$(\mathcal{J}_{\pm\pm},\mathcal{Y}_{\pm})$. In this case,
$\mathcal{J}_{\pm\pm}$ turns out to be associated to the axial $U(1)_{A}$
$R$-symmetry current and satisfies the conservation equation

$\partial_{--}{\cal J}_{++}-\partial_{++}{\cal J}_{--}=0~{}.$ (6)

This multiplet, which has $4+4$ real components, is the dimensionally-reduced
version of the $D=4$ ${\cal N}=1$ FZ-multiplet [Ferrara:1974pz]; see Appendix
7.C for more details. All of the models we consider in section 3 have the
property that $\chi_{\pm}$ can be improved to zero; that is, they all have a
well-defined FZ-multiplet.

In Chapter 2, we obtained the components of the supercurrent superfield by
finding the Noether currents associated with translations in superspace. One
could use a similar Noether procedure in the case $(2,2)$ superspace, as is
done for $4D\,\mathcal{N}=1$ in [Magro:2001aj]. However, we will find it more
convenient to avoid the Noether procedure in this chapter and instead use the
supersymmetrized version of the Hilbert definition of the stress tensor. Just
as the bosonic Hilbert stress tensor $T_{\mu\nu}$ represents the response
function of the Lagrangian to a linearized perturbation $h_{\mu\nu}$ of the
metric, the supercurrent multiplets correspond to linearized couplings to
supergravity. Therefore, we can compute the components of the supercurrent by
coupling our $(2,2)$ theories to one of the formulations of $2D$ supergravity,
which we now briefly review.

Different formulations of off-shell supergravity couple to different
supercurrent multiplets. If a theory has a well-defined FZ-multiplet, as is
the case for all the examples found in section 3, then the theory can be
consistently coupled to the old-minimal supergravity prepotentials
$H^{\pm\pm}$ and $\sigma$. The nomenclature “old-minimal” is again inherited
from $D=4$ ${\cal N}=1$ supergravity; see [Gates:1983nr, Buchbinder:1998qv]
for pedagogical reviews and references. Here $H^{\pm\pm}$ is the conformal
supergravity prepotential—the analogue of the traceless part of the metric—and
$\sigma$ is a chiral conformal compensator.

We refer the reader to [Grisaru:1994dm, Grisaru:1995dr, Grisaru:1995kn,
Grisaru:1995py, Gates:1995du] and references therein for an exhaustive
description of $D=2$ $\,{\cal N}=(2,2)$ off-shell supergravity in superspace,
which we will use in our analysis; see also Appendix 7.D. For the scope of
this work, it will be enough to know the structure of linearized old-minimal
supergravity. For instance, at the linearized level the gauge symmetry of the
supergravity prepotentials $H^{\pm\pm},\,\sigma$ and $\bar{\sigma}$, can be
parameterized as follows

$\displaystyle\delta H^{++}$ $\displaystyle=$
$\displaystyle\frac{i}{2}\left(\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{-}L^{+}-D_{-}\mkern
1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu^{+}\right)~{},$ (7a)
$\displaystyle\delta H^{--}$ $\displaystyle=$
$\displaystyle\frac{i}{2}\left(\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{+}L^{-}-D_{+}\mkern
1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu^{-}\right)~{},$ (7b)
$\displaystyle\delta\sigma$ $\displaystyle=$ $\displaystyle-\frac{i}{2}\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{+}\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{-}\left(D_{+}L^{+}-D_{-}L^{-}\right)~{},$ (7c)
$\displaystyle\delta\bar{\sigma}$ $\displaystyle=$
$\displaystyle-\frac{i}{2}D_{-}D_{+}\left(\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{+}\mkern
1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu^{+}-\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{-}\mkern
1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu^{-}\right)~{},$ (7d)

in terms of unconstrained spinor superfields $L^{\pm}$ and their complex
conjugates.

The conservation law (5) for the FZ-multiplet can be derived by using the
previous gauge transformations. The linearized supergravity couplings for a
given model are written as

$\displaystyle\mathcal{L}_{\text{linear}}$ $\displaystyle=\int
d^{4}\theta\,\left(H^{++}\mathcal{J}_{++}+H^{--}\mathcal{J}_{--}\right)-\int
d^{2}\theta\,\sigma\,\mathcal{V}-\int
d^{2}\bar{\theta}\,\bar{\sigma}\,\overline{\mathcal{V}}~{},$ (8)

with ${\cal V}$ a chiral superfield and ${\overline{\cal V}}$ its complex
conjugate. Assuming the matter superfields satisfy their equations of motion,
the change in the Lagrangian (8) under the gauge transformation (7) is

$\displaystyle\delta\mathcal{L}_{\text{linear}}$ $\displaystyle=\int
d^{4}\theta\,\left(\delta H^{++}\mathcal{J}_{++}+\delta
H^{--}\mathcal{J}_{--}\right)-\int d^{2}\theta\,\delta\sigma\,\mathcal{V}-\int
d^{2}\bar{\theta}\,\delta\bar{\sigma}\,\overline{\mathcal{V}}$
$\displaystyle=\frac{i}{2}\int d^{4}\theta\,\Big{\\{}\left(\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{-}L^{+}-D_{-}\mkern
1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern
1.5mu^{+}\right)\mathcal{J}_{++}+\left(\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{+}L^{-}-D_{+}\mkern
1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern
1.5mu^{-}\right)\mathcal{J}_{--}$
$\displaystyle\qquad~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\left(D_{+}L^{+}-D_{-}L^{-}\right)\mathcal{V}-\left(\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{+}\mkern
1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu^{+}-\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{-}\mkern
1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern
1.5mu^{-}\right)\overline{\mathcal{V}}\Big{\\}}$
$\displaystyle=\frac{i}{2}\int d^{4}\theta\,\Big{\\{}L^{+}\left(\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{-}\mathcal{J}_{++}+D_{+}\mathcal{V}\right)+L^{-}\left(\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{+}\mathcal{J}_{--}-D_{-}\mathcal{V}\right)+\text{c.c.}\Big{\\}}~{},$
(9)

where we have integrated by parts. Demanding that the variation vanishes for
any gauge parameter $L^{\pm}$ gives

$\mkern 1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{-}\mathcal{J}_{++}+D_{+}\mathcal{V}=0~{},~{}~{}~{}~{}~{}~{}\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{+}\mathcal{J}_{--}-D_{-}\mathcal{V}=0~{}.$ (10)

This matches the constraints (5) for the FZ-multiplet if we identify

$\displaystyle\mathcal{Y}_{\pm}=D_{\pm}\mathcal{V}~{},$ (11)

and set $k^{\prime}=0$.

As we will soon see, studying $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformations requires
consideration of a composite operator constructed out of the square of the
supercurrent multiplet. Hence to solve the $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ flow equations we need
to be able to calculate the supercurrent multiplet explicitly. The coupling to
supergravity provides a straightforward prescription for computing the FZ-
multiplet for matter models that can be coupled to old-minimal
supergravity.111Though we will not need it in this thesis, it is worth
mentioning that the non-minimal supergravity results of [Grisaru:1994dm,
Grisaru:1995dr, Grisaru:1995kn, Grisaru:1995py, Gates:1995du] allow the
computation of the supercurrent multiplet for more general classes of models.
In particular, for a given ${\cal N}=(2,2)$ matter theory we will:

  1. 1.

Begin with an undeformed superspace Lagrangian $\mathcal{L}$ in flat ${\cal
N}=(2,2)$ superspace.

  2. 2.

Minimally couple $\mathcal{L}$ to the supergravity superfield prepotentials
$H^{\pm\pm}$, $\sigma$ and $\bar{\sigma}$.

  3. 3.

Extract the superfields $\mathcal{J}^{\pm\pm}$, $\mathcal{V}$ and
$\overline{\mathcal{V}}$ which couple linearly to $H^{\pm\pm}$, $\sigma$ and
$\bar{\sigma}$, respectively, in the D- and F-terms of (8).

Thanks to the analysis given above, the superfields $\mathcal{J}^{\pm\pm}$,
$\mathcal{V}$ and $\overline{\mathcal{V}}$ will automatically satisfy the FZ-
multiplet constraints (10). A detailed description of the computation of the
FZ-multiplet for the models relevant for our discussion is given in Appendix
7.D.

###  2 The $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$
Operator and ${\cal N}=(2,2)$ Supersymmetry

After having reviewed in the previous section the structure of the ${\cal
S}$-multiplet, we are ready to describe ${\cal N}=(2,2)$ $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformations.

####  1 The $\mathcal{T}\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{T}\mkern-1.5mu}\mkern 1.5mu$ operator

Given a $D=2$ ${\cal N}=(2,2)$ supersymmetric theory with an ${\cal
S}$-multiplet, we define the supercurrent-squared deformation of this theory,
denoted $\mathcal{T}\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{T}\mkern-1.5mu}\mkern 1.5mu$ in analogy
with $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$, by the
flow equation

$\displaystyle\partial_{\lambda}\mathcal{L}$
$\displaystyle=-\frac{1}{8}\mathcal{T}\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{T}\mkern-1.5mu}\mkern 1.5mu~{},$ (1)

where $\mathcal{T}\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{T}\mkern-1.5mu}\mkern 1.5mu$ is
constructed from current bilinears with

$\displaystyle\mathcal{T}\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{T}\mkern-1.5mu}\mkern 1.5mu$
$\displaystyle\equiv-\int
d^{4}\theta\,\mathcal{S}_{++}\mathcal{S}_{--}-\left(\int
d\theta^{-}d\theta^{+}\,\chi_{+}\chi_{-}+\int
d\overline{\theta}^{-}d\theta^{+}\,\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{Y}\mkern-1.5mu}\mkern
1.5mu_{+}\mathcal{Y}_{-}+\text{c.c.}\right)~{},$ (2)

and where the factor of $\frac{1}{8}$ is chosen for later convenience. This
deformation generalizes the results we recently obtained for $D=2$ theories
possessing ${\cal N}=(0,1)$, ${\cal N}=(1,1)$ and ${\cal N}=(0,2)$
supersymmetry [Chang:2018dge, Baggio:2018rpv, Jiang:2019hux] to theories with
${\cal N}=(2,2)$ supersymmetry.

Let us recall the form of the $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ composite operator
[Zamolodchikov:2004ce], which we denote

$\displaystyle T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu(x)=T_{++++}(x)\,T_{----}(x)-\big{[}\Theta(x)\big{]}^{2}~{}.$ (3)

An important property of the ${\cal N}=(0,1)$, ${\cal N}=(1,1)$ and ${\cal
N}=(0,2)$ cases is that the $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ operator turns out to
be the bottom component of a long supersymmetric multiplet. This is true up to
both total vector derivatives ($\partial_{++}$ and $\partial_{--}$), and terms
that vanish upon using the supercurrent conservation equations (Ward
identities). For this reason, in the supersymmetric cases studied previously,
the original $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$
deformation of [Zamolodchikov:2004ce] is manifestly supersymmetric and
equivalent to the deformations constructed in terms of the full superspace
integrals of primary supercurrent-squared composite operators [Chang:2018dge,
Baggio:2018rpv, Jiang:2019hux].

Remarkably, despite the much more involved structure of the $(2,2)$ ${\cal
S}$-multiplet compared to theories with fewer supersymmetries, it is possible
to prove that the following relation holds:

$\displaystyle\mathcal{T}\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{T}\mkern-1.5mu}\mkern 1.5mu(x)=T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu(x)+{\rm
EOM^{\prime}s}+\partial_{++}(\cdots)+\partial_{--}(\cdots)~{}.$ (4)

In (4), we use ${\rm EOM^{\prime}s}$ to denote terms that are identically zero
when (1) are used. Showing (4) requires using (6)–(12), along with several
cancellations, integration by parts and the use of the $(2,2)$ ${\cal
S}$-multiplet conservation equations (1).

In fact, the specific combination of current superfields given in (2) was
chosen precisely for (4) to hold. The combination (4) is also singled out by
being invariant under the improvement transformation (4). The important
implication of (4) is that the $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformation for an
${\cal N}=(2,2)$ supersymmetric quantum field theory is manifestly
supersymmetric and equivalent to the $\mathcal{T}\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{T}\mkern-1.5mu}\mkern 1.5mu$ deformation
of eq. (2).

Note that in the ${\cal N}=(2,2)$ case the deformation we have introduced in
(2) is conceptually different from the cases with less supersymmetry.
Specifically, the deformation is not given by the descendant of a _single_
composite superfield. On the other hand, suppose the ${\cal S}$-multiplet is
such that $C^{(\pm)}=k=k^{\prime}=0$ and it is possible to improve the
superfields $\chi_{\pm}$ and ${\cal Y}_{\pm}$ to a case where

$\displaystyle{\cal Y}_{\pm}=D_{\pm}{\cal V}~{},~{}~{}~{}\overline{{\cal
Y}}_{\pm}=\mkern 1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{\pm}\overline{{\cal V}}~{},$ (5a) $\displaystyle\chi_{+}=i\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{+}\overline{\cal
B}~{},~{}~{}~{}\chi_{-}=i\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{-}{\cal
B}~{},~{}~{}~{}\bar{\chi}_{+}=-iD_{+}{\cal
B}~{},~{}~{}~{}\bar{\chi}_{-}=-iD_{-}\overline{\cal B}~{},$ (5b)

with ${\cal V}$ chiral and ${\cal B}$ twisted-chiral:

$\displaystyle\mkern 1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{\pm}{\cal V}=0~{},~{}~{}~{}D_{\pm}\overline{\cal V}=0~{},$ (6a)
$\displaystyle\mkern 1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{+}{\cal B}=D_{-}{\cal B}=0~{},~{}~{}~{}D_{+}\overline{\cal B}=\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{-}\overline{\cal
B}=0~{}.$ (6b)

In this case (2) simplifies to

$\displaystyle\mathcal{T}\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{T}\mkern-1.5mu}\mkern 1.5mu$
$\displaystyle=$ $\displaystyle-\int
d^{4}\theta\,\mathcal{S}_{++}\mathcal{S}_{--}+\left(\int
d\theta^{-}d\theta^{+}\,\mkern 1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{+}\overline{\cal B}\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{-}\overline{\cal
B}-\int d\overline{\theta}^{-}\,d\theta^{+}\,\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{+}\overline{\mathcal{V}}D_{-}\mathcal{V}+\text{c.c.}\right)$ (7)
$\displaystyle=$ $\displaystyle-\int
d^{4}\theta\,\left(\mathcal{S}_{++}\mathcal{S}_{--}-2\mathcal{B}\overline{\mathcal{B}}-2\mathcal{V}\overline{\mathcal{V}}\right)~{},$

and we see that, up to EOM’s, $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu(x)$ is the bottom
component of a long supersymmetric multiplet. In this situation, once we
define the composite superfield

$\displaystyle{\mathcal{O}}(\zeta):=-\mathcal{S}_{++}(\zeta)\mathcal{S}_{--}(\zeta)+2\mathcal{B}(\zeta)\overline{\mathcal{B}}(\zeta)+2\mathcal{V}(\zeta)\overline{\mathcal{V}}(\zeta)~{},$
(8)

eq. (4) turns into the equivalent result222In the subsequent discussion by
$\theta=0$ we will always mean $\theta^{\pm}={\bar{\theta}}^{\pm}=0$.

$\displaystyle\int d^{4}\theta\,{\mathcal{O}}(\zeta)=D_{-}D_{+}\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{+}\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{-}{\mathcal{O}}(\zeta)|_{\theta=0}=T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu(x)+{\rm
EOM^{\prime}s}+\partial_{++}(\cdots)+\partial_{--}(\cdots)~{},$ (9)

stating that the D-term of the operator ${\mathcal{O}}(\zeta)$ is equivalent
to the standard $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu(x)$ operator.

For a matter theory that can be coupled to old-minimal supergravity, leading
to the FZ-multiplet described by (10), the operator ${\mathcal{O}}(\zeta)$
further simplifies thanks to the fact that the twisted-(anti-)chiral operators
${\cal B}$ and $\overline{{\cal B}}$ disappear. For these cases, the $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ flow turns into the
following equation

$\partial_{\lambda}\mathcal{L}=\frac{1}{8}\int
d^{4}\theta\,\left(\mathcal{J}_{++}\mathcal{J}_{--}-2\mathcal{V}\overline{\mathcal{V}}\right)~{}.$
(10)

This will be our starting point in analyzing ${\cal N}=(2,2)$ deformed models
in section 3.

####  2 Point-splitting and well-definedness

The $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu(x)$
operator (3) is quite magical because it is a well-defined irrelevant
composite local operator, free of short distance divergences
[Zamolodchikov:2004ce]. In fact, this property generalizes to the larger class
of operators

$\displaystyle[A_{s}(x)\,A^{\prime}_{s^{\prime}}(x)-B_{s+2}(x)\,B_{s^{\prime}-2}(x)]$
(11)

where $(A_{s},B_{s+2})$ and
$(A^{\prime}_{s^{\prime}},B^{\prime}_{s^{\prime}-2})$ are two pairs of
conserved currents with spins $s$ and $s^{\prime}$. The operator $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu(x)$ is a particular
example with $s=s^{\prime}=0$. As proven in [Smirnov:2016lqw], these composite
operators of “Smirnov-Zamolodchikov”-type have a well-defined point splitting
which is free of short-distance divergences. In the case of ${\cal N}=(0,1)$
and ${\cal N}=(1,1)$ supersymmetric $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformations, the
entire supermultiplet whose bottom component is $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu(x)$ is comprised of
well-defined Smirnov-Zamolodchikov-type operators [Baggio:2018rpv,
Chang:2018dge]. In the ${\cal N}=(0,2)$ case, the primary333We denote as
primary operator the top component of a supersymmetric multiplet even when the
theory is not superconformal. operator whose bottom component is $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu(x)$ is not of Smirnov-
Zamolodchikov-type. Nevertheless, also in this case it was recently shown
that, thanks to supersymmetry, the whole multiplet is well-defined
[Jiang:2019hux].

In the ${\cal N}=(2,2)$ case it is clear that the situation is more
complicated than any of the cases mentioned above. First, in the general
situation, according to (2), the $\mathcal{T}\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{T}\mkern-1.5mu}\mkern 1.5mu$ deformation
is a linear combination of a D-term together with chiral and twisted-chiral
F-terms contributions. Though the F-terms might be protected by standard
perturbative non-renormalization theorems (see, for example, [Gates:1983nr,
Buchbinder:1998qv] for the $D=4$ ${\cal N}=1$ case which dimensionally reduces
to $D=2$ ${\cal N}=(2,2)$), the D-term associated to the ${\cal S}_{++}{\cal
S}_{--}$ operator has no clear reason to be protected in general from short-
distance divergences in point-splitting regularization, and hence has no
obvious reason to be well-defined. This indicates that there might be a clash
between supersymmetry and a point-splitting procedure in the general setting.

We will not attempt to analyze this issue in full generality in the current
work; instead our aim is to describe a subclass of models for which the
$\mathcal{T}\mkern 1.5mu\overline{\mkern-1.5mu\mathcal{T}\mkern-1.5mu}\mkern
1.5mu$ deformation turns out to be well-defined. A natural restriction to
impose is that the ${\cal S}$-multiplet is constrained by (5) and the
$\mathcal{T}\mkern 1.5mu\overline{\mkern-1.5mu\mathcal{T}\mkern-1.5mu}\mkern
1.5mu$ deformation is therefore described by the D-term (7). By trivially
extending the arguments used in [Jiang:2019hux] for the ${\cal N}=(0,2)$ case,
it is not difficult to show that these restrictions are sufficient to imply
that the multiplet described by the ${\cal N}=(2,2)$ primary operator
${\mathcal{O}}(\zeta)$ of (8) is indeed well-defined despite not being of
Smirnov-Zamolodchikov-type. As in the ${\cal N}=(0,2)$, unbroken ${\cal
N}=(2,2)$ supersymmetry turns out to be the reason for this to happen.

Let us quickly explain how this works for the FZ-multiplet and the deformation
(10), which are the main players in this chapter. Note, however, that the same
argument extends to more general cases where both chiral and twisted-chiral
current superfields, $\chi_{\pm}$ and ${\cal Y}_{\pm}$, satisfying (5) are
turned on. We also refer to [Jiang:2019hux] for details that we will skip in
the following discussion, which are trivial extensions from the $(0,2)$ to the
$(2,2)$ case.

A first indication of the well-definedness of the multiplet associated to
${\mathcal{O}}(\zeta)$ comes by looking at the vacuum expectation value of its
lowest component. Define the primary composite operator

$\displaystyle
O(x):=-j_{--}(x)j_{++}(x)+2v(x)\bar{v}(x)={\mathcal{O}}(\zeta)|_{\theta=0}~{},$
(12)

and its point-split version

$\displaystyle
O(x,x^{\prime}):=-j_{--}(x)j_{++}(x^{\prime})+v(x)\bar{v}(x^{\prime})+\bar{v}(x)v(x^{\prime})~{},$
(13)

where

$\displaystyle j_{\pm\pm}(x):={\cal
J}_{\pm\pm}(\zeta)|_{\theta=0}~{},~{}~{}~{}v(x):={\cal
V}(\zeta)|_{\theta=0}~{},~{}~{}~{}\bar{v}(x):=\overline{{\cal
V}}(\zeta)|_{\theta=0}~{}.$ (14)

Note that equation (10) implies the following relation among the component
operators

$\displaystyle\big{[}\overline{Q}_{\pm},j_{\mp\mp}(x)\big{]}=\pm\big{[}Q_{\mp},v(x)\big{]}~{},~{}~{}~{}\big{[}Q_{\pm},j_{\mp\mp}(x)\big{]}=\pm\big{[}\overline{Q}_{\mp},\bar{v}(x)\big{]}~{},$
(15)

with $Q_{\pm}$ and $\overline{Q}_{\pm}$ denoting the ${\cal N}=(2,2)$
supercharges.444Given an operator $F(x)$ defined as the $\theta=0$ component
of the superfield ${\mathcal{F}}(\zeta)$,
$F(x):={\mathcal{F}}(\zeta)|_{\theta=0}$, then its supersymmetry
transformations are such that
$\big{[}Q_{\pm},F(x)\big{\\}}={\mathcal{Q}}_{\pm}{\mathcal{F}}(\zeta)|_{\theta=0}=D_{\pm}{\mathcal{F}}(\zeta)|_{\theta=0}$
and
$\big{[}\overline{Q}_{\pm},F(x)\big{\\}}=\overline{{\mathcal{Q}}}_{\pm}{\mathcal{F}}(\zeta)|_{\theta=0}=\overline{D}_{\pm}{\mathcal{F}}(\zeta)|_{\theta=0}$.
By then using $\partial_{\pm\pm}=i\\{Q_{\pm},\overline{Q}_{\pm}\\}$,
$\\{Q_{+},Q_{-}\\}=\\{\overline{Q}_{+},\overline{Q}_{-}\\}=0$,
$[\overline{Q}_{\pm},v(x)]=[Q_{\pm},\bar{v}(x)]=0$, super-Jacobi identities,
together with the conservation equations (15), and the assumption that the
vacuum is invariant under supersymmetry, it is straightforward to show that
vacuum expectation value of $O(x,x^{\prime})$ satisfies

$\displaystyle\partial_{++}\left\langle
j_{--}(x)j_{++}(x^{\prime})\right\rangle$ $\displaystyle=$ $\displaystyle
i\left\langle\left[\\{Q_{+},\overline{Q}_{+}\\},j_{--}(x)\right]j_{++}(x^{\prime})\right\rangle$
(16) $\displaystyle=$ $\displaystyle
i\left\langle\left\\{Q_{+},[Q_{-},v(x)]\right\\}j_{++}(x^{\prime})+\left\\{\overline{Q}_{+},[\overline{Q}_{-},\bar{v}(x)]\right\\}j_{++}(x^{\prime})\right\rangle$
$\displaystyle=$
$\displaystyle-i\left\langle[Q_{+},v(x)][Q_{-},j_{++}(x^{\prime})]+[\overline{Q}_{+},\bar{v}(x)][\overline{Q}_{-},j_{++}(x^{\prime})]\right\rangle$
$\displaystyle=$ $\displaystyle
i\left\langle[Q_{+},v(x)][\overline{Q}_{+},\bar{v}(x^{\prime})]+[\overline{Q}_{+},\bar{v}(x)][Q_{+},v(x^{\prime})]\right\rangle$
$\displaystyle=$
$\displaystyle\left\langle\left[i\\{\overline{Q}_{+},Q_{+}\\},v(x)\right\\}\bar{v}(x^{\prime})+\left[i\\{Q_{+},\overline{Q}_{+}\\},v(x)\right\\}\bar{v}(x^{\prime})\right\rangle$
$\displaystyle=$ $\displaystyle\partial_{++}\left\langle
v(x)\bar{v}(x^{\prime})+v(x)\bar{v}(x^{\prime})\right\rangle~{},$

and, after performing a similar calculation for
$\left\langle\partial_{--}j_{--}(x)j_{++}(x^{\prime})\right\rangle=-\left\langle
j_{--}(x)\partial^{\prime}_{--}j_{++}(x^{\prime})\right\rangle$, it is clear
that the relation

$\partial_{\pm\pm}\left\langle O(x,x^{\prime})\right\rangle=0$ (17)

holds. Therefore, $\left\langle O(x,x^{\prime})\right\rangle$ is independent
of the positions and free of short distance divergences. It is worth noting
that similarly to the argument showing that the two point function of two
chiral or twisted-chiral operators is independent of the positions $x$ and
$x^{\prime}$, the previous analysis for $\left\langle
O(x,x^{\prime})\right\rangle$ necessarily relies on unbroken ${\cal N}=(2,2)$
supersymmetry.

The argument given above can be generalized to a statement about operators in
superspace in complete analogy to the ${\cal N}=(0,2)$ case of
[Jiang:2019hux]. Let us investigate the short distance singularities in the
bosonic coordinates by defining a point-split version of the ${\cal N}=(2,2)$
primary $\mathcal{T}\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{T}\mkern-1.5mu}\mkern 1.5mu$ operator,

$\displaystyle{\mathcal{O}}(x,x^{\prime},\theta):=-{\cal
J}_{--}(x,\theta)\,{\cal J}_{++}(x^{\prime},\theta)+{\cal
V}(x,\theta)\,\overline{{\cal V}}(x^{\prime},\theta)+\overline{{\cal
V}}(x,\theta)\,{\cal V}(x^{\prime},\theta)~{}.$ (18)

We want to show that the preceding bilocal superfield is free of short
distance divergences in the limit $x\to x^{\prime}$. A straightforward
calculation shows that

$\displaystyle\partial_{++}{\mathcal{O}}(x,x^{\prime},\theta)$
$\displaystyle=$ $\displaystyle-\Big{\\{}iD_{+}{\cal
V}(\zeta)\left[D^{\prime}_{-}{\mathcal{J}}_{++}(\zeta^{\prime})+\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu^{\prime}_{+}\overline{{\cal V}}(\zeta^{\prime})\right]+i\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{+}\overline{{\cal
V}}(\zeta)\left[\mkern 1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu^{\prime}_{-}{\mathcal{J}}_{++}(\zeta^{\prime})+D^{\prime}_{+}{\cal
V}(\zeta^{\prime})\right]$ (19)
$\displaystyle+i({\mathcal{Q}}_{+}+{\mathcal{Q}}^{\prime}_{+})\left[(\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{+}\overline{{\cal
V}}(\zeta)){\cal
V}(\zeta^{\prime})\right]+i(\overline{{\mathcal{Q}}}_{+}+\overline{{\mathcal{Q}}}^{\prime}_{+})\left[(D_{+}{\cal
V}(\zeta))\overline{{\cal V}}(\zeta^{\prime})\right]$
$\displaystyle+i({\mathcal{Q}}_{-}+{\mathcal{Q}}^{\prime}_{-})\left[(D_{+}{\cal
V}(\zeta)){\mathcal{J}}_{++}(\zeta^{\prime})\right]+i(\overline{{\mathcal{Q}}}_{-}+\overline{{\mathcal{Q}}}^{\prime}_{-})\left[(\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{+}\overline{{\cal
V}}(\zeta)){\mathcal{J}}_{++}(\zeta^{\prime})\right]$
$\displaystyle+(\partial_{++}+\partial^{\prime}_{++})\left[{\bar{\theta}}^{+}(\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{+}\overline{{\cal
V}}(\zeta)){\cal V}(\zeta^{\prime})+{\bar{\theta}}^{-}(D_{+}{\cal
V}(\zeta)){\mathcal{J}}_{++}(\zeta^{\prime})\right]$
$\displaystyle-(\partial_{++}+\partial^{\prime}_{++})\left[\theta^{+}(D_{+}{\cal
V}(\zeta))\overline{{\cal V}}(\zeta^{\prime})+\theta^{-}(\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{+}\overline{{\cal
V}}(\zeta)){\mathcal{J}}_{++}(\zeta^{\prime})\right]\Big{\\}}\Big{|}_{\theta=\theta^{\prime}}~{}.$

Note that the first line in the preceding expression is zero because of the FZ
conservation equations (10), which hold up to contact terms in correlation
functions. The other lines are either total vector derivatives or
supersymmetry transformations of bilocal operators. A similar equation holds
for $\partial_{--}{\mathcal{O}}(x,x^{\prime},\theta)$ showing that the
operator ${\mathcal{O}}(x,x^{\prime},\theta)$ satisfies

$\displaystyle\partial_{\pm\pm}{\mathcal{O}}(x,x^{\prime},\theta)$
$\displaystyle=$ $\displaystyle
0\,+\,\textrm{EOM's}\,+\,[P,\cdots]\,+\,[Q,\cdots]~{},$ (20)

where $[P,\cdots]$ and $[Q,\cdots]$ schematically indicate a translation and
supersymmetry transformation of some bilocal superfield operator.555See
Appendix A of [Jiang:2019hux] for the relation between the operators
$({\mathcal{Q}}_{\pm}+{\mathcal{Q}}^{\prime}_{\pm})$,
$(\overline{{\mathcal{Q}}}_{\pm}+\overline{{\mathcal{Q}}}^{\prime}_{\pm})$ and
the generators of supersymmetry transformations on bilocal superfields such as
${\mathcal{O}}(x,x^{\prime},\theta)$. The extension of that analysis from
${\cal N}=(0,2)$ to ${\cal N}=(2,2)$ is straightforward. To conclude, by
employing an OPE argument completely analogous to the one originally given by
Zamolodchikov in [Zamolodchikov:2004ce] and extended to the ${\cal N}=(0,2)$
supersymmetric case in [Jiang:2019hux], one can show that eq. (20) implies

${\mathcal{O}}(x,x^{\prime},\theta)={\mathcal{O}}(\zeta)\,+\,{\rm
derivative~{}terms}~{}.$ (21)

Here “derivative terms” indicate superspace covariant derivatives
$D_{A}=(\partial_{\pm\pm},D_{\pm},\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{\pm})$ acting on local
superfield operators while ${\mathcal{O}}(\zeta)$ arises from the regular,
non-derivative part of the OPE of ${\mathcal{O}}(x,x^{\prime},\theta)$. As a
result the integrated operator

$\displaystyle S_{{\mathcal{O}}}=\int
d^{2}x\,d^{4}\theta\,\lim_{\varepsilon\to
0}{\mathcal{O}}(x,x+\varepsilon,\theta)=\int
d^{2}x\,d^{4}\theta:{\mathcal{O}}(x,x,\theta):~{},$ (22)

which can be considered as a definition of the integrated $\mathcal{T}\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{T}\mkern-1.5mu}\mkern 1.5mu(x)$
operator,666Note that, consistently, one can show that
$\\{Q_{+},[\overline{Q}_{+},\\{Q_{-},[\overline{Q}_{-},O(x,x^{\prime})]\\}]\\}=T_{----}(x)T_{++++}(x^{\prime})-\Theta(x)\Theta(x^{\prime})+\textrm{EOM's}+[P,\cdots]$
(23) implying that the descendant of the point-split primary operator $O(x)$
is equivalent, up to Ward identities and total vector derivatives
($\partial_{\pm\pm}$), to the point-split version of the descendant $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu(x)$ operator. is free of
short distance divergences and well-defined in complete analogy to the non-
supersymmetric case [Zamolodchikov:2004ce] and the ${\cal N}=(0,1)$, ${\cal
N}=(1,1)$, and ${\cal N}=(0,2)$ cases [Baggio:2018rpv, Chang:2018dge,
Jiang:2019hux].

###  3 Deformed $(2,2)$ Models

In this section, we will apply our supercurrent-squared deformation (10) to a
few examples of $\mathcal{N}=(2,2)$ supersymmetric theories for a chiral
multiplet $\Phi$. The superfield $\Phi$ can be written in components as

$\displaystyle\Phi$ $\displaystyle=$
$\displaystyle\phi+\theta^{+}\psi_{+}+\theta^{-}\psi_{-}+\theta^{+}\theta^{-}F-i\theta^{+}\overline{\theta}^{+}\partial_{++}\phi-i\theta^{-}\overline{\theta}^{-}\partial_{--}\phi$
(1)
$\displaystyle-i\theta^{+}\theta^{-}\overline{\theta}^{-}\partial_{--}\psi_{+}-i\theta^{-}\theta^{+}\overline{\theta}^{+}\partial_{++}\psi_{-}-\theta^{+}\theta^{-}\overline{\theta}^{-}\overline{\theta}^{+}\partial_{++}\partial_{--}\phi~{},$

where $\phi$ is a complex scalar field, $\psi_{\pm}$ are Dirac fermions, and
$F$ is a complex auxiliary field. The multiplet $\Phi$ satisfies the chirality
constraint $\mkern 1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{\pm}\Phi=0$.

We denote the physical Lagrangian by $\mathcal{L}$ and the superspace D-term
Lagrangian by $\mathcal{A}$, so that

$\displaystyle S=\int d^{2}x\,\mathcal{L}=\int
d^{2}x\,d^{4}\theta\,\mathcal{A}~{}.$ (2)

A broad class of two-derivative theories for a chiral superfield can be
described by superspace Lagrangians of the form

$\displaystyle\mathcal{L}=\int d^{4}\theta\,K(\Phi,\overline{\Phi})+\int
d^{2}\theta\,W(\Phi)+\int d^{2}\overline{\theta}\mkern
1.5mu\overline{\mkern-1.5muW\mkern-1.5mu}\mkern 1.5mu(\overline{\Phi})~{},$
(3)

where $K(\Phi,\overline{\Phi})$ is a real function called the Kähler potential
and $W(\Phi)$ is a holomorphic function called the superpotential. These are
${\cal N}=(2,2)$ Landau-Ginzburg models. In order for the kinetic terms of the
component fields of $\Phi$ to have the correct sign, we will assume that
$K_{\Phi\overline{\Phi}}=\frac{\partial^{2}K}{\partial\Phi\partial\overline{\Phi}}$
is positive.

Although we will not expand on this point in detail, all the results found in
this section can be derived almost identically for the case of a generic model
of a single scalar twisted-chiral superfield $\mathcal{Y}$, $\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{+}\mathcal{Y}=D_{-}\mathcal{Y}=0$, and its conjugate. This is not
surprising since theories containing only chiral superfields are physically
equivalent to theories formulated in terms of twisted-chiral superfields; see,
for example, [Grisaru:1994dm, Grisaru:1995dr, Grisaru:1995kn, Grisaru:1995py,
Gates:1995du] for a discussion of this equivalence in models with global and
local supersymmetry. There are also many more involved $(2,2)$ theories that
one might also want to study involving chiral, twisted-chiral and semi-chiral
superfields; see, for example, [Caldeira:2018ynv] for a recent discussion and
references. For this analysis, we have chosen to consider only models based on
a single chiral multiplet.

####  1 Kähler potential

First we will set the superpotential $W$ to zero and begin with an undeformed
superspace Lagrangian of the form

$\displaystyle\mathcal{L}=\int d^{4}\theta\,K(\Phi,\overline{\Phi})$ (4)

for some Kähler potential $K$. To leading order around this undeformed theory,
the FZ supercurrents are

$\displaystyle\mathcal{J}_{\pm\pm}$ $\displaystyle=$ $\displaystyle
2K_{\Phi\overline{\Phi}}D_{\pm}\Phi\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{\pm}\overline{\Phi}~{},$ (5a) $\displaystyle\mathcal{V}$
$\displaystyle=$ $\displaystyle 0~{},$ (5b)

where $K_{\Phi}=\frac{\partial
K}{\partial\Phi},K_{\Phi\overline{\Phi}}=\frac{\partial^{2}K}{\partial\Phi\partial\overline{\Phi}}$,
etc. Therefore, at first order the supercurrent-squared deformation driven by
${\mathcal{O}}=\left(-\mathcal{S}_{++}\mathcal{S}_{--}+2\mathcal{V}\overline{\mathcal{V}}\right)$
will source a four-fermion contribution in the D-term, giving

$\displaystyle\mathcal{L}^{(1)}=\mathcal{L}^{(0)}+\frac{1}{2}\lambda
K_{\Phi\overline{\Phi}}^{2}D_{+}\Phi\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{+}\overline{\Phi}D_{-}\Phi\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{-}\overline{\Phi}~{}.$
(6)

Next, we would like to find the all-orders solution for the deformed theory.
We make the ansatz that, at finite deformation parameter $\lambda$, the
Lagrangian takes the form

$\displaystyle\mathcal{L}_{\lambda}=\int
d^{4}\theta\Big{\\{}K(\Phi,\overline{\Phi})+f(\lambda,x,\overline{x},y)K_{\Phi\overline{\Phi}}^{2}D_{+}\Phi\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{+}\overline{\Phi}D_{-}\Phi\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{-}\overline{\Phi}\Big{\\}}~{},$ (7)

where we define the combinations

$\displaystyle
x=K_{\Phi\overline{\Phi}}\partial_{++}\Phi\partial_{--}\overline{\Phi}~{},\qquad
y=K_{\Phi\overline{\Phi}}\left(D_{+}D_{-}\Phi\right)\left(\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{+}\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{-}\overline{\Phi}\right)~{}.$ (8)

Using the results in Appendix 7.D, one finds that the superfields
$\mathcal{J}_{\pm\pm}$ and $\mathcal{V}$ appearing in our supercurrent-squared
deformation, computed for the Lagrangian (7), are given by

$\displaystyle\mathcal{J}_{++}$
$\displaystyle=2K_{\Phi\overline{\Phi}}D_{+}\Phi\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{+}\overline{\Phi}\left[1+f(x+\overline{x}-3y)+x\frac{\partial
f}{\partial x}(\overline{x}-y)+\overline{x}\frac{\partial
f}{\partial\overline{x}}(x-y)+y\frac{\partial f}{\partial
y}(x+\overline{x}-2y)\right]$
$\displaystyle\quad+2K_{\Phi\overline{\Phi}}^{2}D_{-}\Phi\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{-}\overline{\Phi}\partial_{++}\Phi\partial_{++}\overline{\Phi}\left[-f-x\frac{\partial
f}{\partial x}-\overline{x}\frac{\partial
f}{\partial\overline{x}}+y\left(\frac{\partial f}{\partial x}+\frac{\partial
f}{\partial\overline{x}}\right)\right]$
$\displaystyle\quad-2iK_{\Phi\overline{\Phi}}^{2}D_{+}\Phi
D_{-}\Phi\partial_{++}\overline{\Phi}\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{+}\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{-}\overline{\Phi}\left[-f+(x-\overline{x})\frac{\partial
f}{\partial\overline{x}}+(x-y)\frac{\partial f}{\partial y}\right]$
$\displaystyle\quad-2iK_{\Phi\overline{\Phi}}^{2}\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{+}\overline{\Phi}\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{-}\overline{\Phi}\partial_{++}\Phi
D_{+}D_{-}\Phi\left[f+(x-\overline{x})\frac{\partial f}{\partial
x}+(y-\overline{x})\frac{\partial f}{\partial y}\right]~{},$ (9)

and

$\displaystyle\mathcal{J}_{--}$
$\displaystyle=2K_{\Phi\overline{\Phi}}D_{-}\Phi\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{-}\overline{\Phi}\left[1+f(x+\overline{x}-3y)+x\frac{\partial
f}{\partial x}(\overline{x}-y)+\overline{x}\frac{\partial
f}{\partial\overline{x}}(x-y)+y\frac{\partial f}{\partial
y}(x+\overline{x}-2y)\right]$
$\displaystyle\quad+2K_{\Phi\overline{\Phi}}^{2}D_{+}\Phi\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{+}\overline{\Phi}\partial_{--}\Phi\partial_{--}\overline{\Phi}\left[-f-x\frac{\partial
f}{\partial x}-\overline{x}\frac{\partial
f}{\partial\overline{x}}+y\left(\frac{\partial f}{\partial x}+\frac{\partial
f}{\partial\overline{x}}\right)\right]$
$\displaystyle\quad-2iK_{\Phi\overline{\Phi}}^{2}D_{+}\Phi
D_{-}\Phi\partial_{--}\overline{\Phi}\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{+}\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{-}\overline{\Phi}\left[-f+(\overline{x}-x)\frac{\partial f}{\partial
x}+(\overline{x}-y)\frac{\partial f}{\partial y}\right]$
$\displaystyle\quad-2iK_{\Phi\overline{\Phi}}^{2}\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{+}\overline{\Phi}\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{-}\overline{\Phi}\partial_{--}\Phi
D_{+}D_{-}\Phi\left[f+(\overline{x}-x)\frac{\partial
f}{\partial\overline{x}}+(y-x)\frac{\partial f}{\partial y}\right]~{},$ (10)

and

$\displaystyle\mathcal{V}$
$\displaystyle=2K_{\Phi\overline{\Phi}}^{2}\left(f+y\frac{\partial f}{\partial
y}+x\frac{\partial f}{\partial x}+\overline{x}\frac{\partial
f}{\partial\overline{x}}\right)\Big{[}-i\partial_{++}\overline{\Phi}(D_{+}D_{-}\Phi)D_{-}\Phi\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{-}\overline{\Phi}+\partial_{++}\overline{\Phi}\partial_{--}\overline{\Phi}D_{-}\Phi
D_{+}\Phi$ $\displaystyle\quad-\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{-}\overline{\Phi}\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{+}\overline{\Phi}\left(D_{+}D_{-}\Phi\right)^{2}-i\partial_{--}\overline{\Phi}(D_{+}D_{-}\Phi)D_{+}\Phi\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{+}\overline{\Phi}\Big{]}~{}.$ (11)

The supercurrent-squared flow then induces a differential equation for the
superspace Lagrangian $\mathcal{A}_{\lambda}$ (where, again,
$\mathcal{L}_{\lambda}=\int d^{4}\theta\,\mathcal{A}_{\lambda}$) given by

$\displaystyle\frac{d}{d\lambda}\mathcal{A}_{\lambda}$
$\displaystyle=-\frac{1}{8}{\mathcal{O}}=\frac{1}{8}\left(\mathcal{J}_{++}\mathcal{J}_{--}-2\mathcal{V}\overline{\mathcal{V}}\right)~{}.$
(12)

Given our ansatz (7), we see that

$\displaystyle\frac{d\mathcal{A}_{\lambda}}{d\lambda}=\frac{df}{d\lambda}K_{\Phi\overline{\Phi}}^{2}\,D_{+}\Phi\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{+}\overline{\Phi}D_{-}\Phi\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{-}\overline{\Phi}~{}.$
(13)

On the other hand, plugging in our expressions (9), (10), and (11) for the
supercurrents into the right hand side of (12) also gives a result
proportional to $K_{\Phi\overline{\Phi}}^{2}D_{+}\Phi\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{+}\overline{\Phi}D_{-}\Phi\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{-}\overline{\Phi}$.
Equating the coefficients, we find a differential equation for $f$:

$\displaystyle\frac{df}{d\lambda}=$
$\displaystyle\,\frac{1}{2}\Bigg{\\{}-\overline{x}y\left[f+(\overline{x}-x)\frac{\partial
f}{\partial\overline{x}}+(y-x)\frac{\partial f}{\partial
y}\right]^{2}-xy\left[f+(x-\overline{x})\frac{\partial f}{\partial
x}+(y-\overline{x})\frac{\partial f}{\partial y}\right]^{2}$
$\displaystyle+2(x-y)(y-\overline{x})\left[f+y\frac{\partial f}{\partial
y}+\overline{x}\frac{\partial f}{\partial\overline{x}}+x\frac{\partial
f}{\partial x}\right]^{2}+x\overline{x}\left[f+(\overline{x}-y)\frac{\partial
f}{\partial\overline{x}}+(x-y)\frac{\partial f}{\partial x}\right]^{2}$
$\displaystyle+\left[1+(x+\overline{x}-3y)f+(x+\overline{x}-2y)y\frac{\partial
f}{\partial y}+\overline{x}(x-y)\frac{\partial
f}{\partial\overline{x}}+x(\overline{x}-y)\frac{\partial f}{\partial
x}\right]^{2}\Bigg{\\}}~{}.$ (14)

In particular, this shows that our ansatz (7) for the finite-$\lambda$
superspace action is consistent: the supercurrent-squared deformation closes
on an action of this form. It could have been otherwise: the flow equation
might have sourced additional terms proportional, say, to two-fermion
combinations $D_{+}\Phi\mkern 1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{+}\overline{\Phi}$, or required dependence on other dimensionless
variables like $\lambda(D_{+}D_{-}\Phi)^{2}$, but these complications do not
arise in the case where the undeformed theory only has a Kähler potential.

On dimensional grounds, $f$ must be proportional to $\lambda$ times a function
of the dimensionless combinations $\lambda x$ and $\lambda y$. Thus, although
the differential equation for $f$ determined by (14) is complicated, one can
solve order-by-order in $\lambda$. The solution to $\mathcal{O}(\lambda^{3})$
is

$\displaystyle f(\lambda,x,\overline{x},y)=$
$\displaystyle\,\frac{\lambda}{2}+\lambda^{2}\left(\frac{x+\overline{x}}{4}-\frac{3}{4}y\right)$
(15)
$\displaystyle+\lambda^{3}\left(\frac{x^{2}+\overline{x}^{2}+3x\overline{x}}{8}+\frac{37}{24}y^{2}-\frac{25}{24}\left(x+\overline{x}\right)y\right)+\cdots~{}.$
(16)

We were unable to find a closed-form expression for $f$ to all orders in
$\lambda$. However, the differential equation simplifies dramatically when we
impose the equations of motion for the theory, and in this case one can write
down an exact formula. This is similar to the $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ flow of the free action
for a real ${\cal N}=(1,1)$ scalar multiplet that was analyzed in
[Baggio:2018rpv, Chang:2018dge].

We claim that, on-shell, one may drop any terms where
$y\sim(D_{+}D_{-}\Phi)(\mkern 1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{+}\mkern 1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{-}\overline{\Phi})$ multiplies the four-fermion term $|D\Phi|^{4}\equiv
D_{+}\Phi\mkern 1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{+}\overline{\Phi}D_{-}\Phi\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{-}\overline{\Phi}$.
This is shown explicitly in Appendix 7.E and follows directly from the
superspace equation of motion and nilpotency of the fermionic terms
$D_{\pm}\Phi$ and $\mkern 1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{\pm}\overline{\Phi}$. It is also an intuitive statement associated to
the fact that for these models, on-shell, ${\cal N}=(2,2)$ supersymmetry is
not broken. In fact, note that the superfields $(D_{+}D_{-}\Phi)$ and $(\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{+}\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{-}\overline{\Phi})$
have as their lowest components the auxiliary fields $F$ and $\overline{F}$.
If supersymmetry is not broken, the vev of $F$ has to be zero, $\langle
F\rangle=0$, which implies that the auxiliary field $F$ is on-shell at least
quadratic in fermions and, more precisely, can be proven to be at least linear
in $\psi_{\pm}=D_{\pm}\Phi|_{\theta=0}$ and $\bar{\psi}_{\pm}=\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{\pm}\overline{\Phi}|_{\theta=0}$. From this argument it follows that
on-shell $(D_{+}D_{-}\Phi)$ is at least linear in $D_{\pm}\Phi$ and $\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{\pm}\overline{\Phi}$,
and then the two conditions $(D_{+}D_{-}\Phi)|D\Phi|^{4}=0$ and
$y|D\Phi|^{4}=0$ follow.

After removing from (14) the $y$-dependent terms which vanish on-shell, we
find a simpler differential equation for the function $f$,

$\displaystyle\frac{df}{d\lambda}$
$\displaystyle=\frac{1}{2}\left\\{-x\overline{x}\left[f+x\frac{\partial
f}{\partial x}+\overline{x}\frac{\partial
f}{\partial\overline{x}}\right]^{2}+\left[1+(x+\overline{x})f+x\overline{x}\left(\frac{\partial
f}{\partial x}+\frac{\partial
f}{\partial\overline{x}}\right)\right]^{2}\right\\}~{},$ (17)

whose solution is

$\displaystyle
f(\lambda,x,\overline{x},y=0)=\frac{\lambda}{1-\frac{\lambda}{2}(x+\overline{x})+\sqrt{1-\lambda(x+\overline{x})+\frac{\lambda^{2}}{4}\left(x-\overline{x}\right)^{2}}}~{}.$
(18)

Thus we have shown that the supercurrent-squared deformed Lagrangian at finite
$\lambda$ is equivalent on-shell to the following superspace Lagrangian

$\displaystyle\mathcal{L}_{\lambda}$ $\displaystyle=\int
d^{4}\theta\,\left(K(\Phi,\overline{\Phi})+\frac{\lambda
K_{\Phi\overline{\Phi}}^{2}D_{+}\Phi\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{+}\overline{\Phi}D_{-}\Phi\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{-}\overline{\Phi}}{1-\frac{1}{2}\lambda
K_{\Phi\overline{\Phi}}^{2}A+\sqrt{1-\lambda
K_{\Phi\overline{\Phi}}^{2}A+\frac{1}{4}\lambda^{2}K_{\Phi\overline{\Phi}}^{4}B^{2}}}\right)~{},$
(19)

where

$\displaystyle
A=\partial_{++}\Phi\partial_{--}\overline{\Phi}+\partial_{++}\overline{\Phi}\partial_{--}\Phi~{},\qquad
B=\partial_{++}\Phi\partial_{--}\overline{\Phi}-\partial_{++}\overline{\Phi}\partial_{--}\Phi~{}.$
(20)

When $K(\Phi,\overline{\Phi})=\overline{\Phi}\Phi$, it is simple to show that
this model represents an $\mathcal{N}=(2,2)$ off-shell supersymmetric
extension of the $D=4$ Nambu-Goto string in an appropriate gauge— often
referred to as a static gauge in presence of a $B$ field, though it can be
more naturally described as uniform light-cone gauge [Arutyunov:2004yx,
Arutyunov:2005hd] (see refs. [Baggio:2018gct, Frolov:2019nrr] for a discussion
of this point). In particular, by setting various component fields to zero and
performing the superspace integrals, one can show that (19) matches the
expected answer for $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu$ deformations in previously known non-supersymmetric cases. For
instance, setting the fermions to zero and integrating out the auxiliary
fields $F$ and $\overline{F}$ gives the $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformation of the
complex free boson $\phi$, whose Lagrangian is

${\cal L}_{\lambda,\text{bos}}=\frac{\sqrt{1+2\lambda
a+\lambda^{2}b^{2}}-1}{4\lambda}=\frac{a}{4}-\lambda\frac{\partial_{++}\phi\partial_{--}\phi\partial_{++}\bar{\phi}\partial_{--}\bar{\phi}}{1+\lambda
a+\sqrt{1+2\lambda a+\lambda^{2}b^{2}}}~{},$ (21)

where

$a=\partial_{++}\phi\partial_{--}\bar{\phi}+\partial_{++}\bar{\phi}\partial_{--}\phi~{},\qquad
b=\partial_{++}\phi\partial_{--}\bar{\phi}-\partial_{++}\bar{\phi}\partial_{--}\phi~{}.$
(22)

The Lagrangian (21) indeed describes the $D=4$ light-cone gauge-fixed Nambu-
Goto string model.

Alternatively, setting all the bosons to zero in (19) can be shown to give the
$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformation of
a complex free fermion. These calculations are similar to those in the case of
the $(0,2)$ supercurrent-squared action, which are presented in
[Jiang:2019hux]. In fact, it can even be easily shown that an ${\cal N}=(0,2)$
truncation of (19) gives precisely the $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformation of a free
${\cal N}=(0,2)$ chiral multiplet that was derived in [Jiang:2019hux].

It is worth highlighting that, unlike the ${\cal N}=(2,2)$ case, an off-shell
$(0,2)$ chiral scalar multiplet contains only physical degrees of freedom and
no auxiliary fields. Interestingly, related to this fact, it turns out that
(up to integration by parts and total derivatives) the ${\cal N}=(0,2)$ off-
shell supersymmetric extension of the $D=4$ Nambu-Goto string action in light-
cone gauge is unique and precisely matches the off-shell $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformation of a free
${\cal N}=(0,2)$ chiral multiplet action [Jiang:2019hux].

In the ${\cal N}=(2,2)$ case, because of the presence of the auxiliary field
$F$ in the chiral multiplet $\Phi$, there are an infinite set of inequivalent
${\cal N}=(2,2)$ off-shell extensions of the Lagrangian (21) that are all
equivalent on-shell. A representative of these equivalent actions is described
by (19) when $K(\Phi,\overline{\Phi})=\overline{\Phi}\Phi$.

The non-uniqueness of dynamical systems described by actions of the form (19)
can also be understood by noticing that, for example, it is possible to
perform a class of redefinitions that leaves the action (19) invariant on-
shell. As a (very particular) example, note that we are free to perform a
shift of the form

$\displaystyle D_{+}\mkern 1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{-}\left(\mkern 1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{+}\overline{\Phi}D_{-}\Phi\right)\longrightarrow D_{+}\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{-}\left(\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{+}\overline{\Phi}D_{-}\Phi\right)+a\left(D_{+}D_{-}\Phi+\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{+}\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{-}\overline{\Phi}\right)^{2}$ (23)

for any real number $a$. In terms of $A$ and $B$, (23) implements the shifts

$\displaystyle\begin{split}A\longrightarrow
A+a\left(\left(D_{+}D_{-}\Phi\right)^{2}+2y+\left(\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{+}\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{-}\overline{\Phi}\right)^{2}\right)~{},\qquad B\longrightarrow
B\end{split}$ (24)

in (19). The resulting Lagrangian would enjoy the same on-shell
simplifications described in Appendix 7.E and would turn out to be on-shell
equivalent to the Lagrangian (19). In this infinite set of on-shell equivalent
actions, a particular choice would represent an exact solution of the $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ flow equation
(12)–(14), whose leading terms in a $\lambda$ series expansion are given in
(16). Another representative in this on-shell equivalence class is the
simplified model described by (19).

These types of redefinition and on-shell equivalentness are not a surprise,
nor really new. In fact, they are of the same nature as redefinitions that
have been studied in detail in [GonzalezRey:1998kh] (see also [Kuzenko:2011tj]
for a description of these types of “trivial symmetries”) in the context of
$D=4$ ${\cal N}=1$ chiral and linear superfield models possessing a non-
linearly realised additional supersymmetry [Bagger:1997pi,
GonzalezRey:1998kh]. As in (24), the field redefinition in this context does
not affect the dynamics of the physical fields— it basically corresponds only
to an arbitrariness in the definition of the auxiliary fields that always
appear quadratically in the action and then are set to zero (up to fermion
terms that will not contribute due to nilpotency in the action) on-shell.
Although here we only focused on discussing the on-shell ambiguity of the
solution of the ${\cal N}=(2,2)$ $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ flow, we expect that
the exact solution of the flow equations with $y$ nonzero (12)–(14) can be
found by a field redefinition of the kind we made in the action (19).

It is also interesting to note that similar freedoms and field redefinitions
are also described in the construction of $D=4$ ${\cal N}=1$ supersymmetric
Born-Infeld actions; see, for example, [Bagger:1996wp]. In fact, as will be
analyzed in more detail elsewhere [SUSYDBITTbar], it can be shown that the
Lagrangian (19) is structurally of the type described by Bagger and Galperin
for the $D=4$ ${\cal N}=1$ supersymmetric Born-Infeld action [Bagger:1996wp].
The equivalence can be formally shown by identifying $W_{+}=\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{+}\overline{\Phi}$,
$W_{-}=D_{-}\Phi$, $W^{2}=\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{+}\overline{\Phi}D_{-}\Phi$, and
$D^{\alpha}W_{\alpha}=D_{+}D_{-}\Phi+\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{-}\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{+}\overline{\Phi}$ to
match their conventions. As a consequence, we can show that our solution for
the $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ flow
possesses a second non-linearly realised ${\cal N}=(2,2)$ supersymmetry,
besides the $(2,2)$ supersymmetry which is made manifest by the superspace
construction, which is discussed in much greater detail in Chapter 4. We note
that the presence of a second supersymmetry is analogous to what happens in
the ${\cal N}=(0,2)$ case [Jiang:2019hux].

####  2 Adding a superpotential

Now suppose we begin with an undeformed theory that has a superpotential
$W(\Phi)$,

$\displaystyle\mathcal{L}^{(0)}=\int
d^{4}\theta\,K(\Phi,\overline{\Phi})+\left(\int
d^{2}\theta\,W(\Phi)\right)+\left(\int
d^{2}\overline{\theta}\,\overline{W}(\overline{\Phi})\right)~{}.$ (25)

As shown in Appendix 7.D, the superpotential F-term gives a contribution
$\delta\mathcal{V}=2W(\Phi)$ to the field $\mathcal{V}$ which appears in
supercurrent-squared. To leading order in the deformation parameter, the
Lagrangian takes the form

$\displaystyle\mathcal{L}^{(0)}$
$\displaystyle\to\mathcal{L}^{(0)}+\mathcal{L}^{(1)}$
$\displaystyle=\mathcal{L}^{(0)}+\lambda\int
d^{4}\theta\,\left(\frac{1}{2}K_{\Phi\overline{\Phi}}^{2}D_{+}\Phi\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{+}\overline{\Phi}D_{-}\Phi\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{-}\overline{\Phi}+W(\Phi)\mkern
1.5mu\overline{\mkern-1.5muW\mkern-1.5mu}\mkern
1.5mu(\overline{\Phi})\right)~{}.$ (26)

In addition to the four-fermion term which we saw in section 1, we see that
the deformation modifies the Kähler potential, adding a term proportional to
$|W(\Phi)|^{2}$.

Next consider the second order term in $\lambda$. For convenience, we use the
combination $|D\Phi|^{4}=D_{+}\Phi\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{+}\overline{\Phi}D_{-}\Phi\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{-}\overline{\Phi}$,
which is the four-fermion combination that appeared at first order. Then

$\displaystyle\mathcal{L}^{(2)}=\frac{\lambda^{2}}{4}\int
d^{4}\theta\,\Big{(}x+\overline{x}-3y-2|W^{\prime}(\Phi)|^{2}+WD_{-}D_{+}+\mkern
1.5mu\overline{\mkern-1.5muW\mkern-1.5mu}\mkern 1.5mu\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{+}\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{-}\Big{)}|D\Phi|^{4}~{}.$ (27)

The new terms involving supercovariant derivatives of $|D\Phi|^{4}$ will
generate contributions with two fermions in the D-term.

As we continue perturbing to higher orders, the form of the superspace
Lagrangian becomes more complicated. It is no longer true that the
supercurrent-squared flow closes on a simple ansatz with one undetermined
function, as it did in the case with only a Kähler potential. Indeed, the
finite-$\lambda$ deformed superspace Lagrangian in the case with a
superpotential will depend not only on the variables $x$, $\overline{x}$, and
$y$ as in section 1, but also, for example, on combinations like
$\partial_{++}\Phi\mkern 1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{+}\mkern 1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{-}\overline{\Phi}$, which can appear multiplying the two-fermion term
$D_{-}\Phi\mkern 1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{-}\overline{\Phi}$ in the superspace Lagrangian. To find the full
solution, one would need to determine several functions contributing to the
D-term—one multiplying the four-fermion term $|D\Phi|^{4}$ as in the Kähler
case; one for the deformed Kähler potential which may now depend on $x$, $y$,
and other combinations; and four functions multiplying the two-fermion terms
$D_{+}\Phi D_{-}\Phi$, $D_{+}\Phi\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{-}\overline{\Phi}$,
etc. Each function can depend on several dimensionless combinations.

In the presence of a superpotential, the situation might further be
complicated by the fact that supersymmetry can be spontaneously broken. This
would make it impossible, for example, to use on-shell simplifications like
$y|D\Phi|^{4}=0$ that we employed in the section 1, where supersymmetry is
never spontaneously broken.

It should be clear that the case with a superpotential is significantly more
involved and rich than just a pure Kähler potential. In this case, we have not
attempted to find a solution of the $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ flow equation in closed
form. However, it is evident from the form of supercurrent-squared eq.
(12)—which is always written as a D-term integral of current bilinears— that
this deformation will only affect the $D$ term and not the ${\cal N}=(2,2)$
superpotential $W$ appearing in the chiral integral. Therefore the
superpotential, besides being protected from perturbative quantum corrections,
is also protected from corrections along the supercurrent-squared flow.

####  3 The physical classical potential

In view of the difficulty of finding the all-orders deformed superspace action
for a theory with a superpotential, we now consider the simpler problem of
finding the local-potential approximation (or zero-momentum potential) for the
bosonic complex scalar $\phi$ contained in the superfield $\Phi$. We stress
that our analysis here is purely classical and we will make a couple of
comments about possible quantum effects later in this section. For simplicity,
we will also restrict to the case in which the Kähler potential is flat,
$K(\Phi,\overline{\Phi})=\overline{\Phi}\Phi$. By “zero-momentum potential” we
mean the physical potential $V(\phi)$ which appears in the Lagrangian after
performing the superspace integral in the deformed theory and then setting
$\partial_{\pm\pm}\phi=0$. For instance, consider the undeformed Lagrangian

$\displaystyle\mathcal{L}^{(0)}=\int d^{4}\theta\,\overline{\Phi}\Phi+\int
d^{2}\theta\,W(\Phi)+\int d^{2}\overline{\theta}\,\mkern
1.5mu\overline{\mkern-1.5muW\mkern-1.5mu}\mkern 1.5mu(\overline{\Phi})~{}.$
(28)

When we ignore all terms involving derivatives and the fermions $\psi_{\pm}$,
the only contributions to the physical Lagrangian (after performing the
superspace integral) come from an $|F|^{2}$ term from the kinetic term, plus
the term $W(\Phi)=W(\phi)+W^{\prime}(\phi)\theta^{+}\theta^{-}F$. This gives
us the zero-momentum, zero-fermion component action

$\displaystyle S=\int d^{2}x\,\left(|F|^{2}+W^{\prime}(\phi)F+\mkern
1.5mu\overline{\mkern-1.5muW\mkern-1.5mu}\mkern 1.5mu^{\prime}(\mkern
1.5mu\overline{\mkern-1.5mu\phi\mkern-1.5mu}\mkern
1.5mu)\overline{F}\right)~{}.$ (29)

We may integrate out the auxiliary field $F$ using its equation of motion
$\overline{F}=-W^{\prime}(\phi)$, which yields

$\displaystyle S=\int d^{2}x\,\left(-|W^{\prime}(\phi)|^{2}\right)~{},$ (30)

so the zero-momentum potential for $\phi$ is $V=|W^{\prime}(\phi)|^{2}$, as
expected. Note that the previous potential might have extrema that breaks
${\cal N}=(2,2)$ supersymmetry while supersymmetric vacua will always set
$\langle F\rangle=\langle W^{\prime}(\phi)\rangle=0$. We will assume
supersymmetry of the undeformed theory not to be spontaneously broken in our
discussion.

Now suppose we deform by the supercurrent-squared operator to second order in
$\lambda$, which gives the superspace expression (27). If we again perform the
superspace integral and discard any terms involving derivatives or fermions,
we now find the physical Lagrangian

$\displaystyle\mathcal{L}\big{|}_{\partial_{\pm\pm}\phi=0}$ $\displaystyle=$
$\displaystyle|F|^{2}+FW^{\prime}+\overline{F}\mkern
1.5mu\overline{\mkern-1.5muW\mkern-1.5mu}\mkern
1.5mu^{\prime}+\lambda\left(\frac{1}{2}|F|^{4}-|F|^{2}|W^{\prime}|^{2}\right)+\frac{1}{4}\lambda^{2}|F|^{4}\left(W^{\prime}F+\mkern
1.5mu\overline{\mkern-1.5muW\mkern-1.5mu}\mkern
1.5mu^{\prime}\overline{F}\right)$ (31)
$\displaystyle-\frac{1}{2}\lambda^{2}|W^{\prime}|^{2}|F|^{4}+\frac{3}{4}\lambda^{2}|F|^{6}~{}.$

Remarkably, the equations of motion for the auxiliary $F$ in (31) admit the
solution $F=-\mkern 1.5mu\overline{\mkern-1.5muW\mkern-1.5mu}\mkern
1.5mu^{\prime}(\mkern 1.5mu\overline{\mkern-1.5mu\phi\mkern-1.5mu}\mkern
1.5mu)$, $\overline{F}=-W^{\prime}(\phi)$, which is the same as the
unperturbed solution. This for instance implies that if we start from a
supersymmetric vacua in the undeformed theory we will remain supersymmetric
along the $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$
flow. On the one hand, this is not a surprise considering that we know the
$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ flow preserves
the structure of the spectrum, and in particular should leave a zero-energy
supersymmetric vacuum unperturbed. On the other hand, it is a reassuring check
to see this property explicitly appearing in our analysis.

Returning to (31) and integrating out the auxiliary fields gives

$\displaystyle\mathcal{L}\big{|}_{\partial_{\pm\pm}\phi=0}=-|W^{\prime}(\phi)|^{2}-\frac{1}{2}\lambda|W^{\prime}(\phi)|^{4}-\frac{1}{4}\lambda^{2}|W^{\prime}(\phi)|^{6}~{}.$
(32)

These are the leading terms in the geometric series
$\frac{-|W^{\prime}|^{2}}{1-\frac{1}{2}\lambda|W^{\prime}|^{2}}$. In fact, up
to conventions for the scaling of $\lambda$, one could have predicted this
outcome from the form of the supercurrent-squared operator and the known
results for $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$
deformations of a bosonic theory with a potential [Cavaglia:2016oda]. We know
that, up to terms which vanish on-shell, the effect of adding supercurrent-
squared to the physical Lagrangian is to deform by the usual $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ operator. However, in
the zero-momentum sector, we see that the $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformation reduces to
deforming by the square of the potential:

$\displaystyle T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu\Big{|}_{\partial_{\pm\pm}\phi=0}=\mathcal{L}^{2}\Big{|}_{\partial_{\pm\pm}\phi=0}=V^{2}~{}.$
(33)

Therefore, it is easy to solve for the deformed potential if we deform a
physical Lagrangian
$\mathcal{L}=f(\lambda,\partial_{\pm\pm}\phi)+V(\lambda,\phi)$ by $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$, since the flow
equation for the potential term is simply

$\displaystyle\partial_{\lambda}\mathcal{L}=\frac{\partial
V}{\partial\lambda}=V^{2}~{},$ (34)

which admits the solution

$\displaystyle V(\lambda,\phi)=\frac{V(0,\phi)}{1-\lambda V(0,\phi)}~{}.$ (35)

We can apply this result to the Lagrangian (29), treating the entire
expression involving the auxiliary field $F$ as a potential (since it is
independent of derivatives). The deformed theory has a zero-momentum piece
which is therefore equivalent to

$\displaystyle S(\lambda)\Big{|}_{\partial_{\pm\pm}\phi=0}=\int
d^{2}x\,\frac{\left(|F|^{2}+W^{\prime}(\phi)F+\mkern
1.5mu\overline{\mkern-1.5muW\mkern-1.5mu}\mkern 1.5mu^{\prime}(\mkern
1.5mu\overline{\mkern-1.5mu\phi\mkern-1.5mu}\mkern
1.5mu)\overline{F}\right)}{1-\lambda\left(|F|^{2}+W^{\prime}(\phi)F+\mkern
1.5mu\overline{\mkern-1.5muW\mkern-1.5mu}\mkern 1.5mu^{\prime}(\mkern
1.5mu\overline{\mkern-1.5mu\phi\mkern-1.5mu}\mkern
1.5mu)\overline{F}\right)}~{},$ (36)

at least on-shell. Integrating out the auxiliary now gives

$\displaystyle S(\lambda)\Big{|}_{\partial_{\pm\pm}\phi=0}=\int
d^{2}x\,\frac{-|W^{\prime}(\phi)|^{2}}{1-\lambda|W^{\prime}(\phi)|^{2}}$ (37)

as the deformed physical potential. This matches the first few terms of (32),
up to a convention-dependent factor of $\frac{1}{2}$ in the scaling of
$\lambda$.

Now one might ask what superspace Lagrangian would yield the physical action
(37) after performing the $d\theta$ integrals. One candidate is

$\displaystyle\mathcal{L}(\lambda)\Big{|}_{\partial_{\pm\pm}\phi=0}\sim\int
d^{4}\theta\,\left(\overline{\Phi}\Phi-\lambda|W(\Phi)|^{2}\right)+\int
d^{2}\theta\,W(\Phi)+\int d^{2}\overline{\theta}\,\mkern
1.5mu\overline{\mkern-1.5muW\mkern-1.5mu}\mkern 1.5mu(\overline{\Phi})~{},$
(38)

where here $\sim$ means “this superspace Lagrangian gives an equivalent zero-
momentum physical potential for the boson $\phi$ on-shell.”

It is important to note that (38) is _not_ the true solution for the deformed
superspace Lagrangian using supercurrent-squared. The genuine solution
involves a four-fermion term, all possible two-fermion terms, and more
complicated dependence on the variable $y=\lambda(D_{+}D_{-}\Phi)(\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{+}\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{-}\overline{\Phi})$ in
the zero-fermion term. However, if one were to perform the superspace integral
in the true solution and then integrate out the auxiliary field $F$ using its
equation of motion, one would obtain the same zero-momentum potential for
$\phi$ as we find by performing the superspace integral in (38) and
integrating out $F$.

The form (38) is interesting because it shows that the effect of supercurrent-
squared on the physical potential for $\phi$ can be interpreted as a change in
the Kähler metric, which for this Lagrangian is

$\displaystyle K_{\Phi\overline{\Phi}}=1-\lambda|W^{\prime}(\Phi)|^{2}~{}.$
(39)

When one performs the superspace integrals in (38), the result is

$\displaystyle\mathcal{L}\Big{|}_{\partial_{\pm\pm}\phi=0}=K_{\Phi\overline{\Phi}}|F|^{2}+W^{\prime}(\phi)F+\mkern
1.5mu\overline{\mkern-1.5muW\mkern-1.5mu}\mkern 1.5mu^{\prime}(\mkern
1.5mu\overline{\mkern-1.5mu\phi\mkern-1.5mu}\mkern 1.5mu)\overline{F}~{},$
(40)

which admits the solution $F=-\frac{\mkern
1.5mu\overline{\mkern-1.5muW\mkern-1.5mu}\mkern 1.5mu^{\prime}(\mkern
1.5mu\overline{\mkern-1.5mu\phi\mkern-1.5mu}\mkern
1.5mu)}{K_{\Phi\overline{\Phi}}}$. Substituting this solution gives

$\displaystyle\mathcal{L}\Big{|}_{\partial_{\pm\pm}\phi=0}=\frac{-|W^{\prime}(\phi)|^{2}}{K_{\Phi\overline{\Phi}}}=\frac{-|W^{\prime}(\phi)|^{2}}{1-\lambda|W^{\prime}(\phi)|^{2}}~{},$
(41)

which agrees with (37).

As already mentioned, supersymmetric vacua of the original, undeformed, theory
are associated with critical points of the superpotential $W(\phi)$. Any
vacuum of the undeformed theory will persist in the deformed theory: near a
point where $W^{\prime}(\phi)=0$, we see that the physical potential
$V(\phi)=\frac{|W^{\prime}|^{2}}{1-\lambda|W^{\prime}|^{2}}$ also vanishes
(away from the pole $|W^{\prime}|^{2}=\frac{1}{\lambda}$, the deformed
potential is a monotonically increasing function of $|W^{\prime}|^{2}$).
Further, the auxiliary field $F$ does not acquire a vacuum expectation value
because $F=-\mkern 1.5mu\overline{\mkern-1.5muW\mkern-1.5mu}\mkern
1.5mu^{\prime}(\mkern 1.5mu\overline{\mkern-1.5mu\phi\mkern-1.5mu}\mkern
1.5mu)$ remains a solution to its equations of motion in the deformed theory.
Once more, this indicates that supersymmetry is unbroken along the whole
$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ flow if is in
the undeformed theory.

However, this classical analysis suggests that the soliton spectrum of the
theory has changed dramatically at any finite deformation parameter $\lambda$.
There are now generically poles in the physical potential $V(\phi)$ at points
where $|W^{\prime}|^{2}=\frac{1}{\lambda}$ which might separate distinct
supersymmetric vacua of the theory. For instance, if the original theory had a
double-well superpotential with two critical points $\phi_{1}$, $\phi_{2}$
where $W^{\prime}(\phi_{i})=0$, then this undeformed theory supports BPS
soliton solutions which interpolate between these two vacua. But if the
superpotential $W$ reaches a value of order $\frac{1}{\lambda}$ at some point
between $\phi_{1}$ and $\phi_{2}$, then this soliton solution appears naively
forbidden in the deformed theory because it requires crossing an infinite
potential barrier. Another way of seeing this is by considering the effective
Kähler potential (39), which would change sign at some point between the two
supersymmetric vacua in the deformed theory and thus give rise to a negative-
definite Kähler metric.

Our discussion has been purely classical. As we emphasised in the
introduction, a fully quantum analysis of this problem is desirable, though
subtle because of the non-local nature of the $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformation.777For an
analysis of correlation functions in theories deformed by supercurrent-squared
operators, which may offer a first step towards understanding the quantum
properties of these deformations, see [Ebert:2020tuy]. The advantage of
performing such an analysis in models with extended supersymmetry is that
holomorphy and associated non-renormalization theorems provide control over
the form of any possible quantum corrections. For example, the superpotential
for the models studied in this work is not renormalized perturbatively along
the flow. It would be interesting to examine the structure of perturbative
quantum corrections along the lines of [Rosenhaus:2019utc], but in superspace
with manifest supersymmetry. It should be possible to absorb any quantum
corrections visible in perturbation theory by a change in the $D$-term Kähler
potential meaning that at least the structure of the supersymmetric vacua
would be preserved.

##  Chapter 4 Non-Linearly Realized Symmetries

In this chapter, we will investigate certain additional symmetries which are
present in some ${T\overline{T}}$ and supercurrent-squared deformed models.
The treatment of this chapter follows the paper “Non-Linear Supersymmetry and
${T\overline{T}}$-like Flows” [Ferko:2019oyv].

It was briefly mentioned in Chapter 3 that our solution (19) for the
supercurrent-squared deformation of a chiral multiplet possesses a second non-
linearly realized supersymmetry – which we will demonstrate in more detail
shortly – but there are already examples of such additional symmetries in the
non-supersymmetric context. For instance, we reviewed in Section 2 that the
${T\overline{T}}$ deformation of a free boson is given by

$\displaystyle\mathcal{L}_{\lambda}=\frac{1}{2\lambda}\left(\sqrt{1+2\lambda\partial_{\mu}\phi\partial^{\mu}\phi+1}-1\right)\,.$
(1)

Ignoring the constant $-\frac{1}{2\lambda}$, this is of the same form as the
Dirac Lagrangian which describes the transverse fluctuations $\phi$ on a
brane. The Dirac action for a $p$-brane is usually written

$\displaystyle S_{\text{Dirac}}=-T_{p}\int
d^{p+1}\xi\,\sqrt{-\det(\gamma)}\,,$ (2)

where $T_{p}$ is the tension of the $Dp$-brane and

$\displaystyle\gamma_{ab}=\frac{\partial
X^{\mu}}{\partial\xi^{a}}\frac{\partial
X^{\nu}}{\partial\xi^{b}}\eta_{\mu\nu}$ (3)

is the pullback of the target space metric $\eta_{\mu\nu}$ onto the brane’s
worldvolume. We can choose static gauge which identifies the $p+1$ worldvolume
coordinates $\xi^{a}$ with the target space coordinates $X^{a}$, and the
remaining worldvolume coordinates are transverse oscillations:

$\displaystyle\begin{split}X^{a}&=\xi^{a}\,,\qquad\qquad\;\,a=0,\cdots,p\,,\\\
X^{I}&=2\pi\alpha^{\prime}\phi^{I}(\xi)\,,\quad I=p+1,\cdots
D-1\,.\end{split}$ (4)

Here $D$ is the total dimension of the target spacetime and $\alpha^{\prime}$
is a parameter which is proportional to the square of the string length. In
the case of a $D1$-string embedded in a three dimensional spacetime, so that
$p=1$ and $D=3$, the Dirac action can be written as

$\displaystyle S_{\text{Dirac}}=-T_{p}\int
d^{2}\xi\sqrt{1+2\pi\alpha^{\prime}\partial^{a}\phi\partial_{a}\phi}\,,$ (5)

which is the same form as the ${T\overline{T}}$ deformed Lagrangian (1). Here
$\alpha^{\prime}$ plays the role of the ${T\overline{T}}$ parameter $\lambda$.

It is well-known that the $Dp$-brane Lagrangian (2) possesses a non-linearly
realized symmetry for any $p$. We will describe this symmetry in the case (5)
of a $D1$-string. In this case, for either fixed index $a\in\\{0,1\\}$, the
action (5) is invariant under the transformation

$\displaystyle\delta\phi=\xi^{a}+\phi\partial^{a}\phi\,.$ (6)

The interpretation of this symmetry is that the presence of the brane as an
embedded hypersurface spontaneously breaks part of the Poincaré symmetry of
the ambient spacetime. The scalars $\phi$ are Nambu-Goldstone bosons of this
spontaneously broken symmetry, which is then non-linearly realized. One can
view the transformation (6) as a rotation which rotates the transverse
direction $\phi$ into the worldvolume direction $\xi^{a}$, breaking static
gauge, along with a compensating worldvolume diffeomorphism which then
restores static gauge. This symmetry, and its extensions to the Dirac-Born-
Infeld action which includes a gauge field, are nicely discussed in
[Gliozzi:2011hj, Casalbuoni:2011fq, Maxfield:2016vpw].

###  1 $D=2\;\,{\cal N}=(2,2)$ Flows and Non-Linear ${\cal N}=(2,2)$
Supersymmetry

In this section, we are going to explore in detail how the non-linear
supersymmetries – which we briefly alluded to in Chapter 3 – arise for the
simplest ${\cal N}=(2,2)$ ${T\overline{T}}$ flows. The undeformed models are
supersymmetrized theories of free scalars, while the deformed models are
${\cal N}=(2,2)$ supersymmetric extensions of the $D=4$ gauge-fixed Nambu-Goto
string.

####  1 $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$
deformations with ${\cal N}=(2,2)$ supersymmetry

In Chapter 3, supersymmetric flows for various theories were studied; the
conclusion of Section 2 was that $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu(x)$ operator of a
supersymmetric theory is related to a supersymmetric descendant operator
$\mathcal{T}\mkern 1.5mu\overline{\mkern-1.5mu\mathcal{T}\mkern-1.5mu}\mkern
1.5mu(x)$,

$\displaystyle\mathcal{T}\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{T}\mkern-1.5mu}\mkern 1.5mu(x)=T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu(x)+{\rm
EOM}+\partial_{++}(\cdots)+\partial_{--}(\cdots)~{}.$ (1)

The previous equation states the equivalence of $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu(x)$ and
$\mathcal{T}\mkern 1.5mu\overline{\mkern-1.5mu\mathcal{T}\mkern-1.5mu}\mkern
1.5mu(x)$ up to total derivatives and terms that vanish on-shell, which we
have indicated with “${\rm EOM}$”.

The simplest cases of deformed models, on which we will focus in this section,
are $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$-deformed
theories of free scalars, fermions and auxiliary fields. In the case of $D=2$
${\cal N}=(2,2)$ supersymmetry, a scalar multiplet can have several different
off-shell representations [Gates:1984nk, Buscher:1987uw, Grisaru:1997pg,
Lindstrom:2005zr]. The two cases we will consider here are chiral and twisted-
chiral supermultiplets, which are the most commonly studied cases.

In ${\cal N}=(2,2)$ superspace, parametrized by coordinates
$\zeta^{M}=(x^{\pm\pm},\theta^{\pm},{\bar{\theta}}^{\pm})$, let the complex
superfields $X(x,\theta)$ and $Y(x,\theta)$ satisfy chiral and twisted-chiral
constraints, respectively,

$\displaystyle\bar{D}_{\pm}X=0~{},\quad\bar{D}_{+}Y=D_{-}Y=0~{}.$ (2)

In this chapter, we use slightly different conventions for the supercovariant
derivatives than in Chapter 3. These were referred to as $D^{\prime}_{\pm}$
and $\mkern 1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu^{\prime}_{\pm}$ in equation (13), but we will drop the primes in the
following discussion. For convenience, the conventions are repeated here:

$\displaystyle
D_{\pm}=\frac{\partial}{\partial\theta^{\pm}}+i\bar{\theta}^{\pm}\partial_{\pm\pm}\,,\;\;\bar{D}_{\pm}=-\frac{\partial}{\partial\bar{\theta}^{\pm}}-i\theta^{\pm}\partial_{\pm\pm}~{}\;\;,$
(3) $\displaystyle
Q_{\pm}=i\frac{\partial}{\partial\theta^{\pm}}+\bar{\theta}^{\pm}\partial_{\pm\pm}~{},\;\;\bar{Q}_{\pm}=-i\frac{\partial}{\partial\bar{\theta}^{\pm}}-\theta^{\pm}\partial_{\pm\pm}\;\;.$
(4)

These satisfy

$\displaystyle
D_{\pm}^{2}=\bar{D}_{\pm}^{2}=0~{},\qquad\\{D_{\pm},\bar{D}_{\pm}\\}=-2i\partial_{\pm\pm}~{},\qquad[D_{\pm},\partial_{\pm\pm}]=[\bar{D}_{\pm},\partial_{\pm\pm}]=0~{},\quad$
(5) $\displaystyle
Q_{\pm}^{2}=\bar{Q}_{\pm}^{2}=0~{},\qquad\\{Q_{\pm},\bar{Q}_{\pm}\\}=-2i\partial_{\pm\pm}~{},\qquad[Q_{\pm},\partial_{\pm\pm}]=[\bar{Q}_{\pm},\partial_{\pm\pm}]=0~{}.\quad$
(6)

There is one more caveat worth mentioning: in much of the ${\cal N}=(2,2)$
literature, twisted-chiral multiplets, often denoted $\Sigma$ in this context,
naturally arise as field strengths for ${\cal N}=(2,2)$ vector superfields
$V$. The lowest component of such a superfield is a complex scalar, but the
top component proportional to $\bar{\theta}^{-}\theta^{+}$ encodes the gauge-
field strength along with a real auxiliary field. On the other hand, there are
twisted chiral superfields denoted $Y$ whose bottom component is a complex
scalar and whose top component is just a complex auxiliary field. In this work
we restrict to the latter case. The free Lagrangians for these supermultiplets
are given by

$\displaystyle\mathcal{L}^{\text{c}}_{0}=\int
d^{4}\theta\,X\bar{X}~{},\qquad\mathcal{L}^{\text{tc}}_{0}=-\int
d^{4}\theta\,Y\bar{Y}~{}.$ (7)

In [Chang:2019kiu] it was shown that the following Lagrangian

$\displaystyle\mathcal{L}^{\text{c}}_{\lambda}$ $\displaystyle=$
$\displaystyle\int d^{4}\theta\left(X\bar{X}+\frac{\lambda D_{+}X\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{+}\bar{X}D_{-}X\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{-}\bar{X}}{1-\frac{1}{2}\lambda A+\sqrt{1-\lambda
A+\frac{1}{4}\lambda^{2}B^{2}}}\right)~{},$ (8a) with $\displaystyle
A=\partial_{++}X\partial_{--}\bar{X}+\partial_{++}\bar{X}\partial_{--}X~{},\quad
B=\partial_{++}X\partial_{--}\bar{X}-\partial_{++}\bar{X}\partial_{--}X~{},$
(8b)

is a solution of the flow equation (10) on-shell, and hence describes the
$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$-deformation of
the free chiral supermultiplet Lagrangian (7).

A simple way to generate the $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$-deformation of the free
twisted-chiral theory is to remember that a twisted-chiral multiplet can be
obtained from a chiral one by acting with a $\mathbb{Z}_{2}$ automorphism on
the Grassmann coordinates of ${\cal N}=(2,2)$ superspace:

$\displaystyle\theta^{+}\leftrightarrow\theta^{+}~{},\quad\theta^{-}\leftrightarrow-\,\bar{\theta}^{-}~{}.$
(9)

This leaves the $D_{+}$ and $\bar{D}_{+}$ derivatives invariant while it
exchanges $D_{-}$ with $\bar{D}_{-}$. As a result, the chiral and twisted-
chiral differential constraints (2) are mapped into each others under the
automorphism (9).111In the literature this ${\mathbb{Z}}_{2}$ automorphism (9)
is often called the “mirror-map” or “mirror-image” because it exchanges the
vector and axial $U(1)$ R-symmetries.

Under the $\mathbb{Z}_{2}$ automorphism (9), the Lagrangian (8a) turns into
the following twisted-chiral Lagrangian

$\displaystyle\mathcal{L}^{\text{tc}}_{\lambda}$ $\displaystyle=$
$\displaystyle-\int d^{4}\theta\left(Y\bar{Y}+\frac{\lambda D_{+}Y\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{+}\bar{Y}\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{-}YD_{-}\bar{Y}}{1-\frac{1}{2}\lambda A+\sqrt{1-\lambda
A+\frac{1}{4}\lambda^{2}B^{2}}}\right)~{},$ (10a) where $\displaystyle
A=\partial_{++}Y\partial_{--}\bar{Y}+\partial_{++}\bar{Y}\partial_{--}Y~{},\quad
B=\partial_{++}Y\partial_{--}\bar{Y}-\partial_{++}\bar{Y}\partial_{--}Y~{}.$
(10b)

Thanks to the map (9), by construction the Lagrangian (10a) is a $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$-deformation and its
superspace Lagrangian ${\mathcal{A}}^{\rm tc}_{\lambda}$, ${\mathcal{L}}^{\rm
tc}_{\lambda}=\int d^{4}\theta\,{\mathcal{A}}^{\rm tc}_{\lambda}$, is an on-
shell solution of the following flow equation

$\displaystyle\frac{d}{d\lambda}\mathcal{A}^{\text{tc}}_{\lambda}=\frac{1}{8}\left(\mathcal{{\mathcal{R}}}_{++}\mathcal{{\mathcal{R}}}_{--}-2\mathcal{B}\bar{\mathcal{B}}\right)~{}.$
(11)

Here ${\mathcal{R}}_{\pm\pm}(x,\theta)$, $\mathcal{B}(x,\theta)$ and its
complex conjugate $\bar{\mathcal{B}}(x,\theta)$ are the local operators
describing the ${\mathcal{R}}$-multiplet of currents for $D=2$ ${\cal
N}=(2,2)$ supersymmetry that arise by applying (9) to the FZ multiplet of the
chiral theory (8a) [Chang:2019kiu]. They satisfy the conservation equations,

$\mkern 1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{+}\mathcal{R}_{--}=i\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{-}{\cal B}~{},\qquad
D_{-}\mathcal{R}_{++}=iD_{+}{\cal B}~{},\qquad\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{+}{\cal B}=D_{-}{\cal
B}=0~{},$ (12)

together with their complex conjugates. Like the case of the FZ-multiplet, the
supercurrent-squared operator

$\mathcal{T}\mkern 1.5mu\overline{\mkern-1.5mu\mathcal{T}\mkern-1.5mu}\mkern
1.5mu(x)=\int
d^{4}\theta\,\mathcal{O}^{{\mathcal{R}}}(x,\theta)~{},\quad{\mathcal{O}}^{{\mathcal{R}}}(x,\theta):=-\mathcal{R}_{++}(x,\theta)\mathcal{R}_{--}(x,\theta)+2\mathcal{B}(x,\theta)\bar{\mathcal{B}}(x,\theta)~{},$
(13)

satisfies (1); namely, $\mathcal{T}\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{T}\mkern-1.5mu}\mkern 1.5mu(x)$ is
equivalent to $T\bar{T}(x)$ up to total derivatives and ${\rm EOM}$, as we
showed in Chapter 3.

Note that the bosonic truncation of both (8a) and (10a) give the Lagrangian

${\cal L}_{\lambda,\text{bos}}=\frac{\sqrt{1+2\lambda
a+\lambda^{2}b^{2}}-1}{4\lambda}=\frac{a}{4}-\lambda\frac{\partial_{++}\phi\partial_{--}\phi\partial_{++}\bar{\phi}\partial_{--}\bar{\phi}}{1+\lambda
a+\sqrt{1+2\lambda a+\lambda^{2}b^{2}}}~{},$ (14)

where

$a=\partial_{++}\phi\partial_{--}\bar{\phi}+\partial_{++}\bar{\phi}\partial_{--}\phi~{},\qquad
b=\partial_{++}\phi\partial_{--}\bar{\phi}-\partial_{++}\bar{\phi}\partial_{--}\phi~{},$
(15)

and $\phi$ is either $\phi=X|_{\theta=0}$ or $\phi=Y|_{\theta=0}$. This is the
Lagrangian for the gauge-fixed Nambu-Goto string in four dimensions
[Cavaglia:2016oda].

The aim of the remainder of this section is to show that the Lagrangians (8a)
and (10a) are structurally identical to the Bagger–Galperin action for the
$D=4$ ${\cal N}=1$ supersymmetric Born-Infeld theory [Bagger:1996wp], which we
will analyse in detail in section 4. Since the Bagger–Galperin action
possesses a second non-linearly realized $D=4$, ${\cal N}=1$ supersymmetry, we
will show that the theories described by (8a) and (10a) also possess an extra
set of non-linearly realized ${\cal N}=(2,2)$ supersymmetries.

####  2 The $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu$-deformed twisted-chiral model and partial-breaking

Let us start with the twisted-chiral Lagrangian (10a) which, as we will show,
is the one more directly related to the $D=4$ Bagger-Galperin action. In
complete analogy to the $D=4$ case, we are going to show that (10a) is a model
for a Nambu-Goldstone multiplet of $D=2$ ${\cal N}=(4,4)\to{\cal N}=(2,2)$
partial supersymmetry breaking. The analysis is similar in spirit to the $D=4$
construction of the Bagger-Galperin action using $D=4$ ${\cal N}=2$ superspace
proposed by Roček and Tseytlin [Rocek:1997hi]; see also [Kuzenko:2015rfx,
Antoniadis:2017jsk, Antoniadis:2019gbd] for more recent analysis.

To describe manifest ${\cal N}=(4,4)$ supersymmetry we can use ${\cal
N}=(4,4)$ superspace which augments the ${\cal N}=(2,2)$ superspace
coordinates $\zeta^{M}=(x^{\pm\pm},\theta^{\pm},\bar{\theta}^{\pm})$ of the
previous section with the following additional complex Grassmann coordinates
$(\eta^{\pm},\bar{\eta}^{\pm})$. The extra supercovariant derivatives and
supercharges are given by

$\displaystyle{\mathcal{D}}_{+}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial\eta^{+}}+i\bar{\eta}^{+}\partial_{++}~{},\quad\bar{\mathcal{D}}_{+}=-\frac{\partial}{\partial\bar{\eta}^{+}}-i\eta^{+}\partial_{++}~{},$
(16a) $\displaystyle{\mathcal{Q}}_{+}$ $\displaystyle=$ $\displaystyle
i\frac{\partial}{\partial\eta^{+}}+\bar{\eta}^{+}\partial_{++}~{},\quad\bar{\mathcal{Q}}_{+}=-i\frac{\partial}{\partial\bar{\eta}^{+}}-\eta^{+}\partial_{++}~{},$
(16b)

with similar expressions for ${\mathcal{D}}_{-}$ and ${\mathcal{Q}}_{-}$. They
satisfy

$\displaystyle{\mathcal{D}}_{\pm}^{2}=\bar{\mathcal{D}}_{\pm}^{2}=0~{},\quad\\{{\mathcal{D}}_{\pm},\bar{\mathcal{D}}_{\pm}\\}=-2i\partial_{\pm\pm}~{},\quad[{\mathcal{D}}_{\pm},\partial_{\pm\pm}]=[\bar{\mathcal{D}}_{\pm},\partial_{\pm\pm}]=0~{},~{}~{}~{}~{}~{}~{}$
(17)
$\displaystyle{\mathcal{Q}}_{\pm}^{2}=\bar{\mathcal{Q}}_{\pm}^{2}=0~{},\quad\\{{\mathcal{Q}}_{\pm},\bar{\mathcal{Q}}_{\pm}\\}=-2i\partial_{\pm\pm}~{},\quad[{\mathcal{Q}}_{\pm},\partial_{\pm\pm}]=[\bar{\mathcal{Q}}_{\pm},\partial_{\pm\pm}]=0~{},~{}~{}~{}~{}~{}~{}$
(18)

while they (anti-)commute with all the usual $D_{\pm}$ and $Q_{\pm}$
operators.

Two-dimensional ${\cal N}=(4,4)$ supersymmetry can also be usefully described
in the language of ${\cal N}=(2,2)$ superspace. In this section, we will
largely refer to [Ivanov:2004yv] for such a description. In this approach from
the full $(4,4)$ supersymmetry, one copy of $(2,2)$ is manifest while a second
$(2,2)$ is hidden. For our goal of describing a model of partial supersymmetry
breaking, we view the hidden $(2,2)$ supersymmetry as broken and non-linearly
realized. We will derive such a description starting from ${\cal N}=(4,4)$
superspace and describe the broken/hidden supersymmetry using the $\eta^{\pm}$
directions.

The hidden supersymmetry transformation of a generic $D=2$ ${\cal N}=(4,4)$
superfield
$U=U(x^{\pm\pm},\theta^{\pm},{\bar{\theta}}^{\pm},\eta^{\pm},\bar{\eta}^{\pm})$
under the hidden $(2,2)$ supersymmetry is

$\delta
U=i(\epsilon^{+}{\mathcal{Q}}_{+}+\epsilon^{-}{\mathcal{Q}}_{-}-\bar{\epsilon}^{+}\bar{\mathcal{Q}}_{+}-\bar{\epsilon}^{-}\bar{\mathcal{Q}}_{-})U~{}.$
(19)

The $(2,2)$ supersymmetry, generated by the $Q_{\pm}$ and $\bar{Q}_{\pm}$
operators, will always be manifest and preserved, so we will not bother to
discuss it in detail. For convenience, we also introduce the chiral coordinate
$y^{\pm\pm}=x^{\pm\pm}+i\eta^{\pm}\bar{\eta}^{\pm}$. Using this coordinate,
the spinor covariant derivatives and supercharges take the form

$\displaystyle{\mathcal{D}}_{\pm}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial\eta^{\pm}}+2i\bar{\eta}^{\pm}\frac{\partial}{\partial
y^{\pm\pm}}~{},\quad\bar{\mathcal{D}}_{\pm}=-\frac{\partial}{\partial\bar{\eta}^{\pm}}~{},$
(20a) $\displaystyle{\mathcal{Q}}_{\pm}$ $\displaystyle=$ $\displaystyle
i\frac{\partial}{\partial\eta^{\pm}}~{},\quad\bar{\mathcal{Q}}_{\pm}=-i\frac{\partial}{\partial\bar{\eta}^{\pm}}-2\eta^{\pm}\frac{\partial}{\partial
y^{\pm\pm}}~{}.$ (20b)

After this technical introduction, let us turn to our main construction.
Consider a $(4,4)$ superfield which is chiral under the hidden $(2,2)$
supersymmetry:

$\bar{\mathcal{D}}_{\pm}{\bm{\mathcal{X}}}=0~{}.$ (21)

We can expand it in terms of hidden fermionic coordinates,

${\bm{\mathcal{X}}}=X+\eta^{+}X_{+}+\eta^{-}X_{-}+\eta^{+}\eta^{-}F~{},$ (22)

where $X=X(y^{\pm\pm},\theta^{\pm},\bar{\theta}^{\pm})$,
$X_{\pm}=X_{\pm}(y^{\pm\pm},\theta^{\pm},\bar{\theta}^{\pm})$ and
$F=F(y^{\pm\pm},\theta^{\pm},\bar{\theta}^{\pm})$ are themselves $(2,2)$
superfields. In the following discussion, we will keep the
$\theta^{\pm},\,\bar{\theta}^{\pm}$ dependence implicit. The hidden $(2,2)$
supersymmetry transformation rules can then be straightforwardly computed
using (19) and (22). They take the form

$\displaystyle\delta X$ $\displaystyle=$
$\displaystyle-\epsilon^{+}X_{+}-\epsilon^{-}X_{-}~{},$ (23a)
$\displaystyle\delta X_{\pm}$ $\displaystyle=$
$\displaystyle\mp\epsilon^{\mp}F-2i\bar{\epsilon}^{\pm}\partial_{\pm\pm}X,$
(23b) $\displaystyle\delta F$ $\displaystyle=$
$\displaystyle-2i\bar{\epsilon}^{-}\partial_{--}X_{+}+2i\bar{\epsilon}^{+}\partial_{++}X_{-}~{}.$
(23c)

The $\bm{\mathcal{X}}$ superfield is still reducible under ${\cal N}=(4,4)$
supersymmetry so we can put additional constraints on the $(2,2)$ superfields
$X$, $X_{\pm}$ and $F$. Here we will consider $(4,4)$ twisted multiplets, and
refer the reader to [Gates:1984nk, Gates:1983py, Gates:1995aj, Gates:1998fr,
Ivanov:1984ht, Ivanov:1984fe, Ivanov:1987mz] for a more detailed analysis. For
this discussion, we will follow the ${\cal N}=(2,2)$ superspace description of
[Ivanov:2004yv]. One type of twisted multiplet with $(4,4)$ supersymmetry can
be defined by setting

$X_{+}=\bar{D}_{+}\bar{Y},\qquad X_{-}=-\bar{D}_{-}Y~{},$ (24)

where $X$ and $Y$ are chiral and twisted-chiral, respectively, under the
manifest $(2,2)$ supersymmetry:

$\bar{D}_{+}X=\bar{D}_{-}X=\bar{D}_{+}Y=D_{-}Y=0~{},\qquad
D_{+}\bar{X}=D_{-}\bar{X}=D_{+}\bar{Y}=\bar{D}_{-}\bar{Y}=0~{}.$ (25)

The superfield (22) becomes

${{\bm{\mathcal{X}}}}=X+\eta^{+}\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{+}\bar{Y}-\eta^{-}\bar{D}_{-}Y+\eta^{+}\eta^{-}F~{}.$ (26)

The supersymmetry transformation rules then become

$\displaystyle\delta X$ $\displaystyle=$
$\displaystyle-\epsilon^{+}\bar{D}_{+}\bar{Y}+\epsilon^{-}\bar{D}_{-}Y~{},$
(27a) $\displaystyle\delta F$ $\displaystyle=$
$\displaystyle-2i\bar{\epsilon}^{-}\partial_{--}\bar{D}_{+}\bar{Y}-2i\bar{\epsilon}^{+}\partial_{++}\bar{D}_{-}Y~{},$
(27b)

while $\delta X_{\pm}$ remains the same as (23b). By using the conjugation
property for two fermions,
$\overline{\chi\xi}=\bar{\xi}\bar{\chi}=-\bar{\chi}\bar{\xi}$, and the
conjugation property $\overline{D_{+}A}=\bar{D}_{+}\bar{A}$ for a bosonic
superfield $A$, it follows that

$\delta\bar{X}=\bar{\epsilon}^{+}D_{+}Y-\bar{\epsilon}^{-}D_{-}\bar{Y}~{}.$
(28)

One can check that

$\delta\bar{D}^{2}\bar{X}=\bar{D}^{2}\delta\bar{X}=\bar{D}^{2}\Big{(}\bar{\epsilon}^{+}D_{+}Y-\bar{\epsilon}^{-}D_{-}\bar{Y}\Big{)}=2i\bar{\epsilon}^{+}\partial_{++}\bar{D}_{-}Y+2i\bar{\epsilon}^{-}\partial_{--}\bar{D}_{+}Y~{},$
(29)

where $\bar{D}^{2}=\bar{D}_{+}\bar{D}_{-}$. Note that in the first equality we
made use of the fact that the manifest and hidden $(2,2)$ supersymmetries are
independent. The supersymmetry transformation rule for $-\bar{D}^{2}X$ is then
exactly that of the auxiliary field $F$. Thus we can consistently set

$\displaystyle F=-\bar{D}^{2}\bar{X}~{},$ (30)

which is the last constraint necessary to describe a version of the $(4,4)$
twisted multiplet in terms of a chiral and twisted-chiral ${\cal N}=(2,2)$
superfields. The resulting $(4,4)$ superfield ${{\bm{\mathcal{X}}}}$, expanded
in terms of the hidden $(2,2)$ fermionic coordinates, takes the form

${{\bm{\mathcal{X}}}}=X+\eta^{+}\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu_{+}\bar{Y}-\eta^{-}\bar{D}_{-}Y-\eta^{+}\eta^{-}\bar{D}^{2}\bar{X}~{},$
(31)

which closely resembles the expansion of a $D=4$ ${\cal N}=2$ vector multiplet
when one identifies the analogue of the $D=4$ ${\cal N}=1$ chiral vector
multiplet field strength $W_{\alpha}$ with the $(2,2)$ chiral superfields
$\mkern 1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{+}\bar{Y}$ and
$\mkern 1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{-}Y$. Note in
particular that ${{\bm{\mathcal{X}}}}$ turns to be $(4,4)$ chiral:

$\bar{D}_{\pm}{\bm{\mathcal{X}}}=0~{},\quad\bar{\mathcal{D}}_{\pm}{\bm{\mathcal{X}}}=0~{}.$
(32)

To summarize: the entire $(4,4)$ off-shell twisted multiplet is described in
terms of one chiral and one twisted-chiral $(2,2)$ superfield, which possess
the following hidden $(2,2)$ supersymmetry transformations:

$\displaystyle\delta X$ $\displaystyle=$
$\displaystyle-\epsilon^{+}\bar{D}_{+}\bar{Y}+\epsilon^{-}\bar{D}_{-}Y~{},$
(33a) $\displaystyle\delta Y$ $\displaystyle=$
$\displaystyle\bar{\epsilon}^{-}D_{-}X+\epsilon^{+}\bar{D}_{+}\bar{X}~{}.$
(33b)

Let us now introduce the action for a free ${\cal N}=(4,4)$ twisted multiplet.
Taking the square of ${{\bm{\mathcal{X}}}}$ in (31) we obtain

${{\bm{\mathcal{X}}}}^{2}=\eta^{+}\eta^{-}\Big{(}-2X\bar{D}^{2}\bar{X}+2\bar{D}_{+}\bar{Y}\bar{D}_{-}Y\Big{)}+\ldots~{},$
(34)

where the ellipses denote terms that are not important for our analysis. Since
${\bm{\mathcal{X}}}$ and therefore ${\bm{\mathcal{X}}}^{2}$ are chiral
superfields, we can consider the chiral integral in the hidden direction

$\int
d\eta^{+}d\eta^{-}{{\bm{\mathcal{X}}}}^{2}=2X\bar{D}^{2}\bar{X}-2\bar{D}_{+}\bar{Y}\cdot\bar{D}_{-}Y~{}.$
(35)

Note also that, since $X$ and $Y$ are chiral and twisted-chiral under the
manifest supersymmetry (25), it follows that

$\displaystyle\int
d^{2}x\,d\theta^{+}d\theta^{-}d\bar{\theta}^{+}d\bar{\theta}^{-}(X\bar{X}-Y\bar{Y})$
$\displaystyle=\int
d^{2}x\,d\theta^{+}d\theta^{-}\bar{D}_{+}\bar{D}_{-}(X\bar{X}-Y\bar{Y})~{},$
$\displaystyle=\int
d^{2}x\,d\theta^{+}d\theta^{-}\Big{(}X\bar{D}_{+}\bar{D}_{-}\bar{X}-\bar{D}_{+}\bar{Y}\cdot\bar{D}_{-}Y\Big{)}\,,$
(36)

which can also be rewritten as

$\int
d^{2}x\,d\theta^{+}d\theta^{-}d\bar{\theta}^{+}d\bar{\theta}^{-}(X\bar{X}-Y\bar{Y})=\int
d^{2}x\,d\bar{\theta}^{+}d\bar{\theta}^{-}\Big{(}\bar{X}D_{+}D_{-}X-D_{+}Y\cdot
D_{-}\bar{Y}\Big{)}~{}.$ (37)

The sum of the two equations above yields

$4\int
d^{2}x\,d\theta^{+}d\theta^{-}d\bar{\theta}^{+}d\bar{\theta}^{-}(X\bar{X}-Y\bar{Y})=\int
d^{2}x\,d\theta^{+}d\theta^{-}d\eta^{+}d\eta^{-}{{\bm{\mathcal{X}}}}^{2}+{}c.c.~{}.$
(38)

The left-hand side has an enhanced ${\cal N}=(4,4)$ supersymmetry as discussed
in [Ivanov:2004yv]. This becomes manifest from our $(4,4)$ superspace
construction on the right-hand side.

To describe ${\cal N}=(4,4)\to{\cal N}=(2,2)$ supersymmetry breaking we can
appropriately deform the $(4,4)$ twisted multiplet. Analogous to the case of a
$D=4$ ${\cal N}=2$ vector multiplet deformed by a magnetic Fayet-Iliopoulos
term [Antoniadis:1995vb] (see also [Antoniadis:2019gbd, Rocek:1997hi,
Ivanov:1998jq, Kuzenko:2015rfx, Antoniadis:2017jsk]), we add a deformation
parameter to the auxiliary field $F$ of ${{\bm{\mathcal{X}}}}$, which is
deformed to

${{\bm{\mathcal{X}}}}_{\text{def}}=X+\eta^{+}D_{+}\bar{Y}-\eta^{-}\bar{D}_{-}Y-\eta^{+}\eta^{-}\Big{(}\bar{D}^{2}\bar{X}+\kappa\Big{)}~{}.$
(39)

Assuming that the auxiliary field $F$ gets a VEV, $\langle F\rangle=\kappa$ or
equivalently $\langle\mkern 1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern
1.5mu^{2}\bar{X}\rangle=0$, then by looking at the supersymmetry
transformations of $X_{\pm}$ for the deformed multiplet

$\displaystyle\delta X_{\pm}$ $\displaystyle=$
$\displaystyle\pm\epsilon^{\mp}\Big{(}\bar{D}^{2}\bar{X}+\kappa\Big{)}-2i\bar{\epsilon}^{\pm}\partial_{\pm\pm}X~{},$
(40)

we can see the ${\cal N}=(4,4)\to{\cal N}=(2,2)$ supersymmetry breaking
pattern arises; specifically, the hidden ${\cal N}=(2,2)$ is spontaneously
broken and non-linearly realized. For later use, it is important to stress
that, though the hidden transformations of $\delta X_{\pm}$ are modified by
the non-linear term proportional to $\kappa$, the hidden transformation of $X$
remains the same as in the undeformed case given in eq. (27a).

In analogy to the $D=4$ case of [Rocek:1997hi, Kuzenko:2015rfx,
Antoniadis:2017jsk], to describe the Goldstone multiplet associated to partial
supersymmetry breaking we impose the following nilpotent constraint on the
deformed $(4,4)$ twisted superfield:

${{\bm{\mathcal{X}}}}_{\text{def}}^{2}=0=-2\eta^{+}\eta^{-}\Big{(}X(\kappa+\bar{D}^{2}\bar{X})-\bar{D}_{+}\bar{Y}\cdot\bar{D}_{-}Y\Big{)}+\ldots~{}.$
(41)

This implies the constraint

$X\Big{(}\kappa+\bar{D}^{2}\bar{X}\Big{)}-\bar{D}_{+}\bar{Y}\cdot\bar{D}_{-}Y=0~{},$
(42)

which requires

$X=\frac{\bar{D}_{+}\bar{Y}\cdot\bar{D}_{-}Y}{\kappa+\bar{D}^{2}\bar{X}}=\frac{W^{2}}{{\kappa}+\bar{D}^{2}\bar{X}}~{},$
(43)

and its conjugate

$\bar{X}=-\frac{D_{+}Y\cdot
D_{-}\bar{Y}}{\kappa+D^{2}X}=-\frac{\bar{W}^{2}}{{\kappa}+D^{2}X}~{}.$ (44)

Here $\bar{D}^{2}=\bar{D}_{+}\bar{D}_{-},D^{2}=-D_{+}D_{-}$ and we have
introduced the superfields:

$\displaystyle W^{2}$ $\displaystyle=$ $\displaystyle-
X_{+}X_{-}=\bar{D}_{+}\bar{Y}\cdot\bar{D}_{-}Y=\bar{D}_{+}\bar{D}_{-}(Y\bar{Y})=\bar{D}^{2}(Y\bar{Y})~{},$
(45a) $\displaystyle\bar{W}^{2}$ $\displaystyle=$
$\displaystyle\bar{X}_{+}\bar{X}_{-}=-D_{+}Y\cdot
D_{-}\bar{Y}=-D_{+}D_{-}(Y\bar{Y})=D^{2}(Y\bar{Y})~{}.$ (45b)

The constraint (42) is the $D=2$ analogue of the Bagger-Galperin constraint
for a Maxwell-Goldstone multiplet for $D=4\,$ ${\cal N}=2\to{\cal N}=1$
supersymmetry breaking [Bagger:1996wp]. Combining (43) and (44) gives

$\kappa
X=\bar{D}^{2}(Y\bar{Y}-X\bar{X})=\bar{D}^{2}\Big{[}Y\bar{Y}-\frac{\bar{D}_{+}\bar{Y}\cdot\bar{D}_{-}Y\cdot
D_{-}\bar{Y}\cdot
D_{+}Y}{({\kappa}+D^{2}X)({\kappa}+\bar{D}^{2}\bar{X})}\Big{]}~{},$ (46)

which is consistent thanks to the $\kappa$ terms in the denominator. Because
of the four fermion coupling in the numerator of the last term, no fermionic
terms can appear in the denominator. So effectively we have the equation

$(\kappa+D^{2}X)_{\text{eff}}=\Big{(}\kappa+D^{2}\frac{W^{2}}{{\kappa}+\bar{D}^{2}\bar{X}}\Big{)}_{\text{eff}}=\kappa+\frac{D^{2}W^{2}}{\kappa+(\bar{D}^{2}\bar{X})_{\text{eff}}}~{},$
(47)

and its conjugate

$(\kappa+\bar{D}^{2}\bar{X})_{\text{eff}}=\kappa+\frac{\bar{D}^{2}\bar{W}^{2}}{\kappa+(D^{2}X)_{\text{eff}}}~{}.$
(48)

Solving them we get

$\displaystyle(D^{2}X)_{\text{eff}}$ $\displaystyle=$
$\displaystyle\frac{B-\kappa^{2}+\sqrt{B^{2}+2\kappa^{2}A+\kappa^{4}}}{2\kappa}~{},$
(49a) $\displaystyle(\bar{D}^{2}\bar{X})_{\text{eff}}$ $\displaystyle=$
$\displaystyle\frac{-B-\kappa^{2}+\sqrt{B^{2}+2\kappa^{2}A+\kappa^{4}}}{2\kappa}~{}.$
(49b)

Substituting these expressions into (46) gives

$X=\frac{1}{\kappa}\bar{D}^{2}\Upsilon~{},\qquad\bar{X}=\frac{1}{\kappa}D^{2}\Upsilon~{},\qquad\Upsilon=\bar{\Upsilon}=Y\bar{Y}-\frac{2W^{2}\bar{W}^{2}}{A+\kappa^{2}+\sqrt{B^{2}+2\kappa^{2}A+\kappa^{4}}}~{},$
(50)

where

$\displaystyle A$
$\displaystyle=D^{2}W^{2}+\bar{D}^{2}\bar{W}^{2}=\\{D^{2},\bar{D}^{2}\\}(Y\bar{Y})=\partial_{++}Y\partial_{--}\bar{Y}+\partial_{++}\bar{Y}\partial_{--}Y~{},$
(51) $\displaystyle B$
$\displaystyle=D^{2}W^{2}-\bar{D}^{2}\bar{W}^{2}=[D^{2},\bar{D}^{2}](Y\bar{Y})=\partial_{++}Y\partial_{--}\bar{Y}-\partial_{++}\bar{Y}\partial_{--}Y~{}.$
(52)

The result is that the ${\cal N}=(2,2)$ chiral part $X$ of the ${\cal
N}=(4,4)$ twisted multiplet is expressed in terms of the $(2,2)$ twisted-
chiral superfield $Y$. Thanks to the linearly realized construction in terms
of $(4,4)$ superfields, it is straightforward to obtain the non-linearly
realized ${\cal N}=(2,2)$ supersymmetry transformations for $Y$. In
particular, it suffices to look at the transformations of $D_{+}Y$ and $\mkern
1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu_{-}Y$ that can be
obtained by substituting back the composite expression for $X=X[Y]$ into the
transformations (40). By construction, these expressions ensure that $\delta
X$ transforms according to (27a).

Since $X$ is chiral under the manifest $(2,2)$ supersymmetry (25), we can
consider the chiral integral

$\displaystyle S_{\kappa^{2}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\kappa\int
d^{2}x\,d\theta^{+}d\theta^{-}X+c.c=-\frac{1}{2}\int
d^{2}x\,d\theta^{+}d\theta^{-}\bar{D}^{2}\Upsilon+c.c.$ (53) $\displaystyle=$
$\displaystyle-\int
d^{2}x\,d\theta^{+}d\theta^{-}d\bar{\theta}^{+}d\bar{\theta}^{-}\Upsilon~{}.$

A remarkable property of this action is that it is invariant under the hidden
non-linearly realized supersymmetry. Using (27a), we see that

$\displaystyle\delta S_{\kappa^{2}}$ $\displaystyle=-\frac{1}{2}\kappa\int
d^{2}xD_{+}D_{-}\delta X\Big{|}_{\theta=\bar{\theta}=0}+c.c.~{},$
$\displaystyle=-\frac{1}{2}\kappa\int
d^{2}x\Big{(}-2i\epsilon^{-}\partial_{--}D_{+}Y-2i\epsilon^{+}\partial_{++}D_{-}\bar{Y}\Big{)}\Big{|}_{\theta=\bar{\theta}=0}+c.c.=0~{},$
(54)

where we used the fact that $Y$ is a twisted-chiral superfield (25).

Explicitly, the action reads

$S_{\kappa^{2}}=-\int
d^{2}x\,d\theta^{+}d\theta^{-}d\bar{\theta}^{+}d\bar{\theta}^{-}\Bigg{(}Y\bar{Y}-\frac{2W^{2}\bar{W}^{2}}{\kappa^{2}+A+\sqrt{\kappa^{4}+2\kappa^{2}A+B^{2}}}\Bigg{)}~{},$
(55)

which precisely matches the model of eq. (10a) if we identify the coupling
constants:

$\lambda=-\frac{2}{\kappa^{2}}~{}.$ (56)

This shows explicitly that the $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$-deformation of the free
twisted-chiral action possesses a non-linearly realized ${\cal N}=(2,2)$
hidden supersymmetry.

####  3 The $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu$-deformed chiral model and partial-breaking

Let us now turn to the $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformation of the free
chiral model of eq. (8a). The construction follows the previous subsection
with the difference that we will start with a different formulation of the
$(4,4)$ twisted multiplet described in terms of $(2,2)$ superfields. Consider
again an ${\cal N}=(4,4)$ superfield which is chiral under the hidden (2,2)
supersymmetry:

$\bar{\mathcal{D}}_{+}{\bm{\mathcal{Y}}}=\bar{\mathcal{D}}_{-}{{\bm{\mathcal{Y}}}}=0~{}.$
(57)

Its expansion in hidden superspace variables is

${\bm{\mathcal{Y}}}=Y+\eta^{+}Y_{+}+\eta^{-}Y_{-}+\eta^{+}\eta^{-}G~{},$ (58)

where $Y=Y(y^{\pm\pm},\theta^{\pm},\bar{\theta}^{\pm})$,
$Y_{\pm}=Y_{\pm}(y^{\pm\pm},\theta^{\pm},\bar{\theta}^{\pm})$ and
$G=G(y^{\pm\pm},\theta^{\pm},\bar{\theta}^{\pm})$ are themselves superfields
with manifest $(2,2)$ supersymmetry. The hidden $(2,2)$ supersymmetry
transformation rules of the components are

$\displaystyle\delta Y$ $\displaystyle=$
$\displaystyle-\epsilon^{+}Y_{+}-\epsilon^{-}Y_{-}~{},$ (59a)
$\displaystyle\delta Y_{\pm}$ $\displaystyle=$
$\displaystyle\mp\epsilon^{\mp}G-2i\bar{\epsilon}^{\pm}\partial_{\pm\pm}Y~{},$
(59b) $\displaystyle\delta G$ $\displaystyle=$
$\displaystyle-2i\bar{\epsilon}^{-}\partial_{--}Y_{+}+2i\bar{\epsilon}^{+}\partial_{++}Y_{-}~{}.$
(59c)

This representation of $(4,4)$ off-shell supersymmetry is again reducible so
we can impose constraints. As in the construction of the previous section, we
impose

$Y_{+}=\bar{D}_{+}\bar{X}~{},\qquad Y_{-}=D_{-}X~{},$ (60)

then

${\bm{\mathcal{Y}}}=Y+\eta^{+}\bar{D}_{+}\bar{X}+\eta^{-}D_{-}X+\eta^{+}\eta^{-}G~{}.$
(61)

Here $X$ and $Y$ are consistently chosen to be chiral and twisted-chiral under
the manifest $(2,2)$ supersymmetry:

$\bar{D}_{+}X=\bar{D}_{-}X=\bar{D}_{+}Y=D_{-}Y=0~{},\qquad
D_{+}\bar{X}=D_{-}\bar{X}=D_{+}\bar{Y}=\bar{D}_{-}\bar{Y}=0~{}.$ (62)

Then we have

$\delta Y=-\epsilon^{+}\bar{D}_{+}\bar{X}-\epsilon^{-}D_{-}X~{},$ (63)

as well as its conjugate

$\delta\bar{Y}=\bar{\epsilon}^{+}D_{+}X+\bar{\epsilon}^{-}\bar{D}_{-}\bar{X}~{}.$
(64)

Hence it follows that

$\delta(\bar{D}_{+}D_{-}\bar{Y})=\bar{D}_{+}D_{-}\delta\bar{Y}=2i\bar{\epsilon}^{+}\partial_{++}D_{-}X-2i\bar{\epsilon}^{-}\partial_{--}\bar{D}_{+}\bar{X}~{}.$
(65)

This should be compared with

$\delta
G=2i\bar{\epsilon}^{+}\partial_{++}D_{-}X-2i\bar{\epsilon}^{-}\partial_{--}\bar{D}_{+}\bar{X}~{},$
(66)

showing that $\bar{D}_{+}D_{-}\bar{Y}$ transforms exactly like the auxiliary
field $G$. This enables us to further constrain the $(4,4)$ multiplet by
setting

$G=\bar{D}_{+}D_{-}\bar{Y}~{}.$ (67)

Imposing these conditions gives a $(4,4)$ twisted superfield

${\bm{\mathcal{Y}}}=Y+\eta^{+}\bar{D}_{+}\bar{X}+\eta^{-}D_{-}X+\eta^{+}\eta^{-}\bar{D}_{+}D_{-}\bar{Y}~{},$
(68)

which by construction is twisted-chiral and chiral with respect to the
manifest and hidden $(2,2)$ supersymmetries, respectively:

$\bar{D}_{+}{\bm{\mathcal{Y}}}=D_{-}{\bm{\mathcal{Y}}}=0~{},\quad\bar{\mathcal{D}}_{\pm}{\bm{\mathcal{Y}}}=0~{}.$
(69)

Its free dynamical action can be easily constructed by considering its square

${\bm{\mathcal{Y}}}^{2}=2\eta^{+}\eta^{-}\Big{(}Y\bar{D}_{+}D_{-}\bar{Y}-\bar{D}_{+}\bar{X}\cdot
D_{-}X\Big{)}+\ldots~{}.$ (70)

In fact, the following relations hold:

$\displaystyle\int
d^{2}x\,d\theta^{+}d\theta^{-}d\bar{\theta}^{+}d\bar{\theta}^{-}(X\bar{X}-Y\bar{Y})$
$\displaystyle=\int
d^{2}x\,d\theta^{+}d\bar{\theta}^{-}\bar{D}_{+}D_{-}(X\bar{X}-Y\bar{Y})~{},$
$\displaystyle=\int
d^{2}x\,d\theta^{+}d\bar{\theta}^{-}\Big{(}D_{+}X\cdot\bar{D}_{-}\bar{X}-\bar{Y}D_{+}\bar{D}_{-}Y\Big{)}\,.$
(71)

Alternatively,

$\displaystyle\int
d^{2}x\,d\theta^{+}d\theta^{-}d\bar{\theta}^{+}d\bar{\theta}^{-}(X\bar{X}-Y\bar{Y})$
$\displaystyle=\int
d^{2}x\,d\bar{\theta}^{+}d\theta^{-}D_{+}\bar{D}_{-}(X\bar{X}-Y\bar{Y})\,,$
$\displaystyle=\int
d^{2}x\,d\bar{\theta}^{+}d\theta^{-}\Big{(}\bar{D}_{+}\bar{X}\cdot
D_{-}X-Y\bar{D}_{+}D_{-}\bar{Y}\Big{)}\,.$ (72)

These relations imply

$4\int
d^{2}x\,d\theta^{+}d\theta^{-}d\bar{\theta}^{+}d\bar{\theta}^{-}(X\bar{X}-Y\bar{Y})=\int
d^{2}x\,d\theta^{+}d\bar{\theta}^{-}d\eta^{+}d\eta^{-}\
{{\bm{\mathcal{Y}}}}^{2}+c.c.~{}.$ (73)

Once again the $(4,4)$ supersymmetry of the left hand side becomes manifest on
the right hand side.

As in the $(4,4)$ twisted multiplet considered in the previous subsection, we
can deform this representation to induce the partial breaking. The deformed
multiplet is described by the following $(4,4)$ superfield:

${\bm{\mathcal{Y}}}_{\text{def}}=Y+\eta^{+}\bar{D}_{+}\bar{X}+\eta^{-}D_{-}X+\eta^{+}\eta^{-}\Big{(}\bar{D}_{+}D_{-}\bar{Y}+\kappa\Big{)}~{}.$
(74)

The hidden supersymmetry transformations of the component $(2,2)$ superfields
can be straightforwardly computed using the arguments of the previous
subsection. For the goal of this section, it is enough to mention that $\delta
Y$ is the same as the undeformed case of eq. (63).

To eliminate half of the degrees of freedom of
${\bm{\mathcal{Y}}}_{\text{def}}$ and describe a Goldstone multiplet for
${\cal N}=(4,4)\to{\cal N}=(2,2)$ partial supersymmetry breaking, we again
impose the nilpotent constraint

${\bm{\mathcal{Y}}}^{2}_{\text{def}}=0=2\eta^{+}\eta^{-}\Big{(}Y(\kappa+\bar{D}_{+}D_{-}\bar{Y})-\bar{D}_{+}\bar{X}\cdot
D_{-}X\Big{)}+\ldots~{}.$ (75)

This yields the following constraint for the $(2,2)$ superfields

$Y(\kappa+\bar{D}_{+}D_{-}\bar{Y})-\bar{D}_{+}\bar{X}\cdot D_{-}X=0~{},$ (76)

which is equivalent to

$Y=\frac{\bar{D}_{+}\bar{X}\cdot
D_{-}X}{\kappa+\bar{D}_{+}D_{-}\bar{Y}}=\frac{\widetilde{W}^{2}}{\kappa+\overline{\widetilde{D}}^{2}\bar{Y}}~{},\qquad\bar{Y}=\frac{\bar{D}_{-}\bar{X}\cdot
D_{+}X}{\kappa+\bar{D}_{+}D_{-}\bar{Y}}=\frac{\bar{\widetilde{W}}^{2}}{\kappa+\widetilde{D}^{2}\bar{Y}}~{}.$
(77)

Here
$\bar{\widetilde{D}}^{2}=\bar{D}_{+}D_{-},\widetilde{D}^{2}=-D_{+}\bar{D}_{-}$
and we have introduced the following bilinears:

$\widetilde{W}^{2}\equiv\bar{D}_{+}\bar{X}\cdot
D_{-}X=\bar{\widetilde{D}}^{2}(X\bar{X}),\qquad\bar{\widetilde{W}}^{2}\equiv\bar{D}_{-}\bar{X}\cdot
D_{+}X=\widetilde{D}^{2}(X\bar{X})~{}.$ (78)

Using exactly the same tricks as before and inspired by the $D=4$ Bagger-
Galperin model, we can solve the constraints (76) to find

$Y=\frac{1}{\kappa}\bar{\widetilde{D}}^{2}\widetilde{\Upsilon}~{},\quad\bar{Y}=\frac{1}{\kappa}\widetilde{D}^{2}\widetilde{\Upsilon}~{},\qquad\widetilde{\Upsilon}=\bar{\widetilde{\Upsilon}}=X\bar{X}-\frac{2\widetilde{W}^{2}\bar{\widetilde{W}}^{2}}{\widetilde{A}+\kappa^{2}+\sqrt{\widetilde{B}^{2}+2\kappa^{2}\widetilde{A}+\kappa^{4}}}~{},$
(79)

where

$\displaystyle\widetilde{A}=\widetilde{D}^{2}\widetilde{W}^{2}+\bar{\widetilde{D}^{2}}\bar{\widetilde{W}^{2}}=\\{\widetilde{D}^{2},\bar{\widetilde{D}^{2}}\\}(X\bar{X})=\partial_{++}X\partial_{--}\bar{X}+\partial_{++}\bar{X}\partial_{--}X~{},$
(80)
$\displaystyle\widetilde{B}=\widetilde{D}^{2}\widetilde{W}^{2}-\bar{\widetilde{D}}^{2}\bar{\widetilde{W}}^{2}=[\widetilde{D}^{2},\bar{\widetilde{D}^{2}}](X\bar{X})=\partial_{++}X\partial_{--}\bar{X}-\partial_{++}\bar{X}\partial_{--}X~{}.$
(81)

Since $Y$ is twisted-chiral under the manifest $(2,2)$ supersymmetry (62), we
can consider the twisted-chiral integral

$S_{\kappa^{2}}=\frac{1}{2}\kappa\int
d^{2}x\,d\theta^{+}d\bar{\theta}^{-}Y+c.c=\frac{1}{2}\int
d^{2}x\,d\theta^{+}d\bar{\theta}^{-}\bar{\widetilde{D}}^{2}\widetilde{\Upsilon}+c.c.=\int
d^{2}x\,d\theta^{+}d\theta^{-}d\bar{\theta}^{+}d\bar{\theta}^{-}\widetilde{\Upsilon}~{}.~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(82)

By using arguments analogous to those around eqs. (54) of the previous
subsection, the action (82) proves to be ${\cal N}=(4,4)$ supersymmetric.

Explicitly, the action reads

$\displaystyle S_{\kappa^{2}}=\int
d^{2}x\,d\theta^{+}d\theta^{-}d\bar{\theta}^{+}d\bar{\theta}^{-}\Bigg{(}X\bar{X}-\frac{2\widetilde{W}^{2}\bar{\widetilde{W}}^{2}}{\kappa^{2}+\widetilde{A}+\sqrt{\kappa^{4}+2\kappa^{2}\widetilde{A}+\widetilde{B}^{2}}}\Bigg{)}~{},$
(83)

which precisely matches the model of eq. (8a) if we identify the coupling
constants:

$\lambda=-\frac{2}{\kappa^{2}}~{}.$ (84)

This shows explicitly that the $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformation of the free
chiral action possesses a non-linearly realized ${\cal N}=(2,2)$
supersymmetry.

###  2 $T^{2}$ Deformations in $D=4$ and Supersymmetric Extensions

In Section 1 we exhibited the non-linear supersymmetry possessed by two $D=2$,
${\cal N}=(2,2)$ models constructed in [Chang:2019kiu] from the $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformation of free
actions. The striking relationship with the $D=4$ supersymmetric Born-Infeld
(BI) theory naturally makes one wonder whether some kind of $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ flow equation is
satisfied by supersymmetric $D=4$ BI, and related actions. We will spend the
rest of the chapter exploring this possibility. In this section, we start with
a few general observations on $T^{2}$ or supercurrent-squared operators in
$D>2$.

####  1 Comments on the $T^{2}$ operator in $D=4$

In two dimensions, by $T\bar{T}$ we mean the operator
$T_{\mu\nu}T^{\mu\nu}-(T_{\mu}^{\mu})^{2}$, which is proportional to
$\det[T_{\mu\nu}]$ [Zamolodchikov:2004ce, Smirnov:2016lqw, Cavaglia:2016oda].
One can attempt to generalize this structure to $D>2$. In general, one could
consider the following stress-tensor squared operator

$O_{T^{2}}^{[r]}=T^{\mu\nu}T_{\mu\nu}-r\,\Theta^{2}~{},\qquad\Theta\equiv\tensor{T}{{}^{\mu}_{\mu}}~{},$
(1)

with $r$ a real constant parameter. In two dimensions, the unique choice $r=1$
yields a well defined operator which is free of short distance singularities
[Smirnov:2016lqw, Zamolodchikov:2004ce]. However, to the best of our
knowledge, there is no analogous argument in higher dimensions that guarantees
a well-defined irrelevant operator $O_{T^{2}}^{[r]}$ at the quantum level.
Nevertheless, in a $D$-dimensional space-time, one possible extension is given
by $O_{T^{2}}^{[r]}$ with $r=1/(D-1)$, which reduces to the $T\bar{T}$
operator in two dimensions.

This operator has received some attention recently since it is motivated by a
particular holographic picture in $D>2$ [Taylor:2018xcy, Hartman:2018tkw]. We
will not enter into a detailed discussion of the physical properties enjoyed
by $O_{T^{2}}^{[1/(D-1)]}$, but simply comment that this combination is
invariant under a set of improvement transformations of the stress-energy
tensor. Indeed it is easy to show that such a $T^{2}$ operator transforms by,

$O_{T^{2}}^{[1/(D-1)]}\rightarrow O_{T^{2}}^{[1/(D-1)]}+\text{total
derivatives}~{},$ (2)

if the (symmetric) stress-energy tensor shifts by the following improvement
transformation,

$T_{\mu\nu}\rightarrow
T_{\mu\nu}+\big{(}\partial_{\mu}\partial_{\nu}-\eta_{\mu\nu}\partial^{2}\big{)}u~{},$
(3)

for an arbitrary scalar field $u$.

In four dimensions, there is another choice of interest, specifically $r=1/2$.
In fact, it was shown in [Conti:2018jho] that the bosonic Born-Infeld action
can be obtained by deforming the free Maxwell theory with the operator
$O_{T^{2}}^{[1/2]}$.222It is worth mentioning that another type of higher-
dimensional generalization of $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$-deformations,
specifically the operator $|\det T|^{1/(D-1)}$, was studied in [Cardy:2018sdv,
Bonelli:2018kik]. In this work, we are going to use $O_{T^{2}}^{[1/2]}$ as our
deforming operator. Once generalized to the supersymmetric case, we will see
that this operator plays a central role for various models possessing non-
linearly realized symmetries.

One interesting property enjoyed by $O_{T^{2}}^{[1/2]}$ is its invariance
under a shift of the Lagrangian density of the theory, or equivalently a shift
of the zero point energy. This can serve as motivation for this particular
combination. Under a constant shift of the Lagrangian density ${\mathcal{L}}$,
and correspondingly its stress-energy tensor,

${\mathcal{L}}\rightarrow{\mathcal{L}}+c~{},\qquad T^{\mu\nu}\rightarrow
T^{\mu\nu}-c\,\eta^{\mu\nu}~{},$ (4)

the composite operator $O_{T^{2}}^{[r]}$ transforms in the following way:

$O_{T^{2}}^{[r]}\rightarrow O_{T^{2}}^{[r]}+2c(2r-1)\Theta+4c^{2}(1-r)~{}.$
(5)

When the theory is not conformal, which is the general situation at an
arbitrary point in the flow since the deformation introduces a scale, and
$r\neq 1/2$, the operator $O_{T^{2}}^{[r]}$ always transforms in a non-trivial
way because of the extra trace term. This implies that under a constant shift
in the Lagrangian, the dynamics is modified which is certainly peculiar since
the shift is trivial in the undeformed theory.333It is worth noting that
${T\overline{T}}$ in $D=2$ shares this peculiarity.

However if $r=\frac{1}{2}$, $O_{T^{2}}^{[r]}$ is unaffected up to an honest
field-independent cosmological constant term. The shift of the vacuum energy
does not affect the dynamics of the theory, as long as the theory is not
coupled to gravity. This property is especially interesting, since the $D=4$
${\cal N}=1$ Goldstino action, which is also given by a ${T\overline{T}}$
flow, is the low-energy description of supersymmetry breaking which can
generate a cosmological constant. For these reasons, we will study the
particular operator quadratic in stress-energy tensors given by

$O_{T^{2}}\equiv T^{\mu\nu}T_{\mu\nu}-\frac{1}{2}\Theta^{2}~{},$ (6)

in the remainder of this chapter. More general $T^{2}$ type deformations of
gauge theories in any spacetime dimension will be considered in Chapter 6.

####  2 ${\cal N}=1$ supercurrent-squared operator in $D=4$

We would like to find the ${\cal N}=1$ supersymmetric extension of the
$O_{T^{2}}$ operator in four dimensions. As reviewed in section 1, in two
dimensions the manifestly supersymmetric $T\bar{T}$ deformation is roughly
given by the square of the supercurrent superfields. One might suspect that a
similar construction holds in four dimensions.

For the remainder of this work, we will assume that the $D=4$, ${\cal N}=1$
supersymmetric theories under our consideration admit a Ferrara-Zumino (FZ)
multiplet of currents [Ferrara:1974pz]. Generalizations of this case involving
the supercurrent multiplets described in [Komargodski:2010rb,
Dumitrescu:2011iu, Gates:1981yc, Ambrosetti:2016ieg, Gates:1983nr,
Dienes:2009td, Kuzenko:2010am] might be possible, but merit separate
investigation. The operator content of the FZ multiplet, which has $12+12$
component fields, includes the conserved supersymmetry current
$S_{\mu\alpha}$, its conjugate $\bar{S}_{\mu}{}^{\dot{\alpha}}$ and the
conserved symmetric energy-momentum tensor $T_{\mu\nu}$:

$T_{\mu\nu}=T_{\nu\mu}~{},\qquad\partial^{\mu}T_{\mu\nu}=0~{},\qquad\partial^{\mu}S_{\mu}=\partial^{\mu}\bar{S}_{\mu}=0~{}.$
(7)

The FZ multiplet also includes a complex scalar field ${\sf x}$, as well as
the $R$-current vector field $j_{\mu}$, which is not necessarily conserved
[Ferrara:1974pz].

In $D=4$, ${\cal N}=1$ superspace, the FZ multiplet is described by a vector
superfield ${\mathcal{J}}_{\mu}$ and a complex scalar scalar superfield
$\mathcal{X}$ satisfying the following constraints:

$\bar{D}^{\dot{\alpha}}\mathcal{J}_{\alpha\dot{\alpha}}=D_{\alpha}\mathcal{X}~{},\qquad\bar{D}_{\dot{\alpha}}\mathcal{X}=0~{}.$
(8)

The constraints can be solved, and the FZ supercurrents expressed in terms of
its $12+12$ independent components read444For convenience, we have rescaled
the supersymmetry current compared to [Dumitrescu:2011iu]:
$S_{\mu}^{\text{here}}=-iS_{\mu}^{\text{there}}$.

$\displaystyle\mathcal{J}_{\mu}(x)$ $\displaystyle=$ $\displaystyle
j_{\mu}+\theta\Big{(}S_{\mu}-\frac{1}{\sqrt{2}}\sigma_{\mu}\bar{\chi}\Big{)}+\bar{\theta}\Big{(}\bar{S}_{\mu}+\frac{1}{\sqrt{2}}\bar{\sigma}_{\mu}\chi\Big{)}+\frac{i}{2}\theta^{2}\partial_{\mu}\bar{\sf
x}-\frac{i}{2}\bar{\theta}^{2}\partial_{\mu}{\sf x}$ (9)
$\displaystyle+\theta\sigma^{\nu}\bar{\theta}\Big{(}2T_{\mu\nu}-\frac{2}{3}\eta_{\mu\nu}\Theta-\frac{1}{2}\epsilon_{\nu\mu\rho\sigma}\partial^{\rho}j^{\sigma}\Big{)}$
$\displaystyle-\frac{i}{2}\theta^{2}\bar{\theta}\Big{(}\bar{\not{\partial}}S_{\mu}+\frac{1}{\sqrt{2}}\bar{\sigma}_{\mu}\not{\partial}\bar{\chi}\Big{)}-\frac{i}{2}\bar{\theta}^{2}\theta\Big{(}{\not{\partial}}{\bar{S}}_{\mu}-\frac{1}{\sqrt{2}}\sigma_{\mu}\bar{\not{\partial}}\chi\Big{)}$
$\displaystyle+\frac{1}{2}\theta^{2}\bar{\theta}^{2}\Big{(}\partial_{\mu}\partial^{\nu}j_{\nu}-\frac{1}{2}\partial^{2}j_{\mu}\Big{)}~{},$

and

$\displaystyle\mathcal{X}(y)$ $\displaystyle=$ $\displaystyle{\sf
x}(y)+\sqrt{2}\theta\chi(y)+\theta^{2}{\sf F}(y)~{},~{}~{}~{}$ (10a)
$\displaystyle\chi_{\alpha}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}}{3}(\sigma^{\mu})_{\alpha\dot{\alpha}}\bar{S}^{\dot{\alpha}}_{\mu}~{},\qquad{\sf
F}=\frac{2}{3}\Theta+i\partial_{\mu}j^{\mu}~{},$ (10b)

where the chiral coordinate is defined by
$y^{\mu}=x^{\mu}+i\theta\sigma^{\mu}\bar{\theta}$, and we used
$\not{\partial}=\sigma^{\mu}\partial_{\mu},\,\bar{\not{\partial}}=\bar{\sigma}^{\mu}\partial_{\mu}$.

If we seek a manifestly supersymmetric completion of the operator (6) by using
combinations of the supercurrent superfields with dimension $4$, it is clear
that the only possibility is the full superspace integral of a linear
combination of ${\mathcal{J}}^{2}$ and $\mathcal{X}\bar{\mathcal{X}}$. Up to
total derivatives and terms that vanish by using the supercurrent conservation
equations, or equivalently that vanish on-shell, the $D$-terms of
${\mathcal{J}}^{2}$ and $\mathcal{X}\bar{\mathcal{X}}$ are given by555The
composite $A$ (and analogously its conjugate $\bar{A}$) is given by
$\displaystyle
A=\Big{(}S_{\mu}-\frac{1}{\sqrt{2}}\sigma_{\mu}\bar{\chi}\Big{)}\Big{(}{\not{\partial}}{\bar{S}}^{\mu}-\frac{1}{\sqrt{2}}\sigma^{\mu}\bar{\not{\partial}}\chi\Big{)}=S_{\mu}{\not{\partial}}{\bar{S}}^{\mu}-\bar{\chi}\bar{\not{\partial}}\chi+\sqrt{2}{\bar{S}}^{\mu}\partial_{\mu}\bar{\chi}+\text{total
derivatives}~{}.$ (11) The equality can be obtained with some algebra. Note
that the last term drops after integration by parts because of the
conservation equation for $S_{\mu}$.

$\displaystyle\mathcal{J}^{2}|_{\theta^{2}\bar{\theta}^{2}}\equiv\eta^{\mu\nu}{\mathcal{J}}_{\mu}{\mathcal{J}}_{\nu}|_{\theta^{2}\bar{\theta}^{2}}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\Big{(}2T_{\mu\nu}-\frac{2}{3}\eta_{\mu\nu}\Theta-\frac{1}{2}\epsilon_{\nu\mu\rho\sigma}\partial^{\rho}j^{\sigma}\Big{)}^{2}+j^{\mu}\Big{(}\partial_{\mu}\partial^{\nu}j_{\nu}-\frac{1}{2}\partial^{2}j_{\mu}\Big{)}$
(12b) $\displaystyle+\frac{1}{2}\partial_{\mu}{\sf
x}\partial^{\mu}\bar{\sf{x}}+\frac{i}{2}\Big{(}A-\bar{A}\Big{)}$
$\displaystyle=$
$\displaystyle-2(T_{\mu\nu})^{2}+\frac{4}{9}\Theta^{2}-\frac{5}{4}\Big{(}\partial_{\mu}j^{\mu}\Big{)}^{2}-\frac{3}{4}j_{\mu}\partial^{2}j^{\mu}+\frac{1}{2}\partial_{\mu}{\sf\bar{x}}\partial^{\mu}{\sf
x}$
$\displaystyle+i\Big{(}S_{\mu}{\not{\partial}}{\bar{S}}^{\mu}-\bar{\chi}\bar{\not{\partial}}\chi\Big{)}+\text{total
derivatives}+{\rm EOM}~{},$

and

$\displaystyle\mathcal{X}\bar{\mathcal{X}}|_{\theta^{2}\bar{\theta}^{2}}$
$\displaystyle=$ $\displaystyle{\sf F}\bar{\sf F}-\partial_{\mu}{\sf
x}\partial^{\mu}\bar{\sf x}-i\bar{\chi}{\bar{\not{\partial}}}\chi+\text{total
derivatives}$ (13a) $\displaystyle=$
$\displaystyle\frac{4}{9}\Theta^{2}+(\partial_{\mu}j^{\mu})^{2}-\partial_{\mu}{\sf
x}\partial^{\mu}\bar{\sf x}-i\bar{\chi}{\bar{\not{\partial}}}\chi+\text{total
derivatives}~{}.$ (13b)

To get a manifestly supersymmetric extension of
$O_{T^{2}}=T^{2}-\frac{1}{2}\Theta^{2}$, we have to consider the following
linear combination

$\mathcal{O}_{T^{2}}=-\frac{1}{2}\Big{(}\eta^{\mu\nu}\mathcal{J}_{\mu}\mathcal{J}_{\nu}+\frac{5}{4}\mathcal{X}\bar{\mathcal{X}}\Big{)}=\frac{1}{16}\mathcal{J}^{\alpha\dot{\alpha}}\mathcal{J}_{\alpha\dot{\alpha}}-\frac{5}{8}\mathcal{X}\bar{\mathcal{X}}~{}.$
(14)

In fact, the supersymmetric descendant of the supercurrent-squared operator
$\mathcal{O}_{T^{2}}$ is

$\displaystyle{\sf O}_{T^{2}}$ $\displaystyle=$ $\displaystyle\int
d^{4}\theta\,\mathcal{O}_{T^{2}}$ (15b) $\displaystyle=$ $\displaystyle
T^{2}-\frac{1}{2}\Theta^{2}+\frac{3}{8}j_{\mu}\partial^{2}j^{\mu}+\frac{3}{8}\partial_{\mu}{\sf
x}\partial^{\mu}\bar{\sf
x}-\frac{i}{2}\Big{(}S_{\mu}{\not{\partial}}{\bar{S}}^{\mu}-\frac{9}{4}\bar{\chi}\bar{\not{\partial}}\chi\Big{)}$
$\displaystyle+\text{total derivatives}+{\rm EOM}~{}.$

This result shows that ${\sf O}_{T^{2}}$ is the natural supersymmetric
extension of $O_{T^{2}}$. However, it is worth emphasizing that in the $D=4$
case, the supersymmetric descendent ${\sf O}_{T^{2}}$ of
${\mathcal{O}}_{T^{2}}$ has extra non-trivial contributions from other
currents. This should be contrasted with the $D=2$ case where ${\sf
O}_{T^{2}}=O_{T^{2}}$ up to ${\rm EOM}$ and total derivatives, see eq. (1).

It actually does not seem possible to find a linear combination of
${\mathcal{J}}^{2}$ and $\mathcal{X}\bar{\mathcal{X}}$ such that an analogue
of eq. (1) holds in $D=4$. This suggests that, in contrast with the $D=2$
case, deformations of a Lagrangian triggered by the operators $O_{T^{2}}$ and
${\sf O}_{T^{2}}$ will in general lead to different flows: one manifestly
supersymmetric, while the other not.

###  3 Bosonic Born-Infeld as a $T^{2}$ Flow

It was shown in [Conti:2018jho] that the $D=4$ Born-Infeld action arises from
a $D>2$ generalization of the $T\bar{T}$ deformation. Specifically, the
operator driving the flow equation was shown to be the $O_{T^{2}}$ defined in
eq. (6) of the preceding section. In this section we review this result in
detail as it is a primary inspiration for our supersymmetric extensions.

The $D=4$ bosonic BI action on a flat background is given by

$\displaystyle S_{\text{BI }}$ $\displaystyle=$
$\displaystyle\frac{1}{\alpha^{2}}\int
d^{4}x\;\Big{[}1-\sqrt{-\det(\eta_{\mu\nu}+\alpha F_{\mu\nu})}\Big{]}~{},$ (1)
$\displaystyle=$ $\displaystyle\frac{1}{\alpha^{2}}\int
d^{4}x\;\Big{[}1-\sqrt{1+\frac{\alpha^{2}}{2}F^{2}-\frac{\alpha^{4}}{16}(F\tilde{F})^{2}}\Big{]}~{},$
$\displaystyle=$ $\displaystyle-\frac{1}{4}\int d^{4}x\;F^{2}+{\rm
higher~{}derivative~{}terms}~{},$

where $F_{\mu\nu}=(\partial_{\mu}v_{\nu}-\partial_{\nu}v_{\mu})$ is the field
strength for an Abelian gauge field $v_{\mu}$, and

$F^{2}\equiv F_{\mu\nu}F^{\mu\nu},\qquad F\tilde{F}\equiv
F_{\mu\nu}\tilde{F}^{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}F^{\mu\nu}F^{\rho\sigma}~{}.$
(2)

The Hilbert stress-energy tensor for the BI action can be computed
straightforwardly and it reads [Rasheed:1997ns]

$T^{\mu\nu}=-\frac{F^{\mu\lambda}F^{\nu}{}_{\lambda}+\frac{1}{\alpha^{2}}\Big{(}\sqrt{1+\frac{\alpha^{2}}{2}F^{2}-\frac{\alpha^{4}}{16}(F\tilde{F})^{2}}-1-\frac{\alpha^{2}}{2}F^{2}\Big{)}\eta^{\mu\nu}}{\sqrt{1+\frac{\alpha^{2}}{2}F^{2}-\frac{\alpha^{4}}{16}(F\tilde{F})^{2}}}~{}.$
(3)

This can be written in the following useful form

$T^{\mu\nu}=\frac{T^{\mu\nu}_{\text{Maxwell}}}{\sqrt{1+2A+B^{2}}}+\frac{\eta^{\mu\nu}}{\alpha^{2}\sqrt{1+2A+B^{2}}}\frac{A^{2}-B^{2}}{1+A+\sqrt{1+2A+B^{2}}}~{},$
(4)

where we used the stress-energy tensor for the Maxwell theory

$T^{\mu\nu}_{\text{Maxwell}}=-F^{\mu\lambda}F^{\nu}{}_{\lambda}+\frac{1}{4}F^{2}\eta^{\mu\nu}~{},$
(5)

while $A$ and $B$ are defined by

$A=\frac{1}{4}\alpha^{2}F^{2}~{},\qquad B=\frac{i}{4}\alpha^{2}F\tilde{F}~{}.$
(6)

It is easy to compute the trace of the stress-energy tensor

$\Theta=T^{\mu\nu}\eta_{\mu\nu}=\frac{4}{\alpha^{2}\sqrt{1+2A+B^{2}}}\frac{A^{2}-B^{2}}{1+A+\sqrt{1+2A+B^{2}}}~{},$
(7)

where, interestingly, the combination $(A^{2}-B^{2})$ proves to be related to
the square of $T^{\mu\nu}_{\text{Maxwell}}$. Using the identity

$(F\tilde{F})^{2}=\frac{1}{4}(\epsilon_{\mu\nu\rho\sigma}F^{\mu\nu}F^{\rho\sigma})^{2}=4F_{\mu\nu}F^{\nu\rho}F_{\rho\sigma}F^{\sigma\mu}-2(F^{2})^{2}~{},$
(8)

we see that

$T^{2}_{\text{Maxwell}}=F_{\mu\nu}F^{\nu\rho}F_{\rho\sigma}F^{\sigma\mu}-\frac{1}{4}(F^{2})^{2}=\frac{1}{4}\Big{(}(F^{2})^{2}+(F\tilde{F})^{2}\Big{)}=\frac{4}{\alpha^{4}}(A^{2}-B^{2})~{}.$
(9)

Using tracelessness of the free Maxwell stress-energy tensor, the $O_{T^{2}}$
operator can be easily computed:

$\displaystyle O_{T^{2}}=T^{2}-\frac{1}{2}\Theta^{2}$ $\displaystyle=$
$\displaystyle\frac{4(A^{2}-B^{2})}{\alpha^{4}\sqrt{1+2A+B^{2}}^{2}}\Big{(}1-\frac{A^{2}-B^{2}}{(1+A+\sqrt{1+2A+B^{2}})^{2}}\Big{)}~{},~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(10a) $\displaystyle=$
$\displaystyle\frac{4(A^{2}-B^{2})}{\alpha^{4}\sqrt{1+2A+B^{2}}^{2}}\Big{(}1-\frac{1+A-\sqrt{1+2A+B^{2}}}{1+A+\sqrt{1+2A+B^{2}}}\Big{)}~{},$
(10b) $\displaystyle=$
$\displaystyle\frac{8(A^{2}-B^{2})}{\alpha^{4}\sqrt{1+2A+B^{2}}}\frac{1}{1+A+\sqrt{1+2A+B^{2}}}~{},$
(10c) $\displaystyle=$
$\displaystyle\frac{8(1+A-\sqrt{1+2A+B^{2})}}{\alpha^{2}\sqrt{1+2A+B^{2}}}~{}.$
(10d)

The variation of the BI Lagrangian with respect to the parameter $\alpha^{2}$
can be readily computed, and it is given by

$\frac{\partial\mathcal{L}_{\alpha}}{\partial\alpha^{2}}=\frac{1+\frac{1}{4}\alpha^{2}F^{2}-\sqrt{1+\frac{1}{2}\alpha^{2}F^{2}-\frac{1}{16}\alpha^{4}(F\tilde{F})^{2}}}{\alpha^{2}\sqrt{1+\frac{1}{2}\alpha^{4}F^{2}-\frac{1}{16}\alpha^{4}(F\tilde{F})^{2}}}~{}.$
(11)

Once we use (6) it is clear that (10a) and (11) have exactly the same
structure and satisfy

$\frac{\partial\mathcal{L}_{\alpha}}{\partial\alpha^{2}}=\frac{1}{8}O_{T^{2}}~{},$
(12)

showing that the BI Lagrangian satisfies a $T^{2}$-flow driven by the operator
$O_{T^{2}}$.

Before turning to $D=4$ supersymmetric analysis, it is worth mentioning that
the structure of the computation relating the $O_{T^{2}}$ operator to the
bosonic BI theory, which we just reviewed, is quite similar to what we saw in
section 1 for the $D=2$ ${\cal N}=(2,2)$ supersymmetric $T\bar{T}$ flows. For
example, in the deformation of the free twisted-chiral multiplet action, the
analogue of the $A$ and $B$ combinations of (6) is given by (51), but the
square root structure of the actions is completely analogous. This fact,
together with the non-linearly realized supersymmetry we investigated in
section 1, naturally lead to the guess that the $D=4$ ${\cal N}=1$
supersymmetric Born-Infeld (BI) theory may also satisfy a $T^{2}$ flow. The
next section is devoted to explaining how this is the case.

###  4 Supersymmetric Born-Infeld from Supercurrent-Squared Deformation

In Section 1 we proved, by analogy and extension of the $D=4$ results of
[Bagger:1996wp], that two $D=2$ supercurrent-squared flows possess additional
non-linearly realized supersymmetry. In this section we reverse the logic. We
will look at a well-studied model, namely the Bagger-Galperin construction
[Bagger:1996wp] of $D=4$, ${\cal N}=1$ Born-Infeld theory [Deser:1980ck,
cecotti:1987], and show that it satisfies a supercurrent-squared flow
equation.

####  1 $D=4$ ${\cal N}=1$ supersymmetric BI and non-linear supersymmetry

Let us review some well known results about the $D=4$ ${\cal N}=1$ Born-Infeld
theory [cecotti:1987], the Bagger-Galperin action [Bagger:1996wp], the non-
linearly realized second supersymmetry, and its precise ${\cal N}=2\to{\cal
N}=1$ supersymmetry breaking pattern. For more detail, we refer to the
following references on the subject [Bagger:1996wp, Rocek:1997hi,