$\\{{\overline{\mathcal{Q}}}_{\alpha},S_{\beta\mu}\\}$ are affected by
improvements, but the fully integrated commutators like
$\\{\mathcal{Q}_{+},\mathcal{Q}_{+}\\}$ and
$\\{{\overline{\mathcal{Q}}}_{+},\mathcal{Q}_{+}\\}$ must be invariant under
them. This is easy to see in pure bulk theories, but with a boundary, it holds
as well. A generic commutator of a charge $\mathcal{Q}$ and a current
$J^{\mu}$ improves as follows:

$\\{\mathcal{Q},J^{\prime\mu}(x)\\}=\\{\mathcal{Q},J^{\mu}(x)\\}+\\{\mathcal{Q},\partial_{\nu}M^{[\mu\nu]}(x)\\}+\delta(x^{n})\\{\mathcal{Q},M^{n{\hat{\mu}}}(x)\\}.$
(165)

Notice that $\mathcal{Q}$ is a charge and thus invariant under improvements.
If $\mathcal{Q}$ commutes with $\partial_{\nu}$, we find that the integrated
algebra $\int\mathrm{d}x\\{\mathcal{Q},J^{\prime\mu}(x)\\}$ is unchanged.

###  5.1 General properties of the supercharge

Let us now restrict to the case $\alpha=+$, $\beta=+$ as the other supercharge
and -current will be broken by the introduction of the boundary. The
supercurrent from (129), (131) (which we repeat for convenience)

$S_{+\mu}(x)=-\sqrt{2}i(\gamma_{\mu}\overline{\psi})_{+}\overline{W}^{\prime}(\overline{\phi})+\sqrt{2}(\psi\gamma_{\mu}\gamma^{\nu})_{+}\partial_{\nu}\overline{\phi}-\sqrt{2}\,\delta(x^{\perp})\delta_{\mu}^{-}(\overline{J}\overline{\eta}-i\overline{E}\eta)$
(166)

integrates to the full supercharge

$\mathcal{Q}_{+}=\int_{\Sigma}\sqrt{2}\big{(}\psi_{-}\partial_{\perp}\overline{\phi}+i\overline{\psi}_{-}\overline{W}^{\prime}(\overline{\phi})-2\psi_{+}\partial_{+}\overline{\phi}\big{)}+\int_{\partial\Sigma}\sqrt{2}(\overline{J}\overline{\eta}-i\overline{E}\eta),$
(167)

where
$\int_{\Sigma}=\int_{\mathbb{R}\times(-\infty,0]}\mathrm{d}x^{2}\mathrm{d}x^{\perp}$
and $\int_{\partial\Sigma}=\int_{\mathbb{R}}\mathrm{d}x^{2}$. Let us start by
studying the action of the bulk part of $\mathcal{Q}_{+}$ on component fields.
Analogous to DiPietro:2015zia , the action of
$[\mathcal{Q}_{+,{\text{bulk}}},\boldsymbol{\cdot}]$ on bulk component fields
($\phi$, $\overline{\phi}$, $\psi_{\pm}$, $\overline{\psi}_{\pm}$) is the same
as in the pure bulk theory with one exception: The commutator with
$\partial_{0}\phi$ receives an extra boundary term

$[\mathcal{Q}_{+,{\text{bulk}}},\partial_{0}\phi(x)]=\sqrt{2}i\partial_{0}\psi_{+}(x)+\sqrt{2}i\delta(x^{\perp})\psi_{-}(x).$
(168)

This result can be derived using the quantization conditions (162) and the
delta distribution rule in appendix D. Notice that for $\nu\neq 0$, the
identity $\frac{\partial}{\partial
x^{\nu}}[\mathcal{Q}_{+},\phi(x)]=[\mathcal{Q}_{+},\partial_{\nu}\phi(x)]$ is
only true if the correct delta distribution rules derived in appendix D are
applied.

The boundary part $\mathcal{Q}_{+,{\text{bdy}}}$ has a trivial action on the
bulk component fields and on $\partial_{\nu}\phi$, $\nu\neq 0$. The action on
the boundary fermions $\eta$, $\overline{\eta}$ is as expected. Again, we get
extra terms for $\partial_{0}\phi$. Overall, the full charge acts as follows
on $\partial_{0}\phi$:

$[\mathcal{Q}_{+},\partial_{0}\phi(x)]=\sqrt{2}i\partial_{0}\psi+\sqrt{2}(i\overline{\eta}\overline{J}^{\prime}+\eta\overline{E}^{\prime}+i\psi_{-})\delta(x^{\perp}).$
(169)

We see that the boundary contribution vanishes under the symmetric boundary
condition (123), which is an independent way of verifying this boundary
condition. However, for reasons outlined above, we will not impose this
condition statically and thus treat this extra term like a genuine new
contribution.

###  5.2 The $\\{{\overline{\mathcal{Q}}},S\\}$ commutator

We now would like to compute $\\{{\overline{\mathcal{Q}}}_{+},S_{+\mu}(x)\\}$
in the presence of a boundary and verify that it integrates to the expected
preserved algebra $\\{{\overline{\mathcal{Q}}}_{+},\mathcal{Q}_{+}\\}=-4P_{+}$
(see section 3.3). We expect the known terms (163) in the bulk, and extra
terms at the boundary. As both ${\overline{\mathcal{Q}}}_{+}$ and
$S_{+\mu}(x)$ have bulk and boundary parts, there are four combinations from
which new terms may arise: bulk-bulk, bulk-boundary, boundary-bulk, and
boundary-boundary.

####  5.2.1 Boundary contributions from bulk-bulk terms

Let us now check how the changes introduced by the boundary affect the half-
integrated algebra: the bulk-bulk term

$\\{{\overline{\mathcal{Q}}}_{\alpha,{\text{bulk}}},S_{\beta\mu,{\text{bulk}}}\\}=\\{{\overline{\mathcal{Q}}}_{\alpha},-\sqrt{2}i(\gamma_{\mu}\overline{\psi})_{\beta}\overline{W}^{\prime}(\overline{\phi})+\sqrt{2}(\psi\gamma_{\mu}\gamma^{\nu})_{\beta}\partial_{\nu}\overline{\phi}\\}$
(170)

is affected by the changed relation (168) in the term
$\partial_{0}\overline{\phi}$, where we get an extra boundary term

$\\{{\overline{\mathcal{Q}}}_{\alpha,{\text{bulk}}},S_{\beta\mu,{\text{bulk}}}(x)\\}=\text{known
bulk terms \eqref{eq:bulk QbS
commutator}}+\delta(x^{\perp})\underbrace{2i(\psi\gamma_{\mu}\gamma^{0})_{\beta}(\gamma^{n}\gamma^{0}\overline{\psi}(x))_{\alpha}}_{=:B_{\alpha\beta,\mu}},$
(171)

which is also consistent with a similar result in DiPietro:2015zia . The
component most relevant to us is $B_{++}^{\phantom{+}\phantom{+}0}$, as it
appears in the integration of the part of the algebra that is unbroken. It can
be rewritten to

$B_{++}^{\phantom{+}\phantom{+}0}=2i\psi_{+}\overline{\psi}_{-}=i\overline{\psi}\psi-i\overline{\psi}\gamma^{\perp}\psi=i\overline{\psi}\psi-
ij^{\perp},$ (172)

where we have inserted a bulk component of the supercurrent multiplet in the
last equality, see appendix C.5. It is noteworthy that this boundary term is
neither real nor imaginary.

####  5.2.2 Contributions from the boundary degrees of freedom

Again using the commutation relations but not boundary conditions, we find

$\displaystyle\\{{\overline{\mathcal{Q}}}_{+,{\text{bdy}}},S_{+,\mu,{\text{bdy}}}(x)\\}$
$\displaystyle=-2\delta_{\mu}^{-}\delta(x^{\perp})(|J|^{2}+|E|^{2}),$ (173a)
$\displaystyle\\{{\overline{\mathcal{Q}}}_{+,{\text{bulk}}},S_{+,{\hat{\mu}},{\text{bdy}}}(x)\\}$
$\displaystyle=-2i\delta(x^{\perp})\delta_{\hat{\mu}}^{-}\overline{\psi}_{+}(x)\big{(}\overline{J}^{\prime}(x)\overline{\eta}(x)-i\overline{E}^{\prime}(x)\eta(x)\big{)},$
(173b)
$\displaystyle\\{{\overline{\mathcal{Q}}}_{+,{\text{bdy}}},S_{+,\mu,{\text{bulk}}}(x)\\}$
$\displaystyle=2i\delta(x^{\perp})\big{(}J^{\prime}(x)\eta(x)+iE^{\prime}(x)\overline{\eta}(x)\big{)}(\psi(x)\gamma_{\mu}\gamma^{0})_{+}.$
(173c)

It is noteworthy that modulo boundary conditions, the third term cancels the
boundary contribution of
$\\{{\overline{\mathcal{Q}}}_{+,{\text{bulk}}},S_{+,\mu,{\text{bulk}}}(x)\\}$.
This is expected since both extra terms have their origins in the changed
relation (169) which is identical to the original bulk relation modulo
boundary conditions.

####  5.2.3 Integrating the algebra

As the integral of $S_{+}^{\phantom{+}0}$ over a constant time slice yields
the supercharge $\mathcal{Q}_{+}$, an integration of the commutator
$\int\\{{\overline{\mathcal{Q}}}_{+},S_{+}^{\phantom{+}0}\\}=\\{{\overline{\mathcal{Q}}}_{+},\mathcal{Q}_{+}\\}$
is a commutator which appears in the preserved supersymmetry algebra (53). We
can thus check (171) and (173) by integrating
$\\{{\overline{\mathcal{Q}}}_{+},S_{+}^{\phantom{+}0}\\}$ and comparing the
result to the known algebra. We will plug in the component expansions from
appendix C.5.

Let us first check the imaginary part, which is zero on the expected right-
hand side of the equation. Interesting contributions come only from
$\\{{\overline{\mathcal{Q}}}_{+,{\text{bulk}}},S_{+,{\text{bulk}}}^{\phantom{+}0}\\}$,
as all other contributions together are trivially real.

$\operatorname{Im}\int_{\Sigma}\\{{\overline{\mathcal{Q}}}_{+},{S_{+}}^{0}\\}=\int_{\Sigma}2(i\partial_{+}j_{0}-i\eta_{+0}\partial^{\rho}j_{\rho})+\int_{\partial\Sigma}\operatorname{Im}(B_{++}^{\phantom{+}\phantom{+}0})=\int_{\Sigma}i\partial_{\perp}j_{\perp}-\int_{\partial\Sigma}ij_{\perp}=0.$
(174)

A similar computation in four dimensions was done in DiPietro:2015zia .

For the real part we get (as expected from (163))

$\\{{\overline{\mathcal{Q}}}_{+},S_{+}^{\phantom{+}0}\\}|_{\text{bulk}}=-\gamma_{22}^{\nu}\big{(}2T_{\nu
0}+\tfrac{1}{4}\epsilon_{\nu
0\rho}H^{\rho}\big{)}=-4({\widehat{T}}^{B})_{+}^{\phantom{+}0},$ (175)

where ${\widehat{T}}^{B}$ is the bulk Noether (non-symmetric) energy momentum
tensor (136). All boundary terms together (eqs. (171) and (173)) yield

$\\{{\overline{\mathcal{Q}}}_{+},S_{+}^{\phantom{+}0}\\}|_{\partial}=i\overline{\psi}\psi+2|J|^{2}+2|E|^{2}+2i\overline{\psi}_{+}\big{(}\overline{J}^{\prime}\overline{\eta}-i\overline{E}^{\prime}\eta\big{)}-2i\big{(}J^{\prime}\eta+iE^{\prime}\overline{\eta}\big{)}\psi_{+}=-4({\widehat{T}}^{\partial})_{+}^{\phantom{+}0}$
(176)

using an explicit comparison to the non-symmetric boundary energy momentum
tensor ${\widehat{T}}^{\partial}$ (137) which belongs to the bulk Noether
energy momentum tensor ${\widehat{T}}^{B}$. Overall, we find that

$\\{{\overline{\mathcal{Q}}}_{+},\mathcal{Q}_{+}\\}=\int_{\Sigma}\\{{\overline{\mathcal{Q}}}_{+},S_{+}^{\phantom{+}0}\\}=-4\int_{\Sigma}\big{(}({\widehat{T}}^{B})_{+}^{\phantom{+}0}+\delta(x^{\perp})({\widehat{T}}^{\partial})_{+}^{\phantom{+}0}\big{)}=-4P_{+},$
(177)

which verifies the algebra. Notice that $P_{+}$ is independent of
improvements, thus we may use improved versions of the energy-momentum tensor
to compute the right-hand side of the equality. However, on the level of the
half-integrated algebra, we see that the Noether ($\mathcal{S}$-frame)
supercurrent generates the non-symmetric Noether energy momentum tensor, and
both are sensitive to improvements.

Let us emphasize again that this argument works _without_ explicitly assuming
boundary conditions and modifying the bulk fields in the presence of a
boundary. Rather, we study the bulk theory without a boundary, then introduce
it, and verify that the supersymmetry algebra is preserved by the equal-time
commutators. Notice that without assuming any boundary conditions, the charges
$\mathcal{Q}_{+}$, ${\overline{\mathcal{Q}}}_{+}$ are not conserved. However,
no reference to the specific choice of boundary conditions was made in this
argument (although in this simple model with one bulk chiral field and one
boundary Fermi, (123) is the only symmetric boundary condition compatible with
stationarity).

###  5.3 The $\\{\mathcal{Q},S\\}$ commutator

In a similar way, we can also verify that $\\{\mathcal{Q}_{+},S_{+\mu}(x)\\}$
integrates to the expected algebra $\\{\mathcal{Q}_{+},\mathcal{Q}_{+}\\}=0$.
In the bulk, we get from (164)

$\\{\mathcal{Q}_{+,{\text{bulk}}},S_{+\mu,{\text{bulk}}}(x)\\}=i\epsilon_{\mu+\rho}(\gamma^{+})_{++}\overline{Y}^{\rho}=-8i\epsilon_{\mu+\rho}\partial^{\rho}\overline{W}.$
(178)

From the commutation relations (162) we find that
$\\{\mathcal{Q}_{\alpha,{\text{bulk}}},S_{\beta\mu,{\text{bdy}}}\\}$ and
$\\{\mathcal{Q}_{\alpha,{\text{bdy}}},S_{\beta\mu,{\text{bulk}}}\\}$ are zero,
but we do get a contribution from

$\\{\mathcal{Q}_{\alpha,{\text{bdy}}},S_{\beta\mu,{\text{bdy}}}(x)\\}=4i\delta_{\mu}^{-}\delta(x^{\perp})\overline{J}(x)\overline{E}(x).$
(179)

Integrating the relation, we find

$\\{\mathcal{Q}_{+},\mathcal{Q}_{+}\\}=\int_{\Sigma}\\{\mathcal{Q}_{+},S_{+}^{\phantom{+}0}(x)\\}=4i\int_{\partial\Sigma}\big{(}\overline{W}-\overline{J}(x)\overline{E}(x)\big{)},$
(180)

so if the factorization condition (120) is met, the algebra is preserved under
the introduction of the boundary.

##  6 Conclusions and Outlook

In this paper we analyzed three-dimensional theories with $\mathcal{N}=2$
supersymmetry from various points of view: We constructed the symmetry
currents by developing an extension of Noether’s procedure, we considered the
supermultiplets in the boundary situation and we commented on the
$\overline{Q}_{+}$-cohomology in a possible half-twisted theory. There are
various ways to extend our results. First of all, while we focused on the case
of $\mathcal{N}=2$ with specific boundary conditions, a similar analysis can
be performed for other dimensions and other supersymmetries. Furthermore, our
examples were limited to theories of chiral multiplets and it would be
interesting to couple these systems to a gauge group.

We considered only flat backgrounds, and it would be very interesting to
extend the discussion to three-dimensional theories with boundaries in curved
backgrounds. The partition function in such backgrounds has been studied
recently in Yoshida:2014ssa ; Jockers:2018sfl ; Cheng:2018vpl . Matrix
factorizations have already been discussed on curved manifolds with boundary
in Yoshida:2014ssa ; Jockers:2018sfl , see Aprile:2016gvn  for a discussion of
A-type boundary conditions on curved manifolds. The formulation of boundary
conditions must be compatible with the constraints imposed by the bulk
supersymmetry on the background geometry. In particular, the manifolds have to
admit transversely holomorphic foliations, see Closset:2012ru  for a
discussion of the geometry compatible with supersymmetry.

In theories without boundary, supercurrent multiplets have been very useful to
formulate theories in curved space. They also helped to understand the
(in-)dependence of the partition function of part of the data Closset:2013vra
. It would be interesting to see whether boundary supermultiplets can help to
draw similar conclusions in situations with boundaries.

In the example of Landau-Ginzburg models, we have exhibited in some detail the
symmetries of models involving matrix factorizations. In the case of _two-
dimensional_ Landau-Ginzburg models, matrix factorizations provided the key to
fully solve the theories in situations with boundaries, in the sense that the
full bulk and boundary spectrum and all correlation functions Kapustin:2003ga
were determined. It would be interesting to see to what extend these features
have analogs in three dimensions. As the theory cannot be fully twisted, one
would expect that a holomorphic dependence has to remain.

Finally, one could generalize boundary conditions to defect gluing conditions,
making potentially contact with Dimofte:2017tpi ; Drukker:2017dgn . In the
Landau-Ginzburg example it is easy to see that defects between theories with
different superpotentials involve factorizations of the difference of the two
superpotentials on the two sides of the defect.

###### Acknowledgements.

We thank Peter Mayr, Pantelis Fragkos, Tomáš Procházka, and Ismail Achmed-Zade
for valuable discussions and input. The authors were supported by the Deutsche
Forschungsgemeinschaft (DFG) grant “Defects and non-perturbative operators in
superconformal theories” and Exzellenzcluster Universe.
test.-1test.-1AppendicesAppendicestest.-1

##  Appendix A Conventions and notation

We mostly follow the notation and conventions of Dumitrescu:2011iu  which we
recall for convenience.

##### Spacetime

Our (half-)spacetime is given by

$M=\Set{(x^{0},x^{1},x^{2})}{x^{1}\leq 0},$ (181)

with mostly-plus metric $\eta_{\mu\nu}=\operatorname{diag}(-1,1,1)$. We most
frequently use light-cone coordinates

$x^{\pm}=x^{0}\pm x^{2},\quad x_{\pm}=\tfrac{1}{2}(x_{0}\pm x_{2}),\quad
x^{1}=x_{1}=x^{\perp},$ (182)

where the metric reads

$\eta_{\mu\nu}=\begin{pmatrix}0&-\tfrac{1}{2}&0\\\ -\tfrac{1}{2}&0&0\\\
0&0&1\end{pmatrix},\quad\eta^{\mu\nu}=\begin{pmatrix}0&-2&0\\\ -2&0&0\\\
0&0&1\end{pmatrix}.$ (183)

The Levi-Civita symbol is defined by $\epsilon_{012}=-1$, $\epsilon^{012}=1$.
In light-cone coordinates it is
$\epsilon_{+-\perp}=-\tfrac{1}{2},\;\epsilon^{+-\perp}=2$. It satisfies

$\displaystyle\epsilon_{\mu\nu\lambda}\epsilon^{\sigma\rho\lambda}$
$\displaystyle=\delta_{\mu}^{\phantom{\mu}\rho}\delta_{\nu}^{\phantom{\nu}\sigma}-\delta_{\mu}^{\phantom{\mu}\sigma}\delta_{\nu}^{\phantom{\nu}\rho},$
(184) $\displaystyle\epsilon_{\mu\rho\lambda}\epsilon^{\nu\rho\lambda}$
$\displaystyle=-2\delta_{\mu}^{\phantom{\mu}\nu}.$

##### Spinors

Spinors in 3D are $SL(2,\mathbb{R})$ fundamental representations, i.e. two
component spinors $\psi_{\alpha}$, $\alpha\in\\{1,2\\}=\\{-,+\\}$. Indices are
raised and lowered by $\epsilon_{\alpha\beta},\epsilon^{\alpha\beta}$, where
$\epsilon_{12}=-1,\epsilon^{12}=1$ according to the rule

$\psi^{\alpha}=\epsilon^{\alpha\beta}\psi_{\beta},\;\psi_{\alpha}=\epsilon_{\alpha\beta}\psi^{\beta}.$
(185)

Explicitly, we have

$\psi^{\alpha}=\begin{pmatrix}\psi^{-}\\\
\psi^{+}\end{pmatrix}=\begin{pmatrix}\psi_{+}\\\
-\psi_{-}\end{pmatrix},\quad\psi_{\alpha}=\begin{pmatrix}\psi_{-}\\\
\psi_{+}\end{pmatrix}=\begin{pmatrix}-\psi^{+}\\\ \psi^{-}\end{pmatrix}.$
(186)

Indices that are contracted “from top to bottom” are omitted:

$\psi\chi\coloneqq\psi^{\alpha}\chi_{\alpha}=\psi^{-}\chi_{-}+\psi^{+}\chi_{+}.$
(187)

Note that $\psi\chi=\chi\psi$. Since Hermitian conjugation flips the order of
spinors without flipping index position, we have that
$\overline{\psi\chi}=-\overline{\psi}\overline{\chi}$.

Some useful identities are given by

$\displaystyle\psi\psi$ $\displaystyle=2\psi^{+}\psi^{-},$ (188)
$\displaystyle\psi^{\alpha}\psi^{\beta}$
$\displaystyle=-\tfrac{1}{2}(\psi\psi)\epsilon^{\alpha\beta},$
$\displaystyle\psi_{\alpha}\psi_{\beta}$
$\displaystyle=\tfrac{1}{2}(\psi\psi)\epsilon_{\alpha\beta}.$

##### Clifford algebra

We use the real gamma matrices

$\gamma^{\mu}_{\alpha\beta}=(\gamma^{0}_{\alpha\beta},\gamma^{1}_{\alpha\beta},\gamma^{2}_{\alpha\beta})=(-\mathds{1},\sigma^{1},\sigma^{3}).$
(189)

In light-cone coordinates these read explicitly

$\gamma^{\mu}_{\alpha\beta}=(\gamma^{+}_{\alpha\beta},\gamma^{-}_{\alpha\beta},\gamma^{\perp}_{\alpha\beta})=\Bigg{(}\begin{pmatrix}0&0\\\
0&-2\end{pmatrix},\begin{pmatrix}-2&0\\\
0&0\end{pmatrix},\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}\Bigg{)}.$ (190)

They are symmetric $\gamma^{\mu}_{\alpha\beta}=\gamma^{\mu}_{\beta\alpha}$,
real and satisfy the Clifford algebra

$(\gamma^{\mu}\gamma^{\nu})_{\alpha}^{\phantom{\alpha}\beta}=\eta^{\mu\nu}\delta_{\alpha}^{\phantom{\alpha}\beta}+\epsilon^{\mu\nu\rho}(\gamma_{\rho})_{\alpha}^{\phantom{\alpha}\beta}.$
(191)

A useful list of identities follows from these:

$\displaystyle(\gamma^{\mu})_{\alpha\beta}(\gamma_{\mu})_{\gamma\delta}$
$\displaystyle=\epsilon_{\alpha\gamma}\epsilon_{\delta\beta}+\epsilon_{\alpha\delta}\epsilon_{\gamma\beta},$
(192) $\displaystyle(\gamma^{\mu}\gamma^{\rho}\gamma_{\mu})_{\alpha\beta}$
$\displaystyle=-(\gamma_{\rho})_{\alpha\beta},$ $\displaystyle
A_{\alpha}^{\phantom{\alpha}\beta}$
$\displaystyle=\tfrac{1}{2}(\operatorname{tr}A)\delta_{\alpha}^{\phantom{\alpha}\beta}+\tfrac{1}{2}\operatorname{tr}(\gamma_{\mu}A)(\gamma^{\mu})_{\alpha}^{\phantom{\alpha}\beta}.$

We may map vectors to bispinors and vice versa by

$\displaystyle v_{\alpha\beta}=-2\gamma^{\mu}_{\alpha\beta}v_{\mu},\quad
v_{\mu}=\tfrac{1}{4}\gamma^{\alpha\beta}_{\mu}v_{\alpha\beta}.$ (193)

These imply in particular that

$\displaystyle v_{\pm}$ $\displaystyle=\tfrac{1}{4}v_{\pm\pm},$ (194)
$\displaystyle v^{\pm}$ $\displaystyle=-\tfrac{1}{2}v^{\pm\pm},$
$\displaystyle v^{\perp}$
$\displaystyle=v_{\perp}=-\tfrac{1}{2}v_{+-}=-\tfrac{1}{2}v^{-+}.$

##### Integration

We fix the integration order/constants by

$\int\mathrm{d}^{2}\theta\;\theta^{2}=1,\quad\int\mathrm{d}^{2}\overline{\theta}\;\overline{\theta}^{2}=-1\quad\text{and}\quad\int\mathrm{d}\theta^{+}\mathrm{d}\overline{\theta}^{+}\overline{\theta}^{+}\theta^{+}=1$
(195)

i.e. adjacent symbols cancel in ${(0,2)}$-integration. These together imply

$\int\mathrm{d}^{2}\theta=\frac{1}{2}\int\mathrm{d}\theta^{-}\mathrm{d}\theta^{+},\quad\int\mathrm{d}^{2}\overline{\theta}=\frac{1}{2}\int\mathrm{d}\overline{\theta}^{+}\mathrm{d}\overline{\theta}^{-},$
(196)

and

$\int\mathrm{d}^{4}\theta=\frac{1}{4}\int\mathrm{d}^{2}\theta^{+}\mathrm{d}^{2}\theta^{-},$
(197)

where we have defined

$\int\mathrm{d}^{2}\theta^{\pm}=\int\mathrm{d}\theta^{\pm}\mathrm{d}\overline{\theta}^{\pm}.$
(198)

##### Supercharges and superderivatives

The supercharges of 3D $\mathcal{N}=2$ supersymmetry in terms of
$(x^{\mu},\theta,\overline{\theta})$-coordinates are

$\displaystyle Q_{\alpha}$
$\displaystyle=\tfrac{\partial}{\partial\theta^{\alpha}}+i(\gamma^{\mu}\overline{\theta})_{\alpha}\partial_{\mu}=\begin{pmatrix}\tfrac{\partial}{\partial\theta^{-}}+2i\overline{\theta}^{-}\partial_{-}-i\overline{\theta}^{+}\partial_{\perp}\\\
\tfrac{\partial}{\partial\theta^{+}}+2i\overline{\theta}^{+}\partial_{+}-i\overline{\theta}^{-}\partial_{\perp}\end{pmatrix},$
(199) $\displaystyle\overline{Q}_{\alpha}$
$\displaystyle=-\tfrac{\partial}{\partial\overline{\theta}^{\alpha}}-i(\gamma^{\mu}\theta)_{\alpha}\partial_{\mu}=\begin{pmatrix}-\tfrac{\partial}{\partial\overline{\theta}^{-}}-2i\theta^{-}\partial_{-}+i\theta^{+}\partial_{\perp}\\\
-\tfrac{\partial}{\partial\overline{\theta}^{+}}-2i\theta^{+}\partial_{+}+i\theta^{-}\partial_{\perp}\end{pmatrix}.$

The covariant derivatives are

$\displaystyle D_{\alpha}$
$\displaystyle=\tfrac{\partial}{\partial\theta^{\alpha}}-i(\gamma^{\mu}\overline{\theta})_{\alpha}\partial_{\mu}=\begin{pmatrix}\tfrac{\partial}{\partial\theta^{-}}-2i\overline{\theta}^{-}\partial_{-}+i\overline{\theta}^{+}\partial_{\perp}\\\
\tfrac{\partial}{\partial\theta^{+}}-2i\overline{\theta}^{+}\partial_{+}+i\overline{\theta}^{-}\partial_{\perp}\end{pmatrix},$
(200) $\displaystyle\overline{D}_{\alpha}$
$\displaystyle=-\tfrac{\partial}{\partial\overline{\theta}^{\alpha}}+i(\gamma^{\mu}\theta)_{\alpha}\partial_{\mu}=\begin{pmatrix}-\tfrac{\partial}{\partial\overline{\theta}^{-}}+2i\theta^{-}\partial_{-}-i\theta^{+}\partial_{\perp}\\\
-\tfrac{\partial}{\partial\overline{\theta}^{+}}+2i\theta^{+}\partial_{+}-i\theta^{-}\partial_{\perp}\end{pmatrix}.$

They satisfy the known algebra

$\displaystyle\\{Q_{\alpha},\overline{Q}_{\beta}\\}=2i\gamma^{\mu}_{\alpha\beta}\partial_{\mu},\quad\\{D_{\alpha},\overline{D}_{\beta}\\}=-2i\gamma^{\mu}_{\alpha\beta}\partial_{\mu}.$
(201)

##  Appendix B Decomposition of 3D $\mathcal{N}=2$

###  B.1 Decomposition to 2D ${\mathcal{N}=(0,2)}$

We may constructively decompose 3D $\mathcal{N}=2$ superfields and operators
in their ${(0,2)}$-components. To do so, we use the _branching coordinate_
$\xi$.181818$\xi^{\perp}$ is also called “invariant coordinate” in
Drukker:2017xrb ; Drukker:2017dgn . It is chosen such that in superspace with
coordinates $(\xi^{\mu},\theta^{+},\theta^{-})$, the representations of the
preserved supercharge operators $Q_{+}$ and $\overline{Q}_{+}$ commute with
$\theta^{-}$ and $\overline{\theta}^{-}$. Then $Q_{+}$ and $\overline{Q}_{+}$
can be restricted to the sub-superspace without $\theta^{-}$. Another property
of the branching coordinate is that the preserved supercharge operators do not
contain or generate $P_{\perp}$.

If we want to preserve the ${(0,2)}$-subalgebra generated by
$Q_{+},\overline{Q}_{+}$, one can easily check that we need

$\displaystyle\xi^{\mu}=\big{(}x^{+},x^{-},x^{\perp}+i(\theta^{+}\overline{\theta}^{-}-\theta^{-}\overline{\theta}^{+})\big{)}.$
(202)

Indeed, one can check that in terms of $\xi^{\mu}$ the operators (199), (200)
take the following form

$\displaystyle Q_{+}$
$\displaystyle=\tfrac{\partial}{\partial\theta^{+}}+2i\overline{\theta}^{+}\tfrac{\partial}{\partial\xi^{+}},$
$\displaystyle Q_{-}$
$\displaystyle=\tfrac{\partial}{\partial\theta^{-}}+2i\overline{\theta}^{-}\tfrac{\partial}{\partial\xi^{-}}-2i\overline{\theta}^{+}\tfrac{\partial}{\partial\xi^{\perp}},$
(203) $\displaystyle\overline{Q}_{+}$
$\displaystyle=-\tfrac{\partial}{\partial\overline{\theta}^{+}}-2i\theta^{+}\tfrac{\partial}{\partial\xi^{+}},$
$\displaystyle\overline{Q}_{-}$
$\displaystyle=-\tfrac{\partial}{\partial\overline{\theta}^{-}}-2i\theta^{-}\tfrac{\partial}{\partial\xi^{-}}+2i\theta^{+}\tfrac{\partial}{\partial\xi^{\perp}},$
$\displaystyle D_{+}$
$\displaystyle=\tfrac{\partial}{\partial\theta^{+}}-2i\overline{\theta}^{+}\tfrac{\partial}{\partial\xi^{+}}+2i\overline{\theta}^{-}\tfrac{\partial}{\partial\xi^{\perp}},$
$\displaystyle D_{-}$
$\displaystyle=\tfrac{\partial}{\partial\theta^{-}}-2i\overline{\theta}^{-}\tfrac{\partial}{\partial\xi^{-}},$
$\displaystyle\overline{D}_{+}$
$\displaystyle=-\tfrac{\partial}{\partial\overline{\theta}^{+}}+2i\theta^{+}\tfrac{\partial}{\partial\xi^{+}}-2i\theta^{-}\tfrac{\partial}{\partial\xi^{\perp}},$
$\displaystyle\overline{D}_{-}$
$\displaystyle=-\tfrac{\partial}{\partial\overline{\theta}^{-}}+2i\theta^{-}\tfrac{\partial}{\partial\xi^{-}}.$

In particular, the $Q_{+},\overline{Q}_{+}$ do not contain any
$\partial_{\perp}$ terms. The ${(0,2)}$-covariant derivatives are defined by

$\displaystyle D_{+}^{(0,2)}$
$\displaystyle=\tfrac{\partial}{\partial\theta^{+}}-2i\overline{\theta}^{+}\tfrac{\partial}{\partial\xi^{+}},$
(204) $\displaystyle\overline{D}_{+}^{(0,2)}$
$\displaystyle=-\tfrac{\partial}{\partial\overline{\theta}^{+}}+2i\theta^{+}\tfrac{\partial}{\partial\xi^{+}},$

so we have that

$\displaystyle D_{+}$
$\displaystyle=D_{+}^{(0,2)}+2i\overline{\theta}^{-}\partial_{\perp},$ (205)
$\displaystyle\overline{D}_{+}$
$\displaystyle=\overline{D}_{+}^{(0,2)}-2i\theta^{-}\partial_{\perp}.$

We often drop the ${(0,2)}$-label when the covariant derivative type is clear
from context.
We may now simply perform a Taylor expansion of an $\mathcal{N}=2$ superfield
$\mathcal{X}$:

$\mathcal{X}(x,\theta,\overline{\theta})=X^{(0)}(\xi,\theta^{+},\overline{\theta}^{+})+\theta^{-}X^{(1a)}(\xi,\theta^{+},\overline{\theta}^{+})+X^{(1b)}(\xi,\theta^{+},\overline{\theta}^{+})\;\overline{\theta}^{-}+\theta^{-}\overline{\theta}^{-}X^{(2)}(\xi,\theta^{+},\overline{\theta}^{+}).$
(206)

It is clear that the ${(0,2)}$-operators
$\\{Q_{+},\overline{Q}_{+},D_{+}^{(0,2)},\overline{D}_{+}^{(0,2)}\\}$ do not
“mix” the coefficients of different orders in
$\theta^{-},\overline{\theta}^{-}$. In other words, the coefficients are
exactly the ${(0,2)}$-subrepresentations of
$\mathcal{X}(x,\theta,\overline{\theta})$.

As an example, let us decompose the 3D chiral field (99) to its
${(0,2)}$-submultiplets. In terms of $\xi$ we find that

$(y^{+},y^{-},y^{\perp})=(\xi^{+}-2i\theta^{+}\overline{\theta}^{-},\xi^{-}-2i\theta^{-}\overline{\theta}^{-},\xi^{\perp}+2i\theta^{-}\overline{\theta}^{+}).$
(207)

The expansion of $\Phi_{3D}$ gives then

$\Phi_{3D}(x,\theta,\overline{\theta})=\Phi(\xi,\theta^{+},\overline{\theta}^{+})-2i\theta^{-}\overline{\theta}^{-}\partial_{-}\Phi(\xi,\theta^{+},\overline{\theta}^{+})+\sqrt{2}\theta^{-}\Psi(\xi,\theta^{+},\overline{\theta}^{+}),$
(208)

where the chiral and Fermi multiplets are

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

in agreement with Witten:1993yc . These satisfy

$\overline{D}_{+}\Phi=0,\quad\overline{D}_{+}\Psi=-i\sqrt{2}\partial_{\perp}\Phi.$
(210)

We can obtain the full expansion using (202) on the right-hand side of (208)

$\Phi_{3D}=\Phi+\sqrt{2}\theta^{-}\Psi+i(\theta^{+}\overline{\theta}^{-}-\theta^{-}\overline{\theta}^{+})\partial_{\perp}\Phi-2i\theta^{-}\overline{\theta}^{-}\partial_{-}\Phi-\sqrt{2}i\theta^{+}\theta^{-}\overline{\theta}^{-}\partial_{\perp}\Psi-\theta^{+}\overline{\theta}^{+}\theta^{-}\overline{\theta}^{-}\partial_{\perp}^{2}\Phi,$
(211)

where now all (super-)functions depend on $x$.

###  B.2 Decomposition to 2D $\mathcal{N}={(1,1)}$

In the presence of a boundary we may also preserve a 2D $\mathcal{N}={(1,1)}$
subalgebra. For completeness, we also present decomposition of 3D
$\mathcal{N}=2$ superfields and operators into this subalgebra.
We want to preserve the ${(1,1)}$-subalgebra generated by191919Note that in
breaking from a 2D $(2,2)$ algebra to 2D $(1,1)$ we have some freedom of
choice in “picking phases”, i.e. we may choose
$\mathbbm{Q}_{i}\coloneqq\tfrac{1}{2}(e^{iv_{i}}Q_{i}+e^{-iv_{i}}\overline{Q}_{i}),$
for $i=1,2$ (see e.g. Hori:2000ic ; Hori:2003ic ). However, in 3D
$\mathcal{N}=2$ to $(1,1)$-subalgebra breaking, we further need to impose that
the charges do not generate any $\partial_{\perp}$-derivatives, i.e. that
$\\{\mathbbm{Q}_{1},\mathbbm{Q}_{2}\\}=\tfrac{i}{2}\operatorname{Re}(e^{i(v_{1}-v_{2})})\partial_{\perp}\overset{!}{=}0,$
which fixes the phase to $v_{1}-v_{2}=\tfrac{\pi}{2}+k\pi$.

$\displaystyle\mathbbm{Q}_{-}$
$\displaystyle\coloneqq\tfrac{1}{2}(Q_{-}+\overline{Q}_{-}),$ (212)
$\displaystyle\mathbbm{Q}_{+}$
$\displaystyle\coloneqq\tfrac{i}{2}(Q_{+}-\overline{Q}_{+}).$

These satisfy the ${(1,1)}$-algebra:

$\displaystyle\\{\mathbbm{Q}_{-},\mathbbm{Q}_{-}\\}$
$\displaystyle=-2i\partial_{-},$ (213)
$\displaystyle\\{\mathbbm{Q}_{+},\mathbbm{Q}_{+}\\}$
$\displaystyle=-2i\partial_{+},$
$\displaystyle\\{\mathbbm{Q}_{-},\mathbbm{Q}_{+}\\}$ $\displaystyle=0.$

Defining rotated, real Grassmann variables

$\displaystyle\vartheta^{-}$
$\displaystyle=-i(\theta^{-}-\overline{\theta}^{-}),$
$\displaystyle\tilde{\vartheta}^{-}$
$\displaystyle=-(\theta^{-}+\overline{\theta}^{-}),$ (214)
$\displaystyle\vartheta^{+}$
$\displaystyle=-(\theta^{+}+\overline{\theta}^{+}),$
$\displaystyle\tilde{\vartheta}^{+}$
$\displaystyle=-i(\theta^{+}-\overline{\theta}^{+}).$

and using the shifted “branching” coordinate

$\displaystyle(\zeta^{+},\zeta^{-},\zeta^{\perp})$
$\displaystyle=\big{(}x^{+},x^{-},x^{\perp}+\tfrac{i}{2}(\vartheta^{-}\tilde{\vartheta}^{+}+\vartheta^{+}\tilde{\vartheta}^{-})\big{)}$
(215)
$\displaystyle=\big{(}y^{+}-\vartheta^{+}\tilde{\vartheta}^{+},y^{-}+\vartheta^{-}\tilde{\vartheta}^{-},y^{\perp}-\tfrac{1}{2}(\vartheta^{-}\vartheta^{+}-i\vartheta^{-}\tilde{\vartheta}^{+}-i\vartheta^{+}\tilde{\vartheta}^{-}-\tilde{\vartheta}^{-}\tilde{\vartheta}^{+})\big{)},$

the supercharges take the form

$\displaystyle\mathbbm{Q}_{-}$
$\displaystyle=-i\tfrac{\partial}{\partial\vartheta^{-}}+\vartheta^{-}\tfrac{\partial}{\partial\zeta^{-}},$
(216) $\displaystyle\mathbbm{Q}_{+}$
$\displaystyle=-i\tfrac{\partial}{\partial\vartheta^{+}}+\vartheta^{+}\tfrac{\partial}{\partial\zeta^{+}}.$

We can then define also the ${(1,1)}$-covariant derivatives

$\displaystyle\mathbbm{D}_{-}$
$\displaystyle=-i\tfrac{\partial}{\partial\vartheta^{-}}-\vartheta^{-}\tfrac{\partial}{\partial\zeta^{-}},$
(217) $\displaystyle\mathbbm{D}_{+}$
$\displaystyle=-i\tfrac{\partial}{\partial\vartheta^{+}}-\vartheta^{+}\tfrac{\partial}{\partial\zeta^{+}}.$

which of course satisfy

$\displaystyle\\{\mathbbm{D}_{\pm},\mathbbm{D}_{\pm}\\}$
$\displaystyle=2i\partial_{\pm},$ (218)
$\displaystyle\\{\mathbbm{D}_{+},\mathbbm{D}_{-}\\}$ $\displaystyle=0.$

Then, ${(1,1)}$-irreducible multiplets are of the form

$\Sigma=A+i\vartheta^{+}B+i\vartheta^{-}C+i\vartheta^{-}\vartheta^{+}D.$ (219)

Using the branching coordinate we find the decomposition

$\Phi_{3D}(x,\theta,\overline{\theta})=\varPhi+\tilde{\vartheta}^{+}\mathbbm{D}_{+}\varPhi-\tilde{\vartheta}^{-}\mathbbm{D}_{-}\varPhi+\tilde{\vartheta}^{-}\tilde{\vartheta}^{+}(\mathbbm{D}_{-}\mathbbm{D}_{+}\varPhi-\partial_{\perp}\varPhi)$
(220)

where on the right-hand side we have super(functions) of
$(\zeta,\vartheta,\tilde{\vartheta})$ and we have defined the
${(1,1)}$-multiplet

$\varPhi\coloneqq\phi-\tfrac{1}{\sqrt{2}}\vartheta^{+}\psi_{+}+\tfrac{i}{\sqrt{2}}\vartheta^{-}\psi_{-}+\tfrac{1}{2}\vartheta^{-}\vartheta^{+}(\partial_{\perp}\phi+iF).$
(221)

Around the same point, the full branching reads

$\displaystyle\Phi_{3D}$
$\displaystyle=\varPhi+\tilde{\vartheta}^{+}\mathbbm{D}_{+}\varPhi-\tilde{\vartheta}^{-}\mathbbm{D}_{-}\varPhi+\tfrac{i}{2}(\vartheta^{-}\tilde{\vartheta}^{+}+\vartheta^{+}\tilde{\vartheta}^{-})\partial_{\perp}\varPhi+\tilde{\vartheta}^{-}\tilde{\vartheta}^{+}(\mathbbm{D}_{-}\mathbbm{D}_{+}\varPhi-\partial_{\perp}\varPhi)$
(222) $\displaystyle\hskip
11.38092pt+\tfrac{i}{2}\tilde{\vartheta}^{-}\tilde{\vartheta}^{+}\vartheta^{+}\partial_{\perp}\mathbbm{D}_{+}\varPhi+\tfrac{i}{2}\tilde{\vartheta}^{-}\tilde{\vartheta}^{+}\vartheta^{-}\partial_{\perp}\mathbbm{D}_{-}\varPhi+\tfrac{1}{4}\vartheta^{4}\partial_{\perp}^{2}\varPhi,$

where
$\vartheta^{4}\coloneqq\vartheta^{+}\tilde{\vartheta}^{+}\vartheta^{-}\tilde{\vartheta}^{-}$.
The ${(1,1)}$-variations are induced by
$\delta_{\text{sym}}=i\varepsilon_{+}\mathbbm{Q}_{+}+i\varepsilon_{-}\mathbbm{Q}_{-}$,
$\varepsilon_{\pm}$ real spinors:

$\displaystyle\delta_{\text{sym}}\phi$
$\displaystyle=-\tfrac{1}{\sqrt{2}}\varepsilon_{+}\psi_{+}+\tfrac{i}{\sqrt{2}}\varepsilon_{-}\psi_{-},$
(223) $\displaystyle\delta_{\text{sym}}\psi_{+}$
$\displaystyle=\sqrt{2}i\varepsilon_{+}\partial_{+}\phi+\tfrac{1}{\sqrt{2}}\varepsilon_{-}(\partial_{\perp}\phi+iF),$
$\displaystyle\delta_{\text{sym}}\psi_{-}$
$\displaystyle=-\tfrac{i}{\sqrt{2}}\varepsilon_{+}(\partial_{\perp}\phi+iF)-\sqrt{2}\varepsilon_{-}\partial_{-}\phi,$
$\displaystyle\delta_{\text{sym}}iF$
$\displaystyle=\sqrt{2}\varepsilon_{+}(\partial_{+}\psi_{-}+\tfrac{1}{2}\partial_{\perp}\psi_{+})-\sqrt{2}i\varepsilon_{-}(\partial_{-}\psi_{+}+\tfrac{1}{2}\partial_{\perp}\psi_{-}).$

The 3D $\mathcal{N}=2$ bulk Lagrangian of one chiral field $\Phi_{3D}$ is
decomposed as follows

$\displaystyle\mathcal{L}$
$\displaystyle=\int\mathrm{d}^{4}\theta\big{(}\overline{\Phi}_{3D}\Phi_{3D}-(\theta\theta)(\overline{\theta}\overline{\theta})\partial_{+}\partial_{-}(\overline{\Phi}_{3D}\Phi_{3D})\big{)}+\int\mathrm{d}^{2}\theta
W(\Phi_{3D})+\int\mathrm{d}^{2}\overline{\theta}\overline{W}(\overline{\Phi}_{3D})$
(224)
$\displaystyle=\int\mathrm{d}^{2}\vartheta\Big{[}2(\mathbbm{D}_{-}\overline{\varPhi}\mathbbm{D}_{+}\varPhi-\mathbbm{D}_{+}\overline{\varPhi}\mathbbm{D}_{-}\varPhi)+\overline{\varPhi}\partial_{\perp}\varPhi-\partial_{\perp}\overline{\varPhi}\varPhi-2i(W+\overline{W})$
$\displaystyle\phantom{{}={}}+\partial_{\perp}\big{[}\tfrac{i}{2}\vartheta^{+}(\mathbbm{D}_{+}\overline{\varPhi}\varPhi-\overline{\varPhi}\mathbbm{D}_{+}\varPhi)+\tfrac{i}{2}\vartheta^{-}(\mathbbm{D}_{-}\overline{\varPhi}\varPhi-\overline{\varPhi}\mathbbm{D}_{-}\varPhi)$
$\displaystyle\phantom{=+\partial_{\perp}\big{[}}+\vartheta^{-}\vartheta^{+}\big{(}\tfrac{1}{4}\partial_{\perp}(\overline{\varPhi}\varPhi)+i(W-\overline{W})\big{)}\big{]}\Big{]}.$

The equations of motion are in superspace

$0=2\mathbbm{D}_{-}\mathbbm{D}_{+}\varPhi-\partial_{\perp}\varPhi+i\overline{W}^{\prime}(\overline{\varPhi}),$
(225)

which in components are the usual bulk equations.

##  Appendix C Supercurrent multiplets in 3D

###  C.1 Decomposition to ${(0,2)}$-multiplets

We decompose the bulk multiplet
$\mathcal{S}_{\alpha\beta}=-2\gamma^{\mu}_{\alpha\beta}\mathcal{S}_{\mu}$
using the branching coordinate $\xi$ according to

$\mathcal{S}_{\alpha\beta}(x,\theta,\overline{\theta})=\mathcal{S}_{\alpha\beta}^{(0)}(\xi,\theta^{+},\overline{\theta}^{+})+\theta^{-}\mathcal{S}_{\alpha\beta}^{(1)}(\xi,\theta^{+},\overline{\theta}^{+})-\overline{\theta}^{-}\overline{\mathcal{S}_{\alpha\beta}^{(1)}}(\xi,\theta^{+},\overline{\theta}^{+})+\theta^{-}\overline{\theta}^{-}\mathcal{S}_{\alpha\beta}^{(2)}(\xi,\theta^{+},\overline{\theta}^{+}).$
(226)

We obtain:

  1. (1)

The $+$-direction

$\displaystyle\mathcal{S}^{(0)}_{++}$
$\displaystyle=4j_{+}-4i\theta^{+}(S_{+})_{+}-4i\overline{\theta}^{+}(\overline{S}_{+})_{+}-16\theta^{+}\overline{\theta}^{+}T_{++},$
(227a)
$\displaystyle\begin{split}\mathcal{S}_{++}^{(1)}&=-4i(S_{+})_{-}-2\sqrt{2}\overline{\omega}_{+}+\overline{\theta}^{+}(4i\partial_{\perp}j_{+}+4K_{+\perp}+4iL_{+})-4i\theta^{+}\overline{Y}_{+}\\\
&\phantom{{}={}}+8\theta^{+}\overline{\theta}^{+}\partial_{+}(S_{+})_{-},\end{split}$
(227b)
$\displaystyle\begin{split}\mathcal{S}_{++}^{(2)}&=-8K_{+-}+8\theta^{+}\partial_{\perp}(S_{+})_{-}-8\overline{\theta}^{+}\partial_{\perp}(\overline{S}_{+})_{-}+8\theta^{+}\partial_{-}(S_{+})_{+}-8\overline{\theta}^{+}\partial_{-}(\overline{S}_{+})_{+}\\\
&\hskip
11.38092pt-4\sqrt{2}i\theta^{+}\partial_{+}\overline{\omega}_{-}-4\sqrt{2}i\overline{\theta}^{+}\partial_{+}\omega_{-}-4\sqrt{2}i\theta^{+}\partial_{\perp}\omega_{+}-4\sqrt{2}i\overline{\theta}^{+}\partial_{\perp}\overline{\omega}_{+}\\\
&\hskip
11.38092pt-4\theta^{+}\overline{\theta}^{+}\partial_{\perp}^{2}j_{+}-8\theta^{+}\overline{\theta}^{+}\partial_{\perp}L_{+}-4\theta^{+}\overline{\theta}^{+}(-2\partial_{+}\partial_{\nu}j^{\nu}+\partial^{2}j_{+}).\end{split}$
(227c)

  2. (2)

The $-$-direction

$\displaystyle\begin{split}\mathcal{S}_{--}^{(0)}&=4j_{-}-4i\theta^{+}(S_{-})_{+}-4i\overline{\theta}^{+}(\overline{S}_{-})_{+}+2\sqrt{2}\theta^{+}\overline{\omega}_{-}\\\
&\phantom{{}={}}-2\sqrt{2}\overline{\theta}^{+}\omega_{-}-8\theta^{+}\overline{\theta}^{+}K_{-+},\end{split}$
(228a)
$\displaystyle\begin{split}\mathcal{S}_{--}^{(1)}&=-4i(S_{-})_{-}-4i\theta^{+}\overline{Y}_{-}+4\overline{\theta}^{+}(K_{-\perp}+i\partial_{\perp}j_{-}+iL_{-})\\\
&\phantom{{}={}}+8\theta^{+}\overline{\theta}^{+}\partial_{+}(S_{-})_{-}+4\sqrt{2}i\theta^{+}\overline{\theta}^{+}\partial_{-}\overline{\omega}_{+},\end{split}$
(228b)
$\displaystyle\begin{split}\mathcal{S}_{--}^{(2)}&=-16T_{--}+8\theta^{+}\partial_{\perp}(S_{-})_{-}+8\theta^{+}\partial_{-}(S_{-})_{+}-8\overline{\theta}^{+}\partial_{-}(\overline{S}_{-})_{+}-8\overline{\theta}^{+}\partial_{\perp}(\overline{S}_{-})_{-}\\\
&\hskip
11.38092pt+4\theta^{+}\overline{\theta}^{+}\partial_{\perp}^{2}j_{-}-8\theta^{+}\overline{\theta}^{+}\partial_{\perp}L_{-}-4\theta^{+}\overline{\theta}^{+}(-2\partial_{-}\partial^{\nu}j_{\nu}+\partial^{2}j_{-}).\end{split}$
(228c)

  3. (3)

The $\perp$-direction

$\displaystyle\begin{split}\mathcal{S}_{-+}^{(0)}&=-2j_{\perp}+2i\theta^{+}(S_{\perp})_{+}+2i\overline{\theta}^{+}(\overline{S}_{\perp})_{+}+\sqrt{2}\theta^{+}\overline{\omega}_{+}\\\
&\phantom{{}={}}-\sqrt{2}\overline{\theta}^{+}\omega_{+}+4\theta^{+}\overline{\theta}^{+}K_{\perp+},\end{split}$
(229a)
$\displaystyle\begin{split}\mathcal{S}_{-+}^{(1)}&=+2i(S_{\perp})_{-}-\sqrt{2}\overline{\omega}_{-}-2\overline{\theta}^{+}(K_{\perp\perp}+i\partial_{\perp}j_{\perp}+iL_{\perp})+2i\theta^{+}\overline{Y}_{\perp}\\\
&\hskip
11.38092pt-4\theta^{+}\overline{\theta}^{+}\partial_{+}(S_{\perp})_{-}-2\sqrt{2}i\theta^{+}\overline{\theta}^{+}\partial_{\perp}\overline{\omega}_{+}-2\sqrt{2}i\theta^{+}\overline{\theta}^{+}\partial_{+}\overline{\omega}_{-},\end{split}$
(229b)
$\displaystyle\begin{split}\mathcal{S}_{-+}^{(2)}&=+4K_{\perp-}-4\theta^{+}\partial_{\perp}(S_{\perp})_{-}+4\overline{\theta}^{+}\partial_{\perp}(\overline{S}_{\perp})_{-}-4\theta^{+}\partial_{-}(S_{\perp})_{+}\\\
&\phantom{{}={}}+4\overline{\theta}^{+}\partial_{-}(\overline{S}_{\perp})_{+}+\sqrt{2}i\theta^{+}\partial_{-}\overline{\omega}_{+}+\sqrt{2}i\overline{\theta}^{+}\partial_{-}\omega_{+}+4\theta^{+}\overline{\theta}^{+}\partial_{\perp}L_{\perp}\\\
&\phantom{{}={}}+2\theta^{+}\overline{\theta}^{+}\partial_{\perp}^{2}j_{\perp}-4\theta^{+}\overline{\theta}^{+}(\partial_{\perp}\partial^{\nu}j_{\nu}-\tfrac{1}{2}\partial^{2}j_{\perp}).\end{split}$
(229c)

where

$\displaystyle K_{\mu\nu}$
$\displaystyle=2T_{\nu\mu}-\eta_{\mu\nu}A-\tfrac{1}{4}\epsilon_{\mu\nu\rho}H^{\rho}$
$\displaystyle=2T_{\nu\mu}-\eta_{\mu\nu}A-\tfrac{1}{4}C_{\mu\nu},$ (230)
$\displaystyle L_{\mu}$
$\displaystyle=\tfrac{1}{4}\epsilon_{\mu\nu\rho}F^{\nu\rho}+\epsilon_{\mu\nu\rho}\partial^{\nu}j^{\rho}$
$\displaystyle=\tfrac{1}{4}C_{\mu}+\epsilon_{\mu\nu\rho}\partial^{\nu}j^{\rho}.$

where we have also defined the brane currents
$C_{\mu\nu}=\epsilon_{\mu\nu\rho}H^{\rho}$ and
$C_{\mu}=\epsilon_{\mu\nu\rho}F^{\nu\rho}$. The decomposition for the
$\mathcal{R}$-multiplet is found simply by setting the multiplet
$\mathcal{Y}_{\alpha}\ni(\omega_{\alpha},A,Y_{\mu})$ to zero.

One may check that these decompositions are indeed ${(0,2)}$-multiplets using
the (${(0,2)}$-restriction of the) supersymmetry variations that follow.

###  C.2 3D $\mathcal{N}=2$ supersymmetry variations of components

Under a variation induced by $\delta_{\text{sym}}=\xi
Q-\overline{\xi}\overline{Q}$, we may compute from (42)

$\displaystyle\delta_{\text{sym}}\lambda_{\alpha}$
$\displaystyle=i\xi_{\alpha}D-i(\gamma^{\mu}\xi)_{\alpha}(H_{\mu}+\tfrac{i}{2}\epsilon_{\mu\nu\rho}F^{\nu\rho})+\tfrac{i}{2}\overline{\xi}_{\beta}C,$
(231a) $\displaystyle\delta_{\text{sym}}D$
$\displaystyle=\xi\gamma^{\mu}\partial_{\mu}\overline{\lambda}-\overline{\xi}\gamma^{\mu}\partial_{\mu}\lambda,$
(231b) $\displaystyle\delta_{\text{sym}}H_{\mu}$
$\displaystyle=-\xi\partial_{\mu}\overline{\lambda}+\overline{\xi}\partial_{\mu}\lambda,$
(231c) $\displaystyle\epsilon_{\mu\nu\rho}\delta_{\text{sym}}F^{\nu\rho}$
$\displaystyle=-2i\epsilon_{\mu\nu\rho}(\xi\gamma^{\rho}\partial^{\nu}\overline{\lambda}+\overline{\xi}\gamma^{\rho}\partial^{\nu}\lambda),$
(231d) $\displaystyle\delta_{\text{sym}}C$ $\displaystyle=0,$ (231e)
$\displaystyle\delta_{\text{sym}}\omega_{\alpha}$
$\displaystyle=\sqrt{2}\xi_{\alpha}B-\tfrac{1}{2\sqrt{2}}\overline{\xi}_{\alpha}C-\sqrt{2}i(\gamma^{\mu}\overline{\xi})_{\alpha}Y_{\mu},$
(231f) $\displaystyle\delta_{\text{sym}}B$
$\displaystyle=-\sqrt{2}i(\overline{\xi}\gamma^{\mu}\partial_{\mu}\omega),$
(231g) $\displaystyle\delta_{\text{sym}}Y_{\mu}$
$\displaystyle=\sqrt{2}(\xi\partial_{\mu}\omega),$ (231h)
$\displaystyle\delta_{\text{sym}}j_{\mu}$
$\displaystyle=\tfrac{1}{\sqrt{2}}(\xi\gamma_{\mu}\overline{\omega})-\tfrac{1}{\sqrt{2}}(\overline{\xi}\gamma_{\mu}\omega)-i(\xi
S_{\mu})-i(\overline{\xi}\overline{S}_{\mu}),$ (231i)
$\displaystyle\begin{split}\delta_{\text{sym}}(S_{\mu})_{\alpha}&=\tfrac{i}{4}(\gamma_{\mu}\xi)_{\alpha}\overline{C}+\epsilon_{\mu\rho\nu}(\gamma^{\nu}\xi)_{\alpha}\overline{Y}^{\rho}+\overline{\xi}_{\alpha}\big{(}\tfrac{1}{4}\epsilon_{\mu\nu\rho}F^{\nu\rho}+\epsilon_{\mu\nu\rho}\partial^{\nu}j^{\rho}\big{)}\\\
&\hskip
11.38092pt-i(\gamma^{\nu}\overline{\xi})_{\alpha}(2T_{\nu\mu}-\tfrac{1}{4}\epsilon_{\mu\nu\rho}H^{\rho}+i\partial_{\nu}j_{\mu}-i\partial_{\rho}j^{\rho}\eta_{\mu\nu}),\end{split}$
(231j)
$\displaystyle\begin{split}\delta_{\text{sym}}K_{\mu\nu}&=-\tfrac{1}{2}\big{[}2\epsilon_{\nu\rho\lambda}\xi\gamma^{\lambda}\partial^{\rho}S_{\mu}-\sqrt{2}i\epsilon_{\nu\mu\rho}\xi\partial^{\rho}\overline{\omega}-\tfrac{i}{\sqrt{2}}\eta_{\mu\nu}\xi\gamma_{\rho}\partial^{\rho}\overline{\omega}\\\
&\hskip
28.45274pt-2\epsilon_{\nu\rho\lambda}\overline{\xi}\gamma^{\lambda}\partial^{\rho}\overline{S}_{\mu}-\sqrt{2}i\epsilon_{\nu\mu\rho}\overline{\xi}\partial^{\rho}\omega-\tfrac{i}{\sqrt{2}}\eta_{\mu\nu}\overline{\xi}\gamma_{\rho}\partial^{\rho}\omega\big{]},\end{split}$
(231k)
$\displaystyle\begin{split}\delta_{\text{sym}}T_{\mu\nu}&=-\tfrac{i}{4\sqrt{2}}\eta_{\mu\nu}(\xi\gamma_{\rho}\partial^{\rho}\overline{\omega})-\tfrac{1}{2}\xi\gamma_{(\nu}\gamma_{\rho}\partial^{\rho}S_{\mu)}+\tfrac{1}{2}\xi\partial_{(\nu}S_{\mu)}\\\
&\hskip
11.38092pt-\tfrac{i}{4\sqrt{2}}\eta_{\mu\nu}(\overline{\xi}\gamma_{\rho}\partial^{\rho}\omega)-\tfrac{1}{2}\overline{\xi}\gamma_{(\nu}\gamma_{\rho}\partial^{\rho}\overline{S}_{\mu)}+\tfrac{1}{2}\overline{\xi}\partial_{(\nu}\overline{S}_{\mu)}.\end{split}$
(231l)

###  C.3 Decomposition of bulk constraints

Using the branching coordinate $\xi$ and the expansions

$\displaystyle\chi_{\alpha}^{B}$
$\displaystyle=\chi_{\alpha}^{B{(0)}}+\theta^{-}\chi_{\alpha}^{B{(1a)}}+\overline{\theta}^{-}\chi_{\alpha}^{B{(1b)}}+\theta^{-}\overline{\theta}^{-}\chi_{\alpha}^{B{(2)}},$
(232) $\displaystyle\mathcal{Y}_{\alpha}^{B}$
$\displaystyle=\mathcal{Y}_{\alpha}^{B{(0)}}+\theta^{-}\mathcal{Y}_{\alpha}^{B{(1a)}}+\overline{\theta}^{-}\mathcal{Y}_{\alpha}^{B{(1b)}}+\theta^{-}\overline{\theta}^{-}\mathcal{Y}_{\alpha}^{B{(2)}}.$

we may rewrite the constraints (41) for the bulk $\mathcal{S}$-multiplet to
the following collection of equations:
From $\overline{D}_{-}\chi_{\alpha}=\tfrac{1}{2}\epsilon_{-\alpha}C$ (recall
$\epsilon_{-+}=-1$ otherwise zero):

$\displaystyle\chi^{B{(1b)}}_{\alpha}$
$\displaystyle=\tfrac{1}{2}\epsilon_{-\alpha}C,$ (233a)
$\displaystyle\chi^{B{(2)}}_{\alpha}$
$\displaystyle=-2i\partial_{-}\chi^{B{(0)}}_{\alpha}.$ (233b)

From $\overline{D}_{+}\chi_{\alpha}=\tfrac{1}{2}\epsilon_{+\alpha}C$:

$\displaystyle\overline{D}_{+}\chi^{B{(0)}}_{\alpha}$
$\displaystyle=\tfrac{1}{2}\epsilon_{+\alpha}C,$ (234a)
$\displaystyle\overline{D}_{+}\chi^{B{(1a)}}_{\alpha}$
$\displaystyle=2i\partial_{\perp}\chi^{B{(0)}}_{\alpha},$ (234b)
$\displaystyle\overline{D}_{+}\chi^{B{(2)}}_{\alpha}$ $\displaystyle=0.$
(234c)

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

$\displaystyle\operatorname{Im}(D_{+}\chi_{-}^{B{(0)}}-\chi_{+}^{B{(1a)}})$
$\displaystyle=0,$ (235a)
$\displaystyle\overline{D}_{+}\overline{\chi_{-}^{B{(1a)}}}+\chi_{+}^{B{(2)}}-2i\partial_{-}\chi_{+}^{B{(0)}}-2i\partial_{\perp}\chi_{-}^{B{(0)}}$
$\displaystyle=0,$ (235b)
$\displaystyle\operatorname{Im}(D_{+}\chi_{-}^{B{(2)}}-2i\partial_{-}\chi_{+}^{B{(1a)}}-2i\partial_{\perp}\chi_{-}^{B{(1a)}})$
$\displaystyle=0.$ (235c)

From $D_{\alpha}\mathcal{Y}_{\beta}+D_{\beta}\mathcal{Y}_{\alpha}=0$ we obtain

$\displaystyle D_{+}\mathcal{Y}_{-}^{B{(0)}}+\mathcal{Y}_{+}^{B{(1a)}}$
$\displaystyle=0,$ (236a) $\displaystyle D_{+}\mathcal{Y}_{-}^{B{(1a)}}$
$\displaystyle=0,$ (236b) $\displaystyle D_{+}\mathcal{Y}_{-}^{B{(1b)}}$
$\displaystyle=2i\partial_{\perp}\mathcal{Y}_{-}^{B{(0)}}+\mathcal{Y}_{+}^{B{(2)}}-2i\partial_{-}\mathcal{Y}_{+}^{B{(0)}},$
(236c) $\displaystyle D_{+}\mathcal{Y}_{+}^{B{(0)}}$ $\displaystyle=0,$ (236d)
$\displaystyle D_{+}\mathcal{Y}_{+}^{B{(1a)}}$ $\displaystyle=0,$ (236e)
$\displaystyle D_{+}\mathcal{Y}_{+}^{B{(1b)}}$
$\displaystyle=2i\partial_{\perp}\mathcal{Y}_{+}^{B{(0)}},$ (236f)
$\displaystyle D_{+}\mathcal{Y}_{+}^{B{(2)}}$
$\displaystyle=-2i\partial_{\perp}\mathcal{Y}_{+}^{B{(1b)}},$ (236g)
$\displaystyle\mathcal{Y}_{-}^{B{(1a)}}$ $\displaystyle=0,$ (236h)
$\displaystyle\mathcal{Y}_{-}^{B{(2)}}$
$\displaystyle=2i\partial_{-}\mathcal{Y}_{-}^{B{(0)}}.$ (236i)

From $\overline{D}^{\alpha}\mathcal{Y}_{\alpha}+C=0$ we obtain

$\displaystyle\overline{D}_{+}\mathcal{Y}_{-}^{B{(0)}}+\mathcal{Y}_{+}^{B{(1b)}}+C$
$\displaystyle=0,$ (237a)
$\displaystyle\overline{D}_{+}\mathcal{Y}_{-}^{B{(1a)}}+2i\partial_{\perp}\mathcal{Y}_{-}^{B{(0)}}+\mathcal{Y}_{+}^{B{(2)}}+2i\partial_{-}\mathcal{Y}_{+}^{B{(0)}}$
$\displaystyle=0,$ (237b)
$\displaystyle\overline{D}_{+}\mathcal{Y}_{-}^{B{(1b)}}$ $\displaystyle=0,$
(237c)
$\displaystyle\overline{D}_{+}\mathcal{Y}_{-}^{B{(2)}}-2i\partial_{\perp}\mathcal{Y}_{-}^{B{(1b)}}-2i\partial_{-}\mathcal{Y}_{+}^{B{(1b)}}$
$\displaystyle=0.$ (237d)

And the relation
$\overline{D}^{\beta}\mathcal{S}_{\alpha\beta}=\chi_{\alpha}+\mathcal{Y}_{\alpha}$:

$\displaystyle\chi^{B{(0)}}_{\alpha}+\mathcal{Y}^{B{(0)}}_{\alpha}$
$\displaystyle=\overline{D}_{+}\mathcal{S}^{B{(0)}}_{\alpha-}-\overline{\mathcal{S}}^{(1)}_{\alpha+},$
(238a) $\displaystyle\chi^{B{(1a)}}_{\alpha}+\mathcal{Y}^{B{(1a)}}_{\alpha}$
$\displaystyle=-\overline{D}_{+}\mathcal{S}^{(1)}_{\alpha-}-\mathcal{S}^{B{(2)}}_{\alpha+}-2i\partial_{\perp}\mathcal{S}^{B{(0)}}_{\alpha-}-2i\partial_{-}\mathcal{S}^{B{(0)}}_{\alpha+},$
(238b)
$\displaystyle\tfrac{1}{2}\epsilon_{-\alpha}C+\mathcal{Y}_{\alpha}^{B{(1b)}}$
$\displaystyle=\overline{D}_{+}\overline{\mathcal{S}}^{(1)}_{\alpha-},$ (238c)
$\displaystyle\chi^{B{(2)}}_{\alpha}+\mathcal{Y}^{B{(2)}}_{\alpha}$
$\displaystyle=\overline{D}_{+}\mathcal{S}^{B{(2)}}_{\alpha-}+2i\partial_{\perp}\overline{\mathcal{S}}^{(1)}_{\alpha-}+2i\partial_{-}\overline{\mathcal{S}}^{(1)}_{\alpha+}.$
(238d)

###  C.4 Decompositions of bulk improvements

Using the branching coordinate and the expansions (226) and (232) we may
compute the improvements of decomposed multiplets:

$\displaystyle\mathcal{S}_{++}^{B{(0)}}$
$\displaystyle\mapsto\mathcal{S}_{++}^{B{(0)}}+[D_{+},\overline{D}_{+}]U^{B{(0)}},$
(239a) $\displaystyle\mathcal{S}_{--}^{B{(0)}}$
$\displaystyle\mapsto\mathcal{S}_{--}^{B{(0)}}-2U^{B{(2)}},$ (239b)
$\displaystyle\mathcal{S}_{+-}^{B{(0)}}$
$\displaystyle\mapsto\mathcal{S}_{+-}^{B{(0)}}+D_{+}\overline{U^{B{(1)}}}-\overline{D}_{+}U^{B{(1)}}.$
(239c)

$\displaystyle\mathcal{S}_{++}^{B{(1)}}$
$\displaystyle\mapsto\mathcal{S}_{++}^{B{(1)}}+[D_{+},\overline{D}_{+}]U^{B{(1)}}+4i\partial_{\perp}D_{+}U^{B{(0)}},$
(240a) $\displaystyle\mathcal{S}_{--}^{B{(1)}}$
$\displaystyle\mapsto\mathcal{S}_{--}^{B{(1)}}-4i\partial_{-}U^{B{(1)}},$
(240b) $\displaystyle\mathcal{S}_{+-}^{B{(1)}}$
$\displaystyle\mapsto\mathcal{S}_{+-}^{B{(1)}}-D_{+}U^{B{(2)}}+2i\partial_{\perp}U^{B{(1)}}-2i\partial_{-}D_{+}U^{B{(0)}}.$
(240c)

$\displaystyle\begin{split}\mathcal{S}_{++}^{B{(2)}}&\mapsto\mathcal{S}_{++}^{B{(2)}}+[D_{+},\overline{D}_{+}]U^{B{(2)}}+4i\partial_{\perp}D_{+}\overline{U^{B{(1)}}}\\\
&\hskip
34.14322pt+4i\partial_{\perp}\overline{D}_{+}U^{B{(1)}}-8\partial_{\perp}^{2}U^{B{(0)}},\end{split}$
(241a) $\displaystyle\mathcal{S}_{--}^{B{(2)}}$
$\displaystyle\mapsto\mathcal{S}_{--}^{B{(2)}}-8\partial_{-}^{2}U^{B{(0)}},$
(241b) $\displaystyle\mathcal{S}_{+-}^{B{(2)}}$
$\displaystyle\mapsto\mathcal{S}_{+-}^{B{(2)}}-2i\partial_{-}D_{+}\overline{U^{B{(1)}}}-2i\partial_{-}\overline{D}_{+}U^{B{(1)}}+8\partial_{-}\partial_{\perp}U^{B{(0)}}.$
(241c)

$\displaystyle\chi_{+}^{B{(0)}}$
$\displaystyle\mapsto\chi_{+}^{B{(0)}}+2\overline{D}_{+}D_{+}\overline{U^{B{(1)}}},+4i\partial_{\perp}\overline{D}_{+}U^{B{(0)}},$
(242a) $\displaystyle\chi_{-}^{B{(0)}}$
$\displaystyle\mapsto\chi_{-}^{B{(0)}}+2\overline{D}_{+}U^{B{(2)}}-4i\partial_{-}\overline{D}_{+}U^{B{(0)}},$
(242b)
$\displaystyle\begin{split}\chi_{+}^{B{(1a)}}&\mapsto\chi_{+}^{B{(1a)}}+2\overline{D}_{+}D_{+}U^{B{(2)}}-4i\partial_{\perp}\overline{D}_{+}U^{B{(1)}}-4i\partial_{\perp}D_{+}\overline{U^{B{(1)}}}\\\
&\hskip
36.98866pt+8\partial_{\perp}^{2}U^{B{(0)}}+4i\partial_{-}\overline{D}_{+}D_{+}U^{B{(0)}},\end{split}$
(242c) $\displaystyle\chi_{-}^{B{(1a)}}$
$\displaystyle\mapsto\chi_{-}^{B{(1a)}}+4i(2\partial_{-}\overline{D}_{+}U^{B{(1)}}-\partial_{\perp}U^{B{(2)}}+2i\partial_{\perp}\partial_{-}U^{B{(0)}}),$
(242d) $\displaystyle\chi_{\alpha}^{B{(1b)}}$
$\displaystyle\mapsto\chi_{\alpha}^{B{(1b)}},$ (242e)
$\displaystyle\chi_{+}^{B{(2)}}$
$\displaystyle\mapsto\chi_{+}^{B{(2)}}-4i\partial_{-}(\overline{D}_{+}D_{+}\overline{U^{B{(1)}}}+2i\partial_{\perp}\overline{D}_{+}U^{B{(0)}}),$
(242f) $\displaystyle\chi_{-}^{B{(2)}}$
$\displaystyle\mapsto\chi_{-}^{B{(2)}}-4i(\partial_{-}\overline{D}_{+}U^{B{(2)}}-2i\partial_{-}^{2}\overline{D}_{+}U^{B{(0)}}).$
(242g)

$\displaystyle\mathcal{Y}_{+}^{B{(0)}}$
$\displaystyle\mapsto\mathcal{Y}_{+}^{B{(0)}}-D_{+}\overline{D}_{+}\overline{U^{B{(1)}}},$
(243a) $\displaystyle\mathcal{Y}_{-}^{B{(0)}}$
$\displaystyle\mapsto\mathcal{Y}_{-}^{B{(0)}}+\overline{D}_{+}U^{B{(2)}}+2i\partial_{\perp}\overline{U^{B{(1)}}}+2i\partial_{-}\overline{D}_{+}U^{B{(0)}},$
(243b) $\displaystyle\mathcal{Y}_{+}^{B{(1a)}}$
$\displaystyle\mapsto\mathcal{Y}_{+}^{B{(1a)}}-D_{+}\overline{D}_{+}U^{B{(2)}}-2i\partial_{\perp}D_{+}\overline{U^{B{(1)}}}-2i\partial_{-}D_{+}\overline{D}_{+}U^{B{(0)}},$
(243c) $\displaystyle\mathcal{Y}_{-}^{B{(1a)}}$
$\displaystyle\mapsto\mathcal{Y}_{-}^{B{(1a)}},$ (243d)
$\displaystyle\mathcal{Y}_{+}^{B{(1b)}}$
$\displaystyle\mapsto\mathcal{Y}_{+}^{B{(1b)}}-2i\partial_{\perp}\overline{D}_{+}\overline{U^{B{(1)}}},$
(243e) $\displaystyle\mathcal{Y}_{-}^{B{(1b)}}$
$\displaystyle\mapsto\mathcal{Y}_{-}^{B{(1b)}}+4i\partial_{-}\overline{D}_{+}\overline{U^{B{(1)}}},$
(243f)
$\displaystyle\begin{split}\mathcal{Y}_{+}^{B{(2)}}&\mapsto\mathcal{Y}_{+}^{B{(2)}}+2i\partial_{-}D_{+}\overline{D}_{+}\overline{U^{B{(1)}}}-2i\partial_{\perp}\overline{D}_{+}U^{B{(2)}}\\\
&\hskip
34.14322pt+4\partial_{\perp}^{2}\overline{U^{B{(1)}}}+4\partial_{\perp}\partial_{-}\overline{D}_{+}U^{B{(0)}},\end{split}$
(243g) $\displaystyle\mathcal{Y}_{-}^{B{(2)}}$
$\displaystyle\mapsto\mathcal{Y}_{-}^{B{(2)}}+2i\partial_{-}\overline{D}_{+}U^{B{(2)}}-4\partial_{-}\partial_{\perp}\overline{U^{B{(1)}}}-4\partial_{-}^{2}\overline{D}_{+}U^{B{(0)}}.$
(243h)

###  C.5 Explicit bulk components of $\mathcal{S}_{\mu}$ for LG model

We may compute the components of the supercurrent multiplets for the Landau-
Ginzburg model where

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

according to the expansions (42). In other words, we are in the
$\mathcal{S}$-frame (cf. page 3) and we obtain:

$\displaystyle R\text{-``current''}:$ $\displaystyle j^{\mu}$
$\displaystyle=(\overline{\psi}\gamma^{\mu}\psi),$ (245a)
$\displaystyle\text{supercurrent}:$ $\displaystyle S_{\mu\alpha}$
$\displaystyle=\sqrt{2}(\gamma^{\nu}\gamma_{\mu}\psi)_{\alpha}\partial_{\nu}\overline{\phi}-\sqrt{2}i(\gamma_{\mu}\overline{\psi})_{\alpha}\overline{W}^{\prime},$
(245b) $\displaystyle\text{lowest in }\chi_{\alpha}:$
$\displaystyle\lambda_{\alpha}$
$\displaystyle=2\sqrt{2}(\gamma^{\mu}\overline{\psi})_{\alpha}\partial_{\mu}\phi+2\sqrt{2}iW^{\prime}\psi_{\alpha},$
(245c) $\displaystyle\text{lowest in }\mathcal{Y}_{\alpha}:$
$\displaystyle\omega_{\alpha}$ $\displaystyle=4W^{\prime}\psi_{\alpha},$
(245d) $\displaystyle\begin{split}\text{EM tensor}:&\\\ &\end{split}$
$\displaystyle\begin{split}T_{\nu\rho}&=(\partial_{\rho}\phi\partial_{\nu}\overline{\phi}+\partial_{\nu}\phi\partial_{\rho}\overline{\phi})\\!-\\!\eta_{\nu\rho}(|\partial\phi|^{2}\\!+|W^{\prime}|^{2})\\\
&\hskip
11.38092pt+\tfrac{i}{2}(\partial_{(\rho}\overline{\psi}\gamma_{\nu)}\psi)-\tfrac{i}{2}(\overline{\psi}\gamma_{(\nu}\partial_{\rho)}\psi),\end{split}$
(245e) $\displaystyle\text{irrelevant auxilliary}:$ $\displaystyle A$
$\displaystyle=-4|W^{\prime}|^{2}+i\partial_{\mu}\overline{\psi}\gamma^{\mu}\psi-i\overline{\psi}\gamma^{\mu}\partial_{\mu}\psi,$
(245f) $\displaystyle\\{Q,S\\}\text{ 1-brane charge}:$ $\displaystyle Y_{\mu}$
$\displaystyle=4\partial_{\mu}W,$ (245g)
$\displaystyle\\{\overline{Q},S\\}\text{ 1-brane charge}:$ $\displaystyle
H^{\mu}$ $\displaystyle=-2i\partial^{\mu}(\overline{\psi}\psi),$ (245h)
$\displaystyle\\{\overline{Q},S\\}\text{ 0-brane charge}:$
$\displaystyle\epsilon_{\rho\mu\nu}F^{\mu\nu}$
$\displaystyle=-4\epsilon_{\rho\mu\nu}\partial^{\mu}j^{\nu}-8i\epsilon_{\rho\mu\nu}\partial^{\mu}\phi\partial^{\nu}\overline{\phi}.$
(245i) 2020footnotetext: This particular $R$-“current” is not conserved for
most superpotentials. If we improve the multiplet to an
$\mathcal{R}$-multiplet, this component of the multiplet is the conserved
$R$-current.

Note that all (Hodge duals to) brane currents are _exact_ forms. This is to be
expected, since we are working on a trivial space, and it only shows local
triviality in general backgrounds. For example, if $W$ is not a properly
defined function, then $Y^{\mu}$ is not identically exact.

##  Appendix D Delta distributions on the boundary

The commutators of quantized operators involve some subtleties with respect to
delta distributions and boundaries. We define

$\int\limits_{\mathclap{(-\infty,0]}}f(x)\delta(x)\coloneqq f(0),$ (246)

with the reasoning that we want the entire boundary to be part of the theory,
thus anything on the boundary is fully part of the system, and that if the
$\delta$-distribution is to be understood as a limit of functions
$g_{n}(x)\to\delta(x)$ which fulfill $\int_{(-\infty,0]}g_{n}(x)=1$, then the
above will follow automatically. This results in one important subtlety:
switching from $\tfrac{\partial}{\partial y}\delta(x-y)$ to
$-\tfrac{\partial}{\partial x}\delta(x-y)$ introduces a boundary term,
specifically

$\int\limits_{\mathclap{(-\infty,0]}}\mathrm{d}y\>\int\limits_{\mathclap{(-\infty,0]}}\mathrm{d}xf(x)g(y)\big{(}\tfrac{\partial}{\partial
x}\delta(x-y)+\tfrac{\partial}{\partial y}\delta(x-y)\big{)}=f(0)g(0).$ (247)

This, in turn, implies

$\displaystyle-\int\limits_{\mathclap{(-\infty,0]}}\mathrm{d}yf(x)g(y)\tfrac{\partial}{\partial
y}\delta(x-y)$
$\displaystyle=-f(0)g(0)+\int\limits_{\mathclap{(-\infty,0]}}\mathrm{d}yf(y)g^{\prime}(y),$
(248)
$\displaystyle\int\limits_{\mathclap{(-\infty,0]}}\mathrm{d}yf(x)g(y)\tfrac{\partial}{\partial
x}\delta(x-y)$
$\displaystyle=\int\limits_{\mathclap{(-\infty,0]}}\mathrm{d}yf(y)g^{\prime}(y),$
(249)

as both boundary terms cancel in the second equation.

## References

  * (1) T. Dimofte, D. Gaiotto and N. M. Paquette, _Dual boundary conditions in 3d SCFT’s_ , _JHEP_ 05 (2018) 060 [1712.07654].
  * (2) T. Okazaki and S. Yamaguchi, _Supersymmetric boundary conditions in three-dimensional N=2 theories_ , _Phys. Rev._ D87 (2013) 125005 [1302.6593].
  * (3) A. Gadde, S. Gukov and P. Putrov, _Fivebranes and 4-manifolds_ , 1306.4320.
  * (4) Z. Komargodski and N. Seiberg, _Comments on Supercurrent Multiplets, Supersymmetric Field Theories and Supergravity_ , _JHEP_ 07 (2010) 017 [1002.2228].
  * (5) T. T. Dumitrescu and N. Seiberg, _Supercurrents and Brane Currents in Diverse Dimensions_ , _JHEP_ 07 (2011) 095 [1106.0031].
  * (6) N. Drukker, D. Martelli and I. Shamir, _The energy-momentum multiplet of supersymmetric defect field theories_ , _JHEP_ 08 (2017) 010 [1701.04323].
  * (7) D. Gaiotto, S. Gukov and N. Seiberg, _Surface Defects and Resolvents_ , _JHEP_ 09 (2013) 070 [1307.2578].
  * (8) M. Dedushenko, _Chiral algebras in Landau-Ginzburg models_ , _JHEP_ 03 (2018) 079 [1511.04372].
  * (9) E. Witten, _On the Landau-Ginzburg description of N=2 minimal models_ , _Int. J. Mod. Phys._ A9 (1994) 4783 [hep-th/9304026].
  * (10) E. Silverstein and E. Witten, _Global U(1) R symmetry and conformal invariance of (0,2) models_ , _Phys. Lett._ B328 (1994) 307 [hep-th/9403054].
  * (11) E. Witten, _Two-dimensional models with (0,2) supersymmetry: Perturbative aspects_ , _Adv. Theor. Math. Phys._ 11 (2007) 1 [hep-th/0504078].
  * (12) Y. Yoshida and K. Sugiyama, _Localization of 3d $\mathcal{N}=2$ Supersymmetric Theories on $S^{1}\times D^{2}$_, 1409.6713.
  * (13) A. Kapustin and Y. Li, _D branes in Landau-Ginzburg models and algebraic geometry_ , _JHEP_ 12 (2003) 005 [hep-th/0210296].
  * (14) I. Brunner, M. Herbst, W. Lerche and B. Scheuner, _Landau-Ginzburg realization of open string TFT_ , _JHEP_ 11 (2006) 043 [hep-th/0305133].
  * (15) L. Di Pietro, N. Klinghoffer and I. Shamir, _On Supersymmetry, Boundary Actions and Brane Charges_ , _JHEP_ 02 (2016) 163 [1502.05976].
  * (16) A. Bilal, _Supersymmetric Boundaries and Junctions in Four Dimensions_ , _JHEP_ 11 (2011) 046 [1103.2280].
  * (17) M. Magro, I. Sachs and S. Wolf, _Superfield Noether procedure_ , _Annals Phys._ 298 (2002) 123 [hep-th/0110131].
  * (18) S. Ferrara and B. Zumino, _Transformation Properties of the Supercurrent_ , _Nucl. Phys._ B87 (1975) 207.
  * (19) S. J. Gates, M. T. Grisaru, M. Rocek and W. Siegel, _Superspace Or One Thousand and One Lessons in Supersymmetry_ , _Front. Phys._ 58 (1983) 1 [hep-th/0108200].
  * (20) N. Drukker, I. Shamir and C. Vergu, _Defect multiplets of $\mathcal{N}=1$ supersymmetry in 4d_, _JHEP_ 01 (2018) 034 [1711.03455].
  * (21) I. V. Melnikov, _An Introduction to Two-Dimensional Quantum Field Theory with (0,2) Supersymmetry_ , _Lect. Notes Phys._ 951 (2019) pp.
  * (22) M. Bullimore, T. Dimofte, D. Gaiotto and J. Hilburn, _Boundaries, Mirror Symmetry, and Symplectic Duality in 3d $\mathcal{N}=4$ Gauge Theory_, _JHEP_ 10 (2016) 108 [1603.08382].
  * (23) A. Kapustin and Y. Li, _Topological correlators in Landau-Ginzburg models with boundaries_ , _Adv. Theor. Math. Phys._ 7 (2003) 727 [hep-th/0305136].
  * (24) M. Herbst, K. Hori and D. Page, _Phases Of N=2 Theories In 1+1 Dimensions With Boundary_ , 0803.2045.
  * (25) C. I. Lazaroiu, _On the boundary coupling of topological Landau-Ginzburg models_ , _JHEP_ 05 (2005) 037 [hep-th/0312286].
  * (26) E. Witten, _Phases of N=2 theories in two-dimensions_ , _Nucl. Phys._ B403 (1993) 159 [hep-th/9301042].
  * (27) H. Jockers and P. Mayr, _A 3d Gauge Theory/Quantum K-Theory Correspondence_ , 1808.02040.
  * (28) M. C. N. Cheng, S. Chun, F. Ferrari, S. Gukov and S. M. Harrison, _3d Modularity_ , 1809.10148.
  * (29) F. Aprile and V. Niarchos, _$\mathcal{N}$ =2 supersymmetric field theories on 3-manifolds with A-type boundaries_, _JHEP_ 07 (2016) 126 [1604.01561].
  * (30) C. Closset, T. T. Dumitrescu, G. Festuccia and Z. Komargodski, _Supersymmetric Field Theories on Three-Manifolds_ , _JHEP_ 05 (2013) 017 [1212.3388].
  * (31) C. Closset, T. T. Dumitrescu, G. Festuccia and Z. Komargodski, _The Geometry of Supersymmetric Partition Functions_ , _JHEP_ 01 (2014) 124 [1309.5876].
  * (32) K. Hori, _Linear models of supersymmetric D-branes_ , in _Symplectic geometry and mirror symmetry. Proceedings, 4th KIAS Annual International Conference, Seoul, South Korea, August 14-18, 2000_ , pp. 111–186, 2000, hep-th/0012179.
  * (33) K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa et al., _Mirror symmetry_ , vol. 1 of _Clay mathematics monographs_. AMS, Providence, USA, 2003.


#  Achieving mouse-level strategic evasion performance using real-time
computational planning

German Espinosa1, Gabrielle E. Wink2, Alexander T. Lai3, Daniel A. Dombeck4
and Malcolm A. MacIver4 *This work was supported by the National Science
Foundation, ECCS-1835389 and IIS-2123725 to Malcolm A. MacIver and Daniel A.
Dombeck1German Espinosa, Department of Computer Science, Northwestern
University, USA<EMAIL_ADDRESS>E. Wink,
Department of Mechanical Engineering, Northwestern University, USA;
<EMAIL_ADDRESS>T. Lai, Department of Biomedical
Engineering, Northwestern University, USA<EMAIL_ADDRESS>A. Dombeck, Department of Neurobiology, Northwestern University, USA;
<EMAIL_ADDRESS>A. MacIver, Center for Robotics and
Biosystems, Northwestern University, USA<EMAIL_ADDRESS>
###### Abstract

Planning is an extraordinary ability in which the brain imagines and then
enacts evaluated possible futures. Using traditional planning models, computer
scientists have attempted to replicate this capacity with some level of
success but ultimately face a reoccurring limitation: as the plan grows in
steps, the number of different possible futures makes it intractable to
determine the right sequence of actions to reach a goal state. Based on prior
theoretical work on how the ecology of an animal governs the value of spatial
planning, we developed a more efficient biologically-inspired planning
algorithm, _TLPPO_. This algorithm allows us to achieve mouse-level predator
evasion performance with orders of magnitude less computation than a
widespread algorithm for planning in the situations of partial observability
that typify predator-prey interactions. We compared the performance of a real-
time agent using _TLPPO_ against the performance of live mice, all tasked with
evading a robot predator. We anticipate these results will be helpful to
planning algorithm users and developers, as well as to areas of neuroscience
where robot-animal interaction can provide a useful approach to studying the
basis of complex behaviors.

##  I INTRODUCTION

Animals equipped with planning abilities present rich and complex behaviors.
From big cats stalking antelopes on the savanna to mice using clutter to
escape foxes in a forest, animals show the extraordinary advantages of this
action-control mechanism when dealing with life-and-death situations, where
trial and error learning easily transitions to trial and death.

Prior theory and simulations work in paleontology and neuroscience have
provided evidence that planning may confer a “selective benefit”—necessary for
the adoption of a trait through a population over evolutionary time—in
predator-prey interactions in patchy savanna-like terrestrial domains [1, 2,
3, 4]. During these encounters, animals need to keep track of the continuously
changing state of the world affected by every move of both animals. They also
must estimate information beyond what their senses can provide, as the
adversary could be temporarily out of sight. Furthermore, they create
predictions about the future to determine the outcome of possible actions.
This process is done in real-time with extraordinary efficiency. All of these
characteristics make animal planning in naturalistic scenarios significantly
more complex than many of the tasks considered in contemporary applications of
online planning within AI [5, 6, 7, 8, 9, 10, 11, 12, 13, 14].

Figure 1: Predator-prey interaction in a configurable environment with a
robotic predator (left) pursuing a live mouse (right top) or a computational
agent (right bottom) – see video https://youtu.be/PpGYyq2HbUY.

The present work aims to reduce the task complexity gap between AI and
biological planning by developing a computational agent capable of strategic
evasion in real-time at the same level as live animals. For this purpose, we
used a novel, ethologically relevant task: a predator-prey interaction where
lab mice play the role of the prey, and a reactive robotic agent plays the
role of a predator. In this task, a mouse traverses a large semi-occluded
arena to reach a reward while avoiding an aversive air puff given by a roaming
robot predator. The mouse’s performance was quantified by the percentage of
runs where the mouse completed the task without receiving the predator’s puff.
This proxy for survival rate served as the target for a computational agent
performing the same task.

The state-of-the-art planning algorithm Partially Observable Monte-Carlo
Planning (_POMCP_) was initially used for the development of the agent [15].
_POMCP_ is a comprehensive framework for planning in partially observable
scenarios. It provides both a mechanism to create and maintain a statistical
representation of the current state beyond the observable world (belief state)
and a configurable high-performance tree-search algorithm that balances the
exploration-exploitation trade-off to determine the best next action. To test
the performance of the agent, a simulation replicating the experiment was
created. However, during this test, it became apparent that the computational
effort required to reach the target performance exceeded, by several orders of
magnitude, the time available available to select moves in the real
experiment. This inefficiency made the _POMCP_ agent not performant enough for
real-time execution and made it evident that a new approach was needed.

To address this problem, we propose a new spatial planning algorithm inspired
by prior evolutionary computational neuroscience results [3], “Trajectories
over Locations where Planning Pays Off” or _TLPPO_. _TLPPO_ uses the
properties of the graph structure of the space where the predator-prey
interactions occur to determine places that might contain opportunities. Then,
during the planning phase of the agent, it forces the tree-search process to
explore branches in the direction of those opportunities. Simulation results
using _TLPPO_ showed a significant decrease in the computational effort
required to compute plans that match animal survival rates. To test the
_TLPPO_ agent in the same conditions that were used to determine animal
performance, we repeated the mice experiment with the exact same configuration
(same average speed as the mice shown in trials, same robotic predator), but
this time we replaced the live animals with the _TLPPO_ agent.

Our results show that controlling the actions of our planning agent through
the _TLPPO_ algorithm enabled the agent to interact with an aggressive robotic
predator in real-time successfully. Its performance was comparable to a live
mammal, for which there is growing neurophysiological and behavioral evidence
of planning.

##  II Previous Work

In the last decade, online planning research has mainly been addressed by
online partially observable Markov Decision problem (POMDP) solvers [5, 6, 7,
8, 9, 10, 11, 12, 13] based on the Partially Observable Monte Carlo Planning
(_POMCP_) [15] algorithm. _POMCP_ was first introduced in 2010 and is still
considered the fastest online planning POMDP solver to date. However, its
implementation comes with some caveats.

The most important challenge of using _POMCP_ is the exponential growth of the
decision tree while performing the forward information search. This problem is
known as the “curse of dimensionality.” Even with relatively small problems,
the decision tree cannot be explored exhaustively. _POMCP_ addresses this
problem by using the Upper Confidence Bound (_UCB1_) algorithm [16]. During
the tree-search phase, _UCB1_ stores statistics (visit count and reward
values) for each branch expanded in the past and uses them to determine what
branches to expand in the future. This algorithm can be configured to balance
“exploration”—expanding branches that had been less visited before—and
“exploitation”—expanding branches that had better reward values in the past.
This makes _UCB1_ effective when rewards can be obtained quickly, and the tree
has a gradient of reward values that grows towards the optimal plan. But in
cases with sparse rewards that require a large number of steps to obtain them,
the computational effort required to effectively plan is extremely large.

A commonly used approach to overcome the curse of dimensionality is
artificially limiting the number of actions available for tree expansion and
increasing it progressively, called Double Progressive Widening (_DPW_) [11,
12, 13]. By doing this, a smaller section of the tree is tested more
frequently, and as a result, planning performs a smaller number of policy
reward estimations but with higher certainty due to the increased repetitions.
_DPW_ can be used in trees with massive expansion (even in continuous spaces),
but it still relies on being able to find a gradient of reward values in every
direction. With sparse rewards, _DPW_ has a small probability of reaching
them, and policy estimation suffers. A different approach to tackle the curse
of dimensionality is to memorize and reuse parts of the plan computed
previously, called Adaptive Belief Tree (_ABT_) [17]. This algorithm stores
the policies computed in previous planning iterations and determines when the
environment has changed enough to motivate a recalculation. _ABT_ is extremely
useful when the planning process occurs faster than the changes in the
environment or when the environment is highly predictable. However, in
unpredictable dynamic environments, like dealing with a mobile adversary,
policies become obsolete faster than they can be computed.

##  III The _TLPPO_ Algorithm

As discussed above, current sub-sampling methods depend mostly on the reward
distribution to provide clues about where the branches of the decision tree
with higher values are. But in a highly volatile environment where the rewards
are sparse, and the required number of steps to obtain them is large, these
approaches fail to build a correct policy value estimation in a reasonable
time. Here we propose a method inspired by neuroscience research to use
additional information that is available to the agent to inform the tree
expansion.

Figure 2: _TLPPO_ ’s selection of locations where planning pays off greatly
reduces the number of locations that need consideration in a decision tree,
enabling real-time performance. (a) Trimming of features in the space. Only
useful features during an adversarial interaction (window, columns) are
highlighted, as identified by the value of their eigencentrality in the 2D
projections shown below. (b) Workflow of _TLPPO_ algorithm. In a 2D map of the
example space, eigencentrality of each square (purple, left) is computed.
Derivatives along each direction is computed, and the product of these
derivatives (blue, center) is calculated. Squares that exceed a threshold (red
line in the colorbar) are selected as locations where planning pays off
(green, right). (c) Example of tree exploration using _TLPPO_. The agent only
evaluates trajectories connecting locations where planning pays off using the
shortest path between them (left). Instead of expanding new trajectories after
every action as occurs in Monte-Carlo tree search, the algorithm only expands
when it reaches a location where planning pays off, and only in the direction
of other such locations visible from its current location (right). (d) Green
hexagons show locations where planning pays off in a schematic of the same
hexagonal arena as the predator-prey experiments. Black cells are occlusions
blocking movement and vision.

Connectedness and the cognitive map. One of the key assumptions used in
testing different hypotheses about biological spatial planning is that the
brain has a representation of the space around it. There is evidence that this
representation is formed in a brain structure called the hippocampus, where
neurons called place cells get associated with specific locations in space,
thereby forming a cognitive map [18]. Using the cognitive map, the brain is
thought to identify connections between locations and possibly provide the
basis for simulating different trajectories in imagination. These trajectories
have been recorded in animal brains during planning behaviors using neural
probes [19, 20].

The mice used in our experiments were allowed “practice rounds” to explore the
arena without the presence of the predator robot to let them memorize the
space and build their cognitive maps. The computational agent receives the
coordinates of the occlusions at the beginning of each experiment to build its
own version of it. Cognitive maps are powerful tools by themselves; however,
evidence suggests that animals might use additional information to enrich
them. Previous research [3] proposed the animal brain might have evolved
mechanisms to identify aspects of the graph structure of its habitat to
determine connectedness of the space. Animals can use this information to
determine the difference between open and closed spaces. This could provide
clues about how safe the current state of the world is (e.g., prey in an open
space can be seen from far away by a predator). Computationally, connectedness
of the space can be estimated using a graph theory metric called eigenvector
centrality (hereafter, eigencentrality) [3]. Eigencentrality shows not only
how a portion of the space is connected to its neighbours but also how well
connected its neighbors are to their neighbors and so on.

In prior work on arbitration between model-free and model-based action
control, we showed that eigencentrality can be used to activate planning when
moving from a closed to an open space without decreasing survival rate
relative to much more expensive full-time planning [3]. This suggests that the
geometry of the space influences the value of planning. Specifically, planning
only generates a benefit when performed in places with a high gradient of
connectedness (transitional spaces from closed to open). We call these places
“locations where planning pays off.” To illustrate how these locations are
distributed in the environment, we will use a scene from an abandoned factory
(Fig. 2a left). The factory contains a broken window, piles of bricks, timber
pieces, rocks, columns and considerable open space between them. However, if a
human entered this factory during a life-and-death situation (e.g. an active
shooter) and had to plan an escape, there are only two useful features: the
broken window and the columns (Fig. 2a right). These features afford two
critical behaviors: fleeing and hiding. In the 2D representation of the
connectedness of the same space (Fig. 2b left), those same areas present the
highest fluctuations of eigencentrality. To isolate the corresponding cells,
we compute these fluctuations as derivatives (one for each possible direction)
and store their product as an intermediate step. This ensures the
eigencentrality change is occurring in every direction (Fig. 2b center).
Finally, it is only necessary to apply a threshold; cells with stored values
above or equal to the threshold are defined as locations where planning pays
off (Fig. 2b right). Figure 2d shows the result of this process with the
habitat configuration used for the experimental work.

Trajectories over locations where planning pays off. After identifying the
locations where planning is beneficial, the agent still has to manage the full
computational burden of performing the planning process. To address this
problem, we use the identified locations during the tree-search phase to
create a new algorithm “Trajectories over Locations where Planning Pays Off”
(_TLPPO_). _TLPPO_ assumes that in locations outside of those with high
potential pay off, according to the definition above, there will also be no
benefit from expanding the tree while planning. This means that tree
expansions should only happen at locations where planning pays off and only in
the direction of other locations where planning pays off using the shortest
path between them (Fig. 2c). This creates a subset of trajectories to explore
that is several orders of magnitude fewer than the total universe of possible
trajectories. Reducing the number of action sequences to simulate during
planning in this manner has the potential benefit of accelerating and
simplifying the process greatly.

##  IV Mammalian-level performance baseline

Assessment of an animal’s strategic evasion behavior and performance was done
in a new naturalistic, predator-prey task whose design was informed by prior
theory, paleontological and evolutionary neuroscience findings, and
simulations [1, 2, 3, 4]. The task consists of a regular hexagonal arena with
a long diagonal length of 2.34 m. The arena is overlaid with a hexagonal grid
system consisting of 330 magnetic hexagon cells (Fig. 1 & 2d). The distance
between two cell centers is 11 cm. Obstacles with magnetic bases are placed in
a configuration that maximizes the utility of strategic behavior from prior
results [3]. A custom robot predator equipped with an air puff mechanism (Fig.
1), serving as an aversive stimulus in lieu of actual prey capture, actively
roams the arena in search of the animal subject. The animal’s task for each
episode is to traverse the arena to a goal location without being captured by
the predator. An episode ends if the animal reaches the goal or is captured. A
capture is recorded if the distance between the robot and the subject falls
below 2.5 cells (27.5 cm), the threshold to trigger an air puff.

Algorithm 1 Autonomous robot predator behavior

1:while experiment is running do

2: Find spawn cell _S_

3: Move robot to _S_

4: while episode is running do

5: if mouse is visible then

6: Move robot to last seen mouse cell

7: else if mouse is not visible then

8: Find cells not visible to robot

9: Randomly select a non-visible cell _N_

10: Move robot to _N_

11: end if

12: end while

13:end while

Figure 3: (a) Survival rate as a function of branches sampled for _TLPPO_ and
_POMCP_ planning algorithms. Red line represents the survival rate calculated
from mice experiments. (b) Branches sampled versus time needed. Planning time
was the average time across both _POMCP_ and _TLPPO_ algorithms, which were
roughly equal in time per branch sampled. Red line represents the maximum
allowable planning time given the speed of the mouse and arena cell size.

The predator starts each episode at a spawn cell: a cell not visible to the
prey, in a region of the arena (1/3 of the total area) furthest from the start
gate (Fig. 2d) through which the animal enters the arena. The robot then
follows the behaviors described in Algorithm 1 until the end of the
experiment. A high-speed four-camera tracking system actively records the
position of the robot and the animal, providing real-time control information
for the predator and recording animal behavior at 30 Hz. Cameras and encoders
are utilized for PID control of the differential drive autonomous predator
robot. The robot follows trajectories created by an A* algorithm [21] used to
find the shortest path to the destination with obstacle avoidance. Occasional
control intervention with a joystick was applied if the robot navigated into
an obstacle and failed to return to its trajectory.

Two male lab mice (_Mus musculus_ , strain C57BL/6) were water restricted and
trained in this setup to traverse the arena while actively avoiding the robot.
Progression through the arena is controlled by automated doors and water
feeders that provide water rewards and motivation for the mouse to run
multiple trajectories for a given experiment. Training consisted of a multi-
step process where the mouse performed one 30-minute experiment each day and
was gradually exposed to the environment without the robot to learn the space
and build a cognitive map. Once the mouse has explored the entirety of the
arena and performed more than 1 trajectory/minute, the mouse was given
multiple days to perform experiments with the robot to learn the robot’s
behavior. Once the number of loops through the arena over an experimental
session stabilized, data collection began. Each mouse performed 5 days of
robot experiments. During both training and experiments, a white noise
generator covered the sound of the robot’s movements and the arena was cleaned
before each experiment to remove olfactory cues, largely limiting the mouse’s
sensory cues to vision. Both mice were control subjects of a parallel
experiment. As controls, these mice had received a craniotomy and an injection
of an inert viral vector before training and received daily intraperitoneal
injections of saline during experiments. All experiments were approved and
conducted in accordance with the Northwestern University Animal Care and Use
Committee.

Average mice and robot speeds during movement were calculated by extracting
the position of both agents at every frame and then calculating a subsequent
array of velocities where ${\Delta x}$ was the change in position and ${\Delta
t}$ was the change in time between each frame. To focus on speed primarily
during movement and not during pauses, only velocities above 0.15 m/s were
selected for both agents. Then, an average across both agents was taken across
all remaining velocities and all experiments. The average speed of the mouse
was found to be 0.76 m/s.

Baseline survival rate was calculated by dividing the number of trajectories
where the mouse successfully evaded the robot without receiving an air puff by
the total number of trajectories across all experiments and mice. The mice ran
a total of 230 trajectories and had a survival rate of 0.86.

##  V Matching mammalian performance

The work explained in the previous section provided insights about two key
metrics that will determine if an agent can perform at the same level than the
mice in the experiments: survival rate and speed.

###  V-A Survival rate

To determine the planning effort needed to match the mammalian-level survival
rate baseline, we developed a simulation of the experiment where the
computational agents are in control of the actions executed by a simulated
mouse. In the simulation, the predator is provided with the same reactive
behaviors as the robotic predator in the real-world experiment. The key
difference between the execution of the simulation and the experiment is that
the simulation “waits” for the agent to make a decision by stopping all world
updates. This allowed us to validate the survival rate of the agents as we
increased the number of branches sampled per iteration until the simulated
rate was similar to the experimentally measured survival rate.

Simulation benchmarks. Figure 3a shows the evolution of the performance on the
tested agents. To reach mouse-level performance, the _POMCP_ agent required
sampling 175,000 branches on each planning iteration while the _TLPPO_ agent
required $\approx$100 branches.

###  V-B Speed

Another critical feature of the planning agent is planning speed due to the
interactive nature of the task (evade adversary in real-time). This limits the
amount of time that agents can use for planning to the interval between moves.
To determine this upper-bound on control delay, we divided the distance
between cells in the habitat (0.11 m) by the average mouse speed from the
recording sessions (0.76 m/s). This sets the total time that can be used for
planning to $\approx$150 ms.

To determine how many branches could be sampled within the time limit, we
designed an experiment using the hardware that later will be used to run the
agent during the real-world trials. In this experiment we increased the number
of planning cycles for both _POMCP_ and _TLPPO_ until we obtained an average
planning time of 75% of this limit to account for additional delays. This
works out to 1000 branches sampled in the real-time version of the agents
(Fig. 3b).

### Comparing mice and computational agents

In the final part of this work, a new set of real-world experiments was
executed using the hexagonal arena and the robotic predator. This time the
live mice were replaced with the real-time computational agents. For both the
_POMCP_ and _TLPPO_ agents, 100 episodes were recorded to establish their
survival rate performance. Results showed that both agents were able to
complete task, however, only the _TLPPO_ agent was able to obtain a survival
rate comparable to the recorded animal performance (Fig. 4).

Figure 4: Survival rate for real-time computational preys performing _TLPPO_
and _POMCP_ planning algorithms and mice in the presence of a robot predator.

##  VI Discussion

We presented a new approach for computational spatial planning in partially
observable scenarios that performs on par with live animals. We have shown the
efficiency of the _TLPPO_ algorithm in comparison to _POMCP_ and tested its
applicability in a real-time complex interactive task with a highly volatile
environment.

Reinforcement learning and model-free agents. Here we discuss our use of a
model-based method, online planning, rather than other available tools, such
as model-free methods (e.g. deep learning approaches). Many current
computational approaches for complex adversarial problems, at least in gaming,
rely on model-free reinforcement learning. One important benefit of these
techniques is that the agent does not have to perform a tree search at the
time it needs to take an action, allowing it to interact with the environment
more rapidly.

One of the major advances in reinforcement learning was the introduction of
Deep Q Learning Networks (DQN) [22]. The principle behind DQN is that the
resulting policy value estimation from a tree exploration done during Q
learning can be expressed as a function and, consequently, it can be
approximated using deep neural networks. This technique has multiple benefits.
One is that DQN allows agents to learn and store information from larger MDP
problems. One example is the game of Go [23, 24], where the number of possible
states ($10^{172}$) is larger than the atoms in the universe, making it
impossible to store that information on a traditional Q-table, where every
single state requires an entry. Another great benefit is that DQN generalizes
the knowledge acquired during training, allowing the agent to make good
decisions even in states that have not been visited before. DQN caused a major
revolution in AI and became the de facto standard for reinforcement learning
with an increasing number of publications using this technique every year [25,
26, 27, 28, 24, 29, 30, 24, 31, 32].

However, all of these solutions require the agent to go through a learning
cycle, where it naively explores a problem to learn from experience. This
makes applying them to naturalistic problems challenging, where the
irreversibility of death looms. In a predator-prey interaction, the prey
cannot learn from a failed episode. AlphaGo [23], for example, played millions
of games before being able to defeat Lee Sedol (world champion of Go at the
time) in 2016. This was only possible because it was allowed to gain
experience from won and lost games during that training. But this result would
have never been possible if the agent had to erase all previous knowledge with
every defeat. For these reasons, here we focus on a model-based method only.
Agents start every episode with nothing gained via experiential learning, but
two components of static knowledge: the location of the occlusions and
boundaries (a cognitive map), and a generative model of the predator behavior
as required for evaluating the decision trees.

There has been a number of prior works exploring interactions between robots
and animals [33, 34]. Our study differs from these in that we offer a novel
method for directly comparing the behavior, strategy, and performance of a
computational planning agent to that of an animal—possibly using similar
functions instantiated neurally—performing an identical task.

## ACKNOWLEDGMENTS

We thank Parker Ryan, Ben Zitzewitz, Lily Browdy, Chris Angeloni, and Sam
Griswold for their contributions to the live animal experiments.

## References

  * [1] M. A. MacIver, “Neuroethology: From morphological computation to planning,” in The Cambridge Handbook of Situated Cognition (P. Robbins and M. Aydede, eds.), pp. 480–504, New York, NY: Cambridge University Press, 2009\.
  * [2] M. A. MacIver, L. Schmitz, U. Mugan, T. D. Murphey, and C. D. Mobley, “Massive increase in visual range preceded the origin of terrestrial vertebrates,” pnas, vol. 114, no. 12, pp. E2375–E2384, 2017.
  * [3] U. Mugan and M. A. MacIver, “Spatial planning with long visual range benefits escape from visual predators in complex naturalistic environments,” Nature Communications, vol. 11, p. 3057, June 2020.  Number: 1 Publisher: Nature Publishing Group.
  * [4] M. A. MacIver and B. L. Finlay, “The neuroecology of the water-to-land transition and the evolution of the vertebrate brain,” philtransb, vol. 377, pp. 1–26, Feb. 2022.  Publisher: Royal Society.
  * [5] Y. Wang, F. Giuliari, R. Berra, A. Castellini, A. Del Bue, A. Farinelli, M. Cristani, and F. Setti, “POMP: POMCP-based online motion planning for active visual search in indoor environments,” arXiv preprint arXiv:2009.08140, 2020.
  * [6] G. Mazzia, A. Castellinia, and A. Farinellia, “Policy interpretation for partially observable Monte-Carlo planning: A rule-based approach,” CEUR Workshop Proceedings, 2020.
  * [7] E. I. Farhi and V. Indelman, “iX-BSP: Incremental belief space planning,” arXiv preprint arXiv:2102.09539, 2021.
  * [8] Y. Xiao, S. Katt, A. ten Pas, S. Chen, and C. Amato, “Online planning for target object search in clutter under partial observability,” in 2019 International Conference on Robotics and Automation (ICRA), pp. 8241–8247, IEEE, 2019.
  * [9] A. Goldhoorn, A. Garrell, R. Alquézar, and A. Sanfeliu, “Continuous real time POMCP to find-and-follow people by a humanoid service robot,” in 2014 IEEE-RAS International Conference on Humanoid Robots, pp. 741–747, IEEE, 2014.
  * [10] S. Fedeler and M. Holzinger, “Monte Carlo tree search methods for telescope tasking,” in AIAA Scitech 2020 Forum, p. 0659, 2020.
  * [11] A. Couëtoux, J.-B. Hoock, N. Sokolovska, O. Teytaud, and N. Bonnard, “Continuous upper confidence trees,” in International Conference on Learning and Intelligent Optimization, pp. 433–445, Springer, 2011.
  * [12] Z. Sunberg and M. Kochenderfer, “Online algorithms for POMDPs with continuous state, action, and observation spaces,” in Proceedings of the International Conference on Automated Planning and Scheduling, vol. 28, 2018\.
  * [13] J. Mern, A. Yildiz, L. Bush, T. Mukerji, and M. J. Kochenderfer, “Improved POMDP tree search planning with prioritized action branching,” arXiv preprint arXiv:2010.03599, 2020.
  * [14] E. Galceran and M. Carreras, “A survey on coverage path planning for robotics,” Robotics and Autonomous systems, vol. 61, no. 12, pp. 1258–1276, 2013.
  * [15] D. Silver and J. Veness, “Monte-Carlo planning in large POMDPs,” in nips, pp. 2164–2172, 2010.
  * [16] C. B. Browne, E. Powley, D. Whitehouse, S. M. Lucas, P. I. Cowling, P. Rohlfshagen, S. Tavener, D. Perez, S. Samothrakis, and S. Colton, “A survey of Monte Carlo tree search methods,” IEEE Transactions on Computational Intelligence and AI in games, vol. 4, no. 1, pp. 1–43, 2012.
  * [17] H. Kurniawati and V. Yadav, “An online POMDP solver for uncertainty planning in dynamic environment,” in Robotics Research, pp. 611–629, Springer, 2016\.
  * [18] J. O’Keefe and L. Nadel, The Hippocampus as a Cognitive Map.  Clarendon Press, 1978.
  * [19] M. H. Schmitt, A. Shuttleworth, D. Ward, and A. M. Shrader, “African elephants use plant odours to make foraging decisions across multiple spatial scales,” anmbehav, vol. 141, pp. 17 – 27, 2018.
  * [20] K. Kay, J. E. Chung, M. Sosa, J. S. Schor, M. P. Karlsson, M. C. Larkin, D. F. Liu, and L. M. Frank, “Constant sub-second cycling between representations of possible futures in the hippocampus,” Cell, vol. 180, no. 3, pp. 552–567, 2020.
  * [21] P. E. Hart, N. J. Nilsson, and B. Raphael, “A Formal Basis for the Heuristic Determination of Minimum Cost Paths,” IEEE Transactions on Systems Science and Cybernetics, vol. 4, pp. 100–107, July 1968\.  Conference Name: IEEE Transactions on Systems Science and Cybernetics.
  * [22] V. Mnih, K. Kavukcuoglu, D. Silver, A. Graves, I. Antonoglou, D. Wierstra, and M. Riedmiller, “Playing Atari with deep reinforcement learning,” arXiv:1312.5602 [cs], 2013.
  * [23] D. Silver, A. Huang, C. J. Maddison, A. Guez, L. Sifre, G. Van Den Driessche, J. Schrittwieser, I. Antonoglou, V. Panneershelvam, M. Lanctot, et al., “Mastering the game of Go with deep neural networks and tree search,” nature, vol. 529, no. 7587, pp. 484–489, 2016.
  * [24] D. Silver, J. Schrittwieser, K. Simonyan, I. Antonoglou, A. Huang, A. Guez, T. Hubert, L. Baker, M. Lai, A. Bolton, Y. Chen, T. Lillicrap, F. Hui, L. Sifre, G. van den Driessche, T. Graepel, and D. Hassabis, “Mastering the game of Go without human knowledge,” Nature, vol. 550, pp. 354–359, Oct. 2017.  Number: 7676 Publisher: Nature Publishing Group.
  * [25] T. Hester, M. Vecerik, O. Pietquin, M. Lanctot, T. Schaul, B. Piot, D. Horgan, J. Quan, A. Sendonaris, I. Osband, et al., “Deep Q-learning from demonstrations,” in Proceedings of the AAAI Conference on Artificial Intelligence, vol. 32, 2018.
  * [26] M. Hessel, J. Modayil, H. Van Hasselt, T. Schaul, G. Ostrovski, W. Dabney, D. Horgan, B. Piot, M. Azar, and D. Silver, “Rainbow: Combining improvements in deep reinforcement learning,” in Proceedings of the AAAI Conference on Artificial Intelligence, vol. 32, 2018.
  * [27] D. Horgan, J. Quan, D. Budden, G. Barth-Maron, M. Hessel, H. Van Hasselt, and D. Silver, “Distributed prioritized experience replay,” arXiv preprint arXiv:1803.00933, 2018.
  * [28] T. Schaul, J. Quan, I. Antonoglou, and D. Silver, “Prioritized experience replay,” arXiv preprint arXiv:1511.05952, 2015.
  * [29] H. Van Hasselt, A. Guez, and D. Silver, “Deep reinforcement learning with double Q-learning,” in Proceedings of the AAAI Conference on Artificial Intelligence, vol. 30, 2016.
  * [30] O. Vinyals, I. Babuschkin, W. M. Czarnecki, M. Mathieu, A. Dudzik, J. Chung, D. H. Choi, R. Powell, T. Ewalds, P. Georgiev, J. Oh, D. Horgan, M. Kroiss, I. Danihelka, A. Huang, L. Sifre, T. Cai, J. P. Agapiou, M. Jaderberg, A. S. Vezhnevets, R. Leblond, T. Pohlen, V. Dalibard, D. Budden, Y. Sulsky, J. Molloy, T. L. Paine, C. Gulcehre, Z. Wang, T. Pfaff, Y. Wu, R. Ring, D. Yogatama, D. Wünsch, K. McKinney, O. Smith, T. Schaul, T. Lillicrap, K. Kavukcuoglu, D. Hassabis, C. Apps, and D. Silver, “Grandmaster level in StarCraft II using multi-agent reinforcement learning,” Nature, vol. 575, pp. 350–354, Nov. 2019.  Number: 7782 Publisher: Nature Publishing Group.
  * [31] J. Pan, X. Wang, Y. Cheng, and Q. Yu, “Multisource transfer double DQN based on actor learning,” IEEE transactions on neural networks and learning systems, vol. 29, no. 6, pp. 2227–2238, 2018.
  * [32] T. Salimans, J. Ho, X. Chen, S. Sidor, and I. Sutskever, “Evolution strategies as a scalable alternative to reinforcement learning,” arXiv preprint arXiv:1703.03864, 2017.
  * [33] D. Bierbach, T. Landgraf, P. Romanczuk, J. Lukas, H. Nguyen, M. Wolf, and J. Krause, “Using a robotic fish to investigate individual differences in social responsiveness in the guppy,” Royal Society Open Science, vol. 5, p. 181026, Aug. 2018.
  * [34] D. Romano, E. Donati, G. Benelli, and C. Stefanini, “A review on animal–robot interaction: from bio-hybrid organisms to mixed societies,” Biological Cybernetics, vol. 113, pp. 201–225, Nov. 2018.