# A Note on the Representation Power of GHHs

Zhou Lu111This work is done during LZ’s visit to SQZ institution.
Princeton University
<EMAIL_ADDRESS>
(January 2021)

###### Abstract

In this note we prove a sharp lower bound on the necessary number of nestings
of nested absolute-value functions of generalized hinging hyperplanes (GHH) to
represent arbitrary CPWL functions. Previous upper bound states that $n+1$
nestings is sufficient for GHH to achieve universal representation power, but
the corresponding lower bound was unknown. We prove that $n$ nestings is
necessary for universal representation power, which provides an almost tight
lower bound. We also show that one-hidden-layer neural networks don’t have
universal approximation power over the whole domain. The analysis is based on
a key lemma showing that any finite sum of periodic functions is either non-
integrable or the zero function, which might be of independent interest.

##  1 Introduction

We consider the complexity of representing continuous piecewise linear
functions using the generalized hinging hyperplane model Wang and Sun (2005).
We begin with a short review on these two notions.

###  1.1 Continuous Piecewise Linear (CPWL) Functions

Continuous piecewise linear (CPWL) functions play an important role in non-
linear function approximation, such as nonlinear circuit or neural networks.
We introduce the definition of CPWL functions borrowed from Chua and Deng
(1988).

######  Definition 1.1 (CPWL function).

A function $f(x):R^{n}\to R$ is said to be a CPWL function iff it satisfies:

1):The domain space $R^{n}$ is divided into a finite number of polyhedral
regions by a finite number of disjunct boundaries. Each boundary is a subset
of a hyperplane and takes non-zero measure (standard lebesgue measure) on the
hyperplane (as $R^{n-1}$).

2):The restriction of $f(x)$ on each polyhedral region is an affine function.

3):$f(x)$ is continuous on $R^{n}$.

###  1.2 Generalized Hinging Hyperplanes (GHH)

The model of hinging hyperplanes (HH) is a sum of hinges like

$\pm\max\\{w_{1}^{\top}x+b_{1},w_{2}^{\top}x+b_{2}\\}$ (1)

where $w_{1},w_{2}\in R^{n}$ and $b_{1},b_{2}\in R$ are parameters. The HH
model (in fact equivalent to a one hidden-layer ReLU network) can approximate
any continuous function over a compact domain to arbitrary precision as the
number of hinges go infinity Breiman (1993).

However, this model can’t exactly represent all CPWL function as pointed out
in He et al. (2018), which brings doubt on its approximation efficiency. To
overcome this problem, Wang and Sun (2005) first proposed a generalization of
HH model, called GHH which allows more than 2 affine functions within the
nested maximum operator:

######  Definition 1.2 ($n$-order hinge).

A $n$-order hinge is a function of the following form:

$\pm\max\\{w_{1}^{\top}x+b_{1},w_{2}^{\top}x+b_{2},\cdots,w_{n+1}^{\top}x+b_{n+1}\\}$
(2)

where $w_{i}\in R^{n}$ and $b_{i}\in R$ are parameters.

A linear combination of a finite number of $n$-order hinges is called a
$n$-order hinging hyperplane ($n$-HH) model. Such model has universal
representation power over all CPWL functions, as formalized in the theorem
below:

######  Theorem 1.3 (Theorem 1 in Wang and Sun (2005)).

For any positive integer $n$ and CPWL function $f(x):R^{n}\to R$, there exists
a $n$-HH which exactly represents $f(x)$.

The question is whether we can give a sharp lower bound on the necessary
number of affine functions within the nested maximum operator. Wang and Sun
(2005) conjected that $(n-1)$-HH can’t represent all CPWL functions, but this
open problem is left unanswered for more than a decade. In the following
section we will prove our main result that $(n-2)$-HH can’t represent all CPWL
functions, yielding an almost tight lower bound.

##  2 Main Result

Observe that any $(n-2)$-order hinge depends on only $n-1$ affine transforms
of $x$, thus there always exists a direction in which the value of the
$(n-2)$-order hinge remains the same. We make such observation precise by
introducing the definition of low-dimensional and periodic functions.

######  Definition 2.1 (Low-dimensional/periodic function).

A function $f(x):R^{n}\to R$ is said to be low-dimensional, if there exists a
vector $v\neq 0$, such that for any $x\in R^{n}$ and $c\in R$, we have that
$f(x)=f(x+cv)$. If we have only $f(x)=f(x+v)$ then $f$ is said to be periodic
(a weaker notion). $v$ is called an invariant vector of $f$.

Any $(n-2)$-order hinge is a low-dimensional function on $R^{n}$, so our
problem is reduced to proving the class of finite sum of low-dimensional
functions has limited representation power. The following key lemma actually
proves (a stronger result) that finite sum of periodic functions can’t
represent any non-trivial integrable functions.

######  Lemma 2.2.

Any finite sum of periodic functions is either non-integrable or the zero
function, i.e. given periodic functions $f_{i}(x)$, $i=1,...,m$, then
$f(x)\triangleq\sum_{i=1}^{m}f_{i}(x)$ satisfies

$\int_{R^{n}}|f|=\infty\quad or\quad f\equiv 0$ (3)

###### Proof.

We prove Lemma 2.2 by induction. Suppose each $f_{i}$ has an invariant vector
$v_{i}$, base case $m=1$ is trivial since if we denote the orthogonal
hyperplane $H_{i}=\\{x|x^{\top}v_{i}=0\\}$, we have

$\int_{R^{n}}|f|=\int_{R}\int_{H_{1}}|f|$ (4)

thus $\int_{R^{n}}|f|<\infty$ if and only if $\int_{H}|f|=0$. Assume
$f=\sum_{i=1}^{m}f_{i}$ is integrable, then $g(x)\triangleq f(x+v_{m})-f(x)$
is also integrable. We make the following decomposition of $g$:

$g(x)=\sum_{i=1}^{m}f_{i}(x+v_{m})-f_{i}(x)=\sum_{i=1}^{m-1}f_{i}(x+v_{m})-f_{i}(x)$
(5)

where each $f_{i}(x+v_{m})-f_{i}(x)$ is periodic (with invariant vector
$v_{i}$) as well. By induction we have $g\equiv 0$ and $f$ is also a periodic
function (with invariant vector $v_{m}$). Using the base case on $f$ again
concludes our proof. ∎

Our main result is a direct corollary of Lemma 2.2, as stated below:

######  Theorem 2.3.

For any positive integer $n\geq 2$, there exists a CPWL function
$g(x):R^{n}\to R$, such that no $(n-2)$-HH can exactly represent $g(x)$.

###### Proof.

Let $g(x)\triangleq\max\\{0,1-||x||_{\infty}\\}$. It’s straightfoward to check
that $g(x)$ is a CPWL function with at most $2^{n+1}$ affine polyhedral
regions, and meanwhile is an integrable function with positive integral. As
any $(n-2)$-HH can be written as a finite sum of low-dimensional functions, it
can’t represent $g(x)$ by Lemma 2.2. ∎

Theorem 2.3 implies that in order to achieve universal representation power
over all CPWL functions, a $(n-1)$-HH model is necessary which provides an
almost tight lower bound corresponding to the upper bound in Theorem 1.3.

##  3 Implications on Universal Approximation of ANNs

Traditional universal approximation theorems of artifical neural networks
(ANN) Cybenko (1989); Hornik et al. (1989); Barron (1994) typically states
that an ANN with one hidden layer and unbounded width can approximate any
measurable function with arbitrary precision on a compact set. Our result
demonstrates that the compact set assumption is indeed necessary for ANNs with
traditional activation (composition of an affine transform and a fixed
univariate function $\sigma$):

######  Corollary 3.1.

Given an integrable function $f$ on $R^{n}$ ($n\geq 2$), for any one-hidden-
layer neural network $g$ with traditional activation $\sigma(w^{\top}x+b)$, we
have that

$\int_{R^{n}}|f-g|=\infty\quad or\quad\int_{R^{n}}|f-g|=\int_{R^{n}}|f|$ (6)

###### Proof.

Any unit $\sigma(w^{\top}x+b)$ is obviously a low-dimensional function when
$n\geq 2$, thus by Lemma 2.2 we finish our proof. ∎

Corollary 3.1 reveals a fundamental gap of representation power between one-
hidden layer neural networks and deeper ones, as Theorem 1.3 indicates a
neural network with $\lceil log_{2}(n+1)\rceil$ hidden layers can represent
any CPWL function He et al. (2018), showing the benefits of depth in universal
approximation Lu et al. (2017).

##  4 Conclusion

In this note we give a sharp lower bound on the necessary number of nestings
of nested absolute-value functions of generalized hinging hyperplanes (GHH) to
represent arbitrary CPWL functions, which is the first non-trivial lower bound
to the best of our knowledge. Our results fully characterizes the
representation power (and limit) of the GHH model.

Our result also has implications on ANNs, a much more popular model in machine
learning. It shows that one-hidden-layer neural networks with traditional
activation can’t control the approximation error on the whole domain despite
existing universal approximation theorems, a fundamental gap between one-
hidden-layer networks and deeper ones. We conject similar depth-separation
results should hold for deeper networks and the $\lceil log_{2}(n+1)\rceil$
bound should be tight in representing CPWL functions. Instead of low-
dimensional (periodic), other properties need to be discovered for deeper
networks.

## Acknowledgements

The author would like to thank Fedor Petrov for giving an elegant proof of
Lemma 2.2 on Mathoverflow.

## References

  * Barron (1994) Andrew R Barron.  Approximation and estimation bounds for artificial neural networks.  _Machine learning_ , 14(1):115–133, 1994.
  * Breiman (1993) Leo Breiman.  Hinging hyperplanes for regression, classification, and function approximation.  _IEEE Transactions on Information Theory_ , 39(3):999–1013, 1993.
  * Chua and Deng (1988) Leon O Chua and A-C Deng.  Canonical piecewise-linear representation.  _IEEE Transactions on Circuits and Systems_ , 35(1):101–111, 1988.
  * Cybenko (1989) George Cybenko.  Approximation by superpositions of a sigmoidal function.  _Mathematics of control, signals and systems_ , 2(4):303–314, 1989.
  * He et al. (2018) Juncai He, Lin Li, Jinchao Xu, and Chunyue Zheng.  Relu deep neural networks and linear finite elements.  _arXiv preprint arXiv:1807.03973_ , 2018.
  * Hornik et al. (1989) Kurt Hornik, Maxwell Stinchcombe, and Halbert White.  Multilayer feedforward networks are universal approximators.  _Neural networks_ , 2(5):359–366, 1989.
  * Lu et al. (2017) Zhou Lu, Hongming Pu, Feicheng Wang, Zhiqiang Hu, and Liwei Wang.  The expressive power of neural networks: A view from the width.  _arXiv preprint arXiv:1709.02540_ , 2017.
  * Wang and Sun (2005) Shuning Wang and Xusheng Sun.  Generalization of hinging hyperplanes.  _IEEE Transactions on Information Theory_ , 51(12):4425–4431, 2005.


aainstitutetext: Department of Physics, University of Mohaghegh Ardabili, P.O.
Box 179, Ardabil, Iranbbinstitutetext: Department of Physics, University of
Guilan, P.O. Box 41335-1914, Rasht, Iranccinstitutetext: School of Physics and
School of Particles and Accelerators, Institute for Research in Fundamental
Sciences (IPM), P.O. Box 19395-5531, Tehran, Iranddinstitutetext: Department
of Physics, Hakim Sabzevari University, P.O. Box 397, Sabzevar, Iran

#  $T\bar{T}$-like Flows in Non-linear Electrodynamic Theories and S-duality

H. Babaei-Aghbolagh, b,c Komeil Babaei Velni d Davood Mahdavian Yekta a and H.
Mohammadzadeh<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We investigate the $T\bar{T}$-like flows for non-linear electrodynamic
theories in $D(=\\!\\!2n)$-dimensional spacetime. Our analysis is restricted
to the deformation problem of the classical free action by employing the
proposed $T\bar{T}$ operator from a simple integration technique. We show that
this flow equation is compatible with $T\bar{T}$ deformation of a scalar field
theory in $D\\!=\\!2$ and of a non-linear Born-Infeld type theory in
$D\\!=\\!4$ dimensions. However, our computation discloses that this kind of
$T\bar{T}$ flow in higher dimensions is essentially different from deformation
that has been derived from the AdS/CFT interpretations. Indeed, the gravity
that may be exist as a holographic dual theory of this kind of effective Born-
Infeld action is not necessarily an AdS space. As an illustrative
investigation in $D\\!=\\!4$, we shall also show that our construction for the
$T\bar{T}$ operator preserves the original $SL(2,R)$ symmetry of a non-
supersymmetric Born-Infeld theory, as well as $\mathcal{N}=2$ supersymmetric
model. It is shown that the corresponding $SL(2,R)$ invariant action fixes the
relationship between the $T\bar{T}$ operator and quadratic form of the energy-
momentum tensor in $D\\!=\\!4$.

##  1 Introduction

One of the fundamental concepts in any quantum field theory (QFT) is the
implication of the renormalization group approach that helps us to better
understand the space of QFTs. About two decades ago, it has been proposed by
Zamolodchikov Zamolodchikov:2004ce  that one can construct this space by
starting from a fixed point of the renormalization group and perturbing the
free or exactly solvable action of a 2D QFT by an integrated local operator.
In this regard, using a relevant operator, the renormalization group flows to
some IR fixed points while for an irrelevant operator, the UV properties of
the theory will change.

In ref. Smirnov:2016lqw , Smirnov and Zamolodchikov discovered a general class
of exactly solvable irrelevant deformations of 2D conformal field theories
(CFT) known as the $T\bar{T}$ deformation (see also Cavaglia:2016oda ). Even
though the $T\bar{T}$ deformations provide laboratories for investigating
various aspects of field theories Guica:2017lia ; Chen:2018keo ;
Aharony:2018bad ; Cardy:2018sdv ; Datta:2018thy ; Bonelli:2018kik ;
Brennan:2019azg , but most of the attempts have an interpretation from the
AdS/CFT correspondence point of view. It was conjectured in refs.
McGough:2016lol ; Kraus:2018xrn  that the $T\bar{T}$ deformed CFT can be
interpreted as the holographic dual of a finite patch (a finite radial cutoff)
of asymptotically AdS spacetime. This result is equivalent to the statement
that the holographic dual at finite radius should be interpreted as a deformed
CFT Heemskerk:2009pn . Other related interesting developments can be found in
refs. Giveon:2017myj ; Giribet:2017imm ; Aharony:2018vux ; Donnelly:2018bef ;
Hartman:2018tkw ; Conti:2018tca .

It has been shown in ref. McGough:2016lol  that many interesting physical
observables can be computed explicitly from such non-local UV theory. Taking
into account a $T\bar{T}$ operator (an irrelevant operator given by the
product of $T=T_{zz}$ and $\bar{T}={T}_{\bar{z}\bar{z}}$ components of the
energy-momentum (EM) tensor), we have a Lorentz invariant deformation as
$S_{QFT}=S_{CFT}+S_{\lambda}$, where $S_{\lambda}=\lambda\int
d^{2}x\,T\bar{T}$. In this respect, some theories emerge by
${dS_{QFT}}/{d\lambda}=\int d^{2}x\,(T\bar{T})_{\lambda}$ for finite
$\lambda$.

In 2D, the $T\bar{T}$ operator is well-defined quantum mechanically by
examining the energy spectrum (it is free of short distance singularities)
McGough:2016lol . In definition of the $T\bar{T}$ deformation proposed by
Zamolodchikov that the variation of the Lagrangian with respect to the
deformation parameter equals the determinant of the deformed EM tensor, all
objects are quantum renormalized and UV finite. Nevertheless, one can treat
the resulting Lagrangian as a classical one, giving rise to classical fields
which is in accordance with some features of the corresponding models.

The generalization to higher dimensions from the field theory point of view
has also been proposed in refs. Cardy:2018sdv ; Bonelli:2018kik . However, in
general dimensions the operator does not share all the special properties
uncovered by Zamolodchikov in two dimensions. In fact, the $T\bar{T}$
deformation at higher than two dimensions is not unique. It was proposed in
Cardy:2018sdv  to use $(det\,T)^{1/\alpha}$ with $\alpha\\!=\\!D-2$ and in
Bonelli:2018kik  it was further generalized to different values of $\alpha$,
or even additional deformations are required if one includes bulk scalars and
gauge fields.

Considering the approach in Kraus:2018xrn , a generalized $T\bar{T}$
deformation in D dimensions was proposed in ref. Taylor:2018xcy  in the
context of holographic realization of $AdS_{D+1}$ spacetime. As the main
purpose of this paper, we recast the $T\bar{T}$ operator given in ref.
Smirnov:2016lqw  for general dimension $D$ as follows

${O}_{T^{2}}^{[r]}=T_{\mu\nu}T^{\mu\nu}-r{T_{\mu}}^{\mu}{T_{\nu}}^{\nu},$ (1)

where $r=1/(D-1)$ from holographic calculations Taylor:2018xcy .

According to the AdS/CFT conjecture Maldacena:1997re , the type IIB
superstring theory on $AdS_{5}\times S^{5}$ is daul to $\mathcal{N}\\!=\\!4$
supersymmetric Yang-Mills theory in four dimensions. Indeed, the latter is a
non-abelian $SU(N)$ guage theory with large $N$ which lives on $N$ coincident
$D3$-branes in string theory. However, the effective action of a single brane
is described by an abelian theory which specially in $D\\!=\\!4$ is a Born-
Infeld (BI) theory and its gravity dual is not necessarily an $AdS$ theory. In
this respect, the $T\bar{T}$ operator in (1) which has been found by using the
Lax pair operator in ref. Conti:2018jho  for BI theory of Maxwell field in
$D\\!=\\!4$, resembles a higher dimensional generalization of $T\bar{T}$
operator except for the factor $r=1/2$ instead of $r=1/(D-1)=1/3$. This
incompatibility motivates us strongly to investigate the $T\bar{T}$-like flows
in higher dimensional gauge theories.

Though the gravity duals of field theories which are not necessarily CFT are
not clear, but recently there are attempts to generalize the $T\bar{T}$
operator for these QFTs in $D\\!=\\!4$. For example, the extension of this
operator for $\mathcal{N}\\!=\\!1$ supersymmetric BI theory in Ferko:2019oyv ,
the emergent gravity from hidden QFT coupled to the standard model via
$T\bar{T}$ deformation in Betzios:2020sro , the $T\bar{T}$-like deformation of
the Skyrme action as generalizations of the Heisenberg model for nucleon-
nucleon scattering in Nastase:2021mgy  and the $T\bar{T}$ deformation of non-
Lorentz invariant models, such as the nonlinear Schrödinger equation, the
Landau-Lifshitz equation and the Gardner equation in Frolov:2021oav .

In particular, we study a class of QFTs which are described by non-linear
electrodynamic BI type Lagrangian in $D\\!=\\!2n\\!=\\!2(p+1)$ dimensions (for
even $n$) and constructed from $p$–form field $A_{\mu_{1}\dots\mu_{p}}$ with
strength
$F_{\mu_{1}\dots\mu_{n}}=n\,\partial_{[\mu_{1}}A_{\mu_{2}\dots\mu_{n}]}$. The
dynamics of this field is described by a Lorentz invariant local Lagrangian
$L(F)$ which entails the field equations

$\partial_{\mu_{1}}\frac{\partial L(F)}{\partial F_{\mu_{1}\dots\mu_{n}}}=0.$
(2)

The proposed $T\bar{T}$ operator deserves the results obtained in refs.
Smirnov:2016lqw ; Conti:2018jho  for 2D QFTs and generalized 4D BI models
Conti:2018jho , respectively.

In general, the EM tensor of any QFT can be obtained as the variational
derivative of the Lagrangian with respect to the gravitational field (i.e. the
spacetime metric). However, it has been found in refs. Gibbons:1995ap ;
Gaillard:1997rt  that for a given Lagrangian in $D\\!=\\!2n$ dimensions which
depends only on the self-interacting, completely antisymmetric field strength,
such as non-linear electrodynamic theories, the EM tensor can be written as

$T_{\mu\nu}=g_{\mu\nu}L(F)+\frac{1}{(n-1)!}(G_{\mu}F_{\nu}),$ (3)

where
$(G_{\mu}F_{\nu})=G_{\mu\alpha_{1}\dots\alpha_{n-1}}\,{F_{\nu}}^{\alpha_{1}\dots\alpha_{n-1}}$
and the field $G$ is defined by differentiation of the Lorentz invariant
Lagrangian with respect to the field strength as

$G_{\mu_{1}\dots\mu_{n}}=-n!\,\frac{\partial L(F)}{\partial
F^{\mu_{1}\dots\mu_{n}}}.$ (4)

Henceforth, we employ the following $T\bar{T}$ operator to deform the
$D\\!=\\!2n$ dimensional electrodynamic theory111In our convention the
$T\bar{T}$ operator is equal to $\frac{1}{8}\,{O}_{T^{2}}$ which corresponds
to the definition in ref. Ferko:2019oyv .

${O}_{T^{2}}^{[r={1}/{n}]}=T_{\mu\nu}T^{\mu\nu}-\frac{1}{n}{T_{\mu}}^{\mu}{T_{\nu}}^{\nu},$
(5)

which is consistent with deformations in $D\\!=\\!2$ and $D\\!=\\!4$
dimensions given by (1) for $n=1$ and $n=2$, respectively. This paper is an
attempt to construct the deformed action from a free Lagrangian with a simple
integration technique, perturbatively. In fact, $L_{free}$ denotes the
Lagrangian for the local fields and the coupling constant, denoted by
$\lambda$, that controls the strength of interaction among the fields as well
as with local sources.

As starting point for our discussion, we compute the EM (3) for a $L_{free}$
and then construct the $T\bar{T}$ operator ${O}_{T^{2}}={O}_{0}$ at the order
of $\lambda^{0}$. According to the relation ${O}_{T^{2}}=8\frac{\partial
L_{\lambda}}{\partial\lambda}$ in ref. Ferko:2019oyv , the integration of this
operator yields the deformed Lagrangian to the first order of $\lambda$

$L^{\prime}_{\lambda}=L_{free}+L_{1}=L_{free}+\frac{1}{8}\int
O_{0}\,d\lambda.$ (6)

Using $L^{\prime}_{\lambda}$ and (5), one can obtain the Lagrangian of order
$\lambda^{2}$ as

$L^{\prime\prime}_{\lambda}=L_{free}+L_{1}+L_{2}=L_{free}+\frac{1}{8}\int
O_{0}\,d\lambda+\frac{1}{8}\int O_{1}\,d\lambda,$ (7)

where ${O}_{T^{2}}={O}_{0}+O_{1}$. By iterating this method, we will deform
the free theory to higher orders of $\lambda$. When the trace of the EM tensor
for $L_{free}$ is zero, the contribution of the second term in (5) would be of
order $\lambda^{2}$, so it deforms the Lagrangian from the third order of
$\lambda$. For instance, in deforming the non-linear electrodynamic theory by
$T\bar{T}$ operator, the contribution of the second term will appear at the
order $\mathcal{O}(F^{8})$ in the action.

Since the $T\bar{T}$ operator given in (5) is proportional to quadratic power
of EM tensor, it is expected that the symmetries of $T_{\mu\nu}$ of the non-
linear electrodynamic BI theory be inherited to $O_{T^{2}}$. In particular,
the EM tensor is invariant under electromagnetic duality ($S$-duality) in
$D\\!=\\!4$. Moreover, as shown in ref. Gibbons:1995ap , this duality can be
generalized to an $SL(2,R)$ symmetry, that is, $T_{\mu\nu}$ is invariant under
$SL(2,R)$ transformation. An $SL(2,R)$ invariant form for $T_{\mu\nu}$ of non-
linear electrodynamic theories is given in ref. BabaeiVelni:2016qea . This
fact is the second motivation that encourages us to investigate this symmetry
for $O_{T^{2}}$, i.e., we search for an $SL(2,R)$ invariant form of $T\bar{T}$
operator.

It has been shown in refs. Aschieri:2008ns ; Aschieri:2013nda  that for each
non-linear electrodynamic action, as a function of some Lorentz invariant
variables, the variation of the action with respect to an $SL(2,R)$ invariant
parameter is also invariant under this transformation, i.e.,

$\frac{\partial S}{\partial\lambda}=-\frac{1}{\lambda}\left(S+\frac{1}{4}\int
d^{4}x\,GF\right)=-\frac{1}{\lambda}S_{inv},$ (8)

where $\lambda$ is a dimensionful parameter typically presents in a non-linear
theory and all indices in the product $GF$ are contracted. Here, $S_{inv}$ is
the invariant action under $SL(2,R)$ symmetry Gaillard:1981rj . Thus from
${O}_{T^{2}}=8\frac{\partial L_{\lambda}}{\partial\lambda}$ we have

$\mathcal{O}_{T^{2}}=\int d^{4}x\,{O}_{T^{2}}=-\frac{8}{\lambda}S_{inv},$ (9)

which implies the $T\bar{T}$ operator inherits $SL(2,R)$ symmetry from the EM
tensor. We show that eq. (9) works for non-linear electrodynamic theories at
any order of coupling expansion in $D\\!=\\!4$ and also for $D\\!=\\!2n$ with
$r=1/n$ in eq. (1).

The non-linear electrodynamic theories emerge as the generalization of Maxwell
theory that depend only on fields $F$ not their derivatives. In fact, it was
demonstrated that the Maxwell Lagrangian is the leading-order term in the
expansion in $F$ of nonlinear electrodynamic theories. These actions satisfy
the Noether-Gaillard-Zumino (NGZ) identity Gaillard:1981rj , however they are
not invariant under the $S$-duality transformation while the equations of
motion and EM tensor are Green:1996qg . In four dimensions, we consider the
$S$-duality transformations in extended version of the Maxwell theory in the
presence of background fields, and then find the corresponding $T\bar{T}$
operators in different orders of coupling constant. Also we reconstruct the
$T\bar{T}$ operator in a manifestly $SL(2,R)$ invariant form.

It has been shown in refs. Green:1996qg ; BabaeiVelni:2016qea  that the EM
tensor of a non-linear theory in which dilaton and axion fields are added to
BI theory, is invariant under the $SL(2,R)$ transformation ($S$-duality
relevant to string theory) at any order in $\alpha^{\prime}$. It was shown
that the EM tensor of such theories is constructed from $SL(2,R)$ structures
at any order in $\alpha^{\prime}$ BabaeiVelni:2016qea .

The other motivation about the implication of $T\bar{T}$ deformation that
could be overlooked is how it affects a supersymmetric theory. Although
several attempts have been made in refs. Baggio:2018rpv ; Chang:2018dge ;
Jiang:2019hux  for $T\bar{T}$ deformation in 2D supersymmetric theories, but
it would be of interest to investigate this concept in higher dimensions.
Specially, this proposal has been discussed in refs. Chang:2019kiu ;
Ferko:2019oyv  for non-linear BI supersymmetric theories, $\mathcal{N}=(2,2)$
models in $D\\!=\\!2$ as well as $\mathcal{N}=1$ models in $D\\!=\\!4$
dimensions. In this paper we will study the $T\bar{T}$ operator of
$\mathcal{N}=2$ supersymmetric theory with $U(1)$-type duality in $D\\!=\\!4$
that has been discussed in refs. Kuzenko:2000uh ; Broedel:2012gf .

The structure of this paper is organized as follows: in section 2, we review
the structure of $T\bar{T}$ operator for a chiral boson in $D\\!=\\!2$ and
show that the deformation of free theory yields a BI-type expression which
exactly agrees with the solution of flow equation obtained in different texts.
In section 3, we investigate the $T\bar{T}$ deformation for a general non-
linear BI theory of electrodynamics in arbitrary dimension $D\\!=\\!2n$ for
even $n$. We show that the deformed Lagrangian constructed from $T\bar{T}$
operator in $D\\!=\\!4$ is compatible with the expansion of non-linear BI
theories while it is inconsistent with related theory for $D\geq 8$. In
section 4, we investigate the $T\bar{T}$ deformation for non-linear extensions
of BI theory in a non-supersymmetric model as well as $\mathcal{N}=2$
supersymmetric theory. In fact, we will find $T\bar{T}$ operators which are
invariant under $SL(2,R)$ duality symmetry. Finally, the section 5 is devoted
to giving a brief summary of results and concluding with some outlook.

##  2 $T\bar{T}$ deformation in $D=2$

At the forefront of current research we shall consider the structure of
$T\bar{T}$ deformation for a 2D scalar field theory and show that how our
proposal, described by the relations (5)-(7) with $n=1$, works in $D=2$. The
results agree with the expression obtained in refs. Smirnov:2016lqw ;
Cavaglia:2016oda ; Bonelli:2018kik  which gives some confidence that the
deformed theory is well defined. Suppose that $\Phi(x)$ is a dynamical 0-form
scalar field (or a chiral boson) whose field strength is the vector field
$F_{\mu}=\partial_{\mu}\Phi(x)$. Henceforth, for convenience we express
function $\Phi(x)$ as the abbreviated form without functionality of $x$. The
initial Lagrangian in undeformed theory is given by the free Lagrangian
$L_{free}=-\frac{1}{2}\partial_{\alpha}\Phi\,\partial^{\alpha}\Phi$, such that
we expect the deformed Lagrangian to depend on the fields only through
$\partial_{\alpha}\Phi\,\partial_{\beta}\Phi$.

On the other hand, because of diffeomorphism invariance we also expect that
the deformed Lagrangian is only a function of two scalar variables $\lambda$
and $\partial_{\alpha}\Phi\,\partial^{\alpha}\Phi$, i.e. $L(\lambda,P_{1})$
where for later convenience we have defined
$P_{1}=\frac{1}{2}\partial_{\alpha}\Phi\,\partial^{\alpha}\Phi$. Using the
$T\bar{T}$ operator in eq. (5) with $n=1$ and the free Lagrangian, we can
deform this theory to first order of $\lambda$ and so on. In this respect,
from eqs. (4) and (3) one finds that $G_{\mu}(\lambda^{0})=\partial_{\mu}\Phi$
and the EM tensor is

$T_{\mu\nu}(\lambda^{0})=-\tfrac{1}{2}g_{\mu\nu}\partial_{\alpha}\Phi\,\partial^{\alpha}\Phi+\partial_{\mu}\Phi\,\partial_{\nu}\Phi,$
(10)

which can be easily verified that it is traceless and therefore, from (5) we
obtain

$O_{T^{2}}(\lambda^{0})=O_{0}=\tfrac{1}{2}(\partial_{\alpha}\Phi\,\partial^{\alpha}\Phi)^{2}.$
(11)

Substituting this operator in (6), the deformed Lagrangian of 2D QFT up to
order $\lambda$ is

$L^{\prime}_{\lambda}=-\tfrac{1}{2}\partial_{\alpha}\Phi\,\partial^{\alpha}\Phi+\tfrac{1}{16}\lambda\,(\partial_{\alpha}\Phi\,\partial^{\alpha}\Phi)^{2}.$
(12)

Now, following a similar prescription we deform the Lagrangian to the next
order of $\lambda$, that is again by starting from Lagrangian (12), the $G$
tensor simply is

$G_{\mu}(\lambda)=\left(1-\tfrac{1}{4}\lambda\,\partial_{\alpha}\Phi\,\partial^{\alpha}\Phi\right)\partial_{\mu}\Phi,$
(13)

and explicitly the EM tensor in this order becomes

$T_{\mu\nu}(\lambda)=\left(-\tfrac{1}{2}\partial_{\alpha}\Phi\,\partial^{\alpha}\Phi+\tfrac{1}{16}\lambda(\partial_{\alpha}\Phi\,\partial^{\alpha}\Phi)^{2}\right)\mathit{g}_{\mu\nu}+\left(1-\tfrac{1}{4}\lambda\,\partial_{\alpha}\Phi\,\partial^{\alpha}\Phi\right)\,\partial_{\mu}\Phi\,\partial_{\nu}\Phi\,.$
(14)

Thus, the contribution of (12) to $T\bar{T}$ operator in the form
$O_{T^{2}}(\lambda)=O_{0}+O_{1}$ is as follows

$O_{T^{2}}(\lambda)=\tfrac{1}{2}(\partial_{\alpha}\Phi\,\partial^{\alpha}\Phi)^{2}-\tfrac{1}{4}\lambda\,(\partial_{\alpha}\Phi\,\partial^{\alpha}\Phi)^{3},$
(15)

where according to relation (7) this operator provides for us the Lagrangian
of order $\lambda^{2}$

$L^{\prime\prime}_{\lambda}=-\tfrac{1}{2}\partial_{\alpha}\Phi\,\partial^{\alpha}\Phi+\tfrac{1}{16}\lambda(\partial_{\alpha}\Phi\,\partial^{\alpha}\Phi)^{2}-\tfrac{1}{64}\lambda^{2}(\partial_{\alpha}\Phi\,\partial^{\alpha}\Phi)^{3}.$
(16)

Iterating these steps respectively, the EM tensor to the order $\lambda^{2}$
is given by

$\displaystyle T_{\mu\nu}(\lambda^{2})$ $\displaystyle=$
$\displaystyle\left(-\tfrac{1}{2}\partial_{\alpha}\Phi\partial^{\alpha}\Phi+\tfrac{1}{16}\lambda(\partial_{\alpha}\Phi\,\partial^{\alpha}\Phi)^{2}-\tfrac{1}{64}\lambda^{2}(\partial_{\alpha}\Phi\,\partial^{\alpha}\Phi)^{3}\right)\mathit{g}_{\mu\nu}$
(17)
$\displaystyle\,+\,\left(1-\tfrac{1}{4}\lambda\,\partial_{\alpha}\Phi\,\partial^{\alpha}\Phi+\tfrac{3}{32}\lambda^{2}(\partial_{\alpha}\Phi\,\partial^{\alpha}\Phi)^{2}\right)\partial_{\mu}\Phi\,\partial_{\nu}\Phi,$

which is not traceless and have a contribution in the second term of
$T\bar{T}$ operator in (5). The operator to order $\lambda^{2}$, i.e.
$O_{T^{2}}(\lambda^{2})=O_{0}+O_{1}+O_{2}$, is given by

$O_{T^{2}}(\lambda^{2})=\tfrac{1}{2}(\partial_{\alpha}\Phi\,\partial^{\alpha}\Phi)^{2}-\tfrac{1}{4}\lambda(\partial_{\alpha}\Phi\,\partial^{\alpha}\Phi)^{3}+\tfrac{1}{256}(34-\frac{4}{n})\lambda^{2}(\partial_{\alpha}\Phi\,\partial^{\alpha}\Phi)^{4}.$
(18)

Thus, if we set $n\\!=\\!1$, the deformed Lagrangian to order $\lambda^{3}$ is
given by $L^{\prime\prime\prime}_{\lambda}=L_{free}+{1}/{8}\int
O_{0}\,d\lambda+{1}/{8}\int O_{1}\,d\lambda+{1}/{8}\int O_{2}\,d\lambda$ as

$L^{\prime\prime\prime}_{\lambda}=-\tfrac{1}{2}\partial_{\alpha}\Phi\,\partial^{\alpha}\Phi+\tfrac{1}{16}\lambda(\partial_{\alpha}\Phi\,\partial^{\alpha}\Phi)^{2}-\tfrac{1}{64}\lambda^{2}(\partial_{\alpha}\Phi\,\partial^{\alpha}\Phi)^{3}+\tfrac{5}{1024}\lambda^{3}(\partial_{\alpha}\Phi\,\partial^{\alpha}\Phi)^{4}.$
(19)

Nonetheless, after not so hard calculations we found that the resultant
deformed Lagrangian to all orders is equivalent to the expansion of a BI type
Lagrangian

$L_{BI}=\frac{2}{\lambda}\left(1-\sqrt{1+\frac{1}{2}\lambda\,\partial_{\alpha}\Phi\,\partial^{\alpha}\Phi}\right).$
(20)

This Lagrangian is exactly equal to the solution of flow equation described by
the deformation operator calculated in refs. Cavaglia:2016oda ;
Bonelli:2018kik  but with the deformation constant $t=\lambda/4$ and overall
minus sign, because their 2D free theory is described by
$L_{free}=\frac{1}{2}\partial_{\alpha}\Phi\,\partial^{\alpha}\Phi$. Here, the
difference in the minus sign refers to our convention for electrodynamic
theories.

Due to this fact, we expect that there is a similar shift in the levels of
energy in the spectrum. According to the perturbative method in (5), we can
determine the energy spectrum for different orders of 2D action. It has been
shown in ref. Brennan:2020dkw  that the $T\bar{T}$ deformation of 2D QFT on
AdS2 background is well-defined and solvable at the quantum level.

The flow equations of the energy spectrum for a free scalar field has been
derived perturbatively in ref. Brennan:2020dkw , in analogy with the flat
space case Rosenhaus:2019utc . The result of our calculation is given by

$E_{m}=-\frac{4\pi a}{\lambda}\left(1-\sqrt{1+\frac{\lambda E_{m}^{(0)}}{2\pi
a}}\right),$ (21)

where $a$ is a parameter of length dimension and $E_{m}^{(0)}$ is the ground
state energy. Note that our notation is different with convention in ref.
Brennan:2020dkw  by a factor $\frac{1}{4}$ in defining the parameter
$\lambda$. In the following, we obtain an expansion for the energy spectrum
given in eq. (21) in the limit $\lambda=0$

$E_{m}=\sum_{p}\lambda^{p}E_{m}^{(p)}=E_{m}^{(0)}-\lambda\frac{(E_{m}^{(0)})^{2}}{8\pi
a}+\lambda^{2}\frac{(E_{m}^{(0)})^{3}}{32\pi^{2}a^{2}}+\dots\,.$ (22)

The deformation of energy obtained from eq. (5) with $n=1$ gives some
confidence that the deformed theory from our proposal is well-defined and that
the contribution of term with coefficient $\frac{1}{n}$ starts from the order
$\lambda^{3}$.

##  3 $T\bar{T}$ operator in arbitrary $D=2n$

As the main purpose of this paper, in this section we attempt to study the
structure of $T\bar{T}$ operator in general $D=2n$ dimensional BI theory of
self-interacting form fields when $n$ is even. For notational convenience, we
introduce two Lorentz invariant variables as

$P_{1}=\frac{1}{2}F_{\mu_{1}\mu_{2}\dots\mu_{n}}F^{\mu_{1}\mu_{2}\dots\mu_{n}},\quad
P_{2}=\frac{1}{16}\left(F_{\mu_{1}\mu_{2}\dots\mu_{n}}\tilde{F}^{\mu_{1}\mu_{2}\dots\mu_{n}}\right)^{2},$
(23)

where $F_{\mu_{1}\mu_{2}\dots\mu_{n}}$ is the totally antisymmetric field
strength denoted in section 1 and its Hodge dual field is defined in a
standard manner by

$\tilde{F}^{\mu_{1}\mu_{2}\dots\mu_{n}}=\frac{1}{n!}\epsilon^{\mu_{1}\dots\mu_{n}\nu_{1}\dots\nu_{n}}F_{\nu_{1}\dots\nu_{n}},$
(24)

which satisfies the following duality rotation

$\tilde{\tilde{F}}_{\mu_{1}\dots\mu_{n}}=-F_{\mu_{1}\dots\mu_{n}}.$ (25)

The invariance under this rotation can be extended to the electromagnetic
fields interacting with the gravitational field, that does not transform under
duality in the Einstein frame. One can write any non-linear electrodynamic
Lagrangian as a function of these local Lorentz invariants Buratti:2019cbm .
For example, the 4D BI theory of electrodynamics is described by the
Lagrangian $L_{BI}=1-\sqrt{-det(\eta_{\mu\nu}+F_{\mu\nu})}$ where
$\eta_{\mu\nu}$ is the Minkowski metric. At the lowest order of the expansion
of $L_{BI}$ we have the free Maxwell theory denoted by
$L_{free}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}=-\frac{1}{2}P_{1}$. The
generalization of this Lagrangian in $D\\!=\\!2n$ dimensions is also
represented by Gaillard:1997rt ; Buratti:2019cbm

$L_{BI}=\frac{2}{\lambda n!}\left(1-\sqrt{1+\lambda
P_{1}-\lambda^{2}P_{2}}\right),$ (26)

where $\lambda$ is a coupling constant or an expansion parameter such that
$L_{free}=-\frac{1}{n!}P_{1}$ is at the order of $\lambda^{0}$.222We obtained
a similar BI-type theory in section 2 for 2D QFT with
$P_{1}\\!=\\!\frac{1}{2}\partial_{\alpha}\Phi\,\partial^{\alpha}\Phi$ and
$P_{2}=\frac{1}{16}\,\partial_{\mu}\Phi\,\epsilon^{\mu\alpha}\,\partial_{\alpha}\Phi\,\partial_{\nu}\Phi\,\epsilon^{\nu\beta}\,\partial_{\beta}\Phi,$
where $\epsilon^{\mu\nu}$ is the antisymmetric Levi-Civita symbol. Therefore,
due to the symmetry of partial derivatives $P_{2}=0$ and the BI Lagrangian is
$L_{BI}=\frac{2}{\lambda}\left(1-\sqrt{1+\lambda\,P_{1}}\right)$ for $n=1$
which is compatible with (20).

It has been shown in ref. Ferko:2019oyv  that the flow equation for the
deformed Lagrangian is proportional to $T\bar{T}$ operator in two dimensions.
In order to investigate this possibility in higher dimensional non-linear BI
theory, we use the definition of EM tensor given in (3). Thus, we obtain

$T_{\mu\nu}T^{\mu\nu}=DL^{2}+\frac{2}{(n-1)!}\,(F^{\mu}G_{\mu})\,L+\frac{1}{[(n-1)!]^{2}}(F^{\mu}G^{\nu})(F_{\mu}G_{\nu}).$
(27)

According to (26) the Lagrangian can be defined as a function of variables
$P_{1}$ and $P_{2}$, and the coupling constant $\lambda$, i.e.
$L_{BI}(P_{1},P_{2},\lambda)$, such that its expansion in the zero coupling
limit becomes

$L_{BI}\\!=\\!\frac{1}{n!}\\!\left[-P_{1}\\!+\\!\frac{\lambda}{4}(P_{1}^{2}\\!+\\!4P_{2})\\!-\\!\frac{\lambda^{2}}{8}P_{1}(P_{1}^{2}\\!+\\!4P_{2})\\!+\\!\frac{\lambda^{3}}{64}(5P_{1}^{2}\\!+\\!4P_{2})(P_{1}^{2}\\!+\\!4P_{2})\\!+\\!\mathcal{O}(F^{10})\right],$
(28)

therefore, to disclose the relation between $O_{T^{2}}$ and $\frac{\partial
L_{\lambda}}{\partial\lambda}$ both ${T_{\mu}}^{\mu}$ and
$T_{\mu\nu}T^{\mu\nu}$ should be functions of these parameters. From (3), the
trace of the EM tensor includes a term $(F_{\mu}G^{\mu})$ beside the
Lagrangian which is also a function of $P_{1}$, $P_{2}$, and $\lambda$. So
from this expression one finds that ${T_{\mu}}^{\mu}$ satisfies the following
equation

$\frac{1}{D}{T_{\mu}}^{\mu}=-\lambda\frac{\partial
L_{\lambda}}{\partial\lambda}.$ (29)

On the other hand, the first term in $O_{T^{2}}$, i.e. $T_{\mu\nu}T^{\mu\nu}$,
have a term described by a tensor $K_{2}=(F^{\mu}G^{\nu})(F_{\mu}G_{\nu})$.
The most striking feature which distinguishes our calculation for lower
dimensional studies in $T\bar{T}$ deformation, is the presence of tensor
$K_{2}$ in this operator. It can be verified that $K_{2}$ is written exactly
as a function of $P_{1}$ and $P_{2}$ in $D\\!=\\!2$ and $D\\!=\\!4$ but in
higher dimensions $D>4$ this does not happen. We postpone this discussion to
the next subsections. In the context of duality invariant formalism in
Gibbons:1995ap ; Gaillard:1997rt , more details about the most general Lorentz
invariant Lagrangian can be found in refs. Buratti:2019cbm ; Buratti:2019guq .

In the rest of this section we consider the proposal

${O}_{T^{2}}\equiv
T_{\mu\nu}T^{\mu\nu}-\frac{1}{n}{T_{\mu}}^{\mu}{T_{\nu}}^{\nu}=8\frac{\partial
L_{\lambda}}{\partial\lambda},$ (30)

in $D\geq 4$ spacetime dimensions with more details. We show that the
coefficient $r$ in four dimensions is equal to 1/2, however in $D>4$, the
tensor $K_{2}$ precludes this possibility. We will also obtain a general form
of deformed Lagrangian in arbitrary $D\\!=\\!2n$ dimensions at each order of
$\lambda$.

###  3.1 $T\bar{T}$ deformation in $D\\!=\\!4$

In 4D where $n=2$, we have a 2-form field strength $F_{\mu\nu}$ and the
Lorentz invariant variables are

$P_{1}=\frac{1}{2}F_{\mu\nu}F^{\mu\nu},\quad
P_{2}=\frac{1}{16}\left(F_{\mu\nu}\tilde{F}^{\mu\nu}\right)^{2}.$ (31)

Using the expansion in (28) for $\lambda$ and the definition (4), we compute
the tensor $G_{\mu\nu}$ as

$\displaystyle G_{\mu\nu}$ $\displaystyle=$ $\displaystyle
F_{\mu\nu}-\lambda(\tfrac{1}{8}\tilde{F}_{\mu\nu}F_{\alpha\beta}\tilde{F}^{\alpha\beta}+\tfrac{1}{2}F_{\mu\nu}P_{1})+\lambda^{2}\bigl{(}\tfrac{1}{16}\tilde{F}_{\mu\nu}F_{\alpha\beta}\tilde{F}^{\alpha\beta}P_{1}+\tfrac{1}{8}F_{\mu\nu}(3P_{1}^{2}+4P_{2})\bigr{)}$
(32) $\displaystyle-$
$\displaystyle\lambda^{3}\bigl{(}\tfrac{1}{64}\tilde{F}_{\mu\nu}F_{\alpha\beta}\tilde{F}^{\alpha\beta}(3P_{1}^{2}+4P_{2})+\tfrac{1}{16}F_{\mu\nu}P_{1}(5P_{1}^{2}+12P_{2})\bigr{)}+...\,,$

then, by substituting this result in eq. (3), the EM tensor becomes

$\displaystyle T_{\mu\nu}$ $\displaystyle=$ $\displaystyle
F_{\mu}{}^{\alpha}F_{\nu\alpha}-\tfrac{1}{2}P_{1}\mathit{g}_{\mu\nu}-\lambda(\tfrac{1}{2}F_{\mu}{}^{\alpha}F_{\nu\alpha}P_{1}-\tfrac{1}{8}P_{1}^{2}\mathit{g}_{\mu\nu})$
$\displaystyle+$
$\displaystyle\lambda^{2}\bigl{(}\tfrac{1}{8}F_{\mu}{}^{\alpha}F_{\nu\alpha}(3P_{1}^{2}+4P_{2})-\tfrac{1}{16}P_{1}^{3}\mathit{g}_{\mu\nu}\bigr{)}$
$\displaystyle-$
$\displaystyle\lambda^{3}\bigl{(}\tfrac{1}{16}F_{\mu}{}^{\alpha}F_{\nu\alpha}P_{1}(5P_{1}^{2}+12P_{2})-(\tfrac{5}{128}P_{1}^{4}-\tfrac{1}{8}P_{2}^{2})\mathit{g}_{\mu\nu}\bigr{)}+...\,,$

where the elliptic terms in both equations represent higher order powers of
$\lambda$ expansion. We have also checked that the trace of (3.1) is
compatible with (29) for $D\\!=\\!4$.

The tensor $K_{2}$ in 4D is written in the form

$K_{2}=F^{\mu\alpha}{G^{\nu}}_{\alpha}\,F_{\mu\beta}{G_{\nu}}^{\beta},$ (34)

where from the relation (32) we can rewrite it as

$\displaystyle K_{2}$ $\displaystyle=$ $\displaystyle
K_{0}-\lambda(K_{0}P_{1}+2P_{1}P_{2})+\lambda^{2}\bigl{(}K_{0}(P_{1}^{2}+P_{2})+P_{2}(2P_{1}^{2}+P_{2})\bigr{)}$
(35) $\displaystyle-$
$\displaystyle\lambda^{3}\Bigl{(}K_{0}P_{1}(P_{1}^{2}+2P_{2})+P_{1}P_{2}\bigl{(}2P_{1}^{2}+3P_{2}\bigr{)}\Bigr{)}+...\,,$

where $K_{0}=F^{\mu\alpha}{F^{\nu}}_{\alpha}\,F_{\mu\beta}{F_{\nu}}^{\beta}$.
If we attempt to redefine $K_{0}$ in terms of Lorentz variables $P_{1}$ and
$P_{2}$ then $T_{\mu\nu}T^{\mu\nu}$ can be expressed only as a function of
$P_{1},P_{2}$, and $\lambda$. It can be written in implicit form
$K_{0}=2(P_{1}^{2}+2P_{2})$. From this expression it is clear that we can test
our suggestion (30) and determine the coefficient $r$ for each order of
$\lambda$ expansion. The result of calculations has forcefully stressed that
$r=1/2$ which is compatible with the results obtained in ref. Conti:2018jho .

Following the perturbative method mentioned in section 1 by eqs. (5) and (6),
we can deform the Maxwell action at each order of $\lambda$ in 4D and give the
expansion of Maxwell-BI theory. This is reminiscent of the fact that we can
expect a similar deformation for the energy spectrum. Thus, one can find a
solvable energy spectrum in $D\\!=\\!4$ according to (28). In other words,
applying the $T\bar{T}$ deformation to the energy spectrum of the Maxwell
action yields the one for BI theory which in spite of Maxwell theory, its
energy spectrum is finite and solvable. It has been shown in ref.
Rasheed:1997ns  that for purely electric configuration in flat space, the
total self-energy of the point charge is finite and integrable. For the
Lagrangian

$L_{BI}={1\over\lambda}\left\\{1-\sqrt{1-\lambda\mbox{\boldmath
E}^{2}}\right\\},$ (36)

there is an upper bound on the electric field strength $\boldmath E$

$\left|\mbox{\boldmath E}\right|\leq{1\over\sqrt{\lambda}}.$ (37)

The total self-energy of the point charge with
$E_{r}={Q\over\sqrt{r^{4}+\lambda Q^{2}}}$ is

${\mathcal{E}}={1\over 16\pi}\int d^{3}xT_{00}={1\over 16\pi}\int
d^{3}x{1\over\lambda r^{2}}\left(\sqrt{r^{4}+\lambda Q^{2}}-r^{2}\right).$
(38)

Integrating this energy yields

${\cal E}={2Q^{2}\over 12}\int_{0}^{\infty}{dr\over\sqrt{r^{4}+\lambda
Q^{2}}}={(\pi Q)^{3\over 2}\over
12\sqrt{\lambda^{1/2}}\,\Gamma\\!\left({3\over 4}\right)^{2}}.$ (39)

Therefore, the BI theory of electrodynamics succeeded in its original goal of
providing a model for point charges with finite self-energy. Note that in the
limit $\lambda\rightarrow 0$, Maxwell theory is reproduced and the self-energy
diverges.

###  3.2 $T\bar{T}$ deformation in $D=8$

Following the general strategy used in the previous subsection, we define two
Lorentz invariant parameters $P_{1}$ and $P_{2}$ in terms of 4-form field
strengths as follows

$P_{1}=\frac{1}{2}F_{\mu\nu\rho\sigma}F^{\mu\nu\rho\sigma},\quad
P_{2}=\frac{1}{16}\left(F_{\mu\nu\rho\sigma}{\tilde{F}}^{\mu\nu\rho\sigma}\right)^{2},$
(40)

where from the expansion (28) and definition (4), the $G$ tensor is given by

$G_{\mu\nu\rho\sigma}=\frac{1}{12}F_{\mu\nu\rho\sigma}-\tfrac{1}{96}\lambda(4F_{\mu\nu\rho\sigma}P_{1}+\tilde{F}_{\mu\nu\rho\sigma}F^{\alpha\beta\epsilon\delta}\tilde{F}_{\alpha\beta\epsilon\delta})+...\,.$
(41)

Substituting the above results in eq. (3) yields the deformed EM tensor of 8D
electrodynamic theory as

$\displaystyle T_{\mu\nu}$ $\displaystyle=$
$\displaystyle\tfrac{1}{6}F_{\mu}{}^{\alpha\beta\gamma}F_{\nu\alpha\beta\gamma}-\tfrac{1}{24}P_{1}\mathit{g}_{\mu\nu}$
(42) $\displaystyle+$
$\displaystyle\lambda\biggl{(}\tfrac{1}{384}(4P_{1}^{2}+16P_{2})\mathit{g}_{\mu\nu}-\tfrac{1}{48}(4F_{\mu}{}^{\alpha\beta\gamma}F_{\nu\alpha\beta\gamma}P_{1}+F^{\eta\rho\sigma\delta}{\tilde{F}}_{\eta\rho\sigma\delta}{\tilde{F}}_{\mu\alpha\beta\gamma}F_{\nu}{}^{\alpha\beta\gamma})\biggr{)}$
$\displaystyle+$ $\displaystyle\mathcal{O}(\lambda^{2}).$

Without a doubt it can be found that the trace of this tensor satisfies the
condition (29) in $D\\!=\\!8$ dimensions. In this case the tensor $K_{2}$ in
the last term of eq. (27) can be written as

$K_{2}=\tfrac{1}{36}K_{0}-\tfrac{1}{288}\lambda
P_{1}(8K_{0}+3P_{1}^{2}+8P_{2})+...\,,$ (43)

where
$K_{0}=F^{\mu\alpha\beta\gamma}{F^{\nu}}_{\alpha\beta\gamma}\,\,F_{\mu\rho\sigma\delta}{F_{\nu}}^{\rho\sigma\delta}$.
Thus, the quadratic power of EM tensor becomes

$T_{\mu\nu}T^{\mu\nu}=\frac{1}{72}(2K_{0}-P_{1}^{2})-\frac{1}{72}\lambda(2K_{0}P_{1}-P_{1}^{3})+...\,.$
(44)

In order to be able to express the covariant variables $T_{\mu\nu}T^{\mu\nu}$
and $(T_{\mu}{}^{\mu})^{2}$ as a function of Lorentz variables we should write
$K_{0}$ in terms of these parameters, however in spite of four dimensionl
theory, it is not possible to do this in $D=8$. This possibility has been
considered in refs. Buratti:2019cbm ; Buratti:2019guq  and it is shown that
$K_{0}$ recasts as

$K_{0}=\frac{1}{4}P_{1}^{2}+\frac{1}{2}P_{2}+\frac{9}{8}K_{3},$ (45)

where
$K_{3}=(F^{\mu\nu}{F^{\alpha\beta}})\,\,(F_{\mu\nu}{F_{\alpha\beta}})=F^{\mu\nu\gamma\eta}{F^{\alpha\beta}}_{\gamma\eta}\,\,F_{\mu\nu\epsilon\delta}{F_{\alpha\beta}}^{\epsilon\delta}$
is an independent tensor in $D=8$ Buratti:2019cbm . Therefore, we are unable
to compare the variation of the Lagrangian (26) with respect to $\lambda$,
which only depends on $P_{1}$ and $P_{2}$, with $O_{T^{2}}$ operator through
eq. (30). In this respect, we cannot write the variation of 8D BI action in
terms of $O_{T^{2}}$ like in (9) (up to $\mathcal{O}(F^{4})$ and
$\mathcal{O}(F^{6})$).

Here, we apply the prescription explained in section 1 to deform the
Lagrangian in favor of $T\bar{T}$ operator. Starting from the free Lagrangian
in $D=8$, i.e. $L_{free}=-\frac{1}{4!}P_{1}=-\frac{1}{2\times
4!}F_{\mu\nu\alpha\beta}F^{\mu\nu\alpha\beta}$, and using (44), we can deform
the free Lagrangian to the first order in $\lambda$ with operator
$O_{T^{2}}=O_{0}=\frac{1}{72}(2K_{0}-P_{1}^{2})$ as the following form

$L^{\prime}_{\lambda}=L_{free}+\frac{1}{8}\int
O_{0}d\lambda=-\frac{1}{4!}P_{1}+\frac{1}{576}\lambda(2K_{0}-P_{1}^{2}).$ (46)

Now, one can employ this Lagrangian and define the EM tensor to obtain
$O_{T^{2}}(\lambda)=O_{0}+O_{1}$ in which

$\displaystyle O_{1}$ $\displaystyle=$
$\displaystyle\frac{\lambda}{72}\bigg{(}-\frac{1}{3}(F_{\alpha}F^{\delta})(F^{\alpha}F^{\beta})(F_{\beta}F_{\delta})-(F^{\alpha}F^{\beta})(F^{\delta}F^{\zeta})(F_{\beta\delta}F_{\alpha\zeta})$
(47) $\displaystyle\hskip
28.45274pt+\frac{1}{3}(F_{\alpha}F_{\beta})(F^{\alpha}F^{\beta})(FF)-\frac{1}{48}(FF)^{3}\bigg{)},$

where all pairs are contractions of 4-form field strengths. If we iterate this
procedure the next order in $\lambda$ is constructed. So from (7), the
Lagrangian upto $\mathcal{O}(F^{6})$ deforms as

$\displaystyle L^{\prime\prime}_{\lambda}$ $\displaystyle=$
$\displaystyle-\frac{1}{48}(FF)+\frac{\lambda}{288}\bigg{(}(F_{\alpha}F_{\beta})(F^{\alpha}F^{\beta})-\frac{1}{8}(FF)^{2}\bigg{)}-\tfrac{\lambda^{2}}{1152}\bigg{(}\frac{1}{3}(F_{\alpha}F^{\delta})(F^{\alpha}F^{\beta})(F_{\beta}F_{\delta})$
(48)
$\displaystyle+(F^{\alpha}F^{\beta})(F^{\delta}F^{\zeta})(F_{\beta\delta}F_{\alpha\zeta})-\frac{1}{3}(F_{\alpha}F_{\beta})(F^{\alpha}F^{\beta})(FF)+\frac{1}{48}(FF)^{3}\bigg{)}.$

Since the Lagrangian $L^{\prime\prime}_{\lambda}$ is derived from $O_{T^{2}}$
operator of order $\lambda$ and the fact that the second term with
$\frac{1}{n}$ coefficient in $T\bar{T}$ deformation comes from $\lambda^{2}$
order, this term does not contribute in the Lagrangian (48). Thus, we should
consider the next order in $O_{T^{2}}$ operator to examine the contribution of
this term, i.e., $O_{T^{2}}(\lambda^{2})=O_{0}+O_{1}+O_{2}$. Due to this fact,
we obtain

$\displaystyle O_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{27648}\lambda^{2}\bigg{(}\tfrac{88}{3}(F_{\alpha}F_{\beta})(F^{\alpha}F^{\beta})(F^{\epsilon}F^{\delta})(F_{\epsilon}F_{\delta})-24\,(F_{\alpha}F^{\delta})(F^{\alpha}F^{\beta})(F_{\beta}F_{\delta})(FF)$
$\displaystyle+$
$\displaystyle\tfrac{46}{3}(F_{\alpha}F_{\beta})(F^{\alpha}F^{\beta})(FF)^{2}-\tfrac{5}{8}(FF)^{4}-\tfrac{64}{3}(F^{\alpha}F^{\beta})(F_{\beta}F^{\epsilon})(F_{\epsilon}F^{\delta})(F_{\alpha}^{\gamma}{}F_{\gamma\delta})$
$\displaystyle-$ $\displaystyle
72\,(F^{\alpha}F^{\beta})(F^{\epsilon}F^{\delta})(F_{\beta\epsilon}F_{\alpha\delta})(FF)+144\,(F^{\alpha}F^{\beta})(F^{\epsilon}F_{\beta})(F^{\delta}F^{\gamma})(F_{\gamma\alpha}F_{\delta\epsilon})$
$\displaystyle+$ $\displaystyle
32\,(F^{\alpha}F^{\beta})(F^{\epsilon}F^{\delta})(F^{\gamma}F^{\kappa})(F_{\beta\epsilon\gamma}F_{\alpha\delta\kappa})+144\,(F^{\alpha}F^{\beta})(F^{\epsilon}F^{\delta})(F_{\beta}{}^{\gamma}{}F_{\alpha}{}^{\kappa})(F_{\gamma}{}_{\epsilon}F_{\kappa\delta})\bigg{)}$
$\displaystyle\\!\\!\\!\\!-\\!\\!\\!\\!$
$\displaystyle\frac{1}{n}\lambda^{2}\bigg{(}\tfrac{1}{1296}(F_{\alpha}F_{\beta})(F^{\alpha}F^{\beta})(F^{\epsilon}F^{\delta})(F_{\epsilon}F_{\delta})-\tfrac{1}{5184}(F_{\alpha}F_{\beta})(F^{\alpha}F^{\beta})(FF)^{2}+\tfrac{1}{82944}(FF)^{4}\bigg{)}.$

Finally, using this result we are ready to generate the Lagrangian of order
$\lambda^{3}$ by taking $n=4$, which leads to

$L^{\prime\prime\prime}_{\lambda}=L^{\prime\prime}_{\lambda}+\frac{1}{8}\int
O_{2}\,d\lambda.$ (50)

It is worth to mention some relevant issues about the $T\bar{T}$ deformation
in diverse dimension and holographic interpretation before closing this
section;

  * •

The generalization to higher dimensions is analogous to the computations done
for 4D and 8D theories. We find a general form of the deformed Lagrangian at
each order of deformation parameter in $D\\!=\\!2n$-dimensional theory of
electrodynamic fields. For instance, the deformation at the order of $\lambda$
is given by

$L_{\lambda}\sim\lambda\left((F_{\alpha}F_{\beta})(F^{\alpha}F^{\beta})-\frac{1}{D}(FF)^{2}\right),$
(51)

and in the next order has the form

$\displaystyle L_{\lambda}$ $\displaystyle\sim$
$\displaystyle\lambda^{2}\bigg{(}-2(F_{\alpha}F^{\gamma})(F^{\alpha}F^{\beta})(F_{\beta}F_{\gamma})-(D-2)(F_{\alpha\gamma}F_{\beta\eta})(F^{\alpha}F^{\beta})(F^{\gamma}F^{\eta})$
(52)
$\displaystyle+2(F_{\alpha}F_{\beta})(F^{\alpha}F^{\beta})(FF)-\tfrac{1}{D}(FF)^{3}\bigg{)},$

where all the field strengths are $n$-form fields which their indices are
contracted appropriately and the equality is obtained by a numerical
coefficient as a function of spacetime dimension.

  * •

From the holography point of view, it has been shown Henningson:1998gx  that
the anomaly of conformal closed algebra of the CFT is compatible with the
$\mathcal{A}$ anomaly of the $D\\!+\\!1$-dimensional bulk theory under a
conformal transformation. For example, in the case of $D\\!=\\!2$ the finite
term of the regularized action is given by $\mathcal{A}=-\frac{c}{24\pi}R$
with $c=3\ell/2G$ which agrees with the value of the conformal anomaly $c$ as
computed in Brown:1986nw  by considering the asymptotic symmetry algebra of
$AdS_{3}$ space, or in $D\\!=\\!4$ the anomaly
$\mathcal{A}=-\frac{N^{2}}{\pi^{2}}(E_{4}+I_{4})$ is compatible with the
$\mathcal{N}=4$ superconformal $SU(N)$ gauge theory with
$a=c=\frac{1}{4}(N^{2}-1)$ in the large-$N$ limit. Similarly in $D\\!=\\!6$
the anomaly is proportional to $N^{3}$ where $N$ is the number of coincident
$M5$ branes in ($0,2$) superconformal theory Bastianelli:2000hi .
Obviously the field content of QFTs are not the same and there are different
matter fields in each sector. However, the $T\bar{T}$ operators obtained in
this section are only related to the non-linear BI theories of $p$-form fields
that may live on a single brane and are not necessary a CFT with a closed
conformal algebra of central charges $a=c$, so it seems that there is no
justification to compare our results to that derived from AdS/CFT in (1).

  * •

Since the duality-symmetric BI theory in $D\\!=\\!4$ and its chiral 2-form
counterpart in $D\\!=\\!6$ are related by dimensional reduction of the latter,
one can conclude that the $T\bar{T}$ deformation of the free chiral 2-form
theory is the same as in $D\\!=\\!4$ for the BI case. In other words, for the
6D chiral 2-form theories there is exactly the same number "two" of
independent Lorentz-invariants as in $D\\!=\\!4$ (e.g. like $P_{1}$ and
$P_{2}$ in this paper) which can be used to construct their consistent non-
linear generalizations, which are always related to those in 4D theory
Bandos:2020hgy .
For the non-chiral 2-form field, the trace of the EM tensor is non-zero, while
for the chiral (self-dual) field it is zero. By the way, this indicates that
the free chiral 2-form theory is conformal Bandos:2020jsw . However, if we try
to get a non-linear chiral p-form theory upon deformation, we should give a
prescription of how the self-duality condition gets deformed under $T\bar{T}$
operator.
On the other hand, chiral theories require auxiliary fields in their actions
only to ensure manifest Lorentz-invariance. Without the use of the auxiliary
fields the chiral theory is still Lorentz-invariant, but not manifestly. For
example, in the $T\bar{T}$ deformation related to the M5-brane, one should
start from a fully-fledged action for the free chiral 2-form and compare it
with the fully-fledged action for the M5-brane. Such actions have a single
scalar auxiliary field to ensure manifest Lorentz-invariance. But this field
can be gauged away directly in the action. Then the resulting action will be
non-manifestly Lorentz invariant but will contain only the physical 2-form
field whose equation of motion produces the self-duality condition for its
field strength and the required traceless EM tensor.

##  4 $T\bar{T}$ operator in BI type theories and S-duality

As discussed in the previous section, according to theorems in refs.
Gaillard:1997rt ; Aschieri:2008ns , the non-linear electrodynamic theories
satisfying the NGZ identity are called $S$-dual which are invariant under
electromagnetic transformation at the level of the equations of motion, the EM
tensor and the S-matrix. This $S$-dual symmetry can be enhanced to invariance
under $SL(2,R)$ transformation. It is shown that this is true for any non-
linear theory of electrodynamics. The BI theories like S-dual electrodynamic
theories, are a particular class of solutions in these theorems which have
exact closed forms Rasheed:1997ns . In this section we will consider the
$T\bar{T}$ deformation for non-supersymmetric BI-type theory, as well as
$\mathcal{N}=2$ supersymmetric model.

###  4.1 Non-supersymmetric model in $D\\!=\\!4$

The Lagrangian of an electrodynamic BI theory in the presence of a dilatonic
field in Einstein frame is given by the following density

$L_{BI}={\lambda}^{-1}\bigg{[}1-\sqrt{-\det(\eta_{\mu\nu}+\lambda^{\frac{1}{2}}e^{-\phi_{0}/2}F_{\mu\nu})}\,\bigg{]},$
(53)

where the dilaton function $e^{\phi_{0}/2}$ is regarded as an effective gauge
coupling constant. The expansion of this Lagrangian in the limit $\lambda=0$
becomes333Our convention here for $Tr[F^{2}]$ is $F_{\mu\nu}F^{\nu\mu}$ and
similarly for higher powers of $F_{\mu\nu}$.

$\displaystyle L_{BI}$ $\displaystyle=$
$\displaystyle\frac{e^{-\phi_{0}}}{4}Tr[F^{2}]+\lambda\frac{e^{-2\phi_{0}}}{8}\bigg{(}Tr[F^{4}]-\frac{1}{4}Tr[F^{2}]^{2}\bigg{)}$
$\displaystyle+$
$\displaystyle\lambda^{2}\frac{e^{-3\phi_{0}}}{12}\bigg{(}Tr[F^{6}]-\frac{3}{8}Tr[F^{2}]Tr[F^{4}]+\frac{1}{32}Tr[F^{2}]^{3}\bigg{)}+\dots\,.$

After computing the antisymmetric tensor $G$ from (4), one can find the EM
tensor as

$\displaystyle T_{\mu\nu}$ $\displaystyle=$ $\displaystyle
e^{\Phi_{0}}\bigg{(}\frac{1}{4}\mathit{g}_{\mu\nu}Tr[F^{2}]-(F^{2})_{\mu\nu}\bigg{)}+\lambda
e^{-2\phi_{0}}\bigg{(}\frac{1}{4}(F^{2})_{\mu\nu}Tr[F^{2}]-(F^{4})_{\mu\nu}-\frac{1}{32}\mathit{g}_{\mu\nu}Tr[F^{2}]$
(55) $\displaystyle+$
$\displaystyle\frac{1}{8}\mathit{g}_{\mu\nu}Tr[F^{4}]\bigg{)}+\lambda^{2}e^{-3\phi_{0}}\bigg{(}\frac{1}{4}(F^{4})_{\mu\nu}Tr[F^{2}]-\frac{1}{32}(F^{2})_{\mu\nu}Tr[F^{2}]^{2}+\frac{1}{8}(F^{2})_{\mu\nu}Tr[F^{4}]$
$\displaystyle-$
$\displaystyle(F^{6})_{\mu\nu}+\frac{1}{384}\mathit{g}_{\mu\nu}Tr[F^{2}]^{3}-\frac{1}{32}\mathit{g}_{\mu\nu}Tr[F^{2}]Tr[F^{4}]+\frac{1}{12}\mathit{g}_{\mu\nu}Tr[F^{6}]\bigg{)}+\dots\,.$

Equipped with this EM tensor, we find the $T\overline{T}$ operator from eq.
(1) given by

$\displaystyle O_{T^{2}}^{[r]}$ $\displaystyle=$
$\displaystyle\frac{1}{16}e^{-2\phi_{0}}\bigg{(}[D-8-r(D-4)^{2}]\,Tr[F^{2}]^{2}+16\,Tr[F^{4}]\bigg{)}$
(56) $\displaystyle+$
$\displaystyle\frac{\lambda}{64}e^{-3\phi_{0}}\bigg{(}[12-D+r(D^{2}-12D+32)]\,Tr[F^{2}]^{3}$
$\displaystyle+4\,[D-20-r(D^{2}-12D+32)]\,Tr[F^{2}]\,Tr[F^{4}]+128\,Tr[F^{6}]\bigg{)}$
$\displaystyle+$
$\displaystyle\frac{\lambda^{2}}{3072}e^{-4\phi_{0}}\bigg{(}[7D-688-r(7D^{2}-112D+384)]\,Tr[F^{2}]^{4}$
$\displaystyle-24\,[3D-160+r(3D^{2}-48D+160)]\,Tr[F^{2}]^{2}\,Tr[F^{4}]$
$\displaystyle+48\,[D+16-r(D-8)^{2}]\,Tr[F^{4}]^{2}$
$\displaystyle+128\,[D-40-r(D^{2}-16D+16)]\,Tr[F^{2}]\,Tr[F^{6}]\bigg{)}+\dots\,.$

The zeroth-order terms, i.e. $\mathcal{O}(\lambda^{0})$ or
$\mathcal{O}(F^{4})$, is the $T\bar{T}$ operator of the free Maxwell theory.
This is the same for the BI action given in (4.1) at order
$\mathcal{O}(\lambda^{1})$. It is obvious that the Maxwell $T\bar{T}$ operator
is independent of $r$ in $D\\!=\\!4$. The same behaviour holds for the second-
order terms of $\mathcal{O}(\lambda^{1})$ in (1). In fact, the $r$ dependence
of $T\bar{T}$ operator of BI theory in $D\\!=\\!4$ starts at order
$\mathcal{O}(\lambda^{2})$.

On the other hand, due to the action (4.1) and the flow eqs. (8) and (9), we
can find the above $T\bar{T}$ operator in $D\\!=\\!4$ by setting
$r=\frac{1}{2}$ as follows

$\displaystyle O_{T^{2}}^{[1/2]}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\,e^{-2\Phi_{0}}\bigg{(}Tr[F^{2}]^{2}-4\,Tr[F^{4}]\bigg{)}$
(57) $\displaystyle-$
$\displaystyle\frac{1}{8}\,e^{-3\Phi_{0}}\bigg{(}Tr[F^{2}]^{3}-4\,Tr[F^{2}]\,Tr[F^{4}]\bigg{)}$
$\displaystyle-$
$\displaystyle\frac{3}{256}\,e^{-4\Phi_{0}}\bigg{(}3\,Tr[F^{2}]^{4}-8\,Tr[F^{2}]^{2}\,Tr[F^{4}]-16\,Tr[F^{4}]^{2}\bigg{)}$
$\displaystyle-$
$\displaystyle\frac{1}{256}\,e^{-5\Phi_{0}}\bigg{(}Tr[F^{2}]^{5}+8\,Tr[F^{2}]^{3}\,Tr[F^{4}]-48\,Tr[F^{2}]\,Tr[F^{4}]^{2}\bigg{)}+\dots\,,$

where we have set $\lambda=1$ without loss of generality and used the
following identity

$Tr[F^{6}]-\frac{3}{4}\,Tr[F^{2}]\,Tr[F^{4}]+\frac{1}{8}\,Tr[F^{2}]^{3}=0.$
(58)

###  4.2 $SL(2,R)$ invariant structure of 4D $T\bar{T}$ operator

In this subsection we review non-linear $SL(2,R)$ transformation of form
fields and then reconstruct the $T\bar{T}$ operator in an $SL(2,R)$ invariant
form. In order to find the $SL(2,R)$ structure of EM tensor of a non-linear
electrodynamic theory in four dimensions, we first consider the behavior of
field strengths $F_{\mu\nu}$ and $G_{\mu\nu}$ under transformation

$\tau\rightarrow\frac{a\tau+b}{c\tau+d},$ (59)

where $\tau$ is a complex scalar field defined by $\tau=C_{0}+ie^{-\phi_{0}}$.
In general, it is referred to an axion-dilaton field such that $C_{0}$ is an
axion field and $\phi_{0}$ is a dilaton.

The field strengths transform as a doublet $\mathcal{F}_{\mu\nu}$ under the
$SL(2,R)$ symmetry group as following Gibbons:1995ap

$\mathcal{F}_{\mu\nu}\rightarrow(\Lambda^{-1})^{T}\mathcal{F}_{\mu\nu},$ (60)

where $\Lambda=\left(\begin{array}[]{cc}a&b\\\ c&d\end{array}\right)$ is an
$SL(2,R)$ matrix. One can rewrite the Lagrangian in an $SL(2,R)$ invariant
form by using the matrix

$\mathcal{M}=e^{\phi_{0}}\left(\begin{array}[]{cc}|\tau|^{2}&-C_{0}\\\
-C_{0}&1\end{array}\right),$ (61)

which under S-duality transforms as
${\mathcal{M}}\rightarrow\Lambda\,{\mathcal{M}}\,\Lambda^{T}$.

Now, from the above consideration we construct an $SL(2,R)$ invariant
structure $\mathcal{F}^{T}\mathcal{M}\mathcal{F}$. By separating the
contribution of the axion coupling in the Lagrangian as $L=L^{\prime}+C_{0}z$,
that yields a decomposition in $G$ as
$G_{\mu\nu}=G^{{}^{\prime}}_{\mu\nu}-C_{0}\tilde{F}_{\mu\nu}$, one can find
the $SL(2,R)$ invariant structure in the form that the axion field does not
appear, i.e.

$(\mathcal{F}^{T})_{\mu}{}^{\rho}\mathcal{M}_{0}\,\mathcal{F}_{\nu\rho}=e^{-\phi_{0}}\,\tilde{F}_{\mu}{}^{\rho}\,\tilde{F}_{\nu\rho}+e^{\phi_{0}}\,G^{{}^{\prime}}_{\mu}{}^{\rho}\,G^{{}^{\prime}}_{\nu\rho}\,,$
(62)

where $z=1/4\,F^{\mu\nu}\tilde{F}_{\mu\nu}$ is a Lorentz invariant parameter
and $G^{\prime}$ comes from (4) for $L^{\prime}$. Assuming that the axion and
dilaton fields are constant and do not fluctuate, henceforth we use
$\mathcal{M}_{0}$ to denote this fact.

We are interested here to study the $T\bar{T}$ operator of some non-linear BI
electrodynamic theories and $\mathcal{N}=2$ supersymmetric BI type theory. The
equations of motion of these theories enjoy duality symmetries, however, the
corresponding Lagrangians are not invariant under the duality transformations.

According to (3), one can find the EM tensor of the Lagrangian $L$ in terms of
the field strengths $F$ and $G$. Using some identities that hold between some
trace structure of $F$’s and by analogy with the $SL(2,R)$ invariant structure
(62), the EM tensor and consequently the $T\bar{T}$ operator could be appeared
in terms of this $SL(2,R)$ invariant structure.

In the following, we would like to find the above operator in terms of
$SL(2,R)$ invariant structures. Considering the approach in ref.
BabaeiVelni:2016qea  that was applied to find the $SL(2,R)$ invariant form of
EM tensor for non-linear electrodynamic theories, the above $T\overline{T}$
operator can be found in the form that is manifestly $SL(2,R)$ invariant. We
should find the non-linear $SL(2,R)$ invariant structure (62) corresponding to
BI theory in terms of the field strengths $F$ and $G$. Therefore, the
calculation yields

$\displaystyle O_{T^{2}}^{[1/2]}$ $\displaystyle=$
$\displaystyle\frac{1}{4}(\mathcal{F}^{T})_{\mu}{}^{\rho}\mathcal{M}_{0}\,\mathcal{F}_{\rho\nu}(\mathcal{F}^{T})^{\mu\sigma}\mathcal{M}_{0}\,\mathcal{F}_{\sigma}{}^{\nu}-\frac{5}{64}{Tr(\mathcal{F}^{T}\mathcal{M}_{0}\,\mathcal{F})}^{2}$
(63)
$\displaystyle-\frac{1}{32}(\mathcal{F}^{T})_{\mu}{}^{\rho}\mathcal{M}_{0}\,\mathcal{F}_{\rho\nu}(\mathcal{F}^{T})^{\mu\sigma}\mathcal{M}_{0}\,\mathcal{F}_{\sigma}{}^{\nu}{Tr(\mathcal{F}^{T}\mathcal{M}_{0}\,\mathcal{F})}+\dots\,.$

As was mentioned earlier, adding the $T\bar{T}$ deformation to the free
Maxwell Lagrangian in four dimensions matches the $F^{4}$ terms of the BI
action expansion. On the other hand, it has been shown Garousi:2011vs  that
the leading order terms of the scattering amplitude of four gauge fields are
reproduced by the $F^{4}$ terms of the BI action. The on-shell BI action is
invariant under the linear S-duality up to $F^{4}$ terms Babaei-
Aghbolagh:2013hia ; Garousi:2017fbe . The $SL(2,R)$ invariant structure of
this amplitude agrees with eq. (63). It could be uniquely written eq. (63) in
the following appropriate trace terms

$O_{T^{2}}^{[1/2]}=\frac{1}{2}{Tr(\mathcal{F}^{T}\mathcal{M}_{0}\,\mathcal{F})}-\frac{1}{64}{Tr(\mathcal{F}^{T}\mathcal{M}_{0}\,\mathcal{F})}^{2}+\dots,$
(64)

where we have used the following identity that holds for $SL(2,R)$ invariant
structure of BI theory

$\displaystyle(\mathcal{F}^{T})_{\mu}{}^{\rho}\mathcal{M}_{0}\,\mathcal{F}_{\rho\nu}(\mathcal{F}^{T})_{\mu}{}^{\sigma}\mathcal{M}_{0}\,\mathcal{F}_{\sigma\nu}=2{Tr(\mathcal{F}^{T}\mathcal{M}_{0}\,\mathcal{F})}+\frac{1}{2}{Tr(\mathcal{F}^{T}\mathcal{M}_{0}\,\mathcal{F})}^{2}.$
(65)

###  4.3 S-duality of electrodynamic theories in $D\\!=\\!2n$

In addition, we are interested in gauge theories of abelian $p$-form
potentials which have duality invariant structures. Though the Lagrangian of
$p$-form potentials, i.e. $L(P)$, is not invariant under duality
transformation, one can introduce a new Lagrangian in $D\\!=\\!2n$ dimensions
(for even $n$) as being invariant under this transformation. We can find this
kind of Lagrangian by using a Legendre transformation Gaillard:1997rt ;
Aschieri:2008ns ; Buratti:2019cbm  that is not a symmetry transformation.

Consider a nonlinear theory of $p$-form potential $F$ with Lagrangian $L(P)$
where $P$ is the Lorentz invariant parameter. The equations of motion and the
Bianchi identity for $F$ can be derived from the Lagrangian

$L_{D}=L(P)+\frac{1}{D}F_{M}G^{M},$ (66)

where $F_{M}\equiv F_{\mu_{1}\dots\mu_{n}}$ and $G_{M}$ is the field strength
of a Lagrange multiplier S-dual potential $A_{\mu_{2}\dots\mu_{n}}$.
Therefore, the $SL(2,R)$ invariant action is given by

$S_{\rm D}=\int
L_{D}\,d^{D}x=\int\left(L(P)+\frac{1}{D}F_{M}G^{M}\right)d^{D}x.$ (67)

Now from the relations (6), (8) and (9) we can determine the flow equation of
the action as

$L_{D}=\frac{1}{D}{T_{\mu}}^{\mu}=-\lambda\frac{\partial
L(P)}{\partial\lambda},$ (68)

where in 4D we have $L_{D}=L_{inv}$ which means the invariance of the
Lagrangian under $SL(2,R)$ symmetry.

According to the relations (62) and (64), we have obtained an $SL(2,R)$ form
for $T\bar{T}$ operators computed in section 3 as following. We are interested
in a similar structure in arbitrary $D\\!=\\!2n$ dimensions which is invariant
under $\tilde{F}\rightarrow G$ and $G\rightarrow-\tilde{F}$ transformations.
In this respect, we introduce an $SL(2,R)$ invariant tensor as follows

$W_{\mu\nu\alpha\beta}=(\tilde{F}_{\mu\nu}\tilde{F}_{\alpha\beta})+(G_{\mu\nu}G_{\alpha\beta}).$
(69)

It is worth mentioning that though we set $\lambda=1$ for 4D case in
subsection 4.1, the trace of $W$ starts from $\lambda$ while $W^{2}$ is at the
order of $\lambda^{2}$. The $T\bar{T}$ operator in $D\\!=\\!2n$ dimensions
could be written as

$O_{T^{2}}=-\frac{1}{n!}{W_{\mu\nu}}^{\mu\nu}+O(W^{2})+\dots,$ (70)

where the deformed Lagrangian for each dimension is given by

$L_{\lambda}=L_{free}+\frac{1}{8}\int\big{(}-\frac{1}{n!}{W_{\mu\nu}}^{\mu\nu}+O(W^{2})+\ldots\big{)}d\lambda.$
(71)

The contribution of the term $W$ in the deformed action starts from order
$\lambda^{2}$ or $\mathcal{O}(F^{6})$, while the term $W^{2}$ gives the
contribution of order $\lambda^{3}$ or $\mathcal{O}(F^{8})$ to the action. It
should be noted that the relation (71) gives the relations in (51) and (52) in
$D\\!=\\!2n$ with exactly specified coefficients in each dimension.

###  4.4 $\mathcal{N}=2$ supersymmetric model

The $\mathcal{N}\\!=\\!2$ supersymmetric extensions of the BI theory and their
duality properties have been found in refs. Ketov:1998ku ; Ketov:1998sx ;
Kuzenko:2000uh ; Kuzenko:2000tg . Since the equations of motion in these
theories receive contributions from the deformation terms, the NGZ identity
appears in some modified form. This implies, at the quantum level, that
duality transformations receive modifications. At the quantum level, it has
been shown Carrasco:2011jv  that by considering higher order deformations,
maintaining the action’s duality covariance, a theory can preserve the
classical duality transformations, at the presence of a duality-invariant
counterterm.

Here, we would like to find the $T\bar{T}$ operator of $\mathcal{N}=2$
supersymmetric theory with $U(1)$-type duality in $D\\!=\\!4$. It was shown in
ref. Ferko:2019oyv  that the certain $\mathcal{N}=2$ deformed theory in
$D\\!=\\!2$ possesses additional non-linearly realized supersymmetries. From
this, the $T\bar{T}$ operator of $\mathcal{N}=1$ BI theory was found in four
dimensions.

The action of $\mathcal{N}=2$ supersymmetric extension of BI theory includes
explicit spacetime superfield derivatives as well as spinorial derivatives of
the superfields. It is described in an effective framework that is
parameterized by four bosonic (Lorentz vectors) and eight fermionic (Lorentz
Weyl spinors) coordinates ${\mathcal{Z}}^{A}$. It has been proposed in ref.
Kuzenko:2000uh  that a BI action which exhibits $D_{3}$-brane type shift
symmetry is described by

$\displaystyle S_{BI}$ $\displaystyle=$ $\displaystyle\frac{1}{8}\left(\int
d^{8}\mathcal{Z}\mathcal{W}^{2}+\int
d^{8}\bar{\mathcal{Z}}\,\bar{\mathcal{W}}^{2}\right)$ (72) $\displaystyle+$
$\displaystyle\frac{1}{8}\int
d^{12}\mathcal{Z}\bigg{\\{}\mathcal{W}^{2}\bar{\mathcal{W}}^{2}\bigg{[}\lambda+\frac{\lambda^{2}}{2}\left({\mathcal{D}}^{4}{\mathcal{W}}^{4}+\bar{\mathcal{D}}^{4}\,\bar{\mathcal{W}}^{4}\right)$
$\displaystyle+$
$\displaystyle\frac{\lambda^{3}}{4}\Big{(}({\mathcal{D}}^{4}{\mathcal{W}}^{2})^{2}+(\bar{\mathcal{D}}^{4}\,\bar{\mathcal{W}}^{2})^{2}+3\,({\mathcal{D}}^{4}\mathcal{W}^{2})(\bar{\mathcal{D}}^{4}\,\bar{\mathcal{W}}^{2})\Big{)}\bigg{]}$
$\displaystyle+$
$\displaystyle\frac{1}{3}\left[\frac{\lambda^{2}}{3}\mathcal{W}^{3}\Box\bar{\mathcal{W}}^{3}+\frac{\lambda^{3}}{2}\Big{(}(\mathcal{W}^{3}\Box\bar{\mathcal{W}}^{3})\bar{\mathcal{D}}^{4}\,\bar{\mathcal{W}}^{2}+(\bar{\mathcal{W}}^{3}\Box\mathcal{W}^{3})\mathcal{D}^{4}\mathcal{W}^{2}+\frac{1}{24}\mathcal{W}^{4}\Box^{2}\bar{\mathcal{W}}^{4}\Big{)}\right]$
$\displaystyle+$ $\displaystyle{\mathcal{O}}({\mathcal{W}}^{10})\bigg{\\}},$

where $\mathcal{W}(\bar{\mathcal{W}})$ and $\mathcal{D}(\bar{\mathcal{D}})$
are chiral (anti-chiral) superfield strength and super derivative,
respectively. This action is a solution of supersymmetric NGZ condition that
is solved perturbatively in the number of fields. The first line in (72) is
$\mathcal{N}=2$ supersymmetric Maxwell action ($S_{free}$) and the other
integral is the interacting action ($S_{int}$). The term at the order of
$\mathcal{O}(\lambda)$ produces the known $F^{4}$ BI action. From the duality
transformation for $\mathcal{N}=2$ supersymmetric theories that proposed in
the path integral as a Legendre transform Broedel:2012gf , the invariant
action of such theories is given by

$\displaystyle S_{inv}=S(\mathcal{W},\bar{\mathcal{W}})-\frac{i}{8}\int
d^{8}{\mathcal{Z}}\mathcal{W}\mathcal{M}+\frac{i}{8}\int
d^{8}\bar{\mathcal{Z}}\,\bar{\mathcal{W}}\,\bar{\mathcal{M}},$ (73)

where $\mathcal{M}$ and $\bar{\mathcal{M}}$ (similar to the field strength $G$
in non-supersymmetric case) can be defined as
$\mathcal{M}=-4i\frac{\delta}{\delta\mathcal{W}}S(\mathcal{W},\bar{\mathcal{W}})$
and
$\bar{\mathcal{M}}=4i\frac{\delta}{\delta\bar{\mathcal{W}}}S(\mathcal{W},\bar{\mathcal{W}})$.
These superfield strengths satisfy the Bianchi identities
$\mathcal{D}\mathcal{W}=\bar{\mathcal{D}}\,\bar{\mathcal{W}}$ and
$\mathcal{D}\mathcal{M}=\bar{\mathcal{D}}\,\bar{\mathcal{M}}$.

Using these considerations, we find the invariant action (73) of a
supersymmetric BI theory in terms of superfield strengths and their
derivatives as a power series in the coupling constant $\lambda$. Then, the
corresponding $T\bar{T}$ operator could be found from eq. (9) like

$\displaystyle\mathcal{O}_{T^{2}}$ $\displaystyle=$ $\displaystyle\int
d^{12}\mathcal{Z}\,\Bigg{\\{}{\mathcal{W}}^{2}\,\bar{\mathcal{W}}^{2}\,\Bigg{[}1+\lambda\,\Big{(}{\mathcal{D}}^{4}{\mathcal{W}}^{2}+\bar{\mathcal{D}}^{4}\bar{\mathcal{W}}^{2}\Big{)}$
(74) $\displaystyle+$
$\displaystyle\frac{3\lambda^{2}}{4}\,\Big{(}({\mathcal{D}}^{4}{\mathcal{W}}^{2})^{2}+(\bar{\mathcal{D}}^{4}\bar{\mathcal{W}}^{2})^{2}+3\,({\mathcal{D}}^{4}{\mathcal{W}}^{2})(\bar{\mathcal{D}}^{4}\bar{\mathcal{W}}^{2})\Big{)}\Bigg{]}$
$\displaystyle+$ $\displaystyle{1\over
3}\,\Bigg{[}\frac{2\lambda}{3}{\mathcal{W}}^{3}\Box\bar{\mathcal{W}}^{3}+\frac{3\lambda^{2}}{2}({\mathcal{W}}^{3}\Box\bar{\mathcal{W}}^{3})\bar{\mathcal{D}}^{4}\bar{\mathcal{W}}^{2}+(\bar{\mathcal{W}}^{3}\Box{\mathcal{W}}^{3}){\mathcal{D}}^{4}{\mathcal{W}}^{2}+{1\over
24}{\mathcal{W}}^{4}\Box^{2}\bar{\mathcal{W}}^{4}\Bigg{]}$ $\displaystyle+$
$\displaystyle{\mathcal{O}}({\mathcal{W}}^{10})\Bigg{\\}}.$

The $\mathcal{N}=2$ supersymmetric action could be presented in the general
formBroedel:2012gf

$S_{\mathcal{N}=2}=S_{free}+\int
d^{12}\mathcal{Z}\mathcal{W}^{2}\bar{\mathcal{W}}^{2}\,\mathcal{Y}(\mathcal{D}^{4}\mathcal{W}^{2},\bar{\mathcal{D}}^{4}\bar{\mathcal{W}}^{2})+\mathcal{O}(\partial_{\mu}\mathcal{W}),$
(75)

where $\mathcal{Y}$ is a BI-type functional. Finding the invariant action up
to $\mathcal{O}(\partial_{\mu}\mathcal{W})$ from
$S_{inv}=-\lambda\frac{\partial S}{\partial\lambda}$ , we can determine the
$T\bar{T}$ operator in general form using eq. (9). Thus, we obtain

$O_{T^{2}}=8\,\mathcal{W}^{2}\,\bar{\mathcal{W}}^{2}\frac{d}{d\lambda}{\mathcal{Y}}\;(\mathcal{D}^{4}{\mathcal{W}}^{2},\
\bar{\mathcal{D}}^{4}\bar{\mathcal{W}}^{2}).$ (76)

It was shown in ref. Chemissany:2012pf  that the higher derivative terms
$\mathcal{O}(\partial_{\mu}\mathcal{W})$ in eq. (75) reduce to the BI action
at order $\mathcal{O}(\partial F)^{4}$ in the non-supersymmetric level that
come from the one-loop amplitude of four gauge fields. These higher derivative
terms respect to NGZ identity and have found in the form that is manifestly
$SL(2,R)$ invariant BabaeiVelni:2019ptj . This implies that one can find the
invariant higher derivative action which corresponds to
$\mathcal{O}(\partial_{\mu}\mathcal{W})$ and obtain the corresponding
$T\bar{T}$ operator that contributes to eq. (76).

##  5 Conclusion and outlook

The interest in studying a class of QFTs perturbed by irrelevant $T\bar{T}$
operators is one of the most important subjects in recent research arena.
Among the infinite number of possible perturbations of a given QFT, the latter
operator displays very special and universal features. These perturbations may
lead to singular RG flows where the UV fixed point is not well-defined.
Although many studies have been done in the case of 2D QFTs, however a little
attention have been dedicated to investigate this deformation in higher-
dimensional QFTs. It is worth to emphasize that there are no conformal
theories at the quantum level for $D>6$, but the generalization of $T\bar{T}$
operator for non-conformal QFTs may open new windows on gauge/gravity duality
regardless of conformal symmetry. Due to this fact, in this paper we studied
the construction of $T\bar{T}$ deformation for some non-linear electrodynamic
BI-type theories in general $D(=2n)$-dimensional spacetime.

We have proposed a $T\bar{T}$ operator $O_{T^{2}}$ given by (5) in $2n$
dimensions for even $n$. This kind of deformation was compatible with the
results reported in 2D and 4D QFTs, but was in conflict with the Taylor’s
proposal Taylor:2018xcy  in general dimensions. With confidence to this
proposal we reproduced the expansion of BI-type theories in (28) by deforming
the free Lagrangian by this $T\bar{T}$ operator in 2 and 4 dimensions. We
observed that not only the EM tensor (3) satisfies the flow equation (29) for
each $2n$-dimensional theory, but also helps us to reconstruct the deformed
action through $O^{[1/n]}_{T^{2}}$ in (5).

But in higher dimensions $D\geq 8$, the compatibility between the BI expansion
and the deformation of free theory is threatened by the existence of a term
like $K_{2}$ which can not be written only in terms of Lorentz invariant
variables $P_{1}$ and $P_{2}$ in each dimension. Therefore, we trusted to our
proposal to deform the free action of higher dimensional electrodynamic theory
at any arbitrary order of the deformation parameter. In this deformation we
showed that the contribution of the second term in $T\bar{T}$ operator, i.e.
$(T_{\mu}{}^{\mu})^{2}$, starts from $\lambda^{3}$ in the deformed Lagrangian.
We have also shown that one can find a general form for the deformed
Lagrangian at each order of $\lambda$ which at the order of $\lambda$ and
$\lambda^{2}$ are given respectively by the relations (51) and (52).

Following the general strategy suggested in the text, one can also deform the
free theory of $p$-form gauge theories in diverse $D=2n+2$ dimensions. For
instance, we deform a 6D free theory of 3-form self-interacting field
strength, i.e. $\frac{1}{12}F_{\mu\nu\rho}F^{\mu\nu\rho}$, but the results is
incompatible with the expansion of the 6D duality-invariant non-linear BI
theory Bandos:2020jsw . It has been shown in ref. Bandos:2020hgy  that in 6D
there is a unique non-linear conformal modification of the free chiral 2-form
theory which is related to a non-linear modification of 4D Maxwell
electrodynamics by dimensional reduction (see also Berman:1997iz ). It would
be of interest to investigate the $T\bar{T}$ deformation for a 6D non-linear
theory of the so-called chiral 2-forms whose 3-form field strength satisfies a
self-duality condition.

According to the Cardy’s proposal in ref. Cardy:2018sdv , one can define the
$T\bar{T}$ operator as $O_{T^{2}}\sim
aT^{\mu\nu}T_{\mu\nu}+b{T_{\mu}}^{\mu}{T_{\nu}}^{\nu}$ where the constant
coefficients $a$ and $b$ in this proposal depend on the corresponding theory.
For example, we observed that the coefficient $b$ is equal to $1/(D-1)$ for
gravitational theory in general dimension Taylor:2018xcy  while from our
calculations in this paper, one finds that it is $1/n$ for
$D(=2n)$-dimensional gauge theories. Also, it has been shown in refs.
Schwarz:1982jn ; Gross:1986iv  that the effective action of type II
superstring theory, which was obtained from deformation of type II
supergravity theory, starts from order $\lambda^{3}\sim\alpha^{\prime 3}$ as

$\displaystyle S_{eff}$ $\displaystyle=$ $\displaystyle S_{0}+\alpha^{\prime
3}S_{3}+\dots\,.$ (77)

In fact, in the $Ramond\\!-\\!Ramond$ sector of type IIB superstring theory,
there is a term similar to Maxwell theory in $D\\!=\\!10$ dimensions which is
constructed from $5$-forms as follows

$\displaystyle S_{0}$ $\displaystyle=$ $\displaystyle-\tfrac{1}{2\times
5!}\int
d^{10}x\,\,\,F^{\mu\alpha\gamma\iota\kappa}F_{\mu\alpha\gamma\iota\kappa}.$
(78)

From the relation (77), the deformation of action (78) in the $Ramond-Ramond$
sector starts from $\lambda^{3}\sim\alpha^{\prime 3}$. Thus, as shown in
section 3, since the contribution of term $T^{\mu\nu}T_{\mu\nu}$ starts from
$\lambda$ and $(T_{\mu}{}^{\mu})^{2}$ from $\lambda^{3}$, it would be logical
to expect that the deformation of type IIB superstring theory with $T\bar{T}$
operator in the $Ramond\\!-\\!Ramond$ sector is consistent with the assumption
$a=0$ in $O_{T^{2}}$.

We have also investigated the structure of $T\bar{T}$ operator for a class of
non-linear electrodynamic BI-type theories in 4D. In particular, we considered
a gauge theory with a dilatonic field as gauge coupling and showed that the
deformation of free theory made by (5) is also consistent with the expansion
(4.1). It was shown that this operator inherits the $SL(2,R)$ invariant
symmetry of the theory though the action is not. We generalized this
implication to higher dimensional electrodynamic theories investigated in
section 3 by introducing an $SL(2,R)$ invariant tensor and recast the
$T\bar{T}$ operator and deformed Lagrangian in terms of this tensor in eqs.
(70) and (71). As a toy model in the context of supersymmetry, we constructed
the $T\bar{T}$ operator for $\mathcal{N}=2$ supersymmetric BI theory in eq.
(76) from proposals (8) and (9) for the functional $\mathcal{Y}$ in the space
of superfields and superderivatives.

It would be of interest to study the $T\bar{T}$ operator in the case of
$\mathcal{N}=4$ supersymmetric theory in 4D. The $\mathcal{N}=4$ action has a
maximal number of supersymmetries that has been proposed in ref.
Bergshoeff:2013pia . It could be found the corresponding $SL(2,R)$ invariant
action and then, the $T\bar{T}$ operator using eq. (9). We leave the details
of this interesting issue for future study.

## Acknowledgment

We would like to thank S. Sethi, M. Alishahiha, M. R. Garousi and G. Jafari
for valuable comments and discussions. We are also grateful to D. Sorokin for
fruitful discussion and helpful comments on the generalized BI theories in
$D=6$ dimensions.

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