${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$-Like Flows
and $3d$ Nonlinear Supersymmetry

Christian Ferko,a Yangrui Hu,b Zejun Huang,c Konstantinos Koutrolikos,d and
Gabriele Tartaglino-Mazzucchellic

a Center for Quantum Mathematics and Physics (QMAP),
Department of Physics & Astronomy, University of California, Davis, CA 95616,
USA

b Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada

c School of Mathematics and Physics, University of Queensland,
St Lucia, Brisbane, Queensland 4072, Australia

d Department of Physics, University of Maryland,
College Park, MD 20742-4111, USA
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>

We show that the $3d$ Born-Infeld theory can be generated via an irrelevant
deformation of the free Maxwell theory. The deforming operator is constructed
from the energy-momentum tensor and includes a novel non-analytic contribution
that resembles root-${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu}$. We find that a similar operator deforms a free scalar into the scalar
sector of the Dirac-Born-Infeld action, which describes transverse
fluctuations of a D-brane, in any dimension. We also analyse trace flow
equations and obtain flows for subtracted models driven by a relevant
operator. In $3d$, the irrelevant deformation can be made manifestly
supersymmetric by presenting the flow equation in $\mathcal{N}=1$ superspace,
where the deforming operator is built from supercurrents. We demonstrate that
two supersymmetric presentations of the D2-brane effective action, the
Maxwell-Goldstone multiplet and the tensor-Goldstone multiplet, satisfy
superspace flow equations driven by this supercurrent combination. To do this,
we derive expressions for the supercurrents in general classes of vector and
tensor/scalar models by directly solving the superspace conservation equations
and also by coupling to $\mathcal{N}=1$ supergravity. As both of these
multiplets exhibit a second, spontaneously broken supersymmetry, this analysis
provides further evidence for a connection between current-squared
deformations and nonlinearly realized symmetries.

###### Contents

  1. 1 Introduction
  2. 2 Bosonic flows
    1. 2.1 Maxwell-Goldstone Multiplet
    2. 2.2 Tensor Multiplet
  3. 3 Supersymmetric flows
    1. 3.1 Supercurrent Multiplet and Stress Tensor
    2. 3.2 Supercurrent and Supertrace of $3d$ Maxwell-Goldstone multiplet
      1. 3.2.1 Warm up: supercurrent multiplet of super-Maxwell
      2. 3.2.2 Supercurrent multiplet of Maxwell-Goldstone
    3. 3.3 Flow Operator as Supercurrent-squared Operator
    4. 3.4 Flow Interpretations for Tensor-Goldstone
  4. 4 Derivation of supercurrents from ${\cal N}=1$ supergravity
    1. 4.1 Building blocks from superspace supergravity
    2. 4.2 Calculation of supercurrents for vector models
    3. 4.3 Calculation of supercurrents for tensor/scalar multiplet models
  5. 5 Conclusion
  6. A Conventions and Identities for Section 3
    1. A.1 Conventions
    2. A.2 Identities
    3. A.3 Off-shell expressions for supercurrent-squared operators in TG
  7. B Conventions for Sections 2 and 4
    1. B.1 Conventions
    2. B.2 Map of conventions between sections 3 & 4
  8. C Off-Shell Flow for Bosonic Tensor-Goldstone Lagrangian

##  1 Introduction

A very useful tool for the exploration of the space of theories within a
specific theoretical framework is that of _flows_. This tool has been used
extensively in physics and mathematics with enormous success. The most famous
application of a flow in theoretical physics is the renormalization group (RG)
flow which is used to explore the space of field theories. In mathematics, a
well-known example is that of Ricci flow [1] which explores the space of
Riemannian manifolds. Specifically, RG flows run from high energy (UV)
theories to low energy (IR) theories and determine the set of coupling
constants of QFTs along the trajectory of the flow in a manner controlled by
the beta function. Similar in spirit, Ricci flows create a trajectory in the
space of Riemannian manifolds, dictated by the Ricci tensor, and determine the
metric of the manifolds along the trajectory. Both flows have interesting
points. RG flows have fixed points that correspond to conformal field theories
and Ricci flows have attractors which correspond to constant curvature
manifolds.

The success of RG flows suggests further exploration of the space of QFTs
using additional flows triggered by operators that deform theories in new and
interesting ways. A special example is the $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ operator, which is
irrelevant in the sense of the RG.111The coupling constant that turns on and
controls the contribution of this operator has negative mass dimension. Such a
coupling is irrelevant at low energies but grows more important at high
energies. Ordinarily, turning on an irrelevant operator subsequently requires
the addition of infinitely many counterterms, and thus leads to a loss of
analytic control. Nonetheless it has been shown [2, 3] that the ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ operator in two
dimensions is an exception to this rule, and in fact the deformation is
“solvable” in the sense that several quantities in the deformed theory can be
computed analytically in terms of the data of the undeformed theory.

The definition of the $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu$ operator is historically given (up to normalisation factors) as

$\displaystyle T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu:=-\det\left(T_{mn}\right)=\frac{1}{2}\left(T^{mn}T_{mn}-\tensor{T}{{}^{m}_{m}}\tensor{T}{{}^{n}_{n}}\right)\,$

where $T_{mn}$ is the energy-momentum tensor of the system. In $2d$, using the
holomorphic and anti-holomorphic coordinates, this operator takes the form
$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu~{}\sim~{}T_{zz}\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu_{\mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu\mkern
1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu}-T_{z\mkern
1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu}^{2}$. For CFTs, the
trace part ($T_{z\mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern
1.5mu}$) vanishes and we are left only with the $T_{zz}$ and $\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu_{\mkern
1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu\mkern
1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu}$ term, hence the name
$T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$.222However,
this name is somewhat misleading since a ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$-deformed CFT at finite
deformation parameter is no longer conformally invariant and thus its stress
tensor has non-vanishing trace. Interestingly, it appears that ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$-deformed CFTs exhibit
an unconventional field-dependent conformal symmetry [4, 5, 6, 7].

There are various viewpoints that provide intuition as to why $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformations are not
the typical irrelevant operators and should be studied in more detail. One
such viewpoint is its property of being an integrable deformation. This means
that if we start with an integrable theory and follow its flow under $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$, then the output will
also be an integrable theory [8, 9]. In other words, the preservation of
integrability by the flow makes $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ very special. A
different viewpoint, not related to integrability, involves connections
between the $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$
deformation and gravity. For instance one can calculate the contribution of an
infinitesimal $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu$-deformation to the partition function of the theory and show that it is
equivalent to an integration over random variations of the underlying geometry
[10]. A second connection involves coupling to flat space JT gravity [11, 12],
which has more recently been understood as a special case of a topological
gauging procedure that can be applied to more general current-current
deformations [13]. A third approach is to geometrize ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ and related
deformations via a dynamical change of coordinates which couples the
undeformed theory to a field-dependent background metric [14, 15, 16, 17, 18].

Unlike RG flows, $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu$ flows run towards higher energies. Nevertheless there are examples of
quantities that have been computed exactly using this flow. In this paper we
will focus on a specific quantity, the deformed classical Lagrangian. This
means that we: (_i_) consider the space of field theories that have a
Lagrangian description $\mathcal{L}$, (_ii_) assume a curve through this space
parametrized by some parameter $\lambda$ such that the point on the curve
corresponding to value $\lambda=\lambda_{0}$ is the field theory with
Lagrangian $\mathcal{L}^{(\lambda_{0})}$ and (_iii_) define the operator
$\mathcal{O}^{(\lambda)}$ that triggers the flow from point $\lambda$ to point
$\lambda+\delta\lambda$ as

$\mathcal{L}^{(\lambda+\delta\lambda)}=\mathcal{L}^{(\lambda)}+\delta\lambda~{}\mathcal{O}^{(\lambda)}~{}.$
(1.1)

For the case of $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu$-flow of Lagrangians in $2d$ we consider the following deformation:

$\frac{\partial\mathcal{L}^{(\lambda)}}{\partial\lambda}=\det(T_{mn}^{(\lambda)})~{}.$
(1.2)

A well-known example of solving the $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$-flow equation for the
deformed Lagrangian is the case of a free boson in two dimensions:
$\mathcal{L}=\partial\phi\mkern
1.5mu\overline{\mkern-1.5mu\partial\mkern-1.5mu}\mkern 1.5mu\phi$. In [3] it
was shown that under a $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$-flow this theory is
mapped to the $3d$ Nambu-Goto action in the static gauge:

$\mathcal{L}^{(0)}=\partial\phi\mkern
1.5mu\overline{\mkern-1.5mu\partial\mkern-1.5mu}\mkern
1.5mu\phi~{}\to~{}\mathcal{L}^{(\lambda)}=\frac{1}{2\lambda}\Big{(}\sqrt{4\lambda\partial\phi\mkern
1.5mu\overline{\mkern-1.5mu\partial\mkern-1.5mu}\mkern
1.5mu\phi+1}-1\Big{)}=-\frac{1}{2\lambda}+\mathcal{L}_{\text{NG}}~{}.$ (1.3)

An interesting feature of the above deformed Lagrangian is the square root.
Usually this type of nonlinearity _(a)_ corresponds to the resummation of
infinitely many terms, which in this case can be thought of as being generated
by iteratively adding and re-computing the controlled ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ combination along the
flow, and _(b)_ appears in many attractive theories. In this example, the
Nambu-Goto action describes the propagation of bosonic strings with the flow
parameter $\lambda$ playing the role of inverse string tension.

A natural question to ask is whether other “square root” Lagrangians have a
similar interpretation: can they be obtained from a $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$-like deformation of a
free theory? Another well-known and extensively studied member of the square
root Lagrangians is the Born-Infeld (BI) theory and its generalizations. The
BI Lagrangian provides a nonlinear extension of Maxwell theory, but most
importantly it is the leading term in the low-energy effective description of
D-branes [19, 20]. Because of this, BI-type actions beautifully capture a
solitonic realization of the partial supersymmetry breaking phenomenon [21,
22, 23, 24, 25, 26].

This can be understood using some basic properties of the theory. As solitonic
solutions of open string theory, D-branes introduce boundaries that break the
translational symmetries in their perpendicular directions. However, because
translational symmetries are the source of supersymmetry, fewer translational
symmetries lead to less supersymmetry. In fact, based on the size of spinors
it is straightforward to see that for every translational symmetry lost, the
number of supercharges of the theory is cut in half.

Nevertheless, it is a fact of physics that the memory of broken symmetries is
never lost. The theory still realizes these symmetries but in a nonlinear
manner. Hence, the nonlinearities of BI Lagrangians may be interpreted as a
manifestation of partial supersymmetry breaking and the existence of a
Goldstino mode. It would be very interesting to investigate whether the same
nonlinearities can also be generated by a $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$-like flow. This has
been shown to be true for certain models in two dimensions up to ${\cal
N}=(2,2)$ supersymmetry as well as $\mathcal{N}$=$1,~{}4d$ theories, see [27,
28, 29]. This suggests that non-linearly realized (super)symmetries might also
arise from $T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu$-like flow equations, at least when the seed theories possess extra
symmetries such as the shift-symmetries of free models.

In this paper, we consider the $3d,~{}\mathcal{N}$=$1$ supersymmetric Born-
Infeld theory [30, 31, 32] which is related to the effective description of
D2-branes in type IIA string theory. This theory was obtained by partially
breaking supersymmetry from $\mathcal{N}=2$ to $\mathcal{N}$=$1$ in such a
manner that the supersymmetric multiplet which manifests the surviving,
linearly realized, supersymmetry remains massless (i.e. it is the Goldstone
multiplet). It was found that the role of the Goldstone multiplet can be
played by either the $3d,~{}\mathcal{N}=1$ vector multiplet or its dual tensor
multiplet.

This work aims to show that these results can also be obtained from a $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$-like flow. The
deformation parameter which labels points along the flow trajectory is related
to the scale of supersymmetry breaking and more precisely to the VEV
($\kappa$) of the auxiliary $\mathcal{N}=1$ superfield of the $\mathcal{N}=2$
supermultiplet. The flow relates the $\kappa\to\infty$ limit of these
theories, which corresponds to the manifestly supersymmetric Lagrangians that
describe the $3d,~{}\mathcal{N}=1$ free Maxwell and free tensor multiplet, to
theories at finite (but not zero) values of $\kappa$, which describe the
$3d,~{}\mathcal{N}=1$ supersymmetric Born-Infeld theory in terms of the
Maxwell-Goldstone and tensor-Goldstone multiplets, respectively.

As a result, the $3d$ BI Lagrangian – which is the bosonic truncation of the
above supersymmetric theories – arises from a ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation. This may
come as a surprise since it was explicitly checked in [33] that a flow
equation of the form

$\displaystyle\frac{\partial\mathcal{L}}{\partial\lambda}=c_{1}T^{ab}T_{ab}+c_{2}\left(\tensor{T}{{}^{a}_{a}}\right)^{2}$
(1.4)

does not lead to Born-Infeld type solutions in any dimension other than $d=4$,
regardless of the choice of coefficients $c_{1}$ and $c_{2}$. To evade this
no-go result, one must introduce some additional ingredient. We will find that
the necessary addition is a new operator $R$ which is non-analytic in the
energy-momentum tensor:

$\displaystyle\begin{split}&R=\sqrt{\frac{1}{d}T^{ab}T_{ab}-\frac{1}{d^{2}}\left(\tensor{T}{{}^{a}_{a}}\right)^{2}}=\frac{1}{\sqrt{d}}\sqrt{\widehat{T}^{ab}\widehat{T}_{ab}}\,,\\\
&\qquad\qquad\quad\widehat{T}_{ab}=T_{ab}-\frac{1}{d}g_{ab}\tensor{T}{{}^{c}_{c}}\,.\end{split}$
(1.5)

This operator $R$ is constructed from the traceless part $\widehat{T}_{ab}$ of
the stress tensor and is the $d$-dimensional analogue of the root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ operator whose two-
dimensional version was studied in [34] (see also [35, 36, 37, 38,
39]).333Although there have been some proposals [10, 40] for ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$-like deformations in
higher dimensions which involve roots of the determinant of the stress-energy
tensor, such as $\left[\det(T)\right]^{1/(d-1)}$, note that $R$ is not
proportional to any power of the determinant of $T_{ab}$. A marginal flow
driven by the operator $R$ can be used to obtain the Modified Maxwell (ModMax)
theory, which was introduced in [41, 42, 43], in four spacetime dimensions
[44]. This root-${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu}$-like flow can also be made manifestly supersymmetric [45], and along
with the $4d$ ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu}$-like flow, forms a $2$-parameter family of commuting deformations
which generate the ModMax-Born-Infeld theory (or its supersymmetric extension,
in the superspace case) [46]. In two dimensions, the analogous root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ flow can be used to
construct a “Modified Scalar” theory, or combined with ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ to yield a
$2$-parameter family of “Modified-Nambu-Goto” theories [34, 17, 47]. Classical
root-${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ flows
in $2d$ seem to enjoy some of the special properties of the ordinary ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation, such as
preserving integrability in certain examples [48], although it is not yet
known whether this deformation can be defined quantum-mechanically.

In the present work, we will not consider such marginal flows driven by the
root-${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$-like
operator $R$. Instead, we will only use $R$ as a tool to construct more
general _irrelevant_ flows. More precisely, we will expand the class of
${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$-like flow
equations (1.4) by including a third term $c_{3}\tensor{T}{{}^{a}_{a}}R$ on
the right side. A deformation which includes this additional term will be
sufficient to generate the $3d$ Born-Infeld theory, or its dual scalar theory,
via a ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$-like
flow, and likewise to generate the supersymmetric extensions of these theories
by appropriate flows driven by supercurrents.444For other work on ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ and supersymmetry, see
[49, 50, 51, 52, 53, 54, 55, 56, 57, 58].

The paper is organized as follows. In Section 2 we focus on the bosonic
truncations of the supersymmetric D2-brane actions of interest. We present a
flow driven by an irrelevant operator which deforms the $3d$ free Maxwell
theory to the Born-Infeld theory, which is the bosonic part of the Maxwell-
Goldstone multiplet. We also show that a similar flow connects the $3d$ free
scalar to the scalar sector of the Dirac-Born-Infeld action, which is a
truncation of the tensor-Goldstone multiplet, and that an analogue of this
result holds in any spacetime dimension $d$. We then analyse and extend the
trace flow equations presenting new classes of $3d$ relevant flows for
subtracted Lagrangians. In Section 3 we turn to the supersymmetrization of
these results. We view the results of [30] as a flow that connects $3d$ super-
Maxwell to supersymmetric Born-Infeld and we define the appropriate superspace
operator $\mathcal{O}_{\kappa^{2}}$ triggering the flow. Furthermore, we
explicitly construct the supercurrent multiplet for these theories by solving
the corresponding superspace conservation equation. This is the multiplet that
contains the energy-momentum tensor and its superpartner, the supersymmetry
current. The supercurrent and supertrace superfields that define the
supercurrent multiplet are then used to establish appropriate $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$-like superspace
operators
$\mathcal{O}_{T^{2}},~{}{\mathcal{O}_{\Theta^{2}}},~{}{\mathcal{O}_{\Theta
R}}$ with the property that their component expansion includes terms
proportional to $\widehat{T}^{ab}\widehat{T}_{ab}$, $(T^{a}{}_{a})^{2}$ and
$(T^{a}{}_{a})R$ respectively. We show that for both descriptions of the $3d$
supersymmetric BI theory —i.e. the Maxwell-Goldstone or the tensor-Goldstone
multiplets— the superspace flow operator $\mathcal{O}_{\kappa^{2}}$ can be
expressed uniquely as a linear combination of the superfields
$\mathcal{O}_{T^{2}},~{}{\mathcal{O}_{\Theta^{2}}},~{}{\mathcal{O}_{\Theta
R}}$ and thus identify this flow as a $T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$-like flow. These flows
are the manifestly supersymmetric extensions of those in Section 2. Indeed we
check that the bosonic truncation of the superspace results of this section is
consistent with the bosonic flows of the previous section. In Section 4, we
derive the above-mentioned supercurrents by coupling a large class of vector
and tensor/scalar models to $\mathcal{N}=1$ supergravity. Finally, in Section
5 we summarize these results and identify a few directions for future
investigation. We have also collected our conventions and details of some
calculations in Appendices A and B. An alternate presentation of one of our
bosonic flows is presented in Appendix C.

##  2 Bosonic flows

In this section, we will truncate the D2-brane effective actions of interest
to the bosonic degrees of freedom and study stress tensor flows for the
resulting theories in components. This analysis provides a warm-up for the
manifestly supersymmetric flow equations, which will be presented in section
3, in a simpler context without the complications of fermions.

Nonetheless, even this simplified setting will reveal that new ingredients are
required in order to write such flow equations for brane actions in three
spacetime dimensions. Unlike the known examples in $d=2$, where a stress
tensor deformation of a free scalar yields the Nambu-Goto Lagrangian [3], and
in $d=4$, where a flow equation deforms the Maxwell Lagrangian into the Born-
Infeld theory [59], we will find that flow equations in $d=3$ require the
introduction of a new Lorentz scalar constructed from the stress tensor. This
new invariant is the three-dimensional version of the root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ operator [34].

###  2.1 Maxwell-Goldstone Multiplet

We begin with the bosonic part of the spacetime action for the supersymmetric
$3d$ Born-Infeld theory, which reads

$\displaystyle S_{B}=\kappa^{2}\int
d^{3}x\,\left(1-\sqrt{1+\frac{1}{2\kappa^{2}}f^{ab}f_{ab}}\right)\,.$ (2.1)

In this section, we follow the same conventions as in Section 4, which are
laid out in Appendix B. In particular, spacetime indices are denoted with
lowercase Latin letters $a,b$, etc. which are raised and lowered with the flat
Minkowski metric $\eta_{ab}=\mathrm{diag}(-1,1,1)$. We denote spinorial
indices with Greek letters $\alpha,\beta$, etc. which are raised and lowered
with $\varepsilon_{\alpha\beta}$ where
$\varepsilon_{12}=-1=-\varepsilon_{21}$.

It will first be convenient to develop some general results for an arbitrary
theory of an Abelian field strength $f_{ab}$ in three spacetime dimensions
before specializing to the Lagrangian (2.1). Abelian gauge theories in $d=3$
are considerably simpler than their counterparts in $d=4$ because of the
smaller number of Lorentz scalars that can be constructed from the field
strength. We recall that, in $d=4$, one can construct two independent Lorentz
invariants from the field strength $f_{ab}$, namely

$\displaystyle-\frac{1}{4}f_{ab}f^{ab}=\frac{1}{2}\left(\big{|}\vec{E}\big{|}^{2}-\big{|}\vec{B}\big{|}^{2}\right)\,,\qquad-\frac{1}{4}f_{ab}\widetilde{f}^{ab}=\vec{E}\cdot\vec{B}\,,\qquad(d=4)\,,$
(2.2)

where $\widetilde{f}_{ab}=\frac{1}{2}\epsilon^{abcd}f_{cd}$ is the Hodge dual
of $f_{ab}$. A general Lagrangian constructed from the field strength can
therefore be written as a function of these two Lorentz scalars. However, in
$d=3$, the analogue of $\widetilde{f}_{ab}$ is a scalar field, and one cannot
construct a second independent invariant from the field strength and its dual
as in four dimensions. Therefore, a general Lagrangian built from $f_{ab}$ in
three dimensions is a function of one variable. We will call this variable
${\mathfrak{t}}$, not to be confused with the time coordinate $t$, since the
combination ${\mathfrak{t}}$ will be proportional to the lowest component of
the superfield $T$ considered555See, for instance, equations (3.6) and (4.25),
although note that the conventions for this superfield differ by a factor of
$\frac{2}{\kappa^{2}}$ between these sections. in Sections 3 and 4:

$\displaystyle\mathcal{L}(f_{ab})=\mathcal{L}({\mathfrak{t}})\,,\qquad{\mathfrak{t}}=-\frac{1}{4}f_{ab}f^{ab}\,,\qquad(d=3)\,.$
(2.3)

The Hilbert stress tensor $T_{ab}$ associated with such a general Lagrangian
$\mathcal{L}({\mathfrak{t}})$ is

$\displaystyle T_{ab}$ $\displaystyle=-2\frac{\partial\mathcal{L}}{\partial
g^{ab}}+g_{ab}\mathcal{L}$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial{\mathfrak{t}}}\tensor{f}{{}_{a}^{c}}f_{bc}+g_{ab}\mathcal{L}\,,$
(2.4)

where we take $g_{ab}=\eta_{ab}$ to be the Minkowski metric.

We will need to compute various scalar contractions of the stress-energy
tensor in order to construct the operator which drives our flow. Using (2.1),
one finds

$\displaystyle\begin{split}T^{ab}T_{ab}&=3\mathcal{L}^{2}-8\mathcal{L}{\mathfrak{t}}\frac{\partial\mathcal{L}}{\partial{\mathfrak{t}}}+8\left(\frac{\partial\mathcal{L}}{\partial{\mathfrak{t}}}\right)^{2}{\mathfrak{t}}^{2}\,,\\\
\tensor{T}{{}^{a}_{a}}&=3\mathcal{L}-4{\mathfrak{t}}\frac{\partial\mathcal{L}}{\partial{\mathfrak{t}}}\,.\end{split}$
(2.5)

We have simplified the first expression using the identity

$\displaystyle
f^{ac}\tensor{f}{{}_{c}^{b}}\tensor{f}{{}_{a}^{d}}f_{db}=\frac{1}{2}\left(f_{ab}f^{ab}\right)^{2}\,,$
(2.6)

which follows from the trace relations for a $3\times 3$ matrix (see [33] or
[46] for the computation of the two scalars (2.5) for a field strength in $d$
spacetime dimensions).

One can now see that there is a special combination of these two scalars:

$\displaystyle
T^{ab}T_{ab}-\frac{1}{3}\left(\tensor{T}{{}^{a}_{a}}\right)^{2}=\frac{8}{3}{\mathfrak{t}}^{2}\left(\frac{\partial\mathcal{L}}{\partial{\mathfrak{t}}}\right)^{2}\,.$
(2.7)

This is an especially nice expression since it depends only on
${\mathfrak{t}}$ and the derivative
$\frac{\partial\mathcal{L}}{\partial{\mathfrak{t}}}$, but not on the bare
Lagrangian $\mathcal{L}$ without any derivatives. Furthermore, since the
combination is a perfect square, it is sensible to take its square root so
long as we are careful about our choice of branch. To do this, we now lay out
our assumptions about the signs of quantities appearing in (2.7). Recall that

$\displaystyle{\mathfrak{t}}=\frac{1}{2}\left(\big{|}\vec{E}\big{|}^{2}-\big{|}\vec{B}\big{|}^{2}\right)\,,$
(2.8)

and that the Maxwell Lagrangian in our conventions is
$\mathcal{L}_{\text{Maxwell}}={\mathfrak{t}}$, so that

$\displaystyle\frac{\partial\mathcal{L}_{\text{Maxwell}}}{\partial{\mathfrak{t}}}=1\,.$
(2.9)

In particular, for the Maxwell theory we have
$\frac{\partial\mathcal{L}}{\partial{\mathfrak{t}}}>0$. We will assume that
this inequality is true for all of the theories considered in this work. To
fix the sign of ${\mathfrak{t}}$, we further assume that

$\displaystyle\big{|}\vec{E}\big{|}^{2}<\big{|}\vec{B}\big{|}^{2}\implies{\mathfrak{t}}<0\,.$
(2.10)

This can be justified, for instance, by focusing on the dynamics of small
fluctuations around a classical background with a fixed large magnetic field.
In some sense, this is the more natural choice from the perspective of the
Born-Infeld theory, where the electric field must still satisfy
$|\vec{E}|^{2}<\kappa^{2}$ so that the argument of the square root remains
positive. However, the strength of the magnetic field in the Born-Infeld
theory is allowed to grow arbitrarily large without affecting the reality of
the Lagrangian.

Under the assumptions that ${\mathfrak{t}}<0$ and
$\frac{\partial\mathcal{L}}{\partial{\mathfrak{t}}}>0$, we are free to take
the square root of the combination (2.7) and define

$\displaystyle R$
$\displaystyle=\sqrt{\frac{3}{8}\left(T^{ab}T_{ab}-\frac{1}{3}\left(\tensor{T}{{}^{a}_{a}}\right)^{2}\right)}$
$\displaystyle=\sqrt{{\mathfrak{t}}^{2}\left(\frac{\partial\mathcal{L}}{\partial{\mathfrak{t}}}\right)^{2}}$
$\displaystyle=-{\mathfrak{t}}\frac{\partial\mathcal{L}}{\partial{\mathfrak{t}}}\,.$
(2.11)

If we had instead assumed ${\mathfrak{t}}>0$ and
$\frac{\partial\mathcal{L}}{\partial{\mathfrak{t}}}>0$, our definition of $R$
would have differed by an overall minus sign, which changes some of the
numerical coefficients in the flow equations which follow.

#### Flow Equation

We are now ready to construct our deforming operator. Consider the flow
equation

$\displaystyle\begin{split}\frac{\partial\mathcal{L}}{\partial\lambda}&=O_{\lambda}\,,\\\
O_{\lambda}&=\frac{1}{6}T^{ab}T_{ab}-\frac{1}{9}\left(\tensor{T}{{}^{a}_{a}}\right)^{2}+\frac{1}{9}\left(\tensor{T}{{}^{a}_{a}}\right)R\,,\end{split}$
(2.12)

with $R$ as in (2.1). Note that we will later use calligraphic symbols like
$\mathcal{O}_{\kappa^{2}}$ for superfield operators, whereas the non-
calligraphic symbols like $O_{\lambda}$ denotes bosonic operators. Here
$\lambda$ is an irrelevant coupling constant with length dimension $3$.

In this paper, we will always assume that $\lambda$ is positive. This is
sometimes referred to as the “good sign” of the deformation parameter because,
in the context of ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu}$ deformations in two spacetime dimensions where the operator is well-
defined quantum mechanically, the finite-volume spectrum of a deformed CFT
remains real for some range of positive $\lambda$, whereas for negative
$\lambda$ all but finitely many of the energy levels become complex.666In some
cases one can cure these complex energies by performing sequential flows, for
instance deforming a collection of theories by negative $\lambda$ and then
deforming their tensor product by a positive value of $\lambda$ which is
sufficiently large [60]. There have been various proposed interpretations for
these complex energies, including holographic pictures such as a finite cutoff
in $\mathrm{AdS}_{3}$ [61] or a change in spacetime signature [62]. The
negative sign of the deformation parameter has also been related to de Sitter
constructions [63, 64, 65]. However, in the present work we will consider only
classical good-sign flows, in part because it is not known whether one can
define a local ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu}$-like operator from the stress tensor in $d>2$ dimensions (see [66] for
a discussion of this point).

Using the explicit expressions (2.5) for $T^{ab}T_{ab}$,
$\tensor{T}{{}^{a}_{a}}$ and (2.1) for $R$, the flow equation (2.12) can be
written as

$\displaystyle\frac{\partial\mathcal{L}}{\partial\lambda}=\mathcal{L}{\mathfrak{t}}\frac{\partial\mathcal{L}}{\partial{\mathfrak{t}}}-\frac{1}{2}\mathcal{L}^{2}\,.$
(2.13)

It is also sometimes convenient to change variables as
$\lambda=\frac{1}{\kappa^{2}}$, where $\kappa^{2}$ has mass dimension $3$, and
write the flow equation as

$\displaystyle\begin{split}\kappa^{4}\frac{\partial\mathcal{L}}{\partial\kappa^{2}}&=O_{\kappa^{2}}\,,\\\
O_{\kappa^{2}}&=-\frac{1}{6}T^{ab}T_{ab}+\frac{1}{9}\left(\tensor{T}{{}^{a}_{a}}\right)^{2}-\frac{1}{9}\left(\tensor{T}{{}^{a}_{a}}\right)R\,.\end{split}$
(2.14)

Because of the change of variables, the operator $O_{\kappa^{2}}=-O_{\lambda}$
which drives the flow in equation (2.14) differs by a sign from the
corresponding operator in equation (2.12).

We now return to the Born-Infeld theory (2.1). In terms of the variable
${\mathfrak{t}}$, this Lagrangian can be written as

$\displaystyle\mathcal{L}_{B}(\kappa^{2},{\mathfrak{t}})$
$\displaystyle=\kappa^{2}\left(1-\sqrt{1-\frac{2}{\kappa^{2}}{\mathfrak{t}}}\right)\,,$
(2.15)

or using the parameter $\lambda=\frac{1}{\kappa^{2}}$,

$\displaystyle\mathcal{L}_{B}(\lambda,{\mathfrak{t}})$
$\displaystyle=\frac{1}{\lambda}\left(1-\sqrt{1-2\lambda{\mathfrak{t}}}\right)\,.$
(2.16)

One can verify by direct computation that the Lagrangian (2.16) satisfies the
differential equation (2.13), or equivalently that
$\mathcal{L}_{B}(\kappa^{2},{\mathfrak{t}})$ of (2.15) solves the equation
(2.14).

#### Modified Trace Flow Equation

The free Maxwell theory in three spacetime dimensions is not a conformal field
theory. Therefore, deformations of the $3d$ Maxwell Lagrangian – such as the
stress tensor flow which produces the Born-Infeld theory that was presented in
the preceding subsection – do not exhibit the usual properties for
deformations of CFTs which are often used when studying perturbations of this
kind.

An especially useful example of such a property is the so-called trace flow
equation, which plays a role in many places in the ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ literature, including
in cutoff $\mathrm{AdS}_{3}$ holography [67, 68, 69, 70, 71, 72] and in $4d$
${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$-like flows
[45]. We now review and extend this result. Let $\mathcal{L}_{0}$ be a
Lagrangian which describes a classically conformal field theory in $d$
spacetime dimensions (in particular, the trace of the Hilbert stress tensor
associated with $\mathcal{L}_{0}$ vanishes). Suppose that we deform
$\mathcal{L}_{0}$ according to the flow equation

$\displaystyle\frac{\partial\mathcal{L}_{\lambda}}{\partial\lambda}={O\left(T_{ab}\left(\lambda\right)\right)}\,$
(2.17)

with the initial condition $\mathcal{L}_{\lambda}\to\mathcal{L}_{0}$ as
$\lambda\to 0$, and where $O$ is an arbitrary scalar function of the stress
tensor $T_{ab}(\lambda)$ associated with $\mathcal{L}_{\lambda}$. We assume
that the deformation parameter $\lambda$ has length dimension $\Delta$.
Because $\mathcal{L}_{0}$ is conformally invariant, there is only a single
energy scale $\Lambda=\lambda^{-1/\Delta}$ in the deformed theory
$\mathcal{L}_{\lambda}$. The response of $\mathcal{L}_{\lambda}$ to an
infinitesimal scale transformation is determined by the trace of the stress
tensor as

$\displaystyle\Lambda\frac{dS}{d\Lambda}=\int
d^{d}x\,\tensor{T}{{}^{a}_{a}}(\lambda)\,.$ (2.18)

Comparing the expression (2.18) to (2.17) and equating the integrands –
neglecting any potential total derivatives which would not contribute in the
integrals – one finds that

$\displaystyle\tensor{T}{{}^{a}_{a}}(\lambda)=-\Delta\lambda{O\left(T_{ab}\left(\lambda\right)\right)}\,,$
(2.19)

which can also be written as

$\displaystyle\lambda\frac{\partial\mathcal{L}_{\lambda}}{\partial\lambda}=-\frac{1}{\Delta}\tensor{T}{{}^{a}_{a}}\,.$
(2.20)

Therefore, the trace of the stress tensor in the deformed theory
$\mathcal{L}_{\lambda}$ is determined in terms of the deforming operator
${O\left(T_{ab}\left(\lambda\right)\right)}$. The condition (2.19) applies,
for instance, to the ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu}$ deformation of a free scalar in two spacetime dimensions, or to the
analogous four-dimensional $T^{2}$ deformation of the free $4d$ Maxwell
Lagrangian.

Because the $3d$ Maxwell Lagrangian is not a CFT, the trace flow equation
(2.19) does not hold for the $3d$ Born-Infeld Lagrangian (2.16), which we have
obtained as such a stress tensor flow with $\Delta=d=3$ and where the operator
$O(T_{ab})$ is given by

$\displaystyle{O\left(T_{ab}\right)}=O_{\lambda}\equiv\frac{1}{6}T^{ab}T_{ab}-\frac{1}{9}\left(\tensor{T}{{}^{a}_{a}}\right)^{2}+\frac{1}{9}\left(\tensor{T}{{}^{a}_{a}}\right)R\,,$
(2.21)

as in equation (2.12). However, one might wonder whether the Born-Infeld
theory satisfies a modified version of such a trace flow equation. Besides the
trace $\tensor{T}{{}^{a}_{a}}$, another natural Lorentz invariant built from
$T_{ab}$ which has the same dimension is the operator

$\displaystyle
R=\sqrt{\frac{3}{8}\left(T^{ab}T_{ab}-\frac{1}{3}\left(\tensor{T}{{}^{a}_{a}}\right)^{2}\right)}$
(2.22)

which we used to construct the flow. In fact, the Born-Infeld Lagrangian
$\mathcal{L}_{B}$ satisfies

$\displaystyle\lambda\frac{\partial\mathcal{L}_{B}}{\partial\lambda}=-\frac{1}{3}\left(\tensor{T}{{}^{a}_{a}}(\lambda)-R(\lambda)\right)\,,$
(2.23)

which takes a similar form as equation (2.20) for a conformal seed theory
except with an “effective trace” of $\tensor{T}{{}^{a}_{a}}-R$ rather than the
usual trace $\tensor{T}{{}^{a}_{a}}$. Equivalently, one finds that the trace
of the stress tensor associated with $\mathcal{L}_{B}$ obeys

$\displaystyle\tensor{T}{{}^{a}_{a}}-R=-3\lambda
O_{\lambda}\left(T_{ab}(\lambda)\right)\,,$ (2.24)

for the function $O_{\lambda}$ in (2.21). Again, comparing to the ordinary
trace flow equation (2.19) which holds for a deformed CFT, we find that the
$3d$ Born-Infeld theory obeys a modified trace flow equation where the role of
the trace is played by the difference between the trace
$\tensor{T}{{}^{a}_{a}}$ and the root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$-like operator $R$.

#### Relevant Flow Equation

In [45], it was pointed out that one version of the four-dimensional Born-
Infeld Lagrangian also satisfies a flow equation driven by a _relevant_
operator constructed from the stress tensor, as opposed to the usual
irrelevant ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu}$-like combination which deforms the $4d$ Maxwell theory to Born-Infeld.
This result relied upon the fact that one can construct a “subtracted” version
of the theory by adding an appropriate $\lambda$-dependent constant to the
Lagrangian which causes the ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$-like operator, which
usually drives the irrelevant flow, to become a constant. It is natural to ask
whether any analogue of these results also holds for the three-dimensional
Born-Infeld theory.

We will see that the answer is yes. We first define the “subtracted” $3d$
Born-Infeld Lagrangian as

$\displaystyle\widetilde{\mathcal{L}}_{B}=\frac{\alpha}{\lambda}\sqrt{1-2\lambda{\mathfrak{t}}}\,,$
(2.25)

where $\alpha$ is a dimensionless constant. We refer to this as a “subtracted”
theory because the usual Born-Infeld theory (2.16) is a sum of two terms, the
constant term $\frac{1}{\lambda}$ and the square root term, so one can simply
subtract off the constant $\frac{1}{\lambda}$ term (and rescale by an overall
factor) to arrive at (2.25). We note that the equations of motion for this
model are not affected by the constant term so the dynamics of
$\mathcal{L}_{B}$ are identical to those of $\widetilde{\mathcal{L}}_{B}$ (so
long as we do not couple the theory to dynamical gravity, in which case the
$\frac{1}{\lambda}$ term plays the role of a cosmological constant).

One can compute the stress tensor $\widetilde{T}_{ab}$ associated with the
subtracted theory $\widetilde{\mathcal{L}}_{B}$, and then assemble the
combination $O_{\lambda}(\widetilde{T}_{ab})$, which turns out to be a
constant:

$\displaystyle O_{\lambda}(\widetilde{T}_{ab})$
$\displaystyle=\frac{1}{6}\widetilde{T}^{ab}\widetilde{T}_{ab}-\frac{1}{9}\left(\tensor{\widetilde{T}}{{}^{a}_{a}}\right)^{2}+\frac{1}{9}\left(\tensor{\widetilde{T}}{{}^{a}_{a}}\right)R$
$\displaystyle=-\frac{\alpha^{2}}{2\lambda^{2}}\,.$ (2.26)

If we assume that $\lambda$ is positive, we can therefore write

$\displaystyle\lambda=\frac{\left|\alpha\right|}{\sqrt{2\left|O_{\lambda}(\widetilde{T}_{ab})\right|}}\,.$
(2.27)

This relation allows us to write a different flow equation for the Lagrangian
$\widetilde{\mathcal{L}}_{B}$. It is most convenient to express this flow in
terms of the variable $\kappa^{2}$ rather than $\lambda=\frac{1}{\kappa^{2}}$.
By first substituting the solution (2.27) into the modified trace flow
equation (2.24), then using the resulting expression for
$\tensor{T}{{}^{a}_{a}}-R$ in equation (2.23), writing the equation in terms
of the stress tensor $\widetilde{T}_{ab}$ for $\widetilde{\mathcal{L}}_{B}$,
and finally changing variables from $\lambda$ to $\kappa^{2}$, one finds

$\displaystyle\frac{\partial\widetilde{\mathcal{L}}_{B}}{\partial\kappa^{2}}=\frac{\left|\alpha\right|\left(\tensor{\widetilde{T}}{{}^{a}_{a}}-\widetilde{R}\right)}{\sqrt{\left|3\widetilde{T}^{ab}\widetilde{T}_{ab}-2\left(\tensor{\widetilde{T}}{{}^{a}_{a}}\right)^{2}+2\left(\tensor{\widetilde{T}}{{}^{a}_{a}}\right)\widetilde{R}\right|}}\,,$
(2.28)

where $\widetilde{R}$ is the root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$-like combination
constructed from $\widetilde{T}_{ab}$. We have therefore written a flow
equation for the subtracted Lagrangian $\widetilde{\mathcal{L}}_{B}$ driven by
a relevant operator constructed from the stress tensor.

The non-trivial feature of this result is that the right side of (2.28)
depends only on $\widetilde{T}_{ab}$ and not on the flow parameter
$\kappa^{2}$. One can always trivially rewrite a flow equation involving an
irrelevant deformation parameter $\lambda=\frac{1}{\kappa^{2}}$ by changing
variables to $\kappa^{2}$. However, such a rewriting will generally produce an
expression for $\frac{\partial\mathcal{L}}{\partial\kappa^{2}}$ which is a
relevant combination of both $\kappa^{2}$ and the original deforming operator.
In contrast, the flow (2.28) is driven by an operator built solely from the
stress tensor itself, which is a special feature of the Lagrangian
$\widetilde{\mathcal{L}}_{B}$ that is possible because the combination
$f(\widetilde{T}_{ab})$ is a constant.

###  2.2 Tensor Multiplet

As explored in [30], one can also interpret the three-dimensional
$\mathcal{N}=1$ tensor multiplet as the Goldstone for a spontaneously broken
second supersymmetry. In this section, we will study stress tensor flows for
the bosonic truncation of this multiplet.

The bosonic field content of the tensor-Goldstone multiplet is a scalar field
$\phi$ and an auxiliary field $H$ which can be assembled into a bispinor

$\displaystyle\hat{f}_{\alpha\beta}=\partial_{\alpha\beta}\phi+i\varepsilon_{\alpha\beta}H\,,$
(2.29)

where we remind the reader that in this section we use the notation of
Appendix B for spinorial indices $\alpha,\beta$.

One could have anticipated that this multiplet, for which the physical bosonic
degree of freedom is the scalar field $\phi$, might have provided an
equivalent description of the Maxwell-Goldstone multiplet from Hodge duality.
Indeed, in three spacetime dimensions a $2$-form field strength is equivalent
to a scalar field through the duality relation

$\displaystyle d\phi=\star f\,,$ (2.30)

where $f=f_{ab}\,dx^{a}\wedge dx^{b}$, $d$ is the exterior derivative, and
$\star$ is the Hodge star.

The undeformed Lagrangian for the bosonic truncation of this multiplet, which
is the large-$\kappa^{2}$ or small-$\lambda$ limit of the solution to the flow
equation which we develop shortly, is proportional to the combination

$\displaystyle\hat{f}^{\alpha\beta}\hat{f}_{\alpha\beta}=\partial_{\alpha\beta}\phi\partial^{\alpha\beta}\phi+2H^{2}$
$\displaystyle=-2\partial^{a}\phi\partial_{a}\phi+2H^{2}\,.$ (2.31)

Note that, for any Lagrangian $\mathcal{L}$ that depends on the field $H$ only
through the combination $H^{2}$ (and which has no kinetic term for $H$), the
equation of motion for $H$ is

$\displaystyle H\frac{\partial\mathcal{L}}{\partial H^{2}}=0\,,$ (2.32)

which admits the solution $H=0$. Therefore, if we are only interested in on-
shell flows where the equation of motion for $H$ is imposed, then it suffices
to eliminate the field $H$ from the Lagrangian and restrict attention to the
physical degree of freedom $\phi$. A common issue when studying stress tensor
flows for theories which include auxiliary fields is that it is often possible
to engineer a flow which reproduces a given Lagrangian on-shell, but not off-
shell. We will encounter precisely this issue in the analysis of this section
and the same behavior arises in the superspace flow of Section 3.

With a particular choice of normalization, the bosonic truncation of the
Lagrangian for the tensor-Goldstone multiplet can be written as

$\displaystyle\mathcal{L}_{\mathrm{TG}}=\frac{1}{\lambda}\left(1-\sqrt{1+2\lambda\left(\partial^{a}\phi\partial_{a}\phi-H^{2}\right)}\right)=\kappa^{2}\left(1-\sqrt{1+\frac{2}{\kappa^{2}}\left(\partial^{a}\phi\partial_{a}\phi-H^{2}\right)}\right)$
(2.33)

in terms of either the variable $\lambda$ or $\kappa^{2}=\frac{1}{\lambda}$.
When the auxiliary field $H$ is set to zero using its equation of motion, this
reduces to

$\displaystyle\mathcal{L}_{\mathrm{TG}}\big{|}_{H=0}=\frac{1}{\lambda}\left(1-\sqrt{1+2\lambda\partial^{a}\phi\partial_{a}}\right)=\kappa^{2}\left(1-\sqrt{1+\frac{2}{\kappa^{2}}\partial^{a}\phi\partial_{a}\phi}\right)\,,$
(2.34)

which is the scalar sector of the Dirac-Born-Infeld action. From this
perspective, $\phi$ can be thought of as a scalar field that describes the
transverse fluctuations to a D2-brane, and the parameter $\kappa^{2}$ controls
the tension of the brane.

#### Flow Equation

For the moment, we will consider off-shell flows where we do not eliminate the
auxiliary field $H$ using its equation of motion. We first compute the stress
tensor for a general Lagrangian $\mathcal{L}(x_{1},x_{2})$ which depends on
the two Lorentz-invariant combinations

$\displaystyle x_{1}=\partial^{a}\phi\partial_{a}\phi\,,\qquad x_{2}=H^{2}\,.$
(2.35)

As $H$ does not couple to the metric, the Hilbert stress tensor is simply

$\displaystyle
T_{ab}=\eta_{ab}\mathcal{L}-2\frac{\partial\mathcal{L}}{\partial
x_{1}}\partial_{a}\phi\partial_{b}\phi\,.$ (2.36)

A direct computation of the contractions of $T_{ab}$ needed to construct our
flow gives

$\displaystyle T^{ab}T_{ab}$
$\displaystyle=3\mathcal{L}^{2}-4\mathcal{L}x_{1}\frac{\partial\mathcal{L}}{\partial
x_{1}}+4x_{1}^{2}\left(\frac{\partial\mathcal{L}}{\partial
x_{1}}\right)^{2}\,,$ $\displaystyle\tensor{T}{{}^{a}_{a}}$
$\displaystyle=3\mathcal{L}-2x_{1}\frac{\partial\mathcal{L}}{\partial
x_{1}}\,.$ (2.37)

As in the gauge theory analysis, there is a special combination of these
contractions:

$\displaystyle
T^{ab}T_{ab}-\frac{1}{3}\left(\tensor{T}{{}^{a}_{a}}\right)^{2}=\frac{8}{3}x_{1}^{2}\left(\frac{\partial\mathcal{L}}{\partial
x_{1}}\right)^{2}\,.$ (2.38)

Conveniently, this combination (2.38) is a perfect square which does not
depend directly on $\mathcal{L}$ but only on its derivative with respect to
$x_{1}$.

In order to consistently take square roots, we will again need to make certain
assumptions about the signs of quantities in the scalar context. The free
Lagrangian is given by $\mathcal{L}=-x_{1}$, where

$\displaystyle
x_{1}=\partial^{a}\phi\partial_{a}\phi=-\left(\partial_{t}\phi\right)^{2}+\left(\partial_{x}\phi\right)^{2}\,.$
(2.39)

We will assume that $x_{1}<0$ and $\frac{\partial\mathcal{L}}{\partial
x_{1}}<0$. Note that this is, in some sense, the opposite convention as that
chosen in section 2.1, where we assumed that
$\big{|}\vec{E}\big{|}^{2}<\big{|}\vec{B}\big{|}^{2}$. Here we are instead
supposing that
$\left(\partial_{t}\phi\right)^{2}>\left(\partial_{x}\phi\right)^{2}$. We will
see that this sign choice is convenient because it will imply that the scalar
theory satisfies the same flow equation, with the same relative coefficients,
as the Born-Infeld Lagrangian; with the opposite sign convention, the sign of
one of the terms would be reversed.

With this choice of sign, we may define the root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$-like operator $R$ via
the square root of (2.38),

$\displaystyle
R=\sqrt{\frac{3}{8}\left(T^{ab}T_{ab}-\frac{1}{3}\left(\tensor{T}{{}^{a}_{a}}\right)^{2}\right)}=x_{1}\frac{\partial
L}{\partial x_{1}}\,,$ (2.40)

where we choose the positive root because $x_{1}\frac{\partial L}{\partial
x_{1}}>0$ in these conventions.

It is worth pointing out that, with either choice of sign, the operator $R$ is
an on-shell total derivative. The equation of motion associated with any
Lagrangian $\mathcal{L}(x_{1})$ is

$\displaystyle
0=\partial_{a}\left(\frac{\partial\mathcal{L}}{\partial\left(\partial_{a}\phi\right)}\right)=2\partial^{a}\left(\partial_{a}\phi\frac{\partial\mathcal{L}}{\partial
x_{1}}\right)\,.$ (2.41)

On the other hand,

$\displaystyle R$ $\displaystyle=x_{1}\frac{\partial\mathcal{L}}{\partial
x_{1}}=\partial^{a}\phi\partial_{a}\phi\frac{\partial\mathcal{L}}{\partial
x_{1}}=\partial^{a}\left(\phi\partial_{a}\phi\frac{\partial\mathcal{L}}{\partial
x_{1}}\right)-\phi\partial^{a}\left(\partial_{a}\phi\frac{\partial\mathcal{L}}{\partial
x_{1}}\right)\,.$ (2.42)

Comparing the final expression of (2.42) to that of (2.41), we see that the
second term of $R$ vanishes on-shell and thus $R$ is a total derivative up to
equations of motion as claimed.

We now propose the flow equation

$\displaystyle\frac{\partial\mathcal{L}}{\partial\lambda}$
$\displaystyle=\frac{1}{6}T^{ab}T_{ab}-\frac{1}{9}\left(\tensor{T}{{}^{a}_{a}}\right)^{2}+\frac{1}{9}\tensor{T}{{}^{a}_{a}}\cdot
R$ $\displaystyle=\mathcal{L}x_{1}\frac{\partial\mathcal{L}}{\partial
x_{1}}-\frac{1}{2}\mathcal{L}^{2}\,,$ (2.43)

which takes the same form as the differential equation (2.12) which deformed
the free Maxwell theory to the Born-Infeld theory. The solution to the
differential equation (2.2) with initial condition

$\displaystyle\mathcal{L}_{0}=-x_{1}+x_{2}=-\partial^{a}\phi\partial_{a}\phi+H^{2}$
(2.44)

is

$\displaystyle\mathcal{L}(\lambda)=\frac{2+2\lambda x_{2}-2\sqrt{1+\lambda
x_{1}(2+\lambda x_{2})}}{\lambda(2+\lambda x_{2})}\,.$ (2.45)

This solution does not match the bosonic sector of the tensor multiplet (2.33)
off-shell. However, if we put the auxiliary on-shell by setting $H=0$, (2.45)
reduces to

$\displaystyle\mathcal{L}(\lambda)\big{|}_{x_{2}=0}=\frac{1}{\lambda}\left(1-\sqrt{1+2\lambda\partial^{a}\phi\partial_{a}\phi}\right)\,,$
(2.46)

where we have replaced $x_{1}=\partial^{a}\phi\partial_{a}\phi$. This _does_
match the expression (2.34) for the tensor-Goldstone Lagrangian when the
auxiliary field is set to zero. Therefore the differential equation (2.2) can
be thought of as an on-shell flow which produces the tensor-Goldstone
Lagrangian from the initial condition
$\mathcal{L}_{0}=-\partial^{a}\phi\partial_{a}\phi+H^{2}$, albeit only when
the auxiliary field is set to zero.

In a sense, the reason that this deformation only yields the desired solution
on-shell is because the two terms in $\mathcal{L}_{0}$ have different metric
dependence: the scalar kinetic term
$x_{1}=g^{ab}\partial_{a}\phi\partial_{b}\phi$ couples explicitly to the
metric, whereas $x_{2}=H^{2}$ is independent of the metric and therefore
contributes to the Hilbert stress tensor only through the variation of the
measure factor $\sqrt{-g}$. It is possible, though somewhat artificial, to
write a Lagrangian which is equivalent to $\mathcal{L}_{0}$ but where the two
terms couple to the metric in a more symmetrical way by introducing an
auxiliary vector field $v^{a}$; this procedure is discussed in Appendix C and
leads to an off-shell flow that yields the tensor-Goldstone Lagrangian.

#### Modified Trace Flow Equation

For simplicity, in this section we restrict to on-shell flows where the
auxiliary field $H$ is set to zero. We focus on the flow for the Lagrangian

$\displaystyle\mathcal{L}=\frac{1}{\lambda}\left(1-\sqrt{1+2\lambda
x_{1}}\right)\,,$ (2.47)

where $x_{1}=\partial^{a}\phi\partial_{a}\phi$, which we have seen satisfies
the flow equation

$\displaystyle\frac{\partial
L}{\partial\lambda}=O_{\lambda}=\frac{1}{6}T^{ab}T_{ab}-\frac{1}{9}\left(\tensor{T}{{}^{a}_{a}}\right)^{2}+\frac{1}{9}\tensor{T}{{}^{a}_{a}}R\,,$
(2.48)

where
$R=\sqrt{\frac{3}{8}\left(T^{ab}T_{ab}-\frac{1}{3}\left(\tensor{T}{{}^{a}_{a}}\right)^{2}\right)}=x_{1}\frac{\partial\mathcal{L}}{\partial
x_{1}}$.

The undeformed limit of (2.47) is the theory of a free scalar in three
spacetime dimensions, which is not scale invariant in the sense of having a
stress tensor with vanishing trace. For
$\mathcal{L}_{0}=-\partial^{a}\phi\partial_{a}\phi$, one finds

$\displaystyle\tensor{T}{{}^{a}_{a}}=-\partial^{a}\phi\partial_{a}\phi\neq
0\,.$ (2.49)

Although one can perform an improvement transformation to eliminate the trace
(2.49) for the free scalar theory (which is a consequence of the fact that
$\partial^{a}\phi\partial_{a}\phi$ is an on-shell total derivative), the
standard Hilbert stress tensor without any improvement term is non-vanishing.
As a result, the deformed theory (2.47) cannot possibly satisfy the trace flow
equation $\tensor{T}{{}^{a}_{a}}(\lambda)=-3\lambda f(T_{ab}(\lambda))$ for a
deformed conformal field theory, where $f(T_{ab}(\lambda))$ is defined by
equation (2.48), since this requires $\tensor{T}{{}^{a}_{a}}=0$ when
$\lambda=0$.

However, as in the analysis of Section 2.1, one might ask whether the deformed
Lagrangian (2.47) satisfies a modified version of the usual trace flow
equation as the flow parameter $\lambda$ is varied. Again we find that the
answer is yes; one finds that this Lagrangian satisfies the relations

$\displaystyle\lambda\frac{\partial\mathcal{L}}{\partial\lambda}=-\frac{1}{3}\left(\tensor{T}{{}^{a}_{a}}-R\right)$
(2.50)

or equivalently

$\displaystyle\lambda
O_{\lambda}\left(T_{ab}(\lambda)\right)=-\frac{1}{3}\left(\tensor{T}{{}^{a}_{a}}-R\right)\,,$
(2.51)

where again
$O_{\lambda}\left(T_{ab}(\lambda)\right)=\frac{1}{6}T^{ab}T_{ab}-\frac{1}{9}\left(\tensor{T}{{}^{a}_{a}}\right)^{2}+\frac{1}{9}\tensor{T}{{}^{a}_{a}}R$
is the deforming operator. Thus we see that this deformation of the three
dimensional free scalar satisfies a modified trace flow equation where the
role of the trace is played by the combination $\tensor{T}{{}^{a}_{a}}-R$.

Equations (2.50) and (2.51) hold as exact off-shell relations. However, we
have noted before around equation (2.42) that the operator $R$ is an on-shell
total derivative for any Lagrangian $\mathcal{L}(x_{1})$. Thus the
contribution from the $R$ term drops out of the integrated expressions of
these modified trace flow equations. For instance, one has

$\displaystyle\int
d^{3}x\,\lambda\frac{\partial\mathcal{L}}{\partial\lambda}=-\frac{1}{3}\int
d^{3}x\,\left(\tensor{T}{{}^{a}_{a}}-R\right)\simeq-\frac{1}{3}\int
d^{3}x\,\tensor{T}{{}^{a}_{a}}\,,$ (2.52)

where in the last step we write $\simeq$ to indicate equivalence up to any
possible boundary terms which arise from integrating a total spacetime
derivative and equations of motion.

This is in accord with the fact that, despite the non-zero trace of the
Hilbert stress tensor for the theory of a free scalar field in $d>2$
dimensions, one can define an improved stress tensor whose trace vanishes. Let
us briefly review this simple observation. In any spacetime dimension $d$, the
Lagrangian $\mathcal{L}=\partial^{a}\phi\partial_{a}\phi$ for a single
massless scalar has the Hilbert stress tensor

$\displaystyle
T_{ab}=-2\partial_{a}\phi\partial_{b}\phi+g_{\mu\nu}\partial^{c}\phi\partial_{c}\phi\,,$
(2.53)

whose trace is $\tensor{T}{{}^{a}_{a}}=(d-2)\partial^{c}\phi\partial_{c}\phi$.
One can always improve the stress tensor as

$\displaystyle
T_{ab}\longrightarrow{T}^{\prime}_{ab}=T_{ab}+\left(\partial_{a}\partial_{b}-\eta_{ab}\partial^{c}\partial_{c}\right)u\,,$
(2.54)

for any function $u$, without affecting the symmetry or conservation of the
stress tensor. In particular, we may choose

$\displaystyle u=\frac{d-2}{2(d-1)}\phi^{2}\,,$ (2.55)

so that the improved stress tensor is

$\displaystyle
T^{\prime}_{ab}=-2\partial_{a}\phi\partial_{b}\phi+g_{\mu\nu}\partial^{c}\phi\partial_{c}\phi+\frac{d-2}{2(d-1)}\left(\partial_{a}\partial_{b}-\eta_{ab}\partial^{c}\partial_{c}\right)\phi^{2}\,.$
(2.56)

The trace of the stress tensor, after dropping all terms proportional to
$\partial^{c}\partial_{c}\phi$ which vanish on-shell, is then

$\displaystyle\tensor{T}{{}^{\prime}{}^{a}_{a}}=\left(-2+d+\frac{d-2}{2(d-1)}\cdot
2\left(1-d\right)\right)\partial^{c}\phi\partial_{c}\phi=0\,.$ (2.57)

Now consider deforming this $d$-dimensional free scalar theory by any function
$f\left(T^{\prime}_{ab}\right)$. Since the improved stress tensor
$T^{\prime}_{ab}$ agrees with the Hilbert stress tensor $T_{ab}$ up to on-
shell total derivatives, it can also be used to generate scale transformations
inside of spacetime integrals. Further, the trace of the improved stress
tensor vanishes at $\lambda=0$ by construction. Thus by the general arguments
presented around equation (2.19), one has

$\displaystyle\lambda\int
d^{3}x\,\frac{\partial\mathcal{L}_{\lambda}}{\partial\lambda}=-\frac{1}{d}\int
d^{3}x\,\tensor{T}{{}^{\prime}{}^{a}_{a}}\,.$ (2.58)

If one were to strip off the integrals in (2.58), equating the integrands and
neglecting total derivatives, one would conclude that
$\lambda\frac{\partial\mathcal{L}_{\lambda}}{\partial\lambda}=-\frac{1}{d}\tensor{T}{{}^{\prime}{}^{a}_{a}}$,
which seems to contradict the result (2.50) because it is missing the term
proportional to $R$. However we now see that the two equations are consistent
inside of a total spacetime integral, because the term $R$ is an on-shell
total derivative, as is the difference between
$\tensor{T}{{}^{\prime}{}^{a}_{a}}$ and $\tensor{T}{{}^{a}_{a}}$.

#### Relevant Flow Equation

For completeness, we now repeat the relevant flow analysis of Section 2.1 for
the case of the deformed scalar theory. We will find that a subtracted version
of this Lagrangian also satisfies a flow equation driven by a relevant
combination of stress tensors, exactly like in the gauge theory case. As in
the preceding analysis, for simplicity we will use the auxiliary field
equation of motion to set $H=0$ and focus purely on the scalar $\phi$.

We first define the subtracted version of the deformed scalar Lagrangian as

$\displaystyle\widetilde{\mathcal{L}}=\frac{\alpha}{\lambda}\sqrt{1+2\lambda
x_{1}}\,,$ (2.59)

where $\alpha$ is a dimensionless constant. Again, the constant term that we
have removed does not affect the dynamics of the theory. However, we note that
for the $2d$ ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$
deformation of free scalars, the corresponding $\frac{1}{\lambda}$ term can be
interpreted as arising from a coupling between the Nambu-Goto string and a
constant target-space $B$ field [73]. A related observation is that the $2d$
${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation
can be obtained by considering the uniform light-cone gauge for the string, as
discussed in [49, 74, 75, 76]. In our case, since we may interpret the scalar
$\phi$ as a transverse fluctuation of a D2-brane, it is natural to instead
view the $\frac{1}{\lambda}$ term as a coupling to a constant target-space
Ramond-Ramond field $C_{3}$. It would be interesting to explore whether an
analogue of the uniform-light cone gauge for a D2-brane yields the deformation
considered in this work.

Computing the stress tensor $\widetilde{T}_{ab}$ associated with the
subtracted Lagrangian $\widetilde{\mathcal{L}}$, one finds that it satisfies

$\displaystyle
O_{\lambda}\left(\widetilde{T}_{ab}(\lambda)\right)=\frac{1}{6}\widetilde{T}^{ab}\widetilde{T}_{ab}-\frac{1}{9}\left(\tensor{\widetilde{T}}{{}^{a}_{a}}\right)^{2}+\frac{1}{9}\tensor{\widetilde{T}}{{}^{a}_{a}}\widetilde{R}=-\frac{\alpha^{2}}{2\lambda}\,,$
(2.60)

where $\widetilde{R}$ is the root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$-like operator for the
subtracted theory $\widetilde{\mathcal{L}}$. In particular, the deforming
operator $O_{\lambda}\left(\widetilde{T}_{ab}(\lambda)\right)$ of equation
(2.60) is a constant, independent of fields.

By an argument analogous to that of section 2.1, one can then show that the
subtracted scalar Lagrangian satisfies

$\displaystyle\frac{\partial\widetilde{\mathcal{L}}}{\partial\kappa^{2}}=\frac{|\alpha|\left(\tensor{\widetilde{T}}{{}^{a}_{a}}-\widetilde{R}\right)}{\sqrt{\left|3T^{ab}T_{ab}-2\left(\tensor{T}{{}^{a}_{a}}\right)^{2}+2\tensor{T}{{}^{a}_{a}}R\right|}}\,,$
(2.61)

which is again a flow equation driven by a relevant combination of stress
tensors.

#### General Dimension

To conclude this section, we point out that the possibility of engineering a
square-root solution to the flow equation for a single free scalar field is
not special to three spacetime dimensions. A straightforward generalization of
the above argument applies in any spacetime dimension $d$.

Let $\phi$ be a scalar field in $d$ dimensions and consider a general
Lagrangian which depends on the Lorentz invariant
$x_{1}=\partial^{a}\phi\partial_{a}\phi$. The Hilbert stress tensor is
identical to the one we considered in $d=3$, namely

$\displaystyle
T_{ab}=\eta_{ab}\mathcal{L}-2\frac{\partial\mathcal{L}}{\partial
x_{1}}\partial_{a}\phi\partial_{b}\phi\,,$ (2.62)

and the contractions we need are

$\displaystyle T^{ab}T_{ab}$
$\displaystyle=d\mathcal{L}^{2}-4\mathcal{L}x_{1}\frac{\partial\mathcal{L}}{\partial
x_{1}}+4x_{1}^{2}\left(\frac{\partial\mathcal{L}}{\partial
x_{1}}\right)^{2}\,,$ $\displaystyle\tensor{T}{{}^{a}_{a}}$
$\displaystyle=d\mathcal{L}-2x_{1}\frac{\partial\mathcal{L}}{\partial
x_{1}}\,.$ (2.63)

Now the special combination of these invariants is

$\displaystyle
T^{ab}T_{ab}-\frac{1}{d}\left(\tensor{T}{{}^{a}_{a}}\right)^{2}=\frac{4(d-1)}{d}x_{1}^{2}\left(\frac{\partial\mathcal{L}}{\partial
x_{1}}\right)^{2}\,.$ (2.64)

If we again assume $x_{1}<0$ and $\frac{\partial\mathcal{L}}{\partial
x_{1}}<0$ then we may take the square root to define

$\displaystyle
R=\sqrt{\frac{d}{4(d-1)}\left(T^{ab}T_{ab}-\frac{1}{d}\left(\tensor{T}{{}^{a}_{a}}\right)^{2}\right)}=x_{1}\frac{\partial\mathcal{L}}{\partial
x_{1}}\,.$ (2.65)

Next we consider the flow equation

$\displaystyle\begin{split}\frac{\partial\mathcal{L}}{\partial\lambda}&=O_{\lambda}^{(d)}\,,\\\
O_{\lambda}^{(d)}&=\frac{1}{2d}T^{ab}T_{ab}-\frac{1}{d^{2}}\left(\tensor{T}{{}^{a}_{a}}\right)^{2}+\frac{d-2}{d^{2}}\tensor{T}{{}^{a}_{a}}R\,,\end{split}$
(2.66)

which simplifies to

$\displaystyle\frac{\partial\mathcal{L}}{\partial\lambda}=-\frac{1}{2}\mathcal{L}^{2}+\mathcal{L}x_{1}\frac{\partial\mathcal{L}}{\partial
x_{1}}\,.$ (2.67)

The solution to this differential equation with initial condition
$\mathcal{L}_{0}=-x_{1}=-\partial^{a}\phi\partial_{a}\phi$ is

$\displaystyle\mathcal{L}(\lambda)=\frac{1}{\lambda}\left(1-\sqrt{1+2\lambda
x_{1}}\right)\,.$ (2.68)

As in the three-dimensional case, this $d$-dimensional Lagrangian satisfies a
modified trace flow equation

$\displaystyle\lambda\frac{\partial\mathcal{L}}{\partial\lambda}=-\frac{1}{d}\left(\tensor{T}{{}^{a}_{a}}-\left(d-2\right)R\right)\,,$
(2.69)

or equivalently

$\displaystyle\lambda
O_{\lambda}^{(d)}\left(T_{ab}(\lambda)\right)=-\frac{1}{d}\left(\tensor{T}{{}^{a}_{a}}-(d-2)R\right)\,.$
(2.70)

By an argument identical to that in equations (2.41) and (2.42), the operator
$R$ is an on-shell total derivative in the $d$-dimensional setting, so this
modified trace flow equation is consistent with the fact that the stress
tensor for the undeformed theory
$\mathcal{L}_{0}=-\partial^{a}\phi\partial_{a}\phi$ can be improved in such a
way to make it traceless.

Finally, one can repeat the analysis of the subtracted version of the scalar
Lagrangian in $d$ spacetime dimensions, letting

$\displaystyle\widetilde{\mathcal{L}}=\frac{\alpha}{\lambda}\sqrt{1+2\lambda
x_{1}}\,,$ (2.71)

which satisfies

$\displaystyle
O_{\lambda}^{(d)}\left(\widetilde{T}_{ab}(\lambda)\right)=\frac{1}{2d}T^{ab}T_{ab}-\frac{1}{d^{2}}\left(\tensor{T}{{}^{a}_{a}}\right)^{2}+\frac{d-2}{d^{2}}\tensor{T}{{}^{a}_{a}}R=-\frac{\alpha^{2}}{2\lambda}\,,$
(2.72)

As a consequence, when written in terms of the variable
$\kappa^{2}=\frac{1}{\lambda}$, the subtracted Lagrangian (2.71) also
satisfies a relevant stress tensor flow:

$\displaystyle\frac{\partial\widetilde{\mathcal{L}}}{\partial\kappa^{2}}=\frac{|\alpha|\left(\tensor{\widetilde{T}}{{}^{a}_{a}}-(d-2)\widetilde{R}\right)}{\sqrt{\left|dT^{ab}T_{ab}-2\left(\tensor{T}{{}^{a}_{a}}\right)^{2}+2(d-2)\tensor{T}{{}^{a}_{a}}R\right|}}\,.$
(2.73)

##  3 Supersymmetric flows

In this section, we consider the ${\cal N}=1$ supersymmetric extension of the
$3d$ Born-Infeld action and investigate its interpretation as a flow that
deforms the free super-Maxwell theory. We begin with the $3d$ ${\cal N}=1$
Maxwell-Goldstone (MG) multiplet [30]. Recall that in three dimensions, the
${\cal N}=1$ vector multiplet is described by the following superspace
action777In this section, we follow the conventions of [30] and Superspace
[77] – see Appendix A for details.

$S_{\rm VM}~{}\sim~{}\int
d^{3}x\,d^{2}\theta~{}W^{2}~{}~{},~{}~{}W^{2}~{}:=~{}\frac{1}{2}\,W^{\alpha}\,W_{\alpha}~{}~{},$
(3.1)

where the superfield strength $W_{\alpha}$ is constrained by the Bianchi
identity

${\rm D}^{\alpha}\,W_{\alpha}~{}=~{}0~{}~{}\Rightarrow~{}~{}{\rm
D}^{2}\,W_{\alpha}~{}=~{}i\partial_{\alpha}{}^{\beta}\,W_{\beta}~{}~{},~{}~{}{\rm
D}_{\alpha}\,W_{\beta}~{}=~{}{\rm D}_{\beta}\,W_{\alpha}~{}~{}.$ (3.2)

This Bianchi identity can be solved by expressing $W_{\alpha}$ in terms of an
unconstrained prepotential superfield $\Gamma_{\beta}$

$W_{\alpha}~{}=~{}\frac{1}{2}\,{\rm D}^{\beta}\,{\rm
D}_{\alpha}\,\Gamma_{\beta}~{}~{}.$ (3.3)

$\Gamma_{\beta}$ is not uniquely determined and is defined modulo the
following gauge transformation

$\delta\,\Gamma_{\alpha}~{}=~{}{\rm D}_{\alpha}\,K~{}~{},$ (3.4)

where $K$ is an arbitrary scalar superfield. This vector supermultiplet will
play the role of the Goldstone multiplet associated with the spontaneous
supersymmetry breaking ${\cal N}=2\to{\cal N}=1$. The dynamics of MG are
determined by requiring the surviving supersymmetry be manifest and the broken
one realized nonlinearly. The resulting Lagrangian is

${\cal L}_{\kappa^{2}}~{}=~{}W^{2}\,f(T)~{}~{},$ (3.5)

where

$f(T)~{}=~{}1+\frac{T}{1-T+\sqrt{1-2T}}~{}~{},~{}~{}T~{}=~{}\frac{2}{\kappa^{2}}\,{\rm
D}^{2}\,W^{2}~{}~{}.$ (3.6)

The dimensionful parameter $\kappa$ ($[\kappa]=3/2$) corresponds to the VEV of
the superfield $F$, which is one of the partners of $W_{\alpha}$ under the
second supersymmetry transformation $\delta^{*}$ [30]:

$\delta^{*}_{\epsilon}\,W_{\alpha}~{}=~{}\epsilon_{\alpha}\,F~{}+~{}\cdots~{}~{},~{}~{}\big{\langle}F\big{\rangle}~{}\sim~{}\kappa~{}~{}.$
(3.7)

In other words, $\kappa$ parametrizes by how much we break the second
supersymmetry. In particular, at the $\kappa\to\infty$ limit, the theory
reduces to the ${\cal N}=1$ free Maxwell theory (3.1). The dependence of the
Lagrangian (3.5) on $\kappa$ defines a curve in the space of supersymmetric
field theories. This curve can be interpreted as a flow. The operator that
triggers this flow is defined as

${\cal O}_{\kappa^{2}}~{}:=~{}\frac{\partial{\cal
L}_{\kappa^{2}}}{\partial\kappa^{2}}~{}=~{}W^{2}\,\frac{\partial
f(T)}{\partial\kappa^{2}}~{}=~{}W^{2}\,f^{\prime}(T)\,\frac{\partial
T}{\partial\kappa^{2}}~{}=~{}-\,\frac{1}{\kappa^{2}}\,W^{2}\,T\,f^{\prime}(T)~{}~{}.$
(3.8)

In Section 2, it was shown that the $3d$ Born-Infeld theory, which is the
bosonic truncation of the MG theory, can be obtained by a flow driven by the
$({\rm stress~{}tensor})^{2}$-type operator. We would like to investigate
whether this statement holds true for its supersymmetric extension given by
(3.5). Namely, can the flow operator (3.8) be written in terms of the
supersymmetric extension of the stress tensor?

We will show explicitly that the answer to the above question is positive. The
proof is organized in the following way. In section 3.1, we introduce the
supercurrent multiplet and give its explicit component field expansion which
includes the stress tensor and its supersymmetric partner. Section 3.2 is
devoted to the derivation of the supercurrent multiplet superfields for the
Maxwell-Goldstone theory and in section 3.3 we explicitly check that the flow
operator (3.8) is indeed a linear combination of $({\rm supercurrent})^{2}$
terms. We also check that the bosonic truncation of this result matches with
section 2.1. As discussed in [30], the role of the Goldstone multiplet can
also be played by the three dimensional projection of the tensor multiplet
which is dual to the vector multiplet. In section 3.4, we repeat the above
calculations for the tensor-Goldstone (TG) description. We confirm that for
this case, the flow operator also takes the $({\rm supercurrent})^{2}$ form
and is consistent with section 2.2.

###  3.1 Supercurrent Multiplet and Stress Tensor

For any field theory, one can use the standard formula for the Hilbert stress
tensor to find $T_{\underline{a}\underline{b}}$, as we did in section 2.
However, in supersymmetric theories the physical degrees of freedom are
organized into supersymmetric multiplets, and therefore the stress tensor
$T_{\underline{a}\underline{b}}$ becomes a member of such a multiplet which we
call the supercurrent multiplet.888Famous examples of $4d$ supercurrent
multiplets are the Ferrara-Zumino multiplet [78], the $\mathcal{R}$-multiplet
(see section 7 of [77]) and their generalization the S-multiplet [79, 80].
Also see [81] and related constructions in [82]. For $3d$, which is relevant
for our paper, the ${\cal N}=2$ supercurrents were discussed in [83] while the
${\cal N}=1$ case is directly related to the $2d$ ${\cal N}=(1,1)$ case of
[50, 49].

The stress tensor of a field theory, by definition, couples the theory to
gravity at cubic level. Similarly, the supercurrent multiplet of a
supersymmetric theory ${\cal L}_{0}[W]$ defines the coupling of the theory
with linearized supergravity in the following manner:

${\cal L}_{0}[W]~{}\mapsto~{}{\cal
L}[W,\Psi_{\alpha\beta\gamma},\varphi]~{}=~{}{\cal
L}_{0}[W]~{}+~{}\Psi^{\alpha\beta\gamma}\,J_{\alpha\beta\gamma}[W]~{}+~{}\varphi\,J[W]~{}+~{}{\cal
L}^{\text{Linearized}}_{SG}[\Psi_{\alpha\beta\gamma},\varphi].$ (3.9)

The superfield $\Psi_{\alpha\beta\gamma}$ is a completely symmetric spinor and
serves as the linearized prepotential for $3d$, ${\cal N}=1$ conformal
supergravity and $\varphi$ is the linearized compensator. The superfields
$J_{\alpha\beta\gamma}[W]$ and $J[W]$ define the supercurrent multiplet of the
theory ${\cal L}_{0}[W]$ and they are called the supercurrent and supertrace,
respectively.

The supergravity superfields $\Psi^{\alpha\beta\gamma}$, $\varphi$ enjoy the
following gauge transformations

$\begin{split}&\delta\,\Psi^{\alpha\beta\gamma}~{}\sim~{}{\rm
D}^{(\alpha}\,\xi^{\beta\gamma)}~{}~{},\\\
&\delta\,\varphi~{}\sim~{}\partial_{\alpha\beta}\,\xi^{\alpha\beta}~{}~{},\end{split}$
(3.10)

where $\xi^{\alpha\beta}$ is an arbitrary, symmetric, superfield parameter for
the above linearized superdiffeomorphism.999 The superfield
$\Psi_{\alpha\beta\gamma}$ is the linearized super-vielbein, hence
$\nabla_{\alpha}~{}=~{}\Psi_{\alpha\beta\gamma}\,\partial^{\beta\gamma}~{}+~{}\cdots$.
This determines the engineering dimensions $[\Psi^{\alpha\beta\gamma}]=-1/2$
and via (3.9) $[J_{\alpha\beta\gamma}]=5/2$. The conservation equation (3.11)
fixes $[J]=2$.

The gauge invariance of (3.9) requires the following superspace conservation
equation of the supercurrent multiplet — modulo equations of motion —

${\rm
D}^{\alpha}J_{\alpha\beta\gamma}(x,\theta)~{}=~{}2i\,\partial_{\beta\gamma}\,J(x,\theta)~{}~{}.$
(3.11)

This is the supersymmetric extension of the stress tensor conservation
equation.

##### Improvement Terms

It is important to emphasize that the above conservation equation does not
uniquely define the supercurrent and supertrace superfields. In fact there is
an infinite family of them. This is understood by the so-called improvement
terms. These are the deformations that identically satisfy (3.11). It is
straightforward to check that the supercurrent multiplet
$(\tilde{J}_{\alpha\beta\gamma},\tilde{J})$ defined as follows

$\begin{split}\tilde{J}_{\alpha\beta\gamma}~{}=&~{}J_{\alpha\beta\gamma}~{}+~{}{\rm
D}_{(\alpha}\partial_{\beta\gamma)}\Lambda~{}~{},\\\
\tilde{J}~{}=&~{}J~{}-~{}4i\,{\rm D}^{2}\,\Lambda~{}~{},\end{split}$ (3.12)

where $\Lambda$ is an arbitrary superfield also satisfies conservation
equation (3.11). These are the supersymmetric extension of (2.54).

##### Components of Supercurrent Multiplet

The $\theta$-expansions for the superfields $J_{\alpha\beta\gamma}(x,\theta)$
and $J(x,\theta)$ are the following

$\begin{split}J_{\alpha\beta\gamma}(x,\theta)~{}=&~{}{s_{\alpha\beta\gamma}(x)}+\theta^{\rho}\,\Bigg{[}\,{t_{\rho\alpha\beta\gamma}(x)}-\frac{i}{4}\,C_{\rho(\alpha}\,\partial_{\beta\gamma)}\,{j}(x)\,\Bigg{]}\\\
&~{}+~{}\theta^{2}\,\Bigg{[}\,\frac{2i}{3!}\,\partial_{(\alpha\beta}\,{s_{\gamma)}(x)}-\frac{i}{3!}\partial_{(\alpha}{}^{\rho}{s_{\beta\gamma)\rho}(x)}\,\Bigg{]}~{}~{},\\\
{J}(x,\theta)~{}=&~{}{j}(x)~{}+~{}\theta^{\alpha}\,{s_{\alpha}(x)}~{}-~{}\frac{1}{3}\theta^{2}\,{\Theta}(x)~{}~{}.\end{split}$
(3.13)

Here the component fields $s_{\alpha\beta\gamma}(x)$ and
$t_{\alpha\beta\gamma\delta}(x)$ are completely symmetric on their spinorial
indices. Moreover, (3.11) implies that the following fields

$\displaystyle
T_{\alpha\beta\gamma\delta}(x)~{}:=~{}t_{\alpha\beta\gamma\delta}(x)-\frac{1}{3}\,C_{\gamma(\alpha}C_{\beta)\delta}\,{\Theta}(x)~{}~{},$
(3.14a) $\displaystyle
S_{\alpha\beta\gamma}(x)~{}:=~{}s_{\alpha\beta\gamma}(x)-\,C_{\gamma(\alpha}\,s_{\beta)}(x)~{}~{},$
(3.14b)

satisfy the conservation equations

$\partial^{\alpha\beta}\,S_{\alpha\beta\gamma}(x)~{}=~{}0~{}~{},~{}~{}\partial^{\rho\alpha}\,T_{\rho\alpha\beta\gamma}(x)~{}=~{}0~{}~{}.$
(3.15)

The field $T_{\alpha\beta\gamma\delta}(x)$ corresponds to the stress tensor
and its spinor indices have the following symmetry properties:
$T_{\alpha\beta\gamma\delta}(x)=T_{\beta\alpha\gamma\delta}(x)=T_{\gamma\delta\alpha\beta}(x)$.
Using (A.28), the conversion between spacetime indices and spinorial indices
is:

$T_{\underline{a}\underline{b}}~{}:=~{}-\frac{1}{4}(\gamma_{\underline{a}})^{\alpha\beta}(\gamma_{\underline{b}})^{\gamma\delta}T_{\alpha\beta\gamma\delta}~{}~{},~{}~{}T_{\alpha\beta\gamma\delta}~{}=~{}T_{\gamma\delta\alpha\beta}~{}=~{}-\,(\gamma^{\underline{a}})_{\alpha\beta}(\gamma^{\underline{b}})_{\gamma\delta}\,T_{\underline{a}\underline{b}}~{}~{}.$
(3.16)

Note that $t_{\alpha\beta\gamma\delta}(x)$ is the completely symmetric part of
the $T_{\alpha\beta\gamma\delta}(x)$ and it appears only in the supercurrent
superfield $J_{\alpha\beta\gamma}$, while ${\Theta}(x)$ is the trace part of
$T_{\underline{a}\underline{b}}$ and it appears only in the supertrace
superfield $J$.

$\begin{split}t_{\alpha\beta\gamma\delta}(x)~{}=&~{}\frac{1}{4!}\,T_{(\alpha\beta\gamma\delta)}(x)~{}=~{}\frac{1}{4!}\,{\rm
D}_{(\alpha}\,J_{\beta\gamma\delta)}(x,\theta)|~{}~{},~{}~{}\\\
\Theta(x)~{}=&~{}T_{\underline{a}}{}^{\underline{a}}~{}=~{}\frac{1}{2}T_{\alpha\beta}{}^{\alpha\beta}~{}=~{}3\,{\rm
D}^{2}\,J(x,\theta)|~{}~{}.\end{split}$ (3.17)

One can also check that the traceless part
$\widehat{T}_{\underline{a}\underline{b}}$ of the stress tensor only depends
on $t_{\alpha\beta\gamma\delta}(x)$,

$\begin{split}\widehat{T}_{\underline{a}\underline{b}}~{}=&~{}T_{\underline{a}\underline{b}}-\frac{1}{3}\eta_{\underline{a}\underline{b}}\,\Theta~{}=~{}-\frac{1}{4}(\gamma_{\underline{a}})^{\alpha\beta}(\gamma_{\underline{b}})^{\gamma\delta}t_{\alpha\beta\gamma\delta}(x)~{}~{},\end{split}$
(3.18)

and using (A.15) the square of the traceless part becomes

$\begin{split}\widehat{T}^{\underline{a}\underline{b}}\widehat{T}_{\underline{a}\underline{b}}~{}=&~{}{T}^{\underline{a}\underline{b}}{T}_{\underline{a}\underline{b}}-\frac{1}{3}\,\Theta^{2}~{}=~{}\frac{1}{4}\,t_{\alpha\beta\gamma\delta}\,t^{\alpha\beta\gamma\delta}~{}~{}.\end{split}$
(3.19)

The operator $S_{\alpha\beta\gamma}(x)=S_{\alpha\gamma\beta}(x)$ is the
supersymmetric partner of $T_{\alpha\beta\gamma\delta}(x)$ and it corresponds
to Noether’s conserved current associated with supersymmetry. Its completely
symmetric part $s_{\alpha\beta\gamma}(x)$ is embedded in the supercurrent,
while its trace $s_{\alpha}(x)$ is embedded in the supertrace.

$\begin{split}s_{\alpha\beta\gamma}(x)~{}=&~{}\frac{1}{3!}\,S_{(\alpha\beta\gamma)}(x)~{}=~{}J_{\alpha\beta\gamma}(x,\theta)|~{}~{},~{}~{}\\\
s_{\alpha}(x)~{}=&~{}\frac{1}{3}\,S_{\alpha\beta}{}^{\beta}(x)~{}=~{}{\rm
D}_{\alpha}\,J(x,\theta)|~{}~{}.\end{split}$ (3.20)

###  3.2 Supercurrent and Supertrace of $3d$ Maxwell-Goldstone multiplet

The explicit calculation of the supercurrent multiplet
$(J_{\alpha\beta\gamma},J)$ for a particular theory can be carried out in
various ways. The typical method, as suggested by (3.9), is to couple the
theory to supergravity and then take the linearized limit in order to read off
the supercurrent $J_{\alpha\beta\gamma}$ and the supertrace $J$. This approach
will be followed in section 4. A different methodology is to use the
superspace conservation equation (3.11) as a consistency condition which can
be solved in order to determine the supercurrent multiplet. This latter
approach will be used in this section for MG theory (3.5). The calculation has
two parts. First note that for $f(T)=1$ ($\kappa\to\infty$), equation (3.5)
reduces to the free super-Maxwell theory (3.1). Therefore we start with a
warm-up calculation of the supercurrent multiplet corresponding to the free
vector supermultiplet in order to build our intuition about the structure of
the supercurrent and supertrace. In the second part, we generalize these
results to arbitrary $f(T)$ (finite $\kappa\neq 0$).

####  3.2.1 Warm up: supercurrent multiplet of super-Maxwell

For the free super-Maxwell theory, the supercurrent multiplet
$\\{\mathsf{J}_{\alpha\beta\gamma},\,\mathsf{J}\\}$ must be quadratic in
$W_{\alpha}$101010The use of bare prepotential superfield $\Gamma_{\alpha}$
(3.3) is not allowed due to the invariance of the cubic vertex (3.9) under the
gauge transformation (3.4).. Moreover, due to the engineering dimensions
$[\mathsf{J}_{\alpha\beta\gamma}]=5/2$, $[\mathsf{J}]=2$, and
$[W_{\alpha}]=1$, the supercurrent will include exactly one spinorial
derivative and the supertrace will have zero spinorial derivatives. Finally,
the index structure of $\mathsf{J}_{\alpha\beta\gamma}$ and $\mathsf{J}$
suggests the following ansatz111111The corresponding supercurrent of the $4d$
vector multiplet was derived in [78], see also [84]. Extensions to higher spin
gauge theories are found in [85, 86].

$\mathsf{J}_{\alpha\beta\gamma}~{}=~{}W_{(\alpha}{\rm
D}_{\beta}W_{\gamma)}~{}~{},~{}~{}\mathsf{J}~{}=~{}W^{2}~{}~{}.$ (3.21)

We will now show that (3.21) indeed satisfies the conservation equation (3.11)
up to the equations of motion:

${\rm D}^{\beta}{\rm
D}_{\alpha}\,W_{\beta}~{}=~{}0~{}~{}\overset{(\ref{equ:WBI})}{\Rightarrow}~{}~{}\partial_{\alpha\beta}\,W^{\beta}~{}=~{}0~{}=~{}{\rm
D}^{2}\,W_{\alpha}~{}~{}.$ (3.22)

Direct computation yields

$\begin{split}{\rm D}^{\alpha}\,\mathsf{J}_{\alpha\beta\gamma}~{}=&~{}({\rm
D}^{\alpha}\,W_{\alpha})\,({\rm D}_{(\beta}W_{\gamma)})+({\rm
D}^{\alpha}\,W_{\beta})\,({\rm D}_{(\gamma}W_{\alpha)})+({\rm
D}^{\alpha}\,W_{\gamma})\,({\rm D}_{(\alpha}W_{\beta)})\\\
&~{}-W_{\alpha}\,{\rm D}^{\alpha}{\rm D}_{(\beta}W_{\gamma)}-W_{\beta}\,{\rm
D}^{\alpha}{\rm D}_{(\gamma}W_{\alpha)}-W_{\gamma}\,{\rm D}^{\alpha}{\rm
D}_{(\alpha}W_{\beta)}~{}~{}.\\\ \end{split}$ (3.23)

Note that (1.) by using (3.2), the first term vanishes; (2.) terms two and
three cancel; (3.) terms five and six vanish independently on-shell by (3.22);
(4.) the fourth term can be simplified in the following way via (A.21) and
(3.22)

$\begin{split}{\rm
D}^{\alpha}\,\mathsf{J}_{\alpha\beta\gamma}~{}=&~{}W^{\alpha}\,{\rm
D}_{\alpha}{\rm
D}_{(\beta}W_{\gamma)}~{}=~{}i\,W^{\alpha}\,\partial_{\alpha(\beta}W_{\gamma)}~{}=~{}i\,\partial_{\alpha(\beta}W^{\alpha}W_{\gamma)}~{}=~{}2i\,\partial_{\beta\gamma}W^{2}~{}~{}.\end{split}$
(3.24)

This is the conservation equation (3.11).

####  3.2.2 Supercurrent multiplet of Maxwell-Goldstone

For the Maxwell-Goldstone theory

$S_{\rm MG}~{}=~{}\int d^{3}x\,d^{2}\theta\,W^{2}\,f(T)~{}~{},$ (3.25)

$f(T)$ (3.6) is no longer a constant and therefore its supercurrent multiplet
$\\{{\cal J}_{\alpha\beta\gamma},\,{\cal J}\\}$ will generalize the super-
Maxwell theory results (3.21) with additional terms depending on the
derivatives of $f(T)$, such that in the $f(T)\to 1$ limit we restore the
results of section 3.2.1.

$\begin{split}\mathcal{J}_{\alpha\beta\gamma}~{}\sim&~{}f(T)\,\mathsf{J}_{\alpha\beta\gamma}+f^{\prime}(T)\,\mathcal{O}^{(1)}_{\alpha\beta\gamma}+f^{\prime\prime}(T)\,\mathcal{O}^{(2)}_{\alpha\beta\gamma}+\cdots~{}~{},\\\
\mathcal{J}~{}\sim&~{}f(T)\,\mathsf{J}+f^{\prime}(T)\,\mathcal{O}^{(1)}+f^{\prime\prime}(T)\,\mathcal{O}^{(2)}+\cdots~{}~{}.\\\
\end{split}$ (3.26)

This is also reflected in the fact that the equations of motion of MG theory
include factors of $f(T)$ and its derivatives.

##### EOM

Indeed, by varying the action (3.25) with respect to the prepotential
superfield $\Gamma_{\beta}$, it is straightforward to find the following
equation of motion

${\rm D}^{\rho}\,{\rm
D}_{\alpha}\,\Big{[}\,W_{\rho}\,g\,\Big{]}~{}=~{}0~{}~{},$ (3.27)

where the factor $g$ is defined as

$\begin{split}g~{}:=&~{}f(T)+\frac{2}{\kappa^{2}}{\rm
D}^{2}\Big{[}W^{2}f^{\prime}(T)\Big{]}\\\
~{}=&~{}f(T)+Tf^{\prime}(T)+\frac{2}{\kappa^{2}}W^{2}{\rm
D}^{2}f^{\prime}(T)+\frac{2}{\kappa^{2}}({\rm D}^{\delta}W^{2})({\rm
D}_{\delta}f^{\prime}(T))~{}~{}.\end{split}$ (3.28)

Equivalently, using (A.21) the equation of motion can also take the form

${\rm
D}^{2}\,\Big{\\{}W_{\alpha}\,g\Big{\\}}~{}=~{}-i\,\partial_{\alpha}{}^{\beta}\Big{\\{}W_{\beta}\,g\Big{\\}}~{}~{}.$
(3.29)

As expected, (3.27) reduces to (3.22) when we take $f\to 1$.

##### Supercurrents

The results of section 3.2.1 in combination with the MG equations of motion
suggest the following ansatz for the MG supercurrent

${\cal J}^{(1)}_{\alpha\beta\gamma}~{}=~{}W_{(\alpha}\Big{(}{\rm
D}_{\beta}W_{\gamma)}\Big{)}\,g~{}=~{}W_{(\alpha}{\rm
D}_{\beta}\,\Big{(}W_{\gamma)}\,g\Big{)}~{}~{},$ (3.30)

where the second equal sign comes from $W_{(\beta}W_{\gamma)}=0$. After some
algebra using properties of supersymmetric covariant derivatives and the
spinorial superfield $W_{\alpha}$, we find that

${\rm D}^{\alpha}{\cal
J}^{(1)}_{\alpha\beta\gamma}~{}=~{}2i\,\partial_{\beta\gamma}\Big{(}W^{2}\,g\Big{)}~{}+~{}4i\,W^{2}\,(\partial_{\beta\gamma}\,g)~{}-~{}({\rm
D}_{(\beta}W^{2})({\rm D}_{\gamma)}g)~{}~{}.$ (3.31)

It is evident that this equation is not consistent with the expected
conservation equation. In particular, the last two terms measure the failure
of ${\cal J}^{(1)}_{\alpha\beta\gamma}$ to satisfy (3.11). In order to
eliminate these two terms, we consider additional contributions to the
supercurrent as illustrated in (3.26). In particular, we examine the following
term

${\cal J}^{(2)}_{\alpha\beta\gamma}~{}=~{}W^{2}\,\partial_{(\alpha\beta}{\rm
D}_{\gamma)}\Big{(}W^{2}\,h\Big{)}~{}=~{}W^{2}\,\Big{(}\partial_{(\alpha\beta}{\rm
D}_{\gamma)}W^{2}\Big{)}\,h~{}~{}.$ (3.32)

The structure of this term is motivated by the fact that the last two terms of
(3.31) depend only on $W^{2}$ and its derivatives. One factor of $W^{2}$ is
written explicitly and the second factor resides inside derivatives of $g$.
Moreover, notice that ${\cal J}^{(2)}_{\alpha\beta\gamma}$ has two equivalent
representations due to the nilpotency condition of $W_{\alpha}$ (i.e.
$W_{\alpha}W_{\beta}W_{\gamma}\equiv 0$). The factor $h$ will be determined by
demanding the cancellation of the last two terms in (3.31).

Direct calculation using (A.25) and (A.26) gives

${\rm D}^{\alpha}{\cal J}^{(2)}_{\alpha\beta\gamma}~{}=~{}2i\,({\rm
D}_{(\beta}W^{2}){\rm D}_{\gamma)}{\rm
D}^{2}\Big{(}W^{2}h\Big{)}+8\,W^{2}\,\partial_{\beta\gamma}\,{\rm
D}^{2}\Big{(}W^{2}h\Big{)}+6\,({\rm
D}^{\alpha}W^{2})\,\partial_{\beta\gamma}\,{\rm
D}_{\alpha}\Big{(}W^{2}h\Big{)}~{}~{}.$

(3.33)

The last two terms are not independent; in fact they are related by the
following identity

$W^{2}\,\partial_{\beta\gamma}{\rm D}^{2}\Big{(}W^{2}h\Big{)}+({\rm
D}^{\alpha}W^{2})\partial_{\beta\gamma}{\rm
D}_{\alpha}\Big{(}W^{2}h\Big{)}~{}=~{}-\,({\rm
D}^{2}W^{2})\partial_{\beta\gamma}\Big{(}W^{2}h\Big{)}$ (3.34)

which follows from the nilpotency of $W_{\alpha}$ (see proof in A.2). Observe
that if we choose

$h~{}=~{}f^{\prime}(T)~{}~{}\Rightarrow~{}~{}\frac{2}{\kappa^{2}}\,{\rm
D}^{2}(W^{2}h)~{}=~{}g-f(T)~{}~{},$ (3.35)

then the first two terms in (3.33) have a similar structure as the anomalous
terms in (3.31). With these substitutions, we find

$\begin{split}\frac{2}{\kappa^{2}}\,{\rm D}^{\alpha}{\cal
J}^{(2)}_{\alpha\beta\gamma}~{}=&~{}-6\,\partial_{\beta\gamma}\,\Big{[}T\,W^{2}\,f^{\prime}(T)\Big{]}+4\,W^{2}\,\partial_{\beta\gamma}f(T)+2\,W^{2}\,\partial_{\beta\gamma}\,g\\\
&~{}+~{}2i\,({\rm D}_{(\beta}W^{2})\Big{(}{\rm
D}_{\gamma)}g\Big{)}~{}-~{}2i\,({\rm D}_{(\beta}W^{2})\Big{(}{\rm
D}_{\gamma)}f(T)\Big{)}~{}~{}.\end{split}$ (3.36)

Surprisingly, most of these terms are associated with each other via the
following identity

$({\rm D}_{(\beta}W^{2})({\rm
D}_{\gamma)}g)~{}=~{}-2i\,W^{2}\,\partial_{\beta\gamma}g+4i\,W^{2}\,\partial_{\beta\gamma}f(T)-2\,\Big{(}{\rm
D}_{(\beta}f(T)\Big{)}\Big{(}{\rm D}_{\gamma)}W^{2}\Big{)}~{}~{}.$ (3.37)

We leave this derivation in Appendix A.2. Using (3.37), (3.36) can be
rewritten as follows:

$\frac{2}{\kappa^{2}}\,{\rm D}^{\alpha}{\cal
J}^{(2)}_{\alpha\beta\gamma}~{}=~{}-6\,\partial_{\beta\gamma}\,\Big{[}T\,W^{2}\,f^{\prime}(T)\Big{]}+4\,W^{2}\,\partial_{\beta\gamma}\,g+i\,({\rm
D}_{(\beta}W^{2})\Big{(}{\rm D}_{\gamma)}g\Big{)}~{}~{}.$ (3.38)

Finally, we combine (3.31) and (3.38) and we find the following conservation
equation

$\begin{split}{\rm D}^{\alpha}\,\Big{[}\,{\cal
J}^{(1)}_{\alpha\beta\gamma}~{}-~{}i\,\frac{2}{\kappa^{2}}\,{\cal
J}^{(2)}_{\alpha\beta\gamma}\,\Big{]}~{}=&~{}2i\,\partial_{\beta\gamma}\Big{[}W^{2}\,f(T)~{}+~{}4\,W^{2}\,T\,f^{\prime}(T)\Big{]}~{}~{}.\\\
\end{split}$ (3.39)

From this we can immediately identify the supercurrent multiplet of the
Maxwell-Goldstone multiplet,

$\displaystyle{\cal J}_{\alpha\beta\gamma}~{}=$
$\displaystyle~{}W_{(\alpha}{\rm
D}_{\beta}\,\Big{(}W_{\gamma)}\,g\Big{)}~{}-~{}i\,\frac{2}{\kappa^{2}}\,W^{2}\,\partial_{(\alpha\beta}{\rm
D}_{\gamma)}\Big{(}W^{2}\,f^{\prime}(T)\Big{)}~{}~{},$ (3.40a)
$\displaystyle{\cal J}~{}=$
$\displaystyle~{}W^{2}\,f(T)~{}+~{}4\,W^{2}\,T\,f^{\prime}(T)~{}~{}.$ (3.40b)

###  3.3 Flow Operator as Supercurrent-squared Operator

The supercurrent multiplet (3.40a, 3.40b) calculated above allows us to answer
the question we asked at the beginning of section 3. In particular, if we view
the MG theory as the result of a flow triggered by the operator ${\cal
O}_{\kappa^{2}}$ of (3.8), then can we give a $T^{2}$-like interpretation to
this flow? In other words, can ${\cal O}_{\kappa^{2}}$ be expressed as a
linear combination of “supercurrent-squared ” terms?

##### Supercurrent-squared Scalars

To answer this question, we search for superspace operators defined in terms
of the supercurrent and supertrace and their derivatives such that their
bosonic truncations will produce $T^{2}$ terms. In general, there are three
types of $T^{2}$ terms: (1.) the square of the traceless part of the stress
tensor
$\widehat{T}^{\underline{a}\underline{b}}\widehat{T}_{\underline{a}\underline{b}}$;
(2.) the square of the trace $\Theta^{2}$; (3.) the mixed term formed from the
product of the trace with the “strength” of the traceless part
$\Theta\sqrt{\widehat{T}^{\underline{a}\underline{b}}\widehat{T}_{\underline{a}\underline{b}}}$.
Recall that the traceless part is a component of the supercurrent, whereas the
trace resides in supertrace (3.17). Therefore, a natural proposal for the
superspace operators in question is the following:

$\begin{split}\mathcal{O}_{T^{2}}~{}=&~{}J^{\alpha\beta\gamma}J_{\alpha\beta\gamma}~{}~{},\\\
\mathcal{O}_{\Theta^{2}}~{}=&~{}J\,{\rm D}^{2}\,J~{}~{},\\\
\mathcal{O}_{\Theta R}~{}=&~{}J\sqrt{{\rm
D}^{(\alpha}J^{\beta\gamma\delta)}{\rm
D}_{(\alpha}J_{\beta\gamma\delta)}}~{}~{}.\end{split}$ (3.41)

These operators are certainly not unique, however they are computationally
convenient. They partition the three types of contributions mentioned above
based on the fact that the traceless and trace parts of the energy-momentum
tensor reside in different superfields. By integrating these operators over
the fermionic directions and using (3.13), we find their corresponding
spacetime components to be

$\begin{split}\int\,d^{2}\theta\,\mathcal{O}_{T^{2}}~{}=&~{}-\,t^{\rho\alpha\beta\gamma}(x)t_{\rho\alpha\beta\gamma}(x)~{}+~{}{\rm
fermions}~{}~{},\\\
\int\,d^{2}\theta\,{\mathcal{O}_{\Theta^{2}}}~{}=&~{}\frac{1}{9}\,\Theta(x)^{2}~{}+~{}{\rm
fermions}~{}~{},\\\ \int\,d^{2}\theta\,{\mathcal{O}_{\Theta
R}}~{}=&~{}\frac{4!}{3}\,\Theta(x)\,\sqrt{t^{\alpha\beta\gamma\delta}(x)t_{\alpha\beta\gamma\delta}(x)}~{}+~{}{\rm
fermions}~{}~{}.\end{split}$ (3.42)

##### Flow Operator

Now we evaluate these operators for MG theory. In (3.41) we plug in (3.40a)
and (3.40b) to find ll O_T^2 =& -4!2 κ^2 W^2 T [ f(T) + Tf’(T) ]^2 , O_Θ^2 =
κ22 W^2 T [ f(T) + 4 T f’(T)]^2 , O_ΘR = 4!26 κ^2 T W^2 [ f(T) + 4 T f’(T)] [
f(T) + Tf’(T) ] , where we have used the nilpotency condition $W^{3}\equiv 0$
repeatedly. It is straightforward to check that $f(T)$ defined as (3.6)
satisfies the following constraint

$f^{\prime}(T)~{}=~{}\frac{1}{2}\,f(T)^{2}+T\,f(T)\,f^{\prime}(T)~{}~{}.$
(3.43)

With that in mind, we can show that the flow operator can be written as

$\begin{split}{\cal
O}_{\kappa^{2}}~{}=&~{}-\,\frac{1}{\kappa^{2}}\,W^{2}\,T\,f^{\prime}(T)\\\
~{}=&~{}\frac{1}{\kappa^{4}}\Big{[}\frac{1}{108}~{}\mathcal{O}_{T^{2}}~{}+~{}\frac{1}{9}~{}{\mathcal{O}_{\Theta^{2}}}~{}-~{}\frac{1}{432\sqrt{6}}~{}{\mathcal{O}_{\Theta
R}}\Big{]}~{}~{}.\\\ \end{split}$ (3.44)

This concludes the proof that the supersymmetric Maxwell-Goldstone theory
satisfies a supercurrent-square like flow given by

$\kappa^{4}\frac{\partial{\cal
L}_{\kappa^{2}}}{\partial\kappa^{2}}~{}=~{}\frac{1}{108}~{}\mathcal{O}_{T^{2}}~{}+~{}\frac{1}{9}~{}{\mathcal{O}_{\Theta^{2}}}~{}-~{}\frac{1}{432\sqrt{6}}~{}{\mathcal{O}_{\Theta
R}}~{}.$ (3.45)

It is important to emphasize that this superspace flow equation holds off-
shell. In other words, equations (3.3) and (3.44) do not require the use of
equations of motion.

##### Bosonic Truncation

${\cal O}_{\kappa^{2}}$ is a superspace operator that drives the flow in the
space of manifestly ${\cal N}=1$ supersymmetric theories. We can use it to
define the operator that controls the flow of the bosonic truncation of the
theory121212The prefactor $\kappa^{4}$ provides the correct engineering
dimensions for a $T^{2}$ deformation, while the numerical factor is a
normalization choice.

$O^{MG}_{\kappa^{2}}~{}:=~{}\kappa^{4}\,\frac{9}{2}\left[\,\int\,d^{2}\theta\,{\cal
O}_{\kappa^{2}}\,\right]_{\text{fermions}\to 0}~{}~{}.$ (3.46)

Using (3.42) and converting the pairs of spinorial indices to spacetime
indices (3.19) we find

$\begin{split}O^{MG}_{\kappa^{2}}~{}=~{}-\frac{1}{6}{T}^{\underline{a}\underline{b}}{T}_{\underline{a}\underline{b}}+\frac{1}{9}\,\Theta^{2}-\frac{1}{9}\,\Theta\,\sqrt{\frac{3}{8}\widehat{T}^{\underline{a}\underline{b}}\widehat{T}_{\underline{a}\underline{b}}}~{}~{},\end{split}$
(3.47)

which matches the operator $O_{\kappa^{2}}$ defined in (2.14).

###  3.4 Flow Interpretations for Tensor-Goldstone

The $3d$, ${\cal N}=1$ vector multiplet has a variant description in terms of
a dual superfield $U_{\alpha}$. This is demonstrated in [30] where this dual
supermultiplet was used to provide an equivalent Goldstone multiplet
corresponding to a nonlinear realization of the second broken supersymmetry.
Therefore, the question of interpreting the nonlinear theory as the flow of a
$T^{2}$-like operator could also be asked for this dual tensor-Goldstone
description. In this section, we repeat the previous analysis in order to:
(1.) find the appropriate supercurrent multiplet
$\\{\mathscr{J}_{\alpha\beta\gamma},\,\mathscr{J}\\}$ corresponding to the TG;
(2.) use it to define the deformations of the theory that include $T^{2}$-like
terms; (3.) show that the TG theory defines a flow operator
$\mathscr{O}_{\tilde{\kappa}^{2}}$ which can be expressed as a linear
combination of the above deformations; (4.) extract the bosonic truncation of
this flow operator.

##### Preliminaries

We start with a brief reminder of the $3d$ variant description. The superspace
action takes the following form.

$S_{\rm TM}~{}\sim~{}-\,\int\,d^{3}x\,d^{2}\theta\,U^{2}~{}~{},$ (3.48)

where the superfield $U_{\alpha}$ is constrained by the following Bianchi
Identity:

${\rm D}^{\alpha}{\rm D}^{\beta}U_{\alpha}~{}=~{}0~{}~{}\Rightarrow~{}~{}{\rm
D}^{2}\,U_{\alpha}~{}=~{}-i\,\partial_{\alpha}{}^{\beta}\,U_{\beta}~{}~{}.$
(3.49)

This can be solved by expressing $U_{\alpha}$ in terms of an unconstrained
scalar superfield $G$.

$U_{\alpha}~{}=~{}{\rm D}_{\alpha}\,G~{}~{}.~{}~{}$ (3.50)

In [30] we promoted this tensor multiplet to a tensor-Goldstone multiplet,
consistent with the spontaneous breaking of ${\cal N}=2\to{\cal N}=1$,
described by the following action:

$S_{\rm
TG}~{}=~{}-\,\int\,d^{3}x\,d^{2}\theta\,U^{2}\,h(\tilde{T})~{}~{},~{}~{}$
(3.51)

where

$h(x)~{}=~{}1-\frac{x}{1+x+\sqrt{1+2x}}~{}=~{}f(-x)~{}~{},~{}~{}\tilde{T}~{}:=~{}\frac{2}{\tilde{\kappa}^{2}}\,{\rm
D}^{2}U^{2}~{}~{}.$ (3.52)

The dimensionful parameter $\tilde{\kappa}$ has the same engineering dimension
as $\kappa$ ($[\tilde{\kappa}]=3/2$) and describes the scale at which we break
the second supersymmetry. Similar to the MG, the $\tilde{\kappa}$ dependence
of the TG Lagrangian defines a curve which can be intepreted as a flow. This
flow is triggered by the following operator

${\mathscr{O}}_{\tilde{\kappa}^{2}}~{}=~{}\frac{\partial{\cal
L}_{TG}}{\partial\tilde{\kappa}^{2}}~{}=~{}\frac{1}{\tilde{\kappa}^{2}}\,U^{2}\,\tilde{T}\,h^{\prime}(\tilde{T})~{}~{}.$
(3.53)

Varying the action (3.51) with respect to the superfield $G$ yields the
following equation of motion and its variants:

$\begin{split}{\rm
D}^{\rho}\Big{\\{}U_{\rho}\,\tilde{g}\Big{\\}}~{}=~{}0~{}~{}\Leftrightarrow&~{}~{}{\rm
D}_{\alpha}\Big{\\{}U_{\beta}\,\tilde{g}\Big{\\}}~{}=~{}{\rm
D}_{\beta}\Big{\\{}U_{\alpha}\,\tilde{g}\Big{\\}}~{}~{},~{}~{}\\\
~{}~{}\Rightarrow&~{}~{}{\rm
D}^{2}\Big{[}U^{\alpha}\,\tilde{g}\Big{]}~{}=~{}i\,\partial^{\alpha\beta}\Big{[}U_{\beta}\,\tilde{g}\Big{]}~{}~{},\end{split}$
(3.54)

where the factor $\tilde{g}$ is defined as

$\tilde{g}~{}:=~{}h(\tilde{T})~{}+~{}\frac{2}{\tilde{\kappa}^{2}}\,{\rm
D}^{2}\,\Big{[}U^{2}\,h^{\prime}(\tilde{T})\Big{]}~{}~{}.$ (3.55)

Moreover, using properties of superspace covariant derivatives and
$U_{\alpha}$ nilpotency condition ($U^{3}\equiv 0$), we find the following
identity:

$({\rm D}_{(\beta}U^{2})({\rm
D}_{\gamma)}\tilde{g})~{}=~{}-2i\,U^{2}\,\partial_{\beta\gamma}\tilde{g}+4i\,U^{2}\,\partial_{\beta\gamma}h(\tilde{T})-2\,\Big{(}{\rm
D}_{(\beta}h(\tilde{T})\Big{)}\Big{(}{\rm D}_{\gamma)}U^{2}\Big{)}~{}~{}.$
(3.56)

This is an analog of (3.37) for the TG.

##### Comparison between MG and TG

Notice that the Lagrangians of the MG and TG have similar structures
($W^{\alpha}\mapsto U^{\alpha}$, $T\mapsto\tilde{T}$, $f(T)\mapsto
h(\tilde{T})$). However, we would like to emphasize the following differences
(see table 1).

MG |  TG
---|---
Bianchi: ${\rm D}_{\alpha}W_{\beta}={\rm D}_{\beta}W_{\alpha}$ | EoM: ${\rm D}_{\alpha}\Big{(}U_{\beta}\tilde{g}\Big{)}={\rm D}_{\beta}\Big{(}U_{\alpha}\tilde{g}\Big{)}$
EoM: ${\rm D}^{\alpha}{\rm D}^{\beta}\Big{(}W_{\alpha}\,g\Big{)}=0$ | Bianchi: ${\rm D}^{\alpha}{\rm D}^{\beta}U_{\alpha}=0$
Table 1: Comparisons of MG and TG Bianchi identities and EoMs

In the free theory limit $\kappa,\tilde{\kappa}\to\infty$ ($g,\tilde{g}\to
1$), the Bianchi identity of one becomes the EoM of the other. This is another
way to demonstrate the duality between these two theories. But for the
interacting theories ($g,\tilde{g}\neq 1$), there is a mismatch. The
conservation of the MG supercurrent multiplet relies on the Bianchi identity
of $W_{\alpha}$ and EoM (left column). However, these identities are modified
in the TG case (right column). In order to compute the supercurrent-squared
operators $\\{{\cal O}_{T^{2}},{\cal O}_{\Theta^{2}},{\cal O}_{\Theta R}\\}$
for the TG multiplet using similar arguments as those in section 3.3, we use
the following property

$U^{2}\,\tilde{g}\,\Big{(}{\rm D}_{\alpha}U_{\beta}\Big{)}~{}=~{}U^{2}\,{\rm
D}_{\alpha}\Big{(}U_{\beta}\tilde{g}\Big{)}~{}=~{}U^{2}\,{\rm
D}_{\beta}\Big{(}U_{\alpha}\tilde{g}\Big{)}~{}=~{}U^{2}\,\tilde{g}\,\Big{(}{\rm
D}_{\beta}U_{\alpha}\Big{)}~{}~{}.$ (3.60)

Namely, in the presence of an additional $U^{2}$ factor, $\tilde{g}$ can move
in and out of the spinorial derivative in the EoM (3.54), due to the
nilpotency property of $U_{\alpha}$.

##### Supercurrents

Motivated by (3.30) and (3.32), we consider the following ansatz131313One can
also motivate this ansatz by starting with (3.9) and performing the duality
procedure described in [30].

$\displaystyle\mathscr{J}^{(1)}_{\alpha\beta\gamma}~{}=$
$\displaystyle~{}U_{(\alpha}\,\tilde{g}\,\Big{(}{\rm
D}_{\beta}U_{\gamma)}\Big{)}~{}\overset{U_{(\beta}U_{\gamma)}=0}{=}~{}U_{(\alpha}{\rm
D}_{\beta}\,\Big{(}U_{\gamma)}\,\tilde{g}\Big{)}~{}~{},$ (3.61)
$\displaystyle\mathscr{J}^{(2)}_{\alpha\beta\gamma}~{}=$
$\displaystyle~{}U^{2}\,\partial_{(\alpha\beta}{\rm
D}_{\gamma)}\Big{(}U^{2}\,h^{\prime}(\tilde{T})\Big{)}~{}\overset{U^{3}\equiv
0}{=}~{}U^{2}\,\Big{(}\partial_{(\alpha\beta}{\rm
D}_{\gamma)}U^{2}\Big{)}\,h^{\prime}(\tilde{T})~{}~{}.$ (3.62)

Using the various properties of $U_{\alpha}$ (3.49, 3.54, 3.56, A.31, A.32)
and the supersymmetric covariant derivative algebra (A.25, A.26), we find the
following superspace conservation equation

${\rm
D}^{\alpha}\Bigg{[}\mathscr{J}^{(1)}_{\alpha\beta\gamma}+i\,\frac{2}{\tilde{\kappa}^{2}}\,\mathscr{J}^{(2)}_{\alpha\beta\gamma}\Bigg{]}~{}=~{}2i\,\partial_{\beta\gamma}\Bigg{(}U^{2}\,h(\tilde{T})-2\,\tilde{T}U^{2}h^{\prime}(\tilde{T})\Bigg{)}~{}~{}.$
(3.63)

From this, we identify the supercurrent multiplet for tensor-Goldstone theory
as follows:

$\displaystyle\mathscr{J}_{\alpha\beta\gamma}~{}=$
$\displaystyle~{}U_{(\alpha}\,\tilde{g}\,\Big{(}{\rm
D}_{\beta}U_{\gamma)}\Big{)}+i\,\frac{2}{\tilde{\kappa}^{2}}\,U^{2}\,\partial_{(\alpha\beta}{\rm
D}_{\gamma)}\Big{(}U^{2}\,h^{\prime}(\tilde{T})\Big{)}~{}~{},$ (3.64a)
$\displaystyle\mathscr{J}~{}=$
$\displaystyle~{}U^{2}\,\Big{[}h(\tilde{T})-2\,\tilde{T}\,h^{\prime}(\tilde{T})\Big{]}~{}~{}.$
(3.64b)

##### Supercurrent-squared Operator

Using the above supercurrent and supertrace, we evaluate the supercurrent-
squared operators (3.41) ll O_T^2 ≈& -4!2 ~κ^2 U^2 ~T [ h(~T) + ~T h’(~T) ]^2
, O_Θ^2 = ~κ22 ~T U^2 [ h(~T) -2 ~T h’(~T) ]^2 , O_ΘR ≈ 4!2 6 ~κ^2 U^2 ~T [
h(~T) -2 ~T h’(~T) ] [ h(~T) + ~T h’(~T) ] . The ‘$\approx$’ symbol signifies
that the equality holds only on-shell. This is because equation (3.60) was
used in the derivation of (3.4). Again, this is a consequence of the duality
between the MG and TG descriptions. What was a Bianchi identity for
$W_{\alpha}$ now becomes an equation of motion for $U_{\alpha}$. Off-shell,
the expressions for $\mathcal{O}_{T^{2}}$ and ${\mathcal{O}_{\Theta R}}$
acquire corrections proportional to ${\cal H}~{}:=~{}{\rm
D}^{\gamma}\,U_{\gamma}$ (for details see section A.3). It is straightforward
to see that the equation of motion of $U_{\alpha}$ [ ${\rm
D}^{\alpha}(U_{\alpha}g)=0$ ] is equivalent to $U^{2}\tilde{g}\,{\cal
H}=0\Rightarrow{\cal H}=0$. As a consequence all these corrections vanish on-
shell and also the auxiliary component $H:={\cal H}|$ vanishes. This is in
agreement with the use of the on-shell equation $H=0$ in section 2.2. Similar
conditions were used in the analysis of supersymmetric flows in other
dimensions; see, for example, [29].

Note that $h(\tilde{T})$ defined in (3.52) satisfies the following relation

$-\,h^{\prime}(\tilde{T})~{}=~{}\frac{1}{2}\,h^{2}(\tilde{T})+\tilde{T}\,h(\tilde{T})\,h^{\prime}(\tilde{T})~{}~{}.$
(3.65)

Therefore, the superspace flow operator ${\mathscr{O}}_{\tilde{\kappa}^{2}}$
defined in (3.53) can be written on-shell as

${\mathscr{O}}_{\tilde{\kappa}^{2}}~{}\approx~{}\frac{1}{\tilde{\kappa}^{4}}\Big{[}~{}\frac{1}{108}\,\mathcal{O}_{T^{2}}~{}+~{}\frac{1}{9}\,{\mathcal{O}_{\Theta^{2}}}~{}-~{}\frac{1}{432\sqrt{6}}\,{\mathcal{O}_{\Theta
R}}\Big{]}~{}~{}.$ (3.66)

This expression confirms that the TG action (3.51) which describes the partial
${\cal N}=2\to{\cal N}=1$ supersymmetry breaking can be understood as the
result of a supercurrent-squared flow. We also stress that the operator above
coincides precisely with the second line of (3.44) up to exchanging
$\tilde{\kappa}^{2}$ with $\kappa^{2}$. However, unlike the MG case, this flow
is an on-shell one.

##### Bosonic Truncation

As mentioned previously, the tensor-Goldstone multiplet is a variant
description, and hence its bosonic truncation will be similar in form to the
truncation of the Maxwell-Goldstone multiplet. In other words, the bosonic
sectors of both Lagrangians take the form of $3d$ Born-Infeld. As a result, we
expect the operators that trigger the corresponding flows to take identical
forms. Indeed, if we define the spacetime flow operator corresponding to the
TG analogously to (3.46), we find

$\begin{split}O^{TG}_{\kappa^{2}}~{}:=&~{}\tilde{\kappa}^{4}\,\frac{9}{2}\,\left[\,\int
d^{2}\theta\,{\mathscr{O}}_{\tilde{\kappa}^{2}}\,\right]_{{\rm fermions}\to
0}\\\
~{}=&~{}-\frac{1}{6}{T}^{\underline{a}\underline{b}}{T}_{\underline{a}\underline{b}}~{}+~{}\frac{1}{9}\,\Theta^{2}~{}-~{}\frac{1}{9}\,\Theta\,\sqrt{\frac{3}{8}\widehat{T}^{\underline{a}\underline{b}}\widehat{T}_{\underline{a}\underline{b}}}~{}~{}.\\\
\end{split}$ (3.67)

As a function of the energy-momentum tensor, $O^{TG}_{\kappa^{2}}$ is
identical to $O^{MG}_{\kappa^{2}}$. However, we remind the reader that the
evaluation of $T_{\underline{a}\underline{b}}$ differs for the two theories.

##  4 Derivation of supercurrents from ${\cal N}=1$ supergravity

In this section, we repeat the derivation of the supercurrents for the models
analysed in the previous section by using the superspace supergravity results
of [87] and [88]. We stress that the notations in this section are the ones of
[87] and differ from the ones employed in the previous section. For the
reader’s convenience, the conventions used in this section together with a map
between the notation in this and the previous section is given in Appendix B.
The analysis in this section is a manifestly supersymmetric extension of the
definition of the Hilbert stress-energy tensor for a matter system. The reader
can find standard background material on the subject in [77, 89]. Before
describing the calculation of the supercurrents for vector and scalar models,
we introduce in the next subsection the necessary building blocks.

###  4.1 Building blocks from superspace supergravity

We are interested in coupling matter systems to $3d$ ${\cal N}=1$ Poincaré
supergravity. The latter can be described as $3d$ ${\cal N}=1$ conformal
supergravity coupled to a conformal compensator scalar multiplet. In
superspace, the conformal supergravity multiplet can efficiently be described
by a curved superspace geometry with the torsion and curvature satisfying an
appropriate set of constraints. In this paper, we will employ the so-called
$SO(1)$ geometry for conformal supergravity – see [77, 87] and also [90, 91,
88] for related approaches. The superspace geometry is described in terms of
covariant derivatives taking the following form

$\displaystyle{\cal D}_{A}$ $\displaystyle=$ $\displaystyle({\cal D}_{a},{\cal
D}_{\alpha})=E_{A}-\Omega_{A}~{},$ (4.1)

where $E_{A}=E_{A}{}^{M}\partial/\partial z^{M}$ defines the inverse vielbein,
and141414The Lorentz generators with two vector indices ($M_{ab}=-M_{ba}$),
with one vector index ($M_{a}$) and with two spinor indices
($M_{\alpha\beta}=M_{\beta\alpha}$) are related to each other by the rules:
$M_{a}=\frac{1}{2}\varepsilon_{abc}M^{bc}$ and
$M_{\alpha\beta}=(\gamma^{a})_{\alpha\beta}M_{a}$. We will also use ${\cal
D}_{\alpha\beta}=(\gamma^{a})_{\alpha\beta}{\cal D}_{a}$ for the vector
covariant derivative.

$\displaystyle\Omega_{A}=\frac{1}{2}\Omega_{A}{}^{bc}M_{bc}=-\Omega_{A}{}^{b}M_{b}=\frac{1}{2}\Omega_{A}{}^{\beta\gamma}M_{\beta\gamma}$
(4.2)

is the Lorentz connection. The Lorentz generators act on the covariant
derivatives as follows:

$\displaystyle{[}M_{\alpha\beta},{\cal
D}_{\gamma}{]}=\varepsilon_{\gamma(\alpha}{\cal
D}_{\beta)}~{},\qquad{[}M_{ab},{\cal D}_{c}{]}=2\eta_{c[a}{\cal D}_{b]}~{}.$
(4.3)

The torsions and curvatures associated with the geometry arising from
(anti-)commutation relations of the covariant derivatives satisfy an
appropriate set of conventional constraints as described in [87]. The
resulting algebra is:

$\displaystyle\\{{\cal D}_{\alpha},{\cal D}_{\beta}\\}$ $\displaystyle=$
$\displaystyle 2i{\cal D}_{\alpha\beta}-4i{\cal S}M_{\alpha\beta}\ ,$ (4.4a)
$\displaystyle\left[{\cal D}_{\alpha\beta},{\cal D}_{\gamma}\right]$
$\displaystyle=$ $\displaystyle-2\varepsilon_{\gamma(\alpha}{\cal S}{\cal
D}_{\beta)}+2\varepsilon_{\gamma(\alpha}{\cal
C}_{\beta)\delta\rho}M^{\delta\rho}+\frac{2}{3}({\cal D}_{\gamma}{\cal
S})M_{\alpha\beta}-\frac{8}{3}({\cal D}_{(\alpha}{\cal S})M_{\beta)\gamma}\ ,$
(4.4b) $\displaystyle\left[{\cal D}_{a},{\cal D}_{b}\right]$ $\displaystyle=$
$\displaystyle-\frac{i}{2}\varepsilon_{abc}(\gamma^{c})^{\alpha\beta}\Big{\\{}{\cal
C}_{\alpha\beta\gamma}{\cal D}^{\gamma}+\frac{4}{3}({\cal D}_{\alpha}{\cal
S}){\cal D}_{\beta}-({\cal D}_{(\alpha}{\cal
C}_{\beta\gamma\delta)})M^{\gamma\delta}$ (4.4c)
$\displaystyle+\frac{2}{3}({\cal D}^{2}{\cal S}-6i{\cal
S}^{2})M_{\alpha\beta}\Big{\\}}\ ,$

where the torsion superfields ${\cal S}$ and ${\cal C}_{\alpha\beta\gamma}$
satisfy the Bianchi identity

$\displaystyle{\cal D}^{\gamma}{\cal C}_{\alpha\beta\gamma}=-\frac{4i}{3}{\cal
D}_{\alpha\beta}{\cal S}\ .$ (4.5)

The gauge group of conformal supergravity consists of (i) superspace general
coordinate and local Lorentz transformations; (ii) super-Weyl transformations.
Covariant general coordinate transformations and local structure group
transformations act on the covariant derivatives and on a generic tensor
superfield $U$ belonging to some representation of the Lorentz groups as

$\displaystyle\delta_{\cal K}{\cal D}_{A}=[{\cal K},{\cal
D}_{A}]~{},~{}~{}~{}{\cal K}=\xi^{B}(z){\cal
D}_{B}+\frac{1}{2}K^{bc}(z)M_{bc}~{},~{}~{}~{}~{}~{}~{}\delta_{\cal K}U={\cal
K}U~{},$ (4.6)

with the gauge parameters $\xi^{B}$ and $K^{bc}$ obeying natural reality
conditions but otherwise arbitrary. Super-Weyl transformations, which are
related to local dilatation and special conformal transformations, arise as an
invariance of the superspace geometry’s conventional constraints. In
particular, the algebra (4.4) is invariant under the following super-Weyl
transformations of the covariant derivatives and torsion superfields

$\displaystyle\delta_{\sigma}{\cal D}_{\alpha}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sigma{\cal D}_{\alpha}+({\cal
D}^{\beta}\sigma)M_{\alpha\beta}~{},$ (4.7a)
$\displaystyle\delta_{\sigma}{\cal D}_{a}$ $\displaystyle=$
$\displaystyle\sigma{\cal D}_{a}+\frac{i}{2}(\gamma_{a})^{\alpha\beta}({\cal
D}_{\alpha}\sigma){\cal D}_{\beta}+\varepsilon_{abc}({\cal
D}^{b}\sigma)M^{c}~{},$ (4.7b) $\displaystyle\delta_{\sigma}{\cal S}$
$\displaystyle=$ $\displaystyle\sigma{\cal S}-\frac{i}{4}\,{\cal
D}^{2}\sigma~{},\qquad\delta_{\sigma}{\cal
C}_{\alpha\beta\gamma}=\frac{3}{2}\sigma{\cal
C}_{\alpha\beta\gamma}-\frac{1}{2}\,{\cal D}_{(\alpha\beta}{\cal
D}_{\gamma)}\sigma~{},$ (4.7c)

with the parameter $\sigma$ being a real but otherwise unconstrained scalar
superfield.

Among tensor superfields $U$, a special role is played by _primary_ ones. A
primary superfield of weight $w$ transforms homogenously under super-Weyl
transformations:

$\displaystyle\delta_{\sigma}U=w\sigma U~{}.$ (4.8)

As mentioned before, Poincaré supergravity can be defined as conformal
supergravity coupled to a Lorentz scalar compensating multiplet. For $3d$
${\cal N}=1$ this can be chosen to be a nowhere vanishing real primary scalar
superfield $\varphi$ such that

$\displaystyle\delta_{\sigma}\varphi=\frac{1}{2}\sigma\varphi~{},~{}~{}~{}~{}~{}~{}\varphi\neq
0~{}.$ (4.9)

Another key ingredient in the formulation of general $3d$ ${\cal N}=1$
supergravity-matter systems is the full superspace action principle. This is
given by151515In the notation of [87] that we employ in this section, the
superspace measure ${\rm d}^{3|2}z:={\rm d}^{3}x{\rm d}^{2}\theta$ is purely
imaginary. For this reason, we use a factor of $i$ in (4.10).

$S=i\int{\rm d}^{3|2}z\,E\,\mathcal{L}\ ,\qquad E^{-1}={\rm
Ber}(E_{A}{}^{M})~{}.$ (4.10)

The superspace Lagrangian ${\cal L}$ is chosen to be a primary, weight $2$
real scalar superfield, $\delta_{\sigma}{\cal L}=2\sigma{\cal L}$. Thanks to
the fact that $\delta_{\sigma}E=-2\sigma E$, the action in eq. (4.10) is
invariant under super-Weyl transformations. As we will see in the examples of
the coming subsections, in general, the Lagrangian ${\cal L}$ will be
constructed out of matter fields, their covariant derivatives, and, for non-
conformal models, the compensator $\varphi$.

As discussed in [77, 88], the constraints associated with the conformal
supergravity geometry of (4.4) and (4.5) can be solved in terms of a single
conformal real prepotential superfield
$\Psi_{\alpha\beta\gamma}=\Psi_{(\alpha\beta\gamma)}$. For the scope of our
paper, it suffices to know the covariant derivatives and torsion superfields
at first order in $\Psi$ about a flat Minkowski superspace geometry. Such
results can be readily read off from the analysis given in Appendix B of [88].
One has

$\displaystyle{\cal D}_{\alpha}$ $\displaystyle=$ $\displaystyle
D_{\alpha}+i\Psi_{\alpha\gamma\delta}\partial^{\gamma\delta}-\frac{1}{4}(D^{2}\Psi_{\alpha\beta\gamma})M^{\beta\gamma}-\frac{i}{2}(\partial_{(\alpha}{}^{\delta}\Psi_{\beta\gamma)\delta})M^{\beta\gamma}-\frac{2i}{3}(\partial_{\beta\gamma}\Psi^{\beta\gamma\delta})M_{\delta\alpha}$
(4.11a) $\displaystyle~{}+~{}{\cal O}(\Psi^{2})~{},$ $\displaystyle{\cal
D}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\partial_{\alpha\beta}+(D_{(\alpha}\Psi_{\beta\gamma\delta)})\partial^{\gamma\delta}+\frac{1}{2}(D^{\delta}\Psi_{\delta\rho(\alpha})\partial_{\beta)}{}^{\rho}$
(4.11b)
$\displaystyle+\Big{[}-\frac{i}{4}D^{2}\Psi_{\alpha\beta\gamma}+\frac{1}{2}\partial_{(\alpha}{}^{\delta}\Psi_{\beta\gamma)\delta}-\frac{1}{3}\partial^{\rho\delta}\Psi_{\rho\delta(\alpha}\varepsilon_{\beta)\gamma}\Big{]}D^{\gamma}$
$\displaystyle+\Big{[}-\frac{5}{4}\partial_{(\alpha}{}^{\rho}D_{\beta}\Psi_{\gamma\delta\rho)}+\frac{1}{4}\partial_{(\alpha\beta}D^{\tau}\Psi_{\gamma\delta)\tau}+\frac{1}{4}\partial^{\rho\tau}\Big{(}D_{(\tau}\Psi_{\gamma\rho\alpha)}\varepsilon_{\beta\delta}+D_{(\tau}\Psi_{\gamma\rho\beta)}\varepsilon_{\alpha\delta}\Big{)}$
$\displaystyle~{}~{}~{}-\frac{1}{12}\varepsilon_{\gamma(\alpha}\varepsilon_{\beta)\delta}\partial^{\rho\epsilon}D^{\tau}\Psi_{\rho\tau\epsilon}+\frac{1}{8}\Big{(}\partial_{(\alpha}{}^{\rho}\varepsilon_{\beta)(\gamma}D^{\tau}\Psi_{\delta)\rho\tau}-\partial_{(\gamma}{}^{\rho}\varepsilon_{\delta)(\alpha}D^{\tau}\Psi_{\beta)\rho\tau}\Big{)}\Big{]}M^{\gamma\delta}$
$\displaystyle~{}+~{}{\cal O}(\Psi^{2})\ ,$ $\displaystyle{{\cal S}}$
$\displaystyle=$
$\displaystyle-\frac{1}{8}D^{(\alpha}\partial^{\beta\gamma)}\Psi_{\alpha\beta\gamma}~{}+~{}{\cal
O}(\Psi^{2})\ ,$ (4.11c) $\displaystyle{\cal C}_{\alpha\beta\gamma}$
$\displaystyle=$
$\displaystyle\frac{1}{4}\partial_{(\alpha}{}^{\delta}D^{2}\Psi_{\beta\gamma)\delta}+\frac{i}{2}\partial_{(\alpha}{}^{\delta}\partial_{\beta}{}^{\rho}\Psi_{\gamma)\delta\rho}~{}+~{}{\cal
O}(\Psi^{2})~{}.$ (4.11d)

Here $D_{A}=(\partial_{a},D_{\alpha})$ denote the flat Minkowski superspace
covariant derivatives satisfying the algebra

$\displaystyle\\{D_{\alpha},D_{\beta}\\}=2i\partial_{\alpha\beta}~{},~{}~{}~{}[\partial_{a},D_{\beta}]=[\partial_{a},\partial_{b}]=0~{}.$
(4.12)

Note that the value of the compensator in the flat background is chosen to be
a constant; in particular, we will choose $\varphi=1$. Another useful result,
which is implied by (4.11), is that the Berezinian around a flat background
satisfies

$\displaystyle E=1~{}+~{}{\cal O}(\Psi^{2})~{}.$ (4.13)

As discussed in [88], to obtain the previous representation, part of the
supergravity gauge group, including super-Weyl transformations, has to be used
and it is fixed. The residual gauge freedom of ${\cal D}_{A}$, ${\cal S}$, and
${\cal C}_{\alpha\beta\gamma}$ in (4.11) is described by transformations of
the form

$\displaystyle\delta{\cal D}_{A}:={[}{\cal K},{\cal
D}_{A}{]}+\delta_{\sigma}{\cal D}_{A}~{},$ (4.14)

where the parameters $\xi^{\alpha}$, $K^{bc}$ and $\sigma$ of (4.6) and (4.7)
should be functions of $\xi^{a}$ and its covariant derivatives:

$\displaystyle\xi_{\alpha}$ $\displaystyle=$
$\displaystyle-\frac{i}{6}D^{\beta}\xi_{\beta\alpha}~{}+~{}{\cal O}(\Psi)~{},$
(4.15a) $\displaystyle K_{\alpha\beta}$ $\displaystyle=$ $\displaystyle
2D_{(\alpha}\xi_{\beta)}~{}+~{}{\cal O}(\Psi)~{},$ (4.15b)
$\displaystyle\sigma$ $\displaystyle=$ $\displaystyle
D_{\alpha}\xi^{\alpha}~{}+~{}{\cal
O}(\Psi)=\frac{1}{3}\partial^{a}\xi_{a}~{}+~{}{\cal O}(\Psi)~{}.$ (4.15c)

The prepotential superfield then transforms as

$\displaystyle\delta\Psi_{\alpha\beta\gamma}=\frac{1}{2}D_{(\alpha}\xi_{\beta\gamma)}~{}+~{}{\cal
O}(\Psi)~{},$ (4.16)

while the compensator $\varphi$ satisfies

$\displaystyle\delta\varphi$ $\displaystyle=$
$\displaystyle-\frac{1}{12}\partial^{\alpha\beta}\xi_{\alpha\beta}~{}+~{}{\cal
O}(\Psi)~{}.$ (4.17)

Consider now a supergravity-matter system described by the full superspace
action $S[U,{\cal D}_{A},\varphi]$, and depending in general upon matter
multiplets (that we denote by $U$), the conformal geometry, and the conformal
compensator $\varphi$. Assuming the equations of motion for the matter
superfields $U$ are satisfied, $\delta S/\delta U=0$, the variation of the
action induced by an infinitesimal deformation of the gravitational
superfields $\Psi_{\alpha\beta\gamma}$ and the compensator $\varphi$ gives

$\displaystyle\delta S[U,{\cal D}_{A},\varphi]=i\int{\rm
d}^{3|2}z\,\Big{(}\delta\Psi^{\alpha\beta\gamma}{J}_{\alpha\beta\gamma}+\delta\varphi
J\Big{)}~{},$ (4.18)

where we are taking a variation around a flat background, and hence we are
dropping the $E$ factor in the final expression.

The superfields ${J}_{\alpha\beta\gamma}$ and $J$ define the supercurrent
multiplet and, as we will see in examples in the coming subsections, can be
computed explicitly by using the building blocks described above. The
variation (4.18) must vanish if $\delta\Psi^{\alpha\beta\gamma}$ and
$\delta\varphi$ are the gauge transformations (4.16) and (4.17). Since the
gauge parameter $\xi_{\beta\gamma}=(\gamma^{a})_{\beta\gamma}\xi_{a}$ in
(4.16) and (4.17) is an arbitrary superfield, one concludes that the following
supercurrent conservation equation must hold

$\displaystyle
D^{\gamma}J_{\alpha\beta\gamma}=\frac{i}{6}\partial_{\alpha\beta}J~{},$ (4.19)

provided the equations of motion for the matter multiplets $U$ derived from
the action $S[U,{\cal D}_{A},\varphi]$ are satisfied. Note that if the model
described by $S$ is superconformal, then the dependence upon the compensator
$\varphi$ should disappear, $S=S[U,{\cal D}_{A}]$. In this case, the trace
multiplet superfield vanishes, $J=0$, and (4.19) simplifies to
$D^{\gamma}J_{\alpha\beta\gamma}=0$.

###  4.2 Calculation of supercurrents for vector models

In this subsection we are going to use the supergravity building block of the
previous subsection to compute the supercurrent multiplet for vector multiplet
models taking the following form

$S=-\frac{1}{2}\int{\rm
d}^{3|2}z\,W^{2}f(T)~{},~{}~{}~{}~{}~{}~{}W^{2}=W^{\alpha}W_{\alpha}~{},~{}~{}~{}T:=\frac{1}{8}D^{2}W^{2}~{},~{}~{}~{}D^{2}=D^{\alpha}D_{\alpha}~{},$
(4.20)

for any function $f(T)$. Here $W_{\alpha}$ is the field strength of an Abelian
vector multiplet. In our notations, an Abelian vector multiplet can be
described by a closed super 2-form $F^{(2)}=\frac{1}{2}E^{B}\wedge
E^{A}F_{AB}$, ${\rm d}F^{(2)}=0$, with components

$\displaystyle F_{\alpha\beta}$ $\displaystyle=$ $\displaystyle 0~{},$ (4.21a)
$\displaystyle F_{a\beta}$ $\displaystyle=$
$\displaystyle-i(\gamma_{a})_{\beta\gamma}W^{\gamma}=-F_{\beta a}~{},$ (4.21b)
$\displaystyle F_{ab}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\varepsilon_{abc}(\gamma^{c})^{\alpha\beta}D_{\alpha}W_{\beta}=-F_{ba}~{},$
(4.21c)

where the field strength $W_{\alpha}$ is real and satisfies

$\displaystyle
D^{\alpha}W_{\alpha}=0~{},~{}~{}~{}~{}~{}~{}(W_{\alpha})^{*}=W_{\alpha}~{}.$
(4.22)

We can write the $\theta$-components of $W_{\alpha}$ as

$\displaystyle W_{\alpha}$ $\displaystyle=$
$\displaystyle\lambda_{\alpha}+\theta^{\beta}f_{\alpha\beta}+\frac{i}{2}\theta^{2}\partial_{\alpha\beta}\lambda^{\beta}~{},$
(4.23a)
$\displaystyle\lambda_{\alpha}~{}:=~{}W_{\alpha}|_{\theta=0}~{},~{}~{}~{}f_{\alpha\beta}$
$\displaystyle:=$ $\displaystyle
D_{\beta}W_{\alpha}|_{\theta=0}~{}=~{}\frac{1}{2}\,\varepsilon^{abc}(\gamma_{a})_{\alpha\beta}f_{bc}~{},~{}~{}~{}\partial_{[a}f_{bc]}~{}=~{}0~{},~{}~{}~{}~{}~{}~{}$
(4.23b)

where note that $f_{ab}=F_{ab}|_{\theta=0}$ with $F_{ab}$ being the top
component of the closed superform $F^{(2)}$ in eq. (4.21). The free
supersymmetric Maxwell theory is described by the following action

$\displaystyle S=-\frac{1}{2}\int{\rm d}^{3|2}z\,W^{\alpha}W_{\alpha}=\int{\rm
d}^{3}x\left(-\frac{1}{4}f^{ab}f_{ab}-\frac{i}{2}\lambda^{\alpha}\partial_{\alpha\beta}\lambda^{\beta}\right)~{},$
(4.24)

which corresponds to (4.20) with $f(T)=1$. Note also that the definition of
the superfield $T$ is such that its lowest component coincides with the free
Maxwell Lagrangian:

$\displaystyle
T|_{\theta=0}=-\frac{1}{4}f^{ab}f_{ab}-\frac{i}{2}\lambda^{\alpha}\partial_{\alpha\beta}\lambda^{\beta}~{}.$
(4.25)

The first step to compute the supercurrent for the previous models is to
couple them to Poincaré supergravity. The covariant vector multiplet is
defined by a weight-3/2 primary real spinor superfield strength ${\cal
W}_{\alpha}$ satisfying

$\displaystyle{\cal D}^{\alpha}{\cal
W}_{\alpha}=0~{},~{}~{}~{}~{}~{}~{}\delta_{\sigma}{\cal
W}_{\alpha}=\frac{3}{2}\sigma{\cal W}_{\alpha}~{},$ (4.26)

which represents the curved extension of $W_{\alpha}$. To lift the model
(4.20) to supergravity we would also like to have a primary version of $T$ and
$f(T)$. A straightforward calculation shows that the following superfield is
primary and weight zero:

$\displaystyle{\cal T}=\frac{1}{8}\Big{(}\,\varphi^{-3}({\cal
D}^{2}\varphi^{-5}{\cal W}^{2})-2i\varphi^{-8}{\cal S}{\cal
W}^{2}\,\Big{)}~{},~{}~{}~{}~{}~{}~{}\delta_{\sigma}{\cal T}=0~{},$ (4.27)

where ${\cal D}^{2}:={\cal D}^{\alpha}{\cal D}_{\alpha}$ and ${\cal
W}^{2}:={\cal W}^{\alpha}{\cal W}_{\alpha}$. This expression for ${\cal T}$
can be obtained by using the fact that, given a weight-1/2 scalar primary
superfield, as for example $\varphi$, the combination $({\cal D}^{2}-2i{\cal
S})\varphi$ transforms homogenously with weight 3/2 under super-Weyl
transformations, see [88]. Then any function $f({\cal T})$ is also a weight-
zero primary superfield which reduces to $f(T)$ when taking the flat
superspace limit, ${\cal D}_{A}=D_{A}$ and $\varphi=1$. Since
$\delta_{\sigma}{\cal W}^{2}=3\sigma{\cal W}^{2}$, it is clear that the
following action is an appropriate curved extension of (4.20)

$S=-\frac{1}{2}\int{\rm d}^{3|2}z\,E\,\varphi^{-2}{\cal W}^{2}f({\cal T})~{}.$
(4.28)

By noticing that the nilpotency condition ${\cal W}^{2}{\cal W}_{\alpha}=0$
holds, it follows that the previous action can be written in the following
simplified form

$S=-\frac{1}{2}\int{\rm d}^{3|2}z\,E\,\varphi^{-2}{\cal W}^{2}f({\bf
T})~{},~{}~{}~{}~{}~{}~{}{\bf T}:=\frac{1}{8}\varphi^{-8}{\cal D}^{2}{\cal
W}^{2}~{}.$ (4.29)

This is our starting point to compute the supercurrent for (4.20).

It is simple to compute the supertrace, $J$. It suffices to take the variation
of the compensator $\varphi$ about the flat Minkowski background (${\cal
D}_{A}=D_{A}$, $\varphi=1$). We will denote this variation as
$\delta_{\varphi}$ where the compensator should be thought as
$\varphi=1+\delta\varphi$. By using the following result

$\displaystyle\delta_{\varphi}f({\bf T})$ $\displaystyle=$
$\displaystyle-8\delta\varphi\,Tf^{\prime}(T)~{},$ (4.30)

one obtains

$\delta_{\varphi}S=-\frac{1}{2}\int{\rm
d}^{3|2}z\,\delta\varphi\,W^{2}\Big{\\{}-2f(T)-8Tf^{\prime}(T)\Big{\\}}~{}.$
(4.31)

By comparing with the compensator variation in (4.18), we obtain

$\displaystyle J=-iW^{2}\Big{(}f(T)+4Tf^{\prime}(T)\Big{)}~{}.$ (4.32)

The variation with respect to $\Psi_{\alpha\beta\gamma}$ is slightly more
involved. First of all, we need to obtain the expression of the covariant
field strength ${\cal W}_{\alpha}$ in terms of $W_{\alpha}$ and the conformal
prepotential. A natural candidate is given by the following Ansatz

$\displaystyle{\cal
W}_{\alpha}=W_{\alpha}+a(D^{\beta}\Psi_{\alpha\beta\gamma})W^{\gamma}+b\Psi_{\alpha\beta\gamma}D^{\beta}W^{\gamma}~{}+~{}{\cal
O}(\Psi^{2})~{}.$ (4.33)

By imposing ${\cal D}^{\alpha}{\cal W}_{\alpha}=0$ up to linear order in
$\Psi$ with ${\cal D}_{\alpha}$ given by (4.11a), and by using
$D^{\alpha}W_{\alpha}=0$, one can check that the constant parameters in (4.33)
are uniquely fixed to be

$\displaystyle a=-b=1~{}~{}~{}\Longrightarrow~{}~{}~{}{\cal
W}_{\alpha}=W_{\alpha}+(D^{\beta}\Psi_{\alpha\beta\gamma})W^{\gamma}-\Psi_{\alpha\beta\gamma}D^{\beta}W^{\gamma}~{}+~{}{\cal
O}(\Psi^{2})~{}.$ (4.34)

This, together with (4.11a), implies

$\displaystyle{\cal W}^{2}$ $\displaystyle=$ $\displaystyle
W^{2}-2\Psi^{\alpha\beta\gamma}W_{\alpha}D_{\beta}W_{\gamma}~{}+~{}{\cal
O}(\Psi^{2})~{},$ (4.35a) $\displaystyle{\bf T}$ $\displaystyle=$
$\displaystyle\varphi^{-8}\Big{[}\,T-\frac{1}{4}D^{2}\big{(}\Psi^{\alpha\beta\gamma}W_{\alpha}D_{\beta}W_{\gamma}\big{)}-\frac{i}{8}D_{\alpha}\big{(}\Psi^{\alpha\beta\gamma}\partial_{\beta\gamma}W^{2}\big{)}$
(4.35b)
$\displaystyle~{}~{}~{}~{}~{}~{}\,+\frac{i}{8}\partial_{\alpha\beta}\big{(}\Psi^{\alpha\beta\gamma}D_{\gamma}W^{2}\big{)}~{}+~{}{\cal
O}(\Psi^{2})\Big{]}~{}.$

At this point, by using the results above together with (4.13), it is
straightforward to take the variation of (4.29) with respect to
$\Psi^{\alpha\beta\gamma}$ around ${\cal D}_{A}=D_{A}$ and $\varphi=1$, which
we denote as $\delta_{\Psi}$. After some integration by parts, one obtains

$\delta_{\Psi}S=\int{\rm
d}^{3|2}z\,\delta\Psi^{\alpha\beta\gamma}\Bigg{\\{}\,i\Big{[}f(T)+\frac{1}{8}D^{2}\big{(}W^{2}f^{\prime}(T)\big{)}\Big{]}W_{(\alpha}D_{\beta}W_{\gamma)}-\frac{i}{8}W^{2}f^{\prime}(T)\partial_{(\alpha\beta}D_{\gamma)}W^{2}\Bigg{\\}}~{}.$
(4.36)

From this one can readily read off the expression for $J_{\alpha\beta\gamma}$.
Summarising, the supergravity analysis leads to the following supercurrent
multiplet for the model (4.20)

$\displaystyle J_{\alpha\beta\gamma}$ $\displaystyle=$ $\displaystyle-
ig(T)W_{(\alpha}D_{\beta}W_{\gamma)}-\frac{1}{8}W^{2}f^{\prime}(T)\,\partial_{(\alpha\beta}D_{\gamma)}W^{2}~{},$
(4.37a) $\displaystyle J$ $\displaystyle=$ $\displaystyle-
iW^{2}\Big{(}f(T)+4Tf^{\prime}(T)\Big{)}~{},$ (4.37b)

with the superfield $g(T)$ defined by

$\displaystyle g(T)=f(T)+\frac{1}{8}D^{2}\big{(}W^{2}f^{\prime}(T)\big{)}~{}.$
(4.38)

Up to mapping notations according to Appendix B, and inserting factors of
$\kappa^{2}$ by field redefinitions, the supercurrent coincides with the
result of the previous section, eqs. (3.40).

###  4.3 Calculation of supercurrents for tensor/scalar multiplet models

We conclude this section by looking at the calculation of the supercurrent for
models based on tensor multiplets. In the notation of this section, we look at
models described by the following action principle

$\displaystyle S=-\frac{1}{2}\int{\rm d}^{3|2}z\,U^{2}h(\tilde{T})~{},$ (4.39)

where the superfields $U_{\alpha}$ and $\tilde{T}$ satisfy

$\displaystyle
D^{\alpha}D_{\beta}U_{\alpha}=0~{},~{}~{}~{}(U_{\alpha})^{*}=U_{\alpha}~{},~{}~{}~{}U^{2}:=U^{\alpha}U_{\alpha}~{},~{}~{}~{}~{}~{}~{}\tilde{T}=\frac{1}{8}D^{2}U^{2}~{}.$
(4.40)

The solution of the constraint for the tensor multiplet is given in terms of a
spinor derivative of a real scalar superfield $\rho$:
$U_{\alpha}=iD_{\alpha}\rho$. We will use this decomposition as the starting
point to couple the multiplet to supergravity. In fact, we promote the scalar
superfield $\rho$ to be a primary of weight zero. This implies that the curved
version of $U_{\alpha}$, which we denote as ${\cal U}_{\alpha}$, is a primary
of weight-1/2. More specifically, we use

$\displaystyle\rho~{},~{}~{}~{}\delta_{\sigma}\rho=0~{},~{}~{}~{}~{}~{}~{}{\cal
U}_{\alpha}:=i{\cal D}_{\alpha}\rho~{},~{}~{}~{}\delta{\cal
U}_{\alpha}=\frac{1}{2}\sigma{\cal U}_{\alpha}~{}.$ (4.41)

A locally supersymmetric and super-Weyl invariant version of the model (4.39)
is described by the action

$S=-\frac{1}{2}\int{\rm d}^{3|2}z\,E\,\varphi^{2}\,{\cal U}^{2}\,h(\tilde{{\bf
T}})~{},$ (4.42)

where ${\cal U}^{2}:={\cal U}^{\alpha}{\cal U}_{\alpha}$ and

$\displaystyle\tilde{{\bf T}}$ $\displaystyle=$
$\displaystyle\frac{1}{8}\varphi^{-4}{\cal D}^{2}{\cal U}^{2}~{}.$ (4.43)

Note that $\tilde{{\bf T}}$ is not a primary superfield, as it transforms
inhomogeneously under super-Weyl transformations. However, it is simple to
show that, due to the nilpotency ${\cal U}^{2}{\cal U}_{\alpha}=0$, it
satisfies ${\cal U}^{2}\delta_{\sigma}\tilde{{\bf T}}=0$ which suffices to
make (4.42) super-Weyl invariant.

By following the same analysis of the previous subsection, it is now simple to
obtain the supercurrent for (4.39) by taking the variation of (4.42) with
respect to the conformal prepotential $\Psi_{\alpha\beta\gamma}$ and the
conformal compensator $\varphi$. First of all, the ${\cal U}_{\alpha}$
superfield is independent of the compensator $\varphi$ and it is simple to
show that, around the flat Minkowski superspace background (${\cal
D}_{A}=D_{A}$, $\varphi=1$) one has $\delta_{\varphi}\tilde{{\bf
T}}=-4\delta\varphi\tilde{T}$. Next, by using (4.11a), the fact that $\rho$ is
a scalar superfield, and $\partial^{\alpha\beta}=-iD^{(\alpha}D^{\beta)}$, one
can immediately obtain the following results

$\displaystyle{\cal U}_{\alpha}$ $\displaystyle=$ $\displaystyle
i\big{(}D_{\alpha}+i\Psi_{\alpha\gamma\delta}\partial^{\gamma\delta}\big{)}\rho+{\cal
O}(\Psi^{2})~{},$ (4.44a) $\displaystyle\delta_{\Psi}{\cal U}_{\alpha}$
$\displaystyle=$
$\displaystyle\delta\Psi_{\alpha\beta\gamma}D^{\beta}U^{\gamma}~{},$ (4.44b)
$\displaystyle\delta_{\Psi}{\cal U}^{2}$ $\displaystyle=$ $\displaystyle
2\delta\Psi^{\alpha\beta\gamma}U_{\alpha}D_{\beta}U_{\gamma}~{},$ (4.44c)
$\displaystyle\delta_{\Psi}\tilde{{\bf T}}$ $\displaystyle=$
$\displaystyle\frac{i}{8}\delta\Psi^{\alpha\beta\gamma}\partial_{\alpha\beta}D_{\gamma}U^{2}+\frac{i}{8}\big{(}\partial_{\alpha\beta}\delta\Psi^{\alpha\beta\gamma}\big{)}D_{\gamma}U^{2}-\frac{i}{8}D_{\alpha}\big{(}\delta\Psi^{\alpha\beta\gamma}\partial_{\beta\gamma}U^{2}\big{)}$
(4.44d)
$\displaystyle+\frac{1}{4}D^{2}\big{(}\delta\Psi^{\alpha\beta\gamma}U_{\alpha}D_{\beta}U_{\gamma}\big{)}~{}.$