††institutetext: Arnold Sommerfeld Center, Ludwig-Maximilians-Universität,
Theresienstraße 37, 80333 München, Germany

# Boundaries and supercurrent multiplets in 3D Landau-Ginzburg models

Ilka Brunner  Jonathan Schulz  and Alexander Tabler<EMAIL_ADDRESS>muenchen.de<EMAIL_ADDRESS><EMAIL_ADDRESS>muenchen.de

###### Abstract

Theories with 3D $\mathcal{N}=2$ bulk supersymmetry may preserve a 2D
$\mathcal{N}=(0,2)$ subalgebra when a boundary is introduced, possibly with
localized degrees of freedom. We propose generalized supercurrent multiplets
with bulk and boundary parts adapted to such setups. Using their structure, we
comment on implications for the $\overline{Q}_{+}$-cohomology. As an example,
we apply the developed framework to Landau-Ginzburg models. In these models,
we study the role of boundary degrees of freedom and matrix factorizations. We
verify our results using quantization.

###### Keywords:

Supersymmetry and Duality, Boundary Quantum Field Theory, Field Theories in
Lower Dimensions

††preprint: LMU-ASC 16/19

##  1 Introduction

In this paper we study $\mathcal{N}=2$ supersymmetric theories in flat
$(2+1)$-dimensional spacetime with (spacelike) boundaries. The boundary
necessarily breaks translational invariance and hence can only preserve part
of the bulk supersymmetry. Explicitly, the $\mathcal{N}=2$ algebra in 3
dimensions is

$\\{Q_{\pm},\overline{Q}_{\pm}\\}=-4P_{\pm},\quad\\{Q_{+},\overline{Q}_{-}\\}=2P_{\perp},$
(1)

where spacetime has coordinates $x^{\pm}$ and $x_{\perp}$, see appendix A for
a summary of our conventions. We want to consider the case with a boundary in
$x_{\perp}$-direction, breaking supersymmetry to a subalgebra of (1) that does
not contain $P_{\perp}$, the generator of translations in that direction. As
has been analyzed before Dimofte:2017tpi ; Okazaki:2013kaa ; Gadde:2013sca ,
in the case of $\mathcal{N}=2$ supersymmetry in $3$ dimensions, there are two
types of supersymmetric boundary conditions, referred to as A-type and B-type.
Each of them is associated to a subalgebra of the initial bulk supersymmetry
algebra, containing two momentum operators and two supersymmetry charges.
A-type boundary conditions preserve $(1,1)$ supersymmetry, whereas B-type
boundary conditions preserve a chiral ${\mathcal{N}=(0,2)}$ subalgebra,
generated by $Q_{+}$ and $\overline{Q}_{+}$. We will focus on $(0,2)$ boundary
conditions and analyze them from two points of view: On the one hand, for
theories defined by a Lagrangian, we employ a Noether procedure. On the other
hand, we discuss the structure of the supercurrent multiplets
Komargodski:2010rb ; Dumitrescu:2011iu  and formulate boundary multiplets. The
two points of view are interrelated, as the (improved) Noether currents form
components of the current multiplets.

From a Lagrangian point of view, the supersymmetric bulk Lagrangian transforms
under SUSY-variations into a total derivative. In the presence of a boundary,
this generically yields a boundary term which must be canceled for the
symmetry to be preserved. This can be achieved by choosing boundary conditions
on the fields, such that the boundary variation vanishes. Alternatively, and
this is the main focus of the present paper, one can cancel the boundary
variation by adding a suitable boundary part to the action, such that the
action is invariant under symmetry variations without reference to the
boundary conditions on the fields. This term can contain extra boundary
degrees of freedom that are not inherited from the bulk. The full invariant
action thus contains a bulk and boundary term.

$S=\int_{M}{\cal L}^{B}+\int_{\partial M}{\cal L}^{\partial}.$ (2)

Given an action which is invariant under a symmetry, Noether’s procedure
yields a set of conserved charges and currents. In the case of supersymmetry,
this includes the (canonical) energy momentum tensor and the supersymmetry
currents. After imposing canonical commutation relations between the fields,
the Noether charges provide a representation of the symmetry algebra in terms
of the fields.

In the case of pure bulk theories, it is very useful to arrange the
supercurrents together with other conserved currents into multiplets. This can
be done independently of a Lagrangian definition of a theory. The
supercurrents form a representation of the supersymmetry algebra, and their
most general form has been discussed recently in Komargodski:2010rb ;
Dumitrescu:2011iu . Supercurrent multiplets are indecomposable SUSY multiplets
that contain the stress energy tensor $T_{\mu\nu}$ and the supercurrents
$S_{\alpha\mu}$ as part of their components. In addition, there are brane
currents, whose integrals yield brane charges. The components of the
supercurrent multiplets appear in a local version of the supersymmetry
algebra, which very schematically takes the form

$\displaystyle\\{\overline{Q}_{\alpha},S_{\beta\mu}\\}$
$\displaystyle=2\gamma_{\alpha\beta}^{\nu}T_{\mu\nu}+\dots,$ (3)
$\displaystyle\\{Q_{\alpha},S_{\beta\mu}\\}$ $\displaystyle=\dots,$ (4)

where the dots indicate various currents that will be explained later in the
text. As is well-known, the stress tensor for a theory is not unique, but can
be modified by improvement transformations. Indeed, the same holds for the
supercurrents, and the notion of improvement transformations can be lifted to
the full multiplet. In any three-dimensional $\mathcal{N}=2$ supersymmetric
theory (and also in other dimensions with the same amount of supersymmetry),
there exists a so-called $\mathcal{S}$-multiplet. Under special conditions,
the $\mathcal{S}$-multiplet can be decomposable, such that there exist smaller
multiplets. Of special interest in the context of the current work is the
$\mathcal{R}$-multiplet, which exists in theories which exhibit an
$R$-symmetry.

The notion of supercurrent multiplets has been extended to theories with
defects in Drukker:2017xrb , where a new so-called defect multiplet was
constructed. As a consequence of the violation of translation symmetry
perpendicular to the defect, the stress tensor is no longer conserved. This
violation is encoded in the displacement operator. The defect multiplet
contains the displacement operator as one of its components Gaiotto:2013sma ;
Drukker:2017xrb .

In the current paper, we consider supermultiplets in situations with boundary,
focusing on the _preserved_ symmetries. As mentioned above, to formulate
boundary conditions means to specify a subalgebra of the SUSY algebra such
that the momentum operator in the direction perpendicular to the boundary is
not contained. The supercurrent multiplet is in particular a representation of
the larger (bulk) algebra and hence also of the smaller algebra. In the case
of the $\mathcal{N}=(0,2)$ subalgebra, we show how the bulk supercurrent
multiplets decompose under the smaller algebra. Of course, due to the presence
of the boundary, the currents contained in the multiplet are no longer
conserved by themselves. To formulate a consistent multiplet for a theory with
boundary, we discuss how to add boundary parts to the $(0,2)$-components of
the initial bulk multiplet. Our ansatz for a full $\mathcal{R}$-multiplet is

$\mathcal{R}_{\mu}^{\text{full}}=\mathcal{R}_{\mu}^{B}+\mathcal{P}_{\mu}^{\phantom{\mu}{\hat{\mu}}}\delta(x_{\perp})\mathcal{R}^{\partial}_{\hat{\mu}},$
(5)

where $\mathcal{R}_{\mu}^{B}$ is the bulk part,
$\mathcal{R}^{\partial}_{\hat{\mu}}$ is the boundary part and
$\mathcal{P}_{\mu}^{\phantom{\mu}{\hat{\mu}}}$ denotes an embedding. Both
parts decompose into $(0,2)$-components. The boundary part is added to the
bulk multiplet in such a way that the initial divergence-freeness of the bulk
currents is completed to bulk-boundary conservation laws. We do not discuss
possible modifications of the bulk currents corresponding to the _broken_
symmetries.

One important feature of supercurrent multiplets is that they fall into short
representations of the supersymmetry algebra. Therefore, they are protected
under RG flow and retain their form Dumitrescu:2011iu . The supercurrents of
the quantum theory can thus be used to constrain the IR behavior of a theory
using the UV information. In the case of _two-dimensional_
${\mathcal{N}=(0,2)}$ models, the supercurrent multiplet for theories with an
$R$-symmetry was used to study renormalization group invariants in
Dedushenko:2015opz . In particular, it was shown that an RG invariant chiral
algebra exists, extending earlier works Witten:1993jg ; Silverstein:1994ih ;
Witten:2005px . The chiral algebra arises as the cohomology of the supercharge
$\overline{Q}_{+}$. Using only the form of the supercurrent multiplet,
Dedushenko:2015opz  shows that there is a half-twisted stress tensor (the
original stress tensor modified by the $R$-current) in the cohomology. As a
consequence, conformal symmetry is part of the chiral algebra.

Given these findings, we consider the consequences of the supercurrent
multiplets for the case that the ${\mathcal{N}=(0,2)}$ supersymmetry is the
symmetry preserved at the _boundary_ of a three-dimensional theory. We do not
find a stress-tensor in the cohomology following the steps in
Dedushenko:2015opz , however, there is a weaker statement. For this, one makes
the $(0,2)$-structure manifest by regarding the three-dimensional
$\mathcal{N}=2$ theory as a two-dimensional theory living on the boundary
$\mathbb{R}^{1,1}$. The bosonic fields of this theory are valued in maps from
$\mathbb{R}_{\leq 0}$ to the original target. The currents are obtained from
the original three-dimensional ones by integrating over the direction
perpendicular to the boundary and are preserved in the boundary theory. In
this theory, we then do have a stress energy tensor that is part of the
cohomology. Formulated in the initial theory, this cohomology element is
obtained by an integration in the perpendicular direction from infinity (or a
second boundary, which we do not discuss here) to the boundary. Note that the
action of any charge computed from the currents applied to an insertion at the
boundary would involve an integration over this direction, as well as all
other spatial directions. In this sense we can also interpret the partially
integrated currents in the original theory.

The integration along the perpendicular direction also gives another
perspective on the boundary multiplets. The theory effectively becomes a two-
dimensional theory with $\mathcal{N}=(0,2)$ supersymmetry, and the integrated
multiplets are genuine $\mathcal{N}=(0,2)$ multiplets.

In the second part of the paper, we study a specific example, namely a theory
of three-dimensional chiral multiplets with a superpotential. We restrict our
explicit discussion to the case of a single chiral field with a monomial
superpotential. However, a generalization to more than one superfield and an
arbitrary superpotential preserving an $R$-symmetry is straightforward and our
discussion applies to this case as well. As has already been shown in
Gadde:2013sca ; Yoshida:2014ssa , the condition of preserving
${\mathcal{N}=(0,2)}$ supersymmetry leads to a three-dimensional
generalization of matrix factorizations Kapustin:2002bi ; Brunner:2003dc . In
this case, boundary terms are canceled by adding fermionic degrees of freedom
and a potential at the boundary. Using Noether’s procedure, we compute the
conserved currents for different boundary conditions. These currents contain
pure bulk as well as boundary pieces and only the combination of both is
conserved. We also discuss the current multiplets, following Dumitrescu:2011iu
for the bulk case. Starting from the Noether currents, one needs to apply
improvement transformations to symmetrize the stress energy tensor and
subsequently organize the currents into supercurrent multiplets. As we
consider the case where $R$-symmetry is preserved, the relevant multiplet is
the $\mathcal{R}$-multiplet. We spell out all components of the bulk-boundary
multiplet in the example following the strategy outlined above.

To complete our understanding of the symmetries in the model, we study the
algebra of the supercharges and supercurrents (3) in the example. We start
with the explicit expressions of the currents in terms of fields and impose
canonical commutation relations on the fields. We then verify the relations
(3) as well as the $\mathcal{N}=(0,2)$ superalgebra in the specific
representation of the example. In the computation, it is essential to make use
of the correct factorization properties of the superpotential to recover the
correct form of the algebra.

This paper is organized as follows: In section 2 we review and elaborate on
various aspects of theories formulated on Minkowski space $\mathbb{R}^{1,N-1}$
with a flat boundary $\mathbb{R}^{1,N-2}$, following and extending
DiPietro:2015zia . In section 3 we discuss supercurrent multiplets. We review
the constraints that have to be satisfied by such a multiplet from
Dumitrescu:2011iu . As explained there, the most general supercurrent
multiplets consists of certain superfields satisfying defining constraint
relations. Their solutions are unique up to improvement transformations. In
special situations, such improvements can be used to formulate shorter
multiplets, in particular the $\mathcal{R}$-multiplet. We decompose the
multiplets and constraints according to the $(0,2)$-substructure and formulate
consistent bulk-boundary multiplets. Here, current conservation of the
combined bulk-boundary system is imposed. We then discuss an integrated
structure, where we integrate in the direction perpendicular to the boundary.
This provides a two-dimensional version of the conservation equations, taking
the familiar form of divergence-freeness of the currents.

We then turn to a discussion of the Landau-Ginzburg example in section 4.
First, in 4.1, we introduce the bulk model, then introduce a boundary in 4.2.
We distinguish the cases with and without superpotential and show that without
specifying boundary conditions, ${\mathcal{N}=(0,2)}$ SUSY can be preserved by
introducing boundary fermions and matrix factorizations. Boundary conditions
do however have to be imposed to make the action stationary and we discuss
Dirichlet and Neumann conditions. In particular, there is a possibility to
make boundary conditions dynamical, imposing them as equations of motion. We
then turn to the formulation of currents in 4.3. Here, Noether’s procedure is
employed to compute the conserved currents in the combined bulk boundary
system. We discuss improvements to symmetrize the stress-energy tensor.
Section 4.4 contains a discussion of the supercurrent multiplets in the
example.

In section 5 we study the realization of the symmetry algebra in the Landau-
Ginzburg model in terms of the fields. By imposing canonical commutation
relations for the bulk and boundary fields, we verify that the supercharges
implement the correct symmetry transformations on the fields and their
derivatives, and we compute the brackets between supercharges and
supercurrents. Special attention is paid to the contributions from the
boundary. We do not impose any explicit boundary conditions on the fields in
our computations. In 5.1 we use the action of the supercharge on the
derivative of the scalars to verify how the bulk fermions decompose into
boundary fermions — a decomposition that was seen in earlier sections from the
point of view of the action. In the following subsections 5.2 and 5.3 we
verify the brackets between supercharges and currents and finally integrate
them to the global supersymmetry algebra. The factorization condition of the
superpotential arises as a consistency condition on the SUSY algebra. Finally,
section 6 contains some conclusions and outlook.

We summarize various technical points in a number of appendices. Here we also
collect some supplementary material for completeness. In particular, we
present some results on $\mathcal{N}=(1,1)$ supersymmetry in appendix B.2.

##  2 Currents and charges in theories with boundaries

In this section we want to study (classical) theories on flat Minkowski half-
space with spacelike boundary. In particular, we will consider

$M=\Set{x^{\mu}=\big{(}x^{0},\ldots,x^{n},\ldots,x^{N-1}\big{)}\in\mathbb{R}^{1,N-1}}{x^{n}\leq
0},\quad\partial M=\big{\\{}x^{n}=0\big{\\}}.$ (6)

On the flat boundary we will denote the tangential coordinates by
$x^{\hat{\mu}}$ while the normal coordinate is $x^{n}$, i.e. ${\hat{\mu}}$
takes all values except $n$. While our main focus will be $N=3$ later on, we
will keep the discussion more general in this section.

###  2.1 Bulk and boundary Lagrangians

We want to study theories that have both bulk and boundary fields with an
action of the form

$S=S^{B}+S^{\partial}=\int_{M}\mathcal{L}^{B}+\int_{\partial
M}\mathcal{L}^{\partial},\quad\mathcal{L}^{B}=\mathcal{L}^{B}[\Phi,\partial_{\mu}\Phi],\quad\mathcal{L}^{\partial}=\mathcal{L}^{\partial}[X,\partial_{\hat{\mu}}X,\Phi|_{\partial},\partial_{\hat{\mu}}\Phi|_{\partial}],$
(7)

where $\Phi$ and $X$ denotes bulk and boundary fields, respectively.
Furthermore, one has to impose _boundary conditions_. We follow the similar
discussion from DiPietro:2015zia , which we generalize here. The most general
boundary conditions will be of the form

$G(\text{fields}|_{\partial},\text{derivatives of fields}|_{\partial})=0.$ (8)

Under a (rigid) variation of the fields and an integration by parts, we get

$\displaystyle\delta S$
$\displaystyle=\int_{M}\delta\mathcal{L}^{B}+\int_{\partial
M}\delta\mathcal{L}^{\partial}$ (9)
$\displaystyle=\int_{M}\Big{[}\underset{\text{Bulk
}\operatorname{EoM}^{B}[\Phi]}{\underbrace{\tfrac{\partial\mathcal{L}^{B}}{\partial\Phi}-\partial_{\mu}\tfrac{\partial\mathcal{L}^{B}}{\partial(\partial_{\mu}\Phi)}}}\Big{]}\delta\Phi$
$\displaystyle\hskip 11.38092pt+\int_{\partial
M}\Big{[}\big{(}\underset{\text{Boundary
}\operatorname{EoM}^{\partial}[X]}{\underbrace{\tfrac{\partial\mathcal{L}^{\partial}}{\partial
X}-\partial_{\hat{\mu}}\tfrac{\partial\mathcal{L}^{\partial}}{\partial(\partial_{\hat{\mu}}X)}}}\big{)}\delta
X+\big{(}\underset{\eqqcolon\mathcal{A}}{\underbrace{\tfrac{\partial\mathcal{L}^{\partial}}{\partial\Phi|_{\partial}}-\partial_{\hat{\mu}}\tfrac{\partial\mathcal{L}^{\partial}}{\partial(\partial_{\hat{\mu}}\Phi|_{\partial})}+\tfrac{\partial\mathcal{L}^{B}}{\partial(\partial_{n}\Phi|_{\partial})}}}\big{)}\delta\Phi|_{\partial}+\partial_{\hat{\mu}}(\cdots)\Big{]}\mathclap{.}$

Stationarity of the action requires that

$\big{[}\mathcal{A}\cdot\delta\Phi|_{\partial}\big{]}_{G=0}=\partial_{\hat{\mu}}A^{\hat{\mu}},$
(10)

for some boundary vector field $A^{\hat{\mu}}$. Note that in any case, the
variations $\delta\Phi|_{\partial}$, $\delta X$ we consider must be
_compatible_ with the chosen boundary condition, i.e. we may only consider
such variations that satisfy

$\delta G|_{G=0}=0.$ (11)

A special kind of boundary condition is the _dynamical boundary condition_ ,
which amounts to requiring

$G\coloneqq\mathcal{A}\overset{!}{=}0.$ (12)

This is equivalent to the paradigm (e.g. found in Dimofte:2017tpi ;
Bilal:2011gp ) not to impose static boundary conditions, but instead adopt the
boundary conditions that are naturally imposed by the tendency of the system
to make the action stationary.

###  2.2 Symmetries, currents and charges in boundary theories

####  2.2.1 Symmetries in boundary theories

Let us try to understand symmetries in theories with a boundary. If we want
the full theory to be invariant under some symmetry transformation of the
fields, the boundary condition must be compatible with the symmetry
transformation, as mentioned in DiPietro:2015zia :

$\delta_{\text{sym}}G|_{G=0}=0.$ (13)

If this is satisfied, $G$ is called a _symmetric_ boundary condition with
respect to $\delta_{\text{sym}}$. Conveniently, this requirement is equivalent
to demanding that the symmetry variation $\delta_{\text{sym}}$ is a _permitted
variation_ in the sense of equation (11).

The definition of a symmetry in the presence of a boundary is analogous to the
case of a pure bulk theory: a symmetry is an off-shell transformation of both
bulk and boundary fields that leaves the action invariant, _possibly_ after
using boundary conditions:

$0=\delta_{\text{sym}}S|_{G=0}=(\delta_{\text{sym}}S^{B}+\delta_{\text{sym}}S^{\partial})|_{G=0}=0.$
(14)

It is natural to restrict to symmetries that arise from a symmetry of the bulk
theory, i.e. $\delta_{\text{sym}}\mathcal{L}^{B}=\partial_{\mu}V^{\mu}$ holds
for some bulk vector fields $V^{\mu}$. Noether’s theorem in the bulk then
ensures that $\partial_{\mu}J^{\mu}_{B}=0$, where
$J_{B}^{\mu}=-\tfrac{\partial\mathcal{L}^{B}}{\partial(\partial_{\mu}\Phi)}\delta_{\text{sym}}\Phi+V^{\mu}$
still holds. In terms of Lagrangians, we can then write

$0=\delta_{\text{sym}}S|_{G=0}=\int_{M}\delta_{\text{sym}}\mathcal{L}^{B}+\int_{\partial
M}\delta_{\text{sym}}\mathcal{L}^{\partial}=\int_{\partial
M}(V^{n}+\delta_{\text{sym}}\mathcal{L}^{\partial})|_{G=0}.$ (15)

If the above condition holds without imposing any boundary condition $G=0$,
one says that the symmetry is preserved _without reference to specific
boundary conditions_. As we start from a bulk theory $\mathcal{L}^{B}$ with a
symmetry, it is interesting to investigate whether a _boundary compensating
term_ $\mathcal{L}^{\partial}$ exists, so that (14) holds without referring to
a boundary condition DiPietro:2015zia ; Bilal:2011gp . However, in general
cases, one must impose specific symmetric boundary conditions _and_ add
boundary terms so that the full action is stationary (10) and symmetric (14).

####  2.2.2 Currents and charges

It is clear that the bulk theory charge $Q_{B}=\int_{\Sigma}J_{B}^{0}$ of the
aforementioned symmetry is, in general, no longer conserved after introducing
a boundary, since the constant-time slice $\Sigma$ now has a boundary
$\partial\Sigma$. As a physical interpretation, the bulk current “leaks” from
the boundary, and this “leakage” must be compensated by a boundary term. More
precisely, what we need in addition to the bulk current $J^{\mu}_{B}$ is a
_boundary current_ $J_{\partial}^{\hat{\mu}}$ which lives on the boundary of
the full theory, such that the equations

$\partial_{\mu}J_{B}^{\mu}=0,\quad\partial_{\hat{\mu}}J_{\partial}^{\hat{\mu}}=J_{B}^{n}|_{\partial}$
(16)

are satisfied. We also introduce the _total_ conserved current222See appendix
D for details on $\delta$-distributions at the boundary.

$J^{\mu}_{\text{full}}=J_{B}^{\mu}+\delta(x^{n})\mathcal{P}^{\mu}_{\phantom{\mu}{\hat{\mu}}}J^{\hat{\mu}}_{\partial},$
(17)

where $\mathcal{P}^{\mu}_{\phantom{\mu}{\hat{\mu}}}$ is a projector/embedding
with $\mathcal{P}^{n}_{\phantom{n}{\hat{\mu}}}\\!=0$,
$\mathcal{P}^{\hat{\nu}}_{\phantom{{\hat{\nu}}}{\hat{\mu}}}\\!=\delta^{\hat{\nu}}_{\phantom{{\hat{\nu}}}{\hat{\mu}}}$.
The conservation equations (16) can be expressed as:

$\partial_{\mu}J^{\mu}_{\text{full}}=\delta(x^{n})J^{n}_{\text{full}}.$ (18)

Note that the (boundary) conservation equation might only hold modulo boundary
conditions. The full conserved charge of the theory is then given by

$Q=\int_{\Sigma}J^{0}_{B}+\int_{\partial\Sigma}J^{0}_{\partial}=\int_{\Sigma}J_{\text{full}}^{0},$
(19)

whose conservation is easy to see from (16).

Just as in pure bulk theories, there is more than one current leading to the
same conserved charge associated to a given symmetry. Transformations of the
currents that preserve the conservation equations and the charges are called
the _improvements_ of the currents.

For a pure bulk theory, an improvement locally takes the form

$J_{B}^{\mu}\mapsto\tilde{J}_{B}^{\mu}=J_{B}^{\mu}+\partial_{\nu}M^{[\mu\nu]},$
(20)

which preserves the conservation equations and charges:
$\partial_{\mu}\tilde{J}_{B}^{\mu}=0$ and
$\tilde{Q}_{B}=\int_{\Sigma}\tilde{J}_{B}^{0}\equiv Q_{B}$.
For a theory with boundary, an improvement takes the form

$\begin{Bmatrix}J_{B}^{\mu}\\\
J_{\partial}^{\hat{\mu}}\end{Bmatrix}\mapsto\begin{Bmatrix}\tilde{J}_{B}^{\mu}=J_{B}^{\mu}+\partial_{\nu}M^{[\mu\nu]}\\\
\tilde{J}_{\partial}^{\hat{\mu}}=J_{\partial}^{\hat{\mu}}+M^{n{\hat{\mu}}}+\partial_{\hat{\nu}}m^{[{\hat{\mu}}{\hat{\nu}}]}\end{Bmatrix},$
(21)

which preserves the conservation equations (16) and the charge (19). The
improvement of the bulk current induces an improvement on the boundary
current, and the boundary current may be further improved by a pure boundary
improvement.

Note that it is sometimes possible to completely “improve away” the boundary
part of the conserved current, in particular if there are no degrees of
freedom on the boundary. In that case the bulk charge $Q_{B}$ is conserved
even in the presence of a boundary, but then it is, in general, sensitive to
bulk improvements. This is the approach of the authors of DiPietro:2015zia .

###  2.3 Noether’s theorem on manifolds with boundary

Now that we have discussed the properties of conserved currents and charges in
boundary theories, let us investigate how we can compute them in a particular
model. We present a modification of Noether’s theorem that yields bulk and
boundary currents in the sense we defined above. The special case of a theory
with boundary (and boundary terms) but without boundary dynamics is discussed
in detail in DiPietro:2015zia .

####  2.3.1 Currents and charges without boundary

For completeness, let us quickly repeat Noether’s theorem in pure bulk
theories. A symmetry is an off-shell transformation of the fields that leaves
the action invariant:

$\delta_{\text{sym}}S=0.$ (22)

If the transformation is rigid (i.e. leaves spacetime invariant) and assuming
that fields “fall off” at infinity the above condition is equivalent to

$\delta_{\text{sym}}\mathcal{L}^{B}=\partial_{\mu}V^{\mu}$ (23)

for some bulk vector field $V^{\mu}$. On the other hand, a generic variation
of the Lagrangian is also given by

$\delta_{\text{sym}}\mathcal{L}^{B}=\operatorname{EoM}^{B}[\Phi]\delta_{\text{sym}}\Phi+\partial_{\mu}\big{(}\tfrac{\partial\mathcal{L}^{B}}{\partial(\partial_{\mu}\Phi|_{\partial})}\delta_{\text{sym}}\Phi\big{)}\overset{\text{\tiny
on-
shell}}{=}\partial_{\mu}\big{(}\tfrac{\partial\mathcal{L}^{B}}{\partial(\partial_{\mu}\Phi|_{\partial})}\delta_{\text{sym}}\Phi\big{)}.$
(24)

We find that on-shell
$0=\partial_{\mu}(V^{\mu}-\tfrac{\partial\mathcal{L}^{B}}{\partial(\partial_{\mu}\Phi)}\delta_{\text{sym}}\Phi)$,
thus the _bulk Noether current_

$J_{B}^{\mu}=-\tfrac{\partial\mathcal{L}^{B}}{\partial(\partial_{\mu}\Phi)}\delta_{\text{sym}}\Phi+V^{\mu},\quad\partial_{\mu}J_{B}^{\mu}=0,$
(25)

is divergence-free (on-shell). Since there is no boundary present, divergence-
freeness implies the conservation of the charge

$Q_{B}=\int_{\Sigma}J_{B}^{0},$ (26)

where $\Sigma$ is a constant-time slice of $M$, since
$\partial_{0}Q_{B}=\int_{\Sigma}\partial_{0}J^{0}=-\int_{\Sigma}\partial_{i}J^{i}=0$.

####  2.3.2 Currents and charges with boundary

As we restricted to symmetries of boundary theories that come from a bulk
theory, $J_{B}^{\mu}$ from (25) is still a valid divergence-free current by
the same argument as above, so we can use it as the bulk part of the full
current (17). The task at hand is to now find a boundary current which
satisfies
$\partial_{\hat{\mu}}J_{\partial}^{\hat{\mu}}=J_{B}^{n}|_{\partial}$. We want
to apply a similar strategy as in the pure bulk theory: compare the (off-
shell) symmetry variation of the action with an on-shell variation. On the
off-shell side we get

$0=\delta_{\text{sym}}S|_{G=0}=\int_{M}\partial_{\mu}V^{\mu}+\int_{\partial
M}\delta_{\text{sym}}\mathcal{L}^{\partial}|_{G=0}=\int_{\partial
M}[V^{n}+\delta_{\text{sym}}\mathcal{L}^{\partial}]_{G=0},$ (27)

which implies that

$[V^{n}+\delta_{\text{sym}}\mathcal{L}^{\partial}]_{G=0}=\partial_{\hat{\mu}}K^{\hat{\mu}}$
(28)

for some boundary vector field $K^{\hat{\mu}}$.

On the on-shell side, we now use equations of motion _and_ the stationarity
condition (10). By varying the boundary Lagrangian directly and assuming
$G[\ldots]|_{\partial}=0$ we get

$\displaystyle\delta_{\text{sym}}\mathcal{L}^{\partial}$
$\displaystyle=\left[\tfrac{\partial\mathcal{L}^{\partial}}{\partial\Phi|_{\partial}}-\partial_{\hat{\mu}}\tfrac{\partial\mathcal{L}^{\partial}}{\partial(\partial_{\hat{\mu}}\Phi|_{\partial})}\right]\delta_{\text{sym}}\Phi|_{\partial}+\operatorname{EoM}^{\partial}[X]\delta_{\text{sym}}X$
(29)
$\displaystyle\phantom{{}={}}+\partial_{\hat{\mu}}\big{(}\tfrac{\partial\mathcal{L}^{\partial}}{\partial(\partial_{\hat{\mu}}X)}\delta_{\text{sym}}X+\tfrac{\partial\mathcal{L}^{\partial}}{\partial(\partial_{\hat{\mu}}\Phi|_{\partial})}\delta_{\text{sym}}\Phi|_{\partial}\big{)}.$

To rewrite the first term, let us plug in $\delta=\delta_{\text{sym}}$ into
(10) (still assuming $G[\ldots]=0$) and use the definition of the bulk Noether
current (25) to rewrite it:

$\partial_{\hat{\mu}}A^{\hat{\mu}}\overset{\text{\tiny on-
shell}}{=}\left[\tfrac{\partial\mathcal{L}^{\partial}}{\partial\Phi|_{\partial}}-\partial_{\hat{\mu}}\tfrac{\partial\mathcal{L}^{\partial}}{\partial(\partial_{\hat{\mu}}\Phi|_{\partial})}\right]\delta_{\text{sym}}\Phi|_{\partial}+[V^{n}-J_{B}^{n}]_{\partial}.$
(30)

Plugging this into the previous equation and going on-shell, we get

$\delta_{\text{sym}}\mathcal{L}^{\partial}\overset{\text{\tiny on-
shell}}{=}[J_{B}^{n}-V^{n}]_{\partial}+\partial_{\hat{\mu}}\big{(}A^{\hat{\mu}}+\tfrac{\partial\mathcal{L}^{\partial}}{\partial(\partial_{\hat{\mu}}X)}\delta_{\text{sym}}X+\tfrac{\partial\mathcal{L}^{\partial}}{\partial(\partial_{\hat{\mu}}\Phi|_{\partial})}\delta_{\text{sym}}\Phi|_{\partial}\big{)}.$
(31)

We can now compare this on-shell variation to the off-shell variation in (28)
and see

$J_{B}^{n}|_{\partial}=\partial_{\hat{\mu}}\big{(}K^{\hat{\mu}}-A^{\hat{\mu}}-\tfrac{\partial\mathcal{L}^{\partial}}{\partial(\partial_{\hat{\mu}}X)}\delta_{\text{sym}}X-\tfrac{\partial\mathcal{L}^{\partial}}{\partial(\partial_{\hat{\mu}}\Phi|_{\partial})}\delta_{\text{sym}}\Phi|_{\partial}).$
(32)

Thus, we can read off the boundary Noether current

$J_{\partial}^{\hat{\mu}}=K^{\hat{\mu}}-A^{\hat{\mu}}-\tfrac{\partial\mathcal{L}^{\partial}}{\partial(\partial_{\hat{\mu}}X)}\delta_{\text{sym}}X-\tfrac{\partial\mathcal{L}^{\partial}}{\partial(\partial_{\hat{\mu}}\Phi|_{\partial})}\delta_{\text{sym}}\Phi|_{\partial},\quad\partial_{\hat{\mu}}J_{\partial}^{\hat{\mu}}=J_{B}^{n}|_{\partial}.$
(33)

Together with the bulk current (25), this forms a conserved boundary theory
current in the sense of (16). We recall that $K^{\hat{\mu}}$ is defined by the
symmetry condition (28) and $A^{\hat{\mu}}$ is defined by the stationarity
condition (10). Notice that through the dependency on $A^{\hat{\mu}}$, the
boundary Noether current may explicitly depend on the boundary condition, even
if the bulk variation is compensated at the boundary in a boundary-condition-
independent way (cf. section 2.2.1).

####  2.3.3 Special case: Energy-momentum tensor

In the presence of a boundary, Noether’s theorem applies to spacetime
translations as well: the total energy-momentum tensor is333Strictly speaking,
the index $\nu$ does not take the value $n$: translations in $x^{n}$-direction
are no longer symmetries. However, we may still consider this part of the
tensor even though it does not lead to a conserved charge.

$T_{\nu}^{\phantom{\nu}\mu}=(T^{B})_{\nu}^{\phantom{\nu}\mu}+\delta(x^{n})\mathcal{P}^{\mu}_{\phantom{\mu}{\hat{\mu}}}\mathcal{P}_{\nu}^{\phantom{\nu}{\hat{\nu}}}(T^{\partial})_{\hat{\nu}}^{\phantom{{\hat{\nu}}}{\hat{\mu}}}.$
(34)

Here, the bulk contribution is

$(T^{B})_{\nu}^{\phantom{\nu}\mu}=-\frac{\partial\mathcal{L}^{B}}{\partial(\partial_{\mu}\Phi)}\partial_{\nu}\Phi+\delta_{\nu}^{\phantom{\nu}\mu}\mathcal{L}^{B},$
(35)

while the boundary contribution is

$(T^{\partial})_{\hat{\nu}}^{\phantom{{\hat{\nu}}}{\hat{\mu}}}=-\frac{\partial\mathcal{L}^{\partial}}{\partial(\partial_{\hat{\mu}}X)}\partial_{\hat{\nu}}X-\frac{\partial\mathcal{L}^{\partial}}{\partial(\partial_{\hat{\mu}}\Phi)}\partial_{\hat{\nu}}\Phi+\delta_{\phantom{{\hat{\mu}}}{\hat{\nu}}}^{\hat{\mu}}\mathcal{L}^{\partial},$
(36)

where summation over fields is implied. The conservation equations are given
by

$\displaystyle\partial_{\mu}(T^{B})_{\nu}^{\phantom{\nu}\mu}$
$\displaystyle=0,$ (37)
$\displaystyle\partial_{\hat{\mu}}(T^{\partial})_{\hat{\nu}}^{\phantom{{\hat{\nu}}}{\hat{\mu}}}$
$\displaystyle=T_{\hat{\nu}}^{\phantom{{\hat{\nu}}}n}|_{\partial},$

and the total tensor satisfies
$\partial_{\mu}T_{\nu}^{\phantom{\nu}\mu}=\delta(x^{n})\mathcal{P}_{\nu}^{\phantom{\nu}{\hat{\nu}}}T_{\hat{\nu}}^{\phantom{{\hat{\nu}}}n}$.
The momenta along the tangential ${\hat{\nu}}$-directions are conserved

$P_{\hat{\nu}}=\int_{\Sigma}(T^{B})_{\hat{\nu}}^{\phantom{{\hat{\nu}}}0}+\int_{\partial\Sigma}(T^{\partial})_{\hat{\nu}}^{\phantom{{\hat{\nu}}}0},\quad\quad\partial_{0}P_{\hat{\nu}}=0,$
(38)

while $P_{n}=\int_{\Sigma}(T^{B})_{n}^{\phantom{n}0}$ is clearly not conserved
in general:
$\partial_{0}P_{n}=-\int_{\Sigma}\partial_{i}(T^{B})_{n}^{\phantom{n}0}=-T_{n}^{\phantom{n}n}|_{\partial}$.
This is consistent with a flat boundary: the theory is only invariant under
spacetime translations tangential to the boundary.

As far as improvements are concerned, the most general improvement takes the
form

$\begin{Bmatrix}(T^{B})_{\mu\nu}\\\
(T^{\partial})_{{\hat{\mu}}{\hat{\nu}}}\end{Bmatrix}\mapsto\begin{Bmatrix}(T^{B})_{\mu\nu}+\partial^{\rho}M_{\nu[\mu\rho]}\\\
(T^{\partial})_{{\hat{\mu}}{\hat{\nu}}}+M_{{\hat{\nu}}n{\hat{\mu}}}+\partial^{\hat{\rho}}m_{{\hat{\nu}}[{\hat{\mu}}{\hat{\rho}}]}\end{Bmatrix},$
(39)

which, as before, leads to the same charges. However, if we restrict to
improvements of symmetric tensors containing up to spin 1 components
Dumitrescu:2011iu , the allowed improvements take the form:

$\begin{Bmatrix}(T^{B})_{\mu\nu}\\\
(T^{\partial})_{{\hat{\mu}}{\hat{\nu}}}\end{Bmatrix}\mapsto\begin{Bmatrix}(T^{B})_{\mu\nu}+\partial_{\nu}U_{\mu}-\eta_{\mu\nu}\partial^{\rho}U_{\rho}\\\
(T^{\partial})_{{\hat{\mu}}{\hat{\nu}}}+\eta_{{\hat{\mu}}{\hat{\nu}}}U_{n}-\eta_{n{\hat{\nu}}}U_{\hat{\mu}}+\partial_{\hat{\nu}}u_{\hat{\mu}}-\eta_{{\hat{\mu}}{\hat{\nu}}}\partial^{\hat{\rho}}u_{\hat{\rho}}\end{Bmatrix},$
(40)

where $U_{\mu}$ is the bulk improvement and $u_{\hat{\mu}}$ the boundary
improvement.

##  3 Boundary supercurrent multiplets in 3D

We want to study supercurrent multiplets of theories on manifolds with
boundary. In particular, we consider the special case of bulk theories with 3D
$\mathcal{N}=2$ supersymmetry, broken to 2D ${\mathcal{N}=(0,2)}$ due to the
boundary. While our discussion is limited to this particular case, the
strategy is expected to work in greater generality. We start by recalling the
definitions and some facts about supercurrent multiplets, following
Dumitrescu:2011iu  (see also Magro:2001aj  for a connection to a superspace
Noether procedure).

The defining properties of a supercurrent multiplet are:

  1. (a)

The energy-momentum tensor $(T^{B})^{\mu\nu}$ should be a component of the
multiplet. It is also the only component with spin 2.

  2. (b)

The supercurrents, i.e. conserved currents associated to supersymmetry, are
components of the multiplet. They are the only components with spin $3/2$. No
component other than the supercurrents and the energy-momentum tensor are
allowed to have spin larger than $1$.

  3. (c)

The supercurrent multiplet is not unique: it allows for (supersymmetrically
complete) improvements of its components.

  4. (d)

The multiplet is _indecomposable_ , so it may have non-trivial submultiplets,
but it may not be decomposed into two independent decoupled multiplets.

The components of a supercurrent multiplet (in particular, the conserved
currents) are only unique up to improvements. However, improving one component
and not the others breaks the structure of the multiplet. Hence, to
consistently improve the supercurrent multiplet, we must restrict to
improvements of all components which are related in a certain way
(specifically, the improvement terms have to form a supersymmetry multiplet
themselves; details are in Dumitrescu:2011iu ). In other words, if one is
given two components (e.g. a supercurrent and an energy-momentum tensor), one
may have to improve one of them such that they can be part of the same
supercurrent multiplet. We say two conserved currents which have been improved
such that they are part of a consistent supercurrent multiplet are in the same
_improvement frame_.

For some theories the supercurrent multiplet may be improved into a smaller
multiplet (e.g. to obtain an $\mathcal{R}$-multiplet or a Ferrara-Zumino
multiplet). There are still improvements that preserve this smaller multiplet
Dumitrescu:2011iu ; Komargodski:2010rb ; Dedushenko:2015opz . We will recall
the case of 3D $\mathcal{N}=2$ theories in more detail.

###  3.1 In bulk theories

We will focus on three-dimensional theories with two-dimensional boundaries.
In this section we recall the defining relations and properties of
supercurrent multiplets in three-dimensional bulk theories with
$\mathcal{N}=2$ supersymmetry from Dumitrescu:2011iu .

The most general supercurrent multiplet satisfying the conditions (a)–(d)
(called the _$\mathcal{S}$ -multiplet_) consists of three superfields,
$\mathcal{S}_{\alpha\beta},\chi_{\alpha},\mathcal{Y}_{\alpha}$ with
$\mathcal{S}_{\alpha\beta}$ real, $\chi_{\alpha},\mathcal{Y}_{\alpha}$
fermionic, and a complex constant $C$. They must satisfy the defining
relations:

$\displaystyle\overline{D}^{\beta}\mathcal{S}_{\alpha\beta}$
$\displaystyle=\chi_{\alpha}+\mathcal{Y}_{\alpha},$ (41)
$\displaystyle\overline{D}_{\alpha}\chi_{\beta}$
$\displaystyle=\tfrac{1}{2}C\epsilon_{\alpha\beta},$ $\displaystyle
D^{\alpha}\chi_{\alpha}+\overline{D}^{\alpha}\overline{\chi}_{\alpha}$
$\displaystyle=0,$ $\displaystyle
D_{\alpha}\mathcal{Y}_{\beta}+D_{\beta}\mathcal{Y}_{\alpha}$
$\displaystyle=0,$ $\displaystyle\overline{D}^{\alpha}\mathcal{Y}_{\alpha}+C$
$\displaystyle=0.$

These defining relations are solved by the following expansions (using
bispinor relations (193)):

$\displaystyle\begin{split}\mathcal{S}_{\mu}&=j_{\mu}-i\theta(S_{\mu}+\tfrac{i}{\sqrt{2}}\gamma_{\mu}\overline{\omega})-i\overline{\theta}(\overline{S}_{\mu}-\tfrac{i}{\sqrt{2}}\gamma_{\mu}\omega)+\tfrac{i}{2}\theta^{2}\overline{Y}_{\mu}+\tfrac{i}{2}\overline{\theta}^{2}Y_{\mu}\\\
&\hskip
11.38092pt-(\theta\gamma^{\nu}\overline{\theta})\big{(}2T_{\nu\mu}-\eta_{\mu\nu}A-\tfrac{1}{4}\epsilon_{\mu\nu\rho}H^{\rho}\big{)}-i\theta\overline{\theta}\big{(}\tfrac{1}{4}\epsilon_{\mu\nu\rho}F^{\nu\rho}+\epsilon_{\mu\nu\rho}\partial^{\nu}j^{\rho}\big{)}\\\
&\hskip
11.38092pt+\tfrac{1}{2}\theta^{2}\overline{\theta}\big{(}\gamma^{\nu}\partial_{\nu}S_{\mu}-\tfrac{i}{\sqrt{2}}\gamma_{\mu}\gamma_{\nu}\partial^{\nu}\overline{\omega}\big{)}+\tfrac{1}{2}\overline{\theta}^{2}\theta\big{(}\gamma^{\nu}\partial_{\nu}\overline{S}_{\mu}+\tfrac{i}{\sqrt{2}}\gamma_{\mu}\gamma_{\nu}\partial^{\nu}\omega\big{)}\\\
&\hskip
11.38092pt-\tfrac{1}{2}\theta^{2}\overline{\theta}^{2}\big{(}\partial_{\mu}\partial^{\nu}j_{\nu}-\tfrac{1}{2}\partial^{2}j_{\mu}\big{)},\end{split}$
(42a)
$\displaystyle\begin{split}\chi_{\alpha}&=-i\lambda_{\alpha}(y)+\theta_{\beta}\big{[}\delta_{\alpha}^{\phantom{\alpha}\beta}D(y)-(\gamma^{\mu})_{\alpha}^{\phantom{\alpha}\beta}\big{(}H_{\mu}(y)-\tfrac{i}{2}\epsilon_{\mu\nu\rho}F^{\nu\rho}(y)\big{)}\big{]}\\\
&\hskip
11.38092pt+\tfrac{1}{2}\overline{\theta}_{\alpha}C-\theta^{2}(\gamma^{\mu}\partial_{\mu}\overline{\lambda})_{\alpha}(y),\end{split}$
(42b)
$\displaystyle\begin{split}\mathcal{Y}_{\alpha}&=\sqrt{2}\omega_{\alpha}+2\theta_{\alpha}B+2i\gamma^{\mu}_{\alpha\beta}\overline{\theta}^{\beta}Y_{\mu}+\sqrt{2}i(\theta\gamma^{\mu}\overline{\theta})\epsilon_{\mu\nu\rho}(\gamma^{\nu}\partial^{\rho}\omega)_{\alpha}\\\
&\hskip
11.38092pt+\sqrt{2}i\theta\overline{\theta}(\gamma^{\mu}\partial_{\mu}\omega)_{\alpha}+i\theta^{2}\gamma^{\mu}_{\alpha\beta}\overline{\theta}^{\beta}\partial_{\mu}B-\overline{\theta}^{2}\theta_{\alpha}\partial_{\mu}Y^{\mu}+\tfrac{1}{2\sqrt{2}}\theta^{2}\overline{\theta}^{2}\partial^{2}\omega_{\alpha},\end{split}$
(42c)

where $(S^{\mu})_{\alpha},(\overline{S}^{\mu})_{\alpha}$ are conserved
supercurrents, $T_{\mu\nu}$ is a symmetric energy-momentum tensor, and

$\displaystyle\lambda_{\alpha}$
$\displaystyle=-2(\gamma^{\mu}\overline{S}_{\mu})_{\alpha}+2\sqrt{2}i\omega_{\alpha},$
(43) $\displaystyle D$ $\displaystyle=-4T^{\mu}_{\phantom{\mu}\mu}+4A,$
$\displaystyle B$ $\displaystyle=A+i\partial_{\mu}j^{\mu},$
$\displaystyle\mathrm{d}H$
$\displaystyle=0,\quad\mathrm{d}Y=0,\quad\mathrm{d}F=0,$

where $H,F,Y$ are forms with components $H_{\mu},F_{\mu\nu},Y_{\mu}$.
Additionally, $y$ is the “chiral” coordinate
$y^{\mu}=x^{\mu}-i\theta\gamma^{\mu}\overline{\theta}$. If the forms $Y$ or
$H$ are exact, the superfields $\mathcal{Y}_{\alpha}$ or $\chi_{\alpha}$ may
be written as covariant derivatives: If $Y_{\mu}=\partial_{\mu}x$, then
$\mathcal{Y}_{\alpha}=D_{\alpha}X$ where $X|_{\theta^{0}}=x$, and if
$H_{\mu}=\partial_{\mu}g$, then $\chi_{\alpha}=i\overline{D}_{\alpha}G$ where
$G|_{\theta^{0}}=g$.

####  3.1.1 Improvements

The expansions (42) together with the relations (43) and the conservation of
currents $\partial_{\mu}(S^{\mu})_{\alpha}=0$, $\partial^{\mu}T_{\mu\nu}=0$ do
not form the only solution of the constraints (41). We may improve without
violating the constraints

$\displaystyle\mathcal{S}_{\mu}$
$\displaystyle\mapsto\mathcal{S}_{\mu}+\tfrac{1}{4}\gamma_{\mu}^{\alpha\beta}[D_{\alpha},\overline{D}_{\beta}]U,$
(44) $\displaystyle\chi_{\alpha}$
$\displaystyle\mapsto\chi_{\alpha}-\overline{D}^{2}D_{\alpha}U,$
$\displaystyle\mathcal{Y}_{\alpha}$
$\displaystyle\mapsto\mathcal{Y}_{\alpha}-\tfrac{1}{2}D_{\alpha}\overline{D}^{2}U,$

where
$U=u+\theta\eta-\overline{\theta}\overline{\eta}+\theta^{2}N-\overline{\theta}^{2}\overline{N}+(\theta\gamma^{\mu}\overline{\theta})V_{\mu}-i\theta\overline{\theta}K+\ldots$
is a real superfield. The improvement transforms

$\displaystyle(S_{\mu})_{\alpha}$
$\displaystyle\mapsto(S_{\mu})_{\alpha}+\epsilon_{\mu\nu\rho}(\gamma^{\nu}\partial^{\rho}\eta)_{\alpha},$
(45) $\displaystyle T_{\mu\nu}$ $\displaystyle\mapsto
T_{\mu\nu}+\tfrac{1}{2}(\partial_{\mu}\partial_{\nu}-\eta_{\mu\nu}\partial^{2})u,$
$\displaystyle H_{\mu}$ $\displaystyle\mapsto H_{\mu}-4\partial_{\mu}K,$
$\displaystyle F_{\mu\nu}$ $\displaystyle\mapsto
F_{\mu\nu}-4(\partial_{\mu}V_{\nu}-\partial_{\nu}V_{\mu}),$ $\displaystyle
Y_{\mu}$ $\displaystyle\mapsto Y_{\mu}-2\partial_{\mu}\overline{N}.$

The multiplet $\mathcal{S}_{\mu}$ may be improved into smaller multiplets. In
particular:

  1. (1)

If $C=0$, $\chi_{\alpha}=i\overline{D}_{\alpha}G$ (i.e. $H$ is exact) and
there exists a well-defined improvement $U$ such that
$G=2i\overline{D}^{\alpha}D_{\alpha}U$, then it sends $\chi_{\alpha}$ to zero
and we obtain a _Ferrara-Zumino multiplet_ Ferrara:1974pz :

$\displaystyle\overline{D}^{\beta}\mathcal{J}_{\alpha\beta}$
$\displaystyle=\mathcal{Y}_{\alpha},$ (46) $\displaystyle
D_{\alpha}\mathcal{Y}_{\beta}+D_{\beta}\mathcal{Y}_{\alpha}$
$\displaystyle=0,\quad\overline{D}^{\alpha}\mathcal{Y}_{\alpha}=0.$

  2. (2)

If $C=0$, $\mathcal{Y}_{\alpha}=D_{\alpha}X$ (i.e. $Y$ is exact) and there
exists a well-defined improvement $U$ such that
$X=\tfrac{1}{2}\overline{D}^{2}U$, then it sends $\mathcal{Y}_{\alpha}$ to
zero and we obtain an _$\mathcal{R}$ -multiplet_ Gates:1983nr :

$\displaystyle\overline{D}^{\beta}\mathcal{R}_{\alpha\beta}$
$\displaystyle=\chi_{\alpha},$ (47)
$\displaystyle\overline{D}_{\alpha}\chi_{\beta}=0,$ $\displaystyle
D^{\alpha}\chi_{\alpha}+\overline{D}^{\alpha}\overline{\chi}_{\beta}=0.$

In this case, the lowest component $j_{\mu}$ of the multiplet
$\mathcal{R}_{\mu}$ (we relabel $\mathcal{S}_{\mu}$ to $\mathcal{R}_{\mu}$) is
a conserved $R$-current (in the general $\mathcal{S}_{\mu}$-multiplet,
$j^{\mu}$ is not conserved; however, we still call it a “non-conserved
$R$-current”). The $\mathcal{R}$-multiplet will be the primary focus of our
example in section 4.

  3. (3)

If $C=0$ and the improvements from (1) and (2) coincide, we can improve both
superfields $\chi_{\alpha},\mathcal{Y}_{\alpha}$ to zero _simultaneously_. In
that case we obtain a _superconformal multiplet_

$\overline{D}^{\beta}\mathcal{S}_{\alpha\beta}=0.$ (48)

Note that even if smaller multiplets exist, they are still not unique: we may
further improve the smaller multiplets without violating the respective
additional constraints. For example, in the case of the
$\mathcal{R}$-multiplet, the improvements that preserve the defining
constraints are transformations

$\displaystyle\mathcal{R}_{\mu}$
$\displaystyle\mapsto\mathcal{R}_{\mu}+\tfrac{1}{4}\gamma_{\mu}^{\alpha\beta}[D_{\alpha},\overline{D}_{\beta}]U,$
(49) $\displaystyle\chi_{\alpha}$
$\displaystyle\mapsto\chi_{\alpha}-\overline{D}^{2}D_{\alpha}U,$
$\displaystyle D_{\alpha}\overline{D}^{2}U=0.$

####  3.1.2 Brane currents

We may associate to the closed forms $F,H,Y,C$ the _brane currents_ defined by
taking their Hodge dual:

$C_{\mu}\sim\epsilon_{\mu\nu\rho}F^{\nu\rho},\quad
C_{\mu\nu}\sim\epsilon_{\mu\nu\rho}H^{\rho},\quad
C^{\prime}_{\mu\nu}\sim\epsilon_{\mu\nu\rho}\overline{Y}^{\rho},\quad
C_{\mu\nu\rho}\sim\epsilon_{\mu\nu\rho}\overline{C}.$ (50)

Note that these are conserved by construction,
$\partial_{\mu}C^{\mu}_{\phantom{\mu}\mu_{1}\ldots\mu_{k}}=0$, since the forms
are closed.444In coordinate-free notation this is written as $\mathrm{d}*C=0$
which follows trivially if $C=*A$, with $\mathrm{d}A=0$. Then, the _brane
charges_ defined by
$Z_{\mu_{1}\ldots\mu_{k}}=\int_{\Sigma}C^{0}_{\phantom{0}\mu_{1}\ldots\mu_{k}}$
are conserved as well. In addition, they are also invariant under the
improvements (45). The brane charges, if they are non-trivial, are central
charges of the supersymmetry algebra (but not of the Poincaré algebra). This
is motivated by studying the explicit commutators Dumitrescu:2011iu  that
follow from the multiplet structure of $\mathcal{S}_{\mu}$:555Recall, the
action of physical supercharges via commutators is related to the action via
(super)-differential operators by
$[\xi^{\alpha}\mathcal{Q}_{\alpha}-\overline{\xi}^{\alpha}\overline{\mathcal{Q}}_{\alpha},X]=i(\xi^{\alpha}Q_{\alpha}-\overline{\xi}^{\alpha}\overline{Q}_{\alpha})X\eqqcolon
i\delta_{\text{sym}}^{\xi,\overline{\xi}}X.$

$\displaystyle\\{{\overline{\mathcal{Q}}}_{\alpha},(S_{\mu})_{\beta}\\}$
$\displaystyle=\gamma^{\nu}_{\alpha\beta}(2T_{\nu\mu}-\tfrac{1}{4}\epsilon_{\mu\nu\rho}H^{\rho})+i\epsilon_{\alpha\beta}\tfrac{1}{4}\epsilon_{\mu\nu\rho}F^{\nu\rho}+\text{total
derivatives,}$ (51)
$\displaystyle\\{\mathcal{Q}_{\alpha},(S_{\mu})_{\beta}\\}$
$\displaystyle=\tfrac{1}{4}(\gamma_{\mu})_{\alpha\beta}\overline{C}+i\epsilon_{\mu\nu\rho}\gamma^{\nu}_{\alpha\beta}\overline{Y}^{\rho},$

where we may find non-trivial central charges in the supersymmetry algebra
upon integration. Each current $C_{\mu\mu_{1}\ldots\mu_{k}}$ and the
corresponding charge $Z_{\mu_{1}\ldots\mu_{k}}$ is associated to a $k$-brane.
Hence, the brane charges form a physical obstruction to improvements into
smaller multiplets. In particular, a non-zero charge associated to $F$ or $H$
obstructs the existence of a Ferrara-Zumino multiplet, and a non-zero charge
associated to $Y$ obstructs the existence of an $\mathcal{R}$-multiplet.

###  3.2 In theories with boundary

The introduction of a boundary affects supercurrent multiplets in two obvious
ways.

First, supersymmetry is broken to a subalgebra, where our main focus will be
2D ${\mathcal{N}=(0,2)}$. The bulk supercurrent multiplets, previously 3D
$\mathcal{N}=2$ superfields, now decompose under the subalgebra to
${(0,2)}$-superfields. We spell out this decomposition in appendix C.1.
Similarly, the constraints (41) now decompose into constraints of the
${(0,2)}$-superfields. We will spell out this decomposition in the next
subsection.

Second, a boundary changes the conserved currents of _remaining_ symmetries by
supplementing the bulk currents (25) with boundary currents (33) satisfying
appropriate conservation equations. The conservation equations must follow
from the constraints that define the supercurrent multiplets, as in bulk
theories, and the _full_ supercurrent multiplets will now consist of bulk and
boundary pieces. The schematic form of full supercurrent multiplets reads

$\displaystyle\mathcal{S}_{\mu}^{\text{full}}$
$\displaystyle=\mathcal{S}_{\mu}^{B}+\delta(x^{n})\mathcal{P}^{\hat{\mu}}_{\phantom{{\hat{\mu}}}\mu}\mathcal{S}_{\hat{\mu}}^{\partial},$
(52) $\displaystyle\chi_{\alpha}^{\text{full}}$
$\displaystyle=\chi_{\alpha}^{B}+\delta(x^{n})\chi_{\alpha}^{\partial},$
$\displaystyle\mathcal{Y}_{\alpha}^{\text{full}}$
$\displaystyle=\mathcal{Y}_{\alpha}^{B}+\delta(x^{n})\mathcal{Y}_{\alpha}^{\partial},$

where once again $\mathcal{P}^{\mu}_{\phantom{\mu}{\hat{\mu}}}$ is an
embedding.

Let us briefly discuss how the conditions (a)–(d) are modified. It is clear
that the new superfields should contain the _full_ conserved currents of
unbroken symmetries, in the sense of section 2.3 (conditions (a), (b)).
Furthermore, improvements of the full conserved currents in the sense of (21)
that form consistent multiplets under the smaller subalgebra are improvements
of the ${\mathcal{N}=(0,2)}$ supercurrent multiplets (condition (c)). However,
under the smaller symmetry algebra, the previously indecomposable (bulk)
multiplet decomposes into possibly several indecomposable multiplets of the
remaining symmetry subalgebra. Therefore condition (d) is not preserved in
general.

###  3.3 Bulk and boundary constraints by decomposition

Let us recall the structure of subalgebras of the 3D $\mathcal{N}=2$ algebra
which may be preserved after the introduction of a boundary. The (unbroken)
symmetry algebra is generated by tangential translations $P_{\hat{\mu}}$,
Lorentz transformations $M_{{\hat{\mu}}{\hat{\nu}}}$ in the unbroken
directions, and one of the following:

  1. (1)

supercharges $Q_{+},\overline{Q}_{+}$ corresponding to a 2D
${(0,2)}$-subalgebra satisfying

$(Q_{+})^{2}=0,\quad\\{Q_{+},\overline{Q}_{+}\\}=-4i\partial_{+},$ (53)

  2. (2)

their left-moving $(2,0)$ counterparts $Q_{-},\overline{Q}_{-}$,

  3. (3)

(real) supercharges $\mathbbm{Q}_{-},\mathbbm{Q}_{+}$ corresponding to a 2D
${(1,1)}$-subalgebra satisfying

$(\mathbbm{Q}_{\pm})^{2}=-i\partial_{\pm},\quad\\{\mathbbm{Q}_{-},\mathbbm{Q}_{+}\\}=0.$
(54)

In this work we consider the only the first case. We want to determine
constraint equations that define supercurrent multiplets in a 3D theory with
boundary and 2D ${\mathcal{N}=(0,2)}$ supersymmetry. To do so, we first
decompose the 3D $\mathcal{N}=2$ bulk constraints into ${\mathcal{N}=(0,2)}$
bulk constraints, and then investigate possible ${\mathcal{N}=(0,2)}$ boundary
constraints.

We supplement the superspace operators $Q_{+},\overline{Q}_{+}$ with covariant
derivatives $D_{+}^{(0,2)}$, $\overline{D}_{+}^{(0,2)}$ defined in appendix
B.1. Let us emphasize that these are _not_ the operators $D_{\alpha}$,
$\overline{D}_{\alpha}$ acting on 3D $\mathcal{N}=2$ superspace, but are
related to them by (205). We will omit the label ${(0,2)}$ from now on; it
should be clear from context whether a 3D $\mathcal{N}=2$ or a
${\mathcal{N}=(0,2)}$ covariant derivative is meant.

####  3.3.1 Bulk constraints

Since the bulk conservation equations remain unchanged, the constraints on the
bulk pieces in our ansatz (52) will remain the same component-wise. We merely
have to decompose the multiplets and their constraints into constraints of
${(0,2)}$-submultiplets. For simplicity we choose to do so for the case where
the supermultiplet is an $\mathcal{R}$-multiplet:666The more general case of
the $\mathcal{S}$-multiplet is quite similar and is given in the appendix, see
(233)–(238). We will decompose the superfields
$(\mathcal{R}_{\mu},\chi_{\alpha})$ and the constraints (47). We can achieve
this using the _branching coordinate_ $\xi^{\mu}$. It has the defining
property that in the coordinates $(\xi^{\mu},\theta^{+},\theta^{-})$, the
preserved supercharges $Q_{+}$ and $\overline{Q}_{+}$ commute with
$\theta^{-}$ and $\overline{\theta}^{-}$; see appendix B.1 for details.
Another property is that $Q_{+}$, $\overline{Q}_{+}$ do not involve a
derivative in $\perp$-direction. In terms of $\xi$ we can decompose

$\displaystyle\mathcal{R}_{\mu}^{B}(x,\theta,\overline{\theta})$
$\displaystyle=\mathcal{R}^{B{(0)}}_{\mu}+\theta^{-}\mathcal{R}^{B{(1)}}_{\mu}-\overline{\theta}^{-}\overline{\mathcal{R}^{B{(1)}}_{\mu}}+\theta^{-}\overline{\theta}^{-}\mathcal{R}^{B{(2)}}_{\mu},$
(55) $\displaystyle\chi^{B}_{\alpha}(x,\theta,\overline{\theta})$
$\displaystyle=\chi^{B{(0)}}_{\alpha}+\theta^{-}\chi^{B{(1a)}}_{\alpha}+\overline{\theta}^{-}\chi^{B{(1b)}}_{\alpha}+\theta^{-}\overline{\theta}^{-}\chi^{B{(2)}}_{\alpha},$

where we now denote bulk fields by a superscript $B$, and boundary fields (to
appear later) with a superscript $\partial$. The bracketed number subscripts
refer to the order in $\theta^{-},\overline{\theta}^{-}$ we have expanded in.
Here, each field on each right-hand side is a (super)function of
$(\xi,\theta^{+},\overline{\theta}^{+})$. Furthermore, because $Q_{+}$,
$\overline{Q}_{+}$ commute with $\theta^{-}$, $\overline{\theta}^{-}$, the
coefficient at each order in $\theta^{-},\overline{\theta}^{-}$ is a
${(0,2)}$-submultiplet — the remaining supersymmetry group acts independently
on each of them. This is a constructive way to decompose 3D $\mathcal{N}=2$
superfields with respect to the 2D ${\mathcal{N}=(0,2)}$ subalgebra. In the
appendix, we write the above decomposition explicitly for the
$\mathcal{S}$-multiplet (227)–(229), from which the $\mathcal{R}$-multiplet
follows by setting appropriate terms to zero.

In terms of the ${(0,2)}$-submultiplets, the constraints (47) are then written
as the following collection of equations, where we use coordinates
$\xi^{+}=\xi^{0}+\xi^{1}$, $\xi^{-}=\xi^{0}-\xi^{1}$ and
$\xi^{\perp}=x^{\perp}+i(\theta^{+}\overline{\theta}^{-}-\theta^{-}\overline{\theta}^{+})$:777Note
that covariant derivatives acting on ${(0,2)}$-superfields are
${(0,2)}$-covariant derivatives.
From $\overline{D}_{-}\chi_{\alpha}=0$:

$\displaystyle\chi^{B{(1b)}}_{\alpha}$ $\displaystyle=0,$ (56a)
$\displaystyle\chi^{B{(2)}}_{\alpha}+2i\partial_{-}\chi^{B{(0)}}_{\alpha}$
$\displaystyle=0.$ (56b)

From $\overline{D}_{+}\chi_{\alpha}=0$:

$\displaystyle\overline{D}_{+}\chi^{B{(0)}}_{\alpha}$ $\displaystyle=0,$ (57a)
$\displaystyle\overline{D}_{+}\chi^{B{(1a)}}_{\alpha}+2i\partial_{\perp}\chi^{B{(0)}}_{\alpha}$
$\displaystyle=0,$ (57b) $\displaystyle\overline{D}_{+}\chi^{B{(2)}}_{\alpha}$
$\displaystyle=0.$ (57c)

From $\operatorname{Im}D^{\alpha}\chi_{\alpha}=0$:

$\displaystyle\operatorname{Im}(D_{+}\chi_{-}^{B{(0)}}-\chi_{+}^{B{(1a)}})$
$\displaystyle=0,$ (58a)
$\displaystyle\overline{D}_{+}\overline{\chi_{-}^{B{(1a)}}}+\chi_{+}^{B{(2)}}-2i\partial_{-}\chi_{+}^{B{(0)}}-2i\partial_{\perp}\chi_{-}^{B{(0)}}$
$\displaystyle=0,$ (58b)
$\displaystyle\operatorname{Im}(D_{+}\chi_{-}^{B{(2)}}-2i\partial_{-}\chi_{+}^{B{(1a)}}-2i\partial_{\perp}\chi_{-}^{B{(1a)}})$
$\displaystyle=0.$ (58c)

Finally, the relation
$\overline{D}^{\beta}\mathcal{R}_{\alpha\beta}=\chi_{\alpha}$ yields:888These
have already been simplified by some relations we found, e.g. (56a).

$\displaystyle\chi^{B{(0)}}_{\alpha}$
$\displaystyle=\overline{D}_{+}\mathcal{R}^{B{(0)}}_{\alpha-}-\overline{\mathcal{R}^{B{(1)}}_{\alpha+}},$
(59a) $\displaystyle-\chi^{B{(1a)}}_{\alpha}$
$\displaystyle=\overline{D}_{+}\mathcal{R}^{B{(1)}}_{\alpha-}+\mathcal{R}^{B{(2)}}_{\alpha+}+2i\partial_{\perp}\mathcal{R}^{B{(0)}}_{\alpha-}+2i\partial_{-}\mathcal{R}^{B{(0)}}_{\alpha+},$
(59b) $\displaystyle 0$
$\displaystyle=\overline{D}_{+}\overline{\mathcal{R}}^{B{(1)}}_{\alpha\beta},$
(59c) $\displaystyle\chi^{B{(2)}}_{\alpha}$
$\displaystyle=\overline{D}_{+}\mathcal{R}^{B{(2)}}_{\alpha-}+2i\partial_{\perp}\overline{\mathcal{R}^{B{(1)}}_{\alpha-}}+2i\partial_{-}\overline{\mathcal{R}^{B{(1)}}_{\alpha+}}.$
(59d)

Note that we have not introduced _any_ new structure here: component-wise,
equations (47) have identical content as (56a)–(59d). In particular, the bulk
conservation equations follow from these constraints. Let us explicitly verify
this in the example of the conservation of the $R$-current $j_{\mu}^{B}$. We
start with equation (59b) setting $\alpha=+$ and taking the imaginary part.
Using the reality of $\mathcal{R}_{\alpha\beta}$ (which implies the reality of
$\mathcal{R}^{B{(0)}}_{\alpha\beta}$ and
$\mathcal{R}^{B{(2)}}_{\alpha\beta}$), we arrive at

$-\operatorname{Im}(\chi^{B{(1a)}}_{+})=\operatorname{Im}(\overline{D}_{+}\mathcal{R}^{B{(1)}}_{+-})+2\partial_{\perp}\mathcal{R}^{B{(0)}}_{+-}+2\partial_{-}\mathcal{R}^{B{(0)}}_{++}.$
(60)

Now consider equation (59a); setting $\alpha=-$, conjugating, applying
$\overline{D}_{+}$ on both sides and finally taking the imaginary part we
obtain

$\operatorname{Im}(\overline{D}_{+}\overline{\chi^{B{(0)}}_{-}})=\operatorname{Im}(\overline{D}_{+}D_{+}\mathcal{R}^{B{(0)}}_{--})-\operatorname{Im}(\overline{D}_{+}\mathcal{R}^{B{(1)}}_{-+}).$
(61)

From the reality of $\mathcal{R}^{B{(0)}}_{\alpha\beta}$ we get
$\operatorname{Im}(\overline{D}_{+}D_{+}\mathcal{R}^{B{(0)}}_{--})=2\partial_{+}\mathcal{R}^{B{(0)}}_{--}$.
Finally, we use (58a) to combine (60) and (61) into the bulk conservation
equation for the $R$-current:

$2\partial_{+}\mathcal{R}^{B{(0)}}_{--}+2\partial_{\perp}\mathcal{R}^{B{(0)}}_{+-}+2\partial_{-}\mathcal{R}^{B{(0)}}_{++}=0.$
(62)

This equation also implies the bulk conservation of
$(S_{\mu}^{B})_{+},(\overline{S}_{\mu}^{B})_{+}$ and $T_{\mu+}^{B}$, as can be
verified by the expansions (227)–(229).

In an analogous fashion, we may derive the bulk conservation for
$\mathcal{R}_{\alpha\beta}^{B{(1)}}$ and $\mathcal{R}_{\alpha\beta}^{B{(2)}}$.
The conservation of $\mathcal{R}_{\alpha\beta}^{B{(1)}}$ follows from
(59a)–(59d) together with (58b), and implies the conservation of bulk
supercurrents $(S_{\mu}^{B})_{-},(\overline{S}_{\mu}^{B})_{-}$ and the tensor
$T_{\mu\perp}^{B}$. The conservation of $\mathcal{R}_{\alpha\beta}^{B{(2)}}$
follows from (58c), (59b) and (59d), and implies the conservation of the bulk
tensor $T_{\mu-}^{B}$.

####  3.3.2 Boundary constraints

We now want to discuss constraints that we need to impose on the boundary
parts $\mathcal{R}_{\mu}^{\partial}$ and $\chi_{\alpha}^{\partial}$ (and
$\mathcal{Y}_{\alpha}^{\partial}$ in the case of the
$\mathcal{S}_{\mu}$-multiplet). Our guiding principle is of course the fact
that the constraints need to impose boundary conservation (33) on the
components of the boundary multiplets. The constraints can, in part, be
deduced from the bulk constraints (56)–(59). Let us elaborate on this point:
bulk and boundary superfields are combined to our _total_ supercurrent
multiplet

$\mathcal{R}_{\mu}^{\text{full}}=\mathcal{R}_{\mu}^{B}+\delta(\xi^{\perp})\mathcal{P}_{\mu}^{\phantom{\mu}{\hat{\mu}}}\mathcal{R}^{\partial}_{\hat{\mu}},$
(63)

where both bulk and boundary pieces can be decomposed into
${(0,2)}$-multiplets:999This is essentially an embedding into 3D
$\mathcal{N}=2$ superspace, see Drukker:2017xrb ; Drukker:2017dgn .

$\displaystyle\mathcal{R}_{\mu}^{B}(x,\theta,\overline{\theta})$
$\displaystyle=\mathcal{R}^{B{(0)}}_{\mu}+\theta^{-}\mathcal{R}^{B{(1)}}_{\mu}-\overline{\theta}^{-}\overline{\mathcal{R}^{B{(1)}}_{\mu}}+\theta^{-}\overline{\theta}^{-}\mathcal{R}^{B{(2)}}_{\mu},$
(64)
$\displaystyle\mathcal{R}_{\hat{\mu}}^{\partial}(x,\theta,\overline{\theta})$
$\displaystyle=\mathcal{R}_{\hat{\mu}}^{\partial{(0)}}+\theta^{-}\overline{\theta}^{-}\mathcal{R}_{\hat{\mu}}^{\partial{(2)}},$

and similarly for auxiliary fields

$\displaystyle\chi^{B}_{\alpha}(x,\theta,\overline{\theta})$
$\displaystyle=\chi^{B{(0)}}_{\alpha}+\theta^{-}\chi^{B{(1a)}}_{\alpha}+\overline{\theta}^{-}\chi^{B{(1b)}}_{\alpha}+\theta^{-}\overline{\theta}^{-}\chi^{B{(2)}}_{\alpha},$
(65) $\displaystyle\chi^{\partial}_{\alpha}(x,\theta,\overline{\theta})$
$\displaystyle=\chi^{\partial{(0)}}_{\alpha}+\theta^{-}\chi^{\partial{(1a)}}_{\alpha}+\overline{\theta}^{-}\chi^{\partial{(1b)}}_{\alpha}+\theta^{-}\overline{\theta}^{-}\chi^{\partial{(2)}}_{\alpha}.$

As mentioned before, the bulk part is unchanged compared to the full 3D
$\mathcal{N}=2$ current multiplet; it is simply decomposed into its
${(0,2)}$-submultiplets. The ansatz for the boundary part is directly
motivated by the expansions (227)–(229) of the bulk multiplet, which show that

$\displaystyle\mathcal{R}_{\mu}^{B{(0)}}$ $\displaystyle=j^{B}_{\mu}+\ldots,$
(66) $\displaystyle\mathcal{R}_{\mu}^{B{(1)}}$
$\displaystyle=-i(S^{B}_{\mu})_{-}+\ldots,$
$\displaystyle\mathcal{R}_{\mu}^{B{(2)}}$ $\displaystyle=-2K_{\mu-}+\ldots,$

where for the $\mathcal{R}$-multiplet,
$K_{\mu\nu}=2T_{\nu\mu}-\tfrac{1}{4}\epsilon_{\mu\nu\rho}H^{\rho}$. We can
conclude

$\displaystyle\mathcal{R}_{\hat{\mu}}^{\partial{(0)}}$
$\displaystyle=j^{\partial}_{\hat{\mu}}+\ldots,$ (67)
$\displaystyle\mathcal{R}_{\hat{\mu}}^{\partial{(2)}}$
$\displaystyle=-2K_{{\hat{\mu}}-}^{\partial}+\ldots,$

as the bulk conserved currents have to be paired with their respective
boundary currents. Note that we do not consider a boundary contribution to the
“broken” $(S_{\mu}^{B})_{-}$ currents, as we have no guiding principle in this
framework. Then, due to the form of the full supercurrent multiplet (63), we
postulate that the constraints applied to the boundary piece must be of
similar form as (56a)–(59d), but instead of imposing divergence-freeness (25),
they should impose (33) on the _remaining, conserved_ boundary currents.

We postulate the following adjustments on the constraints obtained from the
bulk (56)–(59), now applied to boundary multiplets
$\mathcal{R}^{\partial(*)}_{\alpha\alpha}$, in order to obtain correct
conservation equations:

  1. (1)

Firstly, since the boundary is two-dimensional with directions $x_{++},x_{--}$
(in bispinor notation, cf. (193)) we only have superfields
$\mathcal{R}^{\partial(*)}_{++},\mathcal{R}^{\partial(*)}_{--}$, and no
superfield $\mathcal{R}^{\partial(*)}_{+-}$.

  2. (2)

Secondly, we do not consider boundary contributions to the “broken”
$(S_{\mu}^{B})_{-}$ currents, and hence no
$\mathcal{R}^{\partial{(1)}}_{\alpha\alpha}$ should appear.

  3. (3)

Lastly, to impose the correct conservation equation on the boundary, we must
replace terms of the form $\partial_{\perp}A^{\partial}$ with
$-A^{B}|_{\partial}$ whenever such terms appear. This transformation precisely
maps divergence-free equations (25) into boundary conservation equations (33).
In addition, this replacement parses well with the fact that derivatives in
the perpendicular direction make little sense when they act on boundary
currents, in particular when the boundary currents are functions of purely
boundary fields.

The preliminary constraints on the boundary pieces then read:
Analogs to (56):

$\displaystyle\chi^{\partial{(1b)}}_{\alpha}$ $\displaystyle=0,$ (68a)
$\displaystyle\chi^{\partial{(2)}}_{-}+2i\partial_{-}\chi^{\partial{(0)}}_{-}$
$\displaystyle=0.$ (68b)

Analogs to (57):

$\displaystyle\overline{D}_{+}\chi_{-}^{\partial{(0)}}$ $\displaystyle=0,$
(69a)
$\displaystyle\overline{D}_{+}\chi^{\partial{(1a)}}_{+}-2i\chi^{B{(0)}}_{+}|_{\partial}$
$\displaystyle=0,$ (69b)
$\displaystyle\overline{D}_{+}\chi^{\partial{(2)}}_{\alpha}$
$\displaystyle=0.$ (69c)

Analogs to (58):

$\displaystyle\operatorname{Im}(D_{+}\chi_{-}^{\partial{(0)}}-\chi_{+}^{\partial{(1a)}})$
$\displaystyle=0,$ (70a)
$\displaystyle\operatorname{Im}(D_{+}\chi_{-}^{\partial{(2)}}-2i\partial_{-}\chi_{+}^{\partial{(1a)}}+2i\chi_{-}^{B{(1a)}}|_{\partial})$
$\displaystyle=0.$ (70b)

Lastly, the analogs to (59):

$\displaystyle\chi^{\partial{(0)}}_{+}$ $\displaystyle=0,$ (71a)
$\displaystyle\chi^{\partial{(0)}}_{-}$
$\displaystyle=\overline{D}_{+}\mathcal{R}^{\partial{(0)}}_{--},$ (71b)
$\displaystyle\chi^{\partial{(1a)}}_{+}$
$\displaystyle=-\mathcal{R}^{\partial{(2)}}_{++}+2i\mathcal{R}^{B{(0)}}_{+-}|_{\partial}-2i\partial_{-}\mathcal{R}^{\partial{(0)}}_{++},$
(71c) $\displaystyle\chi^{\partial{(1a)}}_{-}$
$\displaystyle=2i\mathcal{R}^{B{(0)}}_{--}|_{\partial},$ (71d)
$\displaystyle\chi^{\partial{(2)}}_{+}$
$\displaystyle=-2i\overline{\mathcal{R}_{+-}^{B{(1)}}}|_{\partial},$ (71e)
$\displaystyle\chi^{\partial{(2)}}_{-}$
$\displaystyle=\overline{D}_{+}\mathcal{R}^{\partial{(2)}}_{--}-2i\overline{\mathcal{R}^{B{(1)}}_{--}}|_{\partial}.$
(71f)

Note that the naive application of adjustments (1)–(3) leads to three further
relations, which we have intentionally omitted above. These are: the analog of
(56b) for $\alpha=+$, which reads

$\chi^{\partial{(2)}}_{+}+2i\partial_{-}\chi^{\partial{(0)}}_{+}=0,$ (72)

the analog of (57b) for $\alpha=-$, which reads

$\overline{D}_{+}\chi^{\partial{(1a)}}_{-}-2i\chi^{B{(0)}}_{-}|_{\partial},$
(73)

and lastly, the analog of (58b) reads

$\overline{D}_{+}\overline{\chi_{-}^{\partial{(1a)}}}+\chi_{+}^{\partial{(2)}}-2i\partial_{-}\chi_{+}^{\partial{(0)}}+2i\chi_{-}^{B{(0)}}|_{\partial}=0.$
(74)

We argue that these relations must be discarded from the set of constraints of
boundary multiplets. To see this, note that the first relation (72) is
compatible with equations (71a) and (71e) _only if_
$\mathcal{R}_{+-}^{B{(1)}}|_{\partial}=0$. Similarly, the second relation (73)
in agreement with equations (71d) and (59a) again _only if_
$\mathcal{R}_{+-}^{B{(1)}}|_{\partial}=0$. Lastly, the third relation (74) is
consistent with (71a), (71d), (71e) and (59a), once more _only if_
$\mathcal{R}_{+-}^{B{(1)}}|_{\partial}=0$. Hence, including any of the three
relations in the constraints of boundary ${(0,2)}$-supermultiplets would
impose

$\mathcal{R}_{+-}^{B{(1)}}|_{\partial}=-i(S^{B}_{\perp})_{-}|_{\partial}+\ldots=0.$
(75)

However, imposing this condition would imply the conservation of the “broken”
charge $Q_{-}$, as the current $(S^{B}_{\mu})_{-}$ would fulfill (16) with a
trivial boundary part. Therefore, if we did not omit relations (72)–(74), we
would impose the conservation of the “broken” charges
$Q_{-},\overline{Q}_{-}$, which is inconsistent with the “breaking” of
$P_{\perp}$ and the explicit conservation of the charges
$Q_{+},\overline{Q}_{+}$. As a last argument, we note that the three omitted
relations are _not_ required to obtain the boundary conservation equations for
the remaining symmetries (${\mathcal{N}=(0,2)}$ supersymmetry, $R$-symmetry,
2D Poincaré symmetry). We will verify this in the remainder of this subsection
by explicitly checking that the boundary conservation equations indeed follow
from constraints (68)–(71).

We spell out the derivation of the boundary conservation equation in the
example of $\mathcal{R}_{\alpha\alpha}^{\partial{(0)}}$: Taking the imaginary
part of (71c) and using the reality of the multiplet, we obtain

$\operatorname{Im}(\chi^{\partial{(1a)}}_{+})=2\mathcal{R}^{B{(0)}}_{+-}|_{\partial}-2\partial_{-}\mathcal{R}^{\partial{(0)}}_{++}.$
(76)

Now we take equation (71b), conjugate it, apply $\overline{D}_{+}$ on both
sides and finally take imaginary part again to obtain:

$\operatorname{Im}(\overline{D}_{+}\overline{\chi^{\partial{(0)}}_{-}})=\operatorname{Im}(\overline{D}_{+}D_{+}\mathcal{R}^{\partial{(0)}}_{--}).$
(77)

Again, due to the reality of $\mathcal{R}^{\partial{(0)}}_{--}$, we have that
$\operatorname{Im}(\overline{D}_{+}D_{+}\mathcal{R}^{\partial{(0)}}_{--})=2\partial_{+}\mathcal{R}^{\partial{(0)}}_{--}$.
Finally, we can combine equations (76) and (77) using (70a) into the
conservation equation for the boundary $R$-current:

$2\mathcal{R}^{B{(0)}}_{+-}|_{\partial}-2\partial_{-}\mathcal{R}^{\partial{(0)}}_{++}-2\partial_{+}\mathcal{R}^{\partial{(0)}}_{--}=0.$
(78)

Like its bulk counterpart, this superfield equation also implies the boundary
conservation of
$(S_{\hat{\mu}}^{\partial})_{+},(\overline{S}_{\hat{\mu}}^{\partial})_{+}$ and
$T_{{\hat{\mu}}+}^{\partial}$.

In a similar fashion, we may derive boundary conservation of
$\mathcal{R}_{\alpha\alpha}^{\partial{(2)}}$ using equations (71f), (70b),
(71c) and (71d). Component-wise it implies the conservation of the boundary
tensor $T_{{\hat{\mu}}_{-}}^{\partial}$. No boundary analog to bulk
conservation of $(S_{\mu}^{B})_{-}$ follows from the boundary constraints,
which is what we expect.

###  3.4 Integrated supercurrent multiplets

####  3.4.1 Comparison to pure 2D theories

The supercurrent multiplets that satisfy the bulk (56)–(59) and boundary
(68)–(71) constraints are ${(0,2)}$-multiplets of a three-dimensional theory
with boundary, with supersymmetry algebra isomorphic to that of a 2D
${(0,2)}$-theory. It is interesting to compare and contrast the structure of
this theory to a pure bulk 2D theory with ${(0,2)}$-supersymmetry. We review
some generic aspects of such bulk theories following Dedushenko:2015opz  (see
also Melnikov:2019tpl  for a comprehensive review on 2D ${\mathcal{N}=(0,2)}$
models).

In the case of a 2D theory with ${\mathcal{N}=(0,2)}$ supersymmetry, the most
general $\mathcal{S}$-multiplet is given by superfields
$(\mathcal{S}_{++}^{(0,2)},\mathcal{W}_{-}^{(0,2)},\mathcal{T}_{----}^{(0,2)},C)$
and corresponding defining constraints in 2D ${(0,2)}$-theories, such that
conditions (a)–(d) are satisfied. For details on their structure see
Dumitrescu:2011iu ; Dedushenko:2015opz . If the ${(0,2)}$-model we consider
has an $R$-symmetry, there exists a corresponding smaller
$\mathcal{R}$-multiplet
$(\mathcal{R}_{\hat{\mu}}^{(0,2)},\mathcal{T}_{----}^{(0,2)})$ containing an
improved energy momentum tensor $T_{\mu\nu}$. Furthermore, the structure of
the multiplet guarantees that we can define the _half-twisted_ energy-momentum
tensor $\widetilde{T}_{\mu\nu}$

$\displaystyle\widetilde{T}_{++}$
$\displaystyle=T_{++}+\tfrac{i}{2}\partial_{+}j_{+},$ (79)
$\displaystyle\widetilde{T}_{+-}$
$\displaystyle=T_{+-}-\tfrac{i}{2}\partial_{-}j_{+},$
$\displaystyle\widetilde{T}_{--}$
$\displaystyle=T_{--}-\tfrac{i}{2}\partial_{-}j_{-},$

which satisfies

$\displaystyle\\{\overline{Q}_{+},\cdots\\}$
$\displaystyle=\widetilde{T}_{+{\hat{\mu}}},$ (80)
$\displaystyle\\{\overline{Q}_{+},\widetilde{T}_{--}\\}$
$\displaystyle=0,\quad\text{but
}\\{\overline{Q}_{+},\cdots\\}\neq\widetilde{T}_{--}.$

In other words, the components of the twisted energy-momentum tensor are
$\overline{Q}_{+}$-cohomology elements, and $\widetilde{T}_{--}$ is a non-
trivial element. Starting from these identities, one can show that the
$\overline{Q}_{+}$-cohomology of observables is invariant under
renormalization group flow and thus carries information about possible IR
fixed points of the model under consideration Dedushenko:2015opz . In
particular, there is an emergent conformal symmetry on the level of
cohomology.

Secondly, it is well known for ${(0,2)}$-theories that the cohomology of
$\overline{Q}_{+}$ as an operator on fields is isomorphic to the cohomology of
$\overline{D}_{+}$ as an operator on superfields. In this language, the non-
trivial components of the twisted energy-momentum tensor are given by an
appropriate $\overline{D}_{+}$-closed linear combination of superfields from
the supercurrent multiplet:

$\overline{D}_{+}(\mathcal{T}_{----}^{(0,2)}-\tfrac{i}{2}\partial_{--}\mathcal{R}_{--}^{(0,2)})=0.$
(81)

####  3.4.2 An energy-momentum tensor (not) in the cohomology

Since our three-dimensional theory with boundary has the same supersymmetry
algebra, we might expect a similar structure as far as
$\overline{Q}_{+}$-cohomology is concerned. Indeed, we can identify the analog
of (81) in 3D: we combine bulk equations (56b),(59a) and (59d) for $\alpha=-$
to:101010The factor discrepancy on the left-hand side of (81) and the above
equations is merely due to switching between spacetime and bispinor notation
(cf. (194)).

$\overline{D}_{+}(\mathcal{R}^{B{(2)}}_{--}+2i\partial_{-}\mathcal{R}^{B{(0)}}_{--})=-2i\partial_{\perp}\overline{\mathcal{R}^{B{(1)}}_{--}},$
(82)

as well as boundary equations (68b), (71b) and (71f) to

$\overline{D}_{+}(\mathcal{R}^{\partial{(2)}}_{--}+2i\partial_{-}\mathcal{R}^{\partial{(0)}}_{--})=2i\overline{\mathcal{R}^{B{(1)}}_{--}}|_{\partial}.$
(83)

Furthermore, we can also combine (57b), (59a) and (59b) for $\alpha=+$ into

$\overline{D}_{+}(\mathcal{R}_{++}^{B{(2)}}+2i\mathcal{R}_{++}^{B{(0)}})=-2i\partial_{\perp}\overline{\mathcal{R}_{++}^{B{(1)}}}.$
(84)

Similarly, we combine boundary equations (69b), (70b) and (59a) into

$\overline{D}_{+}(\mathcal{R}^{\partial{(2)}}_{++}+2i\partial_{-}\mathcal{R}^{\partial{(0)}}_{++})=2i\overline{\mathcal{R}^{B{(1)}}_{++}}|_{\partial}.$
(85)

We can rewrite these relations using the full $\mathcal{R}$-multiplet
$\mathcal{R}_{\alpha\beta}^{\text{full}(*)}=\mathcal{R}_{\alpha\beta}^{B(*)}+\delta(\xi^{\perp})\mathcal{R}_{\alpha\beta}^{\partial(*)}$:

$\displaystyle\overline{D}_{+}(\mathcal{R}^{\text{full}{(2)}}_{--}+2i\partial_{-}\mathcal{R}^{\text{full}{(0)}}_{--})$
$\displaystyle=-2i\partial_{\perp}\overline{\mathcal{R}^{B{(1)}}_{--}}+2i\delta(\xi^{\perp})\overline{\mathcal{R}^{B{(1)}}_{--}}|_{\partial},$
(86)
$\displaystyle\overline{D}_{+}(\mathcal{R}^{\text{full}{(2)}}_{++}+2i\partial_{-}\mathcal{R}^{\text{full}{(0)}}_{++})$
$\displaystyle=-2i\partial_{\perp}\overline{\mathcal{R}^{B{(1)}}_{++}}+2i\delta(\xi^{\perp})\overline{\mathcal{R}^{B{(1)}}_{++}}|_{\partial}.$

The first equation is the analog of (81) in 3D, as we already stated. The
second equation is the analog of the $\overline{Q}$-closedness (equivalent to
$\overline{D}_{+}$-closedness) of the half-twisted tensor
$T_{+-}-\tfrac{i}{2}\partial_{-}j_{+}$, which in 2D follows from
$\overline{Q}$-exactness.

####  3.4.3 Integrated currents and multiplets

Equation (86) shows that one cannot, in general, repeat the pure 2D argument
to produce a local energy-momentum tensor twisted by the $R$-symmetry such
that it is a $\overline{Q}_{+}$-cohomology element. However, there is a
different point of view which is helpful here: A three-dimensional quantum
field theory with a finite number of fields can instead be regarded as a two-
dimensional quantum field theory with an infinite number of fields. More
precisely, instead of viewing bulk fields, loosely speaking, as maps $\partial
M\times\mathbb{R}_{\leq 0}\to T$, we view them as maps $\partial
M\to\\{\text{maps: }\mathbb{R}_{\leq 0}\to T\\}$ Dimofte:2017tpi ;
Bullimore:2016nji . Now, instead of considering separate bulk and boundary
actions, we can write a single Lagrangian for the full theory:

$\mathcal{L}^{\text{int.}}\coloneqq\mathcal{L}^{\partial}+\int_{\mathbb{R}_{\leq
0}}\mathrm{d}x^{n}\mathcal{L}^{B};\quad S=\int_{\partial
M}\mathrm{d}x^{N-1}\mathcal{L}^{\text{int.}}.$ (87)

We see that the action is the same as before, but integration along $x^{n}$ is
now a conceptually different operation: before, it used to be an integral on
the spacetime on which the field theory is defined; now it is an operation on
the new target space (i.e. a functional). The integration along $x^{n}$ also
translates to conserved currents: As the theory is now formally two-
dimensional, applying Noether’s theorem to the above Lagrangian yields a two-
dimensional current of the form

$J_{\hat{\mu}}^{\text{int.}}=J_{\hat{\mu}}^{\partial}+\int_{\mathbb{R}_{\leq
0}}\mathrm{d}x^{n}J_{\hat{\mu}}^{B}.$ (88)

We see that its conserved charge

$Q=\int_{\partial\Sigma}J_{\text{int.}}^{0}$ (89)

is identical to the one belonging to the local current (19). This also
provides an argument why the integrated currents are “natural” from the point
of view of the three-dimensional QFT: To find the conserved charge of a
current, one has to integrate all spatial directions, and the integrated
current is an “intermediate step” of this integration. The conservation
equations (33) now take the familiar form

$\partial^{\hat{\mu}}J_{\hat{\mu}}^{\text{int.}}=0,$ (90)

where boundary conditions are possibly used. Extending supersymmetrically, we
introduce the _integrated supercurrent multiplets_ , in our conventions
(recall, mixed indices correspond to the $\perp$-direction, which does not
appear here, see (193))

$\mathcal{R}_{\alpha\alpha}^{\text{int.}}=\mathcal{R}_{\alpha\alpha}^{\partial}+\int_{\mathbb{R}_{\leq
0}}\mathrm{d}x^{\perp}\mathcal{R}_{\alpha\alpha}^{B}.$ (91)

In terms of integrated currents, the right-hand side of (86) cancels exactly
due to the integral:

$\displaystyle\overline{D}_{+}(\mathcal{R}^{\text{int.}{(2)}}_{--}+2i\partial_{-}\mathcal{R}^{\text{int.}{(0)}}_{--})$
$\displaystyle=0,$ (92)
$\displaystyle\overline{D}_{+}(\mathcal{R}^{\text{int.}{(2)}}_{++}+2i\partial_{-}\mathcal{R}^{\text{int.}{(0)}}_{++})$
$\displaystyle=0.$

The general arguments from Dedushenko:2015opz  presented in section 3.4.2 then
imply that the lowest component
$-16(T_{--}^{\text{int.}}-\tfrac{i}{2}\partial_{-}j_{-}^{\text{int.}})$ of the
first equation is a non-trivial $\overline{Q}_{+}$-cohomology element and the
lowest component
$-16(T_{-+}^{\text{int.}}-\tfrac{i}{2}\partial_{-}j_{+}^{\text{int.}})$ of the
second equation is a trivial $\overline{Q}_{+}$-cohomology element in the
integrated 2D theory. In fact, a stronger statement holds: the integrated
multiplets are genuine 2D ${\mathcal{N}=(0,2)}$ supersymmetry multiplets.
Setting

$\displaystyle\mathcal{R}_{\alpha\alpha}^{(0,2)}$
$\displaystyle\coloneqq\mathcal{R}_{\alpha\alpha}^{\text{int.}{(0)}},$ (93)
$\displaystyle\mathcal{T}_{----}^{(0,2)}$
$\displaystyle\coloneqq-\mathcal{R}_{--}^{\text{int.}{(2)}},$

the constraints (56)–(59) and (68)–(71) imply that these are indeed 2D
${(0,2)}$-supercurrent multiplets in the sense of Dumitrescu:2011iu ;
Dedushenko:2015opz .

###  3.5 Summarizing the results on boundary multiplets

We look at three-dimensional theories with $\mathcal{N}=2$ supersymmetry,
broken to a 2D ${(0,2)}$-subalgebra due to a boundary. Currents associated to
(remaining) symmetries now consist of bulk and boundary pieces.

First we study 3D bulk supercurrent multiplets, in particular the
$\mathcal{R}$-multiplet. Its structure remains unchanged, as the bulk parts of
conserved currents are still divergence-free; we merely decompose the bulk
multiplets into their ${(0,2)}$-submultiplets

$\displaystyle\mathcal{R}_{\mu}^{B}(x,\theta,\overline{\theta})$
$\displaystyle=\mathcal{R}^{B{(0)}}_{\mu}+\theta^{-}\mathcal{R}^{B{(1)}}_{\mu}-\overline{\theta}^{-}\overline{\mathcal{R}^{B{(1)}}_{\mu}}+\theta^{-}\overline{\theta}^{-}\mathcal{R}^{B{(2)}}_{\mu},$
(94) $\displaystyle\chi^{B}_{\alpha}(x,\theta,\overline{\theta})$
$\displaystyle=\chi^{B{(0)}}_{\alpha}+\theta^{-}\chi^{B{(1a)}}_{\alpha}+\overline{\theta}^{-}\chi^{B{(1b)}}_{\alpha}+\theta^{-}\overline{\theta}^{-}\chi^{B{(2)}}_{\alpha}.$

The defining constraints (47) now decompose under the
${\mathcal{N}=(0,2)}$-subalgebra into equations (56)–(59).

We investigate possible defining constraints for the boundary parts, using as
guiding principles that

  * •

bulk and boundary pieces combine into a full multiplet
$\mathcal{R}_{\mu}^{\text{full}}=\mathcal{R}_{\mu}^{B}+\delta(\xi^{\perp})\mathcal{P}_{\mu}^{\phantom{\mu}{\hat{\mu}}}\mathcal{R}^{\partial}_{\hat{\mu}}$,
where the boundary pieces are also decomposed as in (64) and (65), hence the
boundary constraints must be of the same form as the bulk constraints
(56)–(59),

  * •

boundary constraints must impose boundary conservation (33) on the _remaining,
conserved_ boundary currents.

We obtain the following list of constraints:

$\displaystyle 0$
$\displaystyle=\chi^{\partial{(2)}}_{-}+2i\partial_{-}\chi^{\partial{(0)}}_{-},$
(95a) $\displaystyle 0$
$\displaystyle=\overline{D}_{+}\chi^{\partial{(1a)}}_{+}-2i\chi^{B{(0)}}_{+}|_{\partial},$
(95b) $\displaystyle 0$
$\displaystyle=\operatorname{Im}(D_{+}\chi_{-}^{\partial{(0)}}-\chi_{+}^{\partial{(1a)}}),$
(95c) $\displaystyle 0$
$\displaystyle=\operatorname{Im}(D_{+}\chi_{-}^{\partial{(2)}}-2i\partial_{-}\chi_{+}^{\partial{(1a)}}+2i\chi_{-}^{B{(1a)}}|_{\partial}),$
(95d) $\displaystyle\chi^{\partial{(0)}}_{-}$
$\displaystyle=\overline{D}_{+}\mathcal{R}^{\partial{(0)}}_{--},$ (95e)
$\displaystyle\chi^{\partial{(1a)}}_{+}$
$\displaystyle=-\mathcal{R}^{\partial{(2)}}_{++}+2i\mathcal{R}^{B{(0)}}_{+-}|_{\partial}-2i\partial_{-}\mathcal{R}^{\partial{(0)}}_{++},$
(95f) $\displaystyle\chi^{\partial{(1a)}}_{-}$
$\displaystyle=2i\mathcal{R}^{B{(0)}}_{--}|_{\partial},$ (95g)
$\displaystyle\chi^{\partial{(2)}}_{-}$
$\displaystyle=\overline{D}_{+}\mathcal{R}^{\partial{(2)}}_{--}-2i\overline{\mathcal{R}^{B{(1)}}_{--}}|_{\partial},$
(95h) $\displaystyle\chi^{\partial{(2)}}_{+}$
$\displaystyle=-2i\overline{\mathcal{R}_{+-}^{B{(1)}}}|_{\partial},$ (95i)
$\displaystyle\chi^{\partial{(0)}}_{+}$ $\displaystyle=0,$ (95j)
$\displaystyle\chi^{\partial{(1b)}}_{\alpha}$ $\displaystyle=0.$ (95k)

The constraints (95a)–(95h) are necessary to derive boundary conservation
equations and equations (83). The last three relations (95i)–(95k) are not
used in any conservation equation and are independent of the rest of the
constraints.

These constraints imply, in particular, equations (86) on full currents:

$\overline{D}_{+}(\mathcal{R}^{\text{full}{(2)}}_{\pm\pm}+2i\partial_{-}\mathcal{R}^{\text{full}{(0)}}_{\pm\pm})=-2i\partial_{\perp}\overline{\mathcal{R}^{B{(1)}}_{\pm\pm}}+2i\delta(\xi^{\perp})\overline{\mathcal{R}^{B{(1)}}_{\pm\pm}}|_{\partial},$
(96)

which motivates the introduction of integrated current multiplets

$\mathcal{R}_{\alpha\alpha}^{\text{int.}}=\mathcal{R}_{\alpha\alpha}^{\partial}+\int\mathrm{d}x^{\perp}\mathcal{R}_{\alpha\alpha}^{B}.$
(97)

These multiplets are genuine 2D ${\mathcal{N}=(0,2)}$ supercurrent multiplets
in the usual sense. In particular, the integration sets the right-hand side of
relation (96) to zero. This implies that
$T_{--}^{\text{int.}}-\tfrac{i}{2}\partial_{-}j_{-}^{\text{int.}}$ is a non-
trivial $\overline{Q}_{+}$-cohomology element and
$T_{-+}^{\text{int.}}-\tfrac{i}{2}\partial_{-}j_{+}^{\text{int.}}$ is a
trivial $\overline{Q}_{+}$-cohomology element in the effective (integrated) 2D
theory.

##  4 Landau-Ginzburg model

###  4.1 Bulk theory

We now study a particular model where the framework we developed above can be
applied. Our bulk theory should be a 3D $\mathcal{N}=2$ Landau-Ginzburg model
which lives on three-dimensional Minkowski space. At a later point, we will
introduce a boundary and restrict the theory to the half-space

$M=\Set{x\in\mathbb{R}^{1,2}}{x^{\smash{\perp}}\coloneqq x^{1}\leq 0}.$ (98)

We will formulate the bulk theory in 3D $\mathcal{N}=2$ superspace. A generic
chiral field is given by

$\Phi_{3D}(x,\theta,\overline{\theta})=\phi(y)+\sqrt{2}\theta\psi(y)+\theta\theta
F(y),\qquad y^{\mu}=x^{\mu}-i\theta\gamma^{\mu}\overline{\theta},$ (99)

where, as usual, $\phi$ is a complex scalar field, $\psi_{\alpha}$ is a
complex fermion, and $F$ is a complex auxiliary field. Under the bulk
supersymmetry, the components transform as follows:

$\displaystyle\delta_{\text{sym}}\phi$ $\displaystyle=\sqrt{2}\epsilon\psi,$
(100) $\displaystyle\delta_{\text{sym}}\psi_{\alpha}$
$\displaystyle=\sqrt{2}\epsilon_{\alpha}F-\sqrt{2}i(\gamma^{\mu}\overline{\epsilon})_{\alpha}\partial_{\mu}\phi,$
$\displaystyle\delta_{\text{sym}}F$
$\displaystyle=-\sqrt{2}i\overline{\epsilon}\gamma^{\mu}\partial_{\mu}\psi.$

Let us now consider the simplest non-trivial theory: a Landau-Ginzburg model
of a single chiral superfield. Its kinetic (Kähler) term is given by

$\mathcal{L}_{\text{kin.}}=\int\mathrm{d}^{4}\theta\,\Phi_{3D}\overline{\Phi}_{3D}=-\partial_{\mu}\overline{\phi}\partial^{\mu}\phi+\tfrac{i}{2}(\overline{\psi}\gamma^{\mu}\partial_{\mu}\psi)-\tfrac{i}{2}(\partial_{\mu}\overline{\psi}\gamma^{\mu}\psi)+\overline{F}F+\tfrac{1}{4}\partial^{2}(\overline{\phi}\phi).$
(101)

At a later point, the term $\tfrac{1}{4}\partial^{2}(\overline{\phi}\phi)$
will be removed; as it is a total derivative, it does not influence the bulk
theory, but will be relevant once a boundary is introduced.
The superpotential is of the well-known form

$\mathcal{L}_{W}=\int\mathrm{d}^{2}\theta\,W(\Phi_{3D})+\text{cc.}=W^{\prime}(\phi)F-\tfrac{1}{2}W^{\prime\prime}(\phi)\psi\psi+\text{cc.}$
(102)

The bulk equations of motion are given by

$\overline{D}^{2}\overline{\Phi}=-4W^{\prime}(\Phi)\Leftrightarrow\left\\{\begin{aligned}
0&=\overline{F}+W^{\prime}(\phi)\\\
0&=\partial_{\mu}\partial^{\mu}\overline{\phi}+W^{\prime\prime}(\phi)F-\tfrac{1}{2}W^{\prime\prime\prime}(\phi)\psi\psi\\\
0&=i(\gamma^{\mu}\partial_{\mu}\overline{\psi})_{\alpha}-W^{\prime\prime}(\phi)\psi_{\alpha}\end{aligned}\right\\}.$
(103)

###  4.2 Introducing a boundary and breaking to ${(0,2)}$

As we already saw in section 3.3, the supersymmetry algebra breaks at least to
a 2D ${\mathcal{N}=(0,2)}$ (or 2D $\mathcal{N}={(1,1)}$, which we do not
consider here) subalgebra when we introduce a boundary. Let us study
sufficient conditions to preserve exactly ${\mathcal{N}=(0,2)}$.

####  4.2.1 Decomposition of the bulk fields

Under the ${\mathcal{N}=(0,2)}$ subalgebra, the chiral field $\Phi_{3D}$
decomposes into a ${(0,2)}$ chiral multiplet and a Fermi multiplet. More
details of this decomposition are written in appendix B.1. The resulting
${(0,2)}$-superfields are

$\displaystyle\Phi$
$\displaystyle=\phi+\sqrt{2}\theta^{+}\psi_{+}-2i\theta^{+}\overline{\theta}^{+}\partial_{+}\phi,$
(104) $\displaystyle\Psi$
$\displaystyle=\psi_{-}-\sqrt{2}\theta^{+}F-2i\theta^{+}\overline{\theta}^{+}\partial_{+}\psi_{-}+\sqrt{2}i\overline{\theta}^{+}\partial_{\perp}\phi-2i\theta^{+}\overline{\theta}^{+}\partial_{\perp}\psi_{+}.$

The Fermi superfield satisfies

$\overline{D}_{+}\Psi=\sqrt{2}E_{\Psi},\quad E_{\Psi}=-i\partial_{\perp}\Phi,$
(105)

and the chirality condition reads

$\overline{D}_{+}\Phi=0.$ (106)

The supersymmetry variation in the smaller algebra is given by
$\delta_{\text{sym}}\coloneqq\epsilon
Q_{+}-\overline{\epsilon}\overline{Q}_{+}$ (cf. equation (203) for the
definition of the superspace operators). The component fields now transform as

$\displaystyle\delta_{\text{sym}}\phi$
$\displaystyle=\sqrt{2}\epsilon\psi_{+},$
$\displaystyle\delta_{\text{sym}}\psi_{+}$
$\displaystyle=-2\sqrt{2}i\overline{\epsilon}\partial_{+}\phi,$ (107)
$\displaystyle\delta_{\text{sym}}F$
$\displaystyle=2\sqrt{2}i\overline{\epsilon}\partial_{+}\psi_{-}+\sqrt{2}i\overline{\epsilon}\partial_{\perp}\psi_{+},$
$\displaystyle\delta_{\text{sym}}\psi_{-}$ $\displaystyle=-\sqrt{2}\epsilon
F+\sqrt{2}i\overline{\epsilon}\partial_{\perp}\phi,$

which is precisely the restriction of (100) to the ${(0,2)}$-subalgebra, given
by choosing $\epsilon^{\alpha}=\left(\begin{smallmatrix}0\\\
\epsilon\end{smallmatrix}\right)$. We may rewrite the Lagrangian in terms of
${(0,2)}$-superspace:

$\displaystyle\mathcal{L}_{\text{kin.}}$
$\displaystyle=\frac{1}{2}\int\mathrm{d}^{2}\theta^{+}\Big{[}i\overline{\Phi}\partial_{-}\Phi-i\partial_{-}\overline{\Phi}\Phi+\overline{\Psi}\Psi$
(108) $\displaystyle\hskip
56.9055pt+\partial_{\perp}\left(\tfrac{1}{2}\theta^{+}\overline{\theta}^{+}\partial_{\perp}(\overline{\Phi}\Phi)+\tfrac{i}{\sqrt{2}}\theta^{+}\overline{\Phi}\Psi+\tfrac{i}{\sqrt{2}}\overline{\theta}^{+}\overline{\Psi}\Phi\right)\Big{]},$
$\displaystyle\mathcal{L}_{W}$
$\displaystyle=-\tfrac{1}{\sqrt{2}}\int\mathrm{d}\theta^{+}\Psi
W^{\prime}(\Phi)+\text{cc.}$

Note that $\mathcal{L}_{\text{kin.}}$ consists of two parts. The first part is
invariant under ${(0,2)}$-supersymmetry even in the presence of a boundary, as
its ${(0,2)}$-variation is just a total $x^{+}$-derivative. The second term
(trivially) transforms into an $x^{\perp}$-derivative, so it breaks
${(0,2)}$-supersymmetry in the presence of a boundary, and hence dictates part
of the “boundary compensating term” (cf. discussion at the end of section
2.2.1).

The equations of motion may again be written as superfield equations, now in
${(0,2)}$-superspace:

$\displaystyle 0$
$\displaystyle=2i\partial_{-}\overline{D}_{+}\overline{\Phi}+\sqrt{2}i\partial_{\perp}\overline{\Psi}-\sqrt{2}W^{\prime\prime}(\Phi)\Psi,$
(109) $\displaystyle 0$
$\displaystyle=\overline{D}_{+}\overline{\Psi}+\sqrt{2}W^{\prime}(\Phi).$

####  4.2.2 Recovering partial supersymmetry

As it stands, the pure bulk action (108) is not even ${(0,2)}$-supersymmetric
in the presence of a boundary:

$\delta_{\text{sym}}S=\int_{M}\delta_{\text{sym}}(\mathcal{L}_{\text{kin.}}+\mathcal{L}_{W})=\int_{M}\partial_{\mu}(V^{\mu}_{\text{kin.}}+V^{\mu}_{W})=\int_{\partial
M}(V^{\perp}_{\text{kin.}}+V^{\perp}_{W}),$ (110)

which, in general, does not vanish. To recover at least ${\mathcal{N}=(0,2)}$
supersymmetry, we must compensate these bulk variations.

For the kinetic term, we can add a boundary compensating term
$\widetilde{\Delta}_{\text{kin.}}$ to the boundary Lagrangian (in a boundary-
condition-independent way) in the spirit of Brunner:2003dc ; DiPietro:2015zia
. The boundary term is precisely minus the total $\perp$-derivative from the
bulk Lagrangian in ${(0,2)}$-superspace (108):

$\widetilde{\Delta}_{\text{kin.}}\coloneqq-\tfrac{1}{4}\partial_{\perp}(\overline{\phi}\phi)-\tfrac{i}{2}\overline{\psi}_{+}\psi_{-}+\tfrac{i}{2}\overline{\psi}_{-}\psi_{+}.$
(111)

We see that the $-\tfrac{1}{4}\partial_{\perp}(\overline{\phi}\phi)$ cancels
the bulk total derivative in $x^{\perp}$ direction when pulled into the bulk.
This means that we can just drop
$\tfrac{1}{4}\partial^{2}(\overline{\phi}\phi)$ from the bulk and
$-\tfrac{1}{4}\partial_{\perp}(\overline{\phi}\phi)$ from the boundary
simultaneously, leaving us with bulk and boundary Lagrangians:

$\displaystyle\mathcal{L}^{B}$
$\displaystyle=-\partial_{\mu}\overline{\phi}\partial^{\mu}\phi+\tfrac{i}{2}(\psi\gamma^{\mu}\partial_{\mu}\overline{\psi})-\tfrac{i}{2}(\partial_{\mu}\psi\gamma^{\mu}\overline{\psi})$
(112) $\displaystyle\hskip
11.38092pt+\overline{F}F+W^{\prime}(\phi)F+\overline{W}^{\prime}(\overline{\phi})\overline{F}-\tfrac{1}{2}W^{\prime\prime}(\phi)\psi\psi+\tfrac{1}{2}\overline{W}^{\prime\prime}(\overline{\phi})\overline{\psi}\overline{\psi},$
$\displaystyle\Delta_{\text{kin.}}$
$\displaystyle=-\tfrac{i}{2}\overline{\psi}_{+}\psi_{-}+\tfrac{i}{2}\overline{\psi}_{-}\psi_{+}=-\tfrac{i}{2}\overline{\psi}\psi.$

For the bulk superpotential term, the supersymmetry variation yields

$\delta_{\text{sym}}\mathcal{L}_{W}=\partial_{\perp}\left(-i\int\mathrm{d}\theta^{+}\overline{\epsilon}W(\Phi)+\text{cc.}\right)+\partial_{+}(\dots)=\partial_{\perp}(-i\overline{\epsilon}\psi_{+}W^{\prime}(\phi)+\text{cc.})+\partial_{+}(\dots),$
(113)

where the right-hand side needs to be compensated. To do this in a boundary-
condition-independent way, one can use a bulk R-symmetry (see DiPietro:2015zia
), or one can add boundary degrees of freedom, which we will discuss in
detail.

####  4.2.3 Boundary Fermi multiplet and factorization

To compensate the superpotential term variation (113), we introduce a 2D
boundary Fermi multiplet with $E$\- and $J$-potential terms, analogously to
Kapustin:2003ga ; Brunner:2003dc ; Herbst:2008jq ; Lazaroiu:2003zi , where a
1D Fermi multiplet was used to compensate bulk 2D superpotential terms (see
also Yoshida:2014ssa ; Gadde:2013sca  for equivalent, three-dimensional
examples). The general superspace expansion of a 2D Fermi multiplet is given
by

$H=\eta-\sqrt{2}\theta^{+}G-2i\theta^{+}\overline{\theta}^{+}\partial_{+}\eta-\sqrt{2}\overline{\theta}^{+}E(\phi)+2\theta^{+}\overline{\theta}^{+}E^{\prime}(\phi)\psi_{+},$
(114)

so it has an $E$-potential of

$\overline{D}_{+}H=\sqrt{2}E(\Phi).$ (115)

The ${(0,2)}$-supersymmetry variation of the components is given by

$\displaystyle\delta_{\text{sym}}\eta$ $\displaystyle=-\sqrt{2}(\epsilon
G+\overline{\epsilon}E),$ (116) $\displaystyle\delta_{\text{sym}}G$
$\displaystyle=\sqrt{2}\overline{\epsilon}(2i\partial_{+}\eta-E^{\prime}\psi_{+}).$

Its (boundary) Lagrangian is

$\displaystyle\mathcal{L}_{H}$
$\displaystyle=\int\mathrm{d}^{2}\theta^{+}\tfrac{1}{2}\overline{H}H-\int\mathrm{d}\theta^{+}\tfrac{i}{\sqrt{2}}J(\Phi)H+\int\mathrm{d}\overline{\theta}^{+}\tfrac{i}{\sqrt{2}}\overline{J}(\overline{\Phi})\overline{H}$
(117)
$\displaystyle=i\overline{\eta}\partial_{+}\eta-i\partial_{+}\overline{\eta}\eta-E^{\prime}\overline{\eta}\psi_{+}-\overline{E}^{\prime}\overline{\psi}_{+}\eta+iJ^{\prime}\eta\psi_{+}-i\overline{J}^{\prime}\overline{\psi}_{+}\overline{\eta}-|E|^{2}-|J|^{2}.$

It consists of a kinetic term, two boundary potentials $E$ and $J$ of the
_bulk_ chiral field $\phi$, and interactions between the boundary and bulk
fermions. The boundary equations of motion are

$\overline{D}_{+}\overline{H}+\sqrt{2}iJ(\Phi)=0\Leftrightarrow\left\\{\begin{aligned}
G&=i\overline{J}\\\
2i\partial_{+}\eta&=E^{\prime}(\phi)\psi_{+}-i\overline{J}^{\prime}(\overline{\phi})\overline{\psi}_{+}\end{aligned}\right\\}.$
(118)

The supersymmetry variation is

$\delta_{\text{sym}}\mathcal{L}_{H}=i\int\mathrm{d}\theta^{+}\overline{\epsilon}J(\Phi)E(\Phi)+\text{cc.}+\partial_{+}(\dots).$
(119)

We thus find that in case of a _matrix factorization_

$W(\Phi)|_{\partial}=E(\Phi)J(\Phi)|_{\partial},$ (120)

the bulk term from (113) will be compensated precisely, and
${(0,2)}$-supersymmetry is preserved. As stated before Witten:1993yc  and can
be seen from (119), a pure 2D ${\mathcal{N}=(0,2)}$ theory must fulfill
$E\cdot J=0$ in order to preserve supersymmetry. However, in our case, the
“failure” of the boundary Fermi multiplet to meet this condition cancels the
failure of the bulk theory to preserve ${\mathcal{N}=(0,2)}$-supersymmetry at
the boundary.

The total action of the factorized Landau-Ginzburg model then reads

$\displaystyle S$ $\displaystyle=\int_{M}\mathcal{L}^{B}+\int_{\partial
M}\mathcal{L}^{\partial}$ (121)
$\displaystyle=\frac{1}{2}\int_{M}\bigg{\\{}\int\mathrm{d}^{2}\theta^{+}\big{[}i\overline{\Phi}\partial_{-}\Phi-i\partial_{-}\overline{\Phi}\Phi+\overline{\Psi}\Psi+\partial_{\perp}\Delta\big{]}-\sqrt{2}\int\mathrm{d}\theta^{+}\Psi
W(\Phi)+\text{cc.}\bigg{\\}}$ $\displaystyle\hskip
11.38092pt+\frac{1}{2}\int_{\partial
M}\bigg{\\{}\int\mathrm{d}^{2}\theta^{+}\big{[}-\Delta|_{\partial}+\overline{H}H\big{]}-\sqrt{2}i\int\mathrm{d}\theta^{+}J(\Phi)H+\text{cc.}\bigg{\\}},$

where
$\tfrac{1}{2}\int\mathrm{d}^{2}\theta^{+}\Delta=\tfrac{i}{2\sqrt{2}}\int\mathrm{d}^{2}\theta^{+}(\theta^{+}\overline{\Phi}\Psi+\overline{\theta}^{+}\overline{\Psi}\Phi)=\tfrac{i}{2}(\overline{\psi}_{+}\psi_{-}-\overline{\psi}_{-}\psi_{+})$
(cf. (112)). After using the algebraic equations of motion, we get the
following component expansions:

$\displaystyle\mathcal{L}^{B}$
$\displaystyle=-\partial_{\mu}\overline{\phi}\partial^{\mu}\phi+\tfrac{i}{2}(\psi\gamma^{\mu}\partial_{\mu}\overline{\psi})-\tfrac{i}{2}(\partial_{\mu}\psi\gamma^{\mu}\overline{\psi})$
(122) $\displaystyle\hskip
11.38092pt-|W(\phi)|^{2}-\tfrac{1}{2}W^{\prime\prime}(\phi)\psi\psi+\tfrac{1}{2}\overline{W}^{\prime\prime}(\overline{\phi})\overline{\psi}\overline{\psi},$
$\displaystyle\mathcal{L}^{\partial}$
$\displaystyle=i\overline{\eta}\partial_{+}\eta-i\partial_{+}\overline{\eta}\eta-|J|^{2}-|E|^{2}-\overline{E}^{\prime}\overline{\psi}_{+}\eta-E^{\prime}\overline{\eta}\psi_{+}$
$\displaystyle\hskip 11.38092pt-
iJ^{\prime}\psi_{+}\eta-i\overline{J}^{\prime}\overline{\psi}_{+}\overline{\eta}-\tfrac{i}{2}(\overline{\psi}_{+}\psi_{-}-\overline{\psi}_{-}\psi_{+})|_{\partial}.$

The ${(0,2)}$-variation of the total action is zero, hence
${\mathcal{N}=(0,2)}$ supersymmetry is preserved in a boundary-condition-
independent way.

####  4.2.4 Symmetric boundary conditions

In any theory with a boundary it is necessary to introduce boundary conditions
such that the action can be made stationary. Requiring the boundary condition
to be compatible with the ${\mathcal{N}=(0,2)}$ subalgebra in the sense of
(13) further restricts the number of options. We now discuss some explicit
boundary conditions for our LG model. We consider boundary conditions without
superpotential (previously discussed in Dimofte:2017tpi ) and with
superpotential separately.

##### Without superpotential

  * •

_(generalized) Dirichlet_ : $\Phi=0$ or more generally $\Phi=c$ (in components
$\phi=c$ and $\psi_{+}=0$) is a symmetric boundary condition (the action may
require the addition of some boundary terms to be symmetric).

  * •

_Neumann_ : $\Psi=0$ (in components $\partial_{\perp}\phi=0$ and $\psi_{-}=0$)
is also symmetric. It is also the dynamical boundary condition in the sense of
(12) for the action (112) without superpotential. Note that one can also
obtain the (generalized) Dirichlet as a dynamical boundary condition by adding
appropriate boundary terms Dimofte:2017tpi .

  * •

_Mixed conditions_ : In models with more than one 3D chiral superfield, we may
assign Dirichlet conditions to some and Neumann conditions to others
Dimofte:2017tpi .

##### With superpotential

  * •

_(generalized) Dirichlet_ : Setting $\Phi=c$ is symmetric and also statically
cancels the supervariation of the potential (113) (albeit in a boundary-
condition-_dependent_ way). However, if $W^{\prime}(c)|_{\partial}\neq 0$,
supersymmetry is broken spontaneously, as the vacuum expectation value of
$\psi_{-}$ then transforms non-trivially under supersymmetry.111111We note
that if there is a bulk $R$-symmetry, one can also compensate the
superpotential variation boundary-condition-independently, see
DiPietro:2015zia . However, in the case of one chiral field, one can then only
impose Dirichlet boundary conditions, as the Neumann condition with $W\neq 0$
is not symmetric.

  * •

_Mixed conditions_ : Setting $\Psi=0$ (Neumann) is only symmetric if
$W^{\prime}(\phi)|_{\partial}=0$. For one bosonic field, this holds only if
$W=0$, as $\phi$ is unconstrained on the boundary. If $W\neq 0$ and the theory
has more than one chiral superfield, one can assign Dirichlet conditions to
some and Neumann conditions to others while maintaining supersymmetry (a
requirement the authors in Dimofte:2017tpi  call “sufficiently Dirichlet”).

  * •

_Factorized Neumann_ : If we introduce additional degrees of freedom on the
boundary as in section 4.2.3, we may again choose dynamical boundary
conditions. In the case without superpotential this lead to the Neumann
boundary condition. For the action (122) the dynamical boundary condition is
the analog of the Neumann boundary condition, now with superpotential:

$\overline{\Psi}=-i\overline{H}E^{\prime}(\Phi)-HJ^{\prime}(\Phi)\Leftrightarrow\left\\{\begin{aligned}
\overline{\psi}_{-}&=-i\overline{\eta}E^{\prime}-\eta J^{\prime},\\\
\partial_{\perp}\overline{\phi}&=-\overline{E}E^{\prime}-\overline{J}J^{\prime}-(\overline{\eta}E^{\prime\prime}-i\eta
J^{\prime\prime})\psi_{+}\end{aligned}\right\\}.$ (123)

One can check that it is indeed symmetric if the factorization condition (120)
is met. We use this boundary condition in our computations for currents and
current multiplets. This choice of boundary condition in fact encodes a
collection of boundary conditions labeled by the choices of _matrix
factorizations_ of $W$ (since the boundary condition depends explicitly on $E$
and $J$).

###  4.3 Currents

Here we present conserved currents associated to the symmetries of the Landau-
Ginzburg theory with one chiral field in the bulk, a Fermi multiplet on the
boundary, and factorized Neumann boundary conditions. We will compute the
currents in various improvement frames in order to place them into consistent
multiplets in the following subsection.

####  4.3.1 $R$-current

If the superpotentials $W,E,J$ are (quasi-)homogeneous functions of $\Phi$ —
in the case of one chiral field, monomials —, then the action is invariant
under the $R$-symmetry transformation121212For multiple chiral fields
$\Phi_{i}$, the condition for quasi-homogeneity reads
$W(\Phi_{1},\ldots,\Phi_{k})=\sum_{i}\alpha_{i}\Phi_{i}\partial_{\Phi_{i}}W$
for some choice of $R$-charges $\alpha_{i}$.

$\displaystyle\theta^{+}$ $\displaystyle\mapsto e^{-i\tau}\theta^{+},$
$\displaystyle\Phi$ $\displaystyle\mapsto e^{-2i\tau\alpha}\Phi,$ (124)
$\displaystyle\Psi$ $\displaystyle\mapsto e^{-i\tau(2\alpha-1)}\Psi,$
$\displaystyle H$ $\displaystyle\mapsto e^{-i\tau(\ell_{E}-\ell_{J})\alpha}H,$

where $\tau$ is the symmetry variation parameter and we have defined

$\alpha\coloneqq(\deg W)^{-1},\quad\ell_{E}\coloneqq\deg
E,\quad\ell_{J}\coloneqq\deg J.$ (125)

Note that factorization implies

$\alpha(\ell_{E}+\ell_{J})=1.$ (126)

The bulk contribution to the $R$-current is given by

$j^{B}_{\mu}=2i\alpha(\overline{\phi}\partial_{\mu}\phi-\partial_{\mu}\overline{\phi}\phi)+(1-2\alpha)\overline{\psi}\gamma_{\mu}\psi,$
(127)

while the boundary contribution is given by

$j^{\partial}_{\hat{\mu}}=\begin{pmatrix}j^{\partial}_{+}\\\
j^{\partial}_{-}\end{pmatrix}=\begin{pmatrix}0\\\
\alpha(\ell_{E}-\ell_{J})\overline{\eta}\eta\end{pmatrix}.$ (128)

####  4.3.2 Supercurrents

After introducing the boundary (with the aforementioned choices), only
${(0,2)}$-supersymmetry is preserved. We may however still discuss the full 3D
$\mathcal{N}=2$ supersymmetry in the bulk, as the ${(0,2)}$-restrictions of
the bulk currents remain identical (and covariant notation can be conveniently
used).

##### Noether frame ($\mathcal{S}$-frame)

The bulk supercurrent induced by $\delta_{\text{sym}}=\epsilon Q$ (full
supersymmetry is $\delta_{\text{sym}}=\epsilon
Q-\overline{\epsilon}\overline{Q}$) in the Noether frame is given by:131313One
finds this supercurrent by applying Noether’s theorem to (112) and improving
the boundary part to zero.

$(S^{B}_{\mu})_{\alpha}=\sqrt{2}(\gamma^{\nu}\gamma_{\mu}\psi)_{\alpha}\partial_{\nu}\overline{\phi}-\sqrt{2}i(\gamma_{\mu}\overline{\psi})_{\alpha}\overline{W}^{\prime}.$
(129)

Its ${(0,2)}$-restriction $\delta_{\text{sym}}=\epsilon^{+}Q_{+}$ is given by
setting $\alpha=+$:

$(S^{B}_{\mu})_{+}=\begin{pmatrix}(S^{B}_{+})_{+}\\\ (S^{B}_{-})_{+}\\\
(S^{B}_{\perp})_{+}\end{pmatrix}=\begin{pmatrix}2\sqrt{2}\psi_{+}\partial_{+}\overline{\phi}\\\
-\sqrt{2}(\psi_{-}\partial_{\perp}\overline{\phi}+i\overline{\psi}_{-}\overline{W}^{\prime})\\\
\sqrt{2}(\psi_{+}\partial_{\perp}\overline{\phi}-2\psi_{-}\partial_{+}\overline{\phi}+i\overline{\psi}_{+}\overline{W}^{\prime})\end{pmatrix}.$
(130)

The boundary contribution is induced by
$\delta_{\text{sym}}=\epsilon^{+}Q_{+}$ and reads in the Noether frame:

$(S_{\hat{\mu}}^{\partial})_{+}=\begin{pmatrix}(S_{+}^{\partial})_{+}\\\
(S_{-}^{\partial})_{+}\end{pmatrix}=\begin{pmatrix}0\\\
-\sqrt{2}(\overline{J}\overline{\eta}-i\overline{E}\eta)\end{pmatrix}.$ (131)

#####  $\mathcal{R}$-frame

If the Lagrangian has an $R$-symmetry (124), we may improve the above
supercurrent to a supercurrent which is part of the $\mathcal{R}$-multiplet.
We call this improvement frame the $\mathcal{R}$-frame. The bulk components
are:

$\displaystyle(S_{\mu}^{B})^{\mathcal{R}}_{\alpha}$
$\displaystyle=(S_{\mu}^{B})^{\mathcal{S}}_{\alpha}-2\sqrt{2}\alpha\epsilon_{\mu\nu\rho}(\gamma^{\nu}\partial^{\rho}(\overline{\phi}\psi))_{\alpha}$
(132)
$\displaystyle=\sqrt{2}(1-2\alpha)\big{(}(\gamma^{\nu}\gamma_{\mu}\psi)_{\alpha}\partial_{\nu}\overline{\phi}+i(\gamma_{\mu}\overline{\psi})_{\alpha}\overline{W}^{\prime}\big{)}+2\sqrt{2}\alpha(\partial_{\mu}\overline{\phi}\psi_{\alpha}-\overline{\phi}\partial_{\mu}\psi_{\alpha}),$

where $(S_{\mu}^{B})^{\mathcal{S}}_{\alpha}$ denotes the supercurrent in the
Noether frame, $\alpha=(\deg W)^{-1}$ and the last equality uses equations of
motion (103) and homogeneity of $W$.

The boundary components are

$(S_{\hat{\mu}}^{\partial})^{\mathcal{R}}_{+}=(S_{\hat{\mu}}^{\partial})^{\mathcal{S}}_{+}+2\sqrt{2}\alpha\epsilon_{{\hat{\mu}}\nu
n}(\gamma^{\nu}\psi)_{+}\overline{\phi}=\begin{pmatrix}0\\\
\sqrt{2}\alpha(\ell_{J}-\ell_{E})(\overline{J}\overline{\eta}+i\overline{E}\eta)\end{pmatrix},$
(133)

where the last equality uses boundary conditions (123).

####  4.3.3 Energy-momentum tensor

Similarly to the case of supercurrents, we stick to covariant notation for the
bulk pieces, even though certain directions are no longer symmetries. Let us
start by simplifying the Lagrangians (122) on-shell:141414Note that the second
equation also uses boundary conditions (123). Without using them, we get
$\mathcal{L}^{\partial}\overset{\text{\tiny on-
shell}}{=}-|E|^{2}-|J|^{2}-\tfrac{1}{2}(\overline{E}^{\prime}\overline{\psi}_{+}\eta+E^{\prime}\overline{\eta}\psi_{+}+iJ^{\prime}\psi_{+}\eta+i\overline{J}^{\prime}\overline{\psi}_{+}\overline{\eta})-\tfrac{i}{2}(\overline{\psi}_{+}\psi_{-}-\overline{\psi}_{-}\psi_{+})|_{\partial}.$

$\displaystyle\mathcal{L}^{B}$ $\displaystyle\overset{\text{\tiny on-
shell}}{=}-\partial_{\rho}\overline{\phi}\partial^{\rho}\phi-|W^{\prime}|^{2},$
(134) $\displaystyle\mathcal{L}^{\partial}$ $\displaystyle\overset{\text{\tiny
on-shell}}{=}-|E|^{2}-|J|^{2}.$ (135)

##### Noether frame ($\mathcal{S}$-frame)

Using the Noether procedure, in the bulk we find the non-symmetric energy-
momentum tensor

$\displaystyle{\widehat{T}}_{\mu\nu}^{B}$
$\displaystyle=\partial_{\mu}\overline{\phi}\partial_{\nu}\phi+\partial_{\nu}\overline{\phi}\partial_{\mu}\phi+\tfrac{i}{2}\partial_{\mu}\overline{\psi}\gamma_{\nu}\psi-\tfrac{i}{2}\overline{\psi}\gamma_{\nu}\partial_{\mu}\psi-\eta_{\mu\nu}(\partial_{\rho}\overline{\phi}\partial^{\rho}\phi+|W^{\prime}|^{2}),$
(136)

and in the boundary we find (using equations of motion but _not_ boundary
conditions)

$\displaystyle{\widehat{T}}^{\partial}_{++}$ $\displaystyle=0,$ (137)
$\displaystyle{\widehat{T}}^{\partial}_{--}$
$\displaystyle=\tfrac{i}{2}\overline{\eta}\partial_{-}\eta-\tfrac{i}{2}\partial_{-}\overline{\eta}\eta,$
$\displaystyle{\widehat{T}}^{\partial}_{+-}$
$\displaystyle=\tfrac{i}{4}\overline{\psi}_{+}\psi_{-}-\tfrac{i}{4}\overline{\psi}_{-}\psi_{+}+\tfrac{1}{2}|E|^{2}+\tfrac{1}{2}|J|^{2}$
$\displaystyle\qquad+\tfrac{1}{2}\big{(}E^{\prime}\overline{\eta}\psi_{+}+\overline{E}^{\prime}\overline{\psi}_{+}\eta-
iJ^{\prime}\eta\psi_{+}+i\overline{J}^{\prime}\overline{\psi}_{+}\overline{\eta}\big{)},$
$\displaystyle{\widehat{T}}^{\partial}_{-+}$
$\displaystyle=\tfrac{i}{4}\overline{\psi}_{+}\psi_{-}-\tfrac{i}{4}\overline{\psi}_{-}\psi_{+}+\tfrac{1}{2}|E|^{2}+\tfrac{1}{2}|J|^{2}$
$\displaystyle\qquad+\tfrac{1}{4}\big{(}E^{\prime}\overline{\eta}\psi_{+}+\overline{E}^{\prime}\overline{\psi}_{+}\eta-
iJ^{\prime}\eta\psi_{+}+i\overline{J}^{\prime}\overline{\psi}_{+}\overline{\eta}\big{)}.$

If we utilize the boundary conditions (123), the expressions simplify to

$\displaystyle{\widehat{T}}_{++}^{\partial}$ $\displaystyle=0,$ (138)
$\displaystyle{\widehat{T}}_{--}^{\partial}$
$\displaystyle=\tfrac{i}{2}\overline{\eta}\partial_{-}\eta-\tfrac{i}{2}\partial_{-}\overline{\eta}\eta,$
$\displaystyle{\widehat{T}}_{+-}^{\partial}$
$\displaystyle=\tfrac{i}{2}\overline{\eta}\partial_{+}\eta-\tfrac{i}{2}\partial_{+}\overline{\eta}\eta+\tfrac{1}{2}(|E|^{2}+|J|^{2}),$
$\displaystyle{\widehat{T}}_{-+}^{\partial}$
$\displaystyle=\tfrac{1}{2}(|E|^{2}+|J|^{2}).$

##### Symmetrization

These can by made symmetric using an improvement. In the bulk we find

$\displaystyle T_{\mu\nu}^{B}$
$\displaystyle=\widehat{T}_{\mu\nu}^{B}-\tfrac{1}{8}\epsilon_{\mu\nu\rho}H^{\rho}$
(139)
$\displaystyle=(\partial_{\mu}\overline{\phi}\partial_{\nu}\phi+\partial_{\nu}\overline{\phi}\partial_{\mu}\phi)-\eta_{\mu\nu}(|\partial\phi|^{2}\\!+|W^{\prime}|^{2})+\tfrac{i}{2}(\partial_{(\mu}\overline{\psi}\gamma_{\nu)}\psi)-\tfrac{i}{2}(\overline{\psi}\gamma_{(\nu}\partial_{\mu)}\psi),$

where $H^{\rho}=-2i\partial^{\rho}(\overline{\psi}\psi)$.151515This is
precisely the brane current from the supercurrent multiplet, see appendix C.5.
To obtain the desired form for $T_{\mu\nu}^{B}$ we use equations of motion and
the Clifford algebra. The induced boundary improvement is
$T_{{\hat{\mu}}{\hat{\nu}}}^{\partial}=\widehat{T}_{{\hat{\mu}}{\hat{\nu}}}^{\partial}-\tfrac{i}{4}\epsilon_{{\hat{\mu}}{\hat{\nu}}n}\overline{\psi}\psi|_{\partial}$,
so

$\displaystyle T_{++}^{\partial}$ $\displaystyle=0,$ (140) $\displaystyle
T_{--}^{\partial}$
$\displaystyle=\tfrac{i}{2}\overline{\eta}\partial_{-}\eta-\tfrac{i}{2}\partial_{-}\overline{\eta}\eta,$
$\displaystyle T_{+-}^{\partial}$
$\displaystyle=\tfrac{i}{2}\overline{\eta}\partial_{+}\eta-\tfrac{i}{2}\partial_{+}\overline{\eta}\eta+\tfrac{1}{2}(|E|^{2}+|J|^{2})-\tfrac{i}{8}(\overline{\psi}_{-}\psi_{+}-\overline{\psi}_{+}\psi_{-})|_{\partial},$
$\displaystyle T_{-+}^{\partial}$
$\displaystyle=\tfrac{1}{2}(|E|^{2}+|J|^{2})+\tfrac{i}{8}(\overline{\psi}_{-}\psi_{+}-\overline{\psi}_{+}\psi_{-})|_{\partial}.$

Note that using boundary conditions (123) and equations of motion for $\eta$
(118) we find that

$\tfrac{i}{2}(\overline{\psi}_{-}\psi_{+}-\overline{\psi}_{+}\psi_{-})|_{\partial}=i\overline{\eta}\partial_{+}\eta-i\partial_{+}\overline{\eta}\eta,$
(141)

which shows that the boundary components are symmetric modulo boundary
conditions in this frame as well.

#####  $\mathcal{R}$-frame

Again, as in the case of the supercurrent, there is an improved energy-
momentum tensor in the $\mathcal{R}$-frame. We find

$\displaystyle(T_{\mu\nu}^{B})^{\mathcal{R}}$
$\displaystyle=(T_{\mu\nu}^{B})^{\mathcal{S}}+\tfrac{1}{2}[\partial_{\mu}\partial_{\nu}-\eta_{\mu\nu}\partial^{2}](-2\alpha\overline{\phi}\phi)$
(142)
$\displaystyle=(1-\alpha)(\partial_{\nu}\phi\partial_{\mu}\overline{\phi}+\partial_{\mu}\phi\partial_{\nu}\overline{\phi})-\alpha(\partial_{\mu}\partial_{\nu}\overline{\phi}\phi+\overline{\phi}\partial_{\mu}\partial_{\nu}\phi)+\tfrac{i}{2}(\partial_{(\nu}\overline{\psi}\gamma_{\mu)}\psi)$
$\displaystyle\phantom{{}={}}-\tfrac{i}{2}(\overline{\psi}\gamma_{(\mu}\partial_{\nu)}\psi)-(1-2\alpha)\eta_{\mu\nu}(|\partial\phi|^{2}-|W^{\prime}|^{2})+\alpha\eta_{\mu\nu}(i\psi\gamma^{\rho}\partial_{\rho}\overline{\psi}-i\partial_{\rho}\psi\gamma^{\rho}\overline{\psi})\big{)},$

where for the last equality we have used equations of motion. The boundary
contributions are given by
$(T^{\partial})_{{\hat{\mu}}{\hat{\nu}}}^{R}=(T^{\partial})_{{\hat{\mu}}{\hat{\nu}}}^{S}+\tfrac{1}{2}\eta_{{\hat{\mu}}{\hat{\nu}}}\partial_{\perp}(-2\alpha\overline{\phi}\phi)$,
hence

$\displaystyle(T^{\partial}_{++})^{\mathcal{R}}$ $\displaystyle=0,$ (143)
$\displaystyle(T^{\partial}_{--})^{\mathcal{R}}$
$\displaystyle=\tfrac{i}{2}\overline{\eta}\partial_{-}\eta-\tfrac{i}{2}\partial_{-}\overline{\eta}\eta,$
$\displaystyle(T^{\partial}_{+-})^{\mathcal{R}}$
$\displaystyle=\tfrac{i}{2}\overline{\eta}\partial_{+}\eta-\tfrac{i}{2}\partial_{+}\overline{\eta}\eta+\tfrac{1}{2}(|E|^{2}+|J|^{2})+\tfrac{\alpha}{2}\partial_{\perp}(\overline{\phi}\phi)|_{\partial}-\tfrac{i}{8}(\overline{\psi}_{-}\psi_{+}-\overline{\psi}_{+}\psi_{-})|_{\partial},$
$\displaystyle(T^{\partial}_{-+})^{\mathcal{R}}$
$\displaystyle=\tfrac{1}{2}(|E|^{2}+|J|^{2})+\tfrac{\alpha}{2}\partial_{\perp}(\overline{\phi}\phi)|_{\partial}+\tfrac{i}{8}(\overline{\psi}_{-}\psi_{+}-\overline{\psi}_{+}\psi_{-})|_{\partial}.$

Note that the symmetry of the boundary stress tensor (modulo boundary
conditions) was preserved by the improvement.

###  4.4 Supercurrent multiplets of the LG model

Let us now assemble the conserved currents of the Landau-Ginzburg model from
the previous subsection into supercurrent multiplets. We first recall the
supercurrent multiplets of a pure bulk theory, as well as its possible smaller
multiplets. After that we will present a valid supercurrent multiplet in the
Landau-Ginzburg model with boundary, and also discuss integrated supercurrent
multiplets.

####  4.4.1 Bulk theory

Here we study a pure bulk theory with Lagrangian
$\mathcal{L}=\mathcal{L}_{\text{kin.}}+\mathcal{L}_{W}$ as in (101), (102). In
such a theory a valid $\mathcal{S}$-multiplet is given by

$\mathcal{S}_{\alpha\beta}=D_{\alpha}\Phi_{3D}\overline{D}_{\beta}\overline{\Phi}_{3D}+D_{\beta}\Phi_{3D}\overline{D}_{\alpha}\overline{\Phi}_{3D}.$
(144)

It contains the supercurrent and energy-momentum tensor (in the
$\mathcal{S}$-frame) in its components. We explicitly compute the components
to verify this in the appendix (cf. (245)). The multiplet satisfies

$\overline{D}^{\alpha}\mathcal{S}_{\alpha\beta}=\underbrace{-D_{\beta}\Phi_{3D}\overline{D}^{2}\overline{\Phi}_{3D}}_{=\mathcal{Y}_{\beta}}+\underbrace{(-\tfrac{1}{2})\overline{D}^{2}D_{\beta}(\Phi_{3D}\overline{\Phi}_{3D})}_{=\chi_{\beta}}.$
(145)

Using the equations of motion (103), one may rewrite
$\mathcal{Y}_{\beta}=4D_{\beta}W(\Phi_{3D})$. The defining equations in (41)
can be verified easily, proving that this is indeed an
$\mathcal{S}$-multiplet. The central charge $C$ is zero. This
$\mathcal{S}$-multiplet can be improved to a Ferrara-Zumino multiplet using
the improvement $U_{\text{FZ}}=-\tfrac{1}{2}\overline{\Phi}_{3D}\Phi_{3D}$
(44), as this implies

$\chi_{\alpha}^{\prime}=-\tfrac{1}{2}\overline{D}^{2}D_{\alpha}(\overline{\Phi}_{3D}\Phi_{3D})-\overline{D}^{2}D_{\alpha}U=0.$
(146)

The multiplet is then given by

$\mathcal{J}_{\alpha\beta}=\tfrac{1}{2}(D_{\alpha}\Phi_{3D}\overline{D}_{\beta}\overline{\Phi}_{3D}+D_{\beta}\Phi_{3D}\overline{D}_{\alpha}\overline{\Phi}_{3D})+\tfrac{1}{2}(i\overline{\Phi}_{3D}\partial_{\alpha\beta}\Phi_{3D}-i\partial_{\alpha\beta}\overline{\Phi}_{3D}\Phi_{3D}).$
(147)

If the $R$-symmetry (cf. section 4.3.1) is present, one can instead apply the
improvement $U_{\mathcal{R}}=-2\alpha\overline{\Phi}_{3D}\Phi_{3D}$
($\alpha=(\deg W)^{-1}$) to the $\mathcal{S}$-multiplet, which sets
$\mathcal{Y}_{\alpha}$ to zero modulo equations of motion:

$\mathcal{Y}_{\alpha}^{\prime}=4D_{\alpha}W(\Phi_{3D})-\tfrac{1}{2}D_{\alpha}\overline{D}^{2}U=4D_{\alpha}W(\Phi_{3D})-4\alpha
D_{\alpha}(\Phi_{3D}W^{\prime}(\Phi_{3D}))=0.$ (148)

Now $\mathcal{S}_{\alpha\beta}$ transforms to

$\mathcal{R}_{\alpha\beta}=(1-2\alpha)(D_{\alpha}\Phi_{3D}\overline{D}_{\beta}\overline{\Phi}_{3D}+D_{\beta}\Phi_{3D}\overline{D}_{\alpha}\overline{\Phi}_{3D})+2\alpha(i\overline{\Phi}_{3D}\partial_{\alpha\beta}\Phi_{3D}-i\partial_{\alpha\beta}\overline{\Phi}_{3D}\Phi_{3D}).$
(149)

We see that the lowest component of this multiplet is exactly the $R$-current
(127), and one can check that the remaining currents in the
$\mathcal{R}$-multiplet are in the $\mathcal{R}$-frame.

####  4.4.2 Adding a boundary

Now that we have studied the bulk, let us go back to our Landau-Ginzburg
theory with a boundary and a boundary Fermi multiplets whose potentials
factorizes the superpotential (120). We want to extend the above bulk
supercurrent multiplet to a full (bulk and boundary) supercurrent multiplet as
described in section 3.3.

We already computed the bulk and boundary conserved currents in the sense of
(16) in various improvements frames in the previous subsection, and now have
to organize the components into admissible ${(0,2)}$-multiplets. We choose to
do so in the case of the $\mathcal{R}$-multiplet.

We consider the embedding Drukker:2017xrb  into 3D $\mathcal{N}=2$ superspace:

$\displaystyle\mathcal{R}^{B}_{\alpha\beta}$
$\displaystyle=\mathcal{R}^{B{(0)}}_{\alpha\beta}+\theta^{-}\mathcal{R}^{B{(1)}}_{\alpha\beta}-\overline{\theta}^{-}\overline{\mathcal{R}^{B{(1)}}_{\alpha\beta}}+\theta^{-}\overline{\theta}^{-}\mathcal{R}^{B{(2)}}_{\alpha\beta},$
(150) $\displaystyle\mathcal{R}^{\partial}_{\alpha\alpha}$
$\displaystyle=\mathcal{R}^{\partial{(0)}}_{\alpha\alpha}+\theta^{-}\underset{=0}{\underbrace{\mathcal{R}^{\partial{(1)}}_{\alpha\alpha}}}-\overline{\theta}^{-}\underset{=0}{\underbrace{\overline{\mathcal{R}^{\partial{(1)}}_{\alpha\alpha}}}}+\theta^{-}\overline{\theta}^{-}\mathcal{R}^{\partial{(2)}}_{\alpha\alpha}.$

First, we decompose the bulk contribution to the $\mathcal{R}$-multiplet into
its ${(0,2)}$-submultiplets. The zeroth-order bulk ${(0,2)}$-superfields are

$\displaystyle\mathcal{R}^{B{(0)}}_{++}$
$\displaystyle=8\alpha(i\overline{\Phi}\partial_{+}\Phi-i\partial_{+}\overline{\Phi}\Phi)-2(1-2\alpha)\overline{D}_{+}\overline{\Phi}D_{+}\Phi$
(151) $\displaystyle=4j_{+}^{B}+\ldots,$
$\displaystyle\mathcal{R}^{B{(0)}}_{--}$
$\displaystyle=8\alpha(i\overline{\Phi}\partial_{-}\Phi-i\partial_{-}\overline{\Phi}\Phi)-4(1-2\alpha)\overline{\Psi}\Psi$
$\displaystyle=4j_{-}^{B}+\ldots,$ $\displaystyle\mathcal{R}^{B{(0)}}_{+-}$
$\displaystyle=-4\alpha(i\overline{\Phi}\partial_{\perp}\Phi-i\partial_{\perp}\overline{\Phi}\Phi)-\sqrt{2}(1-2\alpha)(\overline{D}_{+}\overline{\Phi}\Psi+\overline{\Psi}D_{+}\Phi)$
$\displaystyle=-2j_{\perp}^{B}+\ldots.$

The first-order bulk ${(0,2)}$-superfields are

$\displaystyle\mathcal{R}_{++}^{B{(1)}}$
$\displaystyle=4(1-2\alpha)\big{(}i\partial_{\perp}\overline{\Phi}D_{+}\Phi-\overline{D}_{+}\overline{\Phi}\overline{W}^{\prime}(\overline{\Phi})\big{)}-8i\sqrt{2}\alpha(\partial_{+}\overline{\Phi}\Psi-\overline{\Phi}\partial_{+}\Psi)$
(152) $\displaystyle=-4i(S_{+}^{B})^{\mathcal{R}}_{-}+\ldots,$
$\displaystyle\mathcal{R}_{--}^{B{(1)}}$
$\displaystyle=-8i\sqrt{2}(1-\alpha)\Psi\partial_{-}\overline{\Phi}+8i\sqrt{2}\alpha\overline{\Phi}\partial_{-}\Psi$
$\displaystyle=-4i(S_{-}^{B})^{\mathcal{R}}_{-}+\ldots,$
$\displaystyle\mathcal{R}_{+-}^{B{(1)}}$
$\displaystyle=2\sqrt{2}i\big{(}\partial_{\perp}\overline{\Phi}\Psi-\overline{\Phi}\partial_{\perp}\Psi\big{)}+2(1-2\alpha)\big{(}iD_{+}\Phi\partial_{-}\overline{\Phi}-\sqrt{2}\overline{\Psi}\overline{W}^{\prime}(\overline{\Phi})+\sqrt{2}i\overline{\Phi}\partial_{\perp}\Psi\big{)}$
$\displaystyle=2i(S_{\perp}^{B})^{\mathcal{R}}_{-}+\ldots.$

The second-order bulk ${(0,2)}$-superfields are lengthy, but are a
straightforward ${(0,2)}$-completion of their lowest components:161616Recall
the general expansions (227c),(228c) and (229c), in particular the definition
of $K_{\mu\nu}$ (230).

$\displaystyle\mathcal{R}_{++}^{B{(2)}}$
$\displaystyle=-16\big{(}\partial_{+}\overline{\Phi}\partial_{-}\Phi+\partial_{-}\overline{\Phi}\partial_{+}\Phi+\alpha\partial_{+}\partial_{-}(\overline{\Phi}\Phi)-\tfrac{\alpha}{2}\partial_{\perp}^{2}(\overline{\Phi}\Phi)$
(153) $\displaystyle\hskip
28.45274pt-\tfrac{i}{4}\partial_{-}\overline{D}_{+}\overline{\Phi}D_{+}\Phi+\tfrac{i}{4}\overline{D}_{+}\overline{\Phi}\partial_{-}D_{+}\Phi-\tfrac{1}{2}\widetilde{\mathcal{L}^{B}}\big{)}$
$\displaystyle=-16\big{(}\alpha\partial_{+}\partial_{-}(\overline{\Phi}\Phi)+\tfrac{1}{2}\partial_{\perp}\overline{\Phi}\partial_{\perp}\Phi-\tfrac{\alpha}{2}\partial_{\perp}^{2}(\overline{\Phi}\Phi)+\tfrac{1}{2}|W^{\prime}(\Phi)|^{2}$
$\displaystyle\hskip
28.45274pt-\tfrac{i}{4}\partial_{-}\overline{D}_{+}\overline{\Phi}D_{+}\Phi+\tfrac{i}{4}\overline{D}_{+}\overline{\Phi}\partial_{-}D_{+}\Phi\big{)}$
$\displaystyle=-16(T_{-+}^{B})^{\mathcal{R}}+2(C^{B}_{+-})^{\mathcal{R}}+\ldots,$
$\displaystyle\mathcal{R}_{--}^{B{(2)}}$
$\displaystyle=-16\big{(}2\partial_{-}\overline{\Phi}\partial_{-}\Phi-\alpha\partial_{-}^{2}(\overline{\Phi}\Phi)-\tfrac{i}{2}\partial_{-}\overline{\Psi}\Psi+\tfrac{i}{2}\overline{\Psi}\partial_{-}\Psi\big{)}$
$\displaystyle=-16(T_{--}^{B})^{\mathcal{R}}+\ldots,$
$\displaystyle\mathcal{R}_{+-}^{B{(2)}}$
$\displaystyle=8\big{(}\partial_{-}\overline{\Phi}\partial_{\perp}\Phi+\partial_{\perp}\overline{\Phi}\partial_{-}\Phi-\alpha\partial_{-}\partial_{\perp}(\overline{\Phi}\Phi)$
$\displaystyle\hskip
11.38092pt+\tfrac{i}{2\sqrt{2}}(\partial_{-}\overline{D}_{+}\overline{\Phi}\Psi+\partial_{-}\Psi
D_{+}\Phi-\overline{D}_{+}\overline{\Phi}\partial_{\Psi}-\overline{\Psi}\partial_{-}D_{+}\Phi)\big{)}$
$\displaystyle=8(T_{-\perp}^{B})^{\mathcal{R}}-(C^{B}_{\perp-})^{\mathcal{R}}\ldots,$

where the lowest components are given by the energy-momentum tensor (142). The
brane current
$(C^{B}_{\mu\nu})^{\mathcal{R}}=\epsilon_{\mu\nu\rho}(H^{\rho})^{\mathcal{R}}$
in the $\mathcal{R}$-frame is given by
$H^{\mathcal{R}}_{\mu}=-2i(1-4\alpha)\partial_{\mu}(\overline{\psi}\psi)$
where we have used (45) and the explicit improvement $U_{\mathcal{R}}$. We
have also ${(0,2)}$-completed the bulk Lagrangian on-shell

$\widetilde{\mathcal{L}^{B}}=2\partial_{+}\overline{\Phi}\partial_{-}\Phi+2\partial_{-}\overline{\Phi}\partial_{+}\Phi-\partial_{\perp}\overline{\Phi}\partial_{\perp}\Phi-|W^{\prime}(\Phi)|^{2}.$
(154)

Note that one may also interpret the sum of the tensor and the brane current
as a non-symmetric energy-momentum tensor ${\widehat{T}}_{\mu\nu}^{B}$ (cf.
(139)).

For the zeroth component $\mathcal{R}^{\partial{(0)}}_{\hat{\mu}}$, we simply
${(0,2)}$-supersymmetrically complete the boundary $R$-current (128), where
again $\alpha=(\deg W)^{-1},\ell_{E}=\deg E$ and $\ell_{J}=\deg J$:

$\displaystyle\mathcal{R}^{\partial{(0)}}_{\hat{\mu}}$
$\displaystyle=\alpha(\ell_{E}-\ell_{J})\delta_{\hat{\mu}}^{-}\overline{H}H,$
(155)

or, in bispinor notation,

$\displaystyle\mathcal{R}^{\partial{(0)}}_{--}$
$\displaystyle=4\alpha(\ell_{E}-\ell_{J})\overline{H}H,$ (156)
$\displaystyle\mathcal{R}^{\partial{(0)}}_{++}$ $\displaystyle=0.$

Note that the ${(0,2)}$-completion $(\mathcal{R}^{\partial{(0)}})_{\hat{\mu}}$
of $(j^{\partial})_{\hat{\mu}}$ does not contain _all_ the boundary
contributions necessary: we need the boundary corrections $T^{\partial}_{--}$
to the energy-momentum tensor, which are not contained in our boundary
multiplet $(\mathcal{R}^{\partial{(0)}})_{++}$, as can be checked.171717This
can be directly verified by the explicit expansions (227)–(229): $T_{+-}$ and
$T_{++}$ are contained in the ${(0)}$-pieces, while $T_{--}$ is contained in
the ${(2)}$-piece. Hence, we must also compute the correction for the second-
order terms $(\mathcal{R}^{\partial{(2)}})_{\alpha\alpha}$:

$\displaystyle\mathcal{R}^{\partial{(2)}}_{++}$
$\displaystyle=8\widetilde{\mathcal{L}^{\partial}}-8\alpha\partial_{\perp}(\overline{\Phi}\Phi)+4\sqrt{2}i\alpha(\overline{D}_{+}\overline{\Phi}\Psi-\overline{\Psi}D_{+}\Phi)|_{\partial}$
(157)
$\displaystyle=-8|J(\Phi)|^{2}-8|E(\Phi)|^{2}-8\alpha\partial_{\perp}(\overline{\Phi}\Phi)+4\sqrt{2}i\alpha(\overline{D}_{+}\overline{\Phi}\Psi-\overline{\Psi}D_{+}\Phi)|_{\partial}$
$\displaystyle=-16(T^{\partial}_{-+})^{\mathcal{R}}+2C^{\partial}_{+-}+\ldots,$
$\displaystyle\mathcal{R}^{\partial{(2)}}_{--}$
$\displaystyle=8i\partial_{-}\overline{H}H-8i\overline{H}\partial_{-}H$
$\displaystyle=-16(T^{\partial}_{--})^{\mathcal{R}}+\ldots,$

where the boundary contribution
$(C^{\partial}_{{\hat{\mu}}{\hat{\nu}}})^{\mathcal{R}}$ to the brane current
$(C^{B}_{\mu\nu})^{\mathcal{R}}$ in the $\mathcal{R}$-frame can be found to be
$(C^{\partial}_{+-})^{\mathcal{R}}=-i(1-4\alpha)\overline{\psi}\psi$. It is
essentially the induced boundary improvement corresponding to symmetrization
of the energy-momentum tensor (cf. page 15), now in the $\mathcal{R}$-frame.
We have also ${(0,2)}$-completed the on-shell boundary Lagrangian:

$\displaystyle\widetilde{\mathcal{L}^{\partial}}$
$\displaystyle=-|J(\Phi)|^{2}-|E(\Phi)|^{2}.$ (158)

####  4.4.3 Integrated supercurrent multiplets

We now discuss integrated supercurrent multiplets as in section 3.4. The
integration along $x^{\perp}$ will make our Landau-Ginzburg model effectively
two-dimensional and we will recover genuine 2D ${\mathcal{N}=(0,2)}$
(integrated) supercurrent multiplets.

We thus find, according to (91):

$\displaystyle\mathcal{R}^{\text{int.}{(0)}}_{++}$
$\displaystyle=\int\mathrm{d}x^{\perp}\;\big{[}8\alpha(i\overline{\Phi}\partial_{+}\Phi-i\partial_{+}\overline{\Phi}\Phi)-2(1-2\alpha)\overline{D}_{+}\overline{\Phi}D_{+}\Phi\big{]},$
(159) $\displaystyle\mathcal{R}^{\text{int.}{(0)}}_{--}$
$\displaystyle=4\alpha(\ell_{E}-\ell_{J})\overline{H}H+\int\mathrm{d}x^{\perp}\;\big{[}8\alpha(i\overline{\Phi}\partial_{-}\Phi-i\partial_{-}\overline{\Phi}\Phi)-4(1-2\alpha)\overline{\Psi}\,\big{]},$

as well as

$\displaystyle\mathcal{R}^{\text{int.}{(2)}}_{++}$
$\displaystyle=-8|J(\Phi)|^{2}-8|E(\Phi)|^{2}+4\sqrt{2}i\alpha(\overline{D}_{+}\overline{\Phi}\Psi-\overline{\Psi}D_{+}\Phi)|_{\partial}$
(160) $\displaystyle\hskip
11.38092pt-16\int\mathrm{d}x^{\perp}\;\big{[}\alpha\partial_{+}\partial_{-}(\overline{\Phi}\Phi)+\tfrac{1}{2}\partial_{\perp}\overline{\Phi}\partial_{\perp}\Phi+\tfrac{1}{2}|W^{\prime}(\Phi)|^{2}$
$\displaystyle\hskip
56.9055pt-\tfrac{i}{4}\partial_{-}\overline{D}_{+}\overline{\Phi}D_{+}\Phi+\tfrac{i}{4}\overline{D}_{+}\overline{\Phi}\partial_{-}D_{+}\Phi\big{]},$
$\displaystyle\mathcal{R}^{\text{int.}{(2)}}_{--}$
$\displaystyle=8i\partial_{-}\overline{H}H-8i\overline{H}\partial_{-}H$
$\displaystyle\hskip
11.38092pt-16\int\mathrm{d}x^{\perp}\;\big{[}2\partial_{-}\overline{\Phi}\partial_{-}\Phi-\alpha\partial_{-}^{2}(\overline{\Phi}\Phi)-\tfrac{i}{2}\partial_{-}\overline{\Psi}\Psi+\tfrac{i}{2}\overline{\Psi}\partial_{-}\Psi\big{]}.$

Note that from a 2D perspective, the superfields
$\mathcal{R}^{\text{int.}{(0)}}_{++},\mathcal{R}^{\text{int.}{(0)}}_{--}$ and
$\mathcal{R}^{\text{int.}{(2)}}_{--}$ are enough to form a 2D supercurrent
multiplet.

After using equations of motion (103), (118), boundary conditions (123),
factorization condition (120) and homogeneity of superpotential terms, we find
that these integrated current multiplets indeed satisfy the relations

$\displaystyle\overline{D}_{+}\big{(}\mathcal{R}^{\text{int.}{(2)}}_{--}+2i\partial_{-}\mathcal{R}^{\text{int.}{(0)}}_{--}\big{)}$
$\displaystyle=0,$ (161)
$\displaystyle\overline{D}_{+}(\mathcal{R}^{\text{int.}{(2)}}_{++}+2i\partial_{-}\mathcal{R}^{\text{int.}{(0)}}_{++})$
$\displaystyle=0.$

which shows that the respective lowest components are
$\overline{Q}_{+}$-cohomology elements (cf. section 3.4).

##  5 Quantization

Similarly to DiPietro:2015zia , we can follow a canonical quantization
approach in this model and verify the supersymmetry conservation, the boundary
conditions, and the factorization condition in an independent way. We impose
the following canonical quantization conditions:

$\displaystyle[\partial_{0}\phi(x),\overline{\phi}(y)]$
$\displaystyle=-i\delta^{(2)}(x-y),$ (162a)
$\displaystyle\\{\overline{\psi}_{\alpha}(x),\psi_{\beta}(y)\\}$
$\displaystyle=-\gamma^{0}_{\alpha\beta}\delta^{(2)}(x-y),$ (162b)
$\displaystyle\\{\overline{\eta}(x),\eta(y)\\}$
$\displaystyle=\delta(x^{2}-y^{2}).$ (162c)

In general, the commutation relations are modified by the introduction of a
boundary. Here, however, we follow the point of view of the authors in
DiPietro:2015zia  and use the “naive” commutators even after introducing the
boundary. This can be justified by considering a full bulk theory first,
quantizing it, then introducing a boundary and studying the effect of the
boundary on the old bulk fields. Using this method, some properties of the
model can be verified independently of the approach in section 4. If we used
static boundary conditions and the respective modified commutators instead,
these properties would hold trivially. Notice that there are no singularities
when moving component fields of chiral multiplets to the boundary as a
consequence of supersymmetry.

The following relations hold in the bulk Dumitrescu:2011iu :

$\displaystyle\\{{\overline{\mathcal{Q}}}_{\alpha},S_{\beta\mu}\\}$
$\displaystyle=\gamma^{\nu}_{\alpha\beta}\big{(}2T_{\nu\mu}+\tfrac{1}{4}\epsilon_{\nu\mu\rho}H^{\rho}+i\partial_{\nu}j_{\mu}-i\eta_{\mu\nu}\partial_{\rho}j^{\rho}\big{)}+i\epsilon_{\alpha\beta}\epsilon_{\mu\nu\rho}\big{(}\tfrac{1}{4}F^{\nu\rho}+\partial^{\nu}j^{\rho}\big{)},$
(163) $\displaystyle\\{\mathcal{Q}_{\alpha},S_{\beta\mu}\\}$
$\displaystyle=\tfrac{1}{4}\overline{C}(\gamma_{\mu})_{\alpha\beta}+i\epsilon_{\mu\nu\rho}\gamma^{\nu}_{\alpha\beta}\overline{Y}^{\rho}.$
(164)

In the absence of a boundary, integration of these relations yields the
expected supersymmetry algebra. However, this changes under the introduction
of a boundary for two reasons: First, there are additional degrees of freedom
at the boundary which appear in $\mathcal{Q}$ and $S$, and second, there are
boundary contributions from pure _bulk_ terms as well.

Notice that the half-integrated commutators like
$\\{\mathcal{Q}_{\alpha},S_{\alpha\mu}\\}$ or