aainstitutetext: Institute of Mathematical Sciences, Homi Bhabha National
Institute (HBNI)
IV Cross Road, C. I. T. Campus, Taramani, Chennai 600 113, Tamil Nadu,
Indiabbinstitutetext: BITS-Pilani, KK Birla Goa Campus
NH 17B, Bypass Road, Zuarinagar 403 726, Goa, Indiaccinstitutetext: Department
of Theoretical Physics, Tata Institute of Fundamental Research
Homi Bhabha Road, Navy Nagar, Colaba, Mumbai 400 005, Maharashtra, India

# Irrelevant Deformations of Chiral Bosons

Subhroneel Chakrabarti b Divyanshu Gupta a Arkajyoti Manna c
and Madhusudhan Raman<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study $\text{T}\overline{\text{T}}$ deformations of chiral bosons using the
formalism due to Sen. For arbitrary numbers of left- and right-chiral bosons,
we find that the $\text{T}\overline{\text{T}}$-deformed Lagrangian can be
computed in closed form, giving rise to a novel non-local action in Sen’s
formalism. We establish that at the limit of infinite
$\text{T}\overline{\text{T}}$ coupling, the equations of motion of deformed
theory exhibits chiral decoupling. We then turn to a discussion of
$\text{T}\overline{\text{T}}$-deformed chiral fermions, and point out that the
stress tensor of the $\text{T}\overline{\text{T}}$-deformed free fermion
coincides with the undeformed seed theory. We explain this behaviour of the
stress tensor by noting that the deformation term in the action is purely
topological in nature and closely resembles the fermionic Wess-Zumino term in
the Green-Schwarz formalism. In turn, this observation also explains a puzzle
in the literature, viz. why the $\text{T}\overline{\text{T}}$ deformation of
multiple free fermions truncate at linear order. We conclude by discussing the
possibility of an interplay between $\text{T}\overline{\text{T}}$ deformations
and bosonisation.

††preprint: TIFR/TH/20-46

##  1 Introduction

The subject of soluble irrelevant deformations of two-dimensional quantum
field theories has, in recent year, enjoyed significant attention. Various
aspects of the so-called $\text{T}\overline{\text{T}}$ deformation — triggered
at each step of the “flow” by the determinant of the stress tensor
Zamolodchikov:2004ce ; Cavaglia:2016oda  — such as the finite-volume spectrum
Smirnov:2016lqw  and the S-matrix of the deformed theory Dubovsky:2012wk ;
Dubovsky:2013ira ; Dubovsky:2017cnj  are known exactly. These deformations
have been studied from a various points of view, for example holography
McGough:2016lol ; Kraus:2018xrn ; Taylor:2018xcy ; Hartman:2018tkw ;
Caputa:2019pam , supersymmetric quantum field theories Baggio:2018rpv ;
Chang:2018dge ; Jiang:2019hux ; Chang:2019kiu ; Ferko:2019oyv ;
Coleman:2019dvf , and string theory Callebaut:2019omt ; Blair:2020ops . An
excellent introduction to this field can be found in Jiang:2019hxb .

The goal of this paper is to study $\text{T}\overline{\text{T}}$ deformations
of chiral bosons. In this introduction, we will motivate our analyses,
situating our own study in the context of recent developments in this and
adjacent fields.

### Chiral Decoupling

In Chakrabarti:2020pxr , we explored a curious aspect of the
$\text{T}\overline{\text{T}}$ deformation: the limit in which the
$\text{T}\overline{\text{T}}$ coupling constant $\lambda$ is sent to infinity.
In order to make sense of the above limit, it was imperative that we studied
the deformed theory within the Hamiltonian formalism. We argued that this
limit generically effects a decoupling of left- and right-chiral degrees of
freedom. Schematically, we may represent our findings as follows:

Free Bosons $\mathrm{T}\overline{\mathrm{T}}$ Nambu-Goto String Chiral Bosons
$\lambda\rightarrow\infty$

A similar progression was observed for the case of free fermions:

Free Fermions $\mathrm{T}\overline{\mathrm{T}}$ Interacting Fermions Free
Fermions $\lambda\rightarrow\infty$

The curious nature of these progressions, from free to interacting to free
theory, naturally motivates us to ask: what next? That is, what happens to a
theory of free chiral bosons under a $\text{T}\overline{\text{T}}$
deformation?

We will argue in this section that the choice to focus on this question places
us at the intersection of various interesting aspects of the
$\text{T}\overline{\text{T}}$ deformation that merit further investigation.

### A Choice of Formalism

Now that we are decided on the study of chiral bosons, we must first decide
how to describe them. In the study of chiral fields, it is well-known that the
constraint of chirality precludes the possibility of their study via the usual
methods of free field theory. This is true more broadly of all self-dual
$(2n+1)$-form field strengths in $4n+2$ spacetime dimensions. (The case $n=0$
is a theory of chiral bosons.)

This subject of self-dual field strengths has an interesting and tortured
history, with theorists caught for decades between Scylla and Charybdis. The
earliest investigations Siegel:1983es ; Henneaux:1988gg ; Schwarz:1993vs  used
actions that were not manifestly Lorentz invariant. More recently, too,
Ouyang:2020rpq  studied $\text{T}\overline{\text{T}}$ deformations of chiral
bosons utilizing the formalism of Floreanini and Jackiw Floreanini:1987as ;
Sonnenschein:1988ug .

The lack of manifest Lorentz invariance was subsequently overcome
McClain:1990sx ; Pasti:1996vs  at the cost of the introduction of auxiliary
fields: either an infinite number of them, or a finite number of auxiliary
field with non-polynomial actions. For the better part of the last two
decades, this was the most widely accepted framework for the discussion of
chiral $p$-form fields despite the lack of off-shell self-duality; this
permitted a classical analysis but complicates the discussion of the
corresponding quantum theory. Another recent proposal for a Lorentz invariant
action for chiral bosons was put forward in Townsend:2019koy .

Following the first successful construction of the quantum Batalin–Vilkovisky
master action for closed superstring field theory Sen:2015uaa , the
aforementioned difficulties were overcome in the context of Type IIB SUGRA by
Sen in Sen:2015nph . The construction was readily extended to theory of self-
dual forms with arbitrary rank in Sen:2019qit . Sen’s formalism, in addition
to the $(2n+1)$-form field strength, also contains an additional $(2n)$-form
field with the wrong sign for the kinetic term. Despite this, it was shown
that the (free) dynamics of this additional field completely decouples — even
from the background metric, and in the quantum theory as well — thereby
forming an autonomous sector inert to the dynamics of other physical
interacting fields.

We have elected to use Sen’s formalism in our study of chiral bosons. This
choice was made in part by the demands of the problem at hand: various aspects
of $\text{T}\overline{\text{T}}$ deformations are quantum mechanically exact,
and it is desirable to work with a formalism that is capable, at least in
principle, of eventually accessing the quantum theory. Chirality is baked into
the theory, i.e. the field variables are self-dual off-shell, and not imposed
as an equation of motion. Additionally, manifest Lorentz invariance and
polynomial nature of the action makes the case for adoption of Sen’s formalism
very strong indeed.111Sen’s formalism has been successfully used in the
context of self-dual $3$-forms in six-dimensional $(2,0)$ theories, see
Lambert:2019diy ; Andriolo:2020ykk ; Gustavsson:2020ugb .

To work with a formalism where the self-duality of the chiral boson is imposed
at the level of an equation of motion (i.e. on-shell) is to foreclose on the
possibility of studying the quantum aspects of this theory within this
formalism. With this perspective, while the demonstration of Ouyang:2020rpq
is elegant, it is not surprising, since at the level of the classical
Lagrangian (an off-shell quantity), we are dealing with a decomposition of the
scalar field that is only permissible on-shell. This is evident in the
relation between the Hamiltonian densities of the deformed free scalar on one
hand and the deformed left-chiral and right-chiral scalar on the other, see
(Ouyang:2020rpq, , eq. (20)).

Another way of seeing this is that for Ouyang:2020rpq , the seed theory
described by the Floreanini-Jackiw Lagrangian is a theory of chiral bosons
_on-shell_ , and that this is no longer the case once the
$\text{T}\overline{\text{T}}$ flow begins. This is because in the Floreanini-
Jackiw formalism, chirality is imposed on-shell by solving an equation of
motion. Once deformations are turned on, the equation of motion that imposes
chirality changes.

Our set-up is quite different. While the on-shell dynamics of the formalisms
of Floreanini-Jackiw and Sen agree, in the latter formalism one works with
fields that are self-dual _off-shell_ , i.e. they continue to be off-shell
once deformations are turned on. This off-shell self-duality imposes severe
constraints on the form of the interactions one can introduce in Sen’s
formalism. We will see that this implies an interesting closed form of the
$\text{T}\overline{\text{T}}$-deformed Lagrangian.

### Lorentzian vs. Euclidean Spacetime

The decision to study chiral bosons in two dimensions _forces_ us to work with
a Lorentzian signature. It is helpful to quickly review this argument, so let
us suppose that we are studying chiral (i.e. self-dual or anti-self-dual)
$p$-forms on $\mathbb{M}$, an $n$-dimensional manifold whose metric has $s$
time coordinates, and derive this conclusion.

Consider the Hodge star operator
$\star:\Omega^{p}(\mathbb{M})\rightarrow\Omega^{n-p}(\mathbb{M})$, which maps
$p$-forms to $(n-p)$-forms. We find that

$\star^{2}=(-1)^{s+p(n-p)}\ ,$ (1)

where $s=0$ for Euclidean space and $s=1$ for Lorentzian spacetimes. A self-
dual $(+)$ or anti-self-dual $(-)$ form $A$ satisfies the property

$A=\pm\star A\ ,$ (2)

which in turn implies $n=2p$, i.e. we can only have chiral forms in even
dimensions. Further, on Hodge dualising the above equation once more, we find
that

$\star^{2}=(-1)^{s+p(n-p)}=+1\ ,$ (3)

which in turn implies that for Euclidean signature, $p$ is even, implying
$n=4k$ for $k\in\mathbb{N}$. However, for the Minkowski signature, we find
$p^{2}+1$ is even, so $p$ is odd and $n=4k+2$. Therefore, in order to study
real chiral bosons in two dimensions, we must work with a Lorentzian
signature.

In contrast, the $\text{T}\overline{\text{T}}$ deformation has been most
rigorously studied in Euclidean space. For example, one typically defines the
composite $\text{T}\overline{\text{T}}$ operator via point-splitting and the
proof of Zamolodchikov’s factorization formula Zamolodchikov:2004ce  uses
Euclidean operator product expansions. In the Lorentzian setting, it is
natural to expect additional lightcone singularities. To the best of our
knowledge, such an analysis has not yet been carried out.

Despite this, various Lorentzian aspects of the $\text{T}\overline{\text{T}}$
deformation — defined in the same way, except on Minkowski spacetime — have
been studied in the literature. As illustrative examples: McGough:2016lol
builds on Cardy:2015xaa  and studies signal propagation speeds; Frolov:2019nrr
interprets the inviscid Burgers equation which determines the spectrum of the
$\text{T}\overline{\text{T}}$-deformed theory as encoding the gauge invariance
of the target spacetime energy and momentum of a non-critical string theory
quantised in a one-parameter family of deformations of the standard lightcone
gauge called the uniform lightcone gauge; finally, Kruthoff:2020hsi  adopt a
Hamiltonian point of view on $\text{T}\overline{\text{T}}$ deformations and
show that their results in Lorentzian spacetime are consistent with known
exact results like the Castillejo-Dalitz-Dyson phase that relates the deformed
and undeformed S-matrices.

Our perspective in this note will be similar.

###  $\text{T}\overline{\text{T}}$ Flow of Chiral Bosons, Fermions, and
Bosonisation

Another interesting aspect of Sen’s formalism is that the classical action is
_not_ of the general form assumed by Bonelli:2018kik , i.e. of the form

$S=\int\text{d}^{2}x\,\sqrt{g}\,\mathcal{L}\ .$ (4)

We will therefore construct the $\text{T}\overline{\text{T}}$-deformed
Lagrangian order-by-order in the $\text{T}\overline{\text{T}}$ coupling
constant, with the deformations sourced by the determinant of the stress
tensor.

Our central result is that the uniform structure of the deformations implies
that the $\text{T}\overline{\text{T}}$-deformed Lagrangian for a pair of
chiral bosons can in fact be computed in closed form. On considering arbitrary
numbers of left- and right-chiral bosons, once again we compute the closed
form of the deformed Lagrangian to all orders in the
$\text{T}\overline{\text{T}}$ coupling constant. This result produces a novel
non-local action in Sen’s formalism, which is intriguing in its own right. We
also observe that the $\text{T}\overline{\text{T}}$-deformed free fermion
Lagrangian, which is known to truncate at linear order Coleman:2019dvf , does
not actually alter the classical stress tensor of the seed theory. To the best
of our knowledge, this fact is noted and explicitly demonstrated for the first
time in the literature. As we will explain, this feature is due to the fact
that the deformation term for free fermion is a purely topological one and
therefore incapable of altering the stress tensor.

We then initiate a study of the possible interplay between bosonisation and
$\text{T}\overline{\text{T}}$ deformations. We make a preliminary conjecture
that should serve as a suitable starting point for this line of investigation.

Of course, bosonisation duality (say, between sine Gordon theory and the
massive Thirring model Coleman:1974bu ) is a quantum mechanically exact
statement about correlation functions. This demonstration is made possible in
part by the relevant nature of deformations, which makes the corresponding
quantum theories well-defined, i.e. renormalisable. In this paper, we have
restricted our attention only to the classical aspects of
$\text{T}\overline{\text{T}}$-deformed theories.

We hope to address quantum mechanical features of this problem, leveraging the
analyses of Aharony:2018vux ; Guica:2019vnb ; Cardy:2019qao ; He:2020udl  on
correlation functions and Rosenhaus:2019utc  on renormalisation, in the near
future.

### Outline

We now summarise the contents of our paper.

In §2, we provide a pedagogical review of Sen’s formalism as it is applied to
chiral bosons in two dimensions. Following this, in §3 we explain our strategy
for computing the stress tensor. We feel this will help acclimatise our
readers to Sen’s formalism, which may be unfamiliar. Our main results are
contained in §4, where we study $\text{T}\overline{\text{T}}$ deformations of
theories with left- and right-chiral bosons, and solve for the closed form of
the deformed Lagrangian. In §5, we begin by pointing out that stress tensor of
the $\text{T}\overline{\text{T}}$-deformed free fermions theory coincides with
the undeformed seed theory. We explain that this is due to the fact that the
deformation terms in the action is purely topological in nature. We then
initiate a study of the possible bosonisation duality relating two
$\text{T}\overline{\text{T}}$-deformed theories, and speculate on the
possibility that bosonisation “commutes” with $\text{T}\overline{\text{T}}$
deformations. For the theory at hand, this conjecture is not intended to
succeed, but to merely provide a stepping stone towards a more concrete
realization of this program. We conclude in §6 by summarising our results and
listing some further directions for research.

##  2 Sen’s Formalism for Chiral Bosons

A chiral boson in two dimensions is nothing but an $1$-form field $A_{g}$
which is self-dual, i.e. $A_{g}=\star_{g}A_{g}$. Typically one thinks of this
$1$-form as a _field strength_ for a scalar, viz. $A_{g}\sim\text{d}\varphi$,
where d is the exterior derivative and $\varphi$ is a scalar field. However,
in Sen’s formalism, as we will shortly see, we need not invoke a rewriting of
the self-dual $1$-form as an exact $1$-form. Further, it is sufficient for us
to confine our analysis to the classical aspects of this action. Nonetheless,
we wish to emphasize that the fully quantum mechanical analysis in the
Hamiltonian formulation of this formalism is known as well. For more detailed
discussions, we refer the reader to the original papers Sen:2015nph ;
Sen:2019qit  and also Andriolo:2020ykk .

Sen’s action is of the following form

$S=\int_{(\mathbb{R}^{(1,1)},\,\eta)}\left[\dfrac{1}{2}\,\text{d}\phi\wedge\star_{\eta}\text{d}\phi-2\,A\wedge\text{d}\phi+A\wedge\mathcal{M}(A)\right]\;.$
(5)

The action describes a theory defined over a Lorentzian $2$-manifold
$\mathbb{M}$ equipped with a metric $g_{\mu\nu}$. However, note that the
integral is defined over the two-dimensional Minkowski spacetime
$\mathbb{R}^{(1,1)}$. The field $\phi$ is a scalar field on
$\mathbb{R}^{(1,1)}$, and the field $A$ is an $1$-form on $\mathbb{R}^{(1,1)}$
which is defined to be _self-dual with respect to the flat metric_ , i.e.
$A=\star_{\eta}A$. These are just some of the several unfamiliar features of
this action.

Notice that due to this unusual self-duality constraint on the $1$-form $A$,
it does not belong to the space of self-dual $1$-forms associated to
$(\mathbb{M},g)$. Following Andriolo:2020ykk , we refer to these objects as
pseudoforms.

The map $\mathcal{M}(A)$ is a linear map on the space of self-dual pseudoforms
with the following properties (all of which are satisfied at the level of
action itself, i.e. off-shell).

  * •

$\mathcal{M}(A)$ maps self-dual pseudoforms to anti-self-dual pseudoforms.
That is, for all $A=\star_{\eta}A$, we have

$\mathcal{M}(A)=-\star_{\eta}\mathcal{M}(A)\;.$ (6)

  * •

$\mathcal{M}$ annihilates anti-self-dual pseudoforms. That is, for all
$B=-\star_{\eta}B$ we have

$\mathcal{M}(B)=0\;.$ (7)

  * •

For any two self-dual pseudoforms $A_{1}$ and $A_{2}$, the map satisfies

$A_{1}\wedge\mathcal{M}(A_{2})=A_{2}\wedge\mathcal{M}(A_{1})\;.$ (8)

  * •

Even though $A$ and $\mathcal{M}(A)$ are pseudoforms, the following specific
linear combination belongs to the space of self-dual forms associated to
$(\mathbb{M},g)$.

$A-\mathcal{M}(A)=\star_{g}\left(A-\mathcal{M}(A)\right)\;.$ (9)

The astute reader will conclude from this last property that the map
$\mathcal{M}(A)$ is necessarily cognizant of the background metric, even
though the pseudoforms themselves are not. This is indeed true. An explicit
construction of the map with the above properties was supplied in Sen:2015nph
(also see Sen:2019qit ; Andriolo:2020ykk ). We will, however, not require the
explicit construction of the map in this paper.

From the action given in eq. 5, one can easily derive the equations of motion
which are

$\displaystyle\text{d}\left[\dfrac{1}{2}\star_{\eta}\text{d}\phi+A\right]$
$\displaystyle=0\;,$ (10)
$\displaystyle\left(1-\star_{\eta}\right)\bigg{[}\text{d}\phi-\mathcal{M}(A)\bigg{]}$
$\displaystyle=0\;.$ (11)

Let us define the following fields

$\displaystyle\hat{A}$
$\displaystyle:=A+\dfrac{1}{2}\left(\text{d}\phi+\star_{\eta}\text{d}\phi\right)\;,$
(12) $\displaystyle A_{g}$ $\displaystyle:=A-\mathcal{M}(A)\;.$ (13)

In terms of these new fields, the equations of motion assume a very simple
form:

$\displaystyle\text{d}\hat{A}$ $\displaystyle=0\;,$ (14)
$\displaystyle\text{d}A_{g}$ $\displaystyle=0\;.$ (15)

Notice that we can rewrite the property in eq. 9 as $A_{g}=\star_{g}A_{g}$.
This is true off-shell. Therefore, we identify the variable $A_{g}$ as the
desired chiral boson on $(\mathbb{M},g)$ which satisfies the free massless
equation of motion eq. 15 as expected.

The field $\hat{A}$ also satisfies the free massless equation of motion and we
can see that at the level of equation of motion at least, it is completely
decoupled from the physical degree of freedom carried by $A_{g}$. It was shown
in Sen:2019qit  that this decoupling continues to hold even at the quantum
level. Since the only term in the action that is cognizant of the background
metric is $\mathcal{M}(A)$, we see that the field $\hat{A}$ does not couple to
the background metric.

The decoupling of $\hat{A}$ is absolute in this regard; it does not couple to
any of the physical fields and is therefore completely inaccessible to any
experiment one might wish to perform. For all intents and purposes, the field
$\hat{A}$ is just an auxiliary aid which enables us to formulate a description
of the dynamics of self-dual form fields which possesses the desirable
properties we outlined in the introduction.

There is, however, one minor trade-off we must accept when adopting Sen’s
formalism: the action is not manifestly invariant under general coordinate
transformations. In fact, under diffeomorphisms, the transformation properties
of the field $\phi$ and the pseudoform $A$ are rather unusual (the details can
be found in Sen:2015nph ; Andriolo:2020ykk ). Despite the lack of manifest
general coordinate invariance, one can verify that the action is indeed
invariant under general coordinate transformations, as it should be.

Before proceeding, a brief word about our notation and nomenclature is in
order. Throughout the rest of the paper the Hodge dual with respect to $\eta$
will be denoted $\star$ and we suppress the $\eta$ subscript for convenience.
For the Hodge dual with respect to the background metric $g$ on $\mathbb{M}$,
we will continue to explicitly write $\star_{g}$. Furthermore, we suppress the
subscripts $\mathbb{R}^{(1,1)}$ and $\eta$ on all integrals over Minkowski
spacetime. Wherever we use the integral over $\mathbb{M}$, we mention it in
the subscript or write the diffeomorphism invariant measure factor explicitly.
Finally, we will use terms left-chiral (respectively, right-chiral) and self-
dual (respectively, anti-self-dual) interchangeably.

##  3 The Stress Tensor

Despite the unfamiliar structure of the Sen’s action, one can compute the
associated stress tensor in an arbitrary background following a rather simple
prescription. For the case of chiral 3-forms in six dimensions this has
already been done in (Andriolo:2020ykk, ); it is evident, however, that this
analysis can be readily extended to any chiral $p$-form in $2p$ dimensions.
For the sake of completeness, we explicitly present the analysis for the case
at hand, i.e. the case of chiral 1-form in two dimensions.

The steps to compute the stress tensor are as follows:

  1. 1.

Consider the standard action for an _arbitrary_ 1-form field $\mathbf{A}$
living on $\mathbb{M}$

$S_{\text{aux.}}=\dfrac{1}{2}\int_{\mathbb{M},\,g}\mathbf{A}\wedge\star_{g}\mathbf{A}=\dfrac{1}{2}\int\text{d}^{2}x\,\sqrt{-g}~{}g^{\mu\nu}\mathbf{A}_{\mu}\mathbf{A}_{\nu}\;.$
(16)

  2. 2.

Evaluate the stress tensor of $S_{\text{aux.}}$ the usual way by varying with
respect to the metric

$\mathbf{T}_{\mu\nu}:=-\dfrac{2}{\sqrt{-g}}\dfrac{\delta
S_{\text{aux.}}}{\delta g^{\mu\nu}}\;.$ (17)

  3. 3.

Let the desired stress tensor of the chiral 1-form $A_{g}$ (i.e.
$A_{g}=\star_{g}A_{g}$) be denoted by $T_{\mu\nu}$. Then we can extract
$T_{\mu\nu}$ from $\mathbf{T}_{\mu\nu}$ as

$T_{\mu\nu}=\mathbf{T}_{\mu\nu}\Big{|}_{\mathbf{A}=A_{g}}\quad.$ (18)

In the rest of this section we derive the steps outlined above. The reader
uninterested in this derivation can safely skip to the next section once they
have taken note of the final expression eq. 40.

###  3.1 Relating Forms and Pseudoforms

Due to the unusual structure of Sen’s action, the only term contributing to
the stress tensor in an arbitrary background is

$S=\int\,\,A\wedge\mathcal{M}(A)\;.$ (19)

Recall that $A$ is an arbitrary self-dual form with respect to the Minkowski
metric, even though the theory is defined on a two dimensional manifold
$\mathbb{M}$ with a non-flat metric $g_{\mu\nu}$. $\mathcal{M}$ is a linear
map from the space of self-dual pseudoforms to space of anti-self-dual
pseudoforms. In component form, the map acts as
$\mathcal{M}(A)_{\nu}=\mathcal{M}_{\nu}^{\;\;\mu}A_{\mu}$. The properties of
$\mathcal{M}(\cdot)$ can then be translated into equivalent statements about
the tensor $\mathcal{M}_{\nu}^{\;\;\mu}$.

Let us now consider a basis of $1$-forms in two dimensions:
$\\{v_{+}\,,\,v_{-}\\}$. In fact, in a coordinate basis we have

$v_{+}=\text{d}x^{+}\quad\text{and}\quad
v_{-}=\text{d}x^{-}\;\implies\;v_{+}\wedge
v_{-}=2\,\text{d}x^{1}\wedge\text{d}x^{2}.$ (20)

It follows that any self-dual $1$-form can be expanded as
$A=a_{\mu}(x^{+})v_{+}^{\mu}$, we will abbreviate this often as
$A=\mathbf{a}\,v_{+}$, with the boldface serving to remind us it is a
component of a $1$-form. We will now try to pin down a precise relation
between the basis of chiral forms with respect to $\star_{g}$ and the basis
$v_{\pm}$.

Let us suppose we have a basis form $\vartheta_{+}$ such that
$\vartheta_{+}=\star_{g}\vartheta_{+}$. Locally, one can always write down a
relation of the form

$\vartheta_{+}^{\mu}=\mathcal{T}^{\mu}_{\;\;\;\nu}\,v^{\nu}_{+}+\mathcal{U}^{\mu}_{\;\;\;\nu}\,v^{\nu}_{-}$
(21)

Now, due to eqs. 9 and 13, we know $v_{+}-\mathcal{M}(v_{+})$ to be a self-
dual 1-form with respect to $\star_{g}$. Thus we expect it to be linearly
related to $\vartheta_{+}$. We can write

$\displaystyle\Big{[}v_{+}-\mathcal{M}(v_{+})\Big{]}^{\nu}$
$\displaystyle=\mathcal{P}^{\nu}_{\;\;\;\mu}\,\vartheta_{+}^{\mu}$
$\displaystyle=\mathcal{P}^{\nu}_{\;\;\;\alpha}\,\mathcal{T}^{\alpha}_{\;\;\;\beta}\,v^{\beta}_{+}+\mathcal{P}^{\nu}_{\;\;\;\alpha}\,\mathcal{U}^{\alpha}_{\;\;\;\beta}\,v^{\beta}_{-}\;,$
(22)

where in the last line we have made use of eq. 21. Comparing the coefficient
of $v_{+}$ on both sides, we get

$\displaystyle\mathcal{P}^{\nu}_{\;\;\;\mu}\mathcal{T}^{\mu}_{\;\;\;\alpha}$
$\displaystyle=\delta^{\nu}_{\;\;\;\alpha}$
$\displaystyle\implies\mathcal{P}^{\nu}_{\;\;\;\mu}$
$\displaystyle=\left(\mathcal{T}^{-1}\right)^{\nu}_{\;\;\;\mu}\;,$ (23)

whereas the coefficient of $v_{-}$ gives (in matrix notation)

$\mathcal{M}=-\mathcal{T}^{-1}\,\mathcal{U}\;.$ (24)

Now note that we can write

$\displaystyle\mathcal{T}^{-1}\vartheta_{+}$
$\displaystyle=v_{+}+\mathcal{T}^{-1}\,\mathcal{U}\,v_{-}$
$\displaystyle=v_{+}\;,$ (25)

where we made use of eqs. 7 and 24.

It is important to note that $A_{g}$ is not just equal to
$\tfrac{1}{2}(1+\star_{g})A$ as one might naively expect. Rather, we have from
the definition in eq. 13

$A_{g}=\underbrace{\dfrac{1}{2}\left(A+\star_{g}A\right)}_{\text{naively
expected}}-\,\dfrac{1}{2}\Big{(}\mathcal{M}(A)+\star_{g}\mathcal{M}(A)\Big{)}\;.$
(26)

A consequence of self-duality condition is that under a variation with respect
to the metric, the anti-self-dual part of the variation is constrained to be

$\dfrac{1}{2}\left(1-\star_{g}\right)\delta_{g}A_{g}=\dfrac{1}{2}(\delta_{g}\star_{g})A_{g}\;.$
(27)

On the other hand, the self-dual part of the variation can be set to zero
(Gran:2014lia, ). The eq. 27 will play an important role when we evaluate the
stress tensor of the $\text{T}\overline{\text{T}}$ deformed Lagrangian.

Finally, note that for $A=\mathbf{a}\,v_{+}$, using section 3.1 we can express
$A_{g}$ as

$A_{g}=\mathcal{T}^{-1}\,\mathbf{a}\,\vartheta_{+}\;.$ (28)

###  3.2 Evaluating the stress tensor

We define the stress tensor the usual way for a QFT coupled to a curved
background:

$T_{\mu\nu}:=-\dfrac{2}{\sqrt{-g}}\dfrac{\partial\mathcal{L}}{\partial
g^{\mu\nu}}\;.$ (29)

In Sen’s action, the only term which will contribute in the action is the term
reproduced in eq. 19. Recall also that $A=\mathbf{a}_{\mu}v_{+}^{\mu}$. We can
then write the stress-tensor as

$T_{\mu\nu}=-\dfrac{2}{\sqrt{-g}}\,\mathbf{a}_{\mu}\,\mathbf{a}_{\nu}\,v_{+}^{\mu}\wedge\dfrac{\partial\mathcal{M}}{\partial
g^{\mu\nu}}(v_{+}^{\nu})\;.$ (30)

Since by construction $v_{+}-\mathcal{M}(v_{+})$ is self dual with respect to
$\star_{g}$ (recall eq. 9), we can vary the self-duality condition with
respect to the metric and obtain

$\left(1-\star_{g}\right)\delta_{g}\mathcal{M}(v_{+})=-(\delta_{g}\star_{g})\Big{(}v_{+}-\mathcal{M}(v_{+})\Big{)}\;.$
(31)

Hence, for $\vartheta_{+}=\star_{g}\vartheta_{+}$, we get

$2\,\vartheta_{+}\wedge\delta_{g}\mathcal{M}(v_{+})=-\vartheta_{+}\wedge(\delta_{g}\star_{g})\Big{(}v_{+}-\mathcal{M}(v_{+})\Big{)}\;.$
(32)

Writing the indices explicitly and using sections 3.1 and 3.1 to rewrite the
right-hand side, we obtain

$2\delta_{g}\,\mathcal{M}^{\alpha}_{\;\;\;\nu}\,\vartheta_{+}^{\mu}\wedge
v_{+}^{\nu}=-(\mathcal{T}^{-1})^{\alpha}_{\;\;\;\beta}\,\vartheta_{+}^{\mu}\wedge(\delta_{g}\star_{g})\,\vartheta_{+}^{\beta}\;.$
(33)

It follows from eq. 21 that $\vartheta_{+}^{\mu}\wedge
v_{+}^{\nu}=\mathcal{T}^{\mu}_{\;\;\;\alpha}\,v_{+}^{\alpha}\wedge
v_{+}^{\nu}$. Substituting this into the left-hand side of eq. 33, we get

$\displaystyle
2\delta_{g}\,\mathcal{M}^{\alpha}_{\;\;\;\nu}\,\mathcal{T}^{\mu}_{\;\;\;\beta}\,v_{+}^{\beta}\wedge
v_{+}^{\nu}$
$\displaystyle=-(\mathcal{T}^{-1})^{\alpha}_{\;\;\;\beta}\,\vartheta_{+}^{\mu}\wedge(\delta_{g}\star_{g})\,\vartheta_{+}^{\beta}$
$\displaystyle\delta_{g}\,\mathcal{M}^{\alpha}_{\;\;\;\nu}\,v_{+}^{\sigma}\wedge
v_{+}^{\nu}$
$\displaystyle=-\dfrac{1}{2}(\mathcal{T}^{-1})^{\sigma}_{\;\;\;\mu}\,(\mathcal{T}^{-1})^{\alpha}_{\;\;\;\beta}\,\vartheta_{+}^{\mu}\wedge(\delta_{g}\star_{g})\,\vartheta_{+}^{\beta}$
$\displaystyle=-\dfrac{1}{2}\Big{(}v_{+}-\mathcal{M}(v_{+})\Big{)}^{\sigma}\wedge(\delta_{g}\star_{g})\Big{(}v_{+}-\mathcal{M}(v_{+})\Big{)}^{\alpha}\;.$
(34)

Contracting the right-hand side with $\mathbf{a}_{\sigma}$ and
$\mathbf{a}_{\alpha}$, the stress tensor simplifies to

$\displaystyle T_{\mu\nu}$
$\displaystyle=\dfrac{1}{\sqrt{-g}}\Big{(}A-\mathcal{M}(A)\Big{)}\wedge\dfrac{\partial\,\star_{g}}{\partial
g^{\mu\nu}}\Big{(}A-\mathcal{M}(A)\Big{)}$
$\displaystyle=\dfrac{1}{\sqrt{-g}}A_{g}\wedge\dfrac{\partial\,\star_{g}}{\partial
g^{\mu\nu}}\,A_{g}\;.$ (35)

This stress tensor has an alternative interpretation, one that is better
suited for carrying out the explicit computation, which we now explain.
Consider the action

$S_{\text{aux.}}=\dfrac{1}{2}\int_{\mathbb{M},~{}g}\mathbf{A}\wedge\star_{g}\mathbf{A}=\dfrac{1}{2}\int\text{d}^{2}x\sqrt{-g}~{}g^{\mu\nu}\mathbf{A}_{\mu}\mathbf{A}_{\nu}\;.$
(36)

where $\mathbf{A}$ is an _arbitrary_ 1-form field defined on $\mathbb{M}$ .
The stress tensor corresponding to this action is readily seen to be

$\displaystyle\mathbf{T}_{\mu\nu}$
$\displaystyle=-\dfrac{1}{\sqrt{-g}}\mathbf{A}\wedge\dfrac{\partial\,\star_{g}}{\partial
g^{\mu\nu}}\mathbf{A}\;$ (37)
$\displaystyle=\dfrac{1}{4}g_{\mu\nu}\mathbf{A}_{\sigma}\mathbf{A}^{\sigma}-\dfrac{1}{2}\mathbf{A}_{\mu}\mathbf{A}_{\nu}\;.$
(38)

Comparing eq. 37 with section 3.2, we see our intended stress tensor can be
read off as

$T_{\mu\nu}=\mathbf{T}_{\mu\nu}\Big{|}_{\mathbf{A}=A_{g}}\quad.$ (39)

This completes the derivation of the prescription outlined in the beginning of
this section.

All that remains is to make use of this prescription to obtain the stress
tensor for the chiral boson. Using the explicit expression eq. 38 and
substituting $\mathbf{A}\rightarrow A_{g}$ we get the desired answer

$T_{\mu\nu}=-\dfrac{1}{2}A^{g}_{\mu}\,A^{g}_{\nu}\;.$ (40)

##  4 $\text{T}\overline{\text{T}}$ Deformations of Chiral Bosons

We now have all the ingredients required to compute the
$\text{T}\overline{\text{T}}$ deformations of chiral bosons. Of course, to
ensure that the deformation does not trivially vanish, we must have both left-
_and_ right-chiral bosons in our undeformed action. We start with simplest
case, when we have one left- and one right-chiral boson, and subsequently
generalise this to an arbitrary number of left- and right-chiral fields.

Let us recall some basics. The $\text{T}\overline{\text{T}}$ “flow” deforms
the classical Lagrangian of the seed theory according to the following linear
differential equation

$\frac{\text{d}\mathcal{L}_{\lambda}}{\text{d}\lambda}=\det
T_{\mu\nu}^{(\lambda)}=\frac{1}{2}\epsilon^{\mu\nu}\epsilon^{\rho\sigma}T_{\mu\rho}^{(\lambda)}T_{\nu\sigma}^{(\lambda)}\
,$ (41)

and the determinant operator that triggers the flow is defined via point-
splitting, which we assume implicitly. We interpret all objects in the above
equation as power series in $\lambda$, the $\text{T}\overline{\text{T}}$
coupling constant. This means:

$\mathcal{L}_{\lambda}=\mathcal{L}_{0}+\lambda\,\mathcal{L}_{1}+\lambda^{2}\,\mathcal{L}_{2}+\cdots\
,$ (42)

and

$T_{\mu\nu}^{(\lambda)}=T_{\mu\nu}^{(0)}+\lambda\,T_{\mu\nu}^{(1)}+\lambda^{2}\,T_{\mu\nu}^{(2)}+\cdots\
.$ (43)

The goal of the following two sections will be to determine the corrections
$\mathcal{L}_{k\geq 1}$ to the classical Lagrangian, using the methods
discussed in the previous sections.

It is natural to expect that the deformed theory will continue to respect
Lorentz invariance. Before explicitly computing them, we might ask: what kinds
of terms might arise as deformations? It is easy to see that the only non-
trivial Lorentz invariant combination of the left- and right-chiral fields one
can construct are inner products between them, so on general grounds we expect
that

$\mathcal{L}_{k}\propto f_{k}\left(A\cdot B\right)\ ,$ (44)

where $f_{k}$ are some arbitrary functions that we will now determine.

###  4.1 A Pair of Chiral Bosons

In this section we study a pair of chiral bosons, i.e. one left- and one
right-chiral boson.

####  4.1.1 The General Form of Deformations

The stress energy tensor for anti-self-dual field $B_{\mu}$ which takes the
same form as self dual form field

$\displaystyle T_{\gamma\delta}^{(a)}$
$\displaystyle=-\frac{1}{2}B^{g}_{\gamma}B^{g}_{\delta}~{},$ (45)

in which case the total stress tensor for a single self-dual and anti-self-
dual field is

$\displaystyle
T_{\mu\nu}^{(0)}=-\frac{1}{2}[A^{g}_{\mu}A^{g}_{\nu}+B^{g}_{\mu}B^{g}_{\nu}]~{}.$
(46)

Using eq. 41, the first deformation of Lagrangian can be computed:

$\displaystyle\mathcal{L}_{1}=\frac{1}{\sqrt{-g}}\det\left(T_{\mu\nu}^{(0)}\right)=\frac{\sqrt{-g}}{4}\big{(}A_{g}\cdot
B_{g}\big{)}^{2}\ .$ (47)

Note that since the $\det T_{\mu\nu}$ is a tensor density of weight $(-2)$,
for the Lagrangian to have the correct measure in curved background it must be
multiplied by a weight $+1$ object, which is naturally the inverse of
$\sqrt{-g}$.222We are deeply grateful to Dmitri Sorokin for pointing this out
to us, as well as correcting an oversight pertaining to this point in an
earlier version of the draft.

Since $A_{g}$ is self-dual, the variation $\delta_{g}A_{g}$ is necessarily
anti-self-dual with respect to $\star_{g}$. Similarly, we can say that
$\delta_{g}B_{g}$ is necessarily self-dual with respect to $\star_{g}$.
Therefore, we have

$\displaystyle\frac{\partial\mathcal{L}_{1}}{\partial g^{\mu\nu}}$
$\displaystyle=\frac{1}{4}\left[\frac{-1}{2\sqrt{-g}}\frac{\partial
g}{\partial g^{\mu\nu}}\left(A_{g}\cdot
B_{g}\right)^{2}+2\sqrt{-g}\left(A_{g}\cdot
B_{g}\right)A^{g}_{\alpha}\,B^{g}_{\beta}\,\frac{\partial
g^{\alpha\beta}}{\partial g^{\mu\nu}}\right]~{},$ (48)

where we have taken

$\frac{\partial A^{g}_{\alpha}}{\partial
g^{\mu\nu}}\,B_{g}^{\alpha}=0\quad\quad\text{and}\quad\quad
A_{g}^{\alpha}\,\frac{\partial B^{g}_{\alpha}}{\partial g^{\mu\nu}}=0~{}.$
(49)

This implies

$\displaystyle\frac{\partial\mathcal{L}_{1}}{\partial g^{\mu\nu}}$
$\displaystyle=-\frac{\sqrt{-g}}{8}\left[\,g_{\mu\nu}\left(A_{g}\cdot
B_{g}\right)^{2}-2\left(A_{g}\cdot
B_{g}\right)\,A^{g}_{(\mu}B^{g}_{\nu)}\right]$ (50)
$\displaystyle=\frac{\sqrt{-g}}{8}\ g_{\mu\nu}\left(A_{g}\cdot
B_{g}\right)^{2}\ .$

The simplification in eq. 50 is made possible using the identity eq. 94,
derived explicitly in Appendix A. The first-order deformation of the stress
tensor is then calculated as

$\displaystyle
T_{\mu\nu}^{(1)}=-\frac{2}{\sqrt{-g}}\frac{\partial\mathcal{L}_{1}}{\partial
g^{\mu\nu}}=-\frac{1}{4}\ g_{\mu\nu}\left(A_{g}\cdot B_{g}\right)^{2}\ .$ (51)

The total stress tensor upto $\mathcal{O}(\lambda)$ is given by
$T_{\mu\nu}^{(\lambda)}=T^{\mu\nu}_{(0)}+\lambda T^{\mu\nu}_{(1)}$. So the
determinant of $T_{\mu\nu}^{(\lambda)}$ becomes

$\displaystyle\det\left(T_{\mu\nu}^{(\lambda)}\right)=\frac{1}{2}\epsilon^{\alpha\mu}\epsilon^{\nu\beta}\left(T_{\mu\nu}^{(0)}+\lambda
T_{\mu\nu}^{(1)}\right)\left(T_{\alpha\beta}^{(0)}+\lambda
T_{\alpha\beta}^{(1)}\right)\ .$ (52)

However to compute the second-order deformation of the Lagrangian, we must
pick up $\mathcal{O}(\lambda)$ terms in the determinant. We find that

$\displaystyle\mathcal{L}_{2}=\frac{1}{4\sqrt{-g}}\epsilon^{\alpha\mu}\epsilon^{\beta\nu}T_{\mu\nu}^{(0)}T_{\alpha\beta}^{(1)}=0\
.$ (53)

Since $\mathcal{L}_{2}=0$, we see $T_{\mu\nu}^{(2)}=0$ as well. In fact, by
induction, it can be shown that all even orders of the deformation vanish,
i.e. $\mathcal{L}_{2n}=0$. They are calculated from

$\sum_{i+j=2n-1}\epsilon^{\alpha\mu}\epsilon^{\beta\nu}\
T_{\mu\nu}^{(i)}T_{\alpha\beta}^{(j)}=\epsilon^{\alpha\mu}\epsilon^{\beta\nu}\
T_{\mu\nu}^{(0)}T_{\alpha\beta}^{(2n-1)}~{},$ (54)

which is identically zero. The odd orders are, however, non-vanishing. To
compute $\mathcal{L}_{3}$ , we pick $\mathcal{O}(\lambda^{2})$ terms in
$\det\left(T_{\mu\nu}^{(\lambda)}\right)$, which gives

$\displaystyle\mathcal{L}_{3}$
$\displaystyle=\frac{1}{3\sqrt{-g}}\det\left(T_{\mu\nu}^{(1)}\right)$ (55)
$\displaystyle=-\frac{\sqrt{-g}}{48}\left(A_{g}\cdot B_{g}\right)^{4}\ .$

and

$\displaystyle T_{\mu\nu}^{(3)}=\frac{1}{16}\,g_{\mu\nu}\left(A_{g}\cdot
B_{g}\right)^{4}~{}.$ (56)

While iterating the algorithm to higher orders, a pattern of deformations
emerges: all odd orders in the $\text{T}\overline{\text{T}}$ coupling constant
$\lambda$ of the stress tensor can be expressed as $g_{\mu\nu}$ multiplied by
even powers of $(A_{g}\cdot B_{g})$, leading us to conclude that

$\mathcal{L}_{2n-1}\propto\sqrt{-g}\ \left(A_{g}\cdot B_{g}\right)^{2n}\ .$
(57)

This uniform pattern allows us to assume an ansatz. In the following section,
we solve eq. 41 to all orders. This will allow us to derive the deformed
Lagrangian in closed form.

####  4.1.2 The Flow Equation

Let us define

$X=\lambda^{2}\left(A_{g}\cdot B_{g}\right)^{2}\ .$ (58)

For the flow equation we assume the ansatz
$\mathcal{L}_{\lambda}=\mathcal{L}_{0}+\widehat{\mathcal{L}}_{(\lambda)}$,
where

$\displaystyle\widehat{\mathcal{L}}_{(\lambda)}=\frac{\sqrt{-g}}{\lambda}F\left(X\right)\
,$ (59)

where $F$ is some function we have to solve for. We will require that $F(X)$
satisfy the boundary condition $F(0)=0$, i.e. that when the deformation is
turned off we recover the seed theory. We now write eq. 41 with the above
ansatz. The left-hand side of the flow equation is

$\frac{\text{d}\mathcal{\widehat{L}}_{\lambda}}{\text{d}\lambda}=-\frac{\sqrt{-g}}{\lambda^{2}}\left[F(X)-2F^{\prime}(X)X\right]\
.$ (60)

We also require the stress tensor, which breaks up into contributions from the
seed theory and the deformation terms

$T_{\mu\nu}=T^{(0)}_{\mu\nu}+T^{(\lambda)}_{\mu\nu}\ ,$ (61)

where

$\displaystyle T^{(\lambda)}_{\mu\nu}$
$\displaystyle=-\frac{2}{\sqrt{-g}}\frac{\partial\widehat{\mathcal{L}}_{(\lambda)}}{\partial
g^{\mu\nu}}$ (62)
$\displaystyle=g_{\mu\nu}\frac{F(X)}{\lambda}-\frac{F^{\prime}(X)}{\lambda}\frac{\partial
X}{\partial g^{\mu\nu}}$
$\displaystyle=\frac{g_{\mu\nu}}{\lambda}\left[F(X)-2F^{\prime}(X)X\right]\ .$

In obtaining the final line of the preceding expression we made use of the
identity eq. 94, derived explicitly in Appendix A.

The determinant of the stress tensor of the deformed theory takes the form:

$\frac{1}{\sqrt{-g}}\det
T_{\mu\nu}=\frac{\sqrt{-g}}{4}\frac{X}{\lambda^{2}}-\frac{\sqrt{-g}}{\lambda^{2}}\left[F(X)-2F^{\prime}(X)X\right]^{2}\
.$ (63)

We will construct the $\text{T}\overline{\text{T}}$ flow equation in the flat
spacetime for which we set $g=-1$. In this limit the flow equation eq. 41
becomes

$\left[F(X)-2F^{\prime}(X)X\right]^{2}-\left[F(X)-2F^{\prime}(X)X\right]-\frac{X}{4}=0$
(64)

which, when completing the square, simplifies to

$F(X)-2XF^{\prime}(X)=\frac{1}{2}\left[1-\sqrt{1+X}\right]\ .$ (65)

Note that when completing the square, we have chosen the positive branch to
ensure consistency with the boundary condition. Now, the above equation can be
solved exactly, and we find

$F(X)=\frac{1}{2}\left[1-\sqrt{1+X}+\sqrt{X}\sinh^{-1}\left(\sqrt{X}\,\right)+2C\,\sqrt{X}\right]$
(66)

and

$\displaystyle
F^{\prime}(X)=\frac{1}{4\sqrt{X}}\left(\sinh^{-1}\left(\sqrt{X}\,\right)+2C\right)\
.$ (67)

Requiring that $F(X)$ has a smooth limit as $X\rightarrow 0$ requires that we
set $C=0$. Consequently, the $\text{T}\overline{\text{T}}$-deformed Lagrangian
is

$\mathcal{L}_{\lambda}=\mathcal{L}_{0}+\frac{1}{2\lambda}\left[1-\sqrt{1+X}+\sqrt{X}\sinh^{-1}\left(\sqrt{X}\,\right)\right]\
.$ (68)

The first two terms in $\hat{\mathcal{L}}$ are strongly reminiscent of the
Nambu-Goto action in static gauge — similar looking terms arise when
describing the $\text{T}\overline{\text{T}}$-deformed free boson Lagrangian.
The last term in the Lagrangian, however, appears novel, and suggests that
this is the first non-local action written in Sen’s formalism.

With a view towards studying the classical dynamics of this theory in the
$\lambda\rightarrow\infty$ limit, we write down the equations of motion for
the $\text{T}\overline{\text{T}}$-deformed theory:

$\displaystyle\text{d}\left(A+\frac{1}{2}\sinh^{-1}\left(\sqrt{X}\,\right)\,B\right)$
$\displaystyle=0\ ,$ (69)
$\displaystyle\text{d}\left(B-\frac{1}{2}\sinh^{-1}\left(\sqrt{X}\,\right)\,A\right)$
$\displaystyle=0\ .$ (70)

Note that we are only describing here the _physical_ sector of the equations
of motion. The decoupled free sector, obviously, remains unaffected by the
deformation.

###  4.2 Infinite Coupling Limit

With the fully resummed $\text{T}\overline{\text{T}}$-deformed Lagrangian in
eq. 68, it is only natural to ask, following our earlier investigations
Chakrabarti:2020pxr , what happens to this theory in the limit of infinite
coupling. In our earlier work, it was important that we work in the
Hamiltonian formalism — this was the only way to sensibly take this limit.

In the present context, the Hamiltonian for Sen’s formulation, as derived in
Sen:2019qit , assumed that all other fields with which the self-dual field
interacts are “ordinary” fields described by more familiar Lagrangians. This
is not the case for the $\text{T}\overline{\text{T}}$-deformed theory at hand,
for which both the interacting (self-dual and anti-self-dual) fields are
described using Sen’s formalism. While we cannot evaluate the Hamiltonian
without a minimal extension of the derivation presented in Sen:2019qit , we
can hope to make some progress at the level of the equations of motion eq. 69.
Note that the $\text{T}\overline{\text{T}}$ coupling $\lambda$ has mass
dimension $-2$. The dimensionless coupling obtained from $\lambda$ is of the
form $m^{2}\lambda=\tilde{\lambda}$, for some mass scale $m$. The limit
$\tilde{\lambda}\rightarrow\infty$ is, of course, well-defined. The
$\lambda\rightarrow\infty$ is to be understood as the scenario where one
adopts units where the mass scale $m$ is set to unity.

Using the following expansion of $\sinh^{-1}(\sqrt{X})$ at large $X$

$\displaystyle\sinh^{-1}\sqrt{X}=\frac{1}{2}\Big{[}2\log 2+\log
X\Big{]}+\mathcal{O}\left(\frac{1}{X}\right)~{},$ (71)

we can write

$\displaystyle\lim_{\lambda\rightarrow\infty}\sinh^{-1}\sqrt{X}=\ln\big{(}\lambda\,A\cdot
B\big{)}~{}=\ln\,\lambda+\ln\big{(}A\cdot B\big{)}~{}.$ (72)

So in the $\lambda\rightarrow\infty$ limit, dropping all terms which are
$\lambda$ independent or subleading, the equation of motions of the
$\text{T}\overline{\text{T}}$-deformed theory simplifies to

$\displaystyle\text{d}\left[\log\lambda~{}B\right]=0\Rightarrow\text{d}B=0~{},$
(73)
$\displaystyle\text{d}\left[\log\lambda~{}A\right]=0\Rightarrow\text{d}A=0~{}.$
(74)

These equations are nothing but the equations of the seed theory on a flat
background. Therefore, we observe chiral decoupling for a
$\text{T}\overline{\text{T}}$-deformed theory of chiral bosons in the limit of
infinite coupling.

###  4.3 Arbitrary Number of Chiral Bosons

A similar calculation can be carried out for an arbitrary (but non-zero)
number of self-dual and anti-self-dual bosons. The stress tensor of the
undeformed Lagrangian can be written as

$\displaystyle
T_{\mu\nu}^{(0)}=-\frac{1}{2}\sum_{i=1}^{N_{c}}A_{(g)\mu}^{i}\,A_{(g)\nu}^{i}\,-\frac{1}{2}\sum_{j=1}^{N_{a}}B_{(g)\mu}^{j}B_{(g)\nu}^{j}~{},$
(75)

where $A_{(g)}^{i}$ and $B_{(g)}^{j}$ are left- and right-chiral bosons, and
$N_{c}$ and $N_{a}$ denote the number of such bosons respectively. As before,
the first deformation is given by

$\displaystyle\mathcal{L}_{1}=\frac{1}{\sqrt{-g}}\det\left(T_{\mu\nu}^{(0)}\right)=\frac{\sqrt{-g}}{4}\sum_{i=1}^{N_{c}}\sum_{j=1}^{N_{a}}\Big{(}A_{(g)}^{i}\cdot
B_{(g)}^{j}\Big{)}^{2}~{},$ (76)

since $A^{i}_{(g)}\cdot A^{j}_{(g)}=0$ and $B^{i}_{(g)}\cdot B^{j}_{(g)}=0$.

The next order of the stress tensor is calculated by varying $\mathcal{L}_{1}$
with respect to the metric, as was done previously. This gives us

$\displaystyle T_{\mu\nu}^{(1)}=-\frac{1}{4}\
g_{\mu\nu}\left[\sum_{i=1}^{N_{c}}\sum_{j=1}^{N_{a}}\left(A_{(g)}^{i}\cdot
B_{(g)}^{j}\right)^{2}\right]~{},$ (77)

The $\mathcal{O}(\lambda^{2})$ deformation $\mathcal{L}_{2}$ is calculated
from the $\mathcal{O}(\lambda)$ term as before, and is given in terms of the
determinant of $T_{\mu\nu}^{(\lambda)}=T_{\mu\nu}^{(0)}+\lambda
T_{\mu\nu}^{(1)}$ as

$\mathcal{L}_{2}=\frac{1}{4\sqrt{-g}}\epsilon^{\alpha\alpha^{\prime}}\epsilon^{\beta\beta^{\prime}}T_{\alpha\beta}^{(0)}T_{\alpha^{\prime}\beta^{\prime}}^{(1)}~{}.\qquad$
(78)

When explicitly evaluated, we see that every term contains terms of the form
$(A^{i}_{(g)}\cdot A_{(g)}^{j})$ or $(B_{(g)}^{i}\cdot B_{(g)}^{j})$ which
vanish identically, implying $\mathcal{L}_{2}=0$.

In fact, from the structure of the deformations, it can be shown that all even
orders of the deformation vanish, i.e. $\mathcal{L}_{2n}=0$ for all $n\geq 1$
for the same reason: they are calculated from the terms of the form

$\epsilon^{\alpha\alpha^{\prime}}\epsilon^{\beta\beta^{\prime}}T_{\alpha\beta}^{(i)}T_{\alpha^{\prime}\beta^{\prime}}^{(j)}$
(79)

with $i+j=2n-1$. Since $T_{\alpha\beta}^{(\text{even})}=0$ the only non-zero
term among these is the one with $(i,j)=(0,2n-1)$. When expanded, this terms
always come with the contractions $(A^{i}_{(g)}\cdot A^{j}_{(g)})$ or
$(B^{i}_{(g)}\cdot B^{j}_{(g)})$. Since these contractions are identically
zero, the conclusion that all even order deformations vanishes follows.

However, the procedure we have outlined suggests that odd orders are non-
vanishing. For example, $\mathcal{O}(\lambda^{3})$ term of the deformed
Lagrangian is given by

$\displaystyle\mathcal{L}_{3}$
$\displaystyle=\frac{1}{3\sqrt{-g}}\det\left(T_{\mu\nu}^{(1)}\right)$ (80)
$\displaystyle=-\frac{\sqrt{-g}}{48}\left[\sum_{i=1}^{N_{c}}\sum_{j=1}^{N_{a}}\left(A_{(g)}^{i}\cdot
B_{(g)}^{j}\right)^{2}\right]^{2}\ .$

Observe that,

$\displaystyle\frac{\partial}{\partial
g^{\mu\nu}}\left[\sum_{i=1}^{N_{c}}\sum_{j=1}^{N_{a}}\left(A_{(g)}^{i}\cdot
B_{(g)}^{j}\right)^{2}\right]$
$\displaystyle=\left[\sum_{i=1}^{N_{c}}\sum_{j=1}^{N_{a}}\left(A_{(g)}^{i}\cdot
B_{(g)}^{j}\right)\left(A_{(g)(\mu}^{i}B_{(g)\nu)}^{j}\right)\right]$ (81)
$\displaystyle=g_{\mu\nu}\left[\sum_{i=1}^{N_{c}}\sum_{j=1}^{N_{a}}\left(A_{(g)}^{i}\cdot
B_{(g)}^{j}\right)^{2}\right]\ ,$

where we have used the identity eq. 94 given in Appendix A. Therefore, by
using the chain rule, one can see that

$\displaystyle\frac{\partial}{\partial
g^{\mu\nu}}\left[\sum_{i=1}^{N_{c}}\sum_{j=1}^{N_{a}}\left(A_{(g)}^{i}\cdot
B_{(g)}^{j}\right)^{2}\right]^{n}=n\,g_{\mu\nu}\left[\sum_{i=1}^{N_{c}}\sum_{j=1}^{N_{a}}\left(A_{(g)}^{i}\cdot
B_{(g)}^{j}\right)^{2}\right]^{n}\ .$ (82)

The term in the square brackets is analogous to $\left(A_{g}\cdot
B_{g}\right)^{2}$ from the case of a pair of chiral bosons. Using eq. 82 we
can easily calculate $T^{(3)}_{\mu\nu}$ as

$\displaystyle
T_{\mu\nu}^{(3)}=\frac{1}{16}\,g_{\mu\nu}\left[\sum_{i=1}^{N_{c}}\sum_{j=1}^{N_{a}}\left(A_{(g)}^{i}\cdot
B_{(g)}^{j}\right)^{2}\right]^{2}~{}.$ (83)

From this discussion, it is clear that a new variable analogous to $X$
appropriate to the case of arbitrary numbers of left- and right-chiral bosons
can be defined as

$\displaystyle
X=\lambda^{2}\left[\sum_{i=1}^{N_{c}}\sum_{j=1}^{N_{a}}\left(A_{(g)}^{i}\cdot
B_{(g)}^{j}\right)^{2}\right]\ .$ (84)

With this $X$, the closed form expression for the
$\text{T}\overline{\text{T}}$-deformed Lagrangian is derived in a manner
similar to the previous section. Thus, the
$\text{T}\overline{\text{T}}$-deformed Lagrangian corresponding to the case of
an arbitrary number of left- and right-chiral bosons is given by eq. 64.

Before proceeding, let us comment on the question of curved spacetime, since
our choice of formalism is naturally developed for any Lorentzian $2$-manifold
$\mathbb{M}$ with an arbitrary metric $g$. There is at present no consensus
regarding status of $\text{T}\overline{\text{T}}$ deformations in an arbitrary
curved background, although some progress has been made for spacetimes with
constant curvature in Jiang:2019tcq . It was found that the paradigmatic
properties of the $\text{T}\overline{\text{T}}$ deformation such as
Zamolodchikov’s factorisation formula no longer continue to hold, and one
finds curvature-induced corrections.

To deform a two-dimensional quantum theory on a curved background, we must
first define the operator $\text{T}\overline{\text{T}}$ as usual, via point-
splitting. For curved backgrounds this involves talking about quantities
defined at different spacetime points — this is, in essence, the source of the
difficulty. It is not our goal to propose a general resolution to this
problem. Rather, we observe that for the specific theories under consideration
in this paper, Sen’s formalism enables us to express the theory _at all
points_ using pseudoforms. The pseudoforms themselves are completely oblivious
to the background metric and therefore allow for the comparison of fields at
different spacetime points. It seems to us that Sen’s formalism is well
positioned to investigate $\text{T}\overline{\text{T}}$ deformations on curved
spacetimes in greater detail. We leave this study for the future.

It is well known that a theory with a chiral anomaly, on being coupled to
gravity produces a theory with a gravitational anomaly. While at the classical
level, these anomalies play no role, a theory with an arbitrary number of
left- and right-chiral bosons coupled to background metric is only consistent
quantum mechanically when the original theory has chiral symmetry, i.e. when
$N_{c}=N_{a}$. Since we focus solely on flat spacetime, these considerations
have no bearing on our analysis.

Before concluding, let us remark that a similar analysis following section 4.2
can be done for the case of arbitrary left and right chiral bosons as well.
The conclusion, as can be easily checked, remains the same. In the general
case also, the equations of motion exhibit chiral decoupling in the limit of
infinite coupling.

##  5 Fermions and Bosonisation

In this section we will solely work in flat space. In this case the pseudoform
coincides with the physical form that is self-dual with respect to $\star$,
the flat space Hodge dual.

###  5.1 Fermions

Let us recall the $\text{T}\overline{\text{T}}$-deformed free fermion
Lagrangian Coleman:2019dvf , which is

$\displaystyle\mathcal{L}_{\lambda}$
$\displaystyle=\mathcal{L}_{0}+\mathcal{L}_{\text{def}.}$ (85)
$\displaystyle=\psi_{-}^{\dagger}\partial\psi_{-}+\psi_{+}^{\dagger}\bar{\partial}\psi_{+}+\lambda\left[\left(\psi_{-}^{\dagger}\partial\psi_{-}\right)\left(\psi_{+}^{\dagger}\bar{\partial}\psi_{+}\right)-\left(\psi_{-}^{\dagger}\bar{\partial}\psi_{-}\right)\left(\psi_{+}^{\dagger}\partial\psi_{+}\right)\right]\
,$

where $\mathcal{L}_{\text{def}.}$ refers to the term linear in $\lambda$. It
was observed that the deformations truncate at linear order in the
$\text{T}\overline{\text{T}}$ coupling. In fact, something more drastic is
true: the first-order (and consequently all subsequent orders) deformation of
the stress tensor vanishes identically! That is,

$T_{\mu\nu}^{(1)}=0\ .$ (86)

To summarise, we have a seed conformal field theory and an irrelevant
deformation thereof, both of which have the same (traceless, of course) stress
tensor, the same degrees of freedom, and the same central charge. To the best
of our knowledge, this has not been remarked on in the literature.

That the stress tensors of the seed theory and the deformed theory coincide,
despite the fact that the deformed theory contains an interaction term, is
highly unusual. The interaction term, it would seem, has no impact on the
energetics of the system.

A possible resolution to this puzzle that immediately comes to mind is that
the free and deformed Lagrangians are related by a nonlinear field
redefinition. If this is the case, one diagnostic in the absence of the
explicit field redefinition would be to compute the S-matrices of the deformed
theory, which should be trivial.

Another possibility that might explain this observation is that the
deformation term is purely topological by nature. In fact, as we will now
explain, this indeed is the correct interpretation. Consider the following
$1$-forms

$S^{+}_{\mu}:=\lambda\,\psi_{+}^{\dagger}\partial_{\mu}\psi_{+}\quad\text{and}\quad
S^{-}_{\mu}:=\lambda\,\psi_{-}^{\dagger}\partial_{\mu}\psi_{-}~{}.$ (87)

The prefactor $\lambda$ ensures that $S^{\pm}$ has the mass dimension expected
of a $1$-form. Then one can readily check that the deformation term of eq. 85
can be expressed as

$\mathcal{L}_{\text{def}.}=\frac{1}{\lambda}\int S^{+}\wedge
S^{-}=\frac{1}{\lambda}\int
d^{2}x\,\epsilon^{\mu\nu}\,S^{+}_{\mu}\,S^{-}_{\nu}~{}.$ (88)

Clearly, this term is purely topological and naturally does not contribute to
the stress tensor. This line of investigation might be able to explain a
puzzle we raised in Chakrabarti:2020pxr : how is it possible for strongly
interacting chiral fermions to, in the infinite coupling limit, decouple? Once
we take the infinite coupling limit such that $S^{\pm}$ is held fixed, we see
that the topological term becomes insignificant, leaving us once again with
the free fermion Lagrangian.

The immutable nature of stress tensor also resolves a puzzle raised in
Coleman:2019dvf : why does the deformation of multiple free fermions truncate
at linear order? The genesis of the puzzle can be traced to the argument that
for a single pair of free fermion the truncation of linear order necessarily
follows due to the lack of non-trivial higher power of bilinears owing to the
grassmann odd nature of $\psi_{\pm}$. Such non-trivial higher power terms, of
course , exists once we introduce multiple numbers of left and right chiral
fermions. As we argued in this section the truncation at linear order follows
due to the fact that the deformed stress tensor is _identical_ to the
undeformed stress tensor. Therefore any subsequent deformations will generate
the same deformation terms and can be re-absorbed in a redefinition of the
coupling $\lambda$. This explanation immediately extends to the case of
multiple fermions hence resolving the puzzle discussed above.

Another observation regarding the deformation term eq. 88, is that it closely
resembles the fermionic Wess-Zumino term in the superstring worldsheet action
as described by Green-Schwarz formalism Green:1983wt . Therefore, the action
for deformed fermions in this sense is indeed a fermionic analogue of the
Nambu-Goto action.

###  5.2 A Bosonisation Map

Let us consider the deformation of the chiral boson theory once again, but
this time we will truncate the deformation to linear order in the
$\text{T}\overline{\text{T}}$ coupling constant. The resulting Lagrangian is
easily seen to be

$\mathcal{L}_{\lambda}=\mathcal{L}_{0}+\frac{\lambda}{4}\left(A\cdot
B\right)^{2}\ .$ (89)

It is tempting to conjecture, based on the observation that both theories have
free chiral degrees of freedom as seed theories, that after
$\text{T}\overline{\text{T}}$ deformation at linear order, there might exist a
bosonisation duality relating the two deformed theories. Indeed, their
respective finite-volume spectra are deformed in a universal manner, dictated
by the inviscid Burgers equation.333We thank Shouvik Datta for emphasising
this point. The fact that for both theories the deformation at linear order
continues naturally generalises to an arbitrary number of left- and right-
chiral fields further strengthens the case for a possible bosonisation
relation. However, as we have already shown, on the bosonic side the
deformation leads to an interacting theory, whereas on the fermionic side, the
deformation does not appear to alter the classical dynamics, as evinced by the
stress tensor remaining unchanged. In this section we speculate on the
possibility of an interplay between $\text{T}\overline{\text{T}}$ deformations
and bosonisation, however far-fetched it may seem, and find that there is some
insight to be gained even with the naive comparison.

Let us focus on the deformation terms. On-shell, the deformation on the
fermion side is

$\lambda\left(\psi_{-}^{\dagger}\bar{\partial}\psi_{-}\right)\left(\psi_{+}^{\dagger}\partial\psi_{+}\right)\
,$ (90)

whereas on the boson side we have

$\frac{\lambda}{4}\left(A\cdot B\right)^{2}\ .$ (91)

Let us write the latter in terms of components, for which we use
$A_{\mu}=(-\mathfrak{a},\mathfrak{a})$ and
$B_{\mu}=(\mathfrak{b},\mathfrak{b})$ as before. We then find the bosonic
interaction term is

$\lambda\,\mathfrak{a}^{2}\,\mathfrak{b}^{2}\ .$ (92)

If we were to naively guess a map, then based on considerations of chirality
and dimension, we might say

$\displaystyle\mathfrak{a}^{2}$
$\displaystyle\longleftrightarrow\psi_{+}^{\dagger}\partial\psi_{+}\ ,$ (93)
$\displaystyle\mathfrak{b}^{2}$
$\displaystyle\longleftrightarrow\psi_{-}^{\dagger}\bar{\partial}\psi_{-}\ .$

While these formula do not look like the usual bosonisation correspondence —
between chiral fermions and vertex operators of chiral bosons — we should keep
in mind that $\mathfrak{a}$ and $\mathfrak{b}$ are really components of the
field strengths.

If we postulate that $\mathfrak{a}\sim\partial_{+}\varphi$ and
$\mathfrak{b}\sim\partial_{-}\varphi$ on-shell, one can readily check the map
proposed above in eq. 93 is related to the more familiar bosonisation
formulas. However, we take the perspective that the identification of
$\mathfrak{a}\sim\partial_{+}\varphi$ etc. is antithetical to the spirit of
Sen’s formalism, not to mention severely limiting any attempt to generalise
this to the corresponding quantum theory. The above map raises an intriguing
question: what do bosonisation formulas look like in Sen’s formalism? The way
forward will be to work entirely within the purview of Sen’s formalism on the
bosonic side of the map. On the fermionic side, the astute reader may have
already noted, the bilinears appearing on RHS of eq. 93 are nothing but self-
dual part of the one-form $S^{+}$ and the anti-self-dual part of the one-form
$S^{-}$ respectively. Another way forward might be to re-write the fermionic
free terms using Sen’s formalism in terms of the (anti) self-dual parts of the
1-forms $S^{\pm}$.

We emphasise that this is merely suggestive of the possibility that
$\text{T}\overline{\text{T}}$ deformations may have an interesting interplay
with bosonisation. More adventurously, one can be tempted to conjecture that
$\text{T}\overline{\text{T}}$-deformations “commute” with bosonisation, i.e.
that bosonisation duality persists under irrelevant deformations. Much more
work would be required to establish that this is in fact the case. For the
case at hand the bosonisation proposed is between a seemingly free theory (of
chiral fermions) and an interacting theory (of chiral bosons), and is
therefore not expected to be entirely accurate. Nonetheless, it is natural to
wonder if a similar bosonisation map will emerge for the cases where
$\text{T}\overline{\text{T}}$ deformation is non-trivial on both sides. For
this question, it might be helpful to start with theories of chiral fermions
which are not free but subjected to some potential.

Indeed, the question of bosonisation can already be asked at the level of the
corresponding free theories itself, where due to the non-standard features of
Sen’s formalism, this already is an interesting question. We expect that the
relations proposed in this manuscript might lead us to the correct map. We
hope to return to these questions — in particular, the construction of chiral
boson vertex operators in Sen’s formalism — in the future.

##  6 Discussion

In this paper, we have studied the $\text{T}\overline{\text{T}}$-deformed
Lagrangian corresponding to a theory with an arbitrary number of left- and
right-chiral bosons. We have established that the deformations persists at all
orders in the $\text{T}\overline{\text{T}}$ coupling and we derived the
deformed Lagrangian in closed form by suitably constructing and exactly
solving the flow equation. The choice to work with Sen’s formalism allowed us
to present the first instance of a $\text{T}\overline{\text{T}}$ deformation
of a theory with a non-standard action.

We also argued that at the limit $\lambda\rightarrow\infty$, the equations of
motion of deformed chiral bosons once again exhibits chiral decoupling. This
presents a very nice completion of the questions raised by the results of
Chakrabarti:2020pxr  and reviewed in the introduction. Schematically, we have
established the following for classical theories of free bosons and their
$\text{T}\overline{\text{T}}$-deformations:

Free
Bosons$\mathrm{T}\overline{\mathrm{T}}$$\mathrm{T}\overline{\mathrm{T}}$$\lambda\rightarrow\infty$$\lambda\rightarrow\infty$
Nambu-Goto String Free Chiral Bosons Novel Non-local Action Chiral Decoupling

The limit $\lambda\rightarrow\infty$ was interpreted as ultra-relativistic
(also called Carollian) limit in Blair:2020ops . There, it was further argued
that the chiral decoupling seen in this limit is a natural expectation based
on the classification given in Morand:2017fnv . Since the non-local action
obtained in eq. 68 also appears to exhibit chiral decoupling in this limit,
this raises the interesting possibility of it being the gauge-fixed form of
some kind of worldsheet theory. In any case, this is the first non-local
action presented in Sen’s formalism, and it is quite interesting on its own
merit. We hope to return to a detailed study of this action and its
quantization in the future.

For the case of $\text{T}\overline{\text{T}}$-deformed free fermions, our
analyses indicate that a resolution to a problem we noted in
Chakrabarti:2020pxr  — why does the infinite coupling limit of the
$\text{T}\overline{\text{T}}$-deformed free fermion yield free chiral
fermions? — is within reach. In this case, the vanishing of corrections to the
stress tensor suggest that the “interacting” chiral fermions were really
always free. This result immediately explains the observed chiral decoupling
of free fermions. Schematically,

Free
Bosons$\mathrm{T}\overline{\mathrm{T}}$$\mathrm{T}\overline{\mathrm{T}}$$\lambda\rightarrow\infty$$\lambda\rightarrow\infty$
Nambu-Goto String Free Chiral Bosons Novel Non-local Action Chiral Decoupling
Free Fermions Free Fermions + Topological Term
$\mathrm{T}\overline{\mathrm{T}}$$\lambda\rightarrow\infty$

We also initiated a study on possible interplay between bosonisation and
$\text{T}\overline{\text{T}}$ deformations. In §5, we looked at the
deformation terms (at linear order) on-shell and proposed a possible
bosonisation relation. Our findings imply that if we can find the bosonisation
map between the free theories of chiral fermions and bosons, then we can
possibly extend such a bosonisation map to the full quantum theory. We also
commented on the more tantalising possibility, that
$\text{T}\overline{\text{T}}$-deformations might “commute” with bosonisation.
However, to shore this correspondence up would require much more work.
Firstly, we need to consider seed theory for fermions whose
$\text{T}\overline{\text{T}}$ deformations indeed deforms the dynamics of the
theory. In particular one must compute and compare deformed correlation
functions. As we have discussed, the standard bosonisation map sends chiral
fermions to vertex operators of chiral bosons. An understanding of how
bosonisation might manifest in Sen’s formalism requires that we first
understand how to construct vertex operators in this formalism. It would be an
interesting exercise to then check that paradigmatic examples of this duality
(for example, the duality between the sine Gordon theory and the massive
Thirring model) can be reproduced. We hope to return to these questions in the
future.

###### Acknowledgements.

We are grateful to Sujay Ashok, Soumangsu Chakraborty, Shouvik Datta, Subham
Dutta Chowdhury, and Ronak Soni for helpful discussions. We especially thank
Dmitri Sorokin for his comments on a previous version of this draft. DG
acknowledges the hospitality of IMSc, Chennai during the initial stages of
this work. MR acknowledges support from the Infosys Endowment for Research
into the Quantum Structure of Spacetime.

##  Appendix A A Useful Identity

In this appendix we derive an useful identity which proves to be very useful
in simplifying the stress tensor for chiral bosons. More specifically, we will
show

$\displaystyle g_{\mu\nu}(A_{g}\cdot
B_{g})=A^{g}_{\mu}B^{g}_{\nu}+A^{g}_{\nu}B^{g}_{\mu}~{},$ (94)

by explicitly evaluating the components on both sides. To avoid clutter, we
drop the subscript $g$ in the rest of this appendix.

The left-hand side of eq. 94 when $(\mu,\nu)=(0,0)$ can be written as

$\displaystyle
g_{00}(A_{0}B^{0}+A_{1}B^{1})=g_{00}(A_{0}B^{0}+A^{0}B_{0})~{},$ (95)

where we have used

$\epsilon^{\mu\nu}A_{\nu}=\sqrt{-g}A^{\mu}\quad\text{and}\quad\epsilon^{\mu\nu}B_{\nu}=-\sqrt{-g}B^{\mu}\
,$ (96)

to rewrite the second term. This equation can be further simplified as

$\displaystyle\quad\>\>A_{0}\left(g_{00}B^{0}\right)+\left(g_{00}A^{0}\right)B_{0}$
$\displaystyle=A_{0}\left(B_{0}-g_{01}B^{1}\right)+\left(A_{0}-g_{01}A^{1}\right)B_{0}$
$\displaystyle=2A_{0}B_{0}-g_{01}\left(A_{0}B^{1}+A^{1}B_{0}\right)$
$\displaystyle=2A_{0}B_{0}~{},$ (97)

which is the R.H.S of (94). In the second last line, we make use of (A).

Similarly, for $(\mu,\nu)=(1,1)$,

$\displaystyle g_{11}\left(A_{0}B^{0}+A_{1}B^{1}\right)$
$\displaystyle=g_{11}(A^{1}B_{1}+A_{1}B^{1})$
$\displaystyle=\left(A_{1}-g_{01}A^{0}\right)B_{1}+A_{1}\left(B_{1}-g_{01}B^{0}\right)$
$\displaystyle=2A_{1}B_{1}-g_{01}\left(A^{0}B_{1}+A_{1}B^{0}\right)$
$\displaystyle=2A_{1}B_{1}~{},$ (98)

and for $(\mu,\nu)=(1,0)$,

$\displaystyle g_{10}\left(A_{0}B^{0}+A_{1}B^{1}\right)$
$\displaystyle=g_{10}\left(A_{0}B^{0}+A^{0}B_{0}\right)$
$\displaystyle=A_{0}\left(B_{1}-g_{11}B^{1}\right)+\left(A_{1}-g_{11}A^{1}\right)B_{0}$
$\displaystyle=A_{0}B_{1}+A_{1}B_{0}-g_{11}\left(A_{0}B^{1}+A^{1}B_{0}\right)$
$\displaystyle=A_{0}B_{1}+A_{1}B_{0}~{}.$ (99)

This establishes eq. 94.

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