text:
\section{Introduction}
We study a certain class of homeomorphic solutions arising in several different context that we call $\sigma$-harmonic mappings.

\begin{definition}\label{def.H}
Consider an open, simply connected set $\Omega\subset
\mathbb R^2$. Given positive constants $\alpha$ and $\beta$, we
say that a measurable function $\sigma$, defined on $\Omega$ with
values into the space of $2\times2$ matrices,  belongs to the
class $\mathcal M (\alpha,\beta,\Omega)$  if one has
\begin{equation}\label{ellTM}
\begin{array}{ccrllll}
\sigma(x) \xi\cdot \xi&\geq& \alpha |\xi|^2&,&\hbox{for every $\xi \in \mathbb R^2$ and for a.e. $x\in\Omega$}\,,\\
\sigma^{-1}(x) \xi \cdot \xi &\geq & \beta^{-1}
|\xi|^2&,&\hbox{for every $\xi \in \mathbb R^2$ and for a.e.
$x\in\Omega$}\,.
\end{array}
\end{equation}
\end{definition}
\noindent
We say that $u$ is a $\sigma$-harmonic function if,
given $\sigma \in {\mathcal M}(\alpha, \beta,\Omega)$,   the function $u\in W^{1,2}_{\rm loc}(\Omega)$ is a weak solution to
\begin{equation}\label{u}
\begin{array}{ll}
{\rm div}(\sigma \nabla u )=0 & \hbox{in $\Omega$}\ .
\end{array}
\end{equation}
Here, and in what follows, we adopt the usual convention to identify $z=x_1 + i\,x_2\in\mathbb C$ with $x=(x_1,x_2)\in \mathbb R^2$.
A $\sigma$-harmonic map $U=(u^1,u^2):\Omega\to \mathbb R^2$ is simply a pair of $\sigma$-harmonic functions. If no conditions are imposed on the relationship between $u^1$ and $u^2$, the notion may degenerate to the uninteresting case when $u^1\equiv u^2$. We are interested to the case when $U$ is a homeomorphism and, from now on, we will always assume that it is sense preserving so that $\det DU\geq 0$ almost everywhere in $\Omega$ . The most classic example is the set of conformal mappings, that is holomorphic functions that are univalent, or, as is the same, injective. It is well know  that for holomorphic injective mappings $f$, the complex derivative $f^{\prime}$ does not vanish, or, in real notations, the Jacobian determinant does not vanish. Hence it is positive in $\Omega$ for conformal mappings, since they preserve the orientation. Another classical and  extensively studied example is the case of planar (univalent) harmonic mappings, \cite{duren}.
 Lewy's celebrated theorem
\cite{Lewy} extends to the harmonic planar mappings  the result which is valid for conformal mappings. So also for planar harmonic injective (sense preserving) mappings  $U$, one has $\det DU>0$  and $U$ is necessarily a diffeomorphism.

\noindent
For harmonic mappings, in particular for holomorphic one, $\sigma$ is the identity matrix, so no regularity issue arises. The map $U$ is real analytic.
Another very classical case arises when $\sigma$ satisfies the additional constraint
\begin{equation}\label{det=1}
\det \sigma=1\ .
\end{equation}
Indeed, let us recall that
given $\sigma \in {\mathcal M}(\alpha, \beta,\Omega)$   and  a weak solution $u\in W^{1,2}_{\rm loc}(\Omega)$  to
\eqref{u},
then there exists $\tilde u \in W^{1,2}_{\rm loc}(\Omega)$, called the \emph{stream function} of $u$, such that, setting
\begin{equation}\label{J}
 J:=\left(
\begin{array}{ccc}
0&-1\\
1&0
\end{array}
\right)\,,
\end{equation}
\noindent
 one has
\begin{equation}\label{e2.3}
\begin{array}{lllll}
 \nabla \tilde u= J \sigma \nabla u &\hbox{in}&\Omega\,.
 \end{array}
\end{equation}
If, in addition, $\sigma$ is symmetric and \eqref{det=1} holds, then one easily verifies that $\tilde{u}$ also satisfies \eqref{u}. Hence $U=(u,\tilde{u})$ is $\sigma$-harmonic.

\noindent
For any solution to \eqref{u},
setting
\begin{equation}
\label{e2.4}
F=u+i \tilde u\,,
\end{equation}
one has
$F=u+i \tilde u \in W^{1,2}_{\rm loc}(\Omega;\mathbb R^2)$ and one writes,  in complex notations,
\begin{equation}\label{1storder}
\begin{array}{ll}
F_{\bar{z}}=\mu F_z +\nu \bar{F_z}\ & \hbox{in $\Omega$}\ ,
\end{array}
\end{equation}
where, the so called complex dilatations $\mu , \nu$ are given by
\begin{equation}\label{SNU}
\begin{array}{llll}
\mu=\frac{\sigma_{22}-\sigma_{11}-i(\sigma_{12}+\sigma_{21})}{1+{\rm
Tr\,}\sigma +\det \sigma}& \ ,&\nu =\frac{1-\det \sigma
+i(\sigma_{12}-\sigma_{21})}{1+{\rm Tr\,}\sigma +\det \sigma}\ ,
\end{array}
\end{equation}
and satisfy the following
ellipticity condition
\begin{equation}\label{ellQC}
|\mu|+|\nu|\leq \frac{K-1}{K+1} \,,
\end{equation}
 for some $K\geq 1$ only depending on
$\alpha, \beta$.
As is well known, \eqref{e2.4}-\eqref{ellQC}, imply that
\begin{equation}\label{dist1}
|DF|^2\leq \left(K+\frac 1 K\right) \det DF\,,
\end{equation}
where $|B|=({\rm Trace}(B B^T))^{1/2}$ denotes the Hilbert-Schmidt norm of the matrix $B$. Any $W^{1,2}_{\rm loc}(\Omega)$ mapping satisfying \eqref{dist1} almost everywhere, is called $K$-quasiregular.
In fact a classical equivalent way of introducing  quasi regular mappings is in terms of ACL (Absolutely Continuous on almost very Line) orientation preserving mappings for which there exist $K\geq 1$ such that \eqref{dist1} holds.

\noindent
Associated to the map $F$ we may associate various types of so-called distortions or dilatations, which turn out to be equivalent.
Let us consider the following one
\begin{equation}\label{dist2}
\begin{array}{ccc}
d(F)=\frac{|DF|^2}{2 \det DF} \,.
\end{array}
\end{equation}
\noindent
The principal object of the present paper is to establish conditions on $\sigma$ under which a $\sigma-$harmonic mapping $U$ is quasiregular, that is $d(U)$ is bounded.
\noindent
For reasons that shall be made transparent later on, it will be convenient to introduce a slight variation of the dilatation function and introduce a new but equivalent dilatation $d^{\sigma}$ as follows. For a sense preserving mapping $U$, that is, for a mapping such that $\det DU\geq0$, we define
\begin{equation}\label{dsigma}
d^{\sigma}(U):=\frac{{\rm Trace} (DU \sigma DU^T)}{2 \det DU}=\frac{\sigma \nabla u^1 \cdot \nabla u^1+\sigma \nabla u^2 \cdot \nabla u^2}{2 \det DU}\,.
\end{equation}
In view of the ellipticity of the matrix $\sigma$, it is clear that $d$ and $d^{\sigma}$ are uniformly bounded one from each other by positive constants.
It must be emphasized that, in general, the second component of a $\sigma$-harmonic mapping need not be the stream function of the first one, already at the level of harmonic mappings. So $\sigma$-harmonic mappings are indeed a much wider class of the set of quasiconformal mappings.

\noindent A very natural question is whether the dilatation $d^{\sigma}(U)$ may blow up for general $\sigma$-harmonic maps.

\noindent In fact,  already in the case of harmonic mappings, i.e. when $\sigma$ is the identity matrix $I$, one may have $d^I(U)$ diverging. To see this, it suffices to consider, in complex notation, the homeomorphism
\begin{equation}
U(z)=z+\frac 1 2 \bar{z}^2
\end{equation}
mapping the closed unit disk $B(0,1)$ univalently onto a closed bounded set the boundary of which is an hypocycloid. One easily checks the Jacobian determinant vanishes identically on $\partial B(0,1)$ and hence the dilatation diverges at any point on it. However, by smoothness and Lewy's Theorem, the dilatation may only blow up at the boundary.

\noindent
For $\sigma$-harmonic mappings, instead, the situation is dramatically different. In fact, not only the dilatation may blow up at interior points but, actually, this is the generic situation.

\noindent
A typical example of $\sigma$ for which one can construct a univalent $\sigma$ harmonic mapping the dilatation of which diverges, is given in \cite{ANJAM}, Example 3.1. A much wider class is given in the same paper, Example 3.2. In the same paper it is proved, roughly speaking, that if for a given $\sigma$ there exists a univalent mapping $U=(u^1,u^2)$ which is a solution to
\begin{equation}\label{u1u2}
\begin{array}{ll}
{\rm div}(\sigma \nabla u^i )=0 & \hbox{in $\Omega$ for $i=1,2$}\,,
\end{array}
\end{equation}
that is such that $d^{\sigma}(f)$ is locally bounded away from the boundary of $\Omega$, then {\it all the univalent mappings $U$ which are solutions to \eqref{u1u2}} are quasiconformal, see also \cite{nf}.
The class of all such nice $\sigma$'s, denoted by $\Sigma_{\rm qc}$ is rather ``small", that is nongeneric. More precisely, $\Sigma_{\rm qc}$ is a set of first Baire category as proved in Theorem~4.1 of \cite{ANJAM}. The identity matrix is one of such nice $\sigma$'s. So, for given $\sigma$, in order to have $\sigma\in \Sigma_{\rm qc}$, it suffices to find a single $\sigma$-harmonic mappings with bounded dilatation.

\noindent
One may wonder how to exhibit large classes of univalent $\sigma$-harmonic mappings. The first result for the case of $\sigma$ not equal to the identity, was established in \cite{bmn}, Theorem~3.1, following previous work by Bauman and Phillips \cite{BP}. The idea, going back to Kneser \cite{k} and Rad\`o \cite{r}, is to deduce the univalence by some informations about the Dirichlet boundary data. In \cite{bmn} one assumes that the boundary data is the restriction to the boundary of a smooth sense preserving diffeomorphism $\Psi=(\Psi^1,\Psi^2)$ of an  open, simply connected set $\Omega$ with smooth boundary, onto a convex set. Under these assumptions one proves that $U$ is itself a self preserving homeomorphism so that $\det DU\geq 0$ almost everywhere. Under more stringent regularity assumptions on $\partial \Omega$, $\Psi$ and $\sigma$, one deduces that $DU$ is smooth and $\det DU>0$.

\noindent
A deeper result, in this direction, was proved in \cite{ANARMA}, Theorem~4. In that paper a Kneser-Rad\`o type of theorem is proved under essentially minimal hypotheses. For the kind of questions we address in the present paper, however, the main result is  a different one. To state it in a a precise form, we recall that, given an  open set $D\subset \mathbb R^2$, $\phi \in L^1_{\rm loc} (D)$ belongs to  ${\rm BMO} (D)$ if
\begin{equation*}
\| \phi \| _{*} = \sup_{Q\subset D} \left( \frac{1}{\mid Q\mid}
\int_Q \mid \phi - \phi_Q\mid \right) <\infty\,,
\end{equation*}
where $Q$ is any square in $D$ and $\phi_Q= \frac{1}{\mid Q\mid} \int_Q \phi$.

\noindent
Recall also that the normed space
 $({\rm BMO} (D), \|\cdot\|_*)$ is in fact a Banach space.
  \begin{theorem}
\label{t3.1}
Let $\Omega$ be an open subset of $\mathbb R^2$, let $\sigma \in {\mathcal M}(\alpha,\beta,\Omega)$  and
let $U \in W^{1,2}_{\rm loc}(\Omega,\mathbb R^2)$ be a $\sigma$-harmonic mapping  which is locally one-to-one and sense preserving. For every $D\subset \subset \Omega$, we have
\begin{equation}
\label{e3.1}
\begin{array}{ll}
\log (\det DU) \in {\rm BMO} (D)\,.
\end{array}
\end{equation}
\end{theorem}
 \noindent
 One obtains by \eqref{e3.1}, that the Jacobian determinant has a negative exponent of integrability and, in particular, that  $\det DU>0$ almost everywhere. This theorem, which relies on deep results due to Reimann \cite{re}, Bauman \cite{bau} and Fabes and Strook \cite{fs}, was proved in \cite{ANARMA}, Theorem~5, in the case when $\sigma$ is symmetric. The extension to the nonsymmetric case elaborates on results in \cite{nf} and can be found in \cite{ANFINNICO}, Theorem~3.1.  In fact, in \cite{ANFINNICO} Theorem~4.1 and Remark~4.3,
an even stronger result is obtained, that is
\begin{equation}\label{peso}
\det DU \in A_{\infty}(D),
\end{equation}
where $A_{\infty}$ is the class of Muckenhoupt weights \cite{cf}. Indeed, in Theorem~\ref{theorem2} below, we shall make use of this result.

\noindent
Applications of these achievements may be found in several context, ranging form establishing sharp bounds for effective conductivity as in \cite{achn} and \cite{acn}, or in establishing a conjecture in the field of quasiconfomal mappings about the stability of Beltrami systems under $H$-convergence,  as in \cite{ANFINNICO}. Other applications have been given in inverse problems, see, for instance,  the review article by \cite{balinside}.

\noindent
In the present paper
we investigate sufficient conditions on $\sigma$ such that $d^{\sigma}(U)$ is locally bounded. In \cite{ANJAM}, Theorem~2.2, we proved that if $\sigma$ is symmetric and $\det \sigma$ is H\"older continuous, then $d^{\sigma}(U)$ is locally bounded, however no quantitative estimate was obtained. Here, assuming that the antisymmetric component $\sigma_{12}-\sigma_{21}$ and the determinant $\det \sigma$ are Lipschitz continuous, we achieve a concrete local upper bound on $d^{\sigma}$ in the form of a Harnack type inequality in Theorem~\ref{theorem1}.

\noindent
Next, since we know from \cite{ANJAM}, as already remarked, that in order to have $\sigma\in \Sigma_{qc}$, it suffices to exhibit a single $\sigma$-harmonic mapping whose dilatation is bounded, we focus on the special class of so-called periodic $\sigma$-harmonic mappings, which are of special relevance in homogenization and which have beee analyzed in depth in \cite{ANARMA} and in \cite{ANCOCV}.
More precisely, denoting by $W^{1,2}_{\sharp}(\mathbb R^2,\mathbb R^2)$ the space of $W^{1,2}_{\rm loc}$ mappings whose components are $1$-periodic in each variable, and assuming that $\sigma \in \mathcal M (\alpha,\beta,\Omega)$ is also $1$-periodic, in each variable, we consider $U=(u^1,u^2)\in W^{1,2}_{\rm loc}(\mathbb R^2,\mathbb R^2)$ such that
\begin{equation}\label{Uper}
\left\{
\begin{array}{lllll}
{\rm div}(\sigma \nabla u^i )= 0&{\rm in}&\mathbb R^2\,,\\
U-x\in W^{1,2}_{\sharp}(\mathbb R^2,\mathbb R^2)&&\,.
\end{array}
\right.
\end{equation}
For the solution to \eqref{Uper},  we prove a global uniform bound on $d^{\sigma}(U)$ in the final Theorem~\ref{theorem2}.

\section{An elliptic equation}
\noindent
Throughout the present Section, we assume that $U$ is a locally univalent,  sense preserving $\sigma$-harmonic mapping $U=(u^1,u^2)$ in $\Omega$. The main motivation for the introduction of the quantity $d^{\sigma}$ comes for the following decomposition.
Denote
\begin{equation}\label{wi}
\begin{array}{ll}
w^i=\frac{\det DU}{\sigma \nabla u^i\cdot \nabla u^i}\,,& \hbox{for $i=1,2$}\,.
\end{array}
\end{equation}
Obviously we have
\begin{equation}\label{wvsd}
d^{\sigma}(U)=\frac 1 2 \left(\frac{1}{w^1}+ \frac{1}{w^2}\right).
\end{equation}
We shall show in the next Lemma that, under few hypotheses on $\sigma$, each $w^i$ solves a divergence structure elliptic equation. In fact is is well known that if $\sigma$ is the identity, then indeed $w^1$ and $w^2$ are harmonic \cite{duren}.

\begin{lemma}\label{lemma1}
Assume, for some $E\in (0,+\infty)$,

\begin{equation}\label{Lip}
||\nabla\det \sigma||_{L^{\infty}(\Omega)}+
||\nabla (\sigma_{12}-\sigma_{21})||_{L^{\infty}(\Omega)}\leq E.
\end{equation}
Then, for any $i=1,2$, $w^i\in W^{1,2}_{\rm loc}(\Omega)$ and it is a weak solution to

\begin{equation}\label{neweq}
\begin{array}{llll}
{\rm div}(\sigma \nabla w^i +w^i B^i)=0&\hbox{\rm{in} $\Omega$}\ ,
\end{array}
\end{equation}
where
\begin{equation}
\begin{array}{lll}
B^i=
\left\{
\begin{array}{lcc}
 \displaystyle{\frac{1}{\sigma \nabla u^i \cdot\nabla u^i}}
\left[ \left(J\nabla u^i \cdot\nabla c\right)
  J^T \nabla u^i + \left(J\nabla u^i \cdot\nabla b\right)
\sigma\nabla u^i
  \right]\,,
  &\hbox{\rm{where}}&\nabla u^i\neq 0
 \\
\\
0\,,&\hbox{\rm{where}}&\nabla u^i= 0
\end{array}
\right.
\end{array}
\end{equation}
and
\begin{equation}
\begin{array}{lll}
b=\sigma_{12}-\sigma_{21}&,& c=\det \sigma\,.
\end{array}
\end{equation}
\end{lemma}
\begin{remark}
Being $U$ locally univalent, we know that $\det DU$ may vanish only on sets of zero measure. Even more so, $|\nabla u^i|$, may vanish only on sets of zero measure. See \cite{AJMAA} for a general discussion of unique continuation for two-dimensional elliptic equations.
Observe that
\begin{equation*}
||B_i||_{L^{\infty}(\Omega)}\leq C_0\,E,
\end{equation*}
where $C_0$ only depends on $\alpha$ and $\beta$. Finally note that $w^i$ are actually $\sigma$-harmonic when $\det \sigma$ and the antisymmetric part of $\sigma$ are constant.
\end{remark}
\noindent
{\bf Proof of Lemma~\ref{lemma1}.} It suffices to provide a formal proof of \eqref{neweq}. Standard smoothing arguments and interior a-priori estimates will guarantee that $w^i\in W^{1,2}_{\rm loc}$ and the validity of \eqref{neweq} with no further assumptions other than \eqref{Lip}.

\noindent
We concentrate on $w^1$, the same arguments shall apply to $w^2$.
It was shown in \cite{ANARMA} and \cite{ANFINNICO} that,  being $U$ injective, setting $f=u^1+ i\,\tilde{u}^1$, $f$ is locally quasiconformal and, introducing
\begin{equation*}
V=U\circ f^{-1}\,,
\end{equation*}
one has
\begin{equation*}
w:=\det DV=\frac{\det DU}{\det DF} \circ f^{-1}=w^1\circ f^{-1}\,.
\end{equation*}
Furthermore, see \cite{ANFINNICO}, $w$ is a distributional solution to the following elliptic equation in adjoint form
\begin{equation}\label{adeq}
w_{x_1x_1}+(b\circ f^{-1}\,w)_{x_1x_2}+(c\circ f^{-1}\,w)_{x_2x_2}=0\,.
\end{equation}
From now on we will simply write, with a slight abuse of notation $b$ and $c$ in \eqref{adeq} instead of
\begin{equation*}
b\circ f^{-1}\hskip0.5cm\hbox{and}\hskip0.5cm c\circ f^{-1}\,.
\end{equation*}
In view of the bound \eqref{Lip}, $b$ and $c$ are weakly differentiable, hence  \eqref{adeq} can be rewritten as a divergence structure equation
\begin{equation}\label{adeqdiv}
w_{x_1x_1}+(b \,w_{x_2} +b_{x_2} \,w)_{x_1}+(c\,w_{x_2} +c_{x_2}\,w)_{x_2}=0\,.
\end{equation}

\noindent
Pulling back through the quasiconformal mapping $f,$ we end up with equation
\eqref{neweq}.
\section{Bounds on the dilatation}
In what follows we are interested in pointwise bounds on the dilatation function $d^{\sigma}(U)$ for a given $\sigma$-harmonic mapping $U$. Therefore, in order to emphasize the dependence on the variable $x\in \Omega$, we shall write for short
\begin{equation}
d^{\sigma}(x)\hskip0.5cm\hbox{instead of}\hskip0.5cm d^{\sigma}(U)(x)\,.
\end{equation}
Also, for the sake of simplicity, but with no loss of generality, we shall assume that the ellipticity constant in \eqref{ellTM} satisfy
\begin{equation}\label{K}
\alpha^{-1}=\beta=K\,.
\end{equation}

\begin{theorem}\label{theorem1}
Let $\sigma$ satisfy \eqref{ellTM} with the convention \eqref{K} and assume that  $U:\Omega\to\mathbb R^2$ be a locally univalent and sense preserving $\sigma$-harmonic mapping. Assume, in addition \eqref{Lip}.
Then $d^{\sigma}<+\infty$ in $\Omega$, it is H\"older continuous and for any $A\subset \subset \Omega$, there exists $H>0$ only depending on $K, E, A$ and $\Omega$, such that the following Harnack's inequality holds

\begin{equation}\label{Harnack}
d^{\sigma}(x)\leq H \,d^{\sigma}(x^{\prime}) \,\,\hbox{for all $x, x^{\prime}\in A$}\,.
\end{equation}
\end{theorem}
\noindent{\bf  Proof}. Recall \eqref{wvsd}. In view of Lemma~\ref{lemma1}, by De Giorgi-Nash-Moser theory \cite{gt}, the thesis follows.

\vskip.1cm
\noindent
As a straightforward consequence, we obtain the following
\begin{corollary}\label{corollary} Under the same assumptions as in Theorem~\ref{theorem1}, we have
\begin{equation}\label{stima1}
||d^{\sigma}||_{L^{\infty}(A)}\leq  H \left(\frac{1}{|A|}\int_A(d^{\sigma})^{\delta}\right)^{\frac{1}{\delta}}
\end{equation}
for any positive $\delta$.
\end{corollary}

\begin{remark}
As already pointed out, under the assumptions of Theorem~\ref{theorem1}, the exists a positive $\epsilon$ such that $(\det DU)^{-\epsilon}$ in integrable. Therefore, by Schwartz inequality, one can check  that the right hand side of \eqref{stima1} is bounded for some small $\delta>0$ even when the condition \eqref{Lip} is not assumed.
\end{remark}

\begin{theorem}\label{theorem2}
Assume, in addition to the hypotheses of Theorem~\ref{theorem1}, that $\sigma$ is 1-periodic in each variable on the whole $\mathbb R^2$. Consider the solution to the periodic problem \eqref{Uper} with ``unit cell" $Q_0=[0,1]\times[0,1]$.
Then there exists $M>0$ only depending on $K$ and $E$ such that
\begin{equation}
||d^{\sigma}||_{L^{\infty}(Q_0)}\leq M\,.
\end{equation}
\end{theorem}
\noindent{\bf Proof.}
The solution $U:\mathbb R^2\to\mathbb R^2$ is globally univalent, as proved in \cite{ANARMA}, Theorem~1. By Theorem~4.1 in \cite{ANFINNICO} and by the  theory of Muckenhoupt weights, \cite{cf}, we can find $p>1$ and $C>0$ only depending on $K$ such that

\begin{equation}\label{disC}
\frac{1}{|R|} \int_R \det DU dx \left(\frac{1}{|R|} \int_R (\det DU)^{-\frac{1}{p-1}}dx \right)^{p-1}\leq C\,,
\end{equation}
for every square $R\subset \mathbb R^2$. By Corollary~\ref{corollary}, one has
\begin{equation}
||d^{\sigma}||_{L^{\infty}(Q_0)}\leq  \frac{H}{2} \left(\int_{Q_0} (|DU|_{\sigma})^{2\delta} (\det DU)^{-\delta}dx\right)^{\frac{1}{\delta}},
\end{equation}
where, for given $\sigma\in \mathcal M (\alpha,\beta,\Omega)$, we set $|B|_{\sigma}:=({\rm Trace}(B \sigma B^t))^{1/2}$, a Hilbert-Schmidt norm of the matrix $B$ ``weighted'' with that matrix $\sigma$.

\noindent
By Schwartz's inequality
\begin{equation*}
||d^{\sigma}||_{L^{\infty}(Q_0)}\leq  \frac{H}{2} \left(\int_{Q_0}(|DU|_{\sigma})^{4\delta} dx\right)^{\frac{1}{2\delta}}
 \left(\int_{Q_0}(\det DU)^{-2 \delta}dx\right)^{\frac{1}{2\delta}}.
  \end{equation*}
 Picking $0<\delta<\min\left(\frac 1 2, \frac{1}{2(p-1)}\right)$ and recalling that $|Q_0|=1$, we deduce by H\"older's inequality that
\begin{equation*}
\int_{Q_0}(|DU|_{\sigma})^{4\delta}dx\leq
 \left(\int_{Q_0}(|DU|_{\sigma})^{4\delta s}dx\right)^{\frac{1}{s}},\,\,\forall s>1\,.
  \end{equation*}
  Hence, picking $s=\frac{1}{2\delta}$, we have
  \begin{equation*}
\left(\int_{Q_0}(|DU|_{\sigma})^{4\delta}dx\right)^{\frac{1}{2\delta}}\leq
\int_{Q_0}(|DU|_{\sigma})^{2}dx\,.
  \end{equation*}
  On the other hand, using again H\"older's inequality and the fact that $|Q_0|=1$, we have
  \begin{equation*}
\int_{Q_0}(\det DU)^{-2\delta}dx\leq
 \left(\int_{Q_0}(\det DU)^{-2\delta r}dx\right)^{\frac{1}{r}},\,\,\forall r>1
  \end{equation*}
  and choosing $r=\frac{1}{2\delta (p-1)}$, we have, by \eqref{disC}
  \begin{equation*}
\left(\int_{Q_0}(\det DU)^{-2\delta}dx\right)^{\frac{1}{2\delta}}\leq
 \left(\int_{Q_0}(\det DU)^{-\frac{1}{p-1}}dx\right)^{p-1}\leq
 C \left(\int_{Q_0}(\det DU)dx\right)^{-1}.
  \end{equation*}
 By standard energy estimates
   \begin{equation*}
\int_{Q_0}(|DU|_{\sigma})^{2}dx\leq  \int_{Q_0}{\rm Trace}\,(\sigma) dx\leq 2K
  \end{equation*}
  and, since $\det DU$ is a null-lagrangian, by the standard Area theorem, one has
   \begin{equation*}
\int_{Q_0}\det DU dx=\int_{Q_0}\det I dx=1.
  \end{equation*}
  Hence
  \begin{equation*}
||d^{\sigma}||_{L^{\infty}(Q_0)}\leq  C H K\,.
\end{equation*}
\vskip0.5cm
\noindent
{\bf Acknowledgements} G.A. was supported by
Universit\`a degli Studi di Trieste FRA 2012, Problemi Inversi.
V.N. was supported by PRIN Project 2010-2011 ``Calcolo delle Variazioni''.


text:
\section{Introduction}

Four-dimensional maximally supersymmetric Yang-Mills theory coupled to a codimension-one defect is one of the most throughly studied systems in the realms of integrability, AdS/CFT, and brane physics.  This system arises in string theory as the field theory on D3-branes intersected by D5-branes, which form a defect.  This brane intersection, and a generalization including NS5-branes, was used in studies of $\mathcal{N}_{\rm 3d} = 4$ supersymmetric theories, where it provides an explanation for the connection between Coulomb branch vacua of such theories and monopole moduli spaces \cite{Hanany:1996ie,Cherkis:1997aa}.  It was also the focus of some of the first works to introduce fundamental flavors in AdS/CFT models and the first example of defect AdS/CFT \cite{Karch:2000gx, DeWolfe:2001pq}, where \cite{Erdmenger:2002ex} showed furthermore that the theory with a single defect is conformal.

In the twenty-some years since these first works, a vast literature has explored this system as the starting point of many applications, from using integrability techniques \cite{deLeeuw:2015hxa, DeLeeuw:2018cal} and localization  \cite{Komatsu:2020sup} to extract CFT data, to the study of supersymmetric boundary conditions in $\mathcal{N}_{\rm 4d} = 4$ supersymmetric Yang-Mills (sYM) \cite{Gaiotto:2008sa}, to models that break all supersymmetry and use the defect theory to approximate 2d condensed matter systems (see \cite{Jensen:2010ga} for seminal work).

Despite the extensive literature on this 4d-ambient plus 3d-defect theory, the complete supersymmetry variations under which it is invariant have not -- to our knowledge -- appeared in the literature.  These results could  be useful for studying topologically twisted versions of an ambient plus defect theory along the lines of \cite{Witten:1988ze}, where R-symmetry plays a key role. The explicit form of the variations also plays an essential role in describing Bogomolny-Prasad-Sommerfield (BPS) solitons in this system, which we address in future work.

We emphasize the difference between vacua, which have zero energy and preserve all supersymmetries, and solitons, which are local (but not global) minimizers of the energy and break some supersymmetry. Indeed, the theory in question features interesting vacua realized as solutions to Nahm-type equations \cite{Cherkis:2011ee,Gaiotto:2008sa}. This structure can be exhibited without the explicit form of the supersymmetry variations, for instance using the D- and F-term equations in a superfield-based formalism \cite{Cherkis:2011ee}.  For soliton states, however, the complete supersymmetry transformations of the ambient plus defect theory given here provide the most direct way to characterize the full spectrum of possible BPS solitons. These solitons may generate new insights, via holography, into the vacua and solitons of the dual gravitating theory explored in \cite{Domokos:2017wlb}. The present work is a necessary first step in that direction.

In this note, we write down the full R-symmetry-covariant supersymmetry variations of $\mathcal{N}_{4d}=4$ sYM with codimension-one defect of D5-brane type, preserving $\mathcal{N}_{\rm 3d} = 4$ supersymmetry. We also give the supercurrents and the algebra of the supercharges following \cite{Witten:1978mh}, deriving central charges which measure both ambient and defect topological charges of field configurations.

In the interest of clarity and brevity, we suppress the details of all calculations, providing only explanations of key steps.\footnote{We are, however, happy to provide step-by-step notes upon request.} Despite being conceptually straightforward, constructing the variations in covariant form and demonstrating the invariance of the action were technically non-trivial. The reason for this is that much of the original literature works in superspace formalisms that obscure part of the full  $\mathrm{SU}(2)_{\rm V} \times \mathrm{SU}(2)_{\rm H}$ R-symmetry. In the process of arriving at our results, we derived maps between the $\mathcal{N}_{\rm{3d}}=2$ superfields of Erdmenger et al. \cite{Erdmenger:2002ex},  the formulation in terms of standard superspace coordinates  \cite{Wess:1992cp, Gates:1983nr}, and the R-symmetry-covariant component-field formulation. We include these in the appendices.

An outline of this paper is as follows.  Conventions are specified in Section \ref{subsec:conventions}.  We give the action and variations in Section \ref{sec:SUSY}.  In Section \ref{sec:boundary} we analyze the boundary terms from the variation and provide the supercurrents, which we use to compute the algebra of supercharges.  In Appendices \ref{app:SO4toSU2SU2} and \ref{app:superspace} we write down explicit maps from the fields we use to the common formulations of $\mathcal{N}_{4d}=4$ sYM, and to previous constructions of the ambient plus defect theory based on superspace.

\section{Conventions}\label{subsec:conventions}
We work in a mostly plus metric convention, $\eta_{\mu\nu}=\text{diag}(-1,1,1,1)$.  The (3+1)-dimensional space housing the ambient sYM theory is parametrized by $x^\mu$ for $\mu = 0,...3$. The defect is localized in the $x^2$ direction, and extended along $x^{\hat{\mu}} = (x^0, x^1, x^3) =: (x^{\hat{0}}, x^{\hat{1}}, x^{\hat{2}})$.

 Our $\mathfrak{u}(N_{\rm c})$ Lie algebra conventions for the field strength and covariant derivative are $F_{\mu\nu} = 2 \partial_{[\mu} A_{\nu]} + [A_{\mu},A_{\nu}]$ and $\mathcal{D}_{\mu} X = \partial_{\mu} X + [A_{\mu},X]$.  ``$ \, \textrm{Tr} \, $'' denotes a positive-definite Killing form on the Lie algebra normalized so that the generators satsify $ \, \textrm{Tr} \, (T^a T^b) = \frac{1}{2} \delta^{ab}$.  Defect fields $q$ transform in the ${\boldsymbol{\rho}} \oplus{\overbar{\brho}}$ representation, with ${\boldsymbol{\rho}}$ denoting the fundemantal representation.  We take the Lie algebra generators to be represented by anti-Hermitian matrices, so that $\mathcal{D}_\mu q = \partial_\mu q + A_{{\boldsymbol{\rho}}\mu} q$.  This also implies, for example, that $(\mathcal{D}_\mu q)^\dagger = \partial_\mu q^\dagger + q^\dagger (A_{{\boldsymbol{\rho}}\mu})^\dagger = \partial_\mu q^\dagger - q^\dagger A_{{\boldsymbol{\rho}}\mu}$.

We denote 4d gamma matrices by $\gamma^\mu$ with ${\overbar{\gamma}} = -i\gamma^0\gamma^1\gamma^2\gamma^3$, and we denote 3d gamma matrices by $\rho^{\hat{\mu}}$.  In both 3d and 4d, we define Dirac and charge conjugation of Dirac spinors $\psi$ in terms of intertwiners $A$ and $C$, with $A=A^\dagger$ and $C = - C^T$ as in \cite{Sohnius:1985qm}.
\begin{align}
\text{Dirac conjugation:} \quad &{\overbar{\psi}} := \psi^\dagger A\cr
\text{Charge conjugation:} \quad &\psi_c := C{\overbar{\psi}}^T = C A^T \psi^\ast ~.
\end{align}
A Majorana spinor $\lambda$ satisfies $\lambda_c = \lambda$.  The intertwiners furnish similarity transformations between unitarily equivalent representations of the Clifford algebra $\gamma^\mu, (\gamma^\mu)^T, (\gamma^\mu)^\dagger$ such that
\begin{align}
A\gamma^\mu A^{-1} = (\gamma^\mu)^\dagger,\qquad C^{-1}(\gamma^\mu) C = - (\gamma^\mu)^T~.
\end{align}
Note that all of these relations hold for the 3d and 4d versions, respectively, of the intertwiners and gamma matrices $A_{(3)},~ C_{(3)},~\rho^{\hat{\mu}}$ and $A,~C,~\gamma^\mu$.

As we work almost entirely in terms of 3d and 4d Majorana spinors, a convenient basis for the $\gamma$ matrices --similar that of  \cite{DeWolfe:2001pq}-- is
\begin{align}\label{DFOlikebasis}
& \rho^0 = - \sigma^2~, \qquad \rho^1 = i \sigma^3~, \qquad \rho^2 = -i \sigma^1~, \cr
& \gamma^{{\hat{\mu}}} = \rho^{{\hat{\mu}}} \otimes \sigma^1 ~, \qquad \gamma^2 = i \mathbbm{1} \otimes \sigma^3~, \qquad {\overbar{\gamma}} = \mathbbm{1} \otimes \sigma^2~.
\end{align}
Since $(\gamma^\mu)^\dag = - (\gamma^{\mu})^T$, we can take
\begin{equation}
C = - A = \sigma^2 \otimes \sigma^1 ~.
\end{equation}
We will interpret the first tensor factor of $C$ as the 3d charge conjugation matrix, so that
\begin{equation}\label{3dchargeconj}
C = C_{(3)} \otimes \sigma^1 = - A_{(3)} \otimes \sigma^1 ~, \qquad C_{(3)} = - A_{(3)} = \sigma^2~.
\end{equation}
The advantage of this basis is that  the 4d Majorana condition reduces block-diagonally to two 3d Majorana conditions and in each case $C A^T = \mathbbm{1}$, so each 4d Majorana spinor decomposes into a doublet 3d Majorana spinors:
\begin{equation}\label{4dMto3dM}
\textrm{Majorana spinor:} \quad \lambda = \left( \begin{array}{c} \boldsymbol{\uplambda} \\[1ex] \tilde{\boldsymbol{\uplambda}} \end{array} \right) ~, \qquad \textrm{with} \quad \boldsymbol{\uplambda}^\ast = \boldsymbol{\uplambda}~,\quad  \tilde{\boldsymbol{\uplambda}}^\ast = \tilde{\boldsymbol{\uplambda}}~.
\end{equation}

\section{Supersymmetry Variations}\label{sec:SUSY}

The field content of the D3/D5 defect field theory consists of
\begin{align}
&\text{Ambient:} \qquad  X^{\rm{V}}_{{\widetilde{r}}},~X^{\rm{H}}_r,~ \lambda_{{\widetilde{m}} m},~A_\mu\cr
&\text{Defect:}\qquad  q^m, {\boldsymbol{\upzeta}}_{{\widetilde{m}}} ~. \nonumber
\end{align}
The fields of $\mathcal{N}_{\text{4d}}=4$ sYM  are grouped to transform under the 3d $\mathrm{SU}(2)_{\rm V}  \times \mathrm{SU}(2)_{\rm H}$ R-symmetry preserved in the presence of the defect, rather than the larger 4d $\mathrm{SU}(4)$ R-symmetry preserved by  $\mathcal{N}_{\text{4d}}=4$ sYM without the defect.  $X^{\rm{V}}_{{\widetilde{r}}},~X^{\rm{H}}_r$ are triplets of real scalars, and $\lambda_{{\widetilde{m}} m}$ are 4d Majorana spinors transforming in the ${\bf (2,2)}$ of  $\mathrm{SU}(2)_{\rm V}  \times \mathrm{SU}(2)_{\rm H}$.  $A_\mu$ is the 4d gauge field, which decomposes into components $A_{\hat{\mu}}$ parallel to the defect and $A_2$ orthogonal to the defect.  The defect fields consist of complex scalars $q^m$ transforming in the fundamental of $\mathrm{SU}(2)_{\rm H}$ and trivially under $\mathrm{SU}(2)_{\rm V}$, and 3d Dirac fermions $\boldsymbol{\upzeta}_{\widetilde{m}}$ transforming in the fundamental of $\mathrm{SU}(2)_{\rm V}$ and  trivially under $\mathrm{SU}(2)_{\rm H}$.  The various index sets we utilize are summarized in Table \ref{indextable}.

\begin{table}
\begin{center}
\begin{tabular}{| c | c | c |} \hline
index type & description & range \\
\hline \hline
$\mu,\nu,\rho,\sigma$ & 4d ambient spacetime & $0,1,2,3$ \\
$ i,j,k,\ell$  & 3d ambient space & $1,2,3$ \\
${\hat{\mu}},{\hat{\nu}},{\hat{\rho}},{\hat{\sigma}}$ & 3d defect spacetime & $\hat{0},\hat{1},\hat{2}$  \\
${\hat{\imath}},{\hat{\jmath}},{\hat{k}},{\hat{\ell}}$ & 2d defect space & $\hat{1},\hat{2}$ \\
$\alpha,\beta,\gamma,\delta$ & 4d positive chirality Weyl spinors & $1,2$ \\
${\dot{\alpha}},{\dot{\beta}},{\dot{\gamma}},{\dot{\delta}}$ & 4d negative chirality Weyl spinors & $1,2$ \\
${\widetilde{\alpha}},{\widetilde{\beta}}\,{\widetilde{\gamma}}, {\widetilde{\delta}}$ & 3d Dirac or Majorana spinors & $1,2$ \\
\hline
$a,b$ & Lie algebra & $1,\ldots, N_{c}^2$ \\
\hline
$r,s,t,u$ & $\mathrm{SU}(2)_{\rm H}$ triplet & $1,2,3$ \\
$m,n,p,q$ &  $\mathrm{SU}(2)_{\rm H}$ doublet  &  $1,2$ \\
${\widetilde{r}},{\widetilde{s}},{\widetilde{t}},{\widetilde{u}}$ & $\mathrm{SU}(2)_{\rm V}$ triplet &  $1,2,3$ \\
${\widetilde{m}},{\widetilde{n}},\widetilde{p}, \widetilde{q}$  &  $\mathrm{SU}(2)_{\rm V}$ doublet  &  $1,2$ \\
$I,J,K,L$ & $\mathrm{SO}(4)$ quartet & $1,2,3,4$ \\
$A,B,C,D$ & $\mathcal{N}_{\rm{4d}}=2$ $\mathrm{SU}(2)_{\rm{R}}$ doublet & $1,2$\\
\hline
\end{tabular}
\end{center}
\caption{Summary of index conventions and ranges. Note that indices on 4d Dirac and Majorana spinors are always suppressed.}
\label{indextable}
\end{table}

The $\mathrm{SU}(2)$ indices are contracted using a Euclidean metric $\delta_{rs}$ or $\delta_{{\widetilde{r}}{\widetilde{s}}}$, while the $m$-type and ${\widetilde{m}}$-type indices are raised and lowered using the Levi-Civit\`{a} symbols $\epsilon_{mn}, \epsilon_{{\widetilde{m}}{\widetilde{n}}}$ with $\epsilon_{12}= -1$ and $\epsilon^{12}= 1$. The fundamental of $\mathrm{SU}(2)$ is pseudo-real so complex conjugation raises/lowers the index.  Thus, in particular, $(q^m)^\dagger = q^\dagger_m$.

The spinors $\lambda_{{\widetilde{m}} m}$ satisfy a Majorana condition that takes into account their R-symmetry transformation properties:
 %
 \begin{align}\label{eq:modifiedMajoranaCond}
 \lambda_{{\widetilde{m}} m}=\epsilon_{{\widetilde{m}} {\widetilde{n}}}\epsilon_{mn}C({\overbar{\lambda}}^{n{\widetilde{n}}})^T~.
 \end{align}
 %
Here the transpose refers to the spinor space only.  This definition holds for their 3d components, $\boldsymbol{\uplambda}_{{\widetilde{m}} m},\tilde{\boldsymbol{\uplambda}}_{{\widetilde{m}} m}$ as well, with the appropriate charge conjugation operator $C_{(3)}$. In what follows, when we say that a spinor with two R-symmetry indices is Majorana, we mean that it is Majorana with respect to this condition.

The transformation properties of the fields with respect to the gauge group are most easily understood from the intersecting D-brane picture.  Since D3-branes can end on D5-branes, there may be a different number of D3-branes to the left of the defect $(x^2 < 0)$, than to the right $(x^2 > 0)$.  See Figure \ref{fig1}.  If these numbers are $N_L$ and $N_R$, then the gauge group of the ambient theory for $x^2 < 0$ is $\mathrm{U}(N_L)$, and is $\mathrm{U}(N_R)$ for $x^2 > 0$.  For definiteness suppose $N_R \geq N_L$.  In what follows, we take the gauge group of the ambient theory to be $\mathrm{U}(N_{\rm  c})$ with $N_{\rm c} = \max(N_L,N_R)$, with the understanding that any adjoint-valued field $\Phi = \Phi^a T^a$ has $\Phi^a = 0$ for $x_2 < 0$ when $a > N_{L}^2$.  This amounts to choosing an embedding $\mathrm{U}(N_L) \subset \mathrm{U}(N_R)$, in which the left gauge group can be thought of as the upper-left block of the right gauge group in the defining representation.

The set of possible boundary conditions on the ambient fields as $x^2 \to 0$, that preserve $\mathcal{N}_{\rm 3d} = 4$ supersymmetry, was specified in \cite{Gaiotto:2008sa}.  In brief, for theories corresponding to the D3/D5 intersection, the triplet $X_{r}^{\rm H}$ and the gauge field $A_2$ should have behavior consistent with a solution to Nahm's equation as $x^2 \to 0$, which might include a Nahm pole when $N_R > N_L + 1$.  A Nahm pole is specified by an embedding $\rho : \mathfrak{su}(2) \to \mathfrak{su}(N_R - N_L)$, and the possible embeddings are in turn determined by the number of coincident D5-branes present, as explained in \cite{Gaiotto:2008sa}.  If only a single D5-brane is present, then $\rho$ is the principal embedding.

\begin{figure}
\begin{center}
\includegraphics[scale=0.5]{D3D5susy_fig1.pdf}
\end{center}
\caption{Cartoon of the intersecting brane system.  The D5-brane stack sits at $x^2 = 0$.  The asymptotic position of the D3-branes in the directions labeled $\vec{x}_{\rm H}$ are determined by the values of the diagonal components of the triplet $X_{r}^{\rm H}$ as $x_2 \to \pm \infty$.}
\label{D3D5system}
\label{fig1}
\end{figure}

The defect fields transform in the (anti-) fundamental of $\mathrm{U}(N_{\rm c,def})$, where $N_{\rm c,def} = \min(N_L,N_R)$, and  $q^m, q_{m}^\dag$ provide the jumping data for the upper left block of $X_{r}^{\rm H}$ as $x_2 = 0$.  If there are $N_f$ coincident D5-branes, then $q^m,\boldsymbol{\upzeta}_{{\widetilde{m}}}$ transform in the bifundamental $(\boldsymbol{N_{\rm c,def}},\overbar{\boldsymbol{N}_{\rm f}})$ of $\mathrm{U}(N_{\rm c,def})\times \mathrm{U}(N_f)$. Bilinears in the defect fields are always contracted to form $\mathrm{U}(N_f)$ singlets, and this structure is suppressed in what follows.  Thus, a quantity of the form $q^\dag q$ is a $\mathrm{U}(N_{\rm c,def})\times \mathrm{U}(N_f)$ singlet, while a quantity of the form $q^\dag T_{{\boldsymbol{\rho}}}^a q$ transforms in the adjoint of $\mathrm{U}(N_{\rm c,def})$.  We will see below how the Nahm boundary conditions emerge from the perspectives of both supersymmetry and finite-energy conditions.

The generalization of the above to multiple separated stacks of D5-branes and/or NS5-branes, leading to multiple parallel defects (of D5 and/or NS5 type) was described in \cite{Cherkis:2011ee}.  We will restrict ourselves here to the case of a single D3/D5-type defect.

With these preliminaries out of the way we can now write down the action in terms of these fields \cite{DeWolfe:2001pq}:
\begingroup
\allowdisplaybreaks
\begin{align}%
S =&~ S_{\rm amb} + S_{\rm def} ~, \qquad \textrm{with}  \cr
S_{\rm amb} =&~ \frac{1}{g_{\rm ym}^2} \int d^4 x  \, \textrm{Tr} \,  \bigg\{ - \frac{1}{2} F_{\mu\nu} F^{\mu\nu} - \frac{i}{2} {\overbar{\lambda}}^{m {\widetilde{m}}} \gamma^{\mu} \mathcal{D}_{\mu} \lambda_{{\widetilde{m}} m} - \mathcal{D}^{\mu} X_{{\widetilde{r}}}^{\rm V} \mathcal{D}_{\mu} X_{{\widetilde{r}}}^{\rm V} - \mathcal{D}^{\mu} X_{r}^{\rm H} \mathcal{D}_{\mu} X_{r}^{\rm H}  \nonumber \\*
&~ \qquad   +  \frac{i}{2}  {\overbar{\lambda}}^{m {\widetilde{m}}} [\lambda_{{\widetilde{m}} n}, X_{r}^{\rm H}] (\sigma^r)^{n}_{~m} - \frac{1}{2} {\overbar{\lambda}}^{m {\widetilde{m}}} (\sigma^{{\widetilde{r}}})_{{\widetilde{m}}}^{~{\widetilde{n}}} {\overbar{\gamma}} [\lambda_{{\widetilde{n}} m}, X_{{\widetilde{r}}}^{\rm V}] + \nonumber \\*
&~ \qquad - [X_{{\widetilde{r}}}^{\rm V}, X_{s}^{\rm H}] [X_{{\widetilde{r}}}^{\rm V}, X_{s}^{\rm H}] - \frac{1}{2}  [X_{{\widetilde{r}}}^{\rm V}, X_{{\widetilde{s}}}^{\rm V}] [X_{{\widetilde{r}}}^{\rm V}, X_{{\widetilde{s}}}^{\rm V}]  - \frac{1}{2} [X_{r}^{\rm H}, X_{s}^{\rm H}] [X_{r}^{\rm H}, X_{s}^{\rm H}] \bigg\} ~,   \label{eq:ambaction} \\
\intertext{and}
S_{\rm def} =&~  \frac{1}{g_{\rm ym}^2} \int d^3x \bigg\{ -\mathcal{D}^{{\hat{\mu}}} q_{m}^\dag \mathcal{D}_{{\hat{\mu}}} q^m - \frac{i}{2} ( \overbar{\boldsymbol{\upzeta}}^{{\widetilde{m}}} \rho^{{\hat{\mu}}} \mathcal{D}_{{\hat{\mu}}} \boldsymbol{\upzeta}_{{\widetilde{m}}} - (\mathcal{D}_{{\hat{\mu}}} \overbar{\boldsymbol{\upzeta}}^{{\widetilde{m}}}) \rho^{{\hat{\mu}}} \boldsymbol{\upzeta}_{{\widetilde{m}}} ) + \nonumber \\*
&~ \qquad \qquad + i \overbar{\boldsymbol{\upzeta}}^{{\widetilde{m}}} (\sigma^{{\widetilde{r}}})_{{\widetilde{m}}}^{~{\widetilde{n}}} (X_{{\boldsymbol{\rho}}}^{\rm V})_{{\widetilde{r}}} \boldsymbol{\upzeta}_{{\widetilde{n}}} + i q_{m}^\dag (\overbar{\boldsymbol{\uplambda}}_{{\boldsymbol{\rho}}})^{m {\widetilde{m}}} \boldsymbol{\upzeta}_{{\widetilde{m}}} + i \overbar{\boldsymbol{\upzeta}}^{{\widetilde{m}}} (\boldsymbol{\uplambda}_{{\boldsymbol{\rho}}})_{ {\widetilde{m}} m} q^m + \nonumber \\*
&~ \qquad \qquad + q_{m}^\dag (X_{{\boldsymbol{\rho}}}^{\rm V})_{{\widetilde{r}}} (X_{{\boldsymbol{\rho}}}^{\rm V})^{{\widetilde{r}}} q^m + i q_{m}^\dag (\sigma^r)^{m}_{~n} \left( \mathcal{D}_2 X^{\rm H}_r + \frac{1}{2} \epsilon_{r}^{~st} [X_{s}^{\rm H}, X_{t}^{\rm H}] \right)_{{\boldsymbol{\rho}}} q^n + \nonumber \\*
&~ \qquad \qquad  + \frac{1}{2} \delta(0) ( q_{m}^\dag (\sigma^r)^{m}_{~n} T^{a}_{{\boldsymbol{\rho}}} q^n ) ( q_{p}^\dag (\sigma^r)^{p}_{~q} T^{a}_{{\boldsymbol{\rho}}} q^{q} ) \bigg\}  \label{eq:defaction}~,
\end{align}
\endgroup
where the 3d spinor $\boldsymbol{\uplambda}_{{\widetilde{m}} m}$ is the piece of the 4d ambient field $\lambda_{{\widetilde{m}} m}$ that interacts with the defect fields (defined in detail below). $T^a_{\boldsymbol{\rho}}$ are the generators of $\mathfrak{u}(N_{\rm c,def})$ in representation ${\boldsymbol{\rho}}$.
The ambient action is the usual $\mathcal{N}_{4d}=4$ sYM action, but  manifesting the  $\mathrm{SU}(2)_{\rm H}\times \mathrm{SU}(2)_{\rm V} \subset \mathrm{SU}(4)$ R-symmetry preserved by the defect. In Appendix \ref{app:SO4toSU2SU2} we describe the map between this form and a more standard form \cite{Sohnius:1985qm}.

While we have grouped the final term in \eqref{eq:defaction} with the defect action, this term proportional to $\delta(0)$ is in a separate category.  Terms of this type were first discussed in \cite{Sethi:1997zza, Kapustin:1998pb}.  The interpretation of the $\delta(0)$ factor as a UV regulator in the specific term in \eqref{eq:defaction} was described in \cite{DeWolfe:2001pq}.  It is an artifact of the low energy limit and would not be present in the full string field theory where branes have a finite thickness.  If one considers classical finite-energy field configurations, this term should be grouped with others to form a complete square; on the minimum energy configuration, the entire squared term vanishes.

The total action $S_{\rm amb} + S_{\rm def}$ is invariant under variations
\begingroup
\allowdisplaybreaks
\begin{align}\label{eq:SUSYvars}
\text{\underline{Ambient fields}}&\cr
\delta A_{\mu} =&~ \frac{i}{2} {\overbar{\varepsilon}}^{m {\widetilde{m}}} \gamma_\mu \lambda_{{\widetilde{m}} m}~, \qquad   \delta X_{{\widetilde{r}}}^{\rm V} = \frac{1}{2} {\overbar{\varepsilon}}^{m {\widetilde{m}}} (\sigma_{{\widetilde{r}}})_{{\widetilde{m}}}^{~{\widetilde{n}}} {\overbar{\gamma}} \lambda_{{\widetilde{n}} m} ~, \qquad  \delta X_{r}^{\rm H} =  \frac{i}{2} {\overbar{\varepsilon}}^{m {\widetilde{m}}} \lambda_{{\widetilde{m}} n} (\sigma_r)^{n}_{~m} ~, \nonumber \\*
\delta \lambda_{{\widetilde{m}} m} =&~  \frac{1}{2} \gamma^{\mu\nu} F_{\mu\nu} \varepsilon_{{\widetilde{m}} m}  +i {\overbar{\gamma}} \gamma^{\mu} \mathcal{D}_{\mu} X_{{\widetilde{r}}}^{\rm V} (\sigma^{{\widetilde{r}}})_{{\widetilde{m}}}^{~{\widetilde{n}}} \varepsilon_{{\widetilde{n}} m} + \gamma^{\mu} \mathcal{D}_{\mu} X_{r}^{\rm H} \varepsilon_{{\widetilde{m}} n} (\sigma^r)^{n}_{~m} + \nonumber \\*
&~   - \frac{i}{2}  [X_{{\widetilde{r}}}^{\rm V}, X_{{\widetilde{s}}}^{\rm V}] \epsilon^{{\widetilde{r}}{\widetilde{s}}{\widetilde{t}}} (\sigma_{{\widetilde{t}}})_{{\widetilde{m}}}^{~{\widetilde{n}}} \varepsilon_{{\widetilde{n}} m} - i  [ X_{{\widetilde{r}}}^{\rm V}, X_{s}^{\rm H}] (\sigma^{{\widetilde{r}}})_{{\widetilde{m}}}^{~{\widetilde{n}}} {\overbar{\gamma}} \varepsilon_{{\widetilde{n}} n} (\sigma^s)^{n}_{~m} + \nonumber \\*
&~ + \frac{i}{2} [X_{r}^{\rm H}, X_{s}^{\rm H}] \epsilon^{rst} \varepsilon_{{\widetilde{m}} n} (\sigma_t)^{n}_{~m} - i \delta(x^2) Q^{r} \varepsilon_{{\widetilde{m}} n} (\sigma_r)^{n}_{~m}  \nonumber \\*
=:&~ {\overbar{\mathcal{F}}}_{{\widetilde{m}} m}^{n {\widetilde{n}}}\varepsilon_{{\widetilde{n}} n} ~,\nonumber \\*
\delta {\overbar{\lambda}}^{m {\widetilde{m}}} =&~   - \frac{1}{2} {\overbar{\varepsilon}}^{m {\widetilde{m}}} \gamma^{\mu\nu} F_{\mu\nu} + i {\overbar{\varepsilon}}^{{\widetilde{n}} m} \gamma^{\mu} {\overbar{\gamma}} \mathcal{D}_{\mu} X_{{\widetilde{r}}}^{\rm V} (\sigma^{{\widetilde{r}}})_{{\widetilde{n}}}^{~{\widetilde{m}}} + {\overbar{\varepsilon}}^{n {\widetilde{m}}} \gamma^{\mu} \mathcal{D}_{\mu} X_{r}^{\rm H} (\sigma^r)^{m}_{~n} + \nonumber \\*
&~ + \frac{i}{2}  {\overbar{\varepsilon}}^{m {\widetilde{n}}} \epsilon^{{\widetilde{r}}{\widetilde{s}}{\widetilde{t}}} [X_{{\widetilde{r}}}^{\rm V}, X_{{\widetilde{s}}}^{\rm V}] (\sigma_{{\widetilde{t}}})_{{\widetilde{n}}}^{~{\widetilde{m}}} - i  {\overbar{\varepsilon}}^{n {\widetilde{n}}} {\overbar{\gamma}} [ X_{{\widetilde{r}}}^{\rm V}, X_{s}^{\rm H}] (\sigma^{{\widetilde{r}}})_{{\widetilde{n}}}^{~{\widetilde{m}}} (\sigma^s)^{m}_{~n} + \nonumber \\*
&~ - \frac{i}{2} {\overbar{\varepsilon}}^{n {\widetilde{m}}} \epsilon^{rst} [X_{r}^{\rm H}, X_{s}^{\rm H}] (\sigma_t)^{m}_{~n} +i  {\overbar{\varepsilon}}^{n{\widetilde{m}}} \delta(x^2) Q^{r} (\sigma_r)^{m}_{~n} \nonumber \\*
=:&~ {\overbar{\varepsilon}}^{n{\widetilde{n}}}\mathcal{F}^{m{\widetilde{m}}}_{{\widetilde{n}} n}~,\nonumber \\
\intertext{and}
\text{\underline{Defect fields}}&\cr
\delta q^{m} =&~  i\overbar{\boldsymbol{\upepsilon}}^{m {\widetilde{m}}} \boldsymbol{\upzeta}_{{\widetilde{m}}} ~,  \qquad   \delta q_{m}^\dag = -i \overbar{\boldsymbol{\upzeta}}^{{\widetilde{m}}} \boldsymbol{\upepsilon}_{{\widetilde{m}}  m}~,  \nonumber \\*
\delta \boldsymbol{\upzeta}_{{\widetilde{m}}} =&~  \rho^{{\hat{\mu}}} \boldsymbol{\upepsilon}_{{\widetilde{m}}  m} \mathcal{D}_{{\hat{\mu}}} q^m + X_{{\widetilde{r}}}^{\rm V} (\sigma^{{\widetilde{r}}})_{{\widetilde{m}}}^{~{\widetilde{n}}} \boldsymbol{\upepsilon}_{{\widetilde{n}} m} q^m =: {\boldsymbol{f}}^{m {\widetilde{n}}}_{{\widetilde{m}}} \boldsymbol{\upepsilon}_{{\widetilde{n}} m}~, \nonumber \\*
\delta \overbar{\boldsymbol{\upzeta}}^{{\widetilde{m}}} =&~ (\mathcal{D}_{{\hat{\mu}}} q_{m}^\dag ) \overbar{\boldsymbol{\upepsilon}}^{m {\widetilde{m}}} \rho^{{\hat{\mu}}} - q_{m}^\dag \overbar{\boldsymbol{\upepsilon}}^{m{\widetilde{n}}} (\sigma^{{\widetilde{r}}})_{{\widetilde{n}}}^{~{\widetilde{m}}} X_{{\widetilde{r}}}^{\rm V} =: \overbar{\boldsymbol{\upepsilon}}^{m{\widetilde{n}}}\overbar{{\boldsymbol{f}}}^{{\widetilde{m}}}_{{\widetilde{n}} m} ~,
\end{align}
\endgroup
with supersymmetry parameters $\varepsilon_{{\widetilde{m}} m}$ and $\boldsymbol{\upepsilon}_{{\widetilde{m}}  m}$, where $\varepsilon_{{\widetilde{m}} m}$ is a 4d Majorana spinor, and $\boldsymbol{\upepsilon}_{{\widetilde{m}}  m}$ a 3d Majorana spinor embedded in $\varepsilon_{{\widetilde{m}} m}$ the same way that $\boldsymbol{\uplambda}_{{\widetilde{m}} m}$ is embedded in $\lambda_{{\widetilde{m}} m}$.   The quantity $Q^r$ in the ambient fermion variations is the combination
\begin{equation}\label{Qrdef}
Q^r := i(q_{m}^\dag (\sigma^r)^{m}_{~n} T_{{\boldsymbol{\rho}}}^a q^n) T^a ~.
\end{equation}
As we are using an antihermitian representation for our generators $T_{{\boldsymbol{\rho}}}^a$, $Q_{r}$ is an $\mathrm{SU}(2)_{\rm H}$ triplet valued in the real Lie algebra $\mathfrak{u}(N_{\rm c, def})$.  Note that we have defined ${\boldsymbol{f}}^{m{\widetilde{n}}}_{{\widetilde{m}}}$ and $\mathcal{F}^{m{\widetilde{m}}}_{{\widetilde{n}} n}$ to represent the groups of terms appearing in the  defect and ambient fermion variations, respectively.

Let us now fill in some details about these results and make a few comments.
\begin{itemize}
\item In the absence of the defect, the action would be 4d $\mathcal{N}_{\text{4d}}=4$ sYM, invariant for any choice of $\varepsilon_{{\widetilde{m}} m}$. The defect breaks the supersymmetry by half.  There is a map $P$ sending the 4d $\varepsilon_{{\widetilde{m}} m}$ to a 3d Majorana spinor $\boldsymbol{\upepsilon}_{{\widetilde{m}}  m}$:
\begin{align}\label{eq:P}
\boldsymbol{\upepsilon}_{{\widetilde{m}}  m} := P\varepsilon_{{\widetilde{m}} m}~.
\end{align}
The same map picks out the piece of the 4d ambient fermions $\lambda_{{\widetilde{m}} m}$ that interacts with the defect:
\begin{align}
\boldsymbol{\uplambda}_{{\widetilde{m}} m} := P\lambda_{{\widetilde{m}} m}~.
\end{align}
For convenience, we also define $P'$, the map that acts on Dirac conjugates of these Majorana spinors using the 4d and 3d intertwiners $A$ and $A_{(3)}$:
\begin{align}\label{eq:PPrime}
\overbar{\boldsymbol{\upepsilon}}^{m {\widetilde{m}}} = {\overbar{\varepsilon}}^{m {\widetilde{m}}} P' \quad \Rightarrow \quad P' := A^{-1} P^\dagger A_{(3)}~.
\end{align}
\item There are in principle many possible ways to embed the space of 3d Majorana spinors in the space of 4d Majorana spinors, equivalent to different choices of $P$. However, requiring that the action \eqref{eq:ambaction}, \eqref{eq:defaction} be invariant under the variations \eqref{eq:SUSYvars} severely restricts the possibilities. The necessary and sufficient conditions on $P$ are:
\begin{align}\label{eq:Pconditions}
& P (1 + i \gamma^2) \varepsilon_{{\widetilde{m}} m} = 0~, \qquad P \gamma^{{\hat{\mu}}} \varepsilon_{{\widetilde{m}} m} = 0~, \cr
& P {\overbar{\gamma}} \gamma^2 \varepsilon_{{\widetilde{m}} m} = 0~, \qquad P \gamma^{{\hat{\mu}} 2} \varepsilon_{{\widetilde{m}} m} = 0 ~\cr
& ( i \rho^{{\hat{\mu}}} P + P{\overbar{\gamma}} \gamma^{{\hat{\mu}}}) \varepsilon_{{\widetilde{m}} m} = 0~, \qquad (\rho^{{\hat{\mu}}{\hat{\nu}}} P - P \gamma^{{\hat{\mu}}{\hat{\nu}}}) \varepsilon_{{\widetilde{m}} m} = 0~.
\end{align}
The first two lines of \eqref{eq:Pconditions} impose conditions on $P$, while the final line imposes conditions on how the 3d Clifford algebra is embedded in the 4d Clifford algebra. The first two lines
are satisfied if half of the original $\mathcal{N}_{\text{4d}}=4$ supersymmetries are eliminated via the projection
\begin{equation}
\frac{1}{2}(1+i\gamma^2)\varepsilon_{{\widetilde{m}} m} =0~,
\end{equation}
and we take $P$ such that $P^\dag P$ gives the orthogonal projection:
\begin{align}\label{eq:PdagP}
P^\dagger P = \frac{1}{2} (1-i\gamma^2)~,
\end{align}
Note then that $\varepsilon_{{\widetilde{m}} m} = P^\dag P \varepsilon_{{\widetilde{m}} m} = P^\dag \boldsymbol{\upepsilon}_{{\widetilde{m}}  m}$, so $P^\dag$ embeds $\boldsymbol{\upepsilon}_{{\widetilde{m}}  m}$ into $\varepsilon_{{\widetilde{m}} m}$.  The corresponding projection on the Dirac conjugate spinor is ${\overbar{\varepsilon}}^{m {\widetilde{m}}} \frac{1}{2} (1 - i \gamma^2) = 0$.  It follows from the definition of $P'$ that $P' P^{\prime \dag} = \frac{1}{2} (1 + i \gamma^2)$ and so we similarly find that ${\overbar{\varepsilon}}^{m {\widetilde{m}}} = \overbar{\boldsymbol{\upepsilon}}^{m {\widetilde{m}}} P^{\prime \dag}$.  We then also have $P^{\prime \dag} \varepsilon_{{\widetilde{m}} m} = 0$.  In contrast, $\lambda_{{\widetilde{m}} m}$ does not satisfy any projection.  Rather $\tilde{\boldsymbol{\uplambda}}_{{\widetilde{m}} m} := P^{\prime \dag} \lambda_{{\widetilde{m}} m}$ is an independent 3d Majorana spinor contained in $\lambda_{{\widetilde{m}} m}$.  All of these statements take an extremely simple form in the basis \eqref{DFOlikebasis} where
 %
 \begin{align}
 P^\dagger P =
\frac{1}{2} (1 - i \gamma_2) = \frac{1}{2}( \mathbbm{1}_4 + \mathbbm{1}_2 \otimes \sigma^3) = \left( \begin{array}{c c} \mathbbm{1} & 0 \\ 0 & 0 \end{array} \right)~,
\end{align}
 %
and we can take
\begin{equation}\label{Pmatrix}
P = \left( \begin{array}{c c} \mathbbm{1} & 0 \end{array} \right) ~,\qquad P' = \left( \begin{array}{c} 0 \\ \mathbbm{1} \end{array} \right) ~.
\end{equation}

\item The ambient field variations are those of $\mathcal{N}_{4d}=4$ -- which can be obtained by translating the Majorana form given in \cite{Sohnius:1985qm} to an $\mathrm{SU}(2)_{\rm V} \times \mathrm{SU}(2)_{\rm H}$ covariant form as described in Appendix \ref{app:SO4toSU2SU2} -- augmented by terms involving the combination of defect fields $Q^r$.  This modification is due to a contribution from the defect to the ambient auxiliary $D$-term equation of motion.  As the auxiliary $D$-field appears only in the ambient fermion variation, it is only this variation that is modified from its defect-free counterpart.
\item  Due to the defect-induced modification of the ambient fermion variation, the ambient action \eqref{eq:ambaction} now varies as
\begin{align}\label{ambientinflow}
g_{\rm ym}^2 \delta S_{\rm amb} =&~  \int d^4 x  \, \textrm{Tr} \,  \bigg\{  -{\overbar{\varepsilon}}^{m {\widetilde{m}}} \delta(x^2) Q^r (\sigma_{r})^{p}_{~m} \times \cr
&~ \qquad  \times  \bigg( - \gamma^{\mu} \mathcal{D}_{\mu} \lambda_{{\widetilde{m}} p} + [\lambda_{{\widetilde{m}} n}, X_{s}^{\rm H}] (\sigma^{s})^{n}_{~p}  - i  (\sigma^{{\widetilde{s}}})_{{\widetilde{m}}}^{~{\widetilde{n}}} {\overbar{\gamma}} [\lambda_{{\widetilde{n}} p}, X_{{\widetilde{s}}}^{\rm V}]  \bigg) \bigg\} ~, \qquad
\end{align}
up to total derivatives which will be discussed in the next section. These inflow terms \eqref{ambientinflow} cancel against those terms in the variation of the defect action that do not involve $\boldsymbol{\upzeta}_{{\widetilde{m}}}$.  Variations of terms in \eqref{eq:defaction} that contain the defect fermions $\boldsymbol{\upzeta}_{{\widetilde{m}}}$ cancel amongst themselves.
\item The variation of the $\delta(0)$ term in the action cancels against terms that result from evaluating the defect-induced modification of the $\lambda_{{\widetilde{m}} m}$ variation on the defect.
\item We can see from the perspective of supersymmetry how the vacua described by solutions to Nahm's equations arise.  Noting that $\gamma^2 \varepsilon_{{\widetilde{m}} m} = i \varepsilon_{{\widetilde{m}} m}$, one sees that three of terms in the ambient fermion variation can be combined as
\begin{equation}
\delta \lambda_{{\widetilde{m}} m} \supset i \left(  \mathcal{D}_{2} X_{r}^{\rm H}  + \frac{1}{2} \epsilon_{r}^{~st} [X_{s}^{\rm H}, X_{t}^{\rm H}]  -  \delta(x^2) Q_{r} \right) (\sigma^r)^{n}_{~m} \varepsilon_{{\widetilde{m}} n}~.
\end{equation}
Thus, by setting the quantity in parentheses to zero and choosing appropriate vacuum conditions on $A_{{\hat{\mu}}}, X_{{\widetilde{r}}}^{\rm V}$, the fermion variation vanishes without imposing any restrictions on the supersymmetry parameter $\boldsymbol{\upepsilon}_{{\widetilde{m}} m}$.  The quantity in parentheses is precisely the Nahm equation, with jumping data specified by $Q_r$.
\end{itemize}

\section{Boundary Terms, Supercurrents, and Supercharges}\label{sec:boundary}

In this section we describe the boundary terms that arise while applying the supersymmetry variations in the action above. We also write down the supercurrents, as well as the supercharges and the resulting algebra.

\subsection{Boundary Terms and Supercurrent}

Under the supersymmetry transformations \eqref{eq:SUSYvars}, with spacetime-varying supersymmetry parameters, the total ambient plus defect action transforms as
\begin{align}\label{BoundaryAndCurrent}
g_{\rm ym}^2 \delta (S_{\rm amb} + S_{\rm def}) =&~ \int d^4 x \left\{ (\partial_{\mu} {\overbar{\varepsilon}}^{m {\widetilde{m}}}) \mathcal{J}_{{\widetilde{m}} m}^{\mu} + \partial_{\mu} \mathcal{B}^{\mu} \right\}   ~.
\end{align}
The $\mathcal{B}_\mu$'s encapsulate the boundary terms, while the $\mathcal{J}^{\mu}_{{\widetilde{m}} m}$ are supercurrents.

The boundary terms that result from varying the action are given by
\begin{align}\label{totalBB}
\int d^4 x \partial_{\mu} \mathcal{B}^{\mu} =&~ \int d^4 x  \partial_{\mu} (\mathcal{B}_{\rm amb})^{\mu} + \int d^3 x \partial_{{\hat{\mu}}} (\mathcal{B}_{\rm def})^{{\hat{\mu}}} + \cr
& \qquad\qquad\qquad+  \frac{i}{g_{\rm ym}^2} \int d^4 x \partial_2  \, \textrm{Tr} \,  \left\{ \delta(x^2) \overbar{\boldsymbol{\upepsilon}}^{n {\widetilde{m}}} Q_r (\sigma^r)^{m}_{~n} \tilde{\boldsymbol{\uplambda}}_{{\widetilde{m}} m} \right\} ~,
\end{align}
with
\begin{align}\label{boundarycurrents}
(\mathcal{B}_{\rm amb})^{\mu} :=&~ \frac{1}{g_{\rm ym}^2}  \, \textrm{Tr} \,  \bigg\{ - 2 F^{\mu\nu} \delta A_{\nu} - 2 (\mathcal{D}^{\mu} X^{{\rm V}{\widetilde{r}}}) \delta X_{{\widetilde{r}}}^{\rm V} - 2 (\mathcal{D}^{\mu} X^{{\rm H}r}) \delta X_{r}^{\rm H} - \frac{i}{2} \delta {\overbar{\lambda}}^{m {\widetilde{m}}} \gamma^{\mu} \lambda_{{\widetilde{m}} m} \bigg\} ~, \cr
(\mathcal{B}_{\rm def})^{{\hat{\mu}}} :=&~   \frac{1}{g_{\rm ym}^2} \left\{- (\mathcal{D}^{{\hat{\mu}}} q_{m}^\dag) \delta q^m - \frac{i}{2} (\delta \overbar{\boldsymbol{\upzeta}}^{{\widetilde{m}}}) \rho^{{\hat{\mu}}} \boldsymbol{\upzeta}_{{\widetilde{m}}}  - (\delta q_{m}^\dag) \mathcal{D}^{{\hat{\mu}}} q^m + \frac{i}{2} \overbar{\boldsymbol{\upzeta}}^{{\widetilde{m}}} \rho^{{\hat{\mu}}} \delta \boldsymbol{\upzeta}_{{\widetilde{m}}} \right\} ~.\raisetag{20pt}
\end{align}
Here we separated out the $\partial_{2} ( \delta(x^2) \cdots)$ term because it is neither an ambient or defect term. It exists only in vacua where the number of colors is different on one side of the defect than the other. Note that, with the exception of this term, the boundary terms can be written entirely in terms of the SUSY variations \eqref{eq:SUSYvars}.

The boundary terms in \eqref{totalBB} and \eqref{boundarycurrents} must vanish for systems preserving the $\mathcal{N}_{\text{3d}}=4$ supersymmetry. This determines the boundary conditions we must impose on the fields at spatial infinity\footnote{Assuming appropriate Neumann or Dirichlet conditions are set up at initial and final times.} and at the defect ($x^2=0$). The boundary terms at spatial infinity vanish with appropriate fall-off condition on the fields. These conditions play an important role in determining the types of soliton states present in the system, and are better discussed in conjunction with the central charges (in Subsection \ref{subsec:supercharges}).

The boundary at $x^2=0$, meanwhile, is trivial if the ambient fields are continuous across the defect. If, however, there are D3-branes ending on the D5's as in \cite{Hanany:1996ie}, the ambient fields can be discontinuous. In this case, a special choice of  boundary conditions is needed to maintain supersymmetry in the presence of  differing gauge groups on either side of the defect. Choosing  boundary conditions for the ambient fields
\begin{align}
&\delta X^{\rm{H}}_r \bigg|_{x^2=0} = 0 \ , \qquad \delta A_2 \bigg|_{x^2=0} = 0 \ , \qquad \tilde{\boldsymbol{\uplambda}}_{{\widetilde{m}} m}\bigg|_{x^2=0}=0 \cr
& F^{2{\hat{\nu}}} \bigg|_{x^2=0}=0 \ , \qquad  \mathcal{D}_2 X^{\rm{V}}_{\widetilde{r}} \bigg|_{x^2=0} = 0 \ ,
\end{align}
guarantees that all boundary terms from the variation of the ambient action vanish. The variations here are general, and the supersymmetry variations, in particular,  should obey these conditions. This result is consistent with \cite{Hanany:1996ie}'s original analysis: $(\boldsymbol{\uplambda}_{{\widetilde{m}} m}, \ A_{\hat{\mu}}, \ X^{\rm{V}}_r)$ form an $\mathcal{N}_{\text{3d}}=4$ vector multiplet satisfying Neumann boundary conditions, while $(X^{\rm{H}}_r, \ A_2, \ \tilde{\boldsymbol{\uplambda}}_{{\widetilde{m}} m})$ form an $\mathcal{N}_{\text{3d}}=4$ hypermultiplet satisfying Dirichlet conditions. (By ``Dirichlet'' we mean that the behavior of the fields is fixed: this thus includes the Nahm pole boundary condition.)

Meanwhile, the supercurrent from \eqref{BoundaryAndCurrent} takes the form
\begin{align}\label{fullsupercurrent}
\mathcal{J}_{{\widetilde{m}} m}^{\mu} = P^{\prime \dag} (\mathcal{J}_{\rm amb})_{{\widetilde{m}} m} + \delta(x^2) \delta^\mu_{~ {\hat{\mu}}} \left((\mathcal{J}_{\rm def})^{{\hat{\mu}}}_{{\widetilde{m}} m} + C_{(3)} (\overbar{\mathcal{J}}_{\rm def}^T)_{{\widetilde{m}} m}^{{\hat{\mu}}} \right)
\end{align}
with
\begin{align}\label{43supercurrents}
(\mathcal{J}_{\rm amb})_{{\widetilde{m}} m}^{\mu} :=&~  \frac{i}{g_{\rm ym}^2}  \, \textrm{Tr} \,  \left\{ \mathcal{F}_{{\widetilde{m}} m}^{n {\widetilde{n}}} \gamma^{\mu} \lambda_{{\widetilde{n}} n} \right\} ~, \cr
(\mathcal{J}_{\rm def})^{{\hat{\mu}}}_{{\widetilde{m}} m} :=&~  \frac{i}{g_{\rm ym}^2} \overbar{{\boldsymbol{f}}}_{{\widetilde{m}} m}^{{\widetilde{n}}} \rho^{{\hat{\mu}}} \boldsymbol{\upzeta}_{\widetilde{n}} ~, \qquad (\overbar{\mathcal{J}}_{\rm def})_{{\hat{\mu}}}^{m {\widetilde{m}}} := -  \frac{i}{g_{\rm ym}^2} \overbar{\boldsymbol{\upzeta}}^{\widetilde{n}} \rho_{{\hat{\mu}}} {\boldsymbol{f}}_{{\widetilde{n}}}^{m {\widetilde{m}}} ~,
\end{align}
where ${\boldsymbol{\FF}}^{m {\widetilde{m}}}_{n {\widetilde{m}}}$ and ${\boldsymbol{f}}^{m {\widetilde{m}}}_{{\widetilde{n}}}$ are the combinations of fields appearing in the ambient and defect fermion variations \eqref{eq:SUSYvars}. Note the appearance of the projector $P^{\prime \dag}$ in the first line, which expresses ${\overbar{\varepsilon}}^{m {\widetilde{m}}}$ in terms of $\overbar{\boldsymbol{\upepsilon}}^{m {\widetilde{m}}}$.  In the absence of the defect, transformations parameterized by any $\varepsilon_{{\widetilde{m}} m}$ are symmetries; with the defect, only the components of $\varepsilon_{{\widetilde{m}} m}$ restricted by the projection \eqref{eq:PPrime} are. The current $\mathcal{J}_{{\widetilde{m}} m}^{\mu}$ satisfies the Majorana condition \eqref{eq:modifiedMajoranaCond}, because $(\mathcal{J}_{\rm amb})_{{\widetilde{m}} m}^{\mu}$ itself satisfies it, and the remaining contribution is a sum of $(\mathcal{J}_{\rm def})^{{\hat{\mu}}}_{{\widetilde{m}} m}$ and its charge conjugate.

It's a nontrivial result that the supercurrent can be written entirely in terms of the field combinations appearing in the fermion variations alone. In particular, the defect-induced modification of the ambient fermion variation that appears in $\mathcal{F}_{{\widetilde{m}} m}^{n {\widetilde{n}}}$ falls out of the variation of the defect action. As a consequence, the split of $\mathcal{J}^\mu_{{\widetilde{m}} m}$ into $(\mathcal{J}_{\rm amb})_{{\widetilde{m}} m}^{\mu}$ and $(\mathcal{J}_{\rm def})^{{\hat{\mu}}}_{{\widetilde{m}} m}$ is a bit deceptive, since   $(\mathcal{J}_{\rm amb})_{{\widetilde{m}} m}^{\mu}$  contains a term that localizes to the defect.

\subsection{Supercharges}\label{subsec:supercharges}

With the supercurrent in hand, we have the field representation of the supercharges and we can compute their algebra. Of particular interest is the form of the central charges that appear there.

The Noether (super)charges are spatial integrals of the time component of the current (at a fixed time):
\begin{align}\label{QNoether1}
\mathcal{Q}_{{\widetilde{\alpha}} {\widetilde{m}} m} =&~ \int d^3 x (\mathcal{J}^0_{{\widetilde{m}} m})_{{\widetilde{\alpha}}}  \cr
=&~ \frac{i}{g_{\rm ym}^2} \int d^3 x  \, \textrm{Tr} \,  \left\{ (P^{\prime \dag} \mathcal{F}_{{\widetilde{m}} m}^{n {\widetilde{n}}} \gamma^0 \lambda_{{\widetilde{n}} n})_{\widetilde{\alpha}} \right\} + \cr &\qquad\qquad + \frac{i}{g_{\rm ym}^2} \int d^2 x \left( \overbar{{\boldsymbol{f}}}^{{\widetilde{n}}}_{{\widetilde{m}} m} \rho^0 \boldsymbol{\upzeta}_{\widetilde{n}} - \epsilon_{{\widetilde{n}} \widetilde{p}} \epsilon_{mn} C_{(3)} ( \overbar{\boldsymbol{\upzeta}}^{\widetilde{n}} \rho^0 {\boldsymbol{f}}_{{\widetilde{n}}}^{n \widetilde{p}})^T \right)_{{\widetilde{\alpha}}} ~,
\end{align}
where we have restored the 3d spinor index ${\widetilde{\alpha}}$ for clarity.

In order to determine the algebra, we compute the canonical momenta from the Lagrangian and impose canonical brackets between coordinates and momenta.\footnote{These are Dirac brackets for the fermions, since they satisfy second-class constraints.} We find that the bracket of two supercharges takes the form
\begin{align}\label{SUSYalgebra}
\{ \mathcal{Q}_{{\widetilde{\alpha}} {\widetilde{m}} m} , \mathcal{Q}_{{\widetilde{\beta}} {\widetilde{n}} n} \} =&~ -i \bigg\{ \epsilon_{{\widetilde{m}} {\widetilde{n}}} \epsilon_{mn} (\rho^{{\hat{\mu}}} C_{(3)})_{{\widetilde{\alpha}}{\widetilde{\beta}}} P_{{\hat{\mu}}}  + \left( (\sigma^r \epsilon)_{{\widetilde{m}} {\widetilde{n}}} \epsilon_{mn} \mathcal{Z}_{r}^{\rm V} + \epsilon_{{\widetilde{m}} {\widetilde{n}}} (\epsilon \sigma^r)_{mn} \mathcal{Z}_{r}^{\rm H} \right) (C_{(3)})_{{\widetilde{\alpha}}{\widetilde{\beta}}}   + \cr
&~ \qquad \qquad + (\sigma^r \epsilon)_{{\widetilde{m}} {\widetilde{n}}} (\epsilon \sigma^s)_{mn} (\mathcal{Z}^{\rm VH})_{rs}^{\hat{\imath}} (\rho_{\hat{\imath}} C_{(3)})_{{\widetilde{\alpha}}{\widetilde{\beta}}} \bigg\}~.  \raisetag{20pt}
\end{align}

The Hamiltonian $H = P^0 = -P_0$ and momenta are the $\mathcal{N}_{\rm 4d}=4$ Hamiltonian and momenta, supplemented by contributions from the defect. The bosonic terms are
\begin{align}
H =&~ H_{ \mathcal{N}_{\rm 4d} = 4} + \frac{1}{g_{\rm ym}^2} \int d^2 x \bigg\{ \mathcal{D}_{0} q_{p}^\dag \mathcal{D}_{0} q^p + \mathcal{D}_{\hat{\imath}} q_{p}^\dag \mathcal{D}^{\hat{\imath}} q^p - q_{p}^\dag (X_{{\boldsymbol{\rho}} r}^{\rm V})^2 q^p + \cr
&~ \qquad  \qquad \qquad \qquad \qquad  -i q_{p}^{\dag} (\sigma^r)^{p}_{~q} \left( \mathcal{D}_2 X_{r}^{\rm H} + \frac{1}{2} \epsilon_{r}^{~st} [X_{s}^{\rm H}, X_{t}^{\rm H}] \right)_{{\boldsymbol{\rho}}} q^q + \cr
&~ \qquad \qquad \qquad \qquad \qquad  - \frac{1}{2} \delta(0) ( q_{m}^\dag (\sigma^r)^{m}_{~n} T_{{\boldsymbol{\rho}}}^a q^n) ( q_{p}^\dag (\sigma_r)^{p}_{~q} T_{{\boldsymbol{\rho}}}^a q^{q}) \bigg\}~,\cr
P^{\hat{\imath}} =&~ P_{\mathcal{N}_{\rm 4d} = 4}^{\hat{\imath}} - \frac{1}{g_{\rm ym}^2} \int d^2 x  \left\{ \mathcal{D}_0 q_{p}^\dag D_{\hat{\imath}} q^p + \mathcal{D}_{\hat{\imath}} q_{p}^\dag \mathcal{D}_0 q^p \right\}
\end{align}
where the $\mathcal{N}_{\rm 4d}=4$ versions are given by
\begin{align}\label{4DHamiltonianAndMom}
H_{ \mathcal{N}_{\rm 4d} = 4} =&~  \frac{1}{g_{\rm ym}^2} \int d^3 x  \, \textrm{Tr} \,  \bigg\{ F_{i 0} F^{i}_{~0} + \frac{1}{2} F^{ij} F_{ij}  + (\mathcal{D}_{0} X_{{\widetilde{r}}}^{\rm V})^2 + (\mathcal{D}_{i} X_{{\widetilde{r}}}^{\rm V})^2 +  (\mathcal{D}_{0} X_{r}^{\rm H})^2 + (\mathcal{D}_{i} X_{r}^{\rm H})^2 + \cr
& \qquad\qquad\qquad \qquad + \frac{1}{2} [X_{{\widetilde{r}}}^{\rm V}, X_{{\widetilde{s}}}^{\rm V}] ^2 +  [X_{{\widetilde{r}}}^{\rm V}, X_{s}^{\rm H}]^2+ \frac{1}{2}  [X_{r}^{\rm H}, X_{s}^{\rm H}]^2 \bigg\}~, \cr
P_{\mathcal{N}_{\rm 4d} = 4}^{i} =&~  -\frac{1}{g_{\rm ym}^2} \int d^3x  \, \textrm{Tr} \,  \bigg\{ 2 F_{j 0} F^{ji} + 2 (\mathcal{D}_0 X_{{\widetilde{r}}}^{\rm V}) (\mathcal{D}^{i} X_{{\widetilde{r}}}^{\rm V}) +2 (\mathcal{D}_0 X_{r}^{\rm H}) (\mathcal{D}^{i} X_{r}^{\rm H}) \bigg\} ~. \raisetag{20pt}
\end{align}
Note that the Hamiltonian \label{4DHamiltonianAndMom} contains the square
\begin{align}
H \supset &~ \frac{1}{g_{\rm ym}^2} \int d^3 x  \, \textrm{Tr} \,  \left( \mathcal{D}_2 X_{r}^{\rm H} + \frac{1}{2} \epsilon_{r}^{~st} [X_{s}^{\rm H}, X_{t}^{\rm H}]  - \delta(x^2) Q_r  \right)^2 ~,
\end{align}
where one must use \eqref{Qrdef} and $ \, \textrm{Tr} \, (T^aT^b) = \frac{1}{2} \delta^{ab}$.  This demonstrates the existence of the Nahm-type vacua mentioned earlier from the perspective of energy.  We emphasize that the Hamiltonian already contains this square without the need to complete the square by adding and subtracting a topological term.  The $\delta(0)$ term is essential for this property.

The central charges appearing in \eqref{SUSYalgebra} are
\begin{align}
\mathcal{Z}_{{\widetilde{r}}}^{\rm V} =&~  \mathcal{Z}^{\rm V}_{{\rm el},{\widetilde{r}}} + \frac{1}{g_{\rm ym}^2} \epsilon_{{\widetilde{r}}}^{~{\widetilde{s}} {\widetilde{t}}} \epsilon^{\hat{\imath}\hat{\jmath}} \int d^3x \partial_{\hat{\imath}}  \, \textrm{Tr} \,  \left\{ X_{{\widetilde{s}}}^{\rm V} \mathcal{D}_{\hat{\jmath}} X_{{\widetilde{t}}}^{\rm V} \right\} ~, \cr
\mathcal{Z}_{r}^{\rm H} =&~ \mathcal{Z}^{\rm H}_{{\rm mag},r} +  \frac{1}{g_{\rm ym}^2} \epsilon_{r}^{~st} \epsilon^{\hat{\imath}\hat{\jmath}} \int d^3x \partial_{\hat{\imath}}  \, \textrm{Tr} \,  \left\{ X_{s}^{\rm H} \mathcal{D}_{\hat{\jmath}} X_{t}^{\rm H} \right\} + \frac{i}{g_{\rm ym}^2} \epsilon^{\hat{\imath}\hat{\jmath}} \int d^2 x \partial_{\hat{\imath}} \left( q_{p}^\dag (\sigma^r)^{p}_{~q} \mathcal{D}_{\hat{\jmath}} q^q \right) ~, \cr
(\mathcal{Z}^{\rm VH})^{\hat{\imath}}_{{\widetilde{r}} s} =&~ - \frac{2}{g_{\rm ym}^2} \epsilon^{\hat{\imath}\hat{\jmath}} \int d^3 x \partial_{\hat{\jmath}}  \, \textrm{Tr} \,  \left\{ X_{{\widetilde{r}}}^{\rm V} \left( \mathcal{D}_2 X_{s}^{\rm H} - \delta(x^2) Q_s \right) \right\} ~,
\end{align}
where $\mathcal{Z}^{\rm V}_{{\rm el},{\widetilde{r}}}$ and $\mathcal{Z}^{\rm H}_{{\rm mag},r}$ comprise the half of the $\mathcal{N}_{\rm 4d}=4$ central charges that survive the projection to $\mathcal{N}_{\rm 3d}=4$:
\begin{align}
&\mathcal{Z}^{\rm V}_{{\rm el},{\widetilde{r}}} = - \frac{2}{g_{\rm ym}^2} \int d^3 x \partial^{i}  \, \textrm{Tr} \,  \{ F_{i 0} X_{{\widetilde{r}}}^{\rm V} \} ~, \qquad \mathcal{Z}^{\rm H}_{{\rm mag},r} = \frac{1}{g_{\rm ym}^2} \int d^3 x \epsilon^{ijk} \partial_{i}  \, \textrm{Tr} \,  \{ F_{jk} X_{r}^{\rm H} \}  ~.
\end{align}
These measure 3d electric and magnetic flux, respectively, along  the $X^{\rm{V}}$ and $X^{\rm{H}}$  Higgs directions.

Let us make some comments about the central charges:
\begin{itemize}
\item The $\mathcal{Z}^{\rm VH}$ term in \eqref{SUSYalgebra} explicitly breaks the Poincar\'{e} covariance of the algebra, and is not expected to contribute for finite-energy field configurations. It may, however, contribute if one considers BPS strings with finite energy per unit length.
 \item Standard fall-off conditions on the asymptotic $S^2$ of the 3d ambient space allow for finite-energy configurations with electric and magnetic charge. In addition, vortex-type boundary conditions on the asymptotic $S^1$ of the 2d defect space will yield finite contributions to $\mathcal{Z}_r^{\rm H}$ from the $q^\dagger \mathcal{D} q$ term.
 %
 \item The $X\mathcal{D} X$-type terms that appear in both $\mathcal{Z}^{\rm V}_{\widetilde{r}}$ and $\mathcal{Z}^{\rm H}_r$ are integrated over the 3d ambient space, but restrict not to an $S^2$ at infinity, but an ${\mathbbm R}_{x^2}\times S^1$. One can imagine a field configuration which has vortex-like asymptotics on the $S^1$, but which falls off along the ${\mathbbm R}_{x^2}$ as one moves away from the defect, thus resulting in finite energy.
 %
\item It is fascinating that the hypermultiplet central charge receives contributions from both 3d monopole charge, and 2d vortex charge. This strongly suggests that there exist BPS field configurations in which a non-abelian vortex string is stretched between a magnetic monopole in the ambient space and the defect.  We leave a detailed examination of soliton configurations to future work.
\end{itemize}

\section{Conclusion}\label{sec:conclusion}
We have written down the full R-symmetry-covariant supersymmetry variations of $\mathcal{N}_{4d}=4$ sYM with a non-abelian, codimension-one defect preserving $\mathcal{N}_{\rm 3d} = 4$ supersymmetry. We also computed the supercurrents, supercharges, and the algebra. This fills a gap in the literature, and opens the door to studies of BPS states in a defect field theory with a holographic dual.

{\bf Acknowledgements:}  The work of ABR is supported by NSF grant number PHY-2112781.  SKD's work is supported by NSF grant number PHY-2014025. ABR thanks Ilarion Melnikov for useful discussions.

redpajama_set_name: RedPajamaArXiv