weights,
or equivalently of the maximal $\mathfrak{s}\mathfrak{u}(3)$ irrep in $(1,1)\otimes \cH_{\L^+}$.}
$\L'=\alpha}          \def\da{{\dot\alpha}-\lambda} \def\L{\Lambda} \def\la{\lambda$ of any $\phi_\alpha}          \def\da{{\dot\alpha} \in \cH_{\L}$, and we denote it by
\begin{align}
 \phi_{\alpha}          \def\da{{\dot\alpha},\L'} \ \in \cH_{\L} , \qquad \L'=\alpha}          \def\da{{\dot\alpha}-\lambda} \def\L{\Lambda} \def\la{\lambda \equiv K(\phi_{\alpha}          \def\da{{\dot\alpha},\L'})  \ .
 \label{zeromodes-weights-Weyl}
\end{align}
Hence the regular zero modes $\phi_{\alpha}          \def\da{{\dot\alpha},\L'}$ have charge
$\L'= \alpha}          \def\da{{\dot\alpha}-\lambda} \def\L{\Lambda} \def\la{\lambda$ under the $K_i$,
corresponding to a point of the $\mathfrak{s}\mathfrak{u}(3)$ weight lattice in
(the interior of) the Weyl chamber of $\alpha}          \def\da{{\dot\alpha}$. The eigenvalue
$\tau=\pm 1$ determined by the parity of the Weyl chamber of $\alpha}          \def\da{{\dot\alpha}=\pm\alpha}          \def\da{{\dot\alpha}_i$.
Clearly there is only one (extremal) state in $\cH_\L$ for any such $\L'$. Since ${\slashed{D}}_{\rm mix}$ preserves $\L'$
but flips the $\tau$-parity (recalling $\tau {\slashed{D}}_{\rm mix} = - {\slashed{D}}_{\rm mix} \tau$
from \cite{Steinacker:2015mia}), it follows again that
$\slashed{D}_{\rm mix}^X \phi = 0$, i.e.\ \eqref{D-mix-vanish} holds. This
provides another way to characterize the regular zero
modes.

We now observe that the regular zero modes form a \emph{ring}, in the
following sense:
For each $\alpha}          \def\da{{\dot\alpha}$, let $V_\alpha}          \def\da{{\dot\alpha}=\{ \phi_{\alpha}          \def\da{{\dot\alpha},\L'} \}$ be the vector space of regular zero modes with polarization $\alpha}          \def\da{{\dot\alpha}$.
According to \eqref{zeromodes-weights-Weyl}, this vector space is graded by the
integral weights in the Weyl chamber corresponding to $\alpha}          \def\da{{\dot\alpha}$.
Moreover, the decoupling
condition \eqref{extremal-char-1} -- or equivalently the extremal weight property -- implies that
if $\phi_\alpha}          \def\da{{\dot\alpha}$, $\phi'_\alpha}          \def\da{{\dot\alpha}$ are regular zero modes with the same polarization $\alpha}          \def\da{{\dot\alpha}$,
then so is their (matrix) product $\phi_\alpha}          \def\da{{\dot\alpha} \phi'_\alpha}          \def\da{{\dot\alpha}$. Hence, each $V_\alpha}          \def\da{{\dot\alpha}$ is a
graded ring. Moreover, the vector space $V = \oplus_\alpha}          \def\da{{\dot\alpha} V_\alpha}          \def\da{{\dot\alpha}$
forms a ring graded by the (integral) weights
of $\mathfrak{s}\mathfrak{u}(3)$ (or rather of $U(1)_{K_i}$),
where we define the product of zero modes in different Weyl chambers to vanish\footnote{This does not mean that
their matrix product vanishes. However, this concept helps to organize the
regular zero modes}.
This structure is respected by the Weyl group $\cW$, which relates the different $V_\alpha}          \def\da{{\dot\alpha}$.
In particular, all ring elements are nilpotent, due to the cutoff in $\L$.

The same analysis goes through for stacks of branes, where analogous zero modes
arise as strings connecting different branes.
As aforementioned, we will denote the space of zero modes as \emph{Higgs
sector}.
Labeling the six-component vector $\phi = (\phi_\alpha}          \def\da{{\dot\alpha})$ of zero modes by the
dominant $U(1)_{K_i}$ weight, we learn that the Higgs sector forms a
\emph{nilpotent ring graded by integral weights of $\mathfrak{s}\mathfrak{u}(3)$}.

\paragraph{Examples.}
Starting from the simplest solution to the decoupling condition
\begin{align}
 \phi_{\alpha}          \def\da{{\dot\alpha}, \alpha}          \def\da{{\dot\alpha}} = T_{-\alpha}          \def\da{{\dot\alpha}}
\end{align}
one can use the ring multiplication of the regular zero modes to
construct modes of the form
\begin{align}
 \phi_{\alpha}          \def\da{{\dot\alpha},(n+1)\alpha}          \def\da{{\dot\alpha}} = (T_{-\alpha}          \def\da{{\dot\alpha}})^{n} ,
 \label{zeromodes-examples}
\end{align}
and any linear combinations of these.
A possible background with such a zero mode would then be
\begin{align}
 Y_\alpha}          \def\da{{\dot\alpha} = X_\alpha}          \def\da{{\dot\alpha} + \varepsilon_\alpha}          \def\da{{\dot\alpha} (T_{-\alpha}          \def\da{{\dot\alpha}})^n .
\end{align}
On  a single squashed $\mathds{C} P^2_N$ brane $\cC[(N,0)]$, these
exhausts all regular zero modes. The ring structure is given by
$V_\alpha}          \def\da{{\dot\alpha} = \mathds{C}[\phi_{\alpha}          \def\da{{\dot\alpha}, \alpha}          \def\da{{\dot\alpha}}]/\langle\phi_{\alpha}          \def\da{{\dot\alpha}, \alpha}          \def\da{{\dot\alpha}}^{N+1}\rangle$.
In particular,  the regular zero modes with maximal $n=N$
 on squashed $\mathds{C} P^2_N$ link the 3 intersecting $\mathds{R}^4$ sheets at the
origin, with polarization along the common $\mathds{R}^2$ \cite{Steinacker:2014lma}.
These \emph{string-modes} are given by $|i-1, \mu\rangle \langle i,\mu|$, where
$\{|1,\mu\rangle,|2,\mu\rangle,|3,\mu\rangle\}$
denote the 3 extremal weight vectors of $\mu=(N,0)$.
These are coherent states located at the origin
on each of the three sheets.
An artist's rendering of such a string mode
 underlying this solution
is given in figure \ref{fig:squashedCP2-Higgs}.
More generally, the regular zero modes can be interpreted as strings linking these sheets,
shifted along their intersection.

\paragraph{Exceptional zero modes.}
In addition to the regular zero modes, there are certain exceptional zero modes
which are described in \cite{Steinacker:2014lma,Steinacker:2015mia}. For the
squashed $\mathds{C} P^2_N$ solutions, the only
exceptional modes are the
six Goldstone bosons corresponding to the spontaneously broken
$SU(3)\slash (U(1)\times U(1))$. These are
easily understood also in the deformed settings considered below.
For the more general branes, the explicit description of the exceptional zero modes is not known.
This is one reason why we focus mostly on the $\mathds{C} P^2$ branes in this paper.

\paragraph{$\mathds{Z}_3$ and generations.}
Note that the $U(3)_R$ label $\alpha}          \def\da{{\dot\alpha}$ for the scalar fields determines a 3-family structure,
which reflects the Weyl group $\mathds{Z}_3 \times \mathds{Z}_2 = \cW$.
This coincides with the  family and the $\tau$-parity as determined by
the  $U(1)_i$ charges of the zero modes.
 The $\mathds{Z}_3 \times \mathds{Z}_2$ structure is indicated by the field labels as in
 $\phi_{i\pm}$. This will be  useful for selection rules etc.
\paragraph{Stringy versus semi-classical modes.}
The above characterization of regular zero modes hides the fact that they come with very different characteristics.
We single out two extreme types: (i) the maximal or stringy zero modes, and
(ii) the semi-classical or almost-commutative zero modes.
The distinction corresponds to the separation of functions on non-squashed $SU(3)$ coadjoint orbits $\cC[\mu]$ into UV and IR sector,
as discussed in \cite{Steinacker:2016nsc}, but they are also distinguished by their coupling strength:

All Higgs modes couple to all other (scalar, gauge and fermionic) fields through
commutators $[\phi_\alpha}          \def\da{{\dot\alpha},\cdot]$. To quantify this strength, we first need to
proper normalize the Higgs modes. This
is dictated by the kinetic term\footnote{the canonical form is obtained by
absorbing $g$ in the  scalar field $\phi_a$.}
\begin{align}
 - \int d^4 x \ \frac 1{2g^2} \mbox{tr} (D^\mu \phi^a D_\mu \phi_a)
 \label{kinetic-phi-g}
\end{align}
i.e.\ all modes should be normalized w.r.t.\ the trace $\mbox{tr}()$.
Now the \emph{maximal Higgs} modes $\phi^{\mathrm{string}}_{\alpha}          \def\da{{\dot\alpha},\L'}$ are the
ones with extremal $\mathfrak{s}\mathfrak{u}(3)_K$ weight $\L'$;
these modes are given by rank one operators
\begin{align}
 \phi^{\mathrm{string}}_{\alpha}          \def\da{{\dot\alpha},\L'} = |x\rangle\langle x'|
 \label{string-modes}
\end{align}
linking  the extremal (i.e.\ coherent) weight states $|x\rangle$, $|x'\rangle$
of $\cH_\L\subset \mathrm{End}(\cH)$ whose weight difference is maximal.
From the brane point of view, these are the extreme UV modes with the maximal
momentum \cite{Steinacker:2016nsc}.
Since the expectation values vanish, $\langle x|X_\alpha}          \def\da{{\dot\alpha}|x\rangle = 0$, these
states are localized at the origin, and $\phi^{\rm string}$ should be
interpreted as strings linking the sheets of squashed $\cC[\mu]$ at the origin.
For large $N$, there are other \emph{almost-maximal} zero modes, for example
with rank 2 such as the modes considered in section \ref{sec:6D-branes}.
These almost-maximal modes have a similar properties as the maximal modes;
hence, this broader class will be called \emph{stringy} modes.
The commutator of these zero modes is of order one,
\begin{align}
 [\phi_\alpha}          \def\da{{\dot\alpha}^{\mathrm{string}},\phi_\beta}           \def\db{{\dot\beta}^{\mathrm{string}}]= O(1)
\end{align}
i.e.\ they are completely non--commutative \emph{functions} on the branes.
Putting back $g$ \eqref{kinetic-phi-g}, we see that
their interaction strength is characterized by $g$.

In contrast, the  zero modes $\phi^{\rm low}_{\alpha}          \def\da{{\dot\alpha},\L'}$ with small weight $\L'$
are matrices with high, typically maximal rank. These modes correspond to slowly-varying, semi-classical functions
$\phi^{\rm low}_{\alpha}          \def\da{{\dot\alpha},\L'}(x)$  on the $\cC[\mu]$ brane, given by polynomials of small degree in the
$X_\alpha}          \def\da{{\dot\alpha}$, the lowest mode being
\begin{align}
 \phi_\alpha}          \def\da{{\dot\alpha} = \frac 1{c_N} X_{-\alpha}          \def\da{{\dot\alpha}} \ \sim \frac 1{c_N} x_{-\alpha}          \def\da{{\dot\alpha}} \ .
\end{align}
The normalization $c_N$  for the minimal zero mode on an
irreducible  $\cC[\mu]$ can be obtained from the quadratic Casimir on $\cH_\mu$,
\begin{align}
 1= \mbox{tr} (\phi_a \phi_a) =  \frac {r^2}{c_N^2}\, \mbox{tr}  (T_a T_a )
  = \frac {r^2}{c_N^2}\, \frac{\dim(\cH_\mu)}{4}(\mu,\mu+2\rho) \quad \text{(no
sum)} \, .
\end{align}
Thus
\begin{align}
 c_N^2 =  \frac{r^2\dim(\cH_\mu)}{4}(\mu,\mu+2\rho)  = O(N^2)\,,
\end{align}
which means that the $\phi_a$ are almost-commutative functions on
$\cC[\mu]$,
\begin{align}
 [\phi_\alpha}          \def\da{{\dot\alpha}^{\mathrm{low}},\phi_\beta}           \def\db{{\dot\beta}^{\mathrm{low}}] = \frac{1}{c_N^2} c_{\alpha}          \def\da{{\dot\alpha}\beta}           \def\db{{\dot\beta}}^a
T_a   \
  &\sim i \{\phi_\alpha}          \def\da{{\dot\alpha}^{\mathrm{low}},\phi_\beta}           \def\db{{\dot\beta}^{\mathrm{low}}\} \ \ll 1
  \label{almost-comm}
\end{align}
where $\{\cdot,\cdot\}$ denotes the Poisson bracket, as usual for
low-energy functions on fuzzy spaces. Hence their interaction strength is small, and
they are become free fields as $N\to\infty$.

To summarize, the vast number of scalar modes for large $N$ becomes gapped on an irreducible $\cC[\mu]$ vacuum, and the zero modes
consist of stringy modes with interaction strength $g$, as well as weakly interacting semi-classical modes.
We will see below that the stringy modes form bound states which are stable at least for a special value of the mass parameter,
in which case the zero modes are further reduced to a small number independent of $N$.

\subsection{Aspects of the Higgs potential}
\label{sec:interactions}
Now consider the interacting potential for the Higgs sector, i.e.\ the zero
modes $\phi_\alpha}          \def\da{{\dot\alpha}$ on a background solution $X$. The linear term in $\phi$
vanishes due to the eom for $X$, so that the effective potential for $\phi$
obtained from \eqref{V-full-expand} is
\begin{equation}
\begin{aligned}
 V_{\mathrm{eff}}[\phi] &\coloneqq V[X+\phi] -V[X] \\
 &\, = V[\phi] + \mbox{tr} \left( \frac{1}{2} \phi^\alpha}          \def\da{{\dot\alpha}  \big(\Box_X + 2 {\slashed{D}}_{\rm diag}
\big) \phi_\alpha}          \def\da{{\dot\alpha}
 + (X^\alpha}          \def\da{{\dot\alpha} + \frac 14\phi^\alpha}          \def\da{{\dot\alpha})\Box_\phi \phi_\alpha}          \def\da{{\dot\alpha}  - \frac{1}{2}f^2\right)   \; .
\end{aligned}
\end{equation}
For the regular zero modes,
the cubic interaction $V_{31}$ arising from the quartic term drops out. To see this, consider
\begin{align}
 V_{31}\ \ni\ \mbox{tr} X^\alpha}          \def\da{{\dot\alpha}\Box_\phi \phi_\alpha}          \def\da{{\dot\alpha}
 &=  \mbox{tr}[X_\alpha}          \def\da{{\dot\alpha},\phi_\beta}           \def\db{{\dot\beta}][\phi^\alpha}          \def\da{{\dot\alpha},\phi^\beta}           \def\db{{\dot\beta}]     \qquad\qquad\mbox{(no sum)}\nonumber\\
  &=  -\mbox{tr} \phi_\beta}           \def\db{{\dot\beta} [[\phi^\alpha}          \def\da{{\dot\alpha},\phi^\beta}           \def\db{{\dot\beta}],X_\alpha}          \def\da{{\dot\alpha}]
  =  \mbox{tr} \phi_\beta}           \def\db{{\dot\beta} \Big([[\phi^\beta}           \def\db{{\dot\beta},X_\alpha}          \def\da{{\dot\alpha}],\phi^\alpha}          \def\da{{\dot\alpha}]  + [[X_\alpha}          \def\da{{\dot\alpha},\phi^\alpha}          \def\da{{\dot\alpha}],\phi^\beta}           \def\db{{\dot\beta}] \Big)\nonumber\\
  &=  -\mbox{tr} \phi_\beta}           \def\db{{\dot\beta} [[\phi_{-\beta}           \def\db{{\dot\beta}},X_\alpha}          \def\da{{\dot\alpha}],\phi^\alpha}          \def\da{{\dot\alpha}]
  \label{V-phi-rewrite}
\end{align}
using the Jacobi identity,  $\phi^\beta}           \def\db{{\dot\beta} = \phi_{-\beta}           \def\db{{\dot\beta}}$, and
the gauge-fixing condition $f = [X_\alpha}          \def\da{{\dot\alpha},\phi^\alpha}          \def\da{{\dot\alpha}] =0$, which follows from the
decoupling condition \eqref{extremal-char-1}.
Since either $\alpha}          \def\da{{\dot\alpha} \pm \beta}           \def\db{{\dot\beta} \in \cI$ or $\alpha}          \def\da{{\dot\alpha}\pm\beta}           \def\db{{\dot\beta}=0$ for any pair of roots $\alpha}          \def\da{{\dot\alpha},\beta}           \def\db{{\dot\beta}$ of
$\mathfrak{s}\mathfrak{u}(3)$,
the $V_{31}$ term \eqref{V-phi-rewrite} vanishes for the regular zero-modes,
again due to the decoupling condition.
Therefore $V_{\mathrm{eff}}[\phi]$ reduces for the regular zero mode sector to
\begin{align}
 V_{\mathrm{eff}}[\phi]
 &=V[\phi]+ \mbox{tr} \left( \frac 12 \phi^\alpha}          \def\da{{\dot\alpha}  \big(\Box_X + 2 {\slashed{D}}_{\rm diag}^X \big)
\phi_\alpha}          \def\da{{\dot\alpha}  \right) \, .
 \label{V-phi-eff}
\end{align}
The second term is nothing else than $\frac{1}{2}\mbox{tr}(\phi^\alpha}          \def\da{{\dot\alpha}
\cO_{V,M_i\equiv0}^X \phi_\alpha}          \def\da{{\dot\alpha}) $ and vanishes since $\phi_\alpha}          \def\da{{\dot\alpha}$ are regular zero
modes, see \eqref{D-mix-vanish}.
Hence, the effective potential for the regular zero modes is given by
\begin{align}
  {\cal V}} \def\cW{{\cal W}} \def\cX{{\cal X}[\phi] &= \frac {m^4}{g^2} V[\phi]  =\frac{m^4}{g^2}  \mbox{tr}\left(
  V_4[\phi]+ V_{\rm soft}[\phi]\right)
 \label{V-reg-zero-noM}
 \end{align}
 and we arrive at
\begin{align}
{\cal V}} \def\cW{{\cal W}} \def\cX{{\cal X}[X+\phi] =  {\cal V}} \def\cW{{\cal W}} \def\cX{{\cal X}[X] + {\cal V}} \def\cW{{\cal W}} \def\cX{{\cal X}[\phi] \ .
\label{V-additivity}
\end{align}
We emphasize that the argument remains valid for any element of the ring of regular
zero modes:  suppose $\phi$ and $\phi'$ are two six-component vectors of
regular zero modes, then
\begin{align}
{\cal V}} \def\cW{{\cal W}} \def\cX{{\cal X}[X+(\phi+\phi')] =  {\cal V}} \def\cW{{\cal W}} \def\cX{{\cal X}[X] + {\cal V}} \def\cW{{\cal W}} \def\cX{{\cal X}[\phi+\phi']
\quad \text{and} \quad
{\cal V}} \def\cW{{\cal W}} \def\cX{{\cal X}[X+(\phi\cdot\phi')] =  {\cal V}} \def\cW{{\cal W}} \def\cX{{\cal X}[X] + {\cal V}} \def\cW{{\cal W}} \def\cX{{\cal X}[\phi\cdot\phi']
\end{align}
holds, with the component-wise multiplication defined in section \ref{sec:regular-zm}.
However, ${\cal V}} \def\cW{{\cal W}} \def\cX{{\cal X}[\phi+\phi']$ or ${\cal V}} \def\cW{{\cal W}} \def\cX{{\cal X}[\phi\cdot\phi']$ do not
necessarily decompose in any linear fashion.
Similarly, one can extend to discussion to Higgs modes connecting stacks
of branes and the argument remains the same.

In particular, the potential for $\phi$ has the same structure as the original
potential \eqref{V-soft} for the model.
Thus the quadratic potential $V_2[\phi]$ vanishes again in the absence of mass
terms, but the cubic term $V_3[\phi]$ may entail some unstable directions.
Some of the $\phi_\alpha}          \def\da{{\dot\alpha}$ are then expected to take a non-trivial VEV, which is
then stabilized by the quartic term. We will indeed find such non-trivial minima
for $ \phi_\alpha}          \def\da{{\dot\alpha}$, which  will be denoted as \emph{Higgs vacua}.
Such a non-trivial Higgs linking different branes leads to a \emph{bound
state} of the branes.

As a further remark, we note that the mixed term quadratic in both $X$ and $\phi$
can be written equivalently as
\begin{align}
 \mbox{tr} \Big( \phi^\alpha}          \def\da{{\dot\alpha}  \big(\Box_X + 2 {\slashed{D}}_{\rm diag}^X \big) \phi_\alpha}          \def\da{{\dot\alpha} \Big)
  = \mbox{tr} \Big(X^\alpha}          \def\da{{\dot\alpha}  \big(\Box_\phi + 2 {\slashed{D}}_{\rm diag}^\phi \big) X_\alpha}          \def\da{{\dot\alpha}  \Big)
  \label{X-phi-quad}
\end{align}
assuming the decoupling condition between $\phi$ and $X$.

The above argument for \eqref{V-phi-rewrite} to vanish does not apply to the
exceptional zero modes.
For single branes of $\mathds{C} P^2$ type, these are precisely
the $SU(3)_R$ Goldstone bosons, which can be studied separately.
However there exist other exceptional zero modes, e.g.\
$\L\in\cW(1,0), \ \L'\in\cW(2,0)$ (or conjugate) connecting $\cC[(0,1)]$ with $\cC[(1,0)]$
which need to be studied separately.

We will see that
in the presence of positive mass terms $M_i^2 > 0$,  the above instability can be stabilized.
However if the $M^2_i$ are not all equal, some of the $\phi^\alpha}          \def\da{{\dot\alpha}$ turn out to acquire a negative mass.
This will also lead to a non-trivial Higgs vacuum.
Quantum corrections\footnote{Another conceivable mechanism
is a rotation of the branes, see \cite{Steinacker:2014eua}.} might also
play an important role, however we will assume that these are subleading if the branes are sufficiently large,
so that the semi-classical description is valid.
\paragraph{Full equations of motion and decoupling.}

We have just seen that  ${\cal V}} \def\cW{{\cal W}} \def\cX{{\cal X}(X+\phi) = {\cal V}} \def\cW{{\cal W}} \def\cX{{\cal X}(X) + {\cal V}} \def\cW{{\cal W}} \def\cX{{\cal X}(\phi)$
for a sum of backgrounds $X$, $\phi$ satisfying the decoupling condition
\eqref{extremal-char-1}.
Now, we would like to know if such a composed background $ Y^\alpha}          \def\da{{\dot\alpha} = X^\alpha}          \def\da{{\dot\alpha} +
\phi^\alpha}          \def\da{{\dot\alpha} $ can be an exact solution of the full action.
To address this, we cannot rely on the above reduced action for the Higgs sector,
because we need $\delta}  \def\D{\Delta}  \def\ddt{\dot\delta S = 0$ for arbitrary fluctuations of $Y^\alpha}          \def\da{{\dot\alpha}$.

Consider combined (static) configurations of the form $X^a + \phi^a$
which satisfies the equation of motion
\begin{align}
  0=(\Box_{X+\phi} + 4 M_{i}^2)\big)(X+\phi)_i^+ + 4 \varepsilon_{ijk}
(X+\phi)_j^- (X+\phi)_k^- .
 \label{eom-comb}
\end{align}
To simplify this, consider the matrix Laplacian $\Box_{X+\phi}$ acting on an
arbitrary $\psi \in \mathrm{End}(\cH)$. We obtain
\begin{align}
  \Box_{X+\phi} \psi &=  (\Box_\phi + \Box_X) \psi
  + [X_\beta}           \def\db{{\dot\beta},[\phi^\beta}           \def\db{{\dot\beta}, \psi]] + [\phi_\beta}           \def\db{{\dot\beta},[X^\beta}           \def\db{{\dot\beta}, \psi]] \nonumber\\
 &=  (\Box_\phi + \Box_X) \psi
  + 2 [\phi_\beta}           \def\db{{\dot\beta},[X^\beta}           \def\db{{\dot\beta}, \psi]] - [\psi,[X_\beta}           \def\db{{\dot\beta},\phi^\beta}           \def\db{{\dot\beta}]] \; .
\end{align}
Applying this on $X+\phi$ leads to
\begin{align}
  \Box_{X+\phi} (X_\alpha}          \def\da{{\dot\alpha} + \phi_\alpha}          \def\da{{\dot\alpha}) &=  (\Box_\phi + \Box_X) (X_\alpha}          \def\da{{\dot\alpha}+\phi_\alpha}          \def\da{{\dot\alpha})
  + 2 [\phi_\beta}           \def\db{{\dot\beta},[X^\beta}           \def\db{{\dot\beta}, X_\alpha}          \def\da{{\dot\alpha}]] +  2 [X_\beta}           \def\db{{\dot\beta},[\phi^\beta}           \def\db{{\dot\beta}, \phi_\alpha}          \def\da{{\dot\alpha}]] \nonumber\\
  & \quad - [\phi_\alpha}          \def\da{{\dot\alpha},[\phi_\beta}           \def\db{{\dot\beta},X^\beta}           \def\db{{\dot\beta}]]  - [X_\alpha}          \def\da{{\dot\alpha},[X_\beta}           \def\db{{\dot\beta},\phi^\beta}           \def\db{{\dot\beta}]]  \nonumber\\
  &=  (\Box_\phi + \Box_X) (X_\alpha}          \def\da{{\dot\alpha}+\phi_\alpha}          \def\da{{\dot\alpha}) + 2 ({\slashed{D}}_{ad}^X \phi)_\alpha}          \def\da{{\dot\alpha} + 2
({\slashed{D}}_{ad}^\phi X)_\alpha}          \def\da{{\dot\alpha}  \nonumber\\
  &\quad  - [\phi_\alpha}          \def\da{{\dot\alpha},[\phi_\beta}           \def\db{{\dot\beta},X^\beta}           \def\db{{\dot\beta}]]  - [X_\alpha}          \def\da{{\dot\alpha},[X_\beta}           \def\db{{\dot\beta},\phi^\beta}           \def\db{{\dot\beta}]]  \ .
  \label{box-x+y}
\end{align}
If $X_\alpha}          \def\da{{\dot\alpha}$ and $\phi_\alpha}          \def\da{{\dot\alpha}$ satisfy the decoupling condition
\eqref{extremal-char-1}, we can replace ${\slashed{D}}_{\mathrm{ad}}^X \phi
={\slashed{D}}_{\mathrm{diag}}^X \phi$ etc., and also $[X_\beta}           \def\db{{\dot\beta},Y^\beta}           \def\db{{\dot\beta}] = 0$ holds.
This further simplifies the Laplacian to
\begin{align}
 \Box_{X+\phi} (X_\alpha}          \def\da{{\dot\alpha} + \phi_\alpha}          \def\da{{\dot\alpha}) &= (\Box_\phi + \Box_X) (X_\alpha}          \def\da{{\dot\alpha}+\phi_\alpha}          \def\da{{\dot\alpha}) + 2
({\slashed{D}}_{\mathrm{diag}}^X \phi)_\alpha}          \def\da{{\dot\alpha} + 2  ({\slashed{D}}_{\mathrm{diag}}^\phi X)_\alpha}          \def\da{{\dot\alpha} \ .
 \label{Box-Sum-zeromodes}
\end{align}
Furthermore, we note that the cubic term simplifies via \eqref{extremal-3} to
\begin{align}
 \varepsilon_{ijk} (X+\phi)_j^- (X+\phi)_k^-
  &= \varepsilon_{ijk} X_j^- X_k^- + \varepsilon_{ijk} \phi_j^- \phi_k^-  +
\varepsilon_{ijk} [X_j^-, \phi_k^-]  \nonumber\\
  &= \varepsilon_{ijk} X_j^- X_k^- + \varepsilon_{ijk} \phi_j^- \phi_k^- \; .
\end{align}
Collecting all the intermediate steps, the full eom reduces to
\begin{align}
 0 &= \big(  \Box_\phi + 2{\slashed{D}}_{\rm diag}^\phi\big) X_i^+ +
 \big(\Box_X  + 4 M_i^2 \big) X_i^+
 +2  \varepsilon_{ijk}[X_j^-,X_k^-]  \nonumber\\
  & + \big( \Box_X + 2{\slashed{D}}_{\rm diag}^X\big) \phi_\alpha}          \def\da{{\dot\alpha}
  + \big(\Box_\phi  + 4M_i^2 \big) \phi_i^+
  + 2 \varepsilon_{ijk}[\phi_j^-,\phi_k^-]
  \label{eom-X+Y}
\end{align}
provided $X$ and $\phi$ satisfy the decoupling conditions
\eqref{extremal-char-1}.
Now recall that
\begin{subequations}
\begin{align}
 (\Box_X + 2{\slashed{D}}_{ad}^X) \phi_i^+ = \cO_{V,M_i\equiv0}^X \phi_i^+ = 0
 \quad \text{and} \quad
 (\Box_\phi + 2{\slashed{D}}_{ad}^\phi) X_i^+ = \cO_{V,M_i\equiv0}^\phi X_i^+ = 0
\end{align}
\end{subequations}
because the symmetry of the decoupling conditions implies that $X$ and $\phi$
are regular zero modes of $\cO_{V,M_i\equiv0}^\phi$ and $\cO_{V,M_i\equiv0}^X$,
respectively.
 As a consequence, the full eom for $X+\phi$ reduce to the sum of the
individual eom for $X$ and $\phi$ separately.
Thus, \emph{$X+\phi$ is an exact solution
if both $X$ and $\phi$ satisfy their individual equations of motion}
\begin{align}
 \big(\Box_\phi  + 4M_i^2 \big) \phi_i^+
  + 2 \varepsilon_{ijk}[\phi_j^-,\phi_k^-] =0
 \label{eom-phi}
\end{align}
\emph{and similar for $X$, as well as the decoupling condition
\eqref{extremal-3}.
}

In fact, \eqref{eom-X+Y} are precisely the eom obtain using the above reduced
potential,
i.e.\ dropping the mixed $V_{31}$ terms such as $\phi\Box_X X$ and the mixed
terms in $V_3$.
We can therefore use the reduced potential for backgrounds composed of decoupled
solutions.
However, note that finding a minimum within the Higgs sector $V(\phi)$ in general implies
\eqref{eom-phi} \emph{only up to massive modes}. This
is the reason why we will find exact solutions only in special cases within the maximal Higgs sector.

Furthermore, from the homogeneity of the full potential in either $X$ or $\phi$, we
can infer that $4V_4 + 3 V_3 +2V_2= 0$ holds at a minimum within the Higgs
sector,
which implies as in \eqref{energy-homogeneous}
\begin{align}
V[\phi]\big|_{\text{sol. eom}} = \frac{1}{4}
V_3[\phi]+\frac{1}{2} V_2[\phi] \ .
 \label{energy-homogeneous-higgs}
\end{align}
This is useful to compare the energy of different solutions.

\subsection{Massless versus massive solutions, Higgs condensate, and stability
at \texorpdfstring{$M^*$}{M*}}
\label{sec:Higgs-stab-crit-M}
So far, we have restricted the attention mostly to the massless case, i.e.\
$M_i=0$.
This has some immediate implications:
\begin{compactenum}[(i)]
 \item The eom \eqref{eq:eom_new} can be
solved immediately by the \emph{first integral} relations
$\widetilde{B}_{ij}=0=D$, which are automatically satisfied on any $SU(3)$
representation $X_j^\pm = \pi(T_j^\pm)$.
\item The operator $\cO_V^X$ governing
the dynamics of the bosonic fluctuations is positive semi-definite by virtue of
representation theory \cite{Steinacker:2014lma}. This means that
there are no instabilities, and moreover there exists a
classification of all regular zero modes.
\item The potential energy of the background can be
computed by noting that $\widetilde{B_{ij}}=0$ implies
$B_{ij}=\frac{1}{3}\varepsilon_{ijk} X_k^-$, so that
\begin{align}
 V[X]= -\frac{2}{3}\sum_i \mbox{tr}(X_i^- X_i^+) <0 \; .
\end{align}
\end{compactenum}
Now we would like to perturb the background $X$ with regular zero modes
$\phi$ to obtain a solution to the full eom for $X+\phi$, keeping $M_i=0$ for
the moment. Unfortunately, this is less accessible in general, because
\begin{compactenum}[(i)]
 \item Since zero modes $\phi$
do not satisfy the $\mathfrak{s}\mathfrak{u}(3)$ Lie algebra,  it is not easy to find exact solutions.
A notable exception are the maximal regular zero modes on $\cC[(n,0)]$.
\item The spectrum of the operator $\cO_V^{X+\phi}$ is not analytically
understood, but numerical studies presented in sections
\ref{sec:4D-branes}--\ref{sec:6D-branes}, and appendix
\ref{app:solutions_spectrum} show the existence of instabilities. Moreover, we
do not have a classification of appearing zero modes.
\item Assuming that both
$X$ and $\phi$ satisfy the first integral relations
$\widetilde{B}_{ij}=0=D$, the combined potential energy can be evaluate to read
\begin{align}
 V[X+\phi] = -\frac{2}{3}\sum_i \mbox{tr}(X_i^- X_i^+) -\frac{2}{3}\sum_i
\mbox{tr}(\phi_i^- \phi_i^+) < V[X] \;.
\end{align}
Hence starting from a $\cC_N[\mu]$ background, the extension to a combined
background definitely reduces the potential energy. This strongly suggests the existence of a
Higgs condensate.

\end{compactenum}
Let us try to circumvent the appearance of instabilities by including uniform
mass parameters $M_i \equiv M >0$ and adjusting the radii $R_i \equiv R$ in
\eqref{basic-branes} accordingly.
Starting with $X_i^\pm$ satisfying $\widetilde{B}_{ij}=0=D$,
the eom \eqref{eq:eom_new} for the ansatz $Y_i^\pm = R X_i^\pm$ reduce to
\begin{align}
0= R \big(R(R-1)+M^2\big) X_i^+ \quad \forall i
\end{align}
Thus there are three solutions
\begin{align}
 R=0\,, \qquad R = \frac{1}{2} (1\pm \sqrt{1-4M^2})
\end{align}
for $|M|\leq \frac{1}{2}$. For these choices of $R$, the eom are satisfied and
we can compute the potential from \eqref{eq:potential_rewritten}, which is
\begin{align}
 V[Y]=2R^2 \left( \left(R-\frac{2}{3}\right)^2 +2 \left(M^2 -\frac{2}{9}\right)
\right) \sum_i \mbox{tr}(X_i^- X_i^+) \; .
\end{align}
As discussed in appendix \ref{app:explicit_sol}, for $0<M\leq
\frac{\sqrt{2}}{3}$ the solution $R(M) = \frac{1}{2} (1+ \sqrt{1-4M^2})$
describes
the minimal energy configuration. This establishes
the following statement (as stated in the introduction):
any solution to $\widetilde{B}_{ij}=0=D$ gives rise to a
solution to the eom with uniform mass parameter $M$, provided one rescales with
$R(M)$.

Now we make the following very important observation: for the special mass value
\begin{align}
M^* \coloneqq \frac{\sqrt{2}}{3}
\label{M-star-def}
\end{align}
the full potential \eqref{eq:potential_rewritten} is
positive semi-definite, and the potential vanishes if and only if
\begin{align}
 B_{ij}=0 = D \ .
\end{align}
Upon rescaling, this is equivalent to
$\widetilde{B}_{ij}=0 = D$ with a radius $R=\frac{2}{3}$.
In particular, the spectrum of
$\cO_V^{Y}$ is guaranteed to be free of instabilities for any such solution.
We will indeed find a number of nontrivial \emph{brane plus Higgs} solution of
this type, which are thereby local minima up to a compact moduli space.
This statement is clearly reminiscent of the situation in supersymmetry,
even though SUSY is explicitly broken by the potential.

\subsection{Gauge fields and mode decomposition}
Now consider the gauge fields $A_\mu(y) \in \mathrm{End}(\cH)$ on $\mathds{R}^4$,
which together with the gauginos constitute the $\cN=1$ vector superfield in
$\cN=4$ SYM.
The gauge fields take values in $\mathfrak{s}\mathfrak{u}(N)$, and accordingly decompose
\begin{align}
 A_\mu = A_{\mu;\L m}(y) Y_{\L m}
 \label{gauge-mode}
\end{align}
into eigenmodes $ Y_{\L m}$ of the matrix Laplacian $\Box_X$ on squashed
$\cC_N[\mu]$ background.
These modes  acquire a mass due to the Higgs effect, given by \cite{Steinacker:2014lma}
\begin{align}
- \mbox{tr}[X^a,Y_{\L m}][X_a,Y_{\L m}] &= \mbox{tr} (Y_{\L m} ^\dagger \Box_X Y_{\L m})
=  m^2_{\L,m} \nonumber\\[1ex]
   m^2_{\L,m}  &=  2 r^2\big((\L,\L+2\rho) - (m,m)\big)
 \label{gaugemodes-mass}
\end{align}
assuming the normalization
\begin{align}
  \mbox{tr} (Y_{\L' m'} ^\dagger Y_{\L m}) = \delta}  \def\D{\Delta}  \def\ddt{\dot\delta_{\L m}^{\L' m'} .
  \label{harmonics-normaliz}
\end{align}
In contrast to the scalar fields, there are no zero modes, and the
gauge symmetry is broken completely on an irreducible brane.
The lowest of these modes are given by $ Y_{\L m} = \frac{1}{c_N} X_\alpha}          \def\da{{\dot\alpha}$,
and the corresponding KK mass scale is of order
\begin{align}
 m^2_{KK}  \ \sim  \ \frac{r^2}{g^2} m^2
 \label{KK-mass-gauge}
\end{align}
re-inserting the YM coupling constant $g^2$ and the cubic coupling $m^2$  in the potential \eqref{V-soft}.

Now consider the coupling of
the above gauge modes  to the Higgs modes or to the fermions.
We recall from section \ref{sec:regular-zm} that the Higgs modes arise in different types,
in particular  (i) maximal (stringy) modes $\phi_\alpha}          \def\da{{\dot\alpha}^{\rm string}$,
and (ii) semi-classical modes $\phi_\alpha}          \def\da{{\dot\alpha}^{\rm low}$.
A similar classification applies to the gauge modes and fermions.
The effective coupling strength depends strongly on the type of modes.
For the semi-classical gauge modes
(these will include $W$-like bosons discussed in section \ref{sec:standardmodel}), the coupling to
a stringy Higgs mode $\phi^{\rm string} = r_s |x\rangle\langle x'|$
\eqref{string-modes}  has the structure
\begin{align}
 [Y_{\L m},\phi^{\rm string}]  \approx (Y_{\L m}(x) - Y_{\L m}(x')) \phi^{\rm string}
\end{align}
leading to the mass term
\begin{align}
 - \mbox{tr}[\phi^{\rm string},Y_{\L m}][\phi^{\rm string},Y_{\L m}] &\approx
 |Y_{\L m}(x)- Y_{\L m}(x')|^2 \mbox{tr}(\phi^{\rm string}\phi^{\rm string})
  = |Y_{\L m}(x)- Y_{\L m}(x')|^2
\end{align}
Hence such stringy Higgs give a  contribution to the mass of nontrivial gauge boson
modes, which is proportional to the difference of the wave function $Y_{\L m}(x)$
at the two ends of the string.
However this contribution is suppressed\footnote{In contrast, the coupling
of the $\phi^{\rm string}$ to the stringy fermions   is not suppressed.}
by the localization (due to \eqref{harmonics-normaliz}),
and the main contribution to the mass of the gauge bosons arises from
the background  \eqref{gaugemodes-mass}.

These observations will apply in particular for the \emph{chiral} $A_\mu \sim
\chi$ gauge mode \eqref{chiral-gauge-mode} on 6-dimensional branes.

\paragraph{Nonabelian case.}

As usual, unbroken nonabelian gauge fields arise on stacks of coincident branes.
E.g. on a stack of two coinciding branes, the gauge modes have the structure
\begin{align}
 A_\mu = A_{\mu;\L m}(x) Y_{\L m} \otimes\sigma_i
\end{align}
where $\sigma_i \in \mathfrak{s}\mathfrak{u}(2)$ act on the two branes.
Using  $Y_{\L m} = \frac{1}{\sqrt{\dim\cH}}\one$ for
the massless modes, the
corresponding $\mathfrak{s}\mathfrak{u}(2)$ coupling constant is found to be smaller than $g$ by a factor
\begin{align}
 \tilde g\sim\frac{g}{\sqrt{\dim\cH}} \ .
\end{align}
This reduction is somewhat related to \eqref{almost-comm}.

\section{4-dimensional branes}
\label{sec:4D-branes}
After the formal discussion of the algebraic properties of the regular zero
modes, it is now time to show the existence of exact solutions of the form
brane background $\cC[\mu]$ plus some non-trivial Higgs modes. We begin with
the 4-dimensional fuzzy branes $\cC[(N,0)]$ in combination with point branes.
\subsection{Single squashed \texorpdfstring{$\mathds{C} P^2$}{CP2} brane \& Higgs}
\label{sec:single-brane-Higgs}
Now we discuss some of these branes in more detail.
We begin with a squashed brane $\cC[\mu]$ with $\mu = (N,0)$ and add
some (regular) zero mode(s)
\begin{align}
  Y_\alpha}          \def\da{{\dot\alpha} =  X_\alpha}          \def\da{{\dot\alpha} +  \phi_\alpha}          \def\da{{\dot\alpha} ,
\end{align}
where $X_\alpha}          \def\da{{\dot\alpha} = R \pi_{\mu}(T_\alpha}          \def\da{{\dot\alpha})$.
First we treat the massless case $M_i^2 = 0$.
\subsubsection{Minimal brane and instability towards \texorpdfstring{$S^2$}{S2}}
For the minimal branes $\cC[\mu]$ with $\mu = (1,0)$, the only non-trivial
(regular) zero modes are given by
\begin{align}
 \phi_\alpha}          \def\da{{\dot\alpha} = r_\alpha}          \def\da{{\dot\alpha}\pi_{\mu}(T_{-\alpha}          \def\da{{\dot\alpha}}) \ \propto \  X_{-\alpha}          \def\da{{\dot\alpha}}
\end{align}
Since the potential \eqref{V-phi-eff} for $\phi_\alpha}          \def\da{{\dot\alpha}$ is the same as for $X_\alpha}          \def\da{{\dot\alpha}$,
there is a non-trivial minimum at $r_i = \pm 1$, see \eqref{R-solution-equal}.
Then the full matrix configuration is
\begin{align}
Y_\alpha}          \def\da{{\dot\alpha} = X_\alpha}          \def\da{{\dot\alpha} \pm  X_{-\alpha}          \def\da{{\dot\alpha}} = \pm Y_{-\alpha}          \def\da{{\dot\alpha}} \ .
\label{minimal-solutions-S2}
\end{align}
This must be an exact solution according to the discussion in section \ref{sec:interactions}, and indeed
\begin{align}
 [Y_1,Y_2] &=  [X_{\alpha}          \def\da{{\dot\alpha}_1} + X_{-\alpha}          \def\da{{\dot\alpha}_1},X_{\alpha}          \def\da{{\dot\alpha}_2} +  X_{-\alpha}          \def\da{{\dot\alpha}_2}]
  = X_{\alpha}          \def\da{{\dot\alpha}_3} + X_{-\alpha}          \def\da{{\dot\alpha}_3} \
  = Y_3
\end{align}
is nothing but a fuzzy sphere $S^2_n$ with $n=3$, explaining related observations in \cite{Steinacker:2014lma}.
Its energy is
\begin{align}
V[X+\phi] = V[X] + V[\phi] = -16
\end{align}
using \eqref{V-minimalCP2} and \eqref{V-additivity}.
This  appears to be the global  minimum of the classical potential for $N=3$.
We will see below that although squashed $\mathds{C} P^2$ is not the global minimum,
it can be (locally) stabilized by  adding a small positive mass term $M_i^2 > 0$.

\subsubsection{Non-minimal brane with maximal Higgs}
\label{non-mini-brane-Higgs}
The situation is similar, but more interesting for $\cC[\mu]$ with $\mu=(N,0)$.
Then the (regular) zero modes are given by $\phi_{\alpha}          \def\da{{\dot\alpha}}^{(l)} \propto
(X_{-\alpha}          \def\da{{\dot\alpha}})^l$ such that the full matrix configuration looks like
\begin{align}
  Y_\alpha}          \def\da{{\dot\alpha} &=  X_\alpha}          \def\da{{\dot\alpha} + \phi_{\alpha}          \def\da{{\dot\alpha}}^{(l)}.
\end{align}
As before, the $l=1$ mode leads to an exact solution given by an
irreducible fuzzy 2-sphere with (negative) energy equal twice that of $\cC[\mu]$.
This explains how the known fuzzy $S^2$ solutions are related to squashed $\mathds{C} P^2$.

However, there are other, more interesting solutions corresponding to small perturbations of the $\mathds{C} P^2$ branes
localized at their intersections.
For $l \gg 1$, the $\phi_\alpha}          \def\da{{\dot\alpha}^{(l)}$ are string-like modes connecting different sheets of  $\cC[\mu]$,
confined to the vicinity of the origin.
The most interesting ones are the \emph{maximal} zero modes with $l=N$,
denoted by
\begin{align}
  \phi_{\alpha}          \def\da{{\dot\alpha}} &\coloneqq  -r_\alpha}          \def\da{{\dot\alpha}  \frac{1}{N!}{(X_{-\alpha}          \def\da{{\dot\alpha}})}^N \ = \  r_\alpha}          \def\da{{\dot\alpha} \tilde \pi(T_{\alpha}          \def\da{{\dot\alpha}}) , \nonumber\\
  K(\phi_{\alpha}          \def\da{{\dot\alpha}}) &= (N+1)\alpha}          \def\da{{\dot\alpha}
\end{align}
dropping the superscript $N$.
Here $\tilde \pi(T_\alpha}          \def\da{{\dot\alpha})$ is the fundamental representation\footnote{This fixes the normalization $\frac{1}{N!}$} of $\mathfrak{s}\mathfrak{u}(3)$ acting on the 3
extremal weight states of $(N,0)$. In other words, the $\phi_{\alpha}          \def\da{{\dot\alpha}}$ are connecting the 3
coherent states on $\mathds{C} P^2$ located at the origin on each of the 3 sheets of
squashed $\mathds{C} P^2$, cf.\ \cite{Steinacker:2014lma}.
More explicitly, denoting these states
as $\{|1,\mu\rangle,|2,\mu\rangle,|3,\mu\rangle\}$, the zero modes have the form
\begin{align}
  \phi_i^+ &= r_i |i-1, \mu\rangle \langle i,\mu|, \qquad i \sim i+3 \; ,
\end{align}
which are depicted in figure \ref{fig:brane-Higgs-edge}.
Since the potential $V[\phi]$ has the same form as that for the full brane
we obtain a new exact solution for $r_i  = \pm 1$, given by
\begin{align}
 Y^\alpha}          \def\da{{\dot\alpha} = X^\alpha}          \def\da{{\dot\alpha} + \phi^a
\end{align}
with energy $V[X+\phi] = V[X] -8$ (assuming $M_i=0)$.
This follows again from the full equations of motion \eqref{eom-X+Y} and
\eqref{V-minimalCP2},
noting that both $X^\alpha}          \def\da{{\dot\alpha}$ and $\phi^\alpha}          \def\da{{\dot\alpha}$ satisfy the eom and the decoupling condition\footnote{
Note that the $X$ can be viewed as regular zero modes w.r.t.\ the $\phi$,
linking the extremal weight states of $\cH_\mu$  to the remaining weight states of $\cH_\mu$
(which are point branes w.r.t.\ $\phi_\alpha}          \def\da{{\dot\alpha}$), cf.\ section
\ref{sec:brane-point-Higgs}.}.
The signs of the
$r_i$ are of course chosen such that the potential is a minimum.
This solution describes
a Higgs condensate in the $\mathds{C} P^2$ background
localized at the intersections, as sketched in figure
\ref{fig:brane-Higgs-edge}
and \ref{fig:squashedCP2-Higgs}. The general solutions for the combined system
of $\cC[\mu]$ brane background with maximal Higgs are discussed in appendix
\ref{subsec:brane_eom}.
\begin{figure}
\centering
\begin{tikzpicture}
  \coordinate (Origin)   at (0,0);
  \coordinate (P1) at (-2,-1);
  \coordinate (P2) at (0,2);
  \coordinate (P3) at (2,-1);
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
  \draw [thick,red,-latex] (P1) -- (P3) node[midway,below] {$\phi_1^{+}$};
  \draw [thick,red,-latex] (P3) -- (P2) node[midway, right]  {$\phi_2^{+}$};
  \draw [thick,red,-latex] (P2) -- (P1) node[midway, left]  {$\phi_3^{+}$};
\end{tikzpicture}%
\caption{$\cC[\mu]$ for $\mu=(N,0)$ with maximal Higgs $\phi_i^+$. The
indicated dots $\bullet$ are the extremal weights of $(N,0)$, which are
connected by the maximal Higgs $\in \mathrm{End}(\cH_\mu)$.}%
\label{fig:brane-Higgs-edge}%
\end{figure}
Again, even though that this solution is not the global minimum, we
will see in section \ref{sec:masses} that it can be locally stabilized by adding a small mass term $M>0$.

We note that for $1 < l< N$, the zero modes $\phi_\alpha}          \def\da{{\dot\alpha}^{(l)}$
do not satisfy the $\mathfrak{s}\mathfrak{u}(3)$ algebra, and there is no obvious solutions of the
form $X + \phi_\alpha}          \def\da{{\dot\alpha}^{(l)}$ at the non-linear level.

\paragraph{Backreaction and exact solution.}
In general, one should worry about the backreaction of the Higgs $\phi$  on the
background brane $X_\alpha}          \def\da{{\dot\alpha}$.
First, it is important to note that the (almost-) maximal Higgs $\phi$ with $r_\alpha}          \def\da{{\dot\alpha}=\pm 1$ are
a small perturbation\footnote{This is in contrast to the minimal Higgs with $l=1$.} located at the origin of the $\mathds{C} P^2$ background,
so that the backreaction is very small for branes $\cC[\mu]$ with large $\mu=(N,0)$.
To see this,
we compare the matrix elements of $X_\alpha}          \def\da{{\dot\alpha}$ and $\phi_\alpha}          \def\da{{\dot\alpha}$ connecting the states $|i, \mu\rangle$.
For $r_\alpha}          \def\da{{\dot\alpha} = R_\alpha}          \def\da{{\dot\alpha} = \pm 1$,
these are easily see to be
\begin{align}
X_\alpha}          \def\da{{\dot\alpha} |_{|i, \mu\rangle} &=
 \pi_\mu(T_\alpha}          \def\da{{\dot\alpha})|_{|i, \mu\rangle} = O(\sqrt{N})  \quad \gg \quad
 \phi_\alpha}          \def\da{{\dot\alpha} |_{|i, \mu\rangle} =  \tilde\pi_\mu( T_\alpha}          \def\da{{\dot\alpha})|_{|i, \mu\rangle} = O(1)
\,.
\end{align}
Hence, the background generators are larger by a factor of $\sqrt{N}$ and
the backreaction of the background becomes negligible for large $N$.
Moreover, the backreaction vanishes \emph{exactly} for the maximal Higgs modes, as discussed above.
\paragraph{Remark.}
The combined solution $Y^\alpha}          \def\da{{\dot\alpha} = X^\alpha}          \def\da{{\dot\alpha} + \phi^a$ is not only a solution of the potential with cubic term, but also a solution
of the basic $\cN=4$ potential with a negative mass term, since
\begin{align}
 \Box_Y Y = 4 Y
\end{align}
cf. \eqref{eom-X+Y} and \eqref{Box-Sum-zeromodes}.
The point is that $(\Box_X + 2{\slashed{D}}_{ad}^X) \phi_\alpha}          \def\da{{\dot\alpha} = 0$ follows from the
decoupling condition for regular zero modes due to \eqref{fluct-equation},
 and similarly $(\Box_\phi + 2{\slashed{D}}_{ad}^\phi) X_\alpha}          \def\da{{\dot\alpha} = 0$ because $X$ is a regular
zero mode w.r.t.\ $\phi$.  This is quite remarkable, and it means that these
solutions might arise even without the cubic terms in the action, if a
negative mass term arises for these modes by quantum fluctuations.
\subsubsection{Mass terms, stabilization and mass-induced Higgs}
\label{sec:masses}
Now consider a single $\mathds{C} P^2$ brane $(N,0)$ in the presence of a
(sufficiently small) positive mass terms $M_i^2 > 0$.
Then the radii $R_i$ change according to \eqref{eom-general}.
\paragraph{Equal masses $M_i^2 = M^2 < \frac{1}{4}$.}
As a first observation, we note that the squashed brane background is
stabilized
for sufficiently small equal masses $M_i^2 = M^2$, with
\begin{align}
 R_M^2 = 1 - M^2 + O(M^4)
\end{align}
due to \eqref{soln-M-equal}. The quadratic potential for such a background is
\begin{align}
V_{2}[\phi] &= \frac 12R_M^2\, \mbox{tr}\, \phi^\alpha}          \def\da{{\dot\alpha} \Big(\Box_X + \frac 4{R_M^2} M^2
+ 2\slashed{D}_{\rm diag} + 2\slashed{D}_{\rm mix}
 - \frac 4{R_M} \slashed{D}_{\rm mix}\Big) \phi_\alpha}          \def\da{{\dot\alpha} \nonumber\\
 &= \frac 12R_M^2 \, \mbox{tr}\, \phi^\alpha}          \def\da{{\dot\alpha} \Big(\Box_X  + \frac 4{R_M^2}M^2 + 2\slashed{D}_{\rm diag} - 2\slashed{D}_{\rm mix}
 +4 \underbrace{\big(1- \frac 1{R_M}\big)}_{O(M^2)} \slashed{D}_{\rm mix}\Big) \phi_\alpha}          \def\da{{\dot\alpha}
 \label{V2-M}
\end{align}
using \eqref{Dmix-explicit},
where $\Box_X$ and $\slashed{D}_{\ldots}$ are the \emph{same} operators as for
$M^2=0$.
For sufficiently small $M^2$,
the massive modes remain massive, because $\slashed{D}_{\rm mix} $ is clearly
bounded.
For the regular zero modes, we observe that the condition ${\slashed{D}}_{\rm mix}
\phi_\alpha}          \def\da{{\dot\alpha}^{(0)} = 0$ of \eqref{D-mix-vanish} is independent of the $R_i$.
It follows that regular zero modes acquire a positive mass from the
explicit $M^2$ contribution.
For the exceptional zero modes, the above argument does not apply;
for instance, on $\mathds{C} P^2$ the exceptional zero modes are the $SU(3)/U(1)\times
U(1)$  Goldstone bosons, which remain massless even in the presence of
explicit (equal) mass terms $M_i^2=M^2$.
As a consequence, a single $\mathds{C} P^2$ brane is stable in the presence of
small equal masses $M_i^2 = M^2$, up to the flat directions due to Goldstone
bosons.

For further verification, we performed a numerical analysis of the spectrum
of the vector Laplacian
$\cO_V^X$ on a $\cC[(N,0)]$ background for $N=1,\ldots,10$. For details, we
refer to appendix \ref{subsec:brane}, and only summarize the results here.
\begin{itemize}
 \item In the massless case $M_i=0$, we observe $6(N+2)$ zero modes for the
gauge-fixed $\cO_V^X$. These correpond to 6 Goldstone bosons plus $6(N+1)$
regular zero modes, due to \eqref{eq:decomposition_End-CP2}.
\item In the massive case $M_i \equiv M>0$, we observed $6$ zero modes and
precisely $6(N+1)$ massive modes with eigenvalue $4 M^2$. See in particular
figure \ref{fig:CP2-brane_0-modes}.
\end{itemize}
This confirms that small masses are sufficient to stabilize $\mathds{C} P^2$ branes up
to Goldstone bosons.

\paragraph{Different (positive) masses $M_i^2 \neq M_j^2$.}
If the $M_i^2$ are positive but distinct, the situation is more interesting.
The massive modes will stay massive for sufficiently small $M_i^2$, such that
we may focus on the Higgs sector.
However, some of the (would-be) zero modes now acquire a \emph{negative}
mass.
Continuing the computation \eqref{bosonic-zeromodes-ri}, the  adjusted
radii lead to an induced mass originating from the vector Laplacian without
explicit mass terms, i.e.
\begin{align}
 \cO_{V,M_i\equiv 0} \phi_{\alpha}          \def\da{{\dot\alpha},\L'}
  &= [H_R,\phi_{\alpha}          \def\da{{\dot\alpha},\L'}]
  = - \lambda} \def\L{\Lambda} \def\la{\lambda(H_R)\,  \phi_{\alpha}          \def\da{{\dot\alpha},\L'} \ \eqqcolon m_\alpha}          \def\da{{\dot\alpha}^2  \phi_{\alpha}          \def\da{{\dot\alpha},\L'}
 \label{bosonic-zeromodes-m}
\end{align}
with $\lambda} \def\L{\Lambda} \def\la{\lambda$  as in \eqref{zeromodes-weights-Weyl}.
This has one or two negative eigenvalues $m_i$ if the $R_i$ are different, depending on the sign of $ \lambda} \def\L{\Lambda} \def\la{\lambda(H_R)$.
In fact since $ \lambda} \def\L{\Lambda} \def\la{\lambda_1+\lambda} \def\L{\Lambda} \def\la{\lambda_2+\lambda} \def\L{\Lambda} \def\la{\lambda_3 = 0$ (from the $\mathds{Z}_3$ symmetry), we obtain the
following sum rule:
\begin{align}
 m_1^2 + m_2^2 + m_3^2 = 0 \ .
 \label{eq:sum_rule_ind-masses}
\end{align}
Note that $m_i$ depends on $R_i$ resp. $M_i$ through \eqref{eom-general}.
Taking into account the bare masses \eqref{V-full-expand}, the quadratic part of
the potential for $\phi$ is
\begin{align}
 V_2[\phi] &= \frac 12 \sum_{i=1}^3 \mbox{tr} \Big( (m_i^2 + 4M_i^2)
\phi_{-\alpha}          \def\da{{\dot\alpha}_3,-\L'} \phi_{\alpha}          \def\da{{\dot\alpha}_3,\L'} + \text{h.c.} \Big) \,.
\end{align}
If the brane and $\L$ are sufficiently large, the $m_i$ will dominate $M_i$
such that one or two pairs of zero modes acquire a negative mass whenever the
$M_i$ are different.
This is independent of the cubic term in the potential, and it works even positive bare masses $M_i^2>0$.
Therefore at least one pair will definitely get a VEV $\langle
\phi_{\pm\alpha}          \def\da{{\dot\alpha}_i}\rangle \neq 0$, which should in turn lead to a partial stabilization of the zero
modes. Even though this scenario is interesting since it breaks the $\mathds{Z}_3$
\emph{generation} symmetry, we will mostly
focus on the case of equal masses in this paper.

For a $\mathds{C} P^2_N$ brane, the weights of the zero modes are $\lambda} \def\L{\Lambda} \def\la{\lambda_i = k \alpha}          \def\da{{\dot\alpha}_i$ with
$k \leq N$, see \eqref{zeromodes-examples}.
Then switching on one $M_3^2 > 0$ leads to two negative induced mass terms
$m_1^2 = m_2^2 = - O(k M_3^2) < 0$, see for instance
\eqref{mi-induced-oneM}.
\subsubsection{Stability of the brane-Higgs system}
\label{sec:Higgs-stability}

Now consider the non-trivial solutions $X + \phi^{(l)}$
found in section \ref{non-mini-brane-Higgs} involving
maximal zero modes $\phi_{\alpha}          \def\da{{\dot\alpha}}^{(l)}$.
An important question is whether these new solutions
are stable, or if there are further zero modes or instabilities.
Consider the brane $\cC[\mu]$ for $\mu = (N,0)$ with
maximal Higgs solution $\phi_{\alpha}          \def\da{{\dot\alpha}} = c (X_{-a})^N$ as above, and add first
some additional zero modes $\phi_{\alpha}          \def\da{{\dot\alpha}}^{(l)}$ which we assume \emph{not} to be
maximal. Thus
\begin{align}
 X_\alpha}          \def\da{{\dot\alpha} + \phi_\alpha}          \def\da{{\dot\alpha} + \phi_\alpha}          \def\da{{\dot\alpha}, \qquad \phi_\alpha}          \def\da{{\dot\alpha} = \sum c_{l,\alpha}          \def\da{{\dot\alpha}} \phi_{\alpha}          \def\da{{\dot\alpha}}^{(l)} , \qquad  l \leq N .
\end{align}
According to \eqref{V-phi-eff}, the effective potential for the combined
perturbation is given by $V(\phi + \phi)$.
Since $\phi$  has the structure of a squashed $\mathds{C} P^2$ brane,
we know that the masses of $\phi$ arising from
this background are non-negative  \cite{Steinacker:2014lma}.
However, we have to admit the most general fluctuations $\phi_\alpha}          \def\da{{\dot\alpha}$ here, not only
zero modes. Then the linear term in $\phi$ still vanishes identically, because
$X+\phi$ is an exact solution of \eqref{eom-X+Y}.
The quadratic part of the potential for $\phi$ is given by\footnote{We can
assume that $f^2$ is canceled by the gauge fixing term.}
\begin{align}
 V_2^{X+\phi}[\phi]
 = \frac{1}{2}\mbox{tr} \Big(\phi\big(\cO_V^X + \cO_V^\phi + 2[X_\beta}           \def\db{{\dot\beta},[\phi^\beta}           \def\db{{\dot\beta},\cdot]]
+ 2\slashed{D}_{\mathrm{ad}}^{X\phi}\big) \phi \Big)
\end{align}
using $[\phi^\beta}           \def\db{{\dot\beta},X_\beta}           \def\db{{\dot\beta}] = 0$.
In general, it is not clear and, according to numerical studies, also not true
that this is a positive semi-definite bilinear form.
However in the presence of suitable masses $M_i=M$, and notably for $M=M^*$,
it follows using the results in section \ref{sec:Higgs-stab-crit-M} that the
(brane +Higgs) solution is indeed a local minimum up to a compact moduli space.

\paragraph{Numerical results.}
We investigated the stability of $\cC[(N,0)]$ branes with maximal Higgs
numerically by analyzing the spectrum of the
vector Laplacian, see appendix \ref{subsec:brane} for the details.
We considered the massless case $M_i=0$ as well as the massive case $M_i=M$.
To summarize, we find the following for the fluctuations around $(X+\phi)$:
\begin{itemize}
 \item For the $\cC[(1,0)]$ brane with maximal regular zero modes, there are no
negative modes, for both $M=0$ and $M>0$. This is clear because it is a fuzzy
sphere. Moreover, for $M=0$ we find $11$ zero modes, while for $M>0$ there are
$5$ zero modes. This is the expected number of Goldstone bosons for the fuzzy
2-sphere as an $SU(2)$ symmetry is manifest, leaving $8-3=5$ broken generators
for $SU(3)\slash SU(2)$.
 \item For the $\cC[(2,0)]$ brane with maximal regular zero modes, there are no
negative modes. This is very remarkable. In addition, there are $20$ zero modes
for $M=0$ and $8$ zero-modes for $M>0$.
  \item For the $\cC[(N,0)]$ brane with $N\geq 3$ with maximal regular zero
modes, there exist $3(N-2)$ negative modes and $20$ zero modes for $M=0$.
Although the number of negative modes increases with $N$, the amplitude of
the negative eigenvalues decreases,
and it appears that the mass $M$ required to lift them approaches zero for large $N$.
This is exemplified in figure
\ref{fig:neg_ev_n-0_brane} for $N =3,\ldots, 10$. Adding a mass term with
$M\gtrsim 0.2$ lifts all negative modes for $N\geq 3$.
\end{itemize}
Therefore, uniform mass terms $M_i\equiv M$ are sufficient to stabilize the $\mathds{C}
P^2$ brane plus maximal Higgs system up to $8=6+2$ zero modes.
Remarkably, the properties such as number of zero modes, seem to be
independent of the brane size in the massive case $M>0$.
Hence the (brane+Higgs) system has  a well-behaved scaling limit $N\to \infty$,
and is stable up to a compact moduli space (which could be lifted by lifting the degeneracy of the $M_i$).
This is clearly a very interesting result, which will be seen throughout this paper.

\subsection{Single squashed \texorpdfstring{$\mathds{C} P^2$}{CP2} brane with a point
brane}
\label{sec:brane-point-Higgs}
Now consider a $\cC[\mu]$ brane and add a point brane ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}\equiv \cC[0]$.
We will show that there are non-trivial new vacua which involve a non-vanishing
Higgs zero modes linking  $\cC[\mu]$ and  ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$.
We choose $\mu=(N,0)$ to be specific; the discussion for $\mu=(0,N)$ would be
analogous.
\begin{figure}[t!]
\centering
\begin{tikzpicture}
  \coordinate (Origin)   at (0,0);
  \coordinate (P1) at (-4,-2);
  \coordinate (P2) at (0,4);
  \coordinate (P3) at (4,-2);
  \coordinate (P1+a3) at (-4-0.2,-2-0.3);
  \coordinate (P1-a1) at (-4-0.5,-2);
  \coordinate (P2+a2) at (0-0.2,4+0.3);
  \coordinate (P2-a3) at (0+0.2,4+0.3);
  \coordinate (P3+a1) at (4+0.5,-2);
  \coordinate (P3-a2) at (4+0.2,-2-0.3);
  \node at (Origin) {$\bigstar$};
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
  \node at (P1+a3) {$\bullet$};
  \node at (P1-a1) {$\bullet$};
  \node at (P2+a2) {$\bullet$};
  \node at (P2-a3) {$\bullet$};
  \node at (P3+a1) {$\bullet$};
  \node at (P3-a2) {$\bullet$};
  \draw [thick,blue,-latex] (P1+a3) -- (Origin) node[midway,below]
{$\tilde{\varphi}_3^{-}$};%
  \draw [thick,orange,-latex] (P3-a2) -- (Origin) node[midway,below]
{$\varphi_2^{+}$};%
  \draw [thick,green,-latex] (P1-a1) -- (Origin)node[midway,above]
{$\varphi_1^{+}$};%
  \draw [thick,orange,-latex] (P2+a2) -- (Origin) node[midway,left]
{$\tilde{\varphi}_2^{-}$};%
  \draw [thick,green,-latex] (P3+a1) -- (Origin) node[midway,above]
{$\tilde{\varphi}_1^{-}$};%
  \draw [thick,blue,-latex] (P2-a3) -- (Origin) node[midway,right]
{$\varphi_3^{+}$};%
  \draw [thick,red,-latex] (P1) -- (P3) node[midway,below] {$\phi_1^{+}$};
  \draw [thick,red,-latex] (P3) -- (P2) node[midway, right]  {$\phi_2^{+}$};
  \draw [thick,red,-latex] (P2) -- (P1) node[midway, left]  {$\phi_3^{+}$};
\end{tikzpicture}%
\caption{Various regular zero modes for
$\cC[(N,0)]$ brane + point brane
${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$. The $\bigstar$ represents ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$, while $\bullet$ denote the extremal
weights of $(N,0)$. The two shifted two points in the vicinity
of each extremal weight state indicate the polarization of the zero
mode. The $\phi_i^+$ are maximal regular zero modes on $\cC[(N,0)]$, while
$\varphi_i^+$ and $\tilde{\varphi}_i^-$ are  regular zero modes
connecting $\cC[(N,0)]$ and ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$. Note that the conjugate modes,
corresponding to arrows pointing the opposite directions, are not depicted
here.}%
\label{fig:CP2-point}%
\end{figure}
The Higgs modes given by the regular
zero modes discussed above are illustrated in figure \ref{fig:CP2-point}.
The inter-brane regular zero modes, originating from $\mathrm{Hom}(\cH_\mu,\mathds{C})$,
separate into 3+3 independent Higgs $\varphi_i^+$
and \emph{mirror} Higgs $\tilde{\varphi}_i^-$, given by
\begin{align}
 \varphi_i^{+} &= |i\rangle\langle 0| \qquad   \in  \  (N,0)\ , \ \tau=+1
\nonumber\\
 \tilde{\varphi}_i^{-} &= |i-1\rangle\langle 0|  \ \in  \  (N,0)\ , \
\tau=-1 \ .
\end{align}
distinguished by the $\tau$-parity \eqref{tau-parity}.
Note that these determine their conjugate modes $(\varphi_i^{+})^\dagger \sim |0\rangle\langle i|$ etc. living in
$\mathrm{Hom}(\mathds{C},\cH_\mu)$.
Taking into account the maximal intra-brane Higgs $\phi_i^\pm$ on $\cC[\mu]$ as
discussed above,
we find non-trivial solutions where such links $\varphi_i^+$
are switched on with different strength, i.e.\ the
point brane is connected to the $\cC[\mu]$ brane, but the energy is the same.
For example, we can switch on one triangle consisting of two $\varphi_i$,
$\tilde{\varphi}_i$
linking ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$ with two corners of $\cC[\mu]$, connected by a maximal Higgs
$\phi_i$ of $\cC[\mu]$ as in figure \ref{fig:sol_CP2-point}.
The existence of such  solutions is explained by an $SU(2)$ symmetry
within the zero mode sector, rotating the point brane and one
corner. This leads to a moduli space parametrized by some linking angles
$\theta$.

\begin{figure}[t!]
\centering
 \begin{subfigure}{0.32\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (Origin)   at (0,0);
  \coordinate (P1) at (-1,-0.5);
  \coordinate (P2) at (0,1);
  \coordinate (P3) at (1,-0.5);
  \node at (Origin) {$\bigstar$};
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
\draw[thick,red] (P1)--(P2)--(P3)--cycle;
\end{tikzpicture}%
\caption{}%
\label{fig:sol_CP2-point_Higgs}%
\end{subfigure}
 \begin{subfigure}{0.32\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (Origin)   at (0,0);
  \coordinate (P1) at (-1,-0.5);
  \coordinate (P2) at (0,1);
  \coordinate (P3) at (1,-0.5);
  \node at (Origin) {$\bigstar$};
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
\draw[thick,red] (P1)--(Origin)--(P2) -- (P3)-- (P1) --(P2) ;
\end{tikzpicture}%
\caption{}%
\label{fig:sol_CP2-point_interpolate}%
\end{subfigure}
 \begin{subfigure}{0.32\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (Origin)   at (0,0);
  \coordinate (P1) at (-1,-0.5);
  \coordinate (P2) at (0,1);
  \coordinate (P3) at (1,-0.5);
  \node at (Origin) {$\bigstar$};
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
\draw[dotted] (P1)--(P3)--(P2);
\draw[thick,red] (P1)--(Origin)--(P2)--cycle;
\end{tikzpicture}%
\caption{}%
 \label{fig:sol_CP2-point_triangle}%
\end{subfigure}
\caption{One-parameter family of solutions interpolating between the maximal
Higgs solution (\subref{fig:sol_CP2-point_Higgs}) and the connected
configuration (\subref{fig:sol_CP2-point_triangle}) between ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$ and
$\cC[\mu$]. The intermediate states (\subref{fig:sol_CP2-point_interpolate})
have different magnitudes for the  fields involved.}
\label{fig:sol_CP2-point}
\end{figure}

The binding energy for all these configuration is $V[\phi] =-8$.
Numerical investigations indicate that these are indeed the global minima of
this sub sector of the full zero mode sector.
In particular there seems to be no solution which respects the $\mathds{Z}_3$ symmetry,
except for $\varphi = 0$.
In other words, the \emph{generation symmetry} $\mathds{Z}_3$ is spontaneously broken.

This means that although the ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$ brane is connected to $\cC[\mu]$ by some
Higgs as in figure \ref{fig:sol_CP2-point_triangle}, the energy is nevertheless
degenerate
to the case where only the maximal intra-brane Higgs on $\cC[\mu]$ are switched
on, see figure
\ref{fig:sol_CP2-point_Higgs}. This degeneracy is manifest in the existence of the
continuous family of solutions of figure
\ref{fig:sol_CP2-point_interpolate}
that interpolates between these configurations. Hence there is no binding
energy, rather there is a flat
direction in moduli space where the point brane may or may not be attached. The
details of the exact solutions to the equations of motions are presented in
appendix \ref{subsec:brane+point_eom}.

\paragraph{Numerical results.}
We studied numerically the stability of the combined solutions
consisting of $\cC[(N,0)]$ brane plus point brane ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$ together with some
Higgs modes, with respect to arbitrary fluctuations.
The details are provided in appendix \ref{subsec:brane+point_spectrum}.
It turns out that for $M=0$ there are typically a number of negative modes, i.e.\ they are unstable
towards some of the (originally massive) deformation modes. This is similar to
the situation in section \ref{sec:Higgs-stability}.
Again, these instabilities can be stabilized by adding a (small) mass term $M$ to
the background, see in particular figure \ref{fig:brane+pt_spectrum}.
Then the above solutions, comprising the $\cC[\mu]$ brane plus
point brane and several Higgs, are still exact solutions with adjusted radii
according to \eqref{eom-general}.
Here, we
observe different qualitative behavior
between the configurations of figure \ref{fig:sol_CP2-point_Higgs} and
\ref{fig:sol_CP2-point_interpolate}, \ref{fig:sol_CP2-point_triangle}.
\begin{itemize}
 \item The triangle of maximal intra-brane Higgs of figure
\ref{fig:sol_CP2-point_Higgs} can be stabilized with small masses, consistent with
the results in section \ref{sec:Higgs-stability}.
The number of zero modes becomes $14$ in the massive case, independent of the brane
size.
  \item The interpolating state of figure \ref{fig:sol_CP2-point_interpolate}
can only be stabilized with masses almost saturating the allowed values (and
only for large enough branes $N\geq5$). In addition, the number of zero-modes
in the massive case becomes $9$, independent of the brane size.
  \item The connected configuration of figure \ref{fig:sol_CP2-point_triangle}
can also be stabilized only with relatively large mass values, but negativity
of the eigenvalues is qualitatively different to the interpolating case. Again,
the number of zero modes stabilizes at $9$ in the massive case.
\end{itemize}

Consistent with the general results in section \ref{sec:Higgs-stab-crit-M},
we observe that there are no instabilities for the critical mass $M^*$.
The number of zero modes exceeds the expected 6
zero modes corresponding to the 6 Goldstone bosons of the broken $SU(3)_R$.
One may hope that these remaining zero modes are
lifted by introducing different masses $M_i$, but we did not verify this
explicitly.
\subsection{Two squashed \texorpdfstring{$\mathds{C} P^2$}{CP2} branes \& Higgs}
\label{sec:two-branes-Higgs}
Now consider a system of two non-identical $\mathds{C} P^2$ branes, such as  $\cC[\mu_L]$ and
$\cC[\mu_R]$.
Then
\begin{equation}
\label{eq:algebra_two_branes}
\begin{aligned}
 \mathrm{End}(\cH_L\oplus\cH_R) &= {\cal A}} \def\cB{{\cal B}} \def\cC{{\cal C}_{LL} \oplus {\cal A}} \def\cB{{\cal B}} \def\cC{{\cal C}_{RR} \oplus {\cal A}} \def\cB{{\cal B}} \def\cC{{\cal C}_{LR}
\oplus{\cal A}} \def\cB{{\cal B}} \def\cC{{\cal C}_{RL} \, ,\\
{\cal A}} \def\cB{{\cal B}} \def\cC{{\cal C}_{LL}&= \mathrm{End}(\cH_{\mu_L}) \, , \quad {\cal A}} \def\cB{{\cal B}} \def\cC{{\cal C}_{LR}= \mathrm{Hom}(\cH_{\mu_L},\cH_{\mu_R})
\, , \quad \text{etc.}
\end{aligned}
\end{equation}
contains various types of zero modes.
Besides the intra-brane zero modes discussed above, there are additional modes
$\phi_{LR} \in {\cal A}} \def\cB{{\cal B}} \def\cC{{\cal C}_{LR}$ etc. connecting $\cC[\mu_L]$ with  $\cC[\mu_R]$;
those are the most interesting ones as we will see. We will explicitly find such
solutions.

Consider the case of parallel branes $\cC[\mu_L]$ and $\cC[\mu_R]$ with
$\mu_L=(N,0)$ and $\mu_R = (l,0)$,
possibly connected by some Higgs, as in figure \ref{fig:Brane+brane}.
This set-up is interesting because we will find maximal Higgs connecting
different branes, which are exact solutions at the non-linear level. These Higgs
clearly breaks the $U(1)\times U(1)$ gauge symmetry on the two branes down to the diagonal $U(1)$,
and lead to various Yukawa couplings of the fermionic zero modes.
This is the mechanism we are interested in, even though  the present background
may not yet be very interesting physically.
Note that $n$ identical branes would lead to $\mathfrak{u}(n)$-valued fields.
\begin{figure}[t!]
\centering
\begin{tikzpicture}
  \coordinate (Q1) at (-1,-0.5);
  \coordinate (Q2) at (0,1);
  \coordinate (Q3) at (1,-0.5);
  \coordinate (P1) at (-4,-2);
  \coordinate (P2) at (0,4);
  \coordinate (P3) at (4,-2);
  \coordinate (P1+a3) at (-4-0.2,-2-0.3);
  \coordinate (P1-a1) at (-4-0.5,-2);
  \coordinate (P2+a2) at (0-0.2,4+0.3);
  \coordinate (P2-a3) at (0+0.2,4+0.3);
  \coordinate (P3+a1) at (4+0.5,-2);
  \coordinate (P3-a2) at (4+0.2,-2-0.3);
  \node at (Q1) {$\bullet$};
  \node at (Q2) {$\bullet$};
  \node at (Q3) {$\bullet$};
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
  \node at (P1+a3) {$\bullet$};
  \node at (P1-a1) {$\bullet$};
  \node at (P2+a2) {$\bullet$};
  \node at (P2-a3) {$\bullet$};
  \node at (P3+a1) {$\bullet$};
  \node at (P3-a2) {$\bullet$};
  \draw [thick,green,-latex] (P1+a3) -- (Q3) node[midway,below]
{$\varphi_1^{+}$};%
  \draw [thick,green,-latex] (P3-a2) -- (Q1) node[midway,below]
{$\tilde{\varphi}_1^{-}$};%
  \draw [thick,blue,-latex] (P1-a1) -- (Q2) node[midway,left]
{$\tilde{\varphi}_3^{-}$};%
  \draw [thick,blue,-latex] (P2+a2) -- (Q1) node[midway,left]
{$\varphi_{3+}$};%
  \draw [thick,orange,-latex] (P3+a1) -- (Q2) node[midway,right]
{$\varphi_2^{+}$};%
  \draw [thick,orange,-latex] (P2-a3) -- (Q3) node[midway,right]
{$\tilde{\varphi}_2^{-}$};%
  \draw [thick,red,-latex] (P1) -- (P3) node[midway,below] {$\phi_1^{+}$};
  \draw [thick,red,-latex] (P3) -- (P2) node[midway, right]  {$\phi_2^{+}$};
  \draw [thick,red,-latex] (P2) -- (P1) node[midway, left]  {$\phi_3^{+}$};
  \draw [thick,-latex] (Q1) -- (Q3) node[midway,below] {$\phi_1^{+}$};
  \draw [thick,-latex] (Q3) -- (Q2) node[midway, left]  {$\phi_2^{+}$};
  \draw [thick,-latex] (Q2) -- (Q1) node[midway, below]  {$\phi_3^{+}$};
\end{tikzpicture}%
\caption{Parallel branes $\cC[\mu_R]$ and $\cC[\mu_L]$, with
$\cC[\mu_L]$ being the outer brane. We only indicate the polarization for the
outermost brane by two additional $\bullet$ next to each extremal weight state.
The maximal intra-brane Higgs are denoted as
$\phi_i^+$ and $\phi_i^+$ for $\cC[\mu_L]$ and $\cC[\mu_R]$, respectively. The
maximal regular inter-brane zero modes are labeled by $\varphi_i^+$ and
$\tilde{\varphi}_i^-$, and we only draw the ${\cal A}} \def\cB{{\cal B}} \def\cC{{\cal C}_{LR}$ sector here.}%
\label{fig:Brane+brane}%
\end{figure}

Besides the algebras of functions  ${\cal A}} \def\cB{{\cal B}} \def\cC{{\cal C}_{LL}$ and ${\cal A}} \def\cB{{\cal B}} \def\cC{{\cal C}_{RR}$  on $\cC[\mu_L]$
and $\cC[\mu_R]$, respectively, the full
algebra of functions contains the following intertwining part:
\begin{align}
 {\cal A}} \def\cB{{\cal B}} \def\cC{{\cal C}_{LR} =  (N,l) \oplus (N-1,l-1)\oplus  \ldots  \oplus (N-l,0)
\end{align}
and similarly ${\cal A}} \def\cB{{\cal B}} \def\cC{{\cal C}_{RL}$.
There is again a special class of $3+3$ \emph{maximal} Higgs  modes  arising
from the extremal states
of $(N,l)$, denoted by
\begin{align}
 \tilde{\varphi}_{i}^- &\in \ (N,l) \subset {\cal A}} \def\cB{{\cal B}} \def\cC{{\cal C}_{LR} \qquad \mbox{mirror
Higgs} \nonumber\\
 \varphi_{i}^+  &\in \ (N,l) \subset {\cal A}} \def\cB{{\cal B}} \def\cC{{\cal C}_{LR} \qquad \mbox{chiral Higgs}
\end{align}
which connect the corners of $\mu_L$ and $\mu_R$. This is displayed in
figure \ref{fig:Brane+brane}. Their conjugate modes are given by
\begin{align}
 (\tilde{\varphi}_{i}^-)^\dagger = \tilde{\varphi}_{i}^+ \in    {\cal A}} \def\cB{{\cal B}} \def\cC{{\cal C}_{RL}
 \quad \text{and} \quad
 (\varphi_{i}^+)^\dagger = \varphi_{i}^- \in    {\cal A}} \def\cB{{\cal B}} \def\cC{{\cal C}_{RL} \; .
\end{align}
We will see below that each of these Higgs $\tilde{\varphi}_{i}^-$,
$\varphi_{i}^+$  give rise to precisely one Yukawa coupling
between fermionic zero modes linking a point brane with either of the branes.

Additionally, there exist the maximal intra-brane Higgs $\phi_i^\pm \in
{\cal A}} \def\cB{{\cal B}} \def\cC{{\cal C}_{LL}$ and
$\phi_{i}^\pm \in {\cal A}} \def\cB{{\cal B}} \def\cC{{\cal C}_{RR}$ along some edge of $\cC[\mu_L]$ and $\cC[\mu_R]$,
respectively.
As summarized in figure \ref{fig:Brane+brane}, altogether we have the
following
maximal Higgs modes
\begin{align}
 \tilde{\varphi}_{i}^- = (\tilde{\varphi}_{i}^+)^\dagger, \quad
 \varphi_{i}^+ = (\varphi_{i}^-)^\dagger , \quad
 \phi_i^+ = (\phi_i^-)^\dagger , \quad
 \phi_i^+ = (\phi_i^-)^\dagger
\end{align}
which connect the extremal weights and therefore form a closed
algebra.
Our aim is to find stable non-trivial solutions on top of $\cC[\mu_L]$ and
$\cC[\mu_R]$, where some of these are switched on.
\paragraph{Solutions.}
It is clear that there are non-trivial solutions within the maximal Higgs
sector involving one closed triangle and we have exemplified such cases in
figure \ref{fig:2branes_1triangle}.
There exist several continuously parametrized solutions that
interpolate between a maximal intra-brane Higgs on one brane and a closed
triangle between the two branes. We refer to figures
\ref{fig:2branes_1triangle_smallMaxHiggs}--\ref{%
fig:2branes_1triangle_smallBound} and
figures \ref{fig:2branes_1triangle_smallMaxHiggs}--\ref{%
fig:2branes_1triangle_smallBound} for two representative cases.

In addition, as shown in figure \ref{fig:2branes_2triangle_maxHiggs}, there
exists an exact solution of the
form $X + \phi +\phi$ involving the full brane background plus maximal
intra-brane Higgs on $\cC[\mu_L]$ and $\cC[\mu_R]$ simultaneously.
Moreover, one can non-trivially combine the triangular configurations of
\ref{fig:2branes_1triangle_smallBound} and \ref{fig:2branes_1triangle_bigBound}
and obtains another continuous family, see
figure \ref{fig:2branes_2triangle}, which always corresponds to a configuration
of
two closed triangles.

The details of the numerical combinations of the
several Higgs fields that give rise to exact solutions to the equations of
motions are presented in appendix \ref{subsec:brane+brane}.
As far as the potential energy is concerned,
the (degenerate) configurations of figure \ref{fig:2branes_1triangle} count only as one closed
triangle, whereas
the (degenerate) configurations of figure \ref{fig:2branes_2triangle} have two
independent triangles, and consequently have lower potential energy.
That being said and recalling the $\mathds{Z}_3$ symmetry, it is clear that the states of lowest
energy, i.e.\ those equivalent to two closed triangles in figure \ref{fig:2branes_2triangle}, are
highly degenerate. These solutions should be interpreted as 2 branes linked by some Higgs,
but the binding energy is again zero due to the degeneracy.

\begin{figure}[t!]
\centering
\begin{subfigure}{0.32\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (Q1) at (-0.5,-0.25);
  \coordinate (Q2) at (0,0.5);
  \coordinate (Q3) at (0.5,-0.25);
  \coordinate (P1) at (-2,-1);
  \coordinate (P2) at (0,2);
  \coordinate (P3) at (2,-1);
  \node at (Q1) {$\bullet$};
  \node at (Q2) {$\bullet$};
  \node at (Q3) {$\bullet$};
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
\draw[red,thick] (Q1)--(Q2)--(Q3)--cycle;
\draw[dotted] (P1)--(P2)--(P3)--cycle;
\end{tikzpicture}%
\caption{}%
\label{fig:2branes_1triangle_smallMaxHiggs}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (Q1) at (-0.5,-0.25);
  \coordinate (Q2) at (0,0.5);
  \coordinate (Q3) at (0.5,-0.25);
  \coordinate (P1) at (-2,-1);
  \coordinate (P2) at (0,2);
  \coordinate (P3) at (2,-1);
  \node at (Q1) {$\bullet$};
  \node at (Q2) {$\bullet$};
  \node at (Q3) {$\bullet$};
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
\draw[blue,thick] (Q1)--(P2)--(Q3);
\draw[red,thick] (Q1)--(Q2)--(Q3);
\draw[magenta,thick] (Q1)--(Q3);
\draw[dotted] (P2)--(P1)--(P3) --cycle;
\end{tikzpicture}%
\caption{}%
\label{fig:2branes_1triangle_smallInt}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (Q1) at (-0.5,-0.25);
  \coordinate (Q2) at (0,0.5);
  \coordinate (Q3) at (0.5,-0.25);
  \coordinate (P1) at (-2,-1);
  \coordinate (P2) at (0,2);
  \coordinate (P3) at (2,-1);
  \node at (Q1) {$\bullet$};
  \node at (Q2) {$\bullet$};
  \node at (Q3) {$\bullet$};
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
\draw[blue,thick] (Q1)--(P2)--(Q3)--cycle;
\draw[dotted] (Q1)--(Q2)--(Q3);
\draw[dotted] (P2)--(P1)--(P3) --cycle;
\end{tikzpicture}%
\caption{}%
\label{fig:2branes_1triangle_smallBound}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (Q1) at (-0.5,-0.25);
  \coordinate (Q2) at (0,0.5);
  \coordinate (Q3) at (0.5,-0.25);
  \coordinate (P1) at (-2,-1);
  \coordinate (P2) at (0,2);
  \coordinate (P3) at (2,-1);
  \node at (Q1) {$\bullet$};
  \node at (Q2) {$\bullet$};
  \node at (Q3) {$\bullet$};
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
\draw[red,thick] (P1)--(P2)--(P3)--cycle;
\draw[dotted] (Q2)--(Q1)--(Q3) --cycle;
\end{tikzpicture}%
\caption{}%
\label{fig:2branes_1triangle_bigMaxHiggs}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (Q1) at (-0.5,-0.25);
  \coordinate (Q2) at (0,0.5);
  \coordinate (Q3) at (0.5,-0.25);
  \coordinate (P1) at (-2,-1);
  \coordinate (P2) at (0,2);
  \coordinate (P3) at (2,-1);
  \node at (Q1) {$\bullet$};
  \node at (Q2) {$\bullet$};
  \node at (Q3) {$\bullet$};
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
\draw[red,thick] (P1)--(P2)--(P3);
\draw[blue,thick] (P1)--(Q2)--(P3);
\draw[magenta,thick] (P1)--(P3);
\draw[dotted] (Q2)--(Q1)--(Q3) --cycle;
\end{tikzpicture}%
\caption{}%
\label{fig:2branes_1triangle_bigInt}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (Q1) at (-0.5,-0.25);
  \coordinate (Q2) at (0,0.5);
  \coordinate (Q3) at (0.5,-0.25);
  \coordinate (P1) at (-2,-1);
  \coordinate (P2) at (0,2);
  \coordinate (P3) at (2,-1);
  \node at (Q1) {$\bullet$};
  \node at (Q2) {$\bullet$};
  \node at (Q3) {$\bullet$};
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
\draw[blue,thick] (P1)--(Q2)--(P3)--cycle;
\draw[dotted] (Q2)--(Q1)--(Q3) --cycle;
\draw[dotted] (P1)--(P2)--(P3);
\end{tikzpicture}%
\caption{}%
\label{fig:2branes_1triangle_bigBound}
\end{subfigure}
\caption{$\cC[\mu_R]$ and $\cC[\mu_L]$ brane plus intra and inter-brane
Higgs. Here, we display example solutions involving only one triangular
configuration. There exists a continuously parametrized solution interpolating
between (\subref{fig:2branes_1triangle_smallMaxHiggs}) and
(\subref{fig:2branes_1triangle_smallBound}), while all intermediate states can
be
thought of as in (\subref{fig:2branes_1triangle_smallInt}). In addition, there
exists
another continuously parametrized solution interpolating between
(\subref{fig:2branes_1triangle_bigMaxHiggs}) and
(\subref{fig:2branes_1triangle_bigBound}), while all intermediate states look
like
(\subref{fig:2branes_1triangle_bigInt}). Legs depicted with the same color
have
the same amplitude. In particular, the intermediate states have five legs with
three different, but related amplitudes.}
\label{fig:2branes_1triangle}
\end{figure}
\begin{figure}[t!]
\centering
\begin{subfigure}{0.32\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (Q1) at (-0.5,-0.25);
  \coordinate (Q2) at (0,0.5);
  \coordinate (Q3) at (0.5,-0.25);
  \coordinate (P1) at (-2,-1);
  \coordinate (P2) at (0,2);
  \coordinate (P3) at (2,-1);
  \node at (Q1) {$\bullet$};
  \node at (Q2) {$\bullet$};
  \node at (Q3) {$\bullet$};
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
\draw[red,thick] (P1)--(P2)--(P3)--cycle;
\draw[red,thick] (Q2)--(Q1)--(Q3) --cycle;
\end{tikzpicture}%
\caption{}%
\label{fig:2branes_2triangle_maxHiggs}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (Q1) at (-0.5,-0.25);
  \coordinate (Q2) at (0,0.5);
  \coordinate (Q3) at (0.5,-0.25);
  \coordinate (P1) at (-2,-1);
  \coordinate (P2) at (0,2);
  \coordinate (P3) at (2,-1);
  \node at (Q1) {$\bullet$};
  \node at (Q2) {$\bullet$};
  \node at (Q3) {$\bullet$};
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
\draw[red,thick] (P1)--(P2)--(P3);
\draw[red,thick] (Q1)--(Q2)--(Q3);
\draw[blue,thick] (P1)--(Q2)--(P3);
\draw[blue,thick] (Q1)--(P2)--(Q3);
\draw[magenta,thick] (P1)--(P3);
\draw[magenta,thick] (Q1)--(Q3);
\end{tikzpicture}%
\caption{}%
\label{fig:2branes_2triangle_intermed}
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (Q1) at (-0.5,-0.25);
  \coordinate (Q2) at (0,0.5);
  \coordinate (Q3) at (0.5,-0.25);
  \coordinate (P1) at (-2,-1);
  \coordinate (P2) at (0,2);
  \coordinate (P3) at (2,-1);
  \node at (Q1) {$\bullet$};
  \node at (Q2) {$\bullet$};
  \node at (Q3) {$\bullet$};
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
\draw[blue,thick] (P1)--(Q2)--(P3)--cycle;
\draw[blue,thick] (Q1)--(P2)--(Q3)--cycle;
\draw[dotted] (Q1)--(Q2)--(Q3) ;
\draw[dotted] (P1)--(P2)--(P3);
\end{tikzpicture}%
\caption{}%
\label{fig:2branes_2triangle_bound}
\end{subfigure}
\caption{$\cC[\mu_R]$ and $\cC[\mu_L]$ brane plus intra and inter-brane
Higgs. Here, we display example solutions involving only two triangular
configuration. There exists a continuously parametrized solution interpolating
between (\subref{fig:2branes_2triangle_maxHiggs}) and
(\subref{fig:2branes_2triangle_bound}), while all intermediate states can be
thought of as in (\subref{fig:2branes_2triangle_intermed}).As before legs
depicted with the same color have the same amplitude.}
\label{fig:2branes_2triangle}
\end{figure}
\paragraph{Spectrum.}
We have addressed questions about the spectrum of the vector Laplacian around
such new background in appendix \ref{subsec:brane+brane}. Although the spectrum
of $\cO_V^X$ in the background $\cC[\mu_L]+\cC[\mu_L]$ is known to be free of
negative modes, in the novel backgrounds $\cO_V^{X+\phi}$ has a
number of negative modes, see for instance figure \ref{fig:2branes_spectrum}.
Turning on uniform masses $M_i \equiv M$  has two effects on the spectrum:
First, the number of negative modes decreases and all of these modes can be
lifted consistently for mass $0.47 \lesssim M \leq \frac{\sqrt{2}}{3}$, and in particular
for $M^*$. This
exemplifies the general results of section \ref{sec:massive_potential}.
Second, the number of zero modes stabilizes again to levels that are independent of
the size of the system.
\subsection{Two squashed \texorpdfstring{$\mathds{C} P^2$}{CP2} branes with a point
brane \& Higgs}
\label{sec:two-branes-point-Higgs}
Now consider again two parallel branes  $\cC[\mu_L]$ and $\cC[\mu_R]$ with
$\mu_L=(N_1,0)$ and $\mu_R = (N_2,0)$, and a point brane ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F} \cong \cC[0]$.
The full algebra of functions $\mathrm{End}(\cH_{\mu_L}\oplus \cH_{\mu_R} \oplus \mathds{C})$
contains \ref{eq:algebra_two_branes}, but also exhibits additional intra-brane
zero modes originating from ${\cal A}} \def\cB{{\cal B}} \def\cC{{\cal C}_{L0}=\mathrm{Hom}(\cH_{\mu_L},\mathds{C})$,
${\cal A}} \def\cB{{\cal B}} \def\cC{{\cal C}_{R0}=\mathrm{Hom}(\cH_{\mu_R},\mathds{C})$ and their conjugates.
The corresponding regular zero modes are labeled  $\varphi_i^+,
\tilde{\varphi}_i^- \in {\cal A}} \def\cB{{\cal B}} \def\cC{{\cal C}_{L0}$ and $\sigma_i^+,\tilde{\sigma}_i^- \in
{\cal A}} \def\cB{{\cal B}} \def\cC{{\cal C}_{R0}$, respectively.
In addition, there exist Higgs modes $H_i^+$, $\tilde{H}_i^-$ linking
$\cC[\mu_L]$ and $\cC[\mu_R]$. The collection of maximal regular zero modes
(Higgs fields)
is summarized in figure \ref{fig:Brane+brane+point}. As usual, the conjugate
fields are
\begin{equation*}
 \varphi_i^- = (\varphi_i^+)^\dagger , \quad
 \tilde{\varphi}_i^+ = (\tilde{\varphi}_i^-)^\dagger , \quad
 \sigma_i^- = (\sigma_i^+)^\dagger , \quad
 \tilde{\sigma}_i^+ = (\tilde{\sigma}_i^-)^\dagger, \quad
 H_i^- = (H_i^+)^\dagger , \quad
 \tilde{H}_i^+ = (\tilde{H}_i^-)^\dagger
\end{equation*}
and correspond to arrows in the opposite directions.
\begin{figure}[t!]
\centering
\begin{tikzpicture}
  \coordinate (Origin) at (0,0);
  \coordinate (Q1) at (-3,-1);
  \coordinate (Q2) at (0,3);
  \coordinate (Q3) at (3,-1);
  \coordinate (P1) at (-6,-3);
  \coordinate (P2) at (0,6);
  \coordinate (P3) at (6,-3);
  \coordinate (P1+a3) at (-6-0.2,-3-0.3);
  \coordinate (P1-a1) at (-6-0.5,-3);
  \coordinate (P2+a2) at (0-0.2,6+0.3);
  \coordinate (P2-a3) at (0+0.2,6+0.3);
  \coordinate (P3+a1) at (6+0.5,-3);
  \coordinate (P3-a2) at (6+0.2,-3-0.3);
  \node at (Origin) {$\bigstar$};
  \node at (Q1) {$\bullet$};
  \node at (Q2) {$\bullet$};
  \node at (Q3) {$\bullet$};
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
  \node at (P1+a3) {$\bullet$};
  \node at (P1-a1) {$\bullet$};
  \node at (P2+a2) {$\bullet$};
  \node at (P2-a3) {$\bullet$};
  \node at (P3+a1) {$\bullet$};
  \node at (P3-a2) {$\bullet$};
  \draw [thick,green,-latex] (P1+a3) -- (Q3) node[midway,below]
{$H_1^+$};%
  \draw [thick,green,-latex] (P3-a2) -- (Q1) node[midway,below]
{$\tilde{H}_1^- $};%
  \draw [thick,blue,-latex] (P1-a1) -- (Q2) node[midway,left]
{$\tilde{H}_3^-$};%
  \draw [thick,blue,-latex] (P2+a2) -- (Q1) node[midway,left]
{$H_3^+$};%
  \draw [thick,orange,-latex] (P3+a1) -- (Q2) node[midway,right]
{$H_2^+$};%
  \draw [thick,orange,-latex] (P2-a3) -- (Q3) node[midway,right]
{$\tilde{H}_2^-$};%
  \draw [dashed,green,-latex] (P1+a3) -- (Origin) node[midway,below]
{$\varphi_1^{+}$};%
  \draw [dashed,green,-latex] (P3-a2) -- (Origin) node[midway,below]
{$\tilde{\varphi}_1^{-}$};%
  \draw [dashed,blue,-latex] (P1-a1) -- (Origin) node[midway,left]
{$\tilde{\varphi}_3^{-}$};%
  \draw [dashed,blue,-latex] (P2+a2) -- (Origin) node[midway,left]
{$\varphi_3^{+}$};%
  \draw [dashed,orange,-latex] (P3+a1) -- (Origin) node[midway,right]
{$\varphi_2^{+}$};%
  \draw [dashed,orange,-latex] (P2-a3) -- (Origin) node[midway,right]
{$\tilde{\varphi}_2^{-}$};%
  \draw [dotted,green,-latex] (Q1) -- (Origin) node[midway,below]
{$\sigma_1^{+}$};%
  \draw [dotted,green,-latex] (Q3) -- (Origin) node[midway,below]
{$\tilde{\sigma}_1^{-}$};%
  \draw [dotted,blue,-latex] (Q1) -- (Origin) node[midway,left]
{$\tilde{\sigma}_3^{-}$};%
  \draw [dotted,blue,-latex] (Q2) -- (Origin) node[midway,left]
{$\sigma_3^{+}$};%
  \draw [dotted,orange,-latex] (Q3) -- (Origin) node[midway,right]
{$\sigma_2^{+}$};%
  \draw [dotted,orange,-latex] (Q2) -- (Origin) node[midway,right]
{$\tilde{\sigma}_2^{-}$};%
  \draw [thick,red,-latex] (P1) -- (P3) node[midway,below] {$\phi_1^{+}$};
  \draw [thick,red,-latex] (P3) -- (P2) node[midway, right]  {$\phi_2^{+}$};
  \draw [thick,red,-latex] (P2) -- (P1) node[midway, left]  {$\phi_3^{+}$};
  \draw [thick,-latex] (Q1) -- (Q3) node[midway,below] {$\phi_1^{+}$};
  \draw [thick,-latex] (Q3) -- (Q2) node[midway, left]  {$\phi_2^{+}$};
  \draw [thick,-latex] (Q2) -- (Q1) node[midway, right]  {$\phi_3^{+}$};
\end{tikzpicture}%
\caption{The various Higgs fields for the background of $\cC[\mu_L]$ and
$\cC[\mu_R]$ brane together with a point brane ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$. Here, we picture
$\cC[\mu_L]$ as the outer most brane and place the point brane in the center.}%
\label{fig:Brane+brane+point}%
\end{figure}%
\paragraph{Solutions.}

In this set-up, there is a novel type of Higgs solution which involve only links between different
branes as indicated in figure \ref{fig:two-branes-Higgs-point-chiral}, due to
cubic terms like $\sigma_i^+ H_j^+  \tilde{\varphi}_k^+$  and $\varphi_i^+ \tilde{H}_j^+ \tilde{\sigma}_k^+$.
This leads to a potentially interesting structure of Yukawa couplings and chiral fermions, as discussed in sections
\ref{sec:fermions} and \ref{sec:standardmodel}.
The two different types of such triangles have opposite ``orientation``.
\begin{figure}[t!]
\centering
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (Origin) at (0,0);
  \coordinate (Q1) at (-1.5,-0.5);
  \coordinate (Q2) at (0,1.5);
  \coordinate (Q3) at (1.5,-0.5);
  \coordinate (P1) at (-3,-1.5);
  \coordinate (P2) at (0,3);
  \coordinate (P3) at (3,-1.5);
  \node at (Origin) {$\bigstar$};
  \node at (Q1) {$\bullet$};
  \node at (Q2) {$\bullet$};
  \node at (Q3) {$\bullet$};
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
\draw[red,thick] (P3)--(Q1)--(Origin)--cycle;
\draw[dotted] (Q1)--(Q2)--(Q3)--cycle;
\draw[dotted] (P1)--(P2)--(P3)--cycle;
\end{tikzpicture}%
\caption{$\tilde{H}_1^+ \ \varphi_2^+ \ \tilde{\sigma}_3^+$}
\label{fig:2branes+point_chiral-Higgs_1}
\end{subfigure}%
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (Origin) at (0,0);
  \coordinate (Q1) at (-1.5,-0.5);
  \coordinate (Q2) at (0,1.5);
  \coordinate (Q3) at (1.5,-0.5);
  \coordinate (P1) at (-3,-1.5);
  \coordinate (P2) at (0,3);
  \coordinate (P3) at (3,-1.5);
  \node at (Origin) {$\bigstar$};
  \node at (Q1) {$\bullet$};
  \node at (Q2) {$\bullet$};
  \node at (Q3) {$\bullet$};
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
\draw[red,thick] (P1)--(Q3)--(Origin)--cycle;
\draw[dotted] (Q1)--(Q2)--(Q3)--cycle;
\draw[dotted] (P1)--(P2)--(P3)--cycle;
\end{tikzpicture}%
\caption{$H_1^+ \ \sigma_2^+ \ \tilde{\varphi}_3^+$}
\label{fig:2branes+point_chiral-Higgs_2}
\end{subfigure}%
\caption{Higgs configuration with non-vanishing cubic potential.}
\label{fig:two-branes-Higgs-point-chiral}
\end{figure}
As further elaborated in appendix \ref{subsec:brane+brane+point}, one can find
various one-parameter solutions that interpolate between the configurations in
figure \ref{fig:two-branes-Higgs-point-chiral} and other solutions with one
closed triangle. We summarize these in figure
\ref{fig:2branes_1point_1triangle}.
Starting from \ref{fig:2branes+point_chiral-Higgs_1} there exist at least three
continuous families of exact solutions which are displayed in
\ref{fig:2branes_1point_1triangle_1}--\ref{fig:2branes_1point_1triangle_3},
\ref{fig:2branes_1point_1triangle_4}--\ref{fig:2branes_1point_1triangle_6},
and
\ref{fig:2branes_1point_1triangle_7}--\ref{fig:2branes_1point_1triangle_9}.
Due to our knowledge of the two parallel brane case, see figure
\ref{fig:2branes_1triangle}, it follows that configuration
\ref{fig:2branes_1point_1triangle_9}
can be deformed into to the maximal intra-brane Higgs configuration
$\phi_i^+$ on $\cC[\mu_R]$. Moreover, inspecting figure \ref{fig:sol_CP2-point}
reveals that configurations
\ref{fig:2branes_1point_1triangle_3} and \ref{fig:2branes_1point_1triangle_6}
can be deformed to closed maximal intra-brane Higgs triangles on $\cC[\mu_L]$ and
$\cC[\mu_R]$, respectively. They all have the same energy.

Repeating the analogous analysis for \ref{fig:2branes+point_chiral-Higgs_2},
one concludes that all closed one triangle solutions are connected by
continuous deformations, which all have the same  energy.

\begin{figure}[t!]
\centering
\begin{subfigure}{0.33\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (Origin) at (0,0);
  \coordinate (Q1) at (-0.8,-0.4);
  \coordinate (Q2) at (0,1);
  \coordinate (Q3) at (0.8,-0.4);
  \coordinate (P1) at (-2,-1);
  \coordinate (P2) at (0,2);
  \coordinate (P3) at (2,-1);
  \node at (Origin) {$\bigstar$};
  \node at (Q1) {$\bullet$};
  \node at (Q2) {$\bullet$};
  \node at (Q3) {$\bullet$};
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
\draw[red,thick] (P3)--(Q1)--(Origin)--cycle;
\draw[dotted] (Q1)--(Q2)--(Q3)--cycle;
\draw[dotted] (P1)--(P2)--(P3)--cycle;
\end{tikzpicture}%
\caption{}
\label{fig:2branes_1point_1triangle_1}
\end{subfigure}%
\begin{subfigure}{0.33\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (Origin) at (0,0);
  \coordinate (Q1) at (-0.8,-0.4);
  \coordinate (Q2) at (0,1);
  \coordinate (Q3) at (0.8,-0.4);
  \coordinate (P1) at (-2,-1);
  \coordinate (P2) at (0,2);
  \coordinate (P3) at (2,-1);
  \node at (Origin) {$\bigstar$};
  \node at (Q1) {$\bullet$};
  \node at (Q2) {$\bullet$};
  \node at (Q3) {$\bullet$};
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
\draw[red,thick] (P3)--(Q1)--(Origin);
\draw[blue,thick] (P3)--(P1)--(Origin);
\draw[magenta,thick] (P3)--(Origin);
\draw[dotted] (Q1)--(Q2)--(Q3)--cycle;
\draw[dotted] (P1)--(P2)--(P3)--cycle;
\end{tikzpicture}%
\caption{}
\label{fig:2branes_1point_1triangle_2}
\end{subfigure}%
\begin{subfigure}{0.33\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (Origin) at (0,0);
  \coordinate (Q1) at (-0.8,-0.4);
  \coordinate (Q2) at (0,1);
  \coordinate (Q3) at (0.8,-0.4);
  \coordinate (P1) at (-2,-1);
  \coordinate (P2) at (0,2);
  \coordinate (P3) at (2,-1);
  \node at (Origin) {$\bigstar$};
  \node at (Q1) {$\bullet$};
  \node at (Q2) {$\bullet$};
  \node at (Q3) {$\bullet$};
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
\draw[blue,thick] (P3)--(P1)--(Origin)--cycle;
\draw[dotted] (Q1)--(Q2)--(Q3)--cycle;
\draw[dotted] (P1)--(P2)--(P3)--cycle;
\end{tikzpicture}%
\caption{}
\label{fig:2branes_1point_1triangle_3}
\end{subfigure}%
\\
\begin{subfigure}{0.33\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (Origin) at (0,0);
  \coordinate (Q1) at (-0.8,-0.4);
  \coordinate (Q2) at (0,1);
  \coordinate (Q3) at (0.8,-0.4);
  \coordinate (P1) at (-2,-1);
  \coordinate (P2) at (0,2);
  \coordinate (P3) at (2,-1);
  \node at (Origin) {$\bigstar$};
  \node at (Q1) {$\bullet$};
  \node at (Q2) {$\bullet$};
  \node at (Q3) {$\bullet$};
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
\draw[red,thick] (P3)--(Q1)--(Origin)--cycle;
\draw[dotted] (Q1)--(Q2)--(Q3)--cycle;
\draw[dotted] (P1)--(P2)--(P3)--cycle;
\end{tikzpicture}%
\caption{}
\label{fig:2branes_1point_1triangle_4}
\end{subfigure}%
\begin{subfigure}{0.33\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (Origin) at (0,0);
  \coordinate (Q1) at (-0.8,-0.4);
  \coordinate (Q2) at (0,1);
  \coordinate (Q3) at (0.8,-0.4);
  \coordinate (P1) at (-2,-1);
  \coordinate (P2) at (0,2);
  \coordinate (P3) at (2,-1);
  \node at (Origin) {$\bigstar$};
  \node at (Q1) {$\bullet$};
  \node at (Q2) {$\bullet$};
  \node at (Q3) {$\bullet$};
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
\draw[red,thick] (Q1)--(P3)--(Origin);
\draw[blue,thick] (Q1)--(Q3)--(Origin);
\draw[magenta,thick] (Q1)--(Origin);
\draw[dotted] (Q1)--(Q2)--(Q3)--cycle;
\draw[dotted] (P1)--(P2)--(P3)--cycle;
\end{tikzpicture}%
\caption{}
\label{fig:2branes_1point_1triangle_5}
\end{subfigure}%
\begin{subfigure}{0.33\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (Origin) at (0,0);
  \coordinate (Q1) at (-0.8,-0.4);
  \coordinate (Q2) at (0,1);
  \coordinate (Q3) at (0.8,-0.4);
  \coordinate (P1) at (-2,-1);
  \coordinate (P2) at (0,2);
  \coordinate (P3) at (2,-1);
  \node at (Origin) {$\bigstar$};
  \node at (Q1) {$\bullet$};
  \node at (Q2) {$\bullet$};
  \node at (Q3) {$\bullet$};
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
\draw[blue,thick] (Q3)--(Q1)--(Origin)--cycle;
\draw[dotted] (Q1)--(Q2)--(Q3)--cycle;
\draw[dotted] (P1)--(P2)--(P3)--cycle;
\end{tikzpicture}%
\caption{}
\label{fig:2branes_1point_1triangle_6}
\end{subfigure}%
\\
\begin{subfigure}{0.33\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (Origin) at (0,0);
  \coordinate (Q1) at (-0.8,-0.4);
  \coordinate (Q2) at (0,1);
  \coordinate (Q3) at (0.8,-0.4);
  \coordinate (P1) at (-2,-1);
  \coordinate (P2) at (0,2);
  \coordinate (P3) at (2,-1);
  \node at (Origin) {$\bigstar$};
  \node at (Q1) {$\bullet$};
  \node at (Q2) {$\bullet$};
  \node at (Q3) {$\bullet$};
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
\draw[red,thick] (P3)--(Q1)--(Origin)--cycle;
\draw[dotted] (Q1)--(Q2)--(Q3)--cycle;
\draw[dotted] (P1)--(P2)--(P3)--cycle;
\end{tikzpicture}%
\caption{}
\label{fig:2branes_1point_1triangle_7}
\end{subfigure}%
\begin{subfigure}{0.33\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (Origin) at (0,0);
  \coordinate (Q1) at (-0.8,-0.4);
  \coordinate (Q2) at (0,1);
  \coordinate (Q3) at (0.8,-0.4);
  \coordinate (P1) at (-2,-1);
  \coordinate (P2) at (0,2);
  \coordinate (P3) at (2,-1);
  \node at (Origin) {$\bigstar$};
  \node at (Q1) {$\bullet$};
  \node at (Q2) {$\bullet$};
  \node at (Q3) {$\bullet$};
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
\draw[red,thick] (Q1)--(Origin)--(P3);
\draw[blue,thick] (P3)--(Q2)--(Q1);
\draw[magenta,thick] (Q1)--(P3);
\draw[dotted] (Q1)--(Q2)--(Q3)--cycle;
\draw[dotted] (P1)--(P2)--(P3)--cycle;
\end{tikzpicture}%
\caption{}
\label{fig:2branes_1point_1triangle_8}
\end{subfigure}%
\begin{subfigure}{0.33\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (Origin) at (0,0);
  \coordinate (Q1) at (-0.8,-0.4);
  \coordinate (Q2) at (0,1);
  \coordinate (Q3) at (0.8,-0.4);
  \coordinate (P1) at (-2,-1);
  \coordinate (P2) at (0,2);
  \coordinate (P3) at (2,-1);
  \node at (Origin) {$\bigstar$};
  \node at (Q1) {$\bullet$};
  \node at (Q2) {$\bullet$};
  \node at (Q3) {$\bullet$};
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
\draw[blue,thick] (Q1)--(Q2)--(P3)--cycle;
\draw[dotted] (Q1)--(Q2)--(Q3)--cycle;
\draw[dotted] (P1)--(P2)--(P3)--cycle;
\end{tikzpicture}%
\caption{}
\label{fig:2branes_1point_1triangle_9}
\end{subfigure}%
\caption{Continuous solutions that relate the closed $\tilde{H}_1^+ \
\varphi_2^+ \ \tilde{\sigma}_3^+$ triangle with other known one triangle
solutions. The three figures in each row are related by an explicit
one-parameter solution.}
\label{fig:2branes_1point_1triangle}
\end{figure}

As in previous cases, solutions with more than one closed triangle have a lower
potential energy, and we immediately recognize the
solution with maximal intra-brane Higgs $\phi_i^+$ and $\phi_i^+$.
Again, one can verify explicitly that
solutions like the ones depicted in figure \ref{fig:2branes_1point_2triangle}
exist, and we find continuous deformations that transform figure
\ref{fig:2branes_1point_2triangle_1} into figure
\ref{fig:2branes_1point_2triangle_2} as well as
figure
\ref{fig:2branes_1point_2triangle_2} into
figure
\ref{fig:2branes_1point_2triangle_3}. By analogous considerations, one infers that all
 other configurations involving two closed triangles are
degenerate in their potential energy and can be deformed into each other.
However, they lead to  distinct patterns of symmetry breaking and Yukawa couplings.

\begin{figure}[t!]
\begin{subfigure}{0.33\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (Origin) at (0,0);
  \coordinate (Q1) at (-0.8,-0.4);
  \coordinate (Q2) at (0,1);
  \coordinate (Q3) at (0.8,-0.4);
  \coordinate (P1) at (-2,-1);
  \coordinate (P2) at (0,2);
  \coordinate (P3) at (2,-1);
  \node at (Origin) {$\bigstar$};
  \node at (Q1) {$\bullet$};
  \node at (Q2) {$\bullet$};
  \node at (Q3) {$\bullet$};
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
\draw[red,thick] (P1)--(P2)--(Q3)--cycle;
\draw[blue,thick] (P3)--(Q1)--(Origin)--cycle;
\draw[dotted] (Q1)--(Q2)--(Q3)--cycle;
\draw[dotted] (P1)--(P3)--(P2);
\end{tikzpicture}%
\caption{}
\label{fig:2branes_1point_2triangle_1}
\end{subfigure}%
\begin{subfigure}{0.33\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (Origin) at (0,0);
  \coordinate (Q1) at (-0.8,-0.4);
  \coordinate (Q2) at (0,1);
  \coordinate (Q3) at (0.8,-0.4);
  \coordinate (P1) at (-2,-1);
  \coordinate (P2) at (0,2);
  \coordinate (P3) at (2,-1);
  \node at (Origin) {$\bigstar$};
  \node at (Q1) {$\bullet$};
  \node at (Q2) {$\bullet$};
  \node at (Q3) {$\bullet$};
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
\draw[red,thick] (P1)--(P2)--(P3)--cycle;
\draw[blue,thick] (Q3)--(Q1)--(Origin)--cycle;
\draw[dotted] (Q1)--(Q2)--(Q3);
\end{tikzpicture}%
\caption{}
\label{fig:2branes_1point_2triangle_2}
\end{subfigure}%
\begin{subfigure}{0.33\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (Origin) at (0,0);
  \coordinate (Q1) at (-0.8,-0.4);
  \coordinate (Q2) at (0,1);
  \coordinate (Q3) at (0.8,-0.4);
  \coordinate (P1) at (-2,-1);
  \coordinate (P2) at (0,2);
  \coordinate (P3) at (2,-1);
  \node at (Origin) {$\bigstar$};
  \node at (Q1) {$\bullet$};
  \node at (Q2) {$\bullet$};
  \node at (Q3) {$\bullet$};
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
\draw[red,thick] (P1)--(P2)--(P3)--cycle;
\draw[blue,thick] (Q3)--(Q1)--(Q2)--cycle;
\end{tikzpicture}%
\caption{}
\label{fig:2branes_1point_2triangle_3}
\end{subfigure}%
\caption{Three two triangle configurations which are equivalent by continuous
deformations among each other.}
\label{fig:2branes_1point_2triangle}
\end{figure}
\paragraph{Spectrum.}
Having established the existence of numerous novel solutions consisting of brane
backgrounds and maximal Higgs, we need to address the spectrum of the vector
Laplacian. We illustrate the typical behavior for the one-parameter family
connecting the configurations of figure \ref{fig:2branes_1point_2triangle_2}
and \ref{fig:2branes_1point_2triangle_1}, and  summarize  the results in
appendix \ref{subsec:brane+brane+point}, in particular figure
\ref{fig:2branes+pt_spectrum}.

As in the previous cases, the combined backgrounds $X+\phi$ suffer from the
presence of negative modes in the spectrum of $\cO_V^{X+\phi}$. Fortunately,
one can lift all of these consistently by inclusion of uniform masses $M_i
\equiv M$ with the choice $0.47 \lesssim M \leq \frac{\sqrt{2}}{3}$, including $M^*$.
As before, these non-trivial masses eliminate a large fraction of the zero
modes, and stabilize their number to a level which is (roughly) independent of
the system size.
\subsection{Three squashed \texorpdfstring{$\mathds{C} P^2$}{CP2} branes \& Higgs}
\label{sec:three-branes}
Finally, consider the case where also ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$ is not a point brane, but a $\mathds{C}
P^2$ brane. Since the sector of maximal regular zero modes is a straight
forward generalization of the previous cases, we refrain from depicting them in
full detail.
\paragraph{Solutions.}
In this set-up, one can again find configurations with three closed Higgs
triangles connecting the different branes as in figures
\ref{fig:3Branes_1}--\ref{fig:3Branes_3}.
The energy of these configurations is conjectured to be minimal, and equal to the case
where only the maximal intra-brane Higgs on the branes are switched on.
Continuous deformations interpolating between these
configurations are indicated in figure \ref{fig:three-CP2-branes-Higgs}. As we have
already seen in the previous cases, the all solutions with fixed number of
closed triangles (here $1$, $2$, or $3$) can be deformed into one another.
Moreover, we know that solutions with the maximal number of closed triangles
have the minimal potential energy. Hence these configurations are highly
degenerate, but
they lead to  distinct patterns of symmetry breaking and Yukawa couplings.
\begin{figure}[t!]
\centering
\begin{subfigure}{0.32\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (R1) at (-0.5,-0.25);
  \coordinate (R2) at (0,0.5);
  \coordinate (R3) at (0.5,-0.25);
  \coordinate (Q1) at (-1.5,-0.75);
  \coordinate (Q2) at (0,1.5);
  \coordinate (Q3) at (1.5,-0.75);
  \coordinate (P1) at (-2.5,-1.25);
  \coordinate (P2) at (0,2.5);
  \coordinate (P3) at (2.5,-1.25);
  \node at (R1) {$\bullet$};
  \node at (R2) {$\bullet$};
  \node at (R3) {$\bullet$};
  \node at (Q1) {$\bullet$};
  \node at (Q2) {$\bullet$};
  \node at (Q3) {$\bullet$};
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
\draw[dotted] (P1)--(P2)--(P3)--cycle;
\draw[dotted] (Q1)--(Q2)--(Q3)--cycle;
\draw[dotted] (R1)--(R2)--(R3)--cycle;
\draw[thick,red] (P1)--(Q2)--(R3)--cycle;
\draw[thick,green] (P2)--(Q3)--(R1)--cycle;
\draw[thick,blue] (P3)--(Q1)--(R2)--cycle;
\end{tikzpicture}%
\caption{}%
\label{fig:3Branes_1}%
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (R1) at (-0.5,-0.25);
  \coordinate (R2) at (0,0.5);
  \coordinate (R3) at (0.5,-0.25);
  \coordinate (Q1) at (-1.5,-0.75);
  \coordinate (Q2) at (0,1.5);
  \coordinate (Q3) at (1.5,-0.75);
  \coordinate (P1) at (-2.5,-1.25);
  \coordinate (P2) at (0,2.5);
  \coordinate (P3) at (2.5,-1.25);
  \node at (R1) {$\bullet$};
  \node at (R2) {$\bullet$};
  \node at (R3) {$\bullet$};
  \node at (Q1) {$\bullet$};
  \node at (Q2) {$\bullet$};
  \node at (Q3) {$\bullet$};
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
\draw[dotted] (P1)--(P2)--(P3)--cycle;
\draw[dotted] (Q1)--(Q2)--(Q3);
\draw[dotted] (R1)--(R2)--(R3)--cycle;
\draw[thick,red] (P1)--(Q2)--(Q3)--cycle;
\draw[thick,green] (P2)--(R3)--(R1)--cycle;
\draw[thick,blue] (P3)--(Q1)--(R2)--cycle;
\end{tikzpicture}%
\caption{}%
\label{fig:3Branes_2}%
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (R1) at (-0.5,-0.25);
  \coordinate (R2) at (0,0.5);
  \coordinate (R3) at (0.5,-0.25);
  \coordinate (Q1) at (-1.5,-0.75);
  \coordinate (Q2) at (0,1.5);
  \coordinate (Q3) at (1.5,-0.75);
  \coordinate (P1) at (-2.5,-1.25);
  \coordinate (P2) at (0,2.5);
  \coordinate (P3) at (2.5,-1.25);
  \node at (R1) {$\bullet$};
  \node at (R2) {$\bullet$};
  \node at (R3) {$\bullet$};
  \node at (Q1) {$\bullet$};
  \node at (Q2) {$\bullet$};
  \node at (Q3) {$\bullet$};
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
\draw[dotted] (P1)--(P2)--(P3)--cycle;
\draw[dotted] (Q1)--(Q2)--(Q3);
\draw[dotted] (R1)--(R2)--(R3)--cycle;
\draw[thick,red] (P1)--(P2)--(Q3)--cycle;
\draw[thick,green] (Q2)--(R3)--(R1)--cycle;
\draw[thick,blue] (P3)--(Q1)--(R2)--cycle;
\end{tikzpicture}%
\caption{}%
\label{fig:3Branes_3}%
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (R1) at (-0.5,-0.25);
  \coordinate (R2) at (0,0.5);
  \coordinate (R3) at (0.5,-0.25);
  \coordinate (Q1) at (-1.5,-0.75);
  \coordinate (Q2) at (0,1.5);
  \coordinate (Q3) at (1.5,-0.75);
  \coordinate (P1) at (-2.5,-1.25);
  \coordinate (P2) at (0,2.5);
  \coordinate (P3) at (2.5,-1.25);
  \node at (R1) {$\bullet$};
  \node at (R2) {$\bullet$};
  \node at (R3) {$\bullet$};
  \node at (Q1) {$\bullet$};
  \node at (Q2) {$\bullet$};
  \node at (Q3) {$\bullet$};
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
\draw[dotted] (Q1)--(Q2)--(Q3);
\draw[dotted] (R1)--(R2)--(R3);
\draw[thick,red] (P1)--(P2)--(P3)--cycle;
\draw[thick,green] (Q2)--(R3)--(R1)--cycle;
\draw[thick,blue] (Q3)--(Q1)--(R2)--cycle;
\end{tikzpicture}%
\caption{}%
\label{fig:3Branes_4}%
\end{subfigure}
\begin{subfigure}{0.32\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (R1) at (-0.5,-0.25);
  \coordinate (R2) at (0,0.5);
  \coordinate (R3) at (0.5,-0.25);
  \coordinate (Q1) at (-1.5,-0.75);
  \coordinate (Q2) at (0,1.5);
  \coordinate (Q3) at (1.5,-0.75);
  \coordinate (P1) at (-2.5,-1.25);
  \coordinate (P2) at (0,2.5);
  \coordinate (P3) at (2.5,-1.25);
  \node at (R1) {$\bullet$};
  \node at (R2) {$\bullet$};
  \node at (R3) {$\bullet$};
  \node at (Q1) {$\bullet$};
  \node at (Q2) {$\bullet$};
  \node at (Q3) {$\bullet$};
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
\draw[thick,red] (P1)--(P2)--(P3)--cycle;
\draw[thick,green] (R2)--(R3)--(R1)--cycle;
\draw[thick,blue] (Q3)--(Q1)--(Q2)--cycle;
\end{tikzpicture}%
\caption{}%
\label{fig:3Branes_5}%
\end{subfigure}
\caption{Configurations of three closed triangles for three parallel $\mathds{C} P^2$
branes. All of the shown configurations can be deformed into each other by a
continuous family of solutions. Hence, all configurations have identical
potential energy, but the fluctuations around each background can differ.}
 \label{fig:three-CP2-branes-Higgs}
\end{figure}%
\paragraph{Spectrum.}
Again, we studied the spectrum of the vector Laplacian in these combined
background $X+\phi$, and provide the details in appendix
\ref{subsec:brane+brane+brane}, see in particular figure
\ref{fig:3branes_spectrum}.
We illustrate for the configurations of figure \ref{fig:3Branes_1} and
\ref{fig:3Branes_5} the existence of negative modes in the massless case, and
their uplifting via masses $M_i \equiv M$ with $0.47 \lesssim M
\leq \frac{\sqrt{2}}{3}$ including $M^*$. Then the increasingly large number of zero
modes in the massless case is reduced and stabilized to a level that appears to
be independent of the systems size.

\subsection{Coinciding branes with Higgs}

Now consider a stack of $n$ identical branes as above.
Then the massless i.e.\ trivial gauge modes constitute an unbroken $U(n)$
(or $SU(n)$) gauge group.
As in the case of distinct branes, there are numerous exact Higgs solutions, which may or
may not link the different branes in various patterns. It is clear that this  leads
to various patterns of (partial or complete) symmetry breaking, and it is straightforward
in principle to work out the masses of the broken gauge bosons from the
Higgs effect, cf.\ \eqref{gaugeboson-mass-phi}.

Taking into account the fermionic zero modes,
the question arises  how these  fermions
couple to the broken and unbroken gauge fields, which Yukawa couplings arise, and whether some kind of chiral
gauge theory  emerges.
This is quite natural as we will see, and
will be discussed in section \ref{sec:fermions}.

\subsection{Flipped minimal branes plus point brane as
\texorpdfstring{$G_2$}{G2} brane.}
\label{sec:flipped-branes}
Now consider the case of two branes $\cC[\mu_{L,R}]$ of conjugate type, i.e.\
with  $\mu_L = (N,0), \ \mu_R=(0,N_R)$, together with a point brane ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$.
Then the full set of inter-brane zero modes between the two squashed $\mathds{C} P^2$
branes can  be obtained from the mode decomposition
\begin{align}
 \mathrm{Hom}(\cH_{(N,0)},\cH_{(0,N_R)})  = \cH_{(N+N_R,0)} \oplus
\cH_{(N+N_R-2,1)} \oplus \ldots \oplus \cH_{(\ldots,N_R)}
\end{align}
assuming $N \geq N_R$. The maximal regular inter-brane modes transform as
$(N+N_R,0)$.
However, for these modes linking the extremal weights by the longest
possible arrow, there exists no configuration with a non-trivial cubic term.
Therefore we have no reason to assume that they acquire a VEV.
\begin{figure}[t!]
\centering
\begin{subfigure}{0.24\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (P1) at (-1.5,-1);
  \coordinate (P2) at (0,2);
  \coordinate (P3) at (1.5,-1);
  \coordinate (P4) at (-1.5,1);
  \coordinate (P5) at (0,-2);
  \coordinate (P6) at (1.5,1);
  \coordinate (P7) at (0,0);
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
  \node at (P4) {$\blacktriangle$};
  \node at (P5) {$\blacktriangle$};
  \node at (P6) {$\blacktriangle$};
  \node at (P7) {$\bigstar$};
\draw [thick,red,-latex] (P1) -- (P3) node[at end,below] {$\phi_1^+$};%
\draw [thick,red,-latex] (P3) -- (P2) node[at end,right] {$\phi_2^+$};%
\draw [thick,red,-latex] (P2) -- (P1) node[at end,left] {$\phi_3^+$};%
\draw [thick,blue,-latex] (P4) -- (P6) node[at end,above] {$\tilde{\phi}_1^+$};%
\draw [thick,blue,-latex] (P5) -- (P4) node[at end,left] {$\tilde{\phi}_2^+$};%
\draw [thick,blue,-latex] (P6) -- (P5) node[at end,right] {$\tilde{\phi}_3^+$};%
\end{tikzpicture}%
\caption{}%
\label{fig:flipped_brane+pt_intra_Higgs}%
\end{subfigure}%
\begin{subfigure}{0.24\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (P1) at (-1.5,-1);
  \coordinate (P2) at (0,2);
  \coordinate (P3) at (1.5,-1);
  \coordinate (P4) at (-1.5,1);
  \coordinate (P5) at (0,-2);
  \coordinate (P6) at (1.5,1);
  \coordinate (P7) at (0,0);
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
  \node at (P4) {$\blacktriangle$};
  \node at (P5) {$\blacktriangle$};
  \node at (P6) {$\blacktriangle$};
  \node at (P7) {$\bigstar$};
\draw [thick,red,-latex] (P1) -- (P6) node[at end,below] {$\varphi_1^+$};%
\draw [thick,blue,-latex] (P3) -- (P4) node[at end,below]
{$\tilde{\varphi}_1^-$};%
\draw [dotted] (P1)--(P2)--(P3)--cycle;
\draw [dotted] (P4)--(P5)--(P6)--cycle;
\end{tikzpicture}%
\caption{}%
\label{fig:flipped_brane+pt_inter_Higgs_max1}%
\end{subfigure}%
\begin{subfigure}{0.24\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (P1) at (-1.5,-1);
  \coordinate (P2) at (0,2);
  \coordinate (P3) at (1.5,-1);
  \coordinate (P4) at (-1.5,1);
  \coordinate (P5) at (0,-2);
  \coordinate (P6) at (1.5,1);
  \coordinate (P7) at (0,0);
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
  \node at (P4) {$\blacktriangle$};
  \node at (P5) {$\blacktriangle$};
  \node at (P6) {$\blacktriangle$};
  \node at (P7) {$\bigstar$};
\draw [thick,red,-latex] (P3) -- (P4) node[at end,below] {$\varphi_2^+$};%
\draw [thick,blue,-latex] (P2) -- (P5) node[at end,right]
{$\tilde{\varphi}_2^-$};%
\draw [dotted] (P1)--(P2)--(P3)--cycle;
\draw [dotted] (P4)--(P5)--(P6)--cycle;
\end{tikzpicture}%
\caption{}%
\label{fig:flipped_brane+pt_inter_Higgs_max2}
\end{subfigure}%
\begin{subfigure}{0.24\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (P1) at (-1.5,-1);
  \coordinate (P2) at (0,2);
  \coordinate (P3) at (1.5,-1);
  \coordinate (P4) at (-1.5,1);
  \coordinate (P5) at (0,-2);
  \coordinate (P6) at (1.5,1);
  \coordinate (P7) at (0,0);
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
  \node at (P4) {$\blacktriangle$};
  \node at (P5) {$\blacktriangle$};
  \node at (P6) {$\blacktriangle$};
  \node at (P7) {$\bigstar$};
\draw [thick,red,-latex] (P2) -- (P5) node[at end,right] {$\varphi_3^+$};%
\draw [thick,blue,-latex] (P1) -- (P6) node[at end,below]
{$\tilde{\varphi}_3^-$};%
\draw [dotted] (P1)--(P2)--(P3)--cycle;
\draw [dotted] (P4)--(P5)--(P6)--cycle;
\end{tikzpicture}%
\caption{}%
\label{fig:flipped_brane+pt_inter_Higgs_max3}
\end{subfigure}%
\\
\begin{subfigure}{0.24\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (P1) at (-1.5,-1);
  \coordinate (P2) at (0,2);
  \coordinate (P3) at (1.5,-1);
  \coordinate (P4) at (-1.5,1);
  \coordinate (P5) at (0,-2);
  \coordinate (P6) at (1.5,1);
  \coordinate (P7) at (0,0);
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
  \node at (P4) {$\blacktriangle$};
  \node at (P5) {$\blacktriangle$};
  \node at (P6) {$\blacktriangle$};
  \node at (P7) {$\bigstar$};
\draw [thick,red,-latex] (P1) -- (P5) node[midway,below] {$\zeta_1^+$};%
\draw [thick,red,-latex] (P2) -- (P6);
\draw [thick,blue,-latex] (P3) -- (P5) node[midway,below]
{$\tilde{\zeta}_1^-$};%
\draw [thick,blue,-latex] (P2) -- (P4);
\draw [dotted] (P1)--(P2)--(P3)--cycle;
\draw [dotted] (P4)--(P5)--(P6)--cycle;
\end{tikzpicture}%
\caption{}%
\label{fig:flipped_brane+pt_inter_Higgs_next-to1}
\end{subfigure}%
\begin{subfigure}{0.24\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (P1) at (-1.5,-1);
  \coordinate (P2) at (0,2);
  \coordinate (P3) at (1.5,-1);
  \coordinate (P4) at (-1.5,1);
  \coordinate (P5) at (0,-2);
  \coordinate (P6) at (1.5,1);
  \coordinate (P7) at (0,0);
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
  \node at (P4) {$\blacktriangle$};
  \node at (P5) {$\blacktriangle$};
  \node at (P6) {$\blacktriangle$};
  \node at (P7) {$\bigstar$};
\draw [thick,red,-latex] (P1) -- (P4) node[midway,left] {$\zeta_2^+$};%
\draw [thick,red,-latex] (P3) -- (P6);
\draw [thick,blue,-latex] (P1) -- (P5) node[midway,below]
{$\tilde{\zeta}_2^-$};%
\draw [thick,blue,-latex] (P2) -- (P6);
\draw [dotted] (P1)--(P2)--(P3)--cycle;
\draw [dotted] (P4)--(P5)--(P6)--cycle;
\end{tikzpicture}%
\caption{}%
\label{fig:flipped_brane+pt_inter_Higgs_next-to2}
\end{subfigure}%
\begin{subfigure}{0.24\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (P1) at (-1.5,-1);
  \coordinate (P2) at (0,2);
  \coordinate (P3) at (1.5,-1);
  \coordinate (P4) at (-1.5,1);
  \coordinate (P5) at (0,-2);
  \coordinate (P6) at (1.5,1);
  \coordinate (P7) at (0,0);
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
  \node at (P4) {$\blacktriangle$};
  \node at (P5) {$\blacktriangle$};
  \node at (P6) {$\blacktriangle$};
  \node at (P7) {$\bigstar$};
\draw [thick,red,-latex] (P2) -- (P4) node[midway,above] {$\zeta_3^+$};%
\draw [thick,red,-latex] (P3) -- (P5);
\draw [thick,blue,-latex] (P1) -- (P4) node[midway,left]
{$\tilde{\zeta}_3^-$};%
\draw [thick,blue,-latex] (P3) -- (P6);
\draw [dotted] (P1)--(P2)--(P3)--cycle;
\draw [dotted] (P4)--(P5)--(P6)--cycle;
\end{tikzpicture}%
\caption{}%
\label{fig:flipped_brane+pt_inter_Higgs_next-to3}
\end{subfigure}%
\caption{Set-up $\cC[(1,0)]+\cC[(0,1)]+{\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$. Extremal weights for $(1,0)$ are
denoted by $\bullet$, while $(0,1)$ extremal weights are indicated by
$\blacktriangle$, and ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$ is represented by $\bigstar$.
The regular intra-brane Higgs
modes $\phi$ on $\cC[(1,0)]$ and $\tilde{\phi}$ on $\cC[(0,1)]$ are shown in
(\subref{fig:flipped_brane+pt_intra_Higgs}). In
(\subref{fig:flipped_brane+pt_inter_Higgs_max1})--(\subref{%
fig:flipped_brane+pt_inter_Higgs_max3}) we display the maximal regular
inter-brane Higgs $\varphi$,
$\tilde{\varphi}$ between $\cC[(1,0)]$  and $\cC[(0,1)]$, while the
next-to-maximal regular inter-brane Higgs $\zeta$, $\tilde{\zeta}$ between
$\cC[(1,0)]$ and $\cC[(0,1)]$ are shown in
(\subref{fig:flipped_brane+pt_inter_Higgs_next-to1})--(\subref{%
fig:flipped_brane+pt_inter_Higgs_next-to3})}
\label{fig:flipped_brane+point_Higgs_partI}
\end{figure}

Specializing to the case of $N=N_R$, the next most interesting regular
zero modes transform in $(0,N)$. These modes connect the parallel edges of
the irreps $(N,0)$ and $(0,N)$.
For minimal branes, they form a closed algebra and lead to an exact
solution. Let us discuss this in more detail:
Let $\cC[(1,0)] +\cC[(1,0)] + \cC[(0,0)]$ be the background solution. There are
 various regular zero modes to be taken into account. To begin with, the
regular intra-brane zero modes on $\cC[(1,0)]$ and $\cC[(0,1)]$ are depicted
in figure \ref{fig:flipped_brane+pt_intra_Higgs}. From earlier arguments, it is
clear that the background plus the intra-brane triangular configuration lead to
exact solutions of the equations of motion.
In addition, the maximal inter-brane regular zero modes between
$\cC[(1,0)]$ and $\cC[(0,1)]$ are shown in figure
\ref{fig:flipped_brane+pt_inter_Higgs_max1}--\ref{%
fig:flipped_brane+pt_inter_Higgs_max3}, while the next-to-maximal intra-brane
Higgs are displayed in figures
\ref{fig:flipped_brane+pt_inter_Higgs_next-to1}--\ref{%
fig:flipped_brane+pt_inter_Higgs_next-to3}.
Lastly, there exists the class of inter-brane regular zero modes between each
squashed $\mathds{C} P^2$ brane and the point brane. The regular zero modes between
$\cC[(1,0)]$ and ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$ are displayed in figure
\ref{fig:flipped_brane+pt_inter_Higgs_1-0_to_Pt1}--\ref{%
fig:flipped_brane+pt_inter_Higgs_1-0_to_Pt3}; whereas the inter-brane Higgs
between $\cC[(0,1)]$ and ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$ are shown in figures
\ref{fig:flipped_brane+pt_inter_Higgs_0-1_to_Pt1}--\ref{%
fig:flipped_brane+pt_inter_Higgs_0-1_to_Pt3}.
\begin{figure}[t!]
\centering
\begin{subfigure}{0.24\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (P1) at (-1.5,-1);
  \coordinate (P2) at (0,2);
  \coordinate (P3) at (1.5,-1);
  \coordinate (P4) at (-1.5,1);
  \coordinate (P5) at (0,-2);
  \coordinate (P6) at (1.5,1);
  \coordinate (P7) at (0,0);
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
  \node at (P4) {$\blacktriangle$};
  \node at (P5) {$\blacktriangle$};
  \node at (P6) {$\blacktriangle$};
  \node at (P7) {$\bigstar$};
\draw [thick,red,-latex] (P1) -- (P7) node[midway,below] {$\varrho_1^+$};%
\draw [thick,blue,-latex] (P3) -- (P7) node[midway,below]
{$\tilde{\varrho}_1^-$};%
\draw [dotted] (P1)--(P2)--(P3)--cycle;
\draw [dotted] (P4)--(P5)--(P6)--cycle;
\end{tikzpicture}%
\caption{}%
\label{fig:flipped_brane+pt_inter_Higgs_1-0_to_Pt1}%
\end{subfigure}%
\begin{subfigure}{0.24\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (P1) at (-1.5,-1);
  \coordinate (P2) at (0,2);
  \coordinate (P3) at (1.5,-1);
  \coordinate (P4) at (-1.5,1);
  \coordinate (P5) at (0,-2);
  \coordinate (P6) at (1.5,1);
  \coordinate (P7) at (0,0);
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
  \node at (P4) {$\blacktriangle$};
  \node at (P5) {$\blacktriangle$};
  \node at (P6) {$\blacktriangle$};
  \node at (P7) {$\bigstar$};
\draw [thick,red,-latex] (P3) -- (P7) node[midway,below] {$\varrho_2^+$};%
\draw [thick,blue,-latex] (P2) -- (P7) node[midway,left]
{$\tilde{\varrho}_2^-$};%
\draw [dotted] (P1)--(P2)--(P3)--cycle;
\draw [dotted] (P4)--(P5)--(P6)--cycle;
\end{tikzpicture}%
\caption{}%
\label{fig:flipped_brane+pt_inter_Higgs_1-0_to_Pt2}
\end{subfigure}%
\begin{subfigure}{0.24\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (P1) at (-1.5,-1);
  \coordinate (P2) at (0,2);
  \coordinate (P3) at (1.5,-1);
  \coordinate (P4) at (-1.5,1);
  \coordinate (P5) at (0,-2);
  \coordinate (P6) at (1.5,1);
  \coordinate (P7) at (0,0);
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
  \node at (P4) {$\blacktriangle$};
  \node at (P5) {$\blacktriangle$};
  \node at (P6) {$\blacktriangle$};
  \node at (P7) {$\bigstar$};
\draw [thick,red,-latex] (P2) -- (P7) node[midway,left] {$\varrho_3^+$};%
\draw [thick,blue,-latex] (P1) -- (P7) node[midway,above]
{$\tilde{\varrho}_3^-$};%
\draw [dotted] (P1)--(P2)--(P3)--cycle;
\draw [dotted] (P4)--(P5)--(P6)--cycle;
\end{tikzpicture}%
\caption{}%
\label{fig:flipped_brane+pt_inter_Higgs_1-0_to_Pt3}
\end{subfigure}%
\\
\begin{subfigure}{0.24\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (P1) at (-1.5,-1);
  \coordinate (P2) at (0,2);
  \coordinate (P3) at (1.5,-1);
  \coordinate (P4) at (-1.5,1);
  \coordinate (P5) at (0,-2);
  \coordinate (P6) at (1.5,1);
  \coordinate (P7) at (0,0);
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
  \node at (P4) {$\blacktriangle$};
  \node at (P5) {$\blacktriangle$};
  \node at (P6) {$\blacktriangle$};
  \node at (P7) {$\bigstar$};
\draw [thick,red,-latex] (P4) -- (P7) node[midway,below] {$\sigma_1^+$};%
\draw [thick,blue,-latex] (P6) -- (P7) node[midway,below]
{$\tilde{\sigma}_1^-$};%
\draw [dotted] (P1)--(P2)--(P3)--cycle;
\draw [dotted] (P4)--(P5)--(P6)--cycle;
\end{tikzpicture}%
\caption{}%
\label{fig:flipped_brane+pt_inter_Higgs_0-1_to_Pt1}
\end{subfigure}%
\begin{subfigure}{0.24\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (P1) at (-1.5,-1);
  \coordinate (P2) at (0,2);
  \coordinate (P3) at (1.5,-1);
  \coordinate (P4) at (-1.5,1);
  \coordinate (P5) at (0,-2);
  \coordinate (P6) at (1.5,1);
  \coordinate (P7) at (0,0);
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
  \node at (P4) {$\blacktriangle$};
  \node at (P5) {$\blacktriangle$};
  \node at (P6) {$\blacktriangle$};
  \node at (P7) {$\bigstar$};
\draw [thick,red,-latex] (P5) -- (P7) node[midway,left] {$\sigma_2^+$};%
\draw [thick,blue,-latex] (P4) -- (P7) node[midway,above]
{$\tilde{\sigma}_2^-$};%
\draw [dotted] (P1)--(P2)--(P3)--cycle;
\draw [dotted] (P4)--(P5)--(P6)--cycle;
\end{tikzpicture}%
\caption{}%
\label{fig:flipped_brane+pt_inter_Higgs_0-1_to_Pt2}
\end{subfigure}%
\begin{subfigure}{0.24\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (P1) at (-1.5,-1);
  \coordinate (P2) at (0,2);
  \coordinate (P3) at (1.5,-1);
  \coordinate (P4) at (-1.5,1);
  \coordinate (P5) at (0,-2);
  \coordinate (P6) at (1.5,1);
  \coordinate (P7) at (0,0);
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
  \node at (P4) {$\blacktriangle$};
  \node at (P5) {$\blacktriangle$};
  \node at (P6) {$\blacktriangle$};
  \node at (P7) {$\bigstar$};
\draw [thick,red,-latex] (P6) -- (P7) node[midway,above] {$\sigma_3^+$};%
\draw [thick,blue,-latex] (P5) -- (P7) node[midway,left]
{$\tilde{\sigma}_3^-$};%
\draw [dotted] (P1)--(P2)--(P3)--cycle;
\draw [dotted] (P4)--(P5)--(P6)--cycle;
\end{tikzpicture}%
\caption{}%
\label{fig:flipped_brane+pt_inter_Higgs_0-1_to_Pt3}
\end{subfigure}%
\caption{Set-up $\cC[(1,0)]+\cC[(0,1)]+{\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$. The inter-brane Higgs
$\varrho$, $\tilde{\varrho}$ between $\cC[(1,0)]$ and point brane ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$ are
displayed in
(\subref{fig:flipped_brane+pt_inter_Higgs_1-0_to_Pt1})--(\subref{%
fig:flipped_brane+pt_inter_Higgs_1-0_to_Pt3}), while the inter-brane Higgs
$\sigma$, $\tilde{\sigma}$
between $\cC[(0,1)]$ and point brane ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$ are displayed in
(\subref{fig:flipped_brane+pt_inter_Higgs_0-1_to_Pt1})--(\subref{%
fig:flipped_brane+pt_inter_Higgs_0-1_to_Pt3}).}
\label{fig:flipped_brane+point_Higgs_partII}
\end{figure}
\paragraph{Solutions.}
As a nice illustration, we can construct  an exact
solution which has the structure of the 7-dimensional irrep of $G_2$. We
accomplish this by combining the rank 2 inter-brane zero modes between
the minimal brane and its conjugate with the inter-brane modes between the
minimal branes and the point brane. Actually there exist two realizations of
such a solution, as shown in figure
\ref{fig:G2-branes}. Following appendix \ref{subsec:flipped_min_brane+point},
one  realizes that the coefficients of the involved zero modes are
such that they realize the  short roots of $G_2$, while the long roots are realized
by the background $X_\alpha}          \def\da{{\dot\alpha}$.
Since $G_2$ is a Lie algebra, this solution
suggests a vast generalization based on  higher representations of $G_2$.
Indeed, the long and the short roots of $G_2$ satisfy the decoupling
condition\footnote{HS, unpublished; useful discussions with G.\ Zoupanos are
acknowledged.}.
This should be elaborated in more detail elsewhere.

For non-minimal branes $\cC[(N,0)]$, it is difficult to find
analogous exact solutions, because the regular zero modes  no longer satisfy
a closed algebra.
Nevertheless, one would expect that similar  solutions might  exist for
flipped non-minimal branes with a point brane. This construction is in a sense dual to the
discussion of 6-dimensional branes in the next section.

\begin{figure}[t!]
\centering
 \begin{subfigure}{0.485\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (P1) at (-1.5,-1);
  \coordinate (P2) at (0,2);
  \coordinate (P3) at (1.5,-1);
  \coordinate (P4) at (-1.5,1);
  \coordinate (P5) at (0,-2);
  \coordinate (P6) at (1.5,1);
  \coordinate (P7) at (0,0);
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
  \node at (P4) {$\blacktriangle$};
  \node at (P5) {$\blacktriangle$};
  \node at (P6) {$\blacktriangle$};
  \node at (P7) {$\bigstar$};
  \draw [dotted] (P1)--(P2)--(P3)--cycle;
  \draw [dotted] (P4)--(P5)--(P6)--cycle;
\draw [thick,red,-latex] (P5) -- (P3);
\draw [thick,red,-latex] (P4) -- (P2) node[midway,above] {$\tilde{\zeta}_1^+$};
\draw [thick,blue,-latex] (P5) -- (P1);
\draw [thick,blue,-latex] (P6) -- (P2) node[midway,above] {$\tilde{\zeta}_2^+$};
\draw [thick,green,-latex] (P4) -- (P1);
\draw [thick,green,-latex] (P6) -- (P3) node[midway,right]
{$\tilde{\zeta}_3^+$};
\draw [thick,dashed,red,-latex] (P1) -- (P7) node[midway,above] {$\varrho_1^+$};
\draw [thick,dashed,red,-latex] (P7) -- (P6) node[midway,below]
{$\tilde{\sigma}_1^+$};
\draw [thick,dashed,blue,-latex] (P3) -- (P7) node[midway,below]
{$\varrho_2^+$};
\draw [thick,dashed,blue,-latex] (P7) -- (P4) node[midway,above]
{$\tilde{\sigma}_2^+$};
\draw [thick,dashed,green,-latex] (P2) -- (P7) node[midway,right]
{$\varrho_3^+$};
\draw [thick,dashed,green,-latex] (P7) -- (P5)node[midway,left]
{$\tilde{\sigma}_3^+$};
\end{tikzpicture}%
\caption{}%
\label{fig:flipped_brane+pt_G2-sol1}
\end{subfigure}%
\begin{subfigure}{0.485\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (P1) at (-1.5,-1);
  \coordinate (P2) at (0,2);
  \coordinate (P3) at (1.5,-1);
  \coordinate (P4) at (-1.5,1);
  \coordinate (P5) at (0,-2);
  \coordinate (P6) at (1.5,1);
  \coordinate (P7) at (0,0);
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
  \node at (P4) {$\blacktriangle$};
  \node at (P5) {$\blacktriangle$};
  \node at (P6) {$\blacktriangle$};
  \node at (P7) {$\bigstar$};
  \draw [dotted] (P1)--(P2)--(P3)--cycle;
  \draw [dotted] (P4)--(P5)--(P6)--cycle;
\draw [thick,red,-latex] (P1) -- (P5);
\draw [thick,red,-latex] (P2) -- (P6) node[midway,above] {$\zeta_1^+$};
\draw [thick,blue,-latex] (P1) -- (P4);
\draw [thick,blue,-latex] (P3) -- (P6) node[midway,right] {$\zeta_2^+$};
\draw [thick,green,-latex] (P3) -- (P5);
\draw [thick,green,-latex] (P2) -- (P4) node[midway,above]
{$\zeta_3^+$};
\draw [thick,dashed,red,-latex] (P4) -- (P7) node[midway,above] {$\sigma_1^+$};
\draw [thick,dashed,red,-latex] (P7) -- (P3) node[midway,below]
{$\tilde{\varrho}_1^+$};
\draw [thick,dashed,blue,-latex] (P5) -- (P7) node[midway,left]
{$\sigma_2^+$};
\draw [thick,dashed,blue,-latex] (P7) -- (P2) node[midway,right]
{$\tilde{\varrho}_2^+$};
\draw [thick,dashed,green,-latex] (P6) -- (P7) node[midway,below]
{$\sigma_3^+$};
\draw [thick,dashed,green,-latex] (P7) -- (P1) node[midway,above]
{$\tilde{\varrho}_3^+$};
\end{tikzpicture}%
\caption{}%
\label{fig:flipped_brane+pt_G2-sol2}
\end{subfigure}%
 \caption{Set-up $\cC[(1,0)]+\cC[(0,1)]+{\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$. By turning on various
intra-brane zero modes, we can construct an exact solution which realizes
the 7 dimensional irrep of $G_2$. }
 \label{fig:G2-branes}
\end{figure}
\paragraph{Stability.}
The new combined solutions of $G_2$-type have been obtained in the massless
case. However, we can transfer them to massive solutions as before. This turns
out to be necessary to obtain a instability-free spectrum of the vector
Laplacian around this combined backgrounds. As in previous cases, a uniform
mass parameter of order $0.45 \lesssim M \leq \frac{\sqrt{2}}{3}$ is sufficient
to achieve this, and details are provided in appendix
\ref{subsec:flipped_min_brane+point}.
\section{6-dimensional branes}
\label{sec:6D-branes}
Now consider the case of general $\cC[(N,M)]$ branes. These are 6-dimensional quantized coadjoint orbits
embedded in $\mathds{R}^6$, which decompose into $3_L+3_R$ chiral sheets with opposite flux measured
by the gauge mode $\chi$ \eqref{chi-def}.
Because these branes have maximal dimension (in contrast to the above $\mathds{C} P^2$
branes), this leads to $3_L+3_R$ zero modes between a point brane and $\cC[(N,M)]$,
whose chirality is measured by $\chi$. This is the basis for a chiral gauge
theory.
\paragraph{Chirality generator and  chiral sheets.}
For 6-dimensional (fuzzy) branes, the operator
\begin{align}
 \chi \coloneqq  & \frac{\mathrm{i}}{8} \varepsilon^{\alpha}          \def\da{{\dot\alpha}\beta}           \def\db{{\dot\beta} \ldots \g\delta}  \def\D{\Delta}  \def\ddt{\dot\delta}[X_\alpha}          \def\da{{\dot\alpha},X_\beta}           \def\db{{\dot\beta}]
\ldots
[X_\g,X_\delta}  \def\D{\Delta}  \def\ddt{\dot\delta] \ \sim  \mathrm{Pf}(\theta^{\mu\nu}) \nonumber\\
  & \quad \in \ (3,0)  + (0,3) \subset    (1,1)^{\otimes_{\mathrm{sym}}^3}
  \label{chi-def}
 \end{align}
reduces to the Pfaffian of the Poisson tensor in the semi-classical limit, and
therefore it should be a good observable to define the chiral $L$ and $R$
sheets, cf.\ \cite{Steinacker:2014lma,Steinacker:2014eua,Steinacker:2015mia}.
It is easy to see \cite{Steinacker:2014lma} that on the six extremal weight states $w|\mu\rangle$
of $\cC[\mu]$ located at the origin,  $\chi$ takes the values
\begin{align}
 \chi =  (-1)^{|w|}(\alpha}          \def\da{{\dot\alpha}_1,\mu) (\alpha}          \def\da{{\dot\alpha}_2,\mu) (\alpha}          \def\da{{\dot\alpha}_3,\mu)  \,,
\end{align}
where $|w|$ is the signature of the appropriate Weyl group element, and $\alpha}          \def\da{{\dot\alpha}_i$ are the
simple roots.
In the semi-classical limit,
$\chi$ can be identified with a function on the brane (more precisely a polynomial of degree 3), which
takes positive or negative values
corresponding to the orientation of the 3+3 sheets of $\cC[\mu]$. This function is odd under $\cW$.
Thus on the extremal weight states, $\chi$ has the simple structure
\begin{align}
 \chi \sim \one_L - \one_R \,
 \label{chi-structure}
\end{align}
because it has weight zero, it is invariant under $\mathds{Z}_3$ and odd under Weyl reflections.
More precisely,
since $\chi$ is a cubic totally symmetric function on $\cC[\mu]$, it lives in
$(1,1)^{\otimes_{\mathrm{sym}}^3}$. On the other hand,
$\chi = \chi^{38}$ is the weight 0 state in $(1,1)^{\wedge^2} = (3,0) + (0,3) +
(1,1)$
of the $\mathfrak{s}\mathfrak{u}(3)$ intertwiner
\begin{align}
 \chi^{gh} &=  \varepsilon^{ab \ldots gh}[T_a,T_b] \ldots [T,T] \in
(1,1)^{\otimes_{\mathrm{sym}}^3} \cap (1,1)^{\wedge^2}\ = (3,0) + (0,3) +
(1,1)\qquad     ,
\nonumber\\
 \chi &= \chi^{38} = - \chi^{83} \ .
\end{align}
The $(1,1)$ contribution can be excluded\footnote{It would have to be $c^{38}_e T^e$ where $c^{ab}_c$ are the structure constants, which vanishes.},
so that $\chi$ is the hermitian weight 0 combination in
\begin{align}
 \chi \ \in \ (0,3) + (3,0) \ .
\end{align}

\paragraph{Chiral Higgs.}
We have seen that the extremal states of 6-dimensional branes decompose into
$3_L + 3_R$ sets of states with definite chirality.
Now we consider the regular Higgs mode on such a brane, and single out those
Higgs
$\tilde\phi_\alpha}          \def\da{{\dot\alpha}$  which respect the chirality, i.e.\
\begin{align}
 [ \tilde\phi^\alpha}          \def\da{{\dot\alpha},\chi] \sim 0 .
 \label{chiral-Higgs}
\end{align}
These are easy to identify for the $(1,1)$ brane in figure
\ref{fig:brane_1-1_Higgs}. While the maximal regular zero modes depicted
in figure \ref{fig:brane_1-1_rank1} are not chiral,  the
next-to-maximal regular zero modes  $\tilde\phi_\alpha}          \def\da{{\dot\alpha}\in \cH_{(3,0)} + \cH_{(0,3)} \subset \mathrm{End}(\cH)$  depicted in figures
\ref{fig:brane_1-1_rank2_1}--\ref{fig:brane_1-1_rank2_3} are in fact chiral, and have a
string-like structure
\begin{align}
 \tilde\phi^\alpha}          \def\da{{\dot\alpha}  \sim |x_L^i\rangle\langle x_L^j| \pm |x_R^i\rangle\langle x_R^j|
 \label{chiral-Higgs-decomp}
\end{align}
 relating extremal states with the same chirality; then \eqref{chiral-Higgs}
follows from \eqref{chi-structure}.

For the simplest case $\cC[(1,1)]$, these chiral Higgs modes form a closed algebra
and will lead to an exact solution, as
discussed in more detail in section \ref{sec:1-1_brane}.
This justifies the hypothesis that these $\tilde\phi^\alpha}          \def\da{{\dot\alpha}$ acquire a VEV.
The implications  for the fermions
and their Yukawa couplings will be discussed in
section \ref{sec:fermions}.

\begin{figure}[t!]
\centering
\begin{subfigure}{0.33\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (P1) at (-2,0);
  \coordinate (P2) at (-1,1);
  \coordinate (P3) at (-1,-1);
  \coordinate (P5) at (1,1);
  \coordinate (P7) at (1,-1);
  \coordinate (P8) at (2,0);
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
  \node at (P5) {$\bullet$};
  \node at (P7) {$\bullet$};
  \node at (P8) {$\bullet$};
\draw [thick,red,-latex] (P1) -- (P8) node[near end,below] {$\phi_1^+$};%
\draw [thick,red,-latex] (P7) -- (P2) node[near end,right] {$\phi_2^+$};%
\draw [thick,red,-latex] (P5) -- (P3) node[near end,left] {$\phi_3^+$};%
\draw[dotted] (P1)--(P3)--(P7)--(P8)--(P5)--(P2)--cycle;
\draw [black] (P1) node[left] {$R$};
\draw [black] (P2) node[above] {$L$};
\draw [black] (P3) node[below] {$L$};
\draw [black] (P5) node[above] {$R$};
\draw [black] (P7) node[below] {$R$};
\draw [black] (P8) node[right] {$L$};
\end{tikzpicture}%
\caption{}%
\label{fig:brane_1-1_rank1}%
\end{subfigure}%
\\
\begin{subfigure}{0.33\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (P1) at (-2,0);
  \coordinate (P2) at (-1,1);
  \coordinate (P3) at (-1,-1);
  \coordinate (P5) at (1,1);
  \coordinate (P7) at (1,-1);
  \coordinate (P8) at (2,0);
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
  \node at (P5) {$\bullet$};
  \node at (P7) {$\bullet$};
  \node at (P8) {$\bullet$};
\draw [thick,red,-latex] (P1) -- (P7) node[midway,above] {$\varphi_1^{+}$};%
\draw [thick,red,-latex] (P2) -- (P8) ;%
\draw [thick,blue,-latex] (P1) -- (P5);%
\draw [thick,blue,-latex] (P3) -- (P8)  node[midway,above] {$\sigma_1^{+}$};%
\draw[dotted] (P1)--(P3)--(P7)--(P8)--(P5)--(P2)--cycle;
\end{tikzpicture}%
\caption{}%
\label{fig:brane_1-1_rank2_1}%
\end{subfigure}%
\begin{subfigure}{0.33\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (P1) at (-2,0);
  \coordinate (P2) at (-1,1);
  \coordinate (P3) at (-1,-1);
  \coordinate (P5) at (1,1);
  \coordinate (P7) at (1,-1);
  \coordinate (P8) at (2,0);
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
  \node at (P5) {$\bullet$};
  \node at (P7) {$\bullet$};
  \node at (P8) {$\bullet$};
\draw [thick,red,-latex] (P3) -- (P2) ;%
\draw [thick,red,-latex] (P7) -- (P5) node[midway,left] {$\varphi_2^{+}$};%
\draw [thick,blue,-latex] (P7) -- (P1) node[midway,above] {$\sigma_2^{+}$};%
\draw [thick,blue,-latex] (P8) -- (P2) ;%
\draw[dotted] (P1)--(P3)--(P7)--(P8)--(P5)--(P2)--cycle;
\end{tikzpicture}%
\caption{}%
\label{fig:brane_1-1_rank2_2}%
\end{subfigure}%
\begin{subfigure}{0.33\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (P1) at (-2,0);
  \coordinate (P2) at (-1,1);
  \coordinate (P3) at (-1,-1);
  \coordinate (P5) at (1,1);
  \coordinate (P7) at (1,-1);
  \coordinate (P8) at (2,0);
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
  \node at (P5) {$\bullet$};
  \node at (P7) {$\bullet$};
  \node at (P8) {$\bullet$};
\draw [thick,red,-latex] (P5) -- (P1) node[midway,below] {$\varphi_3^{+}$};%
\draw [thick,red,-latex] (P8) -- (P3) ;%
\draw [thick,blue,-latex] (P2) -- (P3) ;%
\draw [thick,blue,-latex] (P5) -- (P7) node[midway,left] {$\sigma_3^{+}$};%
\draw[dotted] (P1)--(P3)--(P7)--(P8)--(P5)--(P2)--cycle;
\end{tikzpicture}%
\caption{}%
\label{fig:brane_1-1_rank2_3}%
\end{subfigure}%
\caption{Regular zero modes of rank $1$ or $2$ for $\cC[(1,1)]$ brane.
The maximal Higgs $\phi_i^+$ of (\subref{fig:brane_1-1_rank1}) are not chiral,
but the rank $2$ Higgs $\varphi_i^+$, $\sigma_i^+$ of
(\subref{fig:brane_1-1_rank2_1})--(\subref{fig:brane_1-1_rank2_3}) are chiral.}
\label{fig:brane_1-1_Higgs}
\end{figure}

\paragraph{Chiral gauge field $A_\mu$ and its mass.}
Among all the gauge fields on the $\cC[(N,M)]$ brane,
consider the $\chi$-valued gauge field
\begin{align}
A_\mu = A_\mu(x) \chi \ .
\label{chiral-gauge-mode}
\end{align}
We call it \emph{chiral}, because it
measures the chirality of the  $L$ and $R$  sheets according to
\eqref{chi-structure}, and
therefore couples accordingly to chiral fermions.
We are going to argue that $A_\mu$ may be the lightest non-trivial gauge mode
in the presence of a chiral Higgs VEV as above,  and describes a  gauge field
in a spontaneously broken $U(1)_L \times U(1)_R$ chiral gauge theory.

With this in mind,
the mass of the gauge boson $A_\mu$ in the presence of some Higgs $\phi_\alpha}          \def\da{{\dot\alpha}$ arises as usual from
\begin{align}
 D_\mu (X^a + \phi^a) D^\mu (X_a + \phi_a)
  &= \partial_\mu \phi_a \partial^\mu \phi^a + A_\mu(x)A^\mu(x) [\chi,X^a +
\phi^a][\chi,X_a + \phi_a] + \ldots  \nonumber\\
  &=  \partial_\mu \phi_a \partial^\mu \phi^a + A_\mu A^\mu \Big( m^2_{(3,0),0}  \chi \chi
   + [\chi,\phi^a][\chi,\phi_a]\Big)
\label{gaugeboson-mass-phi}
\end{align}
assuming $[X^a,\phi_a] =0$  (see \cite{Aschieri:2006uw}).
Since $\chi$ is a weight zero mode in $\cH_{(3,0)}$, the contribution form the
 brane $X$ is obtained from \eqref{gaugemodes-mass}
\begin{align}
 m^2_{(3,0),0} = 2(\L,\L+2\rho)
 = 2 (5,2)^T G (3,0) = 2\cdot 27 \ .
\end{align}
Assuming that a chiral  Higgs $\tilde\phi$ takes a VEV (as justified below),
it is natural to expect that $A_\mu \sim \chi$
will be the lightest gauge boson,
because  the mass contribution $[\chi,\tilde\phi^a][\chi, \tilde\phi_a]$ vanishes
due to its defining property \eqref{chiral-Higgs}.
All other non-chiral gauge modes $A'_\mu \sim \rho$  acquire an extra  mass
 $\mbox{tr} [\tilde\phi_\alpha}          \def\da{{\dot\alpha},\rho]^2$.

Explicitly, the masses of the lightest gauge modes (with zero weight)
due to the background are as follows:
\begin{align}
 m^2_{(3,0),0} &= 2 (5,2)^T G (3,0) = 2\cdot 36 ,  \nonumber\\
 m^2_{(1,1),0} &= 2(3,3)^T G (1,1) = 2\cdot 18 , \nonumber\\
 m^2_{(2,2),0} &= 2(4,4)^T G (2,2) = 2\cdot 48,
 \label{masses-gauge-modes}
\end{align}
using $\rho=(1,1)$. Here $G={\scriptsize \begin{pmatrix}
                           2 & 1 \\1 & 2
                          \end{pmatrix}}$ is the metric on $\mathfrak{s}\mathfrak{u}(3)$ weight space.
We restrict ourselves to gauge fields with weight zero
here\footnote{The modes with nonzero weight acquire additional
 mass terms from $\tilde\phi$ beyond the ones discussed here,
 which are difficult to evaluate. Moreover for large branes,
 we expect that the currents of fermionic links to a point brane
 have either large or zero weight, and therefore do not couple to light
gauge modes with  nonzero weight.};
note that e.g.\ $\L=(2,0)$ contains no weight zero modes.

For the minimal  $\cC[(1,1)]$ brane, we can also compute  the mass contribution from the chiral Higgs
$\tilde\phi_\alpha}          \def\da{{\dot\alpha}$ solution, which lives on the chiral triangles of the
$(1,1)$ branes as in figure \ref{fig:brane_1-1_plus_Higgs}.
To this end, we decompose the $(1,1)$ brane and its Hilbert space
into the two triangles consisting of the $L$ and $R$ states, which can be viewed as $(1,0)$ and $(0,1)$
branes defined by the chiral Higgs $\tilde\phi_\alpha}          \def\da{{\dot\alpha}$.
Then all gauge modes with weight zero live in  $End(1,0) = (1,1) + (0,0)$ and $End(0,1) =
(1,1) + (0,0)$
w.r.t.\ these triangles.
Hence the mass contribution for the chiral gauge field $A_\mu\sim\chi$ with $\L=(3,0)$ vanishes  (because $[\chi,\tilde\phi] = 0$),
while the mass contribution for the weight zero $\L=(1,1)$ mode is
\begin{align}
 m^2_{(1,1),0} &= 2(3,3)^T G (1,1) = 2\cdot 18  .
\end{align}
Adding this to \eqref{masses-gauge-modes}, we see that indeed the chiral boson $A_\mu \sim\chi$
as well as the two $\L=(1,1)$ gauge bosons are the
lightest, degenerate gauge bosons  (within weight zero), with mass $m^2 = 2\cdot36$.
Note that  the contribution from the chiral Higgs  to the
non-chiral gauge bosons such as $m^2_{(1,1),0} $ should be larger on larger branes, while
 \eqref{masses-gauge-modes} is universal.
It is then plausible that  the chiral gauge mode $A_\mu$ \eqref{chiral-gauge-mode}  becomes the lightest mode,
which would entail a chiral gauge theory
with a spontaneously broken $U(1)_L \times U(1)_R$ gauge field and 3 generations.

On larger branes, the details are more complicated, because there are many
chiral Higgs which may contribute to the mass.
However, the underlying geometrical mechanism is very clear: the $3_L+3_R$ sheets at the origin lead a priori to a
$U(3)_L \times U(3)_R$ gauge theory, which
is broken not only by the global connectedness of the brane leading to
\eqref{masses-gauge-modes}, but also by the chiral
Higgs modes which link the L and R sheets among themselves. These in turn
break the symmetry to $U(1)_L\times U(1)_R$ with 3 generations, and hopefully leave
the chiral gauge field $A_\mu$ as lightest non-trivial gauge boson.
Note also that in general, non-trivial Higgs configurations may lead to some back-reaction on the brane
(cf. the discussion in section \ref{sec:rigged-CP2}),  which may lead  to a relative shift between the $L$ and $R$
branes in target space, thus
amplifying the effects on the symmetry breaking. This should be kept in mind in
the discussion about approaching the Standard Model in section
\ref{sec:standardmodel}.
\subsection{\texorpdfstring{$\cC[(1,1)]$}{(1,1)} brane with chiral Higgs
solution}
\label{sec:1-1_brane}
On the  $\cC[(1,1)]$ brane, we have indeed an exact brane plus chiral Higgs
solution.
The underlying rank two regular zero modes have already been presented in
figures
\ref{fig:brane_1-1_rank2_1}--\ref{fig:brane_1-1_rank2_3}.
Because they form a closed algebra, we can combine these to form new exact
solutions of the form $X +\varphi$ and $X + \sigma$, which are depicted in
figure \ref{fig:brane_1-1_plus_Higgs}. The details of how to arrange this to
get an exact solution to the equations of motion are delegated to appendix
\ref{subsec:1-1_brane}.

\begin{figure}[t!]
\centering
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (P1) at (-2,0);
  \coordinate (P2) at (-1,1);
  \coordinate (P3) at (-1,-1);
  \coordinate (P5) at (1,1);
  \coordinate (P7) at (1,-1);
  \coordinate (P8) at (2,0);
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
  \node at (P5) {$\bullet$};
  \node at (P7) {$\bullet$};
  \node at (P8) {$\bullet$};
\draw [thick,red,-latex] (P5) -- (P1);%
\draw [thick,red,-latex] (P8) -- (P3);%
\draw [thick,red,-latex] (P3) -- (P2);%
\draw [thick,red,-latex] (P7) -- (P5);%
\draw [thick,red,-latex] (P1) -- (P7);%
\draw [thick,red,-latex] (P2) -- (P8);%
\draw[dotted] (P1)--(P3)--(P7)--(P8)--(P5)--(P2)--cycle;
\draw [black] (P1) node[left] {$R$};
\draw [black] (P2) node[above] {$L$};
\draw [black] (P3) node[below] {$L$};
\draw [black] (P5) node[above] {$R$};
\draw [black] (P7) node[below] {$R$};
\draw [black] (P8) node[right] {$L$};
\end{tikzpicture}%
\caption{}%
\label{fig:brane_1-1_config_1}%
\end{subfigure}%
\begin{subfigure}{0.48\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (P1) at (-2,0);
  \coordinate (P2) at (-1,1);
  \coordinate (P3) at (-1,-1);
  \coordinate (P5) at (1,1);
  \coordinate (P7) at (1,-1);
  \coordinate (P8) at (2,0);
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
  \node at (P5) {$\bullet$};
  \node at (P7) {$\bullet$};
  \node at (P8) {$\bullet$};
\draw [thick,blue,-latex] (P2) -- (P3);%
\draw [thick,blue,-latex] (P5) -- (P7);%
\draw [thick,blue,-latex] (P7) -- (P1);%
\draw [thick,blue,-latex] (P8) -- (P2);%
\draw [thick,blue,-latex] (P1) -- (P5);%
\draw [thick,blue,-latex] (P3) -- (P8);%
\draw[dotted] (P1)--(P3)--(P7)--(P8)--(P5)--(P2)--cycle;
\draw [black] (P1) node[left] {$R$};
\draw [black] (P2) node[above] {$L$};
\draw [black] (P3) node[below] {$L$};
\draw [black] (P5) node[above] {$R$};
\draw [black] (P7) node[below] {$R$};
\draw [black] (P8) node[right] {$L$};
\end{tikzpicture}%
\caption{}%
\label{fig:brane_1-1_config_2}%
\end{subfigure}%
\caption{$\cC[(1,1)]$ brane with non-maximal regular zero modes. (a) Brane
background together with a configuration involving $\varphi_i^{\pm}$. (b)
Brane background together with a configuration consisting of $\sigma_i^{\pm}$.}
\label{fig:brane_1-1_plus_Higgs}
\end{figure}
Having found two such equivalent solutions, we analyzed the spectrum of the
vector Laplacian around these exact solutions. As it turns out, there are a
number of negative modes, indicating potential instabilities. However
by including a mass terms $M_i\equiv M$, one can again eliminate all instabilities
for  $0.47\lesssim M \leq \frac{\sqrt{2}}{3}$, see for instance figure
\ref{fig:spectrum_1-1_brane}. In this massive case there remain 8 zero modes of
$\cO_V^{X+\varphi}$ or $\cO_V^{X+\sigma}$, which are again understood in terms of a compact moduli space,
which could presumably  be lifted by introducing different masses.
\subsection{Rigged \texorpdfstring{$\mathds{C} P^2$}{CP2} branes}
\label{sec:rigged-CP2}

Now consider $(N,1)$ branes. These can be viewed as a stack of
two $\mathds{C} P^2$ branes linked by a minimal fuzzy sphere $S^2_2$, see for instance
\cite{Grosse:2004wm}. We illustrate this set-up in figure
\ref{fig:inst-diagram}.
\begin{figure}[t!]
\centering
\begin{tikzpicture}
  \coordinate (P1) at (1,-1);
  \coordinate (P3) at (-1,-1);
  \coordinate (P4) at (1.5,0);
  \coordinate (P7) at (-1.5,0);
  \coordinate (P14) at (0.5,2);
  \coordinate (P15) at (-0.5,2);
  \node at (P1) {$\bullet$};
  \node at (P3) {$\bullet$};
  \node at (P4) {$\bullet$};
  \node at (P7) {$\bullet$};
  \node at (P14) {$\bullet$};
  \node at (P15) {$\bullet$};
  \draw[dotted] (P1)--(P3)--(P7)--(P15)--(P14)--(P4)--cycle;
\draw [thick,blue,-latex] (P1) -- (P7) -- (P14) -- cycle;%
\draw [thick,red,-latex]  (P3) -- (P4) -- (P15) -- cycle;%
\draw [black] (P3) node[left] {$L$};
\draw [black] (P4) node[right] {$L$};
\draw [black] (P15) node[left] {$L$};
\draw [black] (P1) node[right] {$R$};
\draw [black] (P7) node[left] {$R$};
\draw [black] (P14) node[right] {$R$};
\end{tikzpicture}%
 \caption{Rigged $(N,1)$ brane as two chiral branes, stabilized by chiral Higgs
(red \& blue).}
 \label{fig:inst-diagram}
\end{figure}
The minimal fuzzy 2-sphere plays the role of a Higgs linking the two $\mathds{C} P^2$
branes.
This can be understood noting that the extra $S^2$ fiber is
embedded transversal to the $\mathds{C} P^2$, leading to a 6-dimensional geometry.
This is also similar to the flipped branes connected by Higgs in section \ref{sec:flipped-branes},
which is now realized as an exact solution with non-trivial intrinsic topology.
A decomposition into $(N,0)$ branes can be obtained explicitly in a suitable basis, see \cite{Grosse:2004wm}.
This also leads to Yukawa couplings  which remove certain
fermions from the massless sector, resulting in the typical structure found on
6-dimensional branes.

\begin{figure}[t!]
\centering
\begin{subfigure}{0.25\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (P1) at (1,-1);
  \coordinate (P2) at (0,-1);
  \coordinate (P3) at (-1,-1);
  \coordinate (P4) at (1.5,0);
  \coordinate (P7) at (-1.5,0);
  \coordinate (P10) at (1,1);
  \coordinate (P12) at (-1,1);
  \coordinate (P14) at (0.5,2);
  \coordinate (P15) at (-0.5,2);
  \node at (P1) {$\bullet$};
  \node at (P2) {$\circ$};
  \node at (P3) {$\bullet$};
  \node at (P4) {$\bullet$};
  \node at (P7) {$\bullet$};
  \node at (P10) {$\circ$};
  \node at (P12) {$\circ$};
  \node at (P14) {$\bullet$};
  \node at (P15) {$\bullet$};
  \draw[dotted] (P1)--(P2)--(P3)--(P7)--(P12)--(P15)--(P14)--(P10)--(P4)--cycle;
\draw [thick,red,-latex] (P1) -- (P15) node[at end,right] {$\phi_1^+$};%
\draw [thick,red,-latex] (P14) -- (P3) node[at end,right] {$\phi_2^+$};%
\draw [thick,red,-latex] (P7) -- (P4) node[at end,above] {$\phi_3^+$};%
\end{tikzpicture}%
\caption{}%
\label{fig:brane_2-1_rank1}%
\end{subfigure}%
\begin{subfigure}{0.25\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (P1) at (1,-1);
  \coordinate (P2) at (0,-1);
  \coordinate (P3) at (-1,-1);
  \coordinate (P4) at (1.5,0);
  \coordinate (P7) at (-1.5,0);
  \coordinate (P10) at (1,1);
  \coordinate (P12) at (-1,1);
  \coordinate (P14) at (0.5,2);
  \coordinate (P15) at (-0.5,2);
  \node at (P1) {$\bullet$};
  \node at (P2) {$\circ$};
  \node at (P3) {$\bullet$};
  \node at (P4) {$\bullet$};
  \node at (P7) {$\bullet$};
  \node at (P10) {$\circ$};
  \node at (P12) {$\circ$};
  \node at (P14) {$\bullet$};
  \node at (P15) {$\bullet$};
  \draw[dotted] (P1)--(P2)--(P3)--(P7)--(P12)--(P15)--(P14)--(P10)--(P4)--cycle;
\draw [thick,red,-latex] (P4) -- (P15) node[midway,left] {$\varphi_1^{+}$};%
\draw [thick,red,-latex] (P1) -- (P12) ;%
\draw [thick,blue,-latex] (P1) -- (P14);%
\draw [thick,blue,-latex] (P2) -- (P15)  node[midway,right] {$\sigma_1^{+}$};%
\end{tikzpicture}%
\caption{}%
\label{fig:brane_2-1_rank2_1}%
\end{subfigure}%
\begin{subfigure}{0.25\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (P1) at (1,-1);
  \coordinate (P2) at (0,-1);
  \coordinate (P3) at (-1,-1);
  \coordinate (P4) at (1.5,0);
  \coordinate (P7) at (-1.5,0);
  \coordinate (P10) at (1,1);
  \coordinate (P12) at (-1,1);
  \coordinate (P14) at (0.5,2);
  \coordinate (P15) at (-0.5,2);
  \node at (P1) {$\bullet$};
  \node at (P2) {$\circ$};
  \node at (P3) {$\bullet$};
  \node at (P4) {$\bullet$};
  \node at (P7) {$\bullet$};
  \node at (P10) {$\circ$};
  \node at (P12) {$\circ$};
  \node at (P14) {$\bullet$};
  \node at (P15) {$\bullet$};
  \draw[dotted] (P1)--(P2)--(P3)--(P7)--(P12)--(P15)--(P14)--(P10)--(P4)--cycle;
\draw [thick,red,-latex] (P14) -- (P2) ;%
\draw [thick,red,-latex] (P15) -- (P3) node[midway,right] {$\varphi_2^{+}$};%
\draw [thick,blue,-latex] (P10) -- (P3) node[midway,left] {$\sigma_2^{+}$};%
\draw [thick,blue,-latex] (P14) -- (P7) ;%
\end{tikzpicture}%
\caption{}%
\label{fig:brane_2-1_rank2_2}%
\end{subfigure}%
\begin{subfigure}{0.25\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (P1) at (1,-1);
  \coordinate (P2) at (0,-1);
  \coordinate (P3) at (-1,-1);
  \coordinate (P4) at (1.5,0);
  \coordinate (P7) at (-1.5,0);
  \coordinate (P10) at (1,1);
  \coordinate (P12) at (-1,1);
  \coordinate (P14) at (0.5,2);
  \coordinate (P15) at (-0.5,2);
  \node at (P1) {$\bullet$};
  \node at (P2) {$\circ$};
  \node at (P3) {$\bullet$};
  \node at (P4) {$\bullet$};
  \node at (P7) {$\bullet$};
  \node at (P10) {$\circ$};
  \node at (P12) {$\circ$};
  \node at (P14) {$\bullet$};
  \node at (P15) {$\bullet$};
  \draw[dotted] (P1)--(P2)--(P3)--(P7)--(P12)--(P15)--(P14)--(P10)--(P4)--cycle;
\draw [thick,red,-latex] (P12) -- (P4) node[midway,below] {$\varphi_3^{+}$};%
\draw [thick,red,-latex] (P7) -- (P1) ;%
\draw [thick,blue,-latex] (P7) -- (P10) node[midway,above right]
{$\sigma_3^{+}$};%
\draw [thick,blue,-latex] (P3) -- (P4);%
\end{tikzpicture}%
\caption{}%
\label{fig:brane_2-1_rank2_3}%
\end{subfigure}%
\caption{The rank 1 and 2 regular zero modes for the $\cC[(2,1)]$ brane,
exemplifying the $(N,1)$ case. The maximal Higgs $\phi_i^+$ of
(\subref{fig:brane_2-1_rank1}) connect extremal weight states, while the
next-to-maximal Higgs $\varphi_i^+$, $\sigma_i^+$ of
(\subref{fig:brane_2-1_rank2_1})--(\subref{fig:brane_2-1_rank2_3}) also relate
extremal and non-extremal weight states. Only the latter are (approximately) chiral.}
\label{fig:brane_2-1_Higgs}
\end{figure}

\paragraph{Equations of motion.}
For these $(N,1)$ branes, the \emph{next-to maximal} Higgs modes play again
the role of
chiral stringy Higgs modes. In view of figure
(\ref{fig:brane_2-1_rank2_1})--(\ref{fig:brane_2-1_rank2_3}),
these modes have the structure
\begin{align}
 \varphi_i^+ &= \tilde{\varphi}_{i}^+ + H_{i}^+
 \label{chiral-Higgs-rigged}
 \end{align}
 where
 \begin{align}
 \label{eq:Higgs_rigged_CP2}
  \tilde{\varphi}_{i}^+ &= \alpha}          \def\da{{\dot\alpha}_N|x_L^i\rangle\langle x_L^j|,   \qquad
   H_{i}^+ = h_N |x_R^i\rangle\langle {x'}^j|,  \qquad
    h_N \sim \frac{1}{\sqrt{N}}\alpha}          \def\da{{\dot\alpha}_N
\end{align}
because
\begin{align}
0 =  [X_i^+ ,\varphi_j^-] = X_i^+ \tilde\varphi_j^- - \tilde\varphi_j^-X_i^+
\sim (\alpha}          \def\da{{\dot\alpha}_N - h_N \sqrt{N}) \phi_j^- \ .
\end{align}
Note that $\varphi_{i}$ relates again extremal states with the same chirality,
but they no longer form a closed algebra
because of the sub-leading $H$ contribution.
Hence the $\varphi_i$ do not yield an exact solution within the Higgs
sector. Nevertheless they lower the energy of the brane, and
we expect that there exist slightly \emph{deformed} solutions with similar
properties. Presumably, such solutions would involve small admixtures of
other zero modes (and possibly massive modes). For large $N$,
this deformation should be negligible.
The argument for the $\sigma_j$ modes is completely analogous.

\subsection{\texorpdfstring{$\cC[(N,N)]$}{C[(N,N)]} branes}
Finally, we briefly consider $\cC[(N,N)]$ branes
as sketched in figure \ref{fig:brane_2-2_Higgs}.
Among the Higgs modes we mention the maximal zero modes
$\phi_i^+$ in $(2N,2N)$, and the $(3N,0)$ modes
$\tilde{\varphi}_i^+$, $\tilde{\sigma}_i^+$ (and similarly $(0,3N)$ modes)
connecting the opposite edges of $(N,N)$. These
are again approximately chiral Higgs modes
(similar to the $(1,1)$ case), which do not quite form a closed algebra,
but clearly lower the energy of the brane.
Hence we expect that there are
nearby deformed solutions.
\begin{figure}[t!]
\centering
\begin{subfigure}{0.25\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (P1) at (1,-2);
  \coordinate (P2) at (0,-2);
  \coordinate (P3) at (-1,-2);
  \coordinate (P4) at (1.5,-1);
  \coordinate (P7) at (-1.5,-1);
  \coordinate (P8) at (2,0);
  \coordinate (P12) at (-2,0);
  \coordinate (P18) at (1.5,1);
  \coordinate (P21) at (-1.5,1);
  \coordinate (P25) at (1,2);
  \coordinate (P26) at (0,2);
  \coordinate (P27) at (-1,2);
  \node at (P1) {$\bullet$};
  \node at (P2) {$\circ$};
  \node at (P3) {$\bullet$};
  \node at (P4) {$\circ$};
  \node at (P7) {$\circ$};
  \node at (P8) {$\bullet$};
  \node at (P12) {$\bullet$};
  \node at (P18) {$\circ$};
  \node at (P21) {$\circ$};
  \node at (P25) {$\bullet$};
  \node at (P26) {$\circ$};
  \node at (P27) {$\bullet$};
  \draw[dotted] (P1)--(P3)--(P12)--(P27)--(P25)--(P8)--cycle;
\draw [thick,red,-latex] (P1) -- (P27) node[at end,right] {$\phi_1^+$};%
\draw [thick,red,-latex] (P25) -- (P3) node[at end,right] {$\phi_2^+$};%
\draw [thick,red,-latex] (P12) -- (P8) node[at end,above] {$\phi_3^+$};%
\end{tikzpicture}%
\caption{}%
\label{fig:brane_2-2_rank1}%
\end{subfigure}%
\begin{subfigure}{0.25\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (P1) at (1,-2);
  \coordinate (P2) at (0,-2);
  \coordinate (P3) at (-1,-2);
  \coordinate (P4) at (1.5,-1);
  \coordinate (P7) at (-1.5,-1);
  \coordinate (P8) at (2,0);
  \coordinate (P12) at (-2,0);
  \coordinate (P18) at (1.5,1);
  \coordinate (P21) at (-1.5,1);
  \coordinate (P25) at (1,2);
  \coordinate (P26) at (0,2);
  \coordinate (P27) at (-1,2);
  \node at (P1) {$\bullet$};
  \node at (P2) {$\circ$};
  \node at (P3) {$\bullet$};
  \node at (P4) {$\circ$};
  \node at (P7) {$\circ$};
  \node at (P8) {$\bullet$};
  \node at (P12) {$\bullet$};
  \node at (P18) {$\circ$};
  \node at (P21) {$\circ$};
  \node at (P25) {$\bullet$};
  \node at (P26) {$\circ$};
  \node at (P27) {$\bullet$};
  \draw[dotted] (P1)--(P3)--(P12)--(P27)--(P25)--(P8)--cycle;
\draw [thick,red,-latex] (P1) -- (P21) node[midway,left] {$\varphi_1^{+}$};%
\draw [thick,red,-latex] (P4) -- (P27) ;%
\draw [thick,blue,-latex] (P1) -- (P26);%
\draw [thick,blue,-latex] (P2) -- (P27)  node[midway,right] {$\sigma_1^{+}$};%
\end{tikzpicture}%
\caption{}%
\label{fig:brane_2-2_rank2_1}%
\end{subfigure}%
\begin{subfigure}{0.25\textwidth}
\centering
\begin{tikzpicture}
    \coordinate (P1) at (1,-2);
  \coordinate (P2) at (0,-2);
  \coordinate (P3) at (-1,-2);
  \coordinate (P4) at (1.5,-1);
  \coordinate (P7) at (-1.5,-1);
  \coordinate (P8) at (2,0);
  \coordinate (P12) at (-2,0);
  \coordinate (P18) at (1.5,1);
  \coordinate (P21) at (-1.5,1);
  \coordinate (P25) at (1,2);
  \coordinate (P26) at (0,2);
  \coordinate (P27) at (-1,2);
  \node at (P1) {$\bullet$};
  \node at (P2) {$\circ$};
  \node at (P3) {$\bullet$};
  \node at (P4) {$\circ$};
  \node at (P7) {$\circ$};
  \node at (P8) {$\bullet$};
  \node at (P12) {$\bullet$};
  \node at (P18) {$\circ$};
  \node at (P21) {$\circ$};
  \node at (P25) {$\bullet$};
  \node at (P26) {$\circ$};
  \node at (P27) {$\bullet$};
  \draw[dotted] (P1)--(P3)--(P12)--(P27)--(P25)--(P8)--cycle
\draw [thick,red,-latex] (P25) -- (P2) ;%
\draw [thick,red,-latex] (P26) -- (P3) node[midway,right] {$\varphi_2^{+}$};%
\draw [thick,blue,-latex] (P25) -- (P7) node[midway,left] {$\sigma_2^{+}$};%
\draw [thick,blue,-latex] (P18) -- (P3) ;%
\end{tikzpicture}%
\caption{}%
\label{fig:brane_2-2_rank2_2}%
\end{subfigure}%
\begin{subfigure}{0.25\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (P1) at (1,-2);
  \coordinate (P2) at (0,-2);
  \coordinate (P3) at (-1,-2);
  \coordinate (P4) at (1.5,-1);
  \coordinate (P7) at (-1.5,-1);
  \coordinate (P8) at (2,0);
  \coordinate (P12) at (-2,0);
  \coordinate (P18) at (1.5,1);
  \coordinate (P21) at (-1.5,1);
  \coordinate (P25) at (1,2);
  \coordinate (P26) at (0,2);
  \coordinate (P27) at (-1,2);
  \node at (P1) {$\bullet$};
  \node at (P2) {$\circ$};
  \node at (P3) {$\bullet$};
  \node at (P4) {$\circ$};
  \node at (P7) {$\circ$};
  \node at (P8) {$\bullet$};
  \node at (P12) {$\bullet$};
  \node at (P18) {$\circ$};
  \node at (P21) {$\circ$};
  \node at (P25) {$\bullet$};
  \node at (P26) {$\circ$};
  \node at (P27) {$\bullet$};
  \draw[dotted] (P1)--(P3)--(P12)--(P27)--(P25)--(P8)--cycle
\draw [thick,red,-latex] (P21) -- (P8) node[midway,below] {$\varphi_3^{+}$};%
\draw [thick,red,-latex] (P12) -- (P4) ;%
\draw [thick,blue,-latex] (P12) -- (P18) node[midway,above right]
{$\sigma_3^{+}$};%
\draw [thick,blue,-latex] (P7) -- (P8);%
\end{tikzpicture}%
\caption{}%
\label{fig:brane_2-2_rank2_3}%
\end{subfigure}%
\\
\begin{subfigure}{0.25\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (P1) at (1,-2);
  \coordinate (P2) at (0,-2);
  \coordinate (P3) at (-1,-2);
  \coordinate (P4) at (1.5,-1);
  \coordinate (P7) at (-1.5,-1);
  \coordinate (P8) at (2,0);
  \coordinate (P12) at (-2,0);
  \coordinate (P18) at (1.5,1);
  \coordinate (P21) at (-1.5,1);
  \coordinate (P25) at (1,2);
  \coordinate (P26) at (0,2);
  \coordinate (P27) at (-1,2);
  \node at (P1) {$\bullet$};
  \node at (P2) {$\circ$};
  \node at (P3) {$\bullet$};
  \node at (P4) {$\circ$};
  \node at (P7) {$\circ$};
  \node at (P8) {$\bullet$};
  \node at (P12) {$\bullet$};
  \node at (P18) {$\circ$};
  \node at (P21) {$\circ$};
  \node at (P25) {$\bullet$};
  \node at (P26) {$\circ$};
  \node at (P27) {$\bullet$};
  \draw[dotted] (P1)--(P3)--(P12)--(P27)--(P25)--(P8)--cycle;
\draw [thick,red,-latex] (P8) -- (P27) node[midway,above]
{$\tilde{\varphi}_1^{+}$};%
\draw [thick,red,-latex] (P4) -- (P21) ;%
\draw [thick,red,-latex] (P1) -- (P12) ;%
\draw [thick,blue,-latex] (P1) -- (P25);%
\draw [thick,blue,-latex] (P2) -- (P26);%
\draw [thick,blue,-latex] (P3) -- (P27)  node[midway,right]
{$\tilde{\sigma}_1^{+}$};%
\end{tikzpicture}%
\caption{}%
\label{fig:brane_2-2_rank3_1}%
\end{subfigure}%
\begin{subfigure}{0.25\textwidth}
\centering
\begin{tikzpicture}
    \coordinate (P1) at (1,-2);
  \coordinate (P2) at (0,-2);
  \coordinate (P3) at (-1,-2);
  \coordinate (P4) at (1.5,-1);
  \coordinate (P7) at (-1.5,-1);
  \coordinate (P8) at (2,0);
  \coordinate (P12) at (-2,0);
  \coordinate (P18) at (1.5,1);
  \coordinate (P21) at (-1.5,1);
  \coordinate (P25) at (1,2);
  \coordinate (P26) at (0,2);
  \coordinate (P27) at (-1,2);
  \node at (P1) {$\bullet$};
  \node at (P2) {$\circ$};
  \node at (P3) {$\bullet$};
  \node at (P4) {$\circ$};
  \node at (P7) {$\circ$};
  \node at (P8) {$\bullet$};
  \node at (P12) {$\bullet$};
  \node at (P18) {$\circ$};
  \node at (P21) {$\circ$};
  \node at (P25) {$\bullet$};
  \node at (P26) {$\circ$};
  \node at (P27) {$\bullet$};
  \draw[dotted] (P1)--(P3)--(P12)--(P27)--(P25)--(P8)--cycle
\draw [thick,red,-latex] (P27) -- (P3) node[midway,left]
{$\tilde{\varphi}_2^{+}$};%
\draw [thick,red,-latex] (P26) -- (P2) ;%
\draw [thick,red,-latex] (P25) -- (P1) ;
\draw [thick,blue,-latex] (P25) -- (P12) ;%
\draw [thick,blue,-latex] (P18) -- (P7) ;%
\draw [thick,blue,-latex] (P8) -- (P3) node[midway,below]
{$\tilde{\sigma}_2^{+}$};%
\end{tikzpicture}%
\caption{}%
\label{fig:brane_2-2_rank3_2}%
\end{subfigure}%
\begin{subfigure}{0.25\textwidth}
\centering
\begin{tikzpicture}
  \coordinate (P1) at (1,-2);
  \coordinate (P2) at (0,-2);
  \coordinate (P3) at (-1,-2);
  \coordinate (P4) at (1.5,-1);
  \coordinate (P7) at (-1.5,-1);
  \coordinate (P8) at (2,0);
  \coordinate (P12) at (-2,0);
  \coordinate (P18) at (1.5,1);
  \coordinate (P21) at (-1.5,1);
  \coordinate (P25) at (1,2);
  \coordinate (P26) at (0,2);
  \coordinate (P27) at (-1,2);
  \node at (P1) {$\bullet$};
  \node at (P2) {$\circ$};
  \node at (P3) {$\bullet$};
  \node at (P4) {$\circ$};
  \node at (P7) {$\circ$};
  \node at (P8) {$\bullet$};
  \node at (P12) {$\bullet$};
  \node at (P18) {$\circ$};
  \node at (P21) {$\circ$};
  \node at (P25) {$\bullet$};
  \node at (P26) {$\circ$};
  \node at (P27) {$\bullet$};
  \draw[dotted] (P1)--(P3)--(P12)--(P27)--(P25)--(P8)--cycle
\draw [thick,red,-latex] (P27) -- (P8) node[midway,above]
{$\tilde{\varphi}_3^{+}$};%
\draw [thick,red,-latex] (P21) -- (P4) ;%
\draw [thick,red,-latex] (P12) -- (P1) ;%
\draw [thick,blue,-latex] (P12) -- (P25) node[midway,below]
{$\tilde{\sigma}_3^{+}$};%
\draw [thick,blue,-latex] (P7) -- (P18);%
\draw [thick,blue,-latex] (P3) -- (P8);%
\end{tikzpicture}%
\caption{}%
\label{fig:brane_2-2_rank3_3}%
\end{subfigure}%
\caption{The rank one, two, and three regular zero modes for the $\cC[(2,2)]$
brane,
exemplifying the $(N,N)$ case. The maximal Higgs $\phi_i^+$ of
(\subref{fig:brane_2-2_rank1}) connect extremal weight states. The
next-to-maximal Higgs $\varphi_i^+$, $\sigma_i^+$ of
(\subref{fig:brane_2-2_rank2_1})--(\subref{fig:brane_2-2_rank2_3}) also relate
extremal
and non-extremal weight states. Neither the rank one nor the rank two zero
modes are chiral. Only the rank three zero modes of
(\subref{fig:brane_2-2_rank3_1})--(\subref{fig:brane_2-2_rank3_3}) are chiral
modes.}
\label{fig:brane_2-2_Higgs}
\end{figure}
\subsection{Nonabelian case: Stacks of 6-dimensional branes and point-branes}
\label{sec:nonabel-Higgs-6D}
Now consider a stack of two identical 6-dimensional branes as above, each with
chiral Higgs $\tilde \phi$ switched on, and add also
an extra point brane ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$ to make it more interesting.
This clearly leads to an unbroken $U(2)$ gauge group. However, the results of
the last sections lead to a more refined statement:
the two branes lead to a $U(2)_L\times U(2)_R$ gauge theory which is
spontaneously broken to
$U(2)_{\mbox{diag}}$, and massive chiral gauge bosons $A_\mu$ taking values in
$\mathfrak{u}(2)_L - \mathfrak{u}(2)_R$.
We will see in section \ref{sec:fermions} that there are also fermionic zero
modes linking ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$ with the $3_L + 3_R$ sheets on each brane,
leading to 3 generations of chiral fermions transforming in the fundamental of
$U(2)_L$ and $U(2)_R$, respectively,
which have opposite charges under $A_\mu$ according to their chirality.
This is a chiral gauge theory in a broken phase, reminiscent of the $SU(2)_L
\times SU(2)_R$ Pati-Salam-type electroweak model.
Further suitable (maximal, non-chiral) Higgs
between the two 6-dimensional branes may break the symmetry to $U(1)$ and lead
to patterns quite close to the Standard Model, which
will be discussed  in section \ref{sec:standardmodel}.
\section{Fermions on branes with Higgs}
\label{sec:fermions}
The Dirac operator on a squashed background $\cC[\mu]$ is given by
\begin{equation}
 \slashed{D}^X = \sqrt{2} \sum_{j=1}^3 \left(\Delta_j^- [X_j^+,\cdot] +
\Delta_j^+ [X_j^-,\cdot]\right)
\end{equation}
acting on $\mathrm{End}(\cH) \otimes {\cal S}} \def\cT{{\cal T}} \def\cU{{\cal U}$, where ${\cal S}} \def\cT{{\cal T}} \def\cU{{\cal U}\cong \mathds{C}^8$
accommodates the spinors.
The $\Delta_j^\pm$ are fermionic ladder operators which satisfy
$\{\Delta_j^-,\Delta_k^+\}=\delta_{j,k}$. For the
squashed background $\cC[\mu]$, the $X_j^\pm$ act as ladder operators for the
preserved $U(1)_{K_i}$ charges. With this input, it was shown in
\cite{Steinacker:2014lma,Steinacker:2014eua,Steinacker:2015mia}
that the fermionic zero modes $\Psi_{\alpha}          \def\da{{\dot\alpha},\L'}$ on $\cC[\mu]$
correspond to the extremal weight states of each
irrep $\cH_\L$ appearing in $\mathrm{End}(\cH)=\oplus_\L \cH_\L$, and are
in one-to-one correspondence to the
regular zero modes $\phi_{\alpha}          \def\da{{\dot\alpha},\L'}$ of the scalar fields discussed in section
\ref{sec:regular-zm}.
The proof in \cite{Steinacker:2014lma} is based on the extremal weight
properties, while a proof in the spirit of index theory was given in
\cite{Steinacker:2015mia}.
Consequently, the zero modes are again labeled by their $U(1)_{K_i}$ quantum
numbers $\L'$, and
their chirality is determined by the
parity $\tau=\pm 1$ of (the Weyl chamber of) $\L'$.
In addition, there exist two trivial gaugino zero modes on each brane.

The results can be summarized by stating that a quiver gauge theory arises on
stacks of squashed branes $\oplus n_i\cC[\mu_i]$, with gauge group $U(n_i)$ on
each node $\mu_i$
and arrows corresponding to chiral superfields $\phi_{\alpha}          \def\da{{\dot\alpha},\L'}$
labeled by the extremal weights $\L'$ corresponding to
the multiplets $\cH_\L \subset \mathrm{Hom}(\cH_{\mu_i},\cH_{\mu_j})$.
The trivial modes $\L=0$ on each node lead to $\cN=4$ supermultiplets.

More specifically, on a given stack of squashed $\cC[\mu_i]$ branes,
fermionic zero modes arise in two ways: first, as \emph{intra-brane fermions}
$\Psi\in \mathrm{End}(\cH_{\mu_i}) \otimes {\cal S}} \def\cT{{\cal T}} \def\cU{{\cal U}$ on some given brane $\cC[\mu_i]$.
The intra-brane fermions are uncharged under the gauge groups arising on the
(stacks of) different branes, but they are chiral and charged under
$U(1)_{K_i}$. The latter two features protect them
from acquiring any mass terms on the $\cC[\mu_i]$ background, because opposite
charges have opposite chirality.
Nevertheless, these modes may acquire masses in the presence of
non-vanishing Higgs modes due to Yukawa couplings discussed below.
In contrast, the two trivial gaugino modes with $\L=0$ are unprotected, and
are therefore expected to acquire a large mass due to the soft SUSY breaking,
either at tree level or through loop corrections.

Second and more interestingly, fermionic zero modes also arise as \emph{inter-brane fermions}
$\Psi\in \mathrm{Hom}(\cH_{\mu_i},\cH_{\mu_j}) \otimes {\cal S}} \def\cT{{\cal T}} \def\cU{{\cal U}$ linking two different
(stacks of) branes.
These are  charged under the gauge groups arising on the
(stacks of) different branes, chiral, and protected by their $U(1)_{K_i}$ charges.
Hence they can acquire a mass only through Yukawa couplings in the presence of
 Higgs modes linking different branes.
Note that due to the 9+1-dimensional Majorana-Weyl condition, the
$\Psi\in \mathrm{Hom}(\cH_{\mu_i},\cH_{\mu_j}) \otimes {\cal S}} \def\cT{{\cal T}} \def\cU{{\cal U}$ are related via charge conjugation
to the $\Psi\in \mathrm{Hom}(\cH_{\mu_j},\cH_{\mu_i}) \otimes {\cal S}} \def\cT{{\cal T}} \def\cU{{\cal U}$. This is important to
avoid over-counting
e.g. in \eqref{fermion-matrix}, and to obtain a chiral gauge theory.

\paragraph{Yukawa couplings.}
Assume that $\cC[\mu_L]$ and $\cC[\mu_R]$ are connected with some Higgs
$\phi_{\alpha}          \def\da{{\dot\alpha}}$. Then Yukawa couplings of two such
fermionic zero modes arise from the $\cN=4$ action \eqref{N=4SYM}, with
structure
\begin{align}
  \mbox{tr} \left( \overline{\Psi} \gamma_5 \Delta^\alpha}          \def\da{{\dot\alpha}[\phi_\alpha}          \def\da{{\dot\alpha},\Psi] \right)\ .
 \label{Yukawa}
\end{align}
This respects the $U(3)_R$ symmetry and the $U(1)_{K_i}$ symmetry.
Consequently, the non-vanishing Yukawa couplings in the zero-mode sector have
the same structure as the cubic term $V_{\mathrm{soft}}$ in the potential,
\begin{align}
 \mbox{tr} \left( \overline{\Psi}_{-\alpha}          \def\da{{\dot\alpha}_i} \gamma_5
\Delta^{\alpha}          \def\da{{\dot\alpha}_j}[\phi_{\alpha}          \def\da{{\dot\alpha}_j},\Psi_{\alpha}          \def\da{{\dot\alpha}_k}]
\right) \ \sim\ \varepsilon_{ijk}
\label{tau-cons-yuk}
\end{align}
and its conjugate. These Yukawa couplings are (non-)vanishing if and only if the
corresponding cubic term $\mbox{tr} ( \phi_{\alpha}          \def\da{{\dot\alpha}_i} [\phi_{\alpha}          \def\da{{\dot\alpha}_j},\phi_{\alpha}          \def\da{{\dot\alpha}_k}])$
(with the same $U(1)_{K_i}$ quantum numbers) is (non-)vanishing.
This requires in particular that the $U(1)^K_i$ charges $\L'$ of $\phi_{\alpha}          \def\da{{\dot\alpha}_j}$
and $\Psi_{\alpha}          \def\da{{\dot\alpha}_k}$ add up to that of $\Psi_{\alpha}          \def\da{{\dot\alpha}_i}$.
In particular, \emph{the $\tau$-parities of $\alpha}          \def\da{{\dot\alpha}_i,\alpha}          \def\da{{\dot\alpha}_j,\alpha}          \def\da{{\dot\alpha}_k$ must be equal}.
\paragraph{Fermionic zero modes on branes with Higgs.}
For a combined brane plus Higgs background $Y=X+\phi$, the above
classification of fermionic (and bosonic) zero modes  does not apply
any more. The reason is that the $Y_j^\pm$ are typically
no longer ladder operators and do not satisfy any Lie algebra relations.
As illustrated in various settings in appendix
\ref{app:solutions_spectrum}, the spectrum of $\cO_V^{Y}$ and ${\slashed{D}}^Y$ may
behave  utterly different compared to their spectrum on the squashed backgrounds.

Nevertheless, one can understand in many cases the fate of the fermionic zero modes, and
obtain a qualitative understanding of the remaining low-energy sector.
We will argue that some of the chiral fermionic zero modes
of ${\slashed{D}}^X$ are coupled by the Yukawa couplings induced by $\phi$ and acquire a
mass. Hence, they disappear from the low-energy spectrum on the combined
background $X+\phi$.
On the other hand, some other fermionic zero modes are protected and remain massless.
Adding a point brane ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$ to the combined background
solution, the inter-brane fermionic zero modes may lead to a very interesting low-energy physics,
reproducing  ingredients of the Standard Model.

Due to the complicated setting, most of these arguments are only qualitative at this point, and
we do not have a complete understanding in all cases.
The detailed numerical results are given in appendix
\ref{app:solutions_spectrum}.
\subsection{Fermions on \texorpdfstring{$\cC[(N,0)]$}{C[N,0]} branes with
maximal Higgs}
For a single $\cC[(N,0)]$ background, there are $6(N+1)+2$ fermionic zero
modes. These consist of $6(N+1)$ modes from the decomposition
$\mathrm{End}(\cH_{(N,0)})=\oplus_{l=0}^N \cH_{(l,l)}$, plus two trivial gaugino
modes. Turning on the maximal regular bosonic zero modes $\phi$, the spectrum
of ${\slashed{D}}^{X+\phi}$ contains only 14 zero modes independent of $N$, see figure
\ref{fig:CP2-brane+Higgs_Dirac-modes}. This reduction clearly arises from the Yukawa couplings
due to the maximal Higgs $\phi$, which leaves the $6+2$ zero modes from $\cH_{(0,0)}$, plus 6 extra zero
modes whose origin is obscure.

Adding a point brane ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$ to this $X+\phi$, the
number of fermionic zero modes is 22 according to figure
\ref{fig:Dirac-modes_brane+point_x=0+1_2},
which differs by 8 from the 14 modes on $\cC[(N,0)]$ with maximal Higgs.
We can understand this as follows:
A priori, there are $2\cdot6$ fermionic inter-brane zero modes between
$\cC[(N,0)]$ and ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$,
which arise from $\mathrm{Hom}(\cH_{(N,0)},\mathds{C}) \cong \cH_{(N,0)}$.
However, upon switching on the maximal Higgs,
Yukawa couplings $\mbox{tr}(\widetilde{\Psi}_3^+ [\phi_2^+,\Psi_1^+])$ arise as
depicted in figure \ref{fig:CP2-brane-center-fermions}, which couple these zero modes and give them a mass.
This leaves only the $6+2$ intra-brane fermions on ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$, which remain massless.
This  explains the numerical findings.
\begin{figure}[t!]
\centering
\begin{tikzpicture}
  \coordinate (Origin)   at (0,0);
  \coordinate (P1) at (-2,-1);
  \coordinate (P2) at (0,2);
  \coordinate (P3) at (2,-1);
  \coordinate (P1+a3) at (-2-0.2,-1-0.3);
  \coordinate (P1-a1) at (-2-0.5,-1);
  \coordinate (P2+a2) at (0-0.2,2+0.3);
  \coordinate (P2-a3) at (0+0.2,2+0.3);
  \coordinate (P3+a1) at (2+0.5,-1);
  \coordinate (P3-a2) at (2+0.2,-1-0.3);
  \node at (Origin) {$\bigstar$};
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
  \draw [thick,-latex] (Origin)--(P1+a3) node[at end,below] {$\psi_3^+$};
  \draw [thick,-latex] (Origin)--(P1-a1) node[at end,left]
{$\widetilde{\Psi}_1^-$};
  \draw [thick,-latex] (Origin)--(P2+a2) node[at end,left] {$\psi_2^+$};
  \draw [thick,-latex] (Origin)--(P2-a3) node[at end,right]
{$\widetilde{\Psi}_3^-$};
  \draw [thick,-latex] (Origin)--(P3+a1) node[at end,right] {$\psi_1^+$};
  \draw [thick,-latex] (Origin)--(P3-a2) node[at end,below]
{$\widetilde{\Psi}_2^-$};
\draw [dashed] (P1)--(P2)--(P3)--cycle;
 \draw [red,thick,-latex] (P3+a1) -- (P2-a3) node[midway,right] {$\phi_2^+$};
 \draw [red,thick,-latex] (P2+a2) -- (P1-a1) node[midway,left] {$\phi_3^+$};
 \draw [red,thick,-latex] (P1+a3) -- (P3-a2) node[midway,below] {$\phi_1^+$};
\end{tikzpicture}%
\caption{Chiral inter-brane fermions $\Psi_{\alpha}          \def\da{{\dot\alpha},\L'}$ and
$\widetilde{\Psi}_{\alpha}          \def\da{{\dot\alpha},\L'}$ linking $\cC[(N,0)]$  with a point-brane ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$,
which is represented by $\bigstar$.
Their chirality is indicated by the
sign $\pm$, which is inherited from the Weyl chamber. Red arrows correspond to
maximal Higgs $\phi_j^+$.}
\label{fig:CP2-brane-center-fermions}
\end{figure}
\subsection{Fermions on \texorpdfstring{$\cC[(1,1)]$}{C[1,1]} brane with
chiral Higgs, and point brane}
\label{sec:ferm-11-chiral}
Now we come to our most interesting solution: start with
one of the two exact solutions $X + \varphi$ or $X+\sigma$
of figure \ref{fig:brane_1-1_plus_Higgs}, corresponding to a $\cC[(1,1)]$ brane with chiral Higgs.
As shown in figure \ref{fig:spectrum_1-1_brane}, the number of
fermionic zero modes is reduced to 20, in comparison to the 38 zero modes of the
$\cC[(1,1)]$ brane without Higgs.

Now, add a  point brane ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$ to the above $\cC[(1,1)]$ with chiral Higgs.
Then on top of the above 20 fermionic zero modes from $\cC[(1,1)]$ with chiral Higgs,
the numerical analysis reveals 20 additional fermionic zero modes. These are understood as follows:
$2 \cdot 6$ zero-modes arise from inter-brane zero modes $\Psi_{\cC,{\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}}$
corresponding to $\mathrm{Hom}(\cH_{(1,1)},\mathds{C})$ and its conjugate.
In addition, there are 8 trivial intra-brane fermions on ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$,
which consist of 6 modes in $\mathrm{End}(\cH_{(0,0)})\cong \mathds{C}$ and 2 gaugino modes.

The  interesting point is that these $2\cdot6$ inter-brane zero modes do
\emph{not} acquire a mass, even in the presence of the
chiral Higgs on $\cC[(1,1)]$. The key word here is \emph{chiral} Higgs, which
by definition link the extremal weight states
of $\cC[(1,1)]$ with the same chirality, see  figure
\ref{fig:brane_1-1_plus_Higgs} and \eqref{chiral-Higgs-decomp}.
Since the $\Psi_{\cC,{\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}}$ linking ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$ with these states have the same
chirality,
they cannot form a mass term, and remain massless.
We view them as (toy-versions of) left-and right-handed leptons,
since they couple with opposite charges to the
chiral gauge field $A_\mu$ of \eqref{chiral-gauge-mode},
\begin{align}
 A_\mu \sim \chi \sim \one_L - \one_R \, \sim \g_5
 \label{chi-structure-g5}
\end{align}
and come in 3 generations.
If some extra Higgs mode is switched on
which links the states with opposite chirality, e.g.\ the maximal Higgs mode,
then these left-and right-handed leptons
would acquire a mass,  as in the Standard Model. Such scenarios will be
discussed
further in section \ref{sec:standardmodel}.
\subsection{Fermions on rigged \texorpdfstring{$\mathds{C} P^2$}{CP2} brane}
\label{sec:ferm-1N-chiral}
The story for the $\cC[(N,1)]$ branes from
section \ref{sec:rigged-CP2} is very similar to the $(1,1)$ case, but should be
even more interesting as far as physics and the scales are concerned.
As before, we are mostly interested  in
a point brane ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$ added to a $\cC[(N,1)]$ brane with Higgs.
The drawback for this rigged $\mathds{C} P^2$ scenario is that we lack an exact solution
which would reflect the configuration of figure \ref{fig:inst-diagram}.
Nonetheless, some qualitative statements can be made.
There are again the $3+3$ fundamental chiral zero modes linking
${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$ to $\cC[(N,1)]$, which are attached to the $3_L+3_R$ extremal weight
(coherent) states, and are viewed as 3 generations of \emph{leptons}.
These leptons will survive in the presence of the chiral Higgs as before, and
couple to
the chiral gauge field $A_\mu \sim \chi \sim \g_5$ of \eqref{chi-structure-g5}.
Due to the large $N$, one may hope that this chiral $A_\mu$ is now indeed the
lightest non-trivial gauge boson on $\cC[(N,1)]$, as discussed in section
\ref{sec:rigged-CP2}.
The resulting physics is that of 3 generations of leptons coupled to $A_\mu \sim\g_5$.
This demonstrates how a chiral gauge theory can arise from softly broken
$\cN=4$ SYM in a suitable vacuum corresponding to space-filling branes with
fluxes.

\subsection{Fermions on flipped minimal branes plus point brane --- the
\texorpdfstring{$G_2$}{G2} brane}
\label{sec:ferm-G2-chiral}
Let us consider the fermionic zero modes around the combined $G_2$-type
backgrounds of figure \ref{fig:G2-branes}.
The analysis presented in appendix \ref{subsec:flipped_min_brane+point} shows
that from the original 84 fermionic zero modes on $\cC[(1,0)]+\cC[(0,1)]+{\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$
(without any Higgs), only 14 remain on the combined background.

Adding an extra point brane ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$ to the $G_2$-type solution,
one might  expect  (i) inter-brane fermions between ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$
and the $G_2$-type solution, and (ii) intra-brane fermions on ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$. The numerical
analysis of the Dirac spectrum, however, shows that the combined system has
only 22 fermion zero modes, which is only 8 more than on the $G_2$ solution.
This means that among the $2\cdot6$ inter-brane fermions
and the $6+2$ trivial fermionic modes from ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$, only 8 remain
massless, while the
remaining ones pair up and form massive states.
The reason is that the $G_2$ orbits have higher dimension, and
the chirality properties of the 6-dimensional solutions no longer apply.
Therefore the present $G_2$ solution is less interesting for the application
in section \ref{sec:standardmodel}, but it may give hints how to find non-trivial Higgs
solutions on several $SU(3)$ branes.
\subsection{Fermions on two squashed \texorpdfstring{$\mathds{C} P^2$}{CP2} branes with
a point brane \& Higgs}
\label{sec:two-CP2-point-Higgs}
Finally consider two parallel branes  $\cC[\mu_L]$ and $\cC[\mu_R]$ with
$\mu_L=(N_L,0)$ and $\mu_R = (N_R,0)$ and a point brane ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$. In
section \ref{sec:two-branes-point-Higgs}, we found various Higgs solutions, linking the
extremal weight states of the various branes, which break the $U(1)\times U(1)$ gauge fields
$A^{L,R}_\mu$ arising from the $\cC[\mu_{L,R}]$.
There are again fermionic zero modes $\Psi_{{\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F} \cC_L}$ and  $\Psi_{{\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F} \cC_R}$ linking ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$ to the two branes,
and we want to see if the Yukawa couplings to  the Higgs may lead to an interesting chiral low-energy
gauge theory, where fermions with different chiralities have different couplings to the gauge fields $A^{L,R}_\mu$
as in sections \ref{sec:ferm-11-chiral} and  \ref{sec:ferm-1N-chiral}.

The task is trickier here, because $\Psi_{{\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F} \cC_L}$, for instance, provides
two fermions connecting to the same
corner of $\cH_{((N_L,0)}$, whose chirality is given by their $\tau$-parity. They will be paired up by the intra-brane Higgs
connecting these corners. However, by inspection, all solutions found in section
\ref{sec:two-branes-point-Higgs}
break the $\mathds{Z}^3$ symmetry, and they also typically involve intra-brane Higgs
(except for the solutions in section \ref{sec:three-branes}).
Hence, even though the surviving massless fermions are chiral and have different
couplings to $A^L_\mu$ and $A^R_\mu$,
the generation symmetry $\mathds{Z}_3$ is not respected. This leads to a somewhat strange low-energy theory far from the
Standard Model. Nevertheless, it is conceivable --- and even reasonable ---
that the inter-brane Higgs acquire by some other mechanism a VEV
which does respect $\mathds{Z}_3$, as indicated in figure \ref{fig:two-CP2-brabes-Yukawas}.
\begin{figure}[t!]
\centering
\begin{tikzpicture}
  \coordinate (Origin) at (0,0);
  \coordinate (Q1) at (-1,-0.5);
  \coordinate (Q2) at (0,1);
  \coordinate (Q3) at (1,-0.5);
  \coordinate (P1) at (-3,-3/2);
  \coordinate (P2) at (0,3);
  \coordinate (P3) at (3,-3/2);
  \coordinate (Q1+a3) at (-1-0.2,-0.5-0.3);
  \coordinate (Q1-a1) at (-1-0.5,-0.5);
  \coordinate (Q2+a2) at (0-0.2,1+0.3);
  \coordinate (Q2-a3) at (0+0.2,1+0.3);
  \coordinate (Q3+a1) at (1+0.5,-0.5);
  \coordinate (Q3-a2) at (1+0.2,-0.5-0.3);

  \coordinate (P1+a3) at (-3-0.2,-3/2-0.3);
  \coordinate (P1-a1) at (-3-0.5,-3/2);
  \coordinate (P2+a2) at (0-0.2,3+0.3);
  \coordinate (P2-a3) at (0+0.2,3+0.3);
  \coordinate (P3+a1) at (3+0.5,-3/2);
  \coordinate (P3-a2) at (3+0.2,-3/2-0.3);
  \node at (Q1) {$\bullet$};
  \node at (Q2) {$\bullet$};
  \node at (Q3) {$\bullet$};
  \node at (P1) {$\bullet$};
  \node at (P2) {$\bullet$};
  \node at (P3) {$\bullet$};
  \draw[-latex,orange] (Origin)--(Q1+a3) node[at end,below]
{$\psi_3^{+}$};
  \draw[-latex] (Origin)--(Q1-a1) node[at end,left]
{$\widetilde{\psi}_1^{-}$};
  \draw[-latex,orange] (Origin)--(Q2+a2) node[at end,left]
{$\psi_2^{+}$};
  \draw[-latex] (Origin)--(Q2-a3) node[at end,right]
{$\widetilde{\psi}_3^{-}$};
  \draw[-latex,orange] (Origin)--(Q3+a1) node[at end,right]
{$\psi_1^{+}$};
  \draw[-latex] (Origin)--(Q3-a2) node[at end,below]
{$\widetilde{\psi}_2^{-}$};
  \draw[-latex] (Origin)--(P1+a3) node[at end,below]
{$\Psi_3^{+}$};
  \draw[-latex,orange] (Origin)--(P1-a1) node[at end,left]
{$\widetilde{\Psi}_1^{-}$};
  \draw[-latex] (Origin)--(P2+a2) node[at end,left]
{$\Psi_2^{+}$};
  \draw[-latex,orange] (Origin)--(P2-a3) node[at end,right]
{$\widetilde{\Psi}_3^{-}$};
  \draw[-latex] (Origin)--(P3+a1) node[at end,right]
{$\Psi_1^{+}$};
  \draw[-latex,orange] (Origin)--(P3-a2) node[at end,below]
{$\widetilde{\Psi}_2^{-}$};
\draw[dotted] (P1)--(P2)--(P3)--cycle;
\draw[dotted] (Q1)--(Q2)--(Q3)--cycle;
\node at (Origin) {$\bigstar$};
  \draw [thick,red,-latex] (P3+a1) -- (Q2-a3) node[midway,above]
{$H_2^{+}$};
  \draw [thick,red,-latex] (P2+a2) -- (Q1-a1) node[midway,left]
{$H_3^{+}$};
  \draw [thick,red,-latex] (P1+a3) -- (Q3-a2) node[midway,below]
{$H_1^{+}$};
\end{tikzpicture}%
 \caption{}
 \label{fig:two-CP2-brabes-Yukawas}
\caption{Set-up $\cC[\mu_L]+\cC[\mu_R] + {\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$ with a $\mathds{Z}_3$-invariant Higgs configuration, where
the point brane ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$ is depicted in the center and $\cC[\mu_L]$ corresponds to
the outermost brane. We sketch the chiral intra-brane fermions $\Psi$,
$\widetilde{\Psi}$ between $\cC[\mu_L]$ and ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$ as well as $\psi$,
$\widetilde{\psi}$ between
$\cC[\mu_R]$ and ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$. The fermions $\Psi$ and $\widetilde{\psi}$ can be
linked by maximal inter-brane Higgs $H$, $\tilde{H}$ between $\cC[\mu_{L,R}]$.
The chirality of the fermionic zero modes is indicated by the sign $\pm$, which
is inherited from the Weyl chamber.}
\end{figure}
Then it is possible to have a situation similar as in sections  \ref{sec:ferm-11-chiral} and  \ref{sec:ferm-1N-chiral},
where e.g. all $\Psi_{{\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F} \cC_L}$ are left-handed and all $\Psi_{{\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F} \cC_R}$ are right-handed.

Since we do not have a dynamical justification for such a $\mathds{Z}_3$-invariant
Higgs configuration,
we will focus on the configurations in sections \ref{sec:ferm-11-chiral} and  \ref{sec:ferm-1N-chiral}
in the following discussion towards the Standard Model.
\section{Approaching the Standard Model}
\label{sec:standardmodel}
At first sight, it may seem impossible to get anything resembling the Standard
Model from deformed $\cN=4$ SYM. After all the Standard Model is chiral, while
$\cN=4$ SYM is not.
In fact, any low-energy gauge theory arising in some vacuum of a
deformation of $\cN=4$, as considered here, will have index zero, cf.\
\cite{Steinacker:2013eya}.
However, the Standard Model extended by right-handed neutrinos $\nu_R$ does
have index zero, and this is what we aim to approach with  \emph{sterile}
$\nu_R$, which are uncharged under the gauge group of the SM.
The scenario to be discussed will be reminiscent of
(a supersymmetric extension of) the Pati-Salam model \cite{Pati:1974yy} in the
broken phase.
This is a refinement of the brane configuration proposed in \cite{Steinacker:2015mia}, using the
results of the previous sections.

\begin{figure}[t!]
\centering
\begin{tikzpicture}
  \coordinate (P1) at (0.7,-0.6);
  \coordinate (P3) at (-1.3,-0.6);
  \coordinate (P4) at (1.5,0);
  \coordinate (P7) at (-1.5,0);
  \coordinate (P14) at (1,1);
  \coordinate (P15) at (0,1);
  \node at (P1) {$\bullet$};
  \node at (P3) {$\bullet$};
  \node at (P4) {$\bullet$};
  \node at (P7) {$\bullet$};
  \node at (P14) {$\bullet$};
  \node at (P15) {$\bullet$};
  \draw[dotted] (P1)--(P3)--(P7)--(P15)--(P14)--(P4)--cycle;
\draw [thick,blue,-latex] (P1) -- (P7) -- (P14) -- cycle;%
\draw [thick,red,-latex]  (P3) -- (P4) -- (P15) -- cycle;%
\draw [black] (P3) node[left] {$L$};
\draw [black] (P4) node[right] {$L$};
\draw [black] (P15) node[left] {$L$};
\draw [black] (P1) node[right] {$R$};
\draw [black] (P7) node[left] {$R$};
\draw [black] (P14) node[right] {$R$};
\draw[decoration={brace,mirror,raise=5pt},decorate,thick]
  (2,-0.5) -- node[right=6pt] {${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_u \equiv \cC[(N,1)]$} (2,1.0);
\coordinate (phiU) at (0,0);
\draw[orange,thick] (P7) .. controls (phiU) .. (P15)
node[midway,right] {$\phi_u$};
  \coordinate (R1) at (0.7,-0.6-2);
  \coordinate (R3) at (-1.3,-0.6-2);
  \coordinate (R4) at (1.5,0-2);
  \coordinate (R7) at (-1.5,0-2);
  \coordinate (R14) at (1,1-2);
  \coordinate (R15) at (0,1-2);
  \node at (R1) {$\bullet$};
  \node at (R3) {$\bullet$};
  \node at (R4) {$\bullet$};
  \node at (R7) {$\bullet$};
  \node at (R14) {$\bullet$};
  \node at (R15) {$\bullet$};
  \draw[dotted] (R1)--(R3)--(R7)--(R15)--(R14)--(R4)--cycle;
\draw [thick,blue,-latex] (R1) -- (R7) -- (R14) -- cycle;%
\draw [thick,red,-latex]  (R3) -- (R4) -- (R15) -- cycle;%
\draw [black] (R3) node[left] {$L$};
\draw [black] (R4) node[right] {$L$};
\draw [black] (R15) node[left] {$L$};
\draw [black] (R1) node[right] {$R$};
\draw [black] (R7) node[left] {$R$};
\draw [black] (R14) node[right] {$R$};
\draw[decoration={brace,mirror,raise=5pt},decorate,thick]
  (2,-2.5) -- node[right=6pt] {${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_d \equiv \cC[(N,1)]$} (2,-1.0);
\coordinate (phiD) at (0,0-2);
\draw[orange,thick] (R7) .. controls (phiD) .. (R15)
node[midway,right] {$\phi_d$};
\coordinate (L) at (2,2);
\node at (L) {$\bigstar$};
\draw [black] (L) node[below right] {${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_l$};
\coordinate (C1) at (-2,2);
\coordinate (C2) at (-2+0.2,2-0.3);
\coordinate (C3) at (-2-0.2,2-0.3);
\node at (C1) {$\bigstar$};
\node at (C2) {$\bigstar$};
\node at (C3) {$\bigstar$};
\draw [black] (C3) node[below left] {$3\times{\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_c$};
\draw[decoration={brace,mirror,raise=5pt},decorate,red,thick]
  (-1.3-1,-0.6) -- node[left=6pt] {$SU(2)_L$} (-1.3-1,-0.6-2);
\coordinate (phi) at (1,2);
\draw[orange,thick] (L) .. controls (phi) .. (P14) node[midway,above]
{$\phi_s$};
\end{tikzpicture}%
 \caption{Brane configuration towards the Standard Model. Both ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_u$ and
${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_d$ can be realized e.g.\ by rigged $(N,1)$ brane solutions, which
decompose into $L$ and $R$ sheets with 3 generations.}
 \label{fig:SM-branes}
\end{figure}
Consider the brane configuration of figure \ref{fig:SM-branes}, which
is constructed in terms of our squashed brane solutions as follows:
The ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_d$ brane is realized by a \emph{rigged brane} $\cC[(N,1)]$,
which decomposes into two chiral ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_{Ld}+{\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_{Rd}$ branes, with chiral Higgs
$\tilde\phi$ switched on as in section \ref{sec:rigged-CP2}.
The ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_u$ has the same structure\footnote{this is
motivated e.g. by the structure of the traceless electric charge generator $Q$ \eqref{Q-def}.} as ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_d$,
which  decomposes into two chiral ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_{Lu}+{\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_{Ru}$ branes, again with chiral Higgs $\tilde\phi$.
Finally add 4 point branes, denoted as $3 \times {\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_c + {\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_l$ for reasons which will become
clear soon.
The overall brane configuration ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_l + 3 \times {\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_c + ({\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_u + {\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_d)$ in
the $SU(N)$ SYM model leads to a $SU(4) \times SU(2) \times U(1)$ gauge group.

Next, we \emph{assume} that there exists a non-vanishing ''Pati-Salam``  Higgs\footnote{Note
that $\phi_S$ corresponds to the $(4,1,2)$ Higgs in the Pati-Salam model
\cite{Pati:1974yy}.
While there are indeed suitable regular zero modes  which link ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_l$ with the
3 extremal $R$ states of ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_u$,
we did not find an exact solution of this type, because they do not form closed
triangles.
It might be possible to find such solutions in conjunction with the Higgs
$H_{u,d}$ \eqref{Higgs-doublets-2},
or along the lines of the $G_2$ solutions in section \ref{sec:flipped-branes}.}
$\phi_S$ linking ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_{Ru}$ with ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_l$, which breaks the gauge group to
\begin{align}
 SU(3)_c \times U(1)_Q \times U(1)_{B'} \ .
 \label{symm-SM-branes}
\end{align}
This breaking might also be achieved or accompanied
by a displacement of the two $R$ branes, while the $L$ branes
remain coincident.
Here $Q$ is the electric charge,  $\cL$ is the lepton number,  $B$ the baryon number, and
\begin{align}
 Q &\coloneqq \frac 12\big(\one_{{u}} -\one_{d}  + \cL - B \big) \ ,  \\
    B &= \frac 13 \one_c, \qquad B' = B - c \one \nonumber\\
    \cL &= \one_l \ ,
\label{Q-def}
\end{align}
such that $B'$ is traceless.
Note that $Q$ is traceless provided $\dim\cH_{u} = \dim\cH_{d}$,
which strongly suggests that the ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_u$ and ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_d$ have the same structure.
We also introduce the weak hypercharge
\begin{align}
 Y \coloneqq \one_{Ru} -\one_{Rd} + \cL - B \ .
 \label{Y-def}
\end{align}
$Q$ and $Y$ will reproduce the correct charge assignments of the Standard Model,
and
we recover the  Gell-Mann-Nishjima formula
\begin{align}
 Q- \frac 12 Y = \frac 12(\one_{Lu} - \one_{Ld}) \eqqcolon  T^3_L \ .
\end{align}
\paragraph{Fermions.}
Now consider the off-diagonal fermions linking these branes,
which arise as  zero modes $\Psi_{ij} \in \mathrm{End}(\cH_i,\cH_j)$ linking the
extremal weight states of the  branes $\cC[\mu_i]$ and $\cC[\mu_j]$.
Consider first the fermions between the point branes
${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_l, {\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_c$ and ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_u, {\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_d$.
Since the former are point branes and
${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_u$ as well as ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_d$ have the structure of $\cC[(N,1)]$ branes (by
assumption),
we can simply apply the results of sections \ref{sec:ferm-11-chiral} and \ref{sec:ferm-1N-chiral}.
Recalling that the extremal weight states of ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_{u}$
separate into $3_L + 3_R$ chiral extremal weight
states $|\mu^i_{Lu}\rangle, |\mu^i_{Ru}\rangle$ and similarly for ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_d$,
we obtain 3 generations of chiral leptons  linking ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_l$ with ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_u, {\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_d$, and
 3 generations of chiral quarks linking ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_l$ with ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_u, {\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_d$.
In the basis $(|\mu^i_{Lu}\rangle,|\mu^i_{Ld}\rangle,|\mu^i_{Ru}\rangle,|\mu^i_{Rd}\rangle,
|0\rangle_l,|0^j\rangle_c)$, we denote the inter-brane fermions as
\begin{align}
 \Psi =
 \begin{pmatrix}
  *_{2} &  \tilde H_u  & \tilde H_d  &  l_L & Q_L \\
   &  *  & e' & \nu_R & u_R \\
   &    & * & e_R & d_R  \\
   & & ~~~ & * &  u' \\
&  & & & *_3
\end{pmatrix} .
\label{fermion-matrix}
\end{align}
Here the left-handed quarks and leptons
\begin{align}
Q_L = \begin{pmatrix} u_L \\ d_L   \end{pmatrix}, \qquad
l_L = \begin{pmatrix} \nu_L \\ e_L   \end{pmatrix},
\end{align}
arise as $SU(2)$ doublet acting on the ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_u,{\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_d$ branes. This $SU(2)$ is
broken by $\phi_S$, which will be discussed in more detail below.
Similarly, the right-handed quarks and leptons also arise as $SU(2)$ doublets
\begin{align}
Q_R = \begin{pmatrix} u_R \\ d_R   \end{pmatrix}, \qquad
l_R = \begin{pmatrix} \nu_R \\ e_R   \end{pmatrix} .
\end{align}
Note that the entries below the diagonal are not independent but related to the upper entries by charge conjugation,
see section \ref{sec:fermions}.
The charge generators \eqref{Q-def}, \eqref{Y-def} are given explicitly by
\begin{equation}
\begin{aligned}
 Q &= \frac{1}{2} \mbox{diag}\left(1,-1,1,-1,1,-\frac{1}{3}\right)\, , \\
  Y &= \mbox{diag}\left(0,0,1,-1,1, - \frac{1}{3}\right)
\end{aligned}
\end{equation}
which results in the following quantum numbers for the off-diagonal modes
\begin{align}
 (Q,Y)|_\Psi =
 \begin{pmatrix}
  * \ \ &   \begin{pmatrix} (0,-1) \\ (-1,-1) \end{pmatrix} & \begin{pmatrix}
(1,1) \\ (0,1) \end{pmatrix}  & \begin{pmatrix} (0,-1) \\ (-1,-1) \end{pmatrix}
& \begin{pmatrix} (\frac{2}{3},\frac{1}{3}) \\
(-\frac{1}{3},\frac{1}{3})\end{pmatrix}  \\[2ex]
   &  *  & (-1,-2)  & (0,0)  & (\frac{2}{3},\frac{4}{3})  \\[1ex]
   &     & *        &  (-1,-2) & (-\frac{1}{3},-\frac{2}{3})  \\[1ex]
   & & ~~~ & * & (\frac 23,\frac 43) \\
&  & & & *
\end{pmatrix}
\label{charges}
\end{align}
dropping the obvious $SU(3)_c$ assignment.
All quantum numbers of the Standard Model are correctly reproduced,
and three families arise automatically due to the $\mathds{Z}_3$ symmetry\footnote{The $\mathds{Z}_3$ symmetry
may of course be broken e.g.\ by the Yukawa couplings
induced by different Higgs VEVs, or by a deformation of the background.}.
There are also some extra modes, including Higgsinos $\tilde H_{u,d}$
(as in the MSSM), the $\nu_R$ which is uncharged under the SM gauge group,
gauginos, Winos, and some sterile diagonal fermionic modes.
Furthermore, there are also extra modes $e'$ with the same quantum numbers as
$e_R$, and
$u'$ with the same quantum numbers as $u_R$. Their fate depends on the
detailed structure of e.g.\ $\phi_S$ and will not be discussed here.
This way of obtaining the correct SM charges is familiar from the context of matrix models
\cite{Grosse:2010zq,Chatzistavrakidis:2011gs,Steinacker:2014fja,Steinacker:2015mia}
and  from intersecting brane constructions in string theory
\cite{Aldazabal:2000cn,Antoniadis:2002qm,Blumenhagen:2005mu}.

\paragraph{Higgs sector.}
In the present background, all the above fermions are exactly massless, even though the
unbroken symmetry is only \eqref{symm-SM-branes}. The reason is that the
$U(2)_L \times U(2)_R$ symmetry of the two $L$ and $R$ sheets
of the 6-dimensional branes ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_u$ and ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_d$ is broken because the sheets
are connected (see section \ref{sec:6D-branes}). This
can be viewed as breaking via some background
''Higgs``,
which however does not couple to the above fermions. On the
other hand, recall that
all the above fermionic zero modes have scalar superpartners,
as discussed in the first part of the paper.
In particular, this includes the \emph{electroweak} Higgs doublets $H_u$, $H_d$
with  $Y(H_d) = 1$ (as in the SM) and
$Y(H_u) = -1$ (as in the MSSM), which fit into the above matrix structure as
\begin{align}
 \phi^a = \begin{pmatrix}
         0_2 & H_u & H_d & 0 & 0 \\
         & 0 & 0 & 0 & 0\\
         && 0 & \varphi_S & 0\\
         &&& 0 & 0 \\
         &&&&0
        \end{pmatrix} \ .
\label{higgs-matrix}
\end{align}
It is reasonable to assume that these are intra-brane Higgs modes within
${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_u$ and within ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_d$, realized by bosonic
zero modes $\varphi_u$  and $\varphi_d$ as discussed in section
\ref{sec:rigged-CP2}.
This means that they take the following VEVs
\begin{align}
 H_d = \begin{pmatrix}
         \varphi_d \\ 0
        \end{pmatrix}, \qquad
 H_u = \begin{pmatrix}
        0 \\ \varphi_u
        \end{pmatrix}
  \label{Higgs-doublets-2}
\end{align}
with $Q=0$.
More explicitly, these Higgs modes have the structure
\begin{align}
 H_d &\sim \sum \big(0\cdot |\mu_{L}\rangle_u + \varphi_{d} \cdot
|\mu_{L}\rangle_d\big)
                   \langle\mu_{Rd}|_d  \cong \begin{pmatrix}
                                         0 \\ \varphi_{d}
                                        \end{pmatrix}  \langle\mu_{Rd}|_d  \nonumber\\
 H_u &\sim \sum \big(\varphi_{u} \cdot|\mu_{L}\rangle_u  +
0 \cdot |\mu_{L}\rangle_d\big)
                \langle\mu_{Ru}|_u \cong \begin{pmatrix}
                                         \varphi_{u} \\ 0
                                        \end{pmatrix} \langle\mu_{Ru}|_u \ .
 \label{Higgs-doublets}
\end{align}
Since they are intra-brane modes, they do not induce any further symmetry
breaking; nonetheless, they do induce the desired Yukawa couplings between the
left-and right-handed
leptons and quarks, as in sections \ref{sec:ferm-11-chiral}, \ref{sec:ferm-1N-chiral}.
Then the low-energy phenomenology should be fairly close
to that of the Standard Model, extended by the various extra fields as above.

These  $H_{u,d}$ should be viewed as part of the combined background,
which gives mass in particular to the $W^\pm$ bosons discussed below,
as explained in section \ref{sec:6D-branes}.
The lowest \emph{fluctuations} of this vacuum will involve all the
constituents and
may behave similar to a SM Higgs, while the detailed composition
of the background will  enter only via the various gauge and Yukawa couplings.
We note that this is somewhat similar to a Pati-Salam model in the broken
phase  \cite{Pati:1974yy}.
Whether or not such a scenario may be realistic is another issue, and perhaps
there is a better background.
The main point here is to show that one may come surprisingly close to SM-like
low-energy physics, with only few reasonable assumptions on the VEVs of the zero
modes.

Now consider the $\nu_R$ in more detail. Its fate is clearly affected by the presence
of the Higgs link $\phi_S$ between ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_l$ and the $R$ states of ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_u$,
which leads to extra Yukawa couplings of the $\nu$ and some
intra-brane fermions.
This would entail a mixture of fermionic states, which
is too complicated to be fully analyzed here.
\paragraph{Gauge bosons.}
It is well-known that
a background consisting of stacks of branes leads to massless $U(n_i)$ gauge
bosons within stacks of $n_i$ coinciding identical branes. This yields
\eqref{symm-SM-branes} in the present background.

The story becomes more interesting if we account for the lightest massive gauge
bosons. Assume for the moment that there is no $\phi_S$.
Then the ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_u + {\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_d$ would define a $U(2)$ gauge symmetry which enhances to
$U(2)_L \times U(2)_R$ by taking the chiral $A_\mu \sim \chi \sim \g_5$ of
\eqref{chi-structure-g5} on the $\cC[(N,1)]$ branes into account.
Together with the $SU(4)$ from the ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_l + 3 \times{\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_c$,
this is reminiscent of a Pati-Salam model in the broken phase.
Clearly the $SU(2)_L$ along with the $Y$
contributes the $W^\pm$ and $Z$ gauge bosons of the electroweak sector, and
the $U(2)_R$ is broken in the presence of $\phi_S$, leading to
\eqref{symm-SM-branes}.
This provides the basic structure of an extended Standard Model.
One may hope that the various extra fields acquire a sufficiently high mass to be
negligible at low energies, but this is beyond the scope of this paper.
\paragraph{Discussion.}
Let us briefly address some of the numerous open questions.
One concern is that we had to assume that the $\phi_S$ and the $H_{u,d}$
acquire the
appropriate VEVs, without having an exact solution. However, if the $H_{u,d}$
are realized as links along the edges of $\cC[(N,1)]$, then there are in fact
non-vanishing cubic terms involving e.g.\ $\mbox{tr}(H_u \phi_S \phi_S')$,
which lower the energy.
It is, therefore, plausible that there is such a solution, but it would
presumably have
a non-vanishing back-reaction on the brane, which we cannot compute. This is a
non-trivial problem, and quantum effects might play an important role.

Another question is whether a suitable hierarchy could arise between the 3 generations.
Even though the background under consideration has an exact $\mathds{Z}_3$  symmetry,
this could easily be broken in the Higgs sector, or possibly by introducing different
mass parameters $M_i$.

A further issue is the extra massless $U(1)_{B'}$
gauge field, which amounts to baryon number.
Even though this protects from proton decay, there should not be any such massless gauge field, and
it is not  clear how to remove this in the present
field-theoretic setting. However, it might
disappear via a St\"uckelberg-type mechanism in an analogous
matrix model setting with an axion, cf.\
\cite{Morelli:2009ev,Coriano:2006xh,Kors:2005uz,Anastasopoulos:2006cz}
and the discussion in \cite{Steinacker:2015mia}.
In fact, all results of the present paper carry over immediately to the
IKKT matrix model, where noncommutative $U(N)$ $\cN=4$ SYM arises on a stack of $N$
$(3+1)$-dimensional noncommutative brane solutions, while the internal
structure
of the present paper are unchanged. Thus, the present paper
can also be seen as a possible way to obtain interesting particle physics
from the IKKT model, cf. \cite{Aoki:2014cya,Steinacker:2014fja}.

Finally, we note that instead of realizing, for instance, the ${\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}_u$
branes via $\cC[(N,1)]$, we could alternatively use separate
$L$ and $R$ branes $\cC[(N_L,0)]$ and $\cC[(N_R,0)]$, linked by suitable Higgs
as in section
\ref{sec:two-CP2-point-Higgs}. These links would give mass to the \emph{mirror
fermions}, which disappear from the
low-energy theory. The remaining discussion follows the logic employed above.
We recall, that this has been the setup proposed in \cite{Steinacker:2015mia}.
The links are essentially an integral part of the
$\cC[(N,1)]$ branes linking their chiral sheets, realized in an exact solution.

\section{Discussion  and conclusion}
\label{sec:conclusion}
Building on previous work \cite{Steinacker:2014lma,Steinacker:2015mia}, we
studied vacua corresponding to squashed fuzzy coadjoint $SU(3)$ branes in
$\cN=4$ SYM, softly broken by a $SU(3)$-invariant cubic potential and masses.
We found a rich class of novel vacua that include non-trivial condensates of
the zero modes. The condensates are interpreted as string-like Higgs modes
linking the self-intersecting branes.
\paragraph{Summary.}
On a formal level, we showed that the potential can be rewritten in terms of complete squares
\eqref{eq:potential_rewritten}, so that the equations of motion follow from a set of
first integral equations \eqref{eq:1st-integral}. These  observations allow to extend
solutions from the massless to the massive case, and to establish the absence of any instabilities for
a preferred mass parameter $M^*$, at the classical level.
Furthermore, we give a useful new characterization of the regular zero modes
(the \emph{Higgs modes}) on the $SU(3)$ branes in terms
of a \emph{decoupling condition} \eqref{extremal-3}. As a consequence, the
full potential and equations of motion decouple completely for a combination
$X+\phi$ of brane plus Higgs mode.
This is the basis for establishing a rich class of new exact solutions of this
type,
which lead to a spontaneously broken gauge theory with non-trivial Higgs VEVs
and corresponding Yukawa couplings,
realizing the ideas put forward in \cite{Steinacker:2015mia}.

The decoupling conditions and the related rewriting of the potential in terms of complete squares are in many ways
reminiscent of supersymmetry. It provides non-trivial moduli spaces of vacua, and establishes their stability.
However there is no underlying symmetry, so that at present we cannot extend these structures to the quantum level.

At a more technical level, the first integral equations are somewhat weaker than
$\mathfrak{s}\mathfrak{u}(3)$ Lie algebra relations \eqref{Cartan-Weyl}. Nonetheless, $\mathfrak{s}\mathfrak{u}(3)$
representations provide a valuable starting point, and most --- but not all ---
of our solutions are based on $\mathfrak{s}\mathfrak{u}(3)$
in some way. Clearly the basic branes $\cC[\mu]$ arise directly from $\mathfrak{s}\mathfrak{u}(3)$. Using the decoupling properties of
the zero modes, we constructed numerous novel combined brane plus Higgs
solutions including the following:
\begin{compactenum}[(i)]
\item The fuzzy 2-sphere $S^2_N$ arises from the minimal regular zero mode $\phi_i^+ =
- X_i^-$ on any $\cC[\mu]$ brane background.
However, these are large perturbations, which completely change the geometry of the brane.
\item The maximal regular zero modes $\phi_i^+ = -\frac{1}{N!}(X_i^-)^N$ on
$\cC[(N,0)]$ furnish the fundamental representation
$(1,0)$ of $\mathfrak{s}\mathfrak{u}(3)$ and correspond the string-like operators. We have
discussed these solutions at length in section \ref{sec:4D-branes}.
\item The next-to-maximal regular zero modes on $\cC[(1,1)]$ furnish interesting
chiral Higgs solutions which stabilize the chiral sheets, as shown in section \ref{sec:1-1_brane}.
A generalization to  $\cC[(N,1)]$ is conjectured.
\item As an elaborate example, various zero modes on
$\cC[(1,0)]+\cC[(0,1)]+{\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}$ are used to form the 7-dimensional fundamental
representation of $\mathfrak{g}_2 \supset \mathfrak{s}\mathfrak{u}(3)$, as discussed in section
\ref{sec:flipped-branes}.
\end{compactenum}
The discussion of the fermion sector in section \ref{sec:fermions} is mostly qualitative.
In the presence of several Yukawa couplings
the situation is complicated, and a full
treatment is beyond our scope. However in many cases we are able to explain the
fermionic zero modes of ${\slashed{D}}^{X+\phi}$, notably for the
$\cC[(1,1)]$ brane with chiral Higgs and point brane, which is used in our
approach to the Standard Model. This is based on the detailed computations of
appendix \ref{app:solutions_spectrum}.

\paragraph{Discussion.}
On a more physical level, two of the most interesting features of the new
brane plus Higgs vacua are the following: (i) they have a mass gap, and
(ii) typically only a small number of zero modes exists. The number
of zero modes is independent of the rank $N \gg 1$ of the underlying $SU(N)$
gauge theory and of the size $\dim \cH_\L$ of the brane $\cC[\L]$.
This observation is  remarkable, since the starting point is a
gauge theory with large $N$, which is typically organized in terms of a t'Hooft
 $\frac 1N$ genus expansion with effective coupling  $\lambda} \def\L{\Lambda} \def\la{\lambda = g^2 N$.
In contrast, the $\cC[(N_1,N_2)]$ vacua  with large $N_i$ behave as
semi-classical, large  branes. The fluctuation spectrum on these vacua
consists of a small number of zero modes with typical coupling strength $g$, and
a large tower of typically weakly interacting KK modes with a finite mass gap independent of $N$.
Hence, the original large $N$ gauge theory reduces to an
effective low-energy theory with few modes and an interesting geometric structure.
This should provide sufficient motivation to study them in more detail.

The most interesting aspect of these vacua is that they can lead to a chiral
gauge theory at low energies, with interesting properties not far from the
Standard Model.
The point is that the underlying branes are locally space-filling in the 6 extra dimensions and carry a flux, so that
bi-fundamental fermions are charged and expected to have chiral zero modes. This is precisely what happens, although the
overall index is  bound to be zero. Elaborating on a previous proposal by using
our new solutions,
we discussed in section \ref{sec:standardmodel} a brane configuration
which comes fairly close to the Standard Model. The set-up is
reminiscent of the Pati-Salam model in the broken phase and of
intersecting brane constructions in string theory. Even though we do not claim
that this is realistic, it certainly provides strong motivation for further
work.

\paragraph{Future directions.}

There are many open issues which should be addressed in future work.
One task is to understand better the
chiral Higgs solution \eqref{chiral-Higgs-rigged} on the $(N,1)$ branes and the associated chiral gauge bosons,
which seem to be particularly interesting for physics.
Another is to justify a non-vanishing ''Pati-Salam`` $\phi_S$  in our Standard Model approach,
and to see how close to real physics one may come in this way.
More generally, it would be desirable to have a more systematic understanding and perhaps
a classification of the solutions of the  first-order equations of motion
\eqref{firstorder-eom}.

At some point of course, these
classical considerations will no longer suffice.
We have argued that a classical treatment should be justifiable to some extent on large branes, due
to the existence of a gap with few remaining low-energy modes, and  the mild UV
behavior of the softly broken $\cN=4$ model.
Nevertheless, quantum effects need to be taken into account at some point.
Due to the global $SU(3)$ symmetry, the relevant terms in the effective potential
must preserve the form \eqref{V-soft}. Moreover, the critical mass $M^*$ of
\eqref{M-star-def} marks the transition
between trivial and non-trivial stable vacua; hence, it should play a special
place in the full quantum theory,
possibly as an RG fixed point.
A direct loop computation by summing over all higher KK modes seems far too
complicated.
For a deformed $\cN=4$ model, one strategy might be to invoke holography;
however, this is questionable since (i) the model is not conformal and (ii) the
vacuum is highly non-trivial.
Fortunately, a suitable alternative technique was recently proposed in
\cite{Steinacker:2016nsc}, which is based on string-like states on the fuzzy brane backgrounds.
This geometric approach has been applied successfully in the
\emph{purely fuzzy} context of \cite{Steinacker:2016nsc,Steinacker:2016vgf},
but not yet in the present field-theoretic setting with
fuzzy extra dimensions. Hence, this needs to be developed elsewhere.

\paragraph{}
The take-home message it that a simple deformation of $\cN=4$ SYM with large $N$
can reduce at low energy to an effective gauge theory with few  string-like
Higgs modes coupled to chiral fermions, interpreted in terms of a gauge theory on
intersecting branes in 6 extra dimensions.
Moreover, the possible quantum numbers  include those of the Standard Model.
This should provide sufficient motivation to study such scenarios in more
detail and at the quantum level, and it will be interesting to see how far
these solutions can reach.

\section*{Acknowledgements}
This work was funded by the Austrian Science
Fund (FWF) grant P28590. The Action MP1405 QSPACE from the European
Cooperation in Science and Technology (COST) also provided  support.
Related discussions with Pascal Anastasopoulos,  Amihay Hanany,
Cumrun Vafa and  George Zoupanos  are gratefully acknowledged.

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