aainstitutetext: Indian Institute of Technology Kanpur, Kanpur 208016, INDIA.
bbinstitutetext: Okinawa Institute of Science & Technology, 1919-1 Tancha,
Onna-son, Okinawa 904-0495, JAPAN.
ccinstitutetext: Center for Geometry and Physics, Institute for Basic Science
(IBS), Pohang 37673, KOREA.

# Boosting to BMS

Arjun Bagchi b Aritra Banerjee c and Hisayoshi Muraki
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Bondi-Metzner-Sachs (BMS) symmetries, or equivalently Conformal Carroll
symmetries, are intrinsically associated to null manifolds and in two
dimensions can be obtained as an Inönü-Wigner contraction of the two-
dimensional ($2d$) relativistic conformal algebra. Instead of performing
contractions, we demonstrate in this paper how this transmutation of
symmetries can be achieved by infinite boosts or degenerate linear
transformations on coordinates. Taking explicit cues from the worldsheet
theory of null strings, we show boosting the system is equivalent to adding a
current-current deformation term to the Hamiltonian. As the strength of this
deformation term reaches a critical value, the classical symmetry algebra
“flows” from two copies of Virasoro to the BMS algebra. We further explore the
situation where the CFT coordinates are asymmetrically transformed, and
degenerate limits lead to chiral theories.

##  1 Introduction

Null surfaces are ubiquitous in nature, and the physics associated to these
have time and again appeared in various circumstances, ranging from Quantum
Gravity and black holes to even certain condensed matter systems. These null
surfaces compel us to look beyond our comfort zone of pseudo-Riemannian
manifolds and known Lorentzian physics. Notable instances of null
hypersurfaces include the event horizons of black holes, boundary of causal
diamonds and hypersurfaces appearing at light-like infinity of asymptotically
flat spacetimes ($\mathscr{I}^{\pm}$).

The Riemannian metric formulation for null surfaces fail since the metric
degenerates in this limit and one has to resort to Carrollian structures that
arise on such surfaces. Carroll group is obtained by a contraction of Poincaré
group where the speed of light $c\to 0$, and the associated kinematical
structures allow us to define locally Carroll manifolds LBLL ; NDS ;
Duval:2014uoa ; Duval:2014uva ; Duval:2014lpa . These manifolds are endowed
with a fibre bundle structure that keeps space and time diffeomorphisms
separate from each other. The theories defined on such Carrollian manifolds
can also be seen as a $c\to 0$ (often called Ultra-Relativistic) limit of
relativistic theories. It is important to note the resultant theories are
different from the diametrically opposite Non-Relativistic ($c\to\infty$), or
Galilean limits, which find interpretation as the ones defined on a Newton-
Cartan manifold Duval:2014uoa . However, there are interesting similarities,
which become most manifest in the two dimensional theories we would
investigate in this paper.

A particular motivation behind the recent resurgence in studies of Carrollian
structures stems from interests in asymptotic symmetries of flat spacetimes.
It has been long known from the work of Bondi, van der Burgh, Metzner and
Sachs Bondi:1 ; Sachs:1962zza  that asymptotic symmetry groups on
$\mathscr{I}^{\pm}$ have an infinite dimensional structure, and are often
called the BMS groups Bondi:1 ; Sachs:1962zza ; Barnich:2006av . Since
$\mathscr{I}^{\pm}$ are explicit examples of null manifolds, it is not
surprising that the $(d+1)$ dimensional BMS algebra is isomorphic to the $d$
dimensional Carrollian Conformal Algebra (CCA). The structure of BMS
symmetries (and hence Carrollian physics) has been instrumental in defining
the building blocks of a formulation of the holographic correspondence for
asymptotically flat spacetimes where putative dual field theories live on the
null boundary which are Carrollian manifolds Barnich:2010eb ; Bagchi:2010zz ;
Bagchi:2012cy . This formulation, more recently called Carroll holography, has
met with success in a $3d$ version of flat holography, where the putative dual
is a $2d$ BMS invariant conformal field theory (BMSFT) Bagchi:2012yk ;
Bagchi:2012xr ; Barnich:2012xq ; Bagchi:2013qva ; Bagchi:2015wna ;
Bagchi:2014iea ; Jiang:2017ecm ; Hijano:2017eii ; Bagchi:2013lma ;
Detournay:2014fva ; Afshar:2013vka ; Gonzalez:2013oaa ; Hartong:2015usd ;
Hartong:2015xda ; Bagchi:2016geg ; Barnich:2014cwa ; Fareghbal:2014qga ;
Grumiller:2019xna ; Ciambelli:2018wre . In another avenue, called Celestial
Holography, holography for asymptotically flat spacetimes formulated as a
correspondence between gravity on $4d$ flat space and a $2d$ CFT living on the
celestial sphere has found a lot of success in terms of new results in
asymptotic symmetries and scattering amplitudes. We refer to the excellent
reviews Strominger:2017zoo ; Pasterski:2021rjz ; Raclariu:2021zjz  for a
detailed exposition of these aspects. There is a very recently discovered
connection between these two formulations Bagchi:2022emh , which uses ideas
from Celestial holography to show how to connect Carroll CFT correlations
naturally to $4d$ scattering amplitudes. For ideas linking the two approaches,
see also Donnay:2022aba .

In a parallel development in the context of string theory, Carrollian strings
have appeared in two distinct, albeit intertwined, situations. These
degenerate metric structures may appear on the worldsheet when one considers
the tension of the string going to zero, i.e. a null string theory. First put
forward by Schild Schild:1976vq , and later reinvented in Isberg:1993av , the
idea of null or tensionless strings have gathered momentum recently
considering the $2d$ null worldsheet is a Carrollian manifold. Consequently
the residual gauge symmetry turns out to be governed by the BMS3 algebra.
There have been a number of studies associated to aspects of bosonic
Bagchi:2013bga ; Bagchi:2015nca ; Bagchi:2020fpr , and supersymmetric
Bagchi:2017cte ; Bagchi:2018wsn  tensionless strings which use explicit
Carrollian formulation to study peculiarities of such string theories both at
classical and quantum levels. One also should note that Carrollian strings can
also appear when the target spacetime has some Carrollian structure or an
embedded null hypersurface. It has been shown very recently Bagchi:2020ats ;
Bagchi:2021ban  that a string worldsheet moving into near horizon spacetime
associated to a black hole inherits an induced Carrollian structure and
effectively turns tensionless.

Carrollian and conformal Carrollian symmetries have also recently arisen in
the context of black hole horizons Donnay:2019jiz , cosmology and dark energy
deBoer:2021jej , and in the study of fractons in condensed matter
Bidussi:2021nmp . More and more intriguing new avenues governed by this exotic
symmetry are being uncovered as one considers interesting corners of
relativistically invariant theories.

Figure 1: Collapsing/Expanding lightcone under Carrollian/Galilean limits.

It is thus evident that Conformal Carroll or equivalently BMS algebras are
important as symmetry principles in various physical situations. Our present
endeavour is to present a more physical perspective on how these symmetries
arise. In this work, we will focus on two dimensional theories that are
invariant under the BMS3 or equivalently the $2d$ Conformal Carroll algebra.
These $2d$ Carrollian Conformal Field Theories (CCFTs) or BMSFTs appear as a
$c\to 0$ Inönü-Wigner contraction of $2d$ relativistic CFTs. This points to
the fact that lightcones associated to such relativistic theories close up as
we take the limit, and effectively become just the time axis in the strict
Carrollian limit ($c=0$). One could also think of the opposite Galilean limit
($c\to\infty$) in $2d$, and associated Galilean Conformal Field Theories
(GCFTs), where lightcones open up and span the whole physical spacetime in the
strict limit, see figure (1) for an idea. It turns out in both of these cases,
the contraction of $2d$ relativistic CFT leads to a classically BMS3 symmetric
theory. This is particular to spacetime dimensions $d=2$ where the Galilean
and Carrollian conformal algebras are isomorphic Bagchi:2010zz .

This brings us to the main question posed in this paper, can we quantify the
closing down (or opening up) of the lightcones associated to a relativistic
system as a result of infinite boosts acting on the theory? In a Lorentzian
world, an evolution in boosts should never change any physics. However funny
things can happen when these boost becomes infinite. Non-invertible infinite
boosts that change the structure of lightcones by singular scalings of the
speed of light, can change physics and that too drastically. Here we
concentrate on explicitly showing this particular phenomenon, and subsequently
reach the conclusion that these particular transformations having degenerate
transformation matrices take a relativistic CFT into a BMS invariant field
theory.

Focussing on the $2d$ case, and constantly alluding to results from
tensionless string theory, we will show how to take a continuous path to
transform CFT (holomorphic-antiholomorphic) quantities into BMS ones by
simultaneously boosting and scaling the speed of light via a $SL(2)$
transformation. For a finite transformation, these effects can be undone for a
conformal theory, however in the extreme limits where the lightcones close
down (or open up), the invertibility is lost, and BMS symmetries naturally
emerge at this point. The reader may wonder how our construction here is
different from the infinite boost limits studied earlier in literature, e.g.
in Balasubramanian:2009bg  where the authors end up with a chiral half of a
$2d$ CFT starting out with a usual $2d$ CFT. Infinite boosts naturally end up
putting the original theory on null surfaces and thereby making these theories
Carrollian. If one starts with a $2d$ CFT one always would end up with a $2d$
Carroll CFT or equivalently a $2d$ field theory with BMS3 invariance. In
certain cases, e.g. Bagchi:2012yk , the symmetries of the $2d$ Carroll CFT
reduce to just a single copy of the Virasoro algebra. Our discussions and
observations are thus not in contradiction to Balasubramanian:2009bg . This
case is just a special one of null surface symmetries reducing from BMS to its
Virasoro subalgebra 111We thank an anonymous referee for bringing up this
point..

Returning to our constructions in this paper, we will show that the “flow”
from 2d CFT to the theory with BMS symmetries is achieved by adding a current-
current bilinear, generated by boost transformations, as a deformation term to
the CFT Hamiltonian. This is again reminiscent of what has already been seen
in tensionless string literature, and grants a sanity check for our formalism.
As a culmination of this, we write down an interpolating algebra that takes
one from the conformal algebra to BMS3 via proper dialling of the boost
parameter.

To show the usefulness of this simple yet powerful procedure, we further
discuss applying asymmetric transformations to (anti)holomorphic quantities
which in general have no relation to Lorentz boosts. In a string theoretic
setting, it turns out that a subsection of these generic transformations are
known in the literature in the garb of an unconventional gauge fixing
procedure, known as the Hohm-Siegel-Zwiebach (HSZ) gauge, that replaces the
usual conformal gauge. As the idea suggests, the theory loses conformal
invariance in this case, but an extreme limit of such a theory also takes one
to a null worldsheet reminiscent of an Ambitwistor string theory
Casali:2016atr ; Casali:2017zkz . We use our formulation to show such theories
also give rise to BMS3 as the symmetry algebra.

The paper is organized as follows. In section (2) we revisit the construction
of tensionless limits of string worldsheets and consequent emergence of BMS3
as residual symmetry algebras in this limit. We will also elucidate how
BMSFT/GCFT appears as contractions of relativistic CFTs. Section (3) is the
main part of the paper, where we introduce a formalism to explicitly transform
conformal field theories into BMS invariant theories by doing linear
transformations that mix left and right sectors of the theory. We will show
how this formalism works at every level including that of currents, stress-
energy tensors, and equations of motion. We will also write down an
interpolating symmetry algebra that connects between the two cases. In section
(4) we will demonstrate the effect of generic, non-equivalent transformations
on left and right sectors of a conformal theory. We will then end with some
discussion of our results and a chart of the road ahead. In the appendices (A)
and (B) we will present some details of our transformations and explicit
calculations.

##  2 Null Strings and Scalar BMSFT

As mentioned in the introduction, tensionless limit of string theory is one
where the string worldsheet becomes a Carrollian (or null) manifold. This can
also be identified as the very high-energy limit of conventional string theory
Gross:1 , which shows high temperature phase structures and Hagedorn
transitions Pisarski:1982cn . For the sake of discussions in this paper, we
will have a quick revision of the central concepts of such tensionless
strings, and associated symmetry structures.

###  2.1 Intrinsic theory

Intrinsic discussion of the theory of tensionless strings starts with the ILST
action Isberg:1993av ,

$S_{\text{ILST}}=\int
d^{2}\sigma\,V^{\alpha}V^{\beta}~{}\partial_{\alpha}X\cdot\partial_{\beta}X.$
(1)

Here $V^{\alpha}$ are certain vector densities, there is an assumed background
Minkowski metric where the string propagates and we have suppressed the target
space indices. To derive this action, one starts with the Polyakov action for
the bosonic tensile strings

${}S_{p}=-\frac{T_{0}}{2}\int
d^{2}\sigma\,\sqrt{-\gamma}\gamma^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\eta_{\mu\nu},~{}~{}~{}T_{0}=\frac{1}{2\pi\alpha^{\prime}}$
(2)

For an explicit theory of tensionless string, the worldsheet metric
$\gamma^{\alpha\beta}$ becomes degenerate and hence has to be replaced by
vector densities: $\sqrt{-\gamma}\gamma^{\alpha\beta}\to V^{\alpha}V^{\beta}$
as $T_{0}\to 0$. The equations of motion (EOM) of this action
$S_{\text{ILST}}$ are given by

$\partial_{\alpha}(V^{\alpha}V^{\beta}\partial_{\beta}X^{\mu})=0,\quad
V^{\beta}G_{\alpha\beta}=0,$ (3)

where
$G_{\alpha\beta}=\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\eta_{\mu\nu}$
is the induced metric on the worldsheet.

Now remember in the conformal gauge, where we choose
$\gamma^{\alpha\beta}=\eta^{\alpha\beta}$, the tensile string action can be
written as,

${}S_{p}=-\frac{T_{0}}{2}\int
dzd{\bar{z}}~{}\partial_{z}X\partial_{{\bar{z}}}X,$ (4)

where on the cylinder the (anti)-holomorphic coordinates are,
$z,{\bar{z}}=\tau\pm\phi$ and the associated (anti)-holomorphic currents are
given by $J^{\mu}=\partial_{z}X^{\mu}$ and
$\bar{J}^{\mu}=\partial_{{\bar{z}}}X^{\mu}$. In this form, we can use CFT
techniques on the worldsheet to map out the symmetries associated to it. It
turns out that after fixing the gauge the residual symmetries are generated by
the vector fields

$\boldsymbol{\ell}_{n}=ie^{inz}\partial_{z},\qquad\bar{\boldsymbol{\ell}}_{n}=ie^{in{\bar{z}}}\partial_{{\bar{z}}},$
(5)

classical constraints corresponding to which generate two copies of the
Virasoro (Witt) algebra

$\\{\mathcal{L}_{n},\mathcal{L}_{m}\\}_{\text{PB}}=-i(n-m)\mathcal{L}_{n+m},\qquad\\{\bar{\mathcal{L}}_{n},\bar{\mathcal{L}}_{m}\\}_{\text{PB}}=-i(n-m)\bar{\mathcal{L}}_{n+m}.$
(6)

This is also two copies of diffeomorphisms on a circle, i.e.
Diff($S^{1}$)$\times$Diff($S^{1}$). The equivalent of conformal gauge
description for the null string action turns out to be the gauge
$V^{\alpha}=(1,0)$. The equivalent action simply reads,

$S_{t}=\frac{1}{4\pi c^{\prime}}\int d\tau d\phi(\partial_{\tau}X)^{2}.$ (7)

This action is also the starting point of discussion for a BMS3 or Carrollian
Conformal invariant scalar field in two dimensions Hao:2021urq , when we strip
the scalar field $X$ of the target space index. For most of this work we will
be interpolating between these two descriptions, but will remind the reader
the subtle differences when they arise. This worldsheet theory enjoys certain
diffeomorphism invariance as well, and some gauge symmetry is still left over
even after this gauge fixing. Like the case of two copies of the Virasoro
symmetries for the conformal gauge-fixed tensile strings, the residual gauge
symmetry algebra here is generated by the following vector fields,

$\ell_{n}=ie^{in\phi}(\partial_{\phi}+in\tau\partial_{\tau}),\qquad
m_{n}=ie^{in\phi}\partial_{\tau},$ (8)

with the associated symmetry algebra having the well known form:

$\\{L_{n},L_{m}\\}_{\text{PB}}=-i(n-m)L_{n+m},\quad\\{L_{n},M_{m}\\}_{\text{PB}}=-i(n-m)M_{n+m},\quad\\{M_{n},M_{m}\\}_{\text{PB}}=0.$
(9)

This precisely is the classical part of the BMS3 algebra Bondi:1 ;
Sachs:1962zza , which as we have stated in the introduction, also arises in
the context of gravitational physics as the asymptotic symmetries of $3d$ flat
spacetimes at the null boundary. One can immediately note that having this
form, the (anti)-holomorphic nature of the algebra is gone in the BMS3 case.
This is the main anchoring point of studying strings with a Carrollian
worldsheet and has been explored in detail in recent years along many
interesting directions Bagchi:2013bga ; Bagchi:2015nca ; Bagchi:2016yyf ;
Bagchi:2017cte ; Bagchi:2018wsn ; Bagchi:2019cay ; Bagchi:2020fpr ;
Bagchi:2020ats  222For other related older literature on null strings, the
reader can look at Karlhede:1986wb ; Lizzi:1986nv ; Gamboa:1989px ;
Gustafsson:1994kr ; Lindstrom:2003mg .

##### Intrinsic mode expansions:

In the gauge $V^{\alpha}=(1,0)$, the EOM for the scalars and the constraints
corresponding to the densities $V^{\alpha}$ are:

$\ddot{X}=0,\quad\dot{X}\cdot X^{\prime}=0,\quad\dot{X}^{2}=0,$ (10)

where dots and primes correspond to derivatives with respect to coordinates on
a cylinder, i.e. $\tau$ and $\phi$ respectively. We also would like the
tensionless worldsheet to maintain the closed string boundary conditions of
the form $X^{\mu}(\tau,\phi)=X^{\mu}(\tau,\phi+2\pi)$, like the tensile one,
then the above EOM can solved by the following mode expansion:

$X^{\mu}(\tau,\phi)=x^{\mu}+\sqrt{\frac{c^{\prime}}{2}}B^{\mu}_{0}\tau+i\sqrt{\frac{c^{\prime}}{2}}\sum_{n\neq
0}\frac{1}{n}\left(A^{\mu}_{n}-in\tau B^{\mu}_{n}\right)e^{-in\phi}.$ (11)

Here the $c^{\prime}$ is a finite constant that has been included to take care
of the dimensions. $A,B$ are certain oscillator modes and the conjugate
momenta is given by $\Pi(\tau,\phi)=\sum_{n}B_{n}e^{-in\phi}.$ Using this
expansion, it is easy to find the form of constraints generating the symmetry
algebra

$L_{n}=\frac{1}{2}\sum_{m}A_{-m}\cdot B_{m+n},\qquad
M_{n}=\frac{1}{2}\sum_{m}B_{-m}\cdot B_{m+n}.$ (12)

Note here $A,B$ are not the harmonic oscillator modes in the usual sense, and
using the canonical equal time Poisson brackets for the theory,

$\displaystyle\\{X(\tau,\phi),\Pi(\tau,\phi^{\prime})\\}_{\rm
PB}=\delta(\phi-\phi^{\prime}),$
$\displaystyle\\{X(\tau,\phi),X(\tau,\phi^{\prime})\\}_{\rm
PB}=\\{\Pi(\tau,\phi),\Pi(\tau,\phi^{\prime})\\}_{\rm PB}=0,$ (13)

we find they satisfy a different poisson bracket structure,

$\displaystyle\\{A_{m}^{\mu},A_{n}^{\nu}\\}_{\rm
PB}=\\{B_{m}^{\mu},B_{n}^{\nu}\\}_{\rm
PB}=0;\qquad\\{A_{m}^{\mu},B_{n}^{\nu}\\}_{\rm
PB}=-2im\delta_{m+n}\eta^{\mu\nu}.$ (14)

The algebra of the constraints, of course, still leads to the BMS3 algebra as
before. To make things more transparent, we need to transform $(A,B)$ into a
harmonic oscillator basis:

$C^{\mu}_{n}=\frac{1}{2}({A}^{\mu}_{n}+B^{\mu}_{n}),\quad\tilde{C}^{\mu}_{n}=\frac{1}{2}(-{A}^{\mu}_{-n}+B^{\mu}_{-n}).$
(15)

It is easy to see that $(C,\tilde{C})$ have the usual structure:

$\displaystyle\\{C_{m}^{\mu},C_{n}^{\nu}\\}_{\rm
PB}=\\{\tilde{C}_{m}^{\mu},\tilde{C}_{n}^{\nu}\\}_{\rm
PB}=-im\delta_{m+n}\eta^{\mu\nu}.$ (16)

One can now also write the mode expansions with respect to these oscillators,
and consequently, the expressions for the classical constraints in this
language becomes

$\displaystyle L_{n}=\sum_{m}(C_{-m}\cdot
C_{m+n}-\tilde{C}_{-m}\cdot\tilde{C}_{m-n})$ (17) $\displaystyle
M_{n}=\sum_{m}(C_{-m}\cdot
C_{m+n}+\tilde{C}_{-m}\cdot\tilde{C}_{m-n}+2C_{-m}\cdot\tilde{C}_{-m-n}).$

The form of these generators will be crucial to the later part of this work.

##### Manifestly Carrollian Conformal field theories:

Recently studying intrinsically Carrollian CFTs on null manifolds has gained
some momentum Gupta:2020dtl ; Henneaux:2021yzg ; Hao:2021urq ; Chen:2021xkw ;
deBoer:2021jej ; Bagchi:2022eav . The null string structure we elucidated
above falls under a certain class of Carrollian actions. Due to the degenerate
nature of the system, under Carroll-diffeomorphisms on two dimensional
cylinder $\sigma^{\mu}\rightarrow\sigma^{\prime\mu}(\sigma^{\nu})$, time and
space coordinates change in different manners viz.
$\tau^{\prime}=\tau^{\prime}(\tau,\phi)$ and
$\phi^{\prime}=\phi^{\prime}(\phi)$. Specifically for an infinitesimal Carroll
Conformal transformation, we will have:

$\delta\tau=f^{\prime}(\phi)\tau+g(\phi),~{}\delta\phi=f(\phi),~{}~{}f~{}\&~{}g~{}\text{arbitrary
functions.}$ (18)

These are the explicit transformations generated by the BMS generators.
Carroll boosts, defined on null manifolds, are different from their
relativistic counterparts in the sense of space-time asymmetry, and these
translate conformal fields on a null manifold. Let us now comment on this
point of view for studying null action without going into gory details.

In Carrollian geometry, the Riemannian metric is replaced by two objects: a
degenerate metric $h_{\alpha\beta}$ and the no-where vanishing vector field
$\tau^{\alpha}$ that gives the manifold a null time direction, also called a
Clock-form Duval:2014uoa . The orthogonality between these two ingredients
signify $h_{\alpha\beta}\tau^{\alpha}\tau^{\beta}=0$. Since the metric in
Riemannian sense is degenerate, taking contractions of these metric fields
$(h,\tau)$ with the derivative of a single scalar field i.e.
$\partial_{\alpha}X$, we can construct different classes of Carroll covariant
second order actions. Especially, contracting with $\tau^{\alpha}$ direction
gives rise to the “timelike” action:

$S_{t}\sim\frac{g}{4\pi}\int d\tau
d\phi\,e\,\tau^{\alpha}\tau^{\beta}\,\partial_{\alpha}X\partial_{\beta}X.$
(19)

Here $X$ are scalar fields and $g$ is a coupling constant. This action by
construction is BMS3 invariant, and can only be defined on a Carrollian
manifold. This is the action comparable to our null string action, and clearly
we can see the map:

$V^{\alpha}=\sqrt{e}\tau^{\alpha},$ (20)

explicitly equates them. We will not be using this geometric language in this
work, but it is important to remember that Carrollian theories can be
intrinsically defined to reside on null surfaces. For a more detailed
discussion on geometric aspects with respect to scalars on $2d$ null
manifolds, the reader is referred to Bagchi:2022eav .

###  2.2 Limiting analysis

As we have seen, intrinsically tensionless strings, or equivalently BMS3
invariant scalar fields are defined with respect to null directions on a two
dimensional null manifold. One can also describe these theories by considering
a systematic limiting procedure starting from the corresponding relativistic
theory. This is essentially a limit where we transform the worldsheet
coordinates to an infinitely boosted frame, i.e. the speed of light on the
string worldsheet goes to zero $(c\to 0)$. As we have discussed earlier, this
particular class of limits are called Carrollian limits and when taken on
relativistic CFTs, they lead to corresponding Carrollian CFTs. In terms of
coordinates on the string worldsheet, taking this limit entails Bagchi:2015nca

$\phi\to\phi,\qquad\tau\to\epsilon\tau,\qquad\alpha^{\prime}\to
c^{\prime}/\epsilon,\qquad\epsilon\to 0.$ (21)

To start with one can see systematically taking this limit on the relativistic
action (4) reproduces the null action (7). This contraction is clearly an
Ultra-Relativistic (UR) one where the worldsheet becomes a degenerate manifold
and lightcones shrink towards being just along the vertical axis (see figure
(1)). Consequently the Riemannian worldsheet is replaced by a Carroll
manifold, which mathematically is a fibre bundle. If one consistently follows
through with the above described limit at every stage of computing symmetries,
it leads to the following contraction of the Virasoro generators
Barnich:2006av ; Bagchi:2012cy

$L_{n}=\mathcal{L}_{n}-\bar{\mathcal{L}}_{-n},\qquad
M_{n}=\epsilon(\mathcal{L}_{n}+\bar{\mathcal{L}}_{-n}),$ (22)

where these contracted generators again give rise exactly to the classical
BMS3 algebra. Note that it is absolutely crucial to take the extreme limit
$\epsilon\to 0$ in this case, or for a finite $\epsilon$ the symmetries of the
system remains that of the relativistic conformal symmetries as the brackets
don’t change. Only at the $\epsilon=0$ the new symmetries emerge out of the
Virasoro symmetries. This makes perfect sense since in an evolution in finite
relativistic boosts, the physics of a system must not change.

One can now remember the tensile oscillator mode expansion:

$X^{\mu}(\phi,\tau)=x^{\mu}+2\sqrt{2\alpha^{\prime}}\alpha^{\mu}_{0}\tau+i\sqrt{2\alpha^{\prime}}\sum_{n\neq
0}\frac{1}{n}[\alpha^{\mu}_{n}e^{-in(\tau+\phi)}+\tilde{\alpha}^{\mu}_{n}e^{-in(\tau-\phi)}]$
(23)

Comparing with the tensionless expansions under the limit, and a few lines of
algebra gives rise to the simple relations between tensile and tensionless
oscillators:

$A_{n}^{\mu}=\frac{1}{\sqrt{\epsilon}}\left(\alpha_{n}^{\mu}-\tilde{\alpha}_{-n}^{\mu}\right),\qquad
B_{n}^{\mu}={\sqrt{\epsilon}}\left(\alpha_{n}^{\mu}+\tilde{\alpha}_{-n}^{\mu}\right).$
(24)

Here $\alpha_{n}$ and $\tilde{\alpha}_{n}$ are the tensile oscillators coming
from the usual mode expansion, and these annihilate the tensile vacuum, which
we can call $|0\rangle_{\alpha}$. All classical physics of the tensionless
string can be reproduced by following the UR limit we described above.

Later in the paper, we will also be encountering the exact opposite of the
Carrollian limit, in the form of the so-called Non-Relativistic (NR) or
Galilean contraction Bagchi:2009pe . In this case the $\epsilon$ dependent
contraction of the generators turn out to be,

$L_{n}=\mathcal{L}_{n}+\bar{\mathcal{L}}_{n},\qquad
M_{n}=\epsilon(\mathcal{L}_{n}-\bar{\mathcal{L}}_{n}),$ (25)

In two dimensions this again leads to the classical part of the BMS3 algebra
starting from two copies of the Virasoro algebra. Written in terms of co-
ordinates on the worldsheet, this could easily be envisaged as the very
opposite limit i.e. $\phi\to\epsilon\phi,\tau\to\tau,~{}\epsilon\to 0$. This
corresponds to the limit where the speed of light on the string worldsheet
goes to infinity $(c\to\infty)$.

##  3 BMSFT as an infinitely boosted CFT

###  3.1 Rationale

We have discussed in the previous section how Carroll contractions of
relativistic worldsheet leads to BMS3 invariant theories. In this section, we
will see how an infinite boost on our worldsheet theories leads to a clear
change in physics.

We start with a special linear ($SL(2)$) transformation of our
(anti)-holomorphic coordinates on our (conformal gauge fixed) worldsheet. Let
the transformed theory be defined by changed coordinates $z_{L/R}$, which
mixes contributions from both holomorphic and anti-holomorphic coordinates:

${}\begin{pmatrix}z_{\rm L}\\\ z_{\rm
R}\end{pmatrix}=\frac{1}{\sqrt{1-\alpha^{2}}}\begin{pmatrix}1&-\alpha\\\
-\alpha&1\end{pmatrix}\begin{pmatrix}z\\\
\overline{z}\end{pmatrix}\qquad\quad$ (26)

For real $\alpha$ this is a relativistic (quasi-) boost in $(z,{\bar{z}})$
coordinates. One should remember that on a cylinder $(z,{\bar{z}})$ are
basically lightcone coordinates, which a timelike observer in $(\tau,\phi)$
coordinates can never reach via finite Lorentz boosts, hence nature of boost
in these coordinates is clearly different. One could actually show that boost
transformations along these $(z,\bar{z})$ coordinates are (quasi-) boosts in
$(\tau,\phi)$ coordinates, albeit up to a rescaling of the speed of light. For
more details on this phenomenon, the reader is directed to Appendix A. This
shows up in the transformed two dimensional metric in
$(\tau,\phi)$-coordinates as well, which reads:

${}\begin{pmatrix}\eta^{\tau\tau}&\eta^{\tau\phi}\\\
\eta^{\phi\tau}&\eta^{\phi\phi}\end{pmatrix}=\begin{pmatrix}\frac{1+\alpha}{1-\alpha}&0\\\
0&-\,\frac{1-\alpha}{1+\alpha}\end{pmatrix}$ (27)

So this transformation in these coordinates still keeps the conformal gauge
invariant upto some rescaling of the coordinates, upon which it becomes the
flat Minkowski metric. However this rescaling ceases to work at the singular
points $\alpha=\pm 1$, which again makes perfect sense as at these points the
transformation (26) stops being invertible, and the metric in (27)
degenerates. Unless one hits these singular point, this set of transformations
should not change the worldsheet physics in any shape or form. We can actually
see that identifying this transformation as a scaling,

$\tau\to\,\frac{1-\alpha}{1+\alpha}\tau=\epsilon\,\tau,\qquad\phi\to\phi,\qquad
T_{0}\to\epsilon\,T_{0},$ (28)

keeps the action (4) invariant, unless $\epsilon=0$. One can compare the above
with the UR contraction (21), to see that $\epsilon=0$ point, i.e. BMS point,
corresponds to $\alpha=1$. Thus it can be expected that when (26) stops being
revertible, the worldsheet CFT structure is taken over by BMSFT. This will be
the main logical point throughout our discussion in this section. One should
also note, the other singular point $\alpha=-1$ corresponds to
$\epsilon\to\infty$, which can be rephrased as the NR contraction of the
worldsheet CFT (25). Reaching these two special points will be our notion of
infinite boosts, and both will correspond to a change in symmetry algebras.
See figure (2) for an illustration of this one parameter evolution between the
UR point and NR point.

Figure 2: A continuous one parameter evolution of lightcone from $\alpha=1$
(UR Case) to $\alpha=-1$ (NR Case) in two dimensions. The dotted lines
represent $\alpha=0$, i.e. the relativistic case.

###  3.2 Virasoro generators and Sugawara construction

Let us first set our CFT notations straight for the rest of this paper. Since
the starting point of our theory is a $2d$ CFT, we assume that there exists a
pair of stress-energy tensors subject to the following Poisson brackets, which
give the classical conformal algebra:

$\displaystyle\\{T(\phi),T(\phi^{\prime})\\}_{\rm PB}$
$\displaystyle=2T(\phi)\partial_{\phi}\delta(\phi-\phi^{\prime})+\partial_{\phi}T(\phi)\delta(\phi-\phi^{\prime})$
(29) $\displaystyle\\{\overline{T}(\phi),\overline{T}(\phi^{\prime})\\}_{\rm
PB}$
$\displaystyle=-2\overline{T}(\phi)\partial_{\phi}\delta(\phi-\phi^{\prime})-\partial_{\phi}\overline{T}(\phi)\delta(\phi-\phi^{\prime})$
$\displaystyle\\{{T}(\phi),\overline{T}(\phi^{\prime})\\}_{\rm PB}$
$\displaystyle=0.$

In the quantum case, the Poisson brackets are as usual replaced with
commutators. These stress tensors come with Fourier expansions in terms of
Virasoro generators,

$T(\phi)=\sum_{m}\mathcal{L}_{m}\,e^{-im\phi}\qquad\quad\overline{T}(\phi)=\sum_{m}\overline{\mathcal{L}}_{m}\,e^{+im\phi},$
(30)

where the $\mathcal{L}$’s generate two disjoint copies of the (classical part
of) Virasoro algebra. Note that the algebra of anti-holomorphic stress tensors
do not change under the transformation
$\overline{T}(\phi)\to-\overline{T}(-\phi)$, which enforces the following
automorphism of the Virasoro algebra
$\overline{\mathcal{L}}_{m}\to-\overline{\mathcal{L}}_{-m}$. 333This
automorphism is in the heart of a duality map between UR and NR contractions
of the Virasoro algebra, shown in (22) and (25).  The current algebra (affine-
Lie algebra) for the theory is given by a set of fields

$J^{a}(\phi)=\sum_{m}J_{m}^{a}e^{-im\phi}$ (31)

which in Dirac bracket sense is equipped with the affine Kac-Moody algebraic
relations

$[J^{a}_{m},J^{b}_{n}]=if^{ab}_{c}J^{c}_{m+n}+\frac{k}{2}m\delta^{ab}\delta_{m+n}.$
(32)

Here $f^{ab}_{c}$ are structure constants with totally anti-symmetric indices
$f^{ab}_{c}=f^{bc}_{a}=-f^{ba}_{c}$, and $k$ is the level. The algebra can
also be represented by the Poisson structure defined by:

$\\{J^{a}(\phi),J^{b}(\phi^{\prime})\\}_{\rm
PB}=\left(f^{ab}_{c}J^{c}(\phi)+\frac{k}{2}\delta^{ab}\partial_{\phi}\right)\delta(\phi-\phi^{\prime}).$
(33)

This is true because, on one hand

$\\{J^{a}(\phi),J^{b}(\phi^{\prime})\\}_{\rm
PB}=\sum_{m,n}\\{J^{a}_{m},J^{b}_{n}\\}_{\rm
PB}\,e^{-i(m\phi+n\phi^{\prime})}$ (34)

on the other hand we can see,

$\left(f^{ab}_{c}J^{c}(\phi)+\frac{k}{2}\delta^{ab}\partial_{\phi}\right)\delta(\phi-\phi^{\prime})=\sum_{m,n}\left(f^{ab}_{c}J^{c}_{m+n}-i\frac{k}{2}m\delta^{ab}\delta_{n,-m}\right)e^{-i(m\phi+n\phi^{\prime})}$
(35)

so that the algebra of the modes turn out to be

$\\{J^{a}_{m},J^{b}_{n}\\}_{\rm
PB}=f^{ab}_{c}J^{c}_{m+n}-i\frac{k}{2}m\delta^{ab}\delta_{m+n},$ (36)

which is the classical version of (32). For later purpose, note that for the
anti-holomorphic current, the Fourier modes are expanded in a form

$\overline{J}^{a}(\phi)=\sum_{m}\overline{J}_{m}^{a}e^{+im\phi}.$ (37)

The Poisson bracket for these currents is conventionally chosen to be

$\\{\overline{J}^{a}(\phi),\overline{J}^{b}(\phi^{\prime})\\}_{\rm
PB}=\left(\bar{f}^{ab}_{c}\overline{J}^{c}(\phi)-\frac{\bar{k}}{2}\delta^{ab}\partial_{\phi}\right)\delta(\phi-\phi^{\prime})$
(38)

so as to give

$\\{\overline{J}^{a}_{m},\overline{J}^{b}_{n}\\}_{\rm
PB}=\bar{f}^{ab}_{c}\overline{J}^{c}_{m+n}-i\frac{\bar{k}}{2}m\delta^{ab}\delta_{m+n}.$
(39)

The Virasoro generators are obtained as the Fourier components of a bilinear
form of the currents known as the Sugawara construction. In the (classical)
regime considered here, inspired by the Sugawara construction, let us
introduce the following fields

$T(\phi)=\frac{1}{k}\sum_{a}J^{a}(\phi)J^{a}(\phi),\qquad\overline{T}(\phi)=\frac{1}{{\bar{k}}}\sum_{{a}}\overline{J}^{{a}}(\phi)\overline{J}^{{a}}(\phi)$
(40)

which are shown to obey the relations, in addition to satisfying (29),

$\displaystyle\\{T(\phi),J^{a}(\phi^{\prime})\\}_{\rm PB}$
$\displaystyle=J^{a}(\phi)\partial_{\phi}\delta(\phi-\phi^{\prime})+\partial_{\phi}J^{a}(\phi)\delta(\phi-\phi^{\prime})$
(41)
$\displaystyle\\{\overline{T}(\phi),\overline{J}^{a}(\phi^{\prime})\\}_{\rm
PB}$
$\displaystyle=-\overline{J}^{a}(\phi)\partial_{\phi}\delta(\phi-\phi^{\prime})-\partial_{\phi}\overline{J}^{a}(\phi)\delta(\phi-\phi^{\prime}).$
(42)

These equations give nothing but a representation of the Virasoro algebra, and
they confirm that the fields $J$ transform like a field with conformal weight
one, i.e. a current:

$\\{\mathcal{L}_{m},J_{n}^{a}\\}_{\rm
PB}=inJ^{a}_{m+n},\qquad\\{\bar{\mathcal{L}}_{m},\bar{J}_{n}^{a}\\}_{\rm
PB}=in\bar{J}^{a}_{m+n}.$ (43)

For details of the calculation of these brackets, the reader can refer to
Appendix B.

For sake of completeness, let us explain how to write the direct product
algebras corresponding to holomorphic and anti-holomorphic sectors in a semi-
direct basis. Let us introduce a pair of combinations of currents weighted by
factors of following form,

$\mathbf{J}^{a}=c_{\mathbf{J}}(J^{a}-\overline{J}^{a}),\qquad\mathbf{P}^{a}=c_{\mathbf{P}}(J^{a}+\overline{J}^{a}).$
(44)

Here $c_{\mathbf{J}}$ and $c_{\mathbf{P}}$ are constants, and a simple
calculation yields the Poisson brackets,

$\displaystyle\\{\mathbf{J}^{a}(\phi),\mathbf{J}^{b}(\phi^{\prime})\\}_{\rm
PB}$
$\displaystyle=c_{\mathbf{J}}^{2}\left(f^{ab}_{c}{J}^{c}(\phi)+\bar{f}^{ab}_{c}\overline{J}^{c}(\phi)+\frac{k_{1}}{2}\delta^{ab}\partial_{\phi}\right)\delta(\phi-\phi^{\prime}),$
(45)
$\displaystyle\\{\mathbf{J}^{a}(\phi),\mathbf{P}^{b}(\phi^{\prime})\\}_{\rm
PB}$
$\displaystyle=c_{\mathbf{J}}c_{\mathbf{P}}\left(f^{ab}_{c}{J}^{c}(\phi)-\bar{f}^{ab}_{c}\overline{J}^{c}(\phi)+\frac{k_{2}}{2}\delta^{ab}\partial_{\phi}\right)\delta(\phi-\phi^{\prime}),$
$\displaystyle\\{\mathbf{P}^{a}(\phi),\mathbf{P}^{b}(\phi^{\prime})\\}_{\rm
PB}$
$\displaystyle=c_{\mathbf{P}}^{2}\left(f^{ab}_{c}{J}^{c}(\phi)+\bar{f}^{ab}_{c}\overline{J}^{c}(\phi)+\frac{k_{1}}{2}\delta^{ab}\partial_{\phi}\right)\delta(\phi-\phi^{\prime}).$

with the modified levels $k_{1}=k-\bar{k}$ and $k_{2}=k+\bar{k}$. Making the
choice of $\bar{f}^{ab}_{c}=-{f}^{ab}_{c}$ makes sure the algebra closes to a
current algebra in this basis, upto the rescalings
$\mathbf{J}\to\mathbf{J}/c_{\mathbf{J}}$ and
$\mathbf{P}\to\mathbf{P}/c_{\mathbf{P}}$. Further assuming the identification
$k=\bar{k}$, which is the case of interest for the present paper, the algebra
above is reduced to:

$\displaystyle\\{\mathbf{J}^{a}(\phi),\mathbf{J}^{b}(\phi^{\prime})\\}_{\rm
PB}$
$\displaystyle=c_{\mathbf{J}}f^{ab}_{c}\mathbf{J}^{c}(\phi)\delta(\phi-\phi^{\prime}),$
(46)
$\displaystyle\\{\mathbf{J}^{a}(\phi),\mathbf{P}^{b}(\phi^{\prime})\\}_{\rm
PB}$
$\displaystyle=c_{\mathbf{J}}\left(f^{ab}_{c}\mathbf{P}^{c}(\phi)+c_{\mathbf{P}}\,k\delta^{ab}\partial_{\phi}\right)\delta(\phi-\phi^{\prime}),$
$\displaystyle\\{\mathbf{P}^{a}(\phi),\mathbf{P}^{b}(\phi^{\prime})\\}_{\rm
PB}$
$\displaystyle=\frac{c_{\mathbf{P}}^{2}}{c_{\mathbf{J}}}f^{ab}_{c}\mathbf{J}^{c}(\phi)\delta(\phi-\phi^{\prime}).$

In particular when the group structure is $U(1)$, i.e. all the structure
constants ${f}^{ab}_{c}$ are zero, the algebra coincides with (14) when we
identify $A,B$ to be the Fourier modes of the currents $\mathbf{J}$ and
$\mathbf{P}$, as evidently only the $\\{\mathbf{J},\mathbf{P}\\}$ survives. In
the limiting sense, referring to the definitions in (24), we can identify the
fourier modes of $J$ and $\overline{J}$ as the tensile oscillators $\alpha$
and $\tilde{\alpha}$, with $c_{\mathbf{J}}=1/\sqrt{\epsilon}$ and
$c_{\mathbf{P}}=\sqrt{\epsilon}$. Note that for $U(1)$ current algebra, any
value of $\epsilon$ gives rise to the same bracket structures as above, as
long as ${f}^{ab}_{c}=0$.

Armed with these definitions, we also note that there is a rephrasing of the
algebra for a set of stress-energy tensors in a different (semi-direct) way:

$\displaystyle\\{\mathcal{J}(\phi),\mathcal{J}(\phi^{\prime})\\}_{\rm PB}$
$\displaystyle=2\mathcal{J}(\phi)\partial_{\phi}\delta(\phi-\phi^{\prime})+\partial_{\phi}\mathcal{J}(\phi)\delta(\phi-\phi^{\prime})$
(47) $\displaystyle\\{\mathcal{J}(\phi),\mathcal{P}(\phi^{\prime})\\}_{\rm
PB}$
$\displaystyle=2\mathcal{P}(\phi)\partial_{\phi}\delta(\phi-\phi^{\prime})+\partial_{\phi}\mathcal{P}(\phi)\delta(\phi-\phi^{\prime})$
$\displaystyle\\{\mathcal{P}(\phi),\mathcal{P}(\phi^{\prime})\\}_{\rm PB}$
$\displaystyle=2\mathcal{J}(\phi)\partial_{\phi}\delta(\phi-\phi^{\prime})+\partial_{\phi}\mathcal{J}(\phi)\delta(\phi-\phi^{\prime})$

where we define

$\displaystyle\mathcal{J}(\phi)$
$\displaystyle=T(\phi)-\overline{T}(\phi)\overset{k=\bar{k}}{=}\frac{1}{k}\sum_{a}\left\\{J^{a}(\phi)J^{a}(\phi)-\overline{J}^{{a}}(\phi)\overline{J}^{{a}}(\phi)\right\\}$
(48) $\displaystyle\mathcal{P}(\phi)$
$\displaystyle=T(\phi)+\overline{T}(\phi)\overset{k=\bar{k}}{=}\frac{1}{k}\sum_{a}\left\\{J^{a}(\phi)J^{a}(\phi)+\overline{J}^{{a}}(\phi)\overline{J}^{{a}}(\phi)\right\\}$

This rephrasing will make it easy to get an interpolation from Virasoro
algebra to BMS algebra as described in the next subsection.

###  3.3 Flowing from CFT to BMS

We now move on to the most crucial point of the paper: appearance of BMS from
degenerate transformations on CFT. In the last section we discussed a manifest
semi-direct product structure of Virasoro-Kac-Moody algebras. The current
algebra case for $U(1)$ (46) turned out to be straightforward to understand
and immediately applicable to our case in (24), where
$\\{\mathbf{P}^{a},\mathbf{P}^{b}\\}_{\rm PB}=0$ is the hallmark of appearance
of the BMS-Kac-Moody algebra. A non abelian generalization of this turns out
to be rather tricky with all structure constants present. In what follows, we
will be mostly talking about generic non-Abelian currents, but switching to
the Abelian case when we make contact will null string theory.

#### Currents and energy-momentum tensors

Now we will be interested to take CFT relations written in (anti)-holomorphic
coordinates and manifestly boost them to the degenerate limit we earlier
talked about. To start with, under the linear transform (26), the
corresponding currents shall be transformed in the following way,

$\begin{pmatrix}J^{a}_{\rm L}\\\ J^{a}_{\rm
R}\end{pmatrix}=\frac{1}{\sqrt{1-\alpha^{2}}}\begin{pmatrix}1&\alpha\\\
\alpha&1\end{pmatrix}\begin{pmatrix}J^{a}(z)\\\
\overline{J}^{a}(\bar{z})\end{pmatrix}$ (49)

owing to the chain rule, e.g. $\partial_{\rm L}=\left(\partial z/\partial_{\rm
L}\right)\partial_{z}+\left(\partial\bar{z}/\partial_{\rm
L}\right)\partial_{\bar{z}}$. Now, under this transformation, (48)
correspondingly gets modified as,

$\displaystyle\mathcal{J}(\alpha;\phi)$
$\displaystyle=\frac{1-\alpha^{2}}{1+\alpha^{2}}\cdot\frac{1}{k}\sum_{a}\left(J^{a}_{\rm
L}(\phi)J^{a}_{\rm L}(\phi)-J^{a}_{\rm R}(\phi)J^{a}_{\rm
R}(\phi)\right)=T(\phi)-\overline{T}(\phi)$ (50)
$\displaystyle\mathcal{P}(\alpha;\phi)$
$\displaystyle=\frac{1-\alpha^{2}}{1+\alpha^{2}}\cdot\frac{1}{k}\sum_{a}\left(J^{a}_{\rm
L}(\phi)J^{a}_{\rm L}(\phi)+J^{a}_{\rm R}(\phi)J^{a}_{\rm R}(\phi)\right)$
$\displaystyle=T(\phi)+\overline{T}(\phi)+\frac{2\widetilde{\alpha}}{k}\sum
J^{a}(\phi)\overline{J}^{a}(\phi)~{}~{}\left(\text{where}~{}\widetilde{\alpha}=\frac{2\alpha}{1+\alpha^{2}}\right)$

Written in this form, these fields give rise to the algebra similar to (47):

$\displaystyle\\{\mathcal{J}(\alpha;\phi),\mathcal{J}(\alpha;\phi^{\prime})\\}_{\rm
PB}$
$\displaystyle=\\{\partial_{\phi},\mathcal{J}(\alpha;\phi)\\}\delta(\phi-\phi^{\prime})$
(51)
$\displaystyle\\{\mathcal{J}(\alpha;\phi),\mathcal{P}(\alpha;\phi^{\prime})\\}_{\rm
PB}$
$\displaystyle=\\{\partial_{\phi},\mathcal{P}(\alpha;\phi)\\}\delta(\phi-\phi^{\prime})$
$\displaystyle\\{\mathcal{P}(\alpha;\phi),\mathcal{P}(\alpha;\phi^{\prime})\\}_{\rm
PB}$
$\displaystyle=(1-\widetilde{\alpha}^{2})\\{\partial_{\phi},\mathcal{J}(\alpha;\phi)\\}\delta(\phi-\phi^{\prime}).$

In the above, we use the shorthand notations
$\\{\partial_{\phi},\mathcal{J}\\}=2\mathcal{J}\partial_{\phi}+\partial_{\phi}\mathcal{J}$
and a similar one for $\mathcal{P}$. For details of the calculation of these
brackets, one can refer to Appendix B.2. From this algebra it is again evident
as long as $\widetilde{\alpha}\neq\pm 1$, equivalently ${\alpha}\neq\pm 1$, we
can always write a rescaled version of $\mathcal{P}$ so that we get back to
(47), or equivalently the conformal algebra (29) via boosted stress tensors

$\displaystyle{T}_{\rm L}(\alpha;\phi)$
$\displaystyle=\frac{1}{2}\left(\frac{1}{\sqrt{1-\widetilde{\alpha}^{2}}}{\mathcal{P}}(\alpha;\phi)+{\mathcal{J}}(\alpha;\phi)\right)$
(52) $\displaystyle{T}_{\rm R}(\alpha;\phi)$
$\displaystyle=\frac{1}{2}\left(\frac{1}{\sqrt{1-\widetilde{\alpha}^{2}}}{\mathcal{P}}(\alpha;\phi)-{\mathcal{J}}(\alpha;\phi)\right).$

This rescaling stops making sense precisely at $\alpha=\pm 1$ and the above
algebra can immediately be identified with the classical BMS3 algebra at that
point. Hence, we can continuously connect the Conformal and BMS algebras via
our boost transformations parameterised by $\alpha$.

Now remember, In terms of the Fourier modes, the currents are written as
$J^{a}(\phi)=\sum_{m}J_{m}^{a}e^{-im\phi}$ and
$\overline{J}^{a}(\phi)=\sum_{m}\overline{J}_{m}^{a}e^{+im\phi}$. Using this
and the expressions in (50), we can write the stress tensor modes in terms of
oscillator Fourier modes,

$\displaystyle\mathcal{J}(\alpha;\phi)$
$\displaystyle=\sum_{n}\mathcal{J}_{n}(\alpha)\,e^{-in\phi}\qquad\mathcal{J}_{n}(\alpha)=\frac{1}{k}\sum_{a,m}\left(J_{-m}^{a}J_{m+n}^{a}-\overline{J}_{-m}^{a}\overline{J}_{m-n}^{a}\right)$
(53) $\displaystyle\mathcal{P}(\alpha;\phi)$
$\displaystyle=\sum_{n}\mathcal{P}_{n}(\alpha)\,e^{-in\phi}\qquad\mathcal{P}_{n}(\alpha)=\frac{1}{k}\sum_{m}\left(J_{-m}^{a}J_{m+n}^{a}+\overline{J}_{-m}^{a}\overline{J}_{m-n}^{a}+2\widetilde{\alpha}J_{-m}^{a}\overline{J}_{-m-n}^{a}\right).$
(54)

For a bosonic string, boost dependent currents are related to the
interpolating oscillators $C,\tilde{C}$ (15). Thus, we can take
$J^{\mu}(\phi)=\partial X^{\mu}=\sum_{m}C_{m}^{\mu}e^{-im\phi}$ and
$\overline{J}^{\mu}(\phi)=\overline{\partial}X^{\mu}=\sum_{m}\tilde{C}_{m}^{\mu}e^{+im\phi}$
with $f^{\mu\nu}_{\rho}=0$ and $k=\overline{k}=2$. Then the above expressions,
at $\alpha=1$, perfectly match with what we computed from an intrinsic null
string theory in (LABEL:LM), i.e. $\mathcal{J}_{n}(\alpha=1)=L_{n}$ and
$\mathcal{P}_{n}(\alpha=1)=M_{n}$. In this sense, the transformation from
$\alpha=0$ to $\alpha=1$ takes the associated string theory from a tensile to
a tensionless one. Consequently, the residual symmetry algebra “flows” from
two copies of Virasoro to the BMS algebra.

Since the basics of the construction are now clear, let us try to alternately
define the set of fields $\mathcal{J}$ and $\mathcal{P}$ by introducing a
particular nonlinear class of deformation:

$\displaystyle\mathcal{J}(\phi)$ $\displaystyle=T(\phi)-\overline{T}(\phi)$
(55a) $\displaystyle\mathcal{P}(\phi)$
$\displaystyle=T(\phi)+\overline{T}(\phi)+2\tilde{\alpha}(T(\phi)\overline{T}(\phi))^{\beta}.$
(55b)

One can now compare these with the ones we derived in (50). It can be easily
seen they satisfy the classical BMS3 algebra when we put the values
$\tilde{\alpha}=1,\beta=1/2$,

$\displaystyle\\{\mathcal{J}(\phi),\mathcal{J}(\phi^{\prime})\\}_{\rm PB}$
$\displaystyle=\\{\partial_{\phi},\mathcal{J}(\phi)\\}\delta(\phi-\phi^{\prime})$
(56a) $\displaystyle\\{\mathcal{J}(\phi),\mathcal{P}(\phi^{\prime})\\}_{\rm
PB}$
$\displaystyle=\\{\partial_{\phi},\mathcal{P}(\phi)\\}\delta(\phi-\phi^{\prime})$
(56b) $\displaystyle\\{\mathcal{P}(\phi),\mathcal{P}(\phi^{\prime})\\}_{\rm
PB}$ $\displaystyle=0.$ (56c)

Curiously, this is the case defined in Rodriguez:2021tcz , where the authors
argue that a $\sqrt{T\overline{T}}$ type marginal deformation to the CFT
hamiltonian deforms the conformal symmetries into that of BMS. These classical
$\sqrt{T\overline{T}}$ deformations are different than their well known
irrelevant cousins given by $T\overline{T}$ deformations444See Jiang:2019epa
(and references therein) for a very well-crafted introduction to such
irrelevant deformations., where the $T,\overline{T}$ explicitly correspond to
the stress tensors of the deformed theory and the energy of the system is
determined by an iterative flow equation.

For our case, using the definitions in (40) for current algebra construction,
it is clear that the $\sqrt{T\overline{T}}$-term is explicitly given by the
current-current interaction:

$\sqrt{T(\phi)\overline{T}(\phi)}\ \simeq\
\frac{1}{k}\sum_{a}J^{a}(\phi)\overline{J}^{a}(\phi),$ (57)

thereby establishing an equivalence of our boost-driven formalism to that in
Rodriguez:2021tcz . Certainly at the level of Poisson brackets, this
equivalence holds well and good. In Rodriguez:2021tcz , the authors define
this term as an ad-hoc deformation, which has to be added by hand to jump into
BMS symmetries 555Also note that in a quantum setting, a square root operator
may turn out to be ill-defined.. We, on the other hand, can physically
interpret this term as an infinite limit of finite boost transformations on
the CFT, and as we have shown, the strength of the term induces a flow
throughout which the theory remains conformal, and at the very end the
Carrollian symmetries emerge.

#### Current algebras

As a further remark, let us point out the algebra concerning following
(normalized) currents

$\displaystyle\mathbf{J}^{a}(\alpha;\phi)$
$\displaystyle=\sqrt{\frac{1+\alpha}{1-\alpha}}(J^{a}_{L}-{J}^{a}_{R})(\phi)=J^{a}(\phi)-\overline{J}^{a}(\phi)\
\left(=\mathbf{J}^{a}(0;\phi)\right)$ (58)
$\displaystyle\mathbf{P}^{a}(\alpha;\phi)$
$\displaystyle=\sqrt{\frac{1-\alpha}{1+\alpha}}(J^{a}_{L}+{J}^{a}_{R})(\phi)=J^{a}(\phi)+\overline{J}^{a}(\phi)\
\left(=\mathbf{P}^{a}(0;\phi)\right).$

Note that they do not explicitly depend on the deformation parameter $\alpha$
and thus hereafter we would omit the argument $\alpha$. As one can easily
deduce, the algebra satisfied by them is (46) with
$c_{\mathbf{J}}=c_{\mathbf{P}}=1$. Some straightforward manipulations show the
current algebra with respect to $\mathcal{J}(\alpha;\phi)$ and
$\mathcal{P}(\alpha;\phi)$ has the form,

$\\{\mathcal{J}(\alpha;\phi),\mathbf{J}^{a}(\phi^{\prime})\\}_{\rm
PB}=\partial_{\phi}\left[\mathbf{J}^{a}(\phi)\delta(\phi-\phi^{\prime})\right]$
(59) $\\{\mathcal{J}(\alpha;\phi),\mathbf{P}^{a}(\phi^{\prime})\\}_{\rm
PB}=\partial_{\phi}\left[\mathbf{P}^{a}(\phi)\delta(\phi-\phi^{\prime})\right]$
(60)

while

$\\{\mathcal{P}(\alpha;\phi),\mathbf{J}^{a}(\phi^{\prime})\\}_{\rm
PB}=(1+\widetilde{\alpha})\partial_{\phi}\left[\mathbf{P}^{a}(\phi)\delta(\phi-\phi^{\prime})\right]$
(61)

and most importantly,

$\\{\mathcal{P}(\alpha;\phi),\mathbf{P}^{a}(\phi^{\prime})\\}_{\rm
PB}=(1-\widetilde{\alpha})\partial_{\phi}\left[\mathbf{J}^{a}(\phi)\delta(\phi-\phi^{\prime})\right]-\frac{2\widetilde{\alpha}}{k}\sum_{b,c}f^{a}_{bc}\,\mathbf{P}^{b}(\phi)\mathbf{J}^{c}(\phi)\delta(\phi-\phi^{\prime}).$
(62)

For the purely Abelian case where $f^{a}_{bc}=0$, the above algebra, together
with (51), nicely describes an interpolation between the Virasoro-Kac-Moody
and BMS-Kac-Moody algebras. Note here, for $\alpha=1$ the BMS-Kac-Moody
structure in $\\{\mathcal{J}(\alpha;\phi),\mathbf{P}^{a}(\phi^{\prime})\\}$
and $\\{\mathcal{P}(\alpha;\phi),\mathbf{P}^{a}(\phi^{\prime})\\}$ was also
spotted in Hao:2021urq , as the reader can notice the Fourier modes of
$\mathbf{P}$ are just the $B_{n}$ modes as defined for null strings (12).

The non-Abelian case still is bit problematic at the current level as it
contains the presence of the last term in (62) involving structure constants.
At this moment a clear understanding of the term for non-Abelian case is
missing. However the term can be carefully treated once the system is
quantized. Naively, in the case of Dirac brackets, we can see this term
involving structure constants gives rise to a contact term

$f^{a}_{bc}\,\mathbf{P}^{b}(\phi)\mathbf{J}^{c}(\phi)=\frac{f^{a}_{bc}}{2}[\mathbf{P}^{b}(\phi),\mathbf{J}^{c}(\phi)]\sim
f^{a}_{bc}f^{bc}_{d}\,\mathbf{P}^{d}(\phi)\delta(0)$ (63)

where the divergent piece $\delta(0)$ should be removed by a suitable
prescription of ordering of operators as well as defining a vacua properly.
Again, we have to note this is precisely the problem of ordering of $A,B$
oscillators for the null string theory case, which lead to the discovery of
three distinct possible vacua Bagchi:2020fpr . However, since we are focusing
only on classical case in the present work, this point is beyond the scope at
this moment. We plan to come back to this soon.

#### Hamiltonian and equations of motion

Let us now elaborate more about our $\alpha$ dependent deformation, this time
focusing on the Hamiltonian and the equations of motion. The boosted
Hamiltonian density in this theory is given by 666We can be very much tempted
to write down the analogue of a flow equation from here, which reads:
$\partial_{\widetilde{\alpha}}\widetilde{\mathcal{H}}(\alpha;\phi)\sim
J\overline{J}$ .

$\displaystyle 2\widetilde{\mathcal{H}}(\alpha;\phi)$ $\displaystyle={T}_{\rm
L}(\alpha;\phi)+{T}_{\rm R}(\alpha;\phi)$ (64)
$\displaystyle=\frac{1}{2\sqrt{1-\widetilde{\alpha}^{2}}}\left(\partial
X^{\mu}\partial
X_{\mu}+\overline{\partial}X^{\mu}\overline{\partial}X_{\mu}+2\widetilde{\alpha}\partial
X^{\mu}\overline{\partial}X_{\mu}\right)$
$\displaystyle=\sqrt{\frac{1+\widetilde{\alpha}}{1-\widetilde{\alpha}}}\dot{X}^{2}+\sqrt{\frac{1-\widetilde{\alpha}}{1+\widetilde{\alpha}}}{X^{\prime}}^{2}$
(65)

The conjugate momenta $\widetilde{\Pi}_{\mu}$ can be calculated as

$\widetilde{\Pi}_{\mu}(\alpha;\phi)=\sqrt{\frac{1+\widetilde{\alpha}}{1-\widetilde{\alpha}}}\dot{X}_{\mu}.$
(66)

Now the Hamiltonian density reads:

$\widetilde{\mathcal{H}}(\alpha;\phi)=\sqrt{\frac{1-\widetilde{\alpha}}{1+\widetilde{\alpha}}}\left(\frac{\widetilde{\Pi}^{2}}{2}+\frac{{X^{\prime}}^{2}}{2}\right)$
(67)

Performing a Legendre transform via
$\partial\widetilde{\mathcal{H}}/\partial\widetilde{\Pi}_{\mu}=\dot{X}_{\mu}$,
the corresponding Lagrangian density is given by

$\widetilde{\mathcal{L}}:=\widetilde{\Pi}_{\mu}\dot{X}^{\mu}-\widetilde{\mathcal{H}}=\frac{1}{\kappa}\frac{\dot{X}^{2}}{2}-\kappa\frac{{X^{\prime}}^{2}}{2}$
(68)

with introducing a parameter

$\kappa=\sqrt{\frac{1-\widetilde{\alpha}}{1+\widetilde{\alpha}}}=\frac{1-\alpha}{1+\alpha}$
(69)

The equations of motion associated to this system reads

$\left(\frac{1}{\kappa^{2}}\frac{\partial}{\partial\tau^{2}}-\frac{\partial^{2}}{\partial\phi^{2}}\right)X^{\mu}=0.$
(70)

Comparing this with wave equation for the pullback field $X$, we can see
$\kappa$ acts as a scaled “speed of light”. Therefore, the limit $\alpha\to 1$
corresponds to the limit where the speed of light’ $\kappa$ approaches to zero
(“ultra-relativistic” limit), while $\alpha\to-1$ corresponds to the “non-
relativistic” limit ($\kappa\to\infty$). Under both of these limits the
symmetries of the system boils down to the (classical part of) BMS3 algebra
due to the BMS/GCA duality in two dimensions Bagchi:2010zz . This matches
perfectly with what we discussed at the very beginning of this section.

This duality can also be seen when one calculates the stress energy tensors
from the Lagrangian density (68). While the diagonal components
$T^{t}_{~{}t}=-T^{\phi}_{~{}\phi}=\widetilde{\mathcal{H}}$, making the stress
tensor traceless, the off diagonal components are not equal anymore , in fact
we can see,

$T^{t}_{~{}\phi}=\frac{1}{\kappa}\dot{X}X^{\prime},~{}~{}~{}T^{\phi}_{~{}~{}t}=\kappa\dot{X}X^{\prime}.$
(71)

This means the limit $\alpha\to 1$ corresponds to $T^{\phi}_{~{}~{}t}=0$, i.e.
no energy flux is possible, signaling the emergence of a Carroll boost
invariant theory deBoer:2021jej , while $\alpha\to-1$ will give rise to
$T^{t}_{~{}\phi}=0$, corresponding to a Galilei boost invariant theory. The
seeming divergence in the other two components can actually be killed off by
the simultaneous scaling of the coupling constant (tension) as in (21).

Although the limit $\alpha\to-1$ ($\kappa\to\infty$) appears fine at the level
of equations of motion, it may seem pathological in a sense that it encounters
a singularity where the Hamiltonian density $\widetilde{\mathcal{H}}$ diverges
at the point. To remove this singularity, a useful procedure is to rescale the
variables as

$\kappa=\frac{1}{\kappa_{D}}\qquad\qquad X^{\mu}\ \longrightarrow\
\sqrt{\kappa_{D}}X_{D}^{\mu}\qquad\qquad\widetilde{\Pi}^{\mu}\
\longrightarrow\
\frac{1}{\sqrt{\kappa_{D}}}\widetilde{\Pi}_{D}^{\mu}\left(=\frac{1}{\sqrt{\kappa_{D}}}\dot{X}_{D}^{\mu}\right)$
(72)

as well as
$\,\widetilde{\mathcal{H}}\,\longrightarrow\,\kappa_{D}^{-2}\widetilde{\mathcal{H}}_{D}$,
so that the Hamiltonian density is replaced by the more suitable expression,

$\widetilde{\mathcal{H}}=\kappa\left(\frac{\widetilde{\Pi}^{2}}{2}+\frac{{X^{\prime}}^{2}}{2}\right)\qquad\longrightarrow\qquad\widetilde{\mathcal{H}}_{D}=\kappa_{D}^{2}\left(\frac{1}{\kappa_{D}^{2}}\frac{\widetilde{\Pi}_{D}^{2}}{2}+\frac{{X^{\prime}}_{D}^{2}}{2}\right)$
(73)

The proportionality constant between $\widetilde{\Pi}_{D}$ and $\dot{X}_{D}$
is defined so as to satisfy
${\partial\widetilde{\mathcal{H}}_{D}}/{\partial\widetilde{\Pi}^{\mu}_{D}}=\dot{X}_{D}^{\mu}$.
The Lagrangian density correspondingly gets replaced by

$\mathcal{L}_{D}=\left.\widetilde{\Pi}_{D}\right._{\mu}\dot{X}_{D}^{\mu}-\widetilde{\mathcal{H}}_{D}=\kappa_{D}\left(\frac{1}{\kappa_{D}}\frac{\dot{X}_{D}^{2}}{2}-\kappa_{D}\frac{{X^{\prime}}_{D}^{2}}{2}\right).$
(74)

The equations of motion now take again the form of the wave propagation
equation

$\left(\frac{1}{\kappa_{D}^{2}}\frac{\partial}{\partial\tau^{2}}-\frac{\partial^{2}}{\partial\phi^{2}}\right)X_{D}^{\mu}=0$
(75)

though the “speed of light” is now given by $\kappa_{D}$, not $\kappa$. Thus
the “non-relativistic” limit of taking $\kappa\to\infty$ is equivalent to the
“ultra-relativistic” limit $\kappa_{D}\to 0$ under the duality. While in this
case the limit $\kappa_{D}\to\infty$ ($\alpha\to 1$) is pathological instead
due to the divergence of the dual Hamiltonian density
$\widetilde{\mathcal{H}}_{D}$ at $\alpha=1$. This duality in “speed of light”
could be thought of as equivalent to the treatise of Duval_2014  where two
different notions of defining “time” has been considered to measure
propagation velocity of electromagnetic waves in Carroll or Galilei frames.

##  4 “Asymmetrical boosts” and null theories

For bosonic string theory in flat space, the classical theory is usually
studied by considering the Polyakov action in the conformal gauge (4). Under
the choice of conformal gauge, the residual symmetries are two copies of the
Virasoro algebra, and important physical properties can be analyzed by
applying well-developed techniques in $2d$ CFT.

Although the conformal gauge is very well studied and offers symmetric
structures, other gauge choices have made their appearance in the literature
as well. For our purpose in this paper, we are also interested in some of the
prominent other examples. Let us first talk about the one parameter family of
gauges, called Hohm, Siegel and Zwiebach (HSZ) gauge abc , which can be
achieved by parametrizing the string worldsheet by a global transformation
777One can note that compared to (26), this is a skewed boost transformation
on the (anti)holomorphic coordinates. These can be compared to Galilean or
Carrollian boosts, upto a scale factor, where either of the two coordinates
remains absolute under these transformations.

$\begin{pmatrix}z_{\rm L}\\\ z_{\rm
R}\end{pmatrix}=\frac{1}{\sqrt{1+\beta}}\begin{pmatrix}1+\beta&0\\\
-\beta&1\end{pmatrix}\begin{pmatrix}z\\\ \overline{z}\end{pmatrix}$ (76)

where $\beta$ is some constant for our purposes, resulting in an action of
form

$\mathcal{L}=\partial_{\rm L}X^{\mu}\partial_{\rm R}X_{\mu}=\partial
X\bar{\partial}X+\beta\left(\bar{\partial}X\right)^{2}$ (77)

The resultant action no longer allows one to apply the conventional techniques
of CFT in a straightforward manner, and instead leads to interesting results
Siegel:2015axg . As one can easily see, these asymmetric transformations on
the (anti)-holomorphic coordinates cannot be called (quasi-) boosts in any way
and they do not conserve the conformal gauge. For finite $\beta$, the usual
symmetries of the worldsheet are broken. The singular limit $\beta\to\infty$,
better known as the HSZ limit, is where a special chiral theory appears.

Taking a cue from the above discussion, we can introduce here a multi-
parameter generalization of the worldsheet parametrization in the following
form

$\begin{pmatrix}z_{\rm L}\\\ z_{\rm
R}\end{pmatrix}=\frac{1}{\sqrt{1+\beta-\alpha\gamma}}\begin{pmatrix}1+\beta&-\alpha\\\
-\gamma&1\end{pmatrix}\begin{pmatrix}z\\\ \overline{z},\end{pmatrix}$ (78)

where $\alpha,\beta,\gamma$ are all numbers, and which includes the HSZ
parameterisation as a special case. Depending on the choice of parameters, it
will turn out that the resultant action produces not only the HSZ gauge
action, but also other classes of interesting actions discussed in literature.

###  4.1 Worldsheet actions

For the class of parametrization on the string worldsheet (78), one may define
the action

$\mathcal{L}=\partial_{\rm L}X^{\mu}\partial_{\rm
R}X_{\mu}=\eta^{AB}\frac{\partial X^{\mu}}{\partial\xi^{A}}\frac{\partial
X_{\mu}}{\partial\xi^{B}}$ (79)

where the coordinates $\\{\xi^{A}\\}=\\{z,\bar{z}\\},\ \\{\tau,\sigma\\}$
($z=\tau+\sigma$ and $\bar{z}=\tau-\sigma$), correspond to the gauge choice
where the worldsheet metric components are fixed to

$\begin{pmatrix}\eta^{zz}&\eta^{z\bar{z}}\\\
\eta^{\bar{z}z}&\eta^{\bar{z}\bar{z}}\end{pmatrix}=\begin{pmatrix}\frac{\alpha}{1+\beta-\alpha\gamma}&\frac{1}{2}\left(\frac{1+\beta+\alpha\gamma}{1+\beta-\alpha\gamma}\right)\\\
\frac{1}{2}\left(\frac{1+\beta+\alpha\gamma}{1+\beta-\alpha\gamma}\right)&\frac{\gamma(1+\beta)}{1+\beta-\alpha\gamma}\end{pmatrix}\qquad\begin{pmatrix}\eta^{\tau\tau}&\eta^{\tau\sigma}\\\
\eta^{\sigma\tau}&\eta^{\sigma\sigma}\end{pmatrix}=\begin{pmatrix}\frac{(1+\gamma)(1+\alpha+\beta)}{4(1+\beta-\alpha\gamma)}&\frac{\alpha-\gamma(1+\beta)}{4(1+\beta-\alpha\gamma)}\\\
\frac{\alpha-\gamma(1+\beta)}{4(1+\beta-\alpha\gamma)}&-\frac{(1-\gamma)(1-\alpha+\beta)}{4(1+\beta-\alpha\gamma)}\end{pmatrix}$
(80)

Turning off all the parameters $\alpha=\beta=\gamma=0$, the action coincides
with the bosonic string action of Polyakov type with conformal gauge. Turning
on $\beta$ and $\gamma$ in such a way that $\beta=\gamma$, while keeping
$\alpha=0$, corresponds to the generic HSZ gauge. It should be emphasised that
the parameters $\beta$ and $\gamma$ can be chosen independently, and more
crucial role is played by $\gamma$, while $\beta$ serves as a normalization
factor. It will be transparent by taking a look into the corresponding metric
for the parameter choice $(\alpha=0,\ \beta,\ \gamma)$ 888Note other choice of
parameters, $(\alpha,\ \beta,\ \gamma=0)$ is essentially same but merely flips
the roles of $z$ and $\bar{z}$.

$\begin{pmatrix}\eta^{zz}&\eta^{z\bar{z}}\\\
\eta^{\bar{z}z}&\eta^{\bar{z}\bar{z}}\end{pmatrix}=\begin{pmatrix}0&\frac{1}{2}\\\
\frac{1}{2}&\gamma\end{pmatrix}\qquad\quad\begin{pmatrix}\eta^{\tau\tau}&\eta^{\tau\sigma}\\\
\eta^{\sigma\tau}&\eta^{\sigma\sigma}\end{pmatrix}=\begin{pmatrix}\frac{1+\gamma}{4}&-\frac{\gamma}{4}\\\
-\frac{\gamma}{4}&-\frac{1-\gamma}{4}\end{pmatrix}$ (81)

where the parameter $\beta$ no longer appears in the metric. The metric on
$(\tau,\sigma)$-coordinates clearly shows the presence of off-diagonal
components once we turn on non-zero $\gamma$, resulting in a metric non-
compliant with conformal gauge, indicating that the conventional techniques of
CFT cannot be applied to the system. There are two interesting points to be
mentioned, first the case of $\gamma=\pm 1$ where one of the diagonal
components vanish. The Lagrangian on these points take the form:

$\mathcal{L}_{\gamma=\pm 1}=\begin{cases}\
\frac{1}{2}\dot{X}\left(\dot{X}-X^{\prime}\right)\qquad(\gamma=1)\\\ \
\frac{1}{2}{X}^{\prime}\left(\dot{X}-X^{\prime}\right)\qquad(\gamma=-1).\end{cases}$
(82)

These are identical to the action (or a space-time flip thereof) known as
Floreanini-Jackiw Lagrangian for a chiral boson Floreanini:1987as . One can
show from a Noether procedure that on-shell the symmetries of these actions
boil down to one copy of the Virasoro algebra Bagchi:2022eav .

Another interesting point in the parameter space is obtained by taking the
limit $\gamma\to\infty$, where the worldsheet metric (81) becomes degenerate:
a telltale sign of a null string limit. Normalizing the action appropriately,
the resulting action takes the form

$\widehat{\mathcal{L}}_{\gamma\to\infty}=\left(\bar{\partial}X\right)^{2}.$
(83)

which coincides with the action of Ambitwistor string Casali:2016atr ;
Casali:2017zkz . This choice is a singular gauge, where all the dependence on
the $z$ direction drops out. Intriguingly, one can check that this action can
also be obtained from the second order ILST action (1) using the gauge choice
$V^{\alpha}=(1,-1)$. It has already been shown Casali:2016atr  that the
Ambitwistor action is classically BMS invariant on shell, as we would expect.

###  4.2 On-shell symmetries for the null case

We now discuss the fate of symmetries for these theories. Since our focus is
on null limits of these “boosted” theories, the class of action corresponding
to (83) is a good choice to focus on. As we can also see, we can generate
these classes of action via a $\beta\to\infty$ limit on the HSZ
parameterisation (76), where again only one chiral sector survives in the
action. Let us start with the transformation of currents corresponding to
(76):

$\begin{pmatrix}J^{a}_{\rm L}\\\ J^{a}_{\rm
R}\end{pmatrix}=\frac{1}{\sqrt{1+\beta}}\begin{pmatrix}1&\beta\\\
0&1+\beta\end{pmatrix}\begin{pmatrix}J^{a}(z)\\\
\overline{J}^{a}(\bar{z})\end{pmatrix}$ (84)

and write the suitably $\beta$ dependent symmetry generators as deformed
versions of (48)

$\displaystyle\mathcal{J}(\beta;\phi)$
$\displaystyle=\frac{1}{k\beta}\sum_{a}\left(J^{a}_{\rm L}(\phi)J^{a}_{\rm
L}(\phi)-J^{a}_{\rm R}(\phi)J^{a}_{\rm R}(\phi)\right)$ (85)
$\displaystyle=\frac{1}{k(1+\beta)}\sum_{a}\Big{[}J^{a}(\phi){J}^{a}(\phi)-\overline{J}^{a}(\phi)\overline{J}^{a}(\phi)-2\beta\left(\overline{J}^{a}(\phi)\overline{J}^{a}(\phi)-J^{a}(\phi)\overline{J}^{a}(\phi)\right)\Big{]}$
$\displaystyle\mathcal{P}(\beta;\phi)$
$\displaystyle=\frac{1}{k\beta}\sum_{a}\left(J^{a}_{\rm L}(\phi)J^{a}_{\rm
L}(\phi)+J^{a}_{\rm R}(\phi)J^{a}_{\rm R}(\phi)\right)$
$\displaystyle=\frac{1}{k\beta(1+\beta)}\sum_{a}\Big{[}J^{a}(\phi){J}^{a}(\phi)+\overline{J}^{a}(\phi)\overline{J}^{a}(\phi)+2\beta(1+\beta)\overline{J}^{a}(\phi)\overline{J}^{a}(\phi)+2\beta
J^{a}(\phi)\overline{J}^{a}(\phi)\Big{]}.$

It is difficult to see the symmetries in this general form of the generators,
but taking a consistent $\beta\to\infty$ limit and a simple rescaling, we can
get the $\beta$ independent generators:

$\mathcal{J}_{\infty}=-\overline{T}+\frac{1}{k}\sum_{a}J^{a}(\phi)\overline{J}^{a}(\phi),~{}~{}~{}\mathcal{P}_{\infty}=\overline{T},$
(86)

so that there are only two independent skewed set of generators
$\frac{1}{k}\sum_{a}J^{a}(\phi)\overline{J}^{a}(\phi)$ and $\overline{T}$. To
find the on-shell symmetries in this limit, the simplest way is to resort to
the equations of motion and mode expansion corresponding to (83):

${}\bar{\partial}^{2}X=0,~{}~{}~{}X(z,\overline{z})=\mathcal{A}(z)+(\overline{z}+z)\mathcal{B}(z)$
(87)

The form of the solution makes sure of periodicity on the infinite cylinder
$\phi\sim\phi+2\pi$. We assume $\mathcal{A},\mathcal{B}$ are arbitrary
functions with Fourier expansions

${}\mathcal{A}(z)=\sum_{n}\mathcal{A}_{n}e^{inz},~{}~{}\mathcal{B}(z)=\sum_{n}\mathcal{B}_{n}e^{inz}.$
(88)

The equal-time Poisson brackets for the modes taking the following form:

${}\\{\mathcal{A}_{n},\mathcal{B}_{m}\\}_{\text{PB}}=\delta_{n+m,0},~{}~{}\\{\mathcal{A}_{n},\mathcal{A}_{m}\\}_{\text{PB}}=\\{\mathcal{B}_{n},\mathcal{B}_{m}\\}_{\text{PB}}=0.$
(89)

A simple calculation yields the symmetry generators in terms of Fourier modes
of $\mathcal{J},\mathcal{P}$:

$\displaystyle\mathcal{J}_{n\infty}$ $\displaystyle=$
$\displaystyle\sum_{m}\Big{(}im\mathcal{A}_{-m}\mathcal{B}_{n+m}-in(\overline{z}+z)\mathcal{B}_{-m}\mathcal{B}_{n+m}\Big{)}=L_{n}-in\tau
M_{n}$ (90) $\displaystyle\mathcal{P}_{n\infty}$ $\displaystyle=$
$\displaystyle\sum_{m}\mathcal{B}_{-m}\mathcal{B}_{n+m}=M_{n}.$ (91)

These modes again satisfy the Poisson brackets corresponding to the BMS3
algebra (9).

##  5 Discussions and Conclusions

### Summary

In this paper, we discussed a mechanism to “flow” from a $2d$ CFT to a $2d$
BMSFT (or GCFT) by explicitly boosting the theory, which mirrors earlier
revelations for tensionless limits on tensile strings, but unlike the latter,
does not impose a direct contraction of the coordinates. We “boosted” the $2d$
CFT quantities written in (anti)holomorphic form by a special linear
transformation that turns out to be equivalent to the contraction procedure.
For a symmetrically defined transformation on both Left and Right sector of
the theory, we found this transformation effectively generates a
$J\overline{J}$-like deformation term in the CFT Hamiltonian, whose strength
is proportional to the boost parameter. As we go to the extreme case where the
boost transformation becomes non-invertible, the CFT quantities undergo a
“phase” transition, and BMS invariance sets in. We also could explicitly see
this in the symmetry algebra which, written in proper basis, smoothly
transitions from two copies of Virasoro algebra to the BMS3 algebra when the
boost parameter touches the extreme allowed values.

This simple way to traverse from one theory to another in the space of
symmetries also turned out to give interesting results when left and right
sectors of the conformal theory were not transformed in the same footing. In a
later section we showed, “asymmetric boosts”, which break explicit Lorentz
invariance, leads one from CFT2 to certain intriguing chiral theories. We
focussed on certain special points in this parameter space, corresponding to
theories well known in literature. Especially one could formulate a degenerate
point, where the intrinsic metric of the theory becomes null, and as expected
from physical considerations, emergent BMS3 invariance (on-shell) can be
found.

### Boost vs Acceleration

As far as this work is concerned, we have stuck to effectively boosting our
CFT coordinates to show a transition into BMS symmetries. This makes concrete
the lore in the literature that BMSFTs are infinitely boosted versions of
CFTs. However, especially in the arena of tensionless strings, it has been
shown over the last couple years, that a similar transition happens when one
“accelerates” the relativistic conformally invariant worldsheet Bagchi:2020ats
; Bagchi:2021ban . While a finitely accelerated $2d$ theory maintains two
copies of Virasoro as classical symmetry algebra, the associated vacuum
structure changes with acceleration due to an analogue of the Unruh effect.
BMS3 then emerges at the limit of infinite acceleration, or to go by the
interpretation of Bagchi:2021ban , when a relativistic string worldsheet falls
into a black hole horizon. This is effectively a quantum mechanical statement
following from (15) and (24), which, when combined gives a relationship
between $C$ (Carrollian) oscillators and $\alpha$ (Relativistic) oscillators:

$\displaystyle C^{\mu}_{n}$
$\displaystyle=\frac{1}{2}\Big{(}\sqrt{\epsilon}+\frac{1}{\sqrt{\epsilon}}\Big{)}\alpha^{\mu}_{n}+\frac{1}{2}\Big{(}\sqrt{\epsilon}-\frac{1}{\sqrt{\epsilon}}\Big{)}\tilde{\alpha}^{\mu}_{-n}$
$\displaystyle\tilde{C}^{\mu}_{n}$
$\displaystyle=\frac{1}{2}\Big{(}\sqrt{\epsilon}-\frac{1}{\sqrt{\epsilon}}\Big{)}\alpha^{\mu}_{-n}+\frac{1}{2}\Big{(}\sqrt{\epsilon}+\frac{1}{\sqrt{\epsilon}}\Big{)}\tilde{\alpha}^{\mu}_{n},$
(92)

which is a Bogoliubov transformation that keeps the structure of canonical
commutation relations unchanged for all possible values of
$\epsilon$.999Although this relation has been written down for $\epsilon\to
0$, one can extrapolate this to realize a connected transformation, at least
near $\epsilon=1$, where $C$ and $\alpha$ oscillators become the same. At the
quantum level this points to a one parameter continuous evolution of vacua,
which takes the CFT one to the BMS one.

Now we can ask the question, whether we can see a similar story in our
formulation as well. Our initial investigations seem to suggest that this is
indeed possible and the clue lies in the observation that the structure of
(49) can be read off as

$\displaystyle
J^{a}_{L}=\cosh\theta~{}J^{a}-\sinh\theta~{}\overline{J}^{a},\quad
J^{a}_{R}=-\sinh\theta~{}J^{a}+\cosh\theta~{}\overline{J}^{a}.$ (93)

Here $\cosh\theta=\frac{1}{\sqrt{1-\alpha^{2}}}$ and
$\sinh\theta=-\frac{\alpha}{\sqrt{1-\alpha^{2}}}$ can be interpreted as
Bogoliubov coefficients. We hope to report on to this interesting problem
soon.

### Future directions

This work was aimed at introducing a straightforward formalism of boosting a
$2d$ CFT to a $2d$ BMSFT, and that out of the way, there are clear things we
aim to look at in the near future. First of all, one should note that this
work has been presented in a way suitable for a $2d$ CFT (or a string
worldsheet theory). However, there must be a generalization of this
formulation into higher dimensions, where one doesn’t have the luxuries
associated to $2d$ and the CCA-GCA isomorphism. Since the kinematical
structure of Carrollian algebras remain the same in all dimensions, and the
contraction from relativistic theories are no different as well, one should be
able to understand them in the “boost” perspective as presented in this work.
This might be more challenging while dealing with more complicated nature of
transformations, but should lead one to insightful details about these
theories.

That being said, there are a plethora of problems just in $2d$ which one can
hope to tackle with our formalism. The very first thing that comes into mind
is to quantify Carrollian spinors. Since the speed of light changes gradually
in our “boosts”, that would indeed mean Clifford algebras would also flow from
that of a relativistic one to a degenerate BMS cousin of it. This would mean
structure of Dirac matrices will also be dependent on the boost parameter.
This has obvious consequences, for example in the null worldsheet theory with
Super-BMS algebra as residual symmetries. It has been shown that the
Carrollian cousin of $\mathcal{N}=1$ supersymmetric string has two distinct
versions of the Super-BMS algebra as worldsheet gauge symmetries, based on how
the spinors are treated under contraction. These two point out two distinct
and inequivalent representations of the degenerate Clifford algebra (called
homogeneous and inhomogeneous). One might envision these two as two separate
fixed points in the space of “boosted” spinors, but this obviously has to be
established mathematically.

Lastly, we were content with the classical version of the symmetry algebras we
used throughout this work. It would be nice to instead work with a full
relativistic Wess-Zumino-Witten model and describe the infinitely boosted
version following methods described in this work. In that case we should be
able to see how physical states flow under these transformations, and e.g. how
the central charges change as we approach the BMS invariant point. In a broad
sense both Abelian and Non-Abelian WZW models can be addressed in this regard,
which would be much more general than our particular focus on worldsheet
string theories in this paper. It would also be interesting to see how modular
transformations and partition functions of relativistic theories get deformed
under such infinite boosts. We hope to discuss these aspects in future work.

## Acknowledgements

We thank Pulastya Parekh for initial discussions.

AB is partially supported by a Swarnajayanti Fellowship of the Department of
Science and Technology and Science and Engineering Research Board (SERB) and
also by the following SERB grants SB/SJF/2019-20/08, CRG/2020/002035. The work
of ArB is supported by the Quantum Gravity Unit of the Okinawa Institute of
Science and Technology Graduate University (OIST). ArB would like to thank
Kyoto University and IIT Kanpur for kind hospitality during the last stages of
this work. The work of HM is supported by the Institute for Basic Science
(IBS-R003-D1).

##  Appendix A Coordinate transforms

###  A.1 Lorentz boosts

Let us first recall the notion of a Lorentz boost, leaving the flat Minkowski
metric $\eta_{ab}={\rm diag}(1,-1)$ invariant. A general form of proper
Lorentz boost (connected component of unity) is given by

$\begin{pmatrix}\tau^{\prime}\\\
\sigma^{\prime}\end{pmatrix}=\begin{pmatrix}\cosh\phi&\sinh\phi\\\
\sinh\phi&\cosh\phi\end{pmatrix}\begin{pmatrix}\tau\\\ \sigma\end{pmatrix}$
(94)

Introducing LC coordinates ($z^{\prime}=\tau^{\prime}+\sigma^{\prime}$ and
$\bar{z}^{\prime}=\tau^{\prime}-\sigma^{\prime}$)

$\begin{pmatrix}z^{\prime}\\\
\bar{z}^{\prime}\end{pmatrix}=L_{\phi}\begin{pmatrix}z\\\
\bar{z}\end{pmatrix}\qquad\qquad L_{\phi}=\begin{pmatrix}e^{\phi}&0\\\
0&e^{-\phi}\end{pmatrix}$ (95)

The relation can be summarized by the following diagram.

$\begin{array}[]{ccc}(\tau,\sigma)&\overset{\text{Lorentz
boost}}{\longrightarrow}&(\tau^{\prime},\sigma^{\prime})\\\
\downarrow&&\downarrow\\\
(z,\bar{z})&\overset{L}{\longrightarrow}&(z^{\prime},\bar{z}^{\prime})\end{array}$
(96)

Any proper Lorentz boosts are represented on LC coordinates by a class of
matrices of form

$\mathbb{L}_{\rm LC}=\left\\{L=\begin{pmatrix}\Lambda&0\\\
0&\Lambda^{-1}\end{pmatrix}\left.\right|\,\Lambda\geq 0\right\\}$ (97)

###  A.2 Boosts beyond Lorentz

Let us relax the conditions so that the resultant metric under said
transformation is invariant up-to a scaling of the coordinates, i.e.
$\tilde{\eta}_{ab}={\rm diag}(a^{2},-b^{2})$ with $(a,b\geq 0)$. We can always
use the conformal symmetries in $2d$ to rescale such a metric to the Minkowski
one, and this does not break the structure of the conformal gauge. The
representation matrix of such transformations can be figured out as follows.

$\begin{pmatrix}\tau\\\ \sigma\end{pmatrix}=\begin{pmatrix}A&B\\\
C&D\end{pmatrix}\begin{pmatrix}\tilde{\tau}\\\ \tilde{\sigma}\end{pmatrix}$
(98)

where $(\tau,\sigma)$ is assumed to be defined on Minkowski metric and then

$\left(d\tau\right)^{2}-\left(d\sigma\right)^{2}=\left(A^{2}-C^{2}\right)\left(d\tilde{\tau}\right)^{2}-\left(D^{2}-B^{2}\right)\left(d\tilde{\sigma}\right)^{2}+2\left(AB-
CD\right)d\tilde{\tau}d\tilde{\sigma}$ (99)

The condition for new metric to be of form $\tilde{\eta}_{ab}={\rm
diag}(a^{2},-b^{2})$ imposes

$\begin{cases}\ A^{2}-C^{2}=a^{2}\\\ \ D^{2}-B^{2}=b^{2}\\\ \ AB-
CD=0\end{cases}$ (100)

The first and second relations are solved by parameterizing

$A=a\,\cosh\phi_{a}\qquad B=b\,\sinh\phi_{b}\qquad C=a\,\sinh\phi_{a}\qquad
D=b\,\cosh\phi_{b}$ (101)

and the third relation imposes $\phi_{a}=\phi_{b}=\tilde{\phi}$. For this
case, since
$\left(d\tau\right)^{2}-\left(d\sigma\right)^{2}=a^{2}\left(d\tilde{\tau}\right)^{2}-b^{2}\left(d\tilde{\sigma}\right)^{2}$,
one may introduce analogues of “LC coordinates” by

$\tilde{z}=a\,\tilde{\tau}+b\,\tilde{\sigma}\qquad\quad\bar{\tilde{z}}=a\,\tilde{\tau}-b\,\tilde{\sigma}$
(102)

Then the representation matrix on “LC coordinates” for the transform is

$\begin{pmatrix}\tilde{z}\\\
\bar{\tilde{z}}\end{pmatrix}=\begin{pmatrix}e^{\tilde{\phi}}&0\\\
0&e^{-\tilde{\phi}}\end{pmatrix}\begin{pmatrix}z\\\ \bar{z}\end{pmatrix}$
(103)

The relation can be summarized by the following diagram.

$\begin{array}[]{ccc}(\tau,\sigma)&\overset{\text{``boost''}}{\longrightarrow}&(\tilde{\tau},\tilde{\sigma})\\\
\downarrow&&\downarrow\\\
(z,\bar{z})&\overset{\tilde{L}}{\longrightarrow}&(\tilde{z},\bar{\tilde{z}})\end{array}$
(104)

The “boosts” are represented on “LC coordinates” by a class of matrices of
form

$\tilde{\mathbb{L}}_{\text{``LC''}}=\left\\{\tilde{L}=\begin{pmatrix}\Lambda&0\\\
0&\Lambda^{-1}\end{pmatrix}\left.\right|\,\Lambda\geq 0\right\\}$ (105)

The separation of variables takes place in the same manner as the
(anti-)holomorphic sectors (or left-/right-movers) of closed strings under the
gauge choice of diagonal metric.

While introducing another pair of coordinates,

${z}_{\rm L}=\tilde{\tau}+\tilde{\sigma}\qquad\quad{z}_{\rm
R}=\tilde{\tau}-\tilde{\sigma}$ (106)

in a similar manner, one finds our notion of “boost” is represented by

$\begin{pmatrix}{z}_{\rm L}\\\ {z}_{\rm
R}\end{pmatrix}=\frac{1}{ab}\begin{pmatrix}\frac{a+b}{2}\,e^{-\tilde{\phi}}&-\frac{a-b}{2}\,e^{\tilde{\phi}}\\\
-\frac{a-b}{2}\,e^{-\tilde{\phi}}&\frac{a+b}{2}\,e^{\tilde{\phi}}\end{pmatrix}\begin{pmatrix}z\\\
\bar{z}\end{pmatrix}$ (107)

The one we have called “symmetric boosts” (26) in the main text, is the
special case when $\tilde{\phi}=0$ and
$a=b^{-1}=\sqrt{\frac{1+\alpha}{1-\alpha}}$, which can also be parametrized by
$a=b^{-1}=e^{\phi}$, giving rise to:

$\begin{pmatrix}{z}_{\rm L}\\\ {z}_{\rm
R}\end{pmatrix}=\begin{pmatrix}\cosh\phi&\sinh\phi\\\
\sinh\phi&\cosh\phi\end{pmatrix}\begin{pmatrix}z\\\ \bar{z}\end{pmatrix}$
(108)

resulting in the same form as (94) but the vectors on which the matrices act
are replaced by $(\tau,\sigma)\to(z,\bar{z})$. Interestingly, from this
viewpoint, the global boost (76) achieving the HSZ gauge can be regarded
essentially as (normalized version of) Galilei transforms

$\begin{pmatrix}\tau^{\prime}\\\
\sigma^{\prime}\end{pmatrix}=\begin{pmatrix}1&0\\\
-\beta&1\end{pmatrix}\begin{pmatrix}\tau\\\ \sigma\end{pmatrix}$ (109)

but replacing formally the coordinates $(\tau,\sigma)\to(z,\bar{z})$. These
observations explain why transformations (26) and (76) are something special,
owing to the fact that the two sets of matrices corresponding to those
transformations respectively form one-parameter subgroups in
$\rm{SL}(2,\mathbb{R})$, along which the speed of light varies:

$\displaystyle\mathbb{M}_{\text{``boosts''}}$
$\displaystyle=\left\\{\left.\begin{pmatrix}\cosh\phi&\sinh\phi\\\
\sinh\phi&\cosh\phi\end{pmatrix}\right|\phi\in(-\infty,+\infty)\right\\}\subset\rm{SL}(2,\mathbb{R})$
(110) $\displaystyle\mathbb{M}_{\text{HSZ}}$
$\displaystyle=\left\\{\left.\frac{1}{\sqrt{1+\beta}}\begin{pmatrix}1+\beta&0\\\
-\beta&1\end{pmatrix}\right|\beta\in(-1,+\infty)\right\\}\subset\rm{SL}(2,\mathbb{R})$

##  Appendix B Explicit calculations

###  B.1 Current algebra brackets

The detailed calculations to prove the relations are as follows. Noting the
point-splitting prescription101010 The differential arose from the Poisson
bracket acts on the other fields possessing the same argument, and the
identification of the arguments by virtue of the delta function can be applied
only after all the differentials evaluated.,

$\displaystyle\\{T(\phi),J^{b}(\phi^{\prime})\\}_{\rm PB}$
$\displaystyle=\frac{2}{k}\sum_{a}\\{J^{a}(\phi),J^{b}(\phi^{\prime})\\}_{\rm
PB}J^{a}(\phi)$
$\displaystyle=\frac{2}{k}\sum_{a}\left(f^{ab}_{c}J^{c}(\phi)+\frac{k}{2}\delta^{ab}\partial_{\phi}\right)\delta(\phi-\phi^{\prime})J^{a}(\phi)$
$\displaystyle=\partial_{\phi}\left(\delta(\phi-\phi^{\prime})J^{b}(\phi)\right)$
$\displaystyle=J^{b}(\phi)\partial_{\phi}\delta(\phi-\phi^{\prime})+\partial_{\phi}J^{b}(\phi)\delta(\phi-\phi^{\prime})$
(111)

and

$\displaystyle\\{T(\phi),T(\phi^{\prime})\\}_{\rm PB}$
$\displaystyle=\frac{4}{k^{2}}\sum_{a,b}\\{J^{a}(\phi),J^{b}(\phi^{\prime})\\}_{\rm
PB}J^{a}(\phi)J^{b}(\phi^{\prime})$
$\displaystyle=\frac{4}{k^{2}}\sum_{a,b}\left(f^{ab}_{c}J^{c}(\phi)+\frac{k}{2}\delta^{ab}\partial_{\phi}\right)\delta(\phi-\phi^{\prime})J^{a}(\phi)J^{b}(\phi^{\prime})$
$\displaystyle=\frac{2}{k}\sum_{a}\partial_{\phi}\left(\delta(\phi-\phi^{\prime})J^{a}(\phi)J^{a}(\phi^{\prime})\right)$
$\displaystyle=\frac{2}{k}\sum_{a}\left(J^{a}(\phi)J^{a}(\phi^{\prime})\partial_{\phi}\delta(\phi-\phi^{\prime})+\partial_{\phi}J^{a}(\phi)J^{a}(\phi^{\prime})\delta(\phi-\phi^{\prime})\right)$
$\displaystyle=2T(\phi)\partial_{\phi}\delta(\phi-\phi^{\prime})+\partial_{\phi}T(\phi)\delta(\phi-\phi^{\prime})$
(112)

For the choice of
$\overline{J}^{a}(\phi)=\sum_{m}\overline{J}_{m}^{a}e^{+im\phi}$ where the
corresponding Poisson bracket is

$\\{\overline{J}^{a}(\phi),\overline{J}^{b}(\phi^{\prime})\\}_{\rm
PB}=\left(f^{ab}_{c}\overline{J}^{c}(\phi)-\frac{k}{2}\delta^{ab}\partial_{\phi}\right)\delta(\phi-\phi^{\prime})$
(113)

the corresponding energy momentum tensor shall be

$\overline{T}(\phi)=\frac{1}{k}\sum_{a}\overline{J}^{a}(\phi)\overline{J}^{a}(\phi)$
(114)

so that in their algebraic relations signs are flipped as follows

$\displaystyle\\{\overline{T}(\phi),\overline{J}^{a}(\phi^{\prime})\\}_{\rm
PB}$
$\displaystyle=-\overline{J}^{a}(\phi)\partial_{\phi}\delta(\phi-\phi^{\prime})-\partial_{\phi}\overline{J}^{a}(\phi)\delta(\phi-\phi^{\prime})$
(115) $\displaystyle\\{\overline{T}(\phi),\overline{T}(\phi^{\prime})\\}_{\rm
PB}$
$\displaystyle=-2\overline{T}(\phi)\partial_{\phi}\delta(\phi-\phi^{\prime})-\partial_{\phi}\overline{T}(\phi)\delta(\phi-\phi^{\prime})$
(116)

###  B.2 BMS brackets

Keeping the point-splitting prescription in mind, one finds, as
$\mathcal{J}=T-\overline{T}$ and
$\mathcal{P}=T+\overline{T}+(2\widetilde{\alpha}/k)\sum_{a}J^{a}\bar{J}^{a}$,

$\displaystyle\\{\mathcal{J}(\phi),\mathcal{P}(\phi^{\prime})\\}_{\rm PB}$
$\displaystyle=\\{T(\phi),T(\phi^{\prime})\\}_{\rm
PB}-\\{\overline{T}(\phi),\overline{T}(\phi^{\prime})\\}_{\rm PB}$
$\displaystyle\qquad+\frac{2\widetilde{\alpha}}{k}\\{T(\phi),J^{a}(\phi^{\prime})\\}_{\rm
PB}\overline{J}^{a}(\phi^{\prime})-\frac{2\widetilde{\alpha}}{k}\\{\overline{T}(\phi),\overline{J}^{a}(\phi^{\prime})\\}_{\rm
PB}J^{a}(\phi^{\prime})$
$\displaystyle=2T(\phi)\partial_{\phi}\delta(\phi-\phi^{\prime})+\partial_{\phi}T(\phi)\delta(\phi-\phi^{\prime})$
$\displaystyle\qquad+2\overline{T}(\phi)\partial_{\phi}\delta(\phi-\phi^{\prime})+\partial_{\phi}\overline{T}(\phi)\delta(\phi-\phi^{\prime})$
$\displaystyle\qquad+\frac{2\widetilde{\alpha}}{k}J^{a}(\phi)\overline{J}^{a}(\phi^{\prime})\partial_{\phi}\delta(\phi-\phi^{\prime})+\frac{2\widetilde{\alpha}}{k}\left(\partial_{\phi}J^{a}(\phi)\right)\overline{J}^{a}(\phi^{\prime})\delta(\phi-\phi^{\prime})$
$\displaystyle\qquad+\frac{2\widetilde{\alpha}}{k}J^{a}(\phi)\overline{J}^{a}(\phi^{\prime})\partial_{\phi}\delta(\phi-\phi^{\prime})+\frac{2\widetilde{\alpha}}{k}J^{a}(\phi)\left(\partial_{\phi}\overline{J}^{a}(\phi^{\prime})\right)\delta(\phi-\phi^{\prime})$
$\displaystyle=\\{\partial_{\phi},\mathcal{P}(\phi)\\}\delta(\phi-\phi^{\prime})$
(117)

for which the value of $\widetilde{\alpha}$ is not significant. While
$\\{\mathcal{P},\mathcal{P}\\}_{\rm PB}$ is calculated as follows

$\displaystyle\\{\mathcal{P}(\phi),\mathcal{P}(\phi^{\prime})\\}_{\rm PB}$
$\displaystyle=\\{T(\phi),T(\phi^{\prime})\\}_{\rm
PB}+\\{\overline{T}(\phi),\overline{T}(\phi^{\prime})\\}_{\rm PB}$
$\displaystyle\qquad+\frac{2\widetilde{\alpha}}{k}\\{T(\phi),J^{a}(\phi^{\prime})\\}_{\rm
PB}\overline{J}^{a}(\phi^{\prime})+\frac{2\widetilde{\alpha}}{k}\\{\overline{T}(\phi),\overline{J}^{a}(\phi^{\prime})\\}_{\rm
PB}J^{a}(\phi^{\prime})$
$\displaystyle\qquad+\frac{2\widetilde{\alpha}}{k}\\{J^{a}({\phi}),T(\phi^{\prime})\\}_{\rm
PB}\overline{J}^{a}({\phi})+\frac{2\widetilde{\alpha}}{k}\\{\overline{J}^{a}({\phi}),\overline{T}(\phi^{\prime})\\}_{\rm
PB}J^{a}({\phi})$
$\displaystyle\qquad+\frac{4\widetilde{\alpha}^{2}}{k^{2}}\\{J^{a}({\phi})\overline{J}^{a}({\phi}),J^{b}(\phi^{\prime})\overline{J}^{b}(\phi^{\prime})\\}_{\rm
PB}$ (118)

where the second and the third lines cancel with each other, whereas the final
line is

$\\{J^{a}({\phi})\overline{J}^{a}({\phi}),J^{b}(\phi^{\prime})\overline{J}^{b}(\phi^{\prime})\\}_{\rm
PB}=-\frac{k^{2}}{4}\left(\\{\overline{T}(\phi),\overline{T}(\phi^{\prime})\\}_{\rm
PB}+\\{T(\phi),T(\phi^{\prime})\\}_{\rm PB}\right)$ (119)

thus in total

$\\{\mathcal{P}(\phi),\mathcal{P}(\phi^{\prime})\\}_{\rm
PB}=(1-\widetilde{\alpha}^{2})\\{\partial_{\phi},\mathcal{J}(\phi)\\}\delta(\phi-\phi^{\prime})$
(120)

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