aainstitutetext: Indian Institute of Technology Kanpur, Kanpur 208016, INDIA.
bbinstitutetext: Okinawa Institute of Science & Technology, 1919-1 Tancha,
Onna-son, Okinawa 904-0495, JAPAN.

# Carroll covariant scalar fields in two dimensions

Arjun Bagchi b Aritra Banerjee a Sudipta Dutta a Kedar S. Kolekar a Punit
Sharma
(abagchi, dsudipta, kedarsk<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Conformal Carroll symmetry generically arises on null manifolds and is
important for holography of asymptotically flat spacetimes, generic black hole
horizons and tensionless strings. In this paper, we focus on two dimensional
($2d$) null manifolds and hence on the $2d$ Conformal Carroll or equivalently
the $3d$ Bondi-Metzner-Sachs (BMS) algebra. Using Carroll covariance, we write
the most general free massless Carroll scalar field theory and discover three
inequivalent actions. Of these, two viz. the time-like and space-like actions,
have made their appearance in literature before. We uncover a third that we
call the mixed-derivative theory. As expected, all three theories enjoy off-
shell BMS invariance. Interestingly, we find that the on-shell symmetry of
mixed derivative theory is a single Virasoro algebra instead of the full BMS.
We discuss potential applications to tensionless strings and flat holography.

##  1 Introduction

Null surfaces are essential for understanding many aspects of gravitational
physics and are crucial for quantum gravity as well. The most important of
these null manifolds are the boundary of asymptotically flat spacetime
$\mathscr{I}^{\pm}$ and the event horizon of generic black holes. In
asymptotically flat spacetimes, the asymptotic in and out states are defined
on $\mathscr{I}^{\pm}$ and hence all scattering processes are linked to this
inexorably. $S$-matrix elements are the only observables for a theory of
quantum gravity in flatspace and hence the understanding of the null boundary
is vital.

The event horizons of black holes on the other hand, provide a description of
the degrees of freedom of the black holes themselves. The Bekenstein-Hawking
formula, famously, gives the entropy of a black hole, not in terms of its
volume, but the area of the event horizon. The seeds of the Holographic
Principle are in this above statement. So for a better understanding of black
holes, and indeed the Holographic Principle in general, understanding the
nature of the event horizon is essential.

Null surfaces are endowed with degenerate metrics. In modern parlance, null
surfaces are described by what is called a Carrollian structure. A $(d+1)$
dimensional Carroll manifold is a fibre-bundle with spatial part forming a
$d$-dimensional base space and the null direction forming the $1d$ fibre.

Figure 1: Collapsing of the lightcone towards the Schwarzschild horizon.

Carroll manifolds can be obtained by taking a Carroll or an ultrarelativistic
(UR) limit $c\rightarrow 0$, _i.e._ sending the speed of light to zero, on
(pseudo-) Riemannian manifolds LBLL ; Henneaux:1979vn ; Duval:2014uoa ;
Duval:2014uva ; Duval:2014lpa . A particularly useful way of visualising this
is to think of the familiar Schwarzschild black hole in the usual
Schwarzschild coordinates:

$ds^{2}=-\left(1-\frac{2m}{r}\right)dt^{2}+\frac{dr^{2}}{\left(1-\frac{2m}{r}\right)}+r^{2}d\Omega^{2}.$

Here $m$ is the mass of the Schwartzschild black hole and $r=2m$ is the event
horizon. Let us focus on lightcones and neglect the angular direction at the
moment. The slope of the lightcone is given by

$\frac{dr}{dt}=\pm\left(1-\frac{2m}{r}\right).$

$r\to\infty$ of course gives one the usual flat spacetimes. The speed of
light, in the natural units, in this case is $c=1$. The lightcone, as one
expects, is at 45 degree angles. As one moves into the bulk, with decreasing
$r$, the lightcone shrinks more and more. At $r=2m$, the horizon is hit. The
lightcone closes up completely, the speed of light $c\to 0$ and we get a
Carrollian manifold endowed with a degenerate metric. See figure 1 for a
depiction of this process. This is of course a coordinate dependent picture in
the Schwarzschild background, but any generic Carrollian limit can be
visualised in this way and the coordinate independent statement is that
whenever a null surface is hit, the lightcones will close up and the effective
velocity of light in these codimension-one surfaces will go to zero.

###  1.1 Carroll and Asymptotically Flat Spacetime

The asymptotic symmetries of flat spacetimes have been the cynosure of a lot
of recent attention. It has been known since the 1960’s following the seminal
work of Bondi, van der Burgh, Metzner and Sachs Bondi:1 ; Sachs:1962zza , that
in $4d$ flat spacetimes, the asymptotic symmetry group on $\mathscr{I}^{\pm}$
extends beyond the expected Poincare group to the infinite dimensional BMS
group. The relevant analysis in $3d$ also exhibits a similar behaviour. The
BMS3 algebra reads Barnich:2006av

$\displaystyle[L_{n},L_{m}]=(n-m)L_{n+m}+\frac{c_{L}}{12}(n^{3}-n)\delta_{n+m,0}$
(1a)
$\displaystyle[L_{n},M_{m}]=(n-m)M_{n+m}+\frac{c_{M}}{12}(n^{3}-n)\delta_{n+m,0}$
(1b) $\displaystyle[M_{n},M_{m}]=0.$ (1c)

Here the generators $M_{n}$ generate angle-dependent translations of the null
direction and are called supertranslations and $L_{n}$’s generate the
$\text{Diff}(S^{1})$ of the circle at infinity. We have already stressed that
e.g. $\mathscr{I}^{+}$ is a null manifold and hence has intrinsic Carroll
structures defined on it. So the question is how Carroll symmetries are
related to BMS symmetries. We are interested in understanding theories that
live on the null surface. So any degrees of freedom (d.o.f) should not leave
the surface and this means that these d.o.f. necessarily have to be massless.
We are hence led to considering conformal Carrollian theories or CFTs that
live on Carroll manifolds. Conformal Carroll symmetries, following this
intuition, turn out to be isomorphic to BMS symmetries and this connection can
be proved rigourously. In a nutshell Duval:2014uva ; Duval:2014lpa

$\mathfrak{Ccarr}(d)\simeq\mathfrak{bms}(d+1),$ (2)

or in words, the conformal Carroll algebra turns out to be isomorphic to the
BMS algebra in one higher dimension 111The full story is a bit more intricate.
The conformal isometries of a $d$ dimensional flat Carroll manifold,
generically, generate the conformal Carroll algebra of level $N$,
$\mathfrak{ccarr}_{N}(d)$. The so called ‘dynamical exponent’ $z=\frac{2}{N}$
captures the relative scaling between space and time. For $N=2$, space and
time dilate homogeneously and it is in this case that the conformal Carroll
algebra turns out to be isomorphic to the BMS algebra in one higher dimension,
i.e. $\mathfrak{ccarr}_{2}(d)\simeq\mathfrak{bms}(d+1)$ Duval:2014uva ;
Duval:2014lpa ..

A word or two about the limiting procedure now. From an algebraic perspective,
the Carroll contraction $c\rightarrow 0$ of the Poincaré group gives the
Carroll group, which is the isometry group of the flat Carroll spacetime. We
have said that the conformal Carroll group and its corresponding algebra are
infinite dimensional. However, its relativistic counterpart for $d>2$ has only
finite number of generators. Systematic contractions from relativistic
conformal algebra yield only the global part of conformal Carroll algebra in
dimensions $d\geq 3$. There is a way of generating the infinite algebra by
rewriting the finite algebra in a suggesting basis and lifting the generators
to their infinite dimensional versions Bagchi:2012cy .

In this paper, we will be interested in $2d$ Carrollian and Conformal
Carrollian structures and hence from the point of view of the limit, the
relativistic system from which the Carroll Conformal algebra would be derived
is of course infinite dimensional as well. The parent symmetry of interest is
two copies of the Virasoro algebra. The algebra contracts into the BMS3
algebra as Barnich:2006av ; Bagchi:2012cy

${\mathcal{L}}_{n}-\bar{{\mathcal{L}}}_{-n}=L_{n},\quad{\mathcal{L}}_{n}+\bar{{\mathcal{L}}}_{-n}=\frac{1}{\epsilon}M_{n}.$
(3)

We will call this the ultra-relativistic (UR) contraction of the algebra.
There is another contraction we will be interested in for the purposes of this
paper. This is the non-relativistic (NR) contraction Bagchi:2009pe  which is

${\mathcal{L}}_{n}+\bar{{\mathcal{L}}}_{n}=L_{n},\quad{\mathcal{L}}_{n}-\bar{{\mathcal{L}}}_{n}=\frac{1}{\epsilon}M_{n}.$
(4)

This also yields the BMS3 from two copies of Virasoro. This magic of two
dimensions makes the NR and UR contractions give isomorphic algebras in the
limit. This is because the corresponding Galilean and Carrollian algebras as
well as their infinite dimensional conformal cousins are isomorphic in $d=2$
Bagchi:2010zz .

We will elaborate especially on the geometric perspective of Carroll and
conformal Carroll symmetries in two dimensions in this work. Below we discuss
where conformal Carroll symmetries find uses in modern theoretical physics.

###  1.2 Flatspace Holography

Borrowing popular wisdom from the AdS/CFT correspondence Maldacena:1997re , we
can postulate that matching of asymptotic symmetries of the bulk gravitational
theory with the global symmetries of the dual boundary quantum field theory is
the building block of the formulation of a holographic correspondence in a
general spacetime.

For asymptotically flat spacetimes, the null boundaries are Carrollian
manifolds, and the associated asymptotic symmetry group (ASG) is the BMS group
Bondi:1 ; Sachs:1962zza ; Barnich:2006av . Following this line of logic, the
formulation of holographic correspondence for asymptotically flat spacetimes
should involve field theories living on null boundaries governed by the BMS
group as global symmetries Barnich:2010eb ; Bagchi:2010zz ; Bagchi:2012cy .

This approach, nowadays known as Carrollian holography, has indeed met with
some success for holography for lower dimensional (specifically $3d$)
asymptotically flat spacetimes. In $3d$ bulk with BMS3 as the ASG, some
concrete developments involve a proposal for a field theory dual Bagchi:2012yk
, matching of entropy Bagchi:2012xr ; Barnich:2012xq ; Bagchi:2013qva , the
computation of stress tensor correlators Bagchi:2015wna , entanglement entropy
Bagchi:2014iea ; Jiang:2017ecm ; Hijano:2017eii , and more. A selected, non-
exhaustive list of papers along these directions are Bagchi:2013lma ;
Detournay:2014fva ; Afshar:2013vka ; Gonzalez:2013oaa ; Hartong:2015usd ;
Hartong:2015xda ; Bagchi:2016geg ; Barnich:2014cwa ; Fareghbal:2014qga ;
Grumiller:2019xna ; Ciambelli:2018wre  and for related higher dimensional
stories, the reader can have a look at Bagchi:2016bcd .

Along another distinct avenue, particularly in four dimensions, following the
initiative by Strominger and collaborators, a lot of progress have been made
in linking up asymptotic symmetries of a $4d$ flat space with a dual $2d$ CFT
living on the Celestial Sphere. This approach, often called Celestial
Holography, has generated a lot of interest and a lot of novel results for 4d
scattering amplitudes and related asymptotic symmetries. For a glimpse of
recent developments one could look at the excellent reviews Strominger:2017zoo
; Pasterski:2021rjz ; Raclariu:2021zjz .

More recently attempts have been made to construct a bridge between these two
formulations Donnay:2022aba ; Bagchi:2022emh  and it has been crucially shown
inspired by ideas from Celestial holography that Carroll CFTs provide a
concrete framework for understanding $S$-matrix elements Bagchi:2022emh , thus
developing a connection that was previously lacking.

Hence a road to holography in asymptotically flat spacetimes is crucially
dependent on understanding conformal Carroll symmetry and the field theories
which realise these symmetries. In this paper, we will be specifically
interested in two dimensional theories and hence the algebra (1).

###  1.3 Horizons, Carroll strings and other things

Having discussed flat holography which is one of our principal motivations
behind studying Carroll CFTs, let us mention a few more important
applications.

As we have mentioned before, another null manifold of interest is the event
horizon of a generic black hole. Indeed, BMS (or “BMS-like”) symmetries have
been found recently on these event horizons Hawking:2016msc ; Hawking:2016sgy
; Donnay:2015abr ; Afshar:2016wfy ; Penna:2017bdn . Geometrically, Carrollian
structures have also been shown to emerge on the event horizons Donnay:2019jiz
. Naturally, it seems conceivable that a better understanding of the dynamics
of black hole event horizons could be achieved by studying BMS invariant field
theories living on the event horizons as putative duals. Some of the recent
success in this direction involves the computation of black hole entropy using
the BMS-Cardy formula Bagchi:2012xr ; Carlip:2017xne ; Carlip:2019dbu .

The discussion in the previous paragraphs involved null manifolds mostly in
holographic scenarios, where the BMS group is a global symmetry. Carrollian
structures also appear in a different context, where the BMS symmetries are
gauge symmetries, _viz_ the tensionless or null strings Schild:1976vq ;
Isberg:1993av ; Bagchi:2013bga ; Bagchi:2015nca . A tensionless string, as the
name suggests, is obtained by taking the tensionless limit $T\rightarrow 0$ on
the tensile string. This is equivalent to the ultra-relativistic limit taking
the speed of light on the worldsheet to zero. In this limit, the worldsheet of
the string becomes null endowed with a degenerate metric, and acquires a
Carrollian structure. On the null worldsheet, the diffeomorphisms are gauge
symmetries. For the tensile string, gauge-fixing the worldsheet
diffeomorphisms by choosing the conformal gauge results in two copies of the
Virasoro algebra as the residual symmetries. Analogously, gauge-fixing of the
diffeomorphisms in an equivalent gauge on the null worldsheet results in the
BMS3 algebra as the residual symmetries. Thus, like $2d$ CFTs for tensile
strings, BMS3 invariant CFTs are inevitable for studying, both classical and
quantum, dynamics of tensionless strings. Some recent advances in quantizing
the tensionless strings include Bagchi:2019cay ; Bagchi:2020fpr ;
Bagchi:2021rfw .

Interestingly, in connection with the discussion of Carroll symmetries on
black hole event horizons, it has been recently shown Bagchi:2020ats ;
Bagchi:2021ban  that strings moving near the horizon of a black hole spacetime
become effectively tensionless. The Carroll structure in the spacetime induces
a Carroll structure on the worldsheet as the string hits the horizon of the
black hole.

From the discussion in the previous paragraphs, we see the essential roles
played by Carrollian structures and BMS symmetries in various physical systems
of importance. It is, then, evident that a crucial ingredient in advancing
these developments further is the systematic study of field theories on null
surfaces with Carrollian and BMS symmetries. Indeed there has already been
some promising progress in this direction. Carrollian field theories for gauge
fields with both Carrollian and conformal Carrollian symmetries have been
analyzed in Duval:2014uoa ; Bagchi:2019xfx ; Bagchi:2019clu . Carrollian field
theories were also analyzed in the context of fluid dynamics Ciambelli:2018wre
, dark energy and inflation deBoer:2021jej . Some other recent developments
include Basu:2018dub ; Henneaux:2021yzg ; Hao:2021urq ; Chen:2021xkw ;
Bidussi:2021nmp . One notices that most of these studies are primarily based
on taking the UR limit of relativistic theories at the level of Lagrangian or
Hamiltonian, or equations of motions. However we should also look for
intrinsic constructions of Carroll covariant field theories, with the belief
that they may provide us with novel insights and widespread applicability for
these field theories. More importantly one can expect that the space of
Carrollian field theories, generically, is larger than those obtained from the
UR limit of relativistic theories. Then thoroughly exploring intrinsic
descriptions of Carrollian field theories becomes necessary. Some progress
towards this has been made: Carroll covariant theories for scalar fields have
been investigated in Ciambelli:2018ojf ; Gupta:2020dtl . The current work aims
to provide a self-consistent and explicit realisation of these theories in
$2d$.

###  1.4 What is in this paper?

Our primary interest, in this paper, is studying scalar field theories on two
dimensional null surfaces. As mentioned before, we want the field theory
degrees of freedom to not leave the null surface. This leads us to restricting
to theories of massless fields on the null surfaces, which naturally have
conformal invariance. Thus we focus on massless field theories on $2d$ Carroll
spacetimes, which have conformal Carroll symmetries and therefore are
invariant under the BMS3 symmetry algebra. In particular, we focus on
classical aspects of these massless scalar field theories in $2d$ and leave
description of quantum theories for future work.

In a nutshell, we find that there are three inequivalent Carrollian scalar
field theory actions in $d=2$. One of them is the well-known “time-like”
action, studied extensively as we describe below. One is a “space-like”
action, which has also appeared in the literature before. We also find a third
and completely new action, which we call the “mixed” action. In what follows,
we will review the time-like action and uncover details of the mixed and the
space-like action in detail. We will treat the theory of the mixed action as a
field theory, with a fixed background, while our explorations of the space-
like action would be in the context of a string theory, where the background
is dynamical. The main source of subtlety of the paper is the fact that fixing
the background in the second order formulation does not fix all vielbeins in
the frame formulation. This leads to various intricacies.

We begin this paper by introducing necessary ingredients of the Carroll
geometry in sec. 2. For Carroll geometry, the metric is replaced by the data:
the degenerate metric $h_{\mu\nu}$ and the no-where vanishing vector field
$\tau^{\mu}$ that gives the (null) time direction Henneaux:1979vn ;
Duval:2014uoa . These degenerate Carroll geometries are generally described by
two formulations in the literature. The frame formulation of Hartong:2015xda ;
Hartong:2015usd ; Bergshoeff_2017  in terms of vielbeins is based on local
Carroll symmetries in the tangent space along with general diffeomorphisms in
the spacetime. The formulation of Ciambelli:2019lap ; Ciambelli:2018wre  is
based on Carroll diffeomorphisms in the spacetime. In this paper, we are
interested in a formulation which is a composite of the preceding two
formulations in a specific way, _i.e._ a formulation having both local Carroll
symmetries in the tangent space and Carroll diffeomorphisms in the spacetime.
In sec. 2, we discuss relevant features of such a composite formulation for
Carroll geometry.

In the composite formulation, taking contractions of the metric fields with
the derivative of the scalar field $\partial_{\mu}\Phi$, we can construct two
Carroll covariant second order actions. Contracting with $\tau^{\mu}$ gives
the “timelike” action. This is a well known action and has been studied quite
extensively in the literature. This action also represents a single scalar
field version of the tensionless string action Schild:1976vq ; Isberg:1993av .
From a field theoretic perspective, this action arises from the
ultrarelativistic limit of the free Klein-Gordon action for free scalar field
on Minkowski spacetime, see _e.g._ Bergshoeff_2017  for the higher dimensional
version. The other action is obtained by contracting $\partial_{\mu}\Phi$ with
$h^{\mu\nu}$, the projective inverse of $h_{\mu\nu}$. We call this as the
“spacelike” action following Ciambelli:2018ojf ; Gupta:2020dtl  where a higher
dimensional version was introduced in the formulation of Ciambelli:2018wre ;
Ciambelli:2019lap  of Carroll geometry.

Interestingly, being in $2$-dimensions and using zweibeins in the composite
formulation to describe the Carroll geometry leads to another novel Carroll
covariant action for the scalar field in $2$-dimensions. Owing to the absence
of rotations in $2$-dimensions, we can use the (spatial) zweibein $e^{\mu}$ to
contract with $\partial_{\mu}\Phi$. We call the resulting action obtained by
using $\tau^{\mu}e^{\nu}$ for contraction with the scalar field as the “mixed-
derivative” action. This looks like the time-space flipped version of the
Floreanini-Jackiw (FJ) action Floreanini:1987as . We will have more to say
about this in Sec 6.

In the composite formulation of the Carroll geometry, the three actions for
the scalar field are Carroll covariant and also BMS-Weyl invariant. For all
these three actions, we further observe that fixing the Carroll geometry to be
torsionless flat Carroll spacetime, the residual symmetries of these gauge-
fixed actions are BMS3 symmetries. A crucial point to note here is that owing
to the degeneracy of the metric, after choosing the geometry to be torsionless
strong flat Carroll spacetime, a component of the zweibein, _i.e._ $e^{t}$
still remains arbitrary. The arbitrariness basically corresponds to the
transformation under local Carroll boost, as we will see in sec. 2.1. Being an
arbitrary function on the spacetime, $e^{t}$ also transforms under spacetime
conformal Carroll transformations. These transformation properties of $e^{t}$
play an essential role in ensuring the offshell BMS3 invariance of the
actions. A detailed construction of these actions and the symmetry analysis is
given in sec. 3.

In sec. 4, we review the well known timelike action. We briefly discuss some
classical aspects of this action from both the (null) string theory
perspective and the field theory perspective. For the spacelike and the mixed-
derivative action, the explicit presence of the unspecified vielbein $e^{t}$
brings non-trivialities in the analysis. In particular, in sec. 2.1 we see
that from a purely geometric perspective, _i.e._ without coupling to any
matter the arbitrariness of $e^{t}_{1}$ (_i.e._ transformations under local
Carroll symmetries as well as spacetime Carroll isometries) doesn’t affect the
geometric quantities of the flat Carroll spacetime, _e.g._ the curvatures are
still vanishing. However when we couple matter (the scalar field in our case)
to the flat Carroll spacetime, we see that $e^{t}_{1}$ plays different roles
depending on how the matter is coupled. This leads us to treating the
spacelike theory and the mixed-derivative theory differently. In sec. 5, we
find that a string theoretic description is aptly suited for analyzing the
spacelike action. We touch upon its residual symmetries and mode expansions.
Sec. 6 is the part of our paper with some novel and unique results, where we
analyze the mixed-derivative theory from a field theoretic perspective on a
fixed flat Carroll spacetime. In particular, we study the symmetries, find
mode expansions and see an emergence of one copy of Virasoro algebra onshell.
In this case, we also see that offshell, we can potentially treat $e^{t}_{1}$
as a gauge field as discussed in sec. 6.1. In sec. 7, we revisit the single
scalar field version of the null string in Isberg-Lindstrom-Sundborg-
Theodoridis (ILST) formalism Isberg:1993av . In particular, we discuss various
gauge choices and comment on comparisons with our actions. Appendix A has some
details on comparing the ILST formalism and the (Carroll) geometric
formulation of the null string. Appendix B discusses the parameter space of
ILST Lagrange multipliers and various particular forms of these actions in
connection to the current work.

##  2 Carroll spacetime in $(d+1)$ dimesions

As we discussed in the introduction, considering Carrollian theories is
equivalent to studying theories on null hypersurfaces characterized by Carroll
structures. A Carroll manifold Henneaux:1979vn ; Duval:2014uoa ; Duval:2014uva
; Duval:2014lpa  in $(d+1)$ dimensions is mostly defined by a degenerate
spacetime metric $h_{\mu\nu}$, whose kernel is generated by a no-where
vanishing vector field $\zeta^{\mu}$ and a symmetric affine connection
$\Gamma^{\rho}_{(\mu\nu)}$. More generally Carroll spacetimes can have torsion
Hartong:2015xda  and are described by a torsion-full connection
$\Gamma^{\rho}_{\mu\nu}$. In this paper, we focus on a composite formulation
of Carroll geometries based on both local Carroll symmetries in the tangent
space and Carroll diffeomorphisms in the spacetime. In this section, we
discuss necessary ingredients for the description of Carroll geometries in the
composite formulation.

In the composite formulation, we describe the Carroll spacetime in an adapted
coordinate system by

$ds^{2}=h_{\mu\nu}dx^{\mu}dx^{\nu}=g_{ij}(t,x^{k})dx^{i}dx^{j},\qquad\zeta=\zeta^{t}(t,x^{k})\frac{\partial}{\partial
t},$ (5)

where $g_{ij}$ is a non-degenerate metric on the $d$-dimensional, exclusively
spatial manifold. Here $\mu,\nu$ are spacetime indices denoting the
coordinates $(t,x^{i})$, where $t$ is a timelike coordinate and $x^{i}$ are
spacelike coordinates $(i=1,...,d)$. In general these are the ingredients that
replace the Riemannian structure associated to non-degenerate spacetimes. The
Carroll structures (5) are invariant under Carroll diffeomorphisms, where time
and space coordinates change in different manners, _i.e._

$t^{\prime}=t^{\prime}(t,x^{j}),\quad x^{\prime i}=x^{\prime i}(x^{j}).$ (6)

We recall that in the frame formulation of Carrollian geometries
Hartong:2015xda ; Hartong:2015usd ; Bergshoeff_2017 , a $(d+1)$ dimensional
Carrollian manifold is described by the vielbeins $\tau_{\mu}(t,x^{i})$ and
$e^{a}_{\mu}(t,x^{i})$, corresponding to time and space translations. Their
projective inverses $\tau^{\mu}(t,x^{i})$ and $e^{\mu}_{a}(t,x^{i})$ are
defined through the relations

$\tau_{\mu}\tau^{\mu}=1,\quad
e^{a}_{\mu}e^{\mu}_{b}=\delta^{a}_{b},\quad\tau_{\mu}e^{\mu}_{a}=0,\quad
e^{a}_{\mu}\tau^{\mu}=0,\quad\tau_{\mu}\tau^{\nu}+e^{a}_{\mu}e^{\nu}_{a}=\delta^{\nu}_{\mu}.$
(7)

Here $a,b=1,\dots,d$ are spatial tangent space indices. In addition to the
vielbeins, we have the boost spin connection $\Omega_{\mu}^{a}$ and the
rotation spin connection $\Omega_{\mu}^{ab}$ in the tangent space as
independent variables. As in the case of Riemannian manifolds, we can
determine these tangent space connections in terms of vielbeins by imposing
curvature constraints, _i.e._ vanishing of the curvatures for local time and
space translations. However owing to the degenerate nature, not all components
of the spin connections are determined in terms of the vielbeins
Bergshoeff_2017 .

Alternately, we can realize the Carroll algebra on a smaller set of fields,
_i.e._ the vielbeins and an independent vector field, where the spin
connections are determined in terms of these fields Hartong:2015xda . In any
case, the degenerate nature of the spacetime explicitly leads to some
independent fields in the tangent space in addition to the vielbeins. We will
not go into these details as we won’t be needing them for our investigations
in this paper. We refer the interested reader to Bergshoeff_2017 ;
Hartong:2015xda  for further reading.

We also recall that in the frame formulation, the vielbeins transform
covariantly under general diffeomorphisms in the spacetime. However, as we
will see below shortly, in the composite formulation, the vielbeins get
restricted to transform covariantly under Carroll diffeomorphisms only.

In the composite formulation, we relate the manifestly Carroll diffeomorphism
invariant metric variables in (5) to the vielbeins in the following way:

$\displaystyle\zeta^{\mu}=\tau^{\mu}\quad$ $\displaystyle\Rightarrow$
$\displaystyle\quad\zeta^{t}=\tau^{t},\ \tau^{i}=0,$ $\displaystyle
h_{\mu\nu}=e^{a}_{\mu}e^{b}_{\nu}\delta_{ab}\quad$ $\displaystyle\Rightarrow$
$\displaystyle\quad g_{ij}=e^{a}_{i}e^{b}_{j}\delta_{ab},\ e^{a}_{t}=0.$ (8)

Solving the vielbein postulates

$\partial_{\mu}\tau_{\nu}-\Gamma^{\rho}_{\mu\nu}\tau_{\rho}-\Omega^{a}_{\mu}e^{b}_{\nu}\delta_{ab}=0,\quad\partial_{\mu}e^{a}_{\nu}-\Gamma^{\rho}_{\mu\nu}e^{a}_{\rho}-\Omega^{ab}_{\mu}e^{c}_{\nu}\delta_{bc}=0,$
(9)

we get the spacetime connection in terms of the boost spin connection
$\Omega_{\mu}^{a}$ and the rotation spin connection $\Omega_{\mu}^{ab}$ as

$\Gamma^{\rho}_{\mu\nu}=\tau^{\rho}\partial_{\mu}\tau_{\nu}-\tau^{\rho}\Omega_{\mu}^{a}e_{\nu}^{b}\delta_{ab}+e^{\rho}_{a}\partial_{\mu}e^{a}_{\nu}-e^{\rho}_{a}\Omega_{\mu}^{ab}e^{c}_{\nu}\delta_{bc}.$
(10)

We can see from (8) that relating vielbeins to manifestly Carroll
diffeomorphism invariant metric variables in (5) restricts the vielbeins to
transform covariantly under Carroll diffeomorphisms only. Under these Carroll-
diffeomorphisms (6) (compactly written as
$x^{\prime\mu}=x^{\prime\mu}(x^{\nu})$), the vielbeins and their (projective)
inverses transform as

$\tau^{\prime\mu}=\frac{\partial x^{\prime\mu}}{\partial
x^{\nu}}\tau^{\nu},\quad e^{\prime\mu}_{a}=\frac{\partial
x^{\prime\mu}}{\partial
x^{\nu}}e^{\nu}_{a},\quad\tau^{\prime}_{\mu}=\frac{\partial x^{\nu}}{\partial
x^{\prime\mu}}\tau_{\nu},\quad e^{\prime a}_{\mu}=\frac{\partial
x^{\nu}}{\partial x^{\prime\mu}}e^{a}_{\nu}.$ (11)

Moreover, a Carroll-Weyl transformation Ciambelli:2019lap , the cousin of
regular Weyl transformations, albeit on null hypersurfaces, is defined as

$\tau_{\mu}\rightarrow e^{z\Omega(t,x)}\tau_{\mu},\quad h_{\mu\nu}\rightarrow
e^{2\Omega(t,x)}h_{\mu\nu}.$ (12)

As one can note, this implies that the metric components associated to the
temporal part and the spatial part scale differently under a Weyl
transformation, which is a crucial notion in Carrollian geometry. The
isomorphism between Carrollian Conformal Algebras (CCA) in $d$ dimensions and
BMS algebras in $d+1$ dimensions requires the exponent value fixed at $z=1$
Duval:2014uva ; Duval:2014lpa . Hence for $z=1$, this Carroll-Weyl
transformation gives rise to the BMS-Weyl transformations of the vielbiens,

$\tau_{\mu}\rightarrow e^{\Omega(t,x)}\tau_{\mu},\quad e^{a}_{\mu}\rightarrow
e^{\Omega(t,x)}e^{a}_{\mu},\quad\tau^{\mu}\rightarrow
e^{-\Omega(t,x)}\tau^{\mu},\quad e^{\mu}_{a}\rightarrow
e^{-\Omega(t,x)}e^{\mu}_{a},$ (13)

signalling the generation of conformal isometries of the Carrollian manifolds
we described earlier.

So far, we discussed Carroll diffeomorphisms on the spacetime. Now we turn to
the second fundamental ingredient of the composite formulation, _i.e._ the
local Carroll symmetries in the tangent space. Under a set of (infinitesimal)
local Carroll boosts and spatial rotations in the tangent space, parametrized
by $\lambda_{a}$ and $\lambda^{a}_{\ b}$ respectively, the vielbeins transform
as

$\delta\tau^{\mu}=0,\quad\delta
e^{\mu}_{a}=-\tau^{\mu}\lambda_{a}+\lambda_{a}^{\
b}e^{\mu}_{b},\quad\delta\tau_{\mu}=e^{a}_{\mu}\lambda_{a},\quad\delta
e^{a}_{\mu}=\lambda^{a}_{\ b}e^{b}_{\mu}.$ (14)

Using $h_{\mu\nu}=e^{a}_{\mu}e^{b}_{\nu}\delta_{ab}$,
$h^{\mu\nu}=e^{\mu}_{a}e^{\nu}_{b}\delta^{ab}$, the transformation of the
degenerate spacetime metric $h_{\mu\nu}$ and the projective inverse
$h^{\mu\nu}$ follows from (14):

$\delta h_{\mu\nu}=0,\quad\delta
h^{\mu\nu}=-(\tau^{\mu}e^{\nu}_{a}+\tau^{\nu}e^{\mu}_{a})\lambda^{a}.$ (15)

We now see how these local transformations of vielbeins are obtained, and
contrast these in the frame formalism with those in the composite formalism.
We begin by recalling that the ordinary derivatives $\partial_{\mu}$
constitute a basis in the tangent space.

In the _frame formalism_ in Hartong:2015xda ; Bergshoeff_2017 , we do a change
of basis in the tangent space as

$e_{0}=\tau^{\mu}\partial_{\mu},\quad e_{a}=e^{\mu}_{a}\partial_{\mu},$ (16)

and likewise $e^{0}=\tau_{\mu}dx^{\mu}$, $e^{a}=e_{\mu}^{a}dx^{\mu}$ in the
co-tangent space. Then under local Carroll transformations in the tangent
space (boosts $\Lambda^{0}_{\ a}=-\lambda_{a}$ and rotations $\Lambda^{a}_{\
b}=\delta^{a}_{b}+\lambda^{a}_{\ b}$), the new basis $(e^{0},e^{a})$,
$(e_{0},e_{a})$ transform like the $t$ and $x^{i}$ directions, _i.e._

$e^{\prime 0}=e^{0}-\lambda_{a}e^{a},\quad e^{\prime a}=\Lambda^{a}_{\
b}e^{b},\quad e^{\prime}_{0}=e_{0},\quad e^{\prime}_{a}=(\Lambda^{-1})^{\
b}_{a}(e_{b}+\lambda_{b}e_{0}).$ (17)

Simplifying these equations, using (16) and $\delta_{\lambda}\partial_{\mu}=0$
gives the local transformations of vielbeins in (14).

In the _composite formalism_ , we do a change of basis from $\partial_{\mu}$
to projected directions in the tangent space:

$\hat{e}_{0}=\hat{\tau}^{\mu}\partial_{\mu},\quad\hat{e}_{a}=\hat{e}^{\mu}_{a}\partial_{\mu},$
(18)

and likewise $\hat{e}^{0}=\hat{\tau}_{\mu}dx^{\mu}$,
$\hat{e}^{a}=\hat{e}_{\mu}^{a}dx^{\mu}$. By projected directions, we mean that
under local Carroll transformations, the basis $(\hat{e}_{0},\hat{e}_{a})$,
$(\hat{e}^{0},\hat{e}^{a})$ transform as “Carrollian” tensors with respect to
the base space (of the tangent space):

$\hat{e}^{\prime 0}=\hat{e}^{0},\quad\hat{e}^{\prime a}=\Lambda^{a}_{\
b}\hat{e}^{b},\quad\hat{e}^{\prime}_{0}=\hat{e}_{0},\quad\hat{e}^{\prime}_{a}=(\Lambda^{-1})^{\
b}_{a}\hat{e}_{b},$ (19)

_i.e._ $\hat{e}^{0}$ is a scalar and $\hat{e}^{a}$ is a spatial vector in the
tangent space, and $\hat{e}_{0}$ is a zero-form and $\hat{e}_{a}$ is a
(spatial) one-form in the co-tangent space. Simplifying the transformations
(19) using (18), we get that the projected vielbeins transform as given in
(14) provided the ordinary derivatives $\partial_{\mu}$ transform as

$\delta_{\lambda}\partial_{t}=0,\quad\delta_{\lambda}\partial_{i}=\lambda_{i}\partial_{t},$
(20)

where $\lambda_{\mu}=e^{a}_{\mu}\lambda_{a}$ and $\lambda_{\mu}\tau^{\mu}=0$.
These local transformations of the (projected) vielbeins and the ordinary
derivatives will be used to show the local Carroll invariance of the actions
in sec. 3.2.

We see that (19) also admits another solution:

$\displaystyle\delta_{\lambda}(\partial_{\mu})=0,\quad\delta_{\lambda}(dx^{\mu})=0;$
$\displaystyle\delta_{\lambda}\hat{\tau}^{\mu}=0,\quad\delta_{\lambda}\hat{\tau}_{\mu}=0,\quad\delta_{\lambda}\hat{e}^{\mu}_{a}=-\lambda^{b}_{\
a}\hat{e}^{\mu}_{b},\quad\delta_{\lambda}\hat{e}_{\mu}^{a}=\lambda^{a}_{\
b}\hat{e}^{b}_{\mu}.$ (21)

It follows that the degenerate metric
$\hat{h}_{\mu\nu}=\hat{e}^{a}_{\mu}\hat{e}^{b}_{\nu}\delta_{ab}$ and its
projective inverse
$\hat{h}^{\mu\nu}=\hat{e}_{a}^{\mu}\hat{e}_{b}^{\nu}\delta^{ab}$ transform as

$\delta_{\lambda}\hat{h}_{\mu\nu}=0,\quad\delta_{\lambda}\hat{h}^{\mu\nu}=0.$
(22)

In this paper, we focus on (20), and leave the investigation of this alternate
solution to a future work.

The reason that allows us to use the projected basis (18) is the degenerate
nature of the Carrollian tangent space. Due to this degenerate nature, we can
change the basis in the tangent space from $\partial_{\mu}$ to either the
standard basis (16) or the projected basis (18). To see this more explicitly,
we note that the metric data of the flat Carroll tangent space looks the same
in standard directions as well as projected directions, _i.e._

$\tau^{A}=(1,0),\quad
h_{AB}=\delta_{ab};\qquad\hat{\tau}^{A}=(1,0),\quad\hat{h}_{AB}=\delta_{ab},$
(23)

where $A=0,a$. To emphasize again, it is because of (23) that we can have the
projected basis (18) and their local transformations (19) in the composite
formalism.

In the rest of the paper, we will use the composite formalism and projected
vielbeins. For the convenience of notation, we will drop the ‘hat’ and call
‘projected vielbeins’ as just ‘vielbeins’.

###  2.1 Flat Carroll spacetime in generic dimensions

A strong flat Carroll spacetime Henneaux:1979vn ; Duval:2014uoa ;
Duval:2014uva ; Duval:2014lpa  in $(d+1)$ dimensions is defined in terms of
one having vanishing (symmetric) affine connection and particular choice of
the spatial metric and vector field,

$ds^{2}=h_{\mu\nu}dx^{\mu}dx^{\nu}=\delta_{ij}dx^{i}dx^{j},\qquad\zeta=\frac{\partial}{\partial
t},\qquad\Gamma^{\rho}_{(\mu\nu)}=0.$ (24)

The Carroll group compositions are made up of those transformations that leave
this flat connection invariant. For the flat Carroll spacetime in (24), using
(8) and (7), we can solve for the values of vielbeins as

$\displaystyle\tau_{t}=1,\quad e^{a}_{t}=0,\quad
e^{a}_{i}=\delta^{a}_{i},\quad\tau_{i}\sim\text{arbitrary},$
$\displaystyle\tau^{t}=1,\quad\tau^{i}=0,\quad e^{i}_{a}=\delta^{i}_{a},\quad
e^{t}_{a}=-\delta^{i}_{a}\tau_{i}.$ (25)

Our focus, in this paper, would be on field theories on a torsionless flat
Carroll background. Imposing that the torsion, _i.e._ the antisymmetric part
of the affine connection vanishes, in addition to the symmetric part, leads to
imposing that the full affine connection summarily vanishes. Then using (10)
to solve $\Gamma^{\rho}_{\mu\nu}=0$ for flat vielbeins (25), we get the
tangent space connections

$\Omega_{\mu}^{ab}=0,\quad\Omega_{\mu}^{a}=\delta^{ai}\partial_{\mu}\tau_{i}.$
(26)

For these expressions of the vielbeins (25) and the spin connections (26) for
the torsionless flat Carroll spacetime (24), the curvatures for the boost
spin-connection and the rotation spin-connection also vanish identically:

$\displaystyle
R^{a}_{\mu\nu}(C)=2\partial_{[\mu}\Omega_{\nu]}^{a}-2\Omega_{[\mu}^{ab}\Omega_{\nu]}^{c}\delta_{bc}=0,\quad
R^{ab}_{\mu\nu}(J)=2\partial_{[\mu}\Omega_{\nu]}^{ab}-2\Omega_{[\mu}^{ac}\Omega_{\nu]}^{db}\delta_{cd}=0,$
(27)

where $C_{a}$ and $J_{ab}$ are generators of local boosts and rotations,
respectively, on the tangent space. This ultimately leads to the vanishing of
Riemann tensor on the spacetime, and this can be seen in terms of the above
curvatures using the vielbein postulates:

$\mathcal{R}_{\mu\nu\sigma}^{\quad\ \rho}=\tau^{\rho}e_{\sigma
a}R_{\mu\nu}^{a}(C)-e_{\sigma a}e^{\rho}_{b}R_{\mu\nu}^{ab}(J)=0.$ (28)

Note that, in our definition of the flat Carroll spacetime in terms of fixing
the spacetime data (24) with vanishing Riemann tensor (28), the spatial
components of the vielbein $\tau_{i}$ remain arbitrary with a non-vanishing
boost spin-connection (26). This arbitrariness of $\tau_{i}$, _i.e._
$e^{t}_{a}$ is a crucial point here, as it allows us to construct novel
actions for fields coupled to the flat Carroll background, as we will
elaborate on in later sections.

### A gauge redundancy in $\tau_{i}$

Under local Carroll boost in the tangent space (14), the components of
$\tau_{\mu}$ transform as

$\delta\tau_{t}=0,\
\delta\tau_{i}=\lambda_{i},\quad\Rightarrow\quad\delta\Omega_{\mu}^{a}=\delta^{ai}\partial_{\mu}\lambda_{i},$
(29)

which leaves the boost curvature (27) invariant. We observe that the
transformation of the boost spin connection here is a gauge transformation, if
we think of $\Omega_{\mu}^{a}$ as a gauge field in the tangent space. An
instructive exercise here is to start with diffeomorphism and Weyl invariant
relativistic field theory on a Randers-Papapetrou type background
Ciambelli:2019lap :

$ds^{2}=-c^{2}(\omega dt-
b_{i}dx^{i})^{2}+a_{ij}dx^{i}dx^{j}=(-c^{2}\tau_{\mu}\tau_{\nu}+e^{a}_{\mu}e^{b}_{\nu}\delta_{ab})dx^{\mu}dx^{\nu},$
(30)

and taking the ultrarelativistic limits ($c\rightarrow 0$) where it produces a
Carroll geometry. In this limit, we can explicitly relate our vielbeins to the
metric variables,

$\omega=\tau_{t},\quad b_{i}=-\tau_{i},\quad
a_{ij}=e^{a}_{i}e^{b}_{j}\delta_{ab}.$ (31)

This identifies Carrollian spacetime as a fiber bundle structure endowed with
an Ehresmann connection $b_{i}$, that allows for separate manners of
transformation for time and space under Carroll diffeomorphisms. We can now
clearly identify the gauge redundancy of the Ehresmann connection $b_{i}$ for
the flat Carroll spacetime (24) introduced in Ciambelli:2019lap  as the
transformation of $\tau_{i}$ under local Carroll boost in the tangent space
(29).

###  2.2 Carroll spacetime in two dimensions

In this work, we will mostly be focussing on scalar fields living on a two
dimensional Carrollian manifold, hence a zoom-in into this particular example
seems useful here. In conformity with our discussion above for general
dimensions, the $2d$ version of the Carrollian geometry is described by the
zweibeins $\tau_{\mu}(t,x)$ and $e^{1}_{\mu}(t,x)$, with their inverses
$\tau^{\mu}(t,x)$ and $e^{\mu}_{1}(t,x)$ defined through the relations

$\tau_{\mu}\tau^{\mu}=1,\quad
e^{1}_{\mu}e^{\mu}_{1}=1,\quad\tau_{\mu}e^{\mu}_{1}=0,\quad
e^{1}_{\mu}\tau^{\mu}=0,\quad\tau_{\mu}\tau^{\nu}+e^{1}_{\mu}e^{\nu}_{1}=\delta^{\nu}_{\mu}.$
(32)

For our discussion in later sections, we can redefine the timelike zweibeins
in these dimensions as

$\tau_{\mu}=e^{0}_{\mu},~{}\tau^{\mu}=e^{\mu}_{0}.$

We will be using the above notations interchangeably throughout this work. Now
we can collectively write the zweibeins and their inverses as $e^{\mu}_{A}$,
($A=0,1$) in terms of which the above relations can be written in a compact
form:

$e^{A}_{\mu}e^{\mu}_{B}=\delta^{A}_{B},\qquad
e^{A}_{\mu}e^{\nu}_{A}=\delta^{\nu}_{\mu}.$ (33)

We further introduce antisymmetric symbols $\epsilon^{\mu\nu}$ and
$\epsilon^{AB}$ defined as $\epsilon^{tx}=1=-\epsilon_{tx}$ and
$\epsilon^{01}=1=-\epsilon_{01}$, which satisfy
$\epsilon_{\mu\rho}\epsilon^{\rho\nu}=\delta_{\mu}^{\nu}$ and
$\epsilon_{AC}\epsilon^{CB}=\delta_{A}^{B}$. Using these antisymmetric
symbols, we can write the determinants $e=\det(e^{A}_{\mu})$,
$\frac{1}{e}=\det(e^{\mu}_{A})$ and the inverse zweibeins in a simple form,

$e=\frac{1}{2}\epsilon_{AB}e^{A}_{\mu}e^{B}_{\nu}\epsilon^{\nu\mu},\qquad\frac{1}{e}=\frac{1}{2}\epsilon^{AB}e^{\mu}_{A}e^{\nu}_{B}\epsilon_{\nu\mu},\qquad
e^{\mu}_{A}=\frac{1}{e}\epsilon^{\mu\nu}e^{B}_{\nu}\epsilon_{BA}.$ (34)

We also need to spell out the transformation of the zweibeins under spacetime
Carroll-diffeomorphisms $x^{\mu}\rightarrow x^{\prime\mu}(x^{\nu})$, _i.e._
$t^{\prime}=t^{\prime}(t,x)$ and $x^{\prime}=x^{\prime}(x)$ in this case,
which can be written in our compact notation as

$e^{\prime\mu}_{A}=\frac{\partial x^{\prime\mu}}{\partial
x^{\nu}}e^{\nu}_{A},\quad e^{\prime A}_{\mu}=\frac{\partial x^{\nu}}{\partial
x^{\prime\mu}}e^{A}_{\nu}.$ (35)

As before a set of Carroll-Weyl transformations Ciambelli:2019lap  are defined
as

$e^{0}_{\mu}\rightarrow e^{z\Omega(t,x)}e^{0}_{\mu},\quad
h_{\mu\nu}\rightarrow e^{2\Omega(t,x)}h_{\mu\nu},$ (36)

with $h_{\mu\nu}=e^{1}_{\mu}e^{1}_{\nu}$. As mentioned earlier, this Carroll-
Weyl transformation gives the BMS-Weyl transformation when $z=1$.

In $2d$ there is only one local Carroll boost and no spatial rotations in the
tangent space. Hence under an infinitesimal local Carroll boost parametrized
by $\lambda_{1}$, the zweibeins transform as

$\delta e^{\mu}_{0}=0,\quad\delta
e^{\mu}_{1}=-e^{\mu}_{0}\lambda_{1},\quad\delta
e^{0}_{\mu}=e^{1}_{\mu}\lambda_{1},\quad\delta e^{1}_{\mu}=0,$ (37)

and the degenerate spacetime metric transforms as expected

$\delta h_{\mu\nu}=0,\quad\delta
h^{\mu\nu}=-(\tau^{\mu}e^{\nu}_{1}+\tau^{\nu}e^{\mu}_{1})\lambda^{1}.$ (38)

###  2.3 Flat Carroll manifold in $2d$

The definition of a torsionless strong flat Carrollian spacetime in $2d$
follows our previous discussion, but simplifies considerably as only one
spatial direction is involved

$ds^{2}=h_{\mu\nu}dx^{\mu}dx^{\nu}=dx^{2},\qquad\zeta=\frac{\partial}{\partial
t},\qquad\Gamma^{\rho}_{\mu\nu}=0.$ (39)

Using the relation (8) (for the index $a=1$) between metric variables in the
second order formalism and the zweibeins, we can identify the values of the
flat zweibeins as

$h_{\mu\nu}=e^{1}_{\mu}e^{1}_{\nu},\quad\zeta^{\mu}=e^{\mu}_{0}=\tau^{\mu}$
(40)

and the flat connection in $(1+1)d$ reads

$\Gamma^{\rho}_{\mu\nu}=e^{\rho}_{0}\partial_{\mu}e^{0}_{\nu}-e^{\rho}_{0}\Omega_{\mu}^{1}e_{\nu}^{1}+e^{\rho}_{1}\partial_{\mu}e^{1}_{\nu},$
(41)

where $\Omega_{\mu}^{1}$ is the boost spin-connection in the tangent space.
Then for (39) we get the values of all zweibeins as

$\displaystyle e^{0}_{t}=1,\quad e^{1}_{t}=0,\quad e^{1}_{x}=1,\quad
e^{0}_{x}\sim\text{arbitrary},$ $\displaystyle e^{t}_{0}=1,\quad
e^{x}_{0}=0,\quad e^{x}_{1}=1,\quad e^{t}_{1}=-e^{0}_{x},$ (42)

and the dependent boost spin-connection has a form
$\Omega_{\mu}^{1}=\partial_{\mu}e^{0}_{x}$, which echoes (2.1), (26) for $2d$.
One again notes here that gauge fixing the spatial metric and the vector field
for a flat Carroll spacetime cannot fully fix the particular vielbein
$e^{t}_{1}=-e^{0}_{x}$. Although gauge fixing this to a constant will make the
boost spin-connection vanish, we don’t elect to choose so at this moment. This
arbitrariness in $e^{t}_{1}$, as alluded to before, will be a central theme in
this work and we will continuously return to this subtlety in the later
sections.

#### More on Flat Carroll manifolds

In the discussion above, the degenerate metric for the flat Carroll manifold
in $2d$ was chosen as $h=\text{diag}(0,1)$. One may be tempted to interpret
this choice as the most obvious one coming from a Carrollian limit of the
Minkowski metric in two dimensions i.e. $\eta=\text{diag}(-c^{2},1)$. However,
one might argue that there are more possible choices for the degenerate metric
which satisfies all tenets of being in the flat Carroll class. Let us discuss
then another choice of $h$ that will be important to us later,

$ds^{2}=h_{\mu\nu}dx^{\mu}dx^{\nu}=(dx^{-})^{2},\qquad\zeta=2\frac{\partial}{\partial
x^{+}},\qquad\Gamma^{\rho}_{\mu\nu}=0.$ (43)

Of course the reader may ask, what is the most general choice for $h$ that is
compliant with all the Carroll postulates? We’ll come back to this question
later, for now concentrating solely on the above case. One may also note that
this particular degenerate metric is not connected to the Minkowski metric via
a Carrollian limit.

For this parametrization of the degenerate metric $h_{\mu\nu}$ and the vector
field $\zeta^{\mu}$ in (43), using $x^{\pm}=t\pm x$ and (32), the components
of the zweibeins are computed,

$\displaystyle\tau^{t}=1,\quad\tau^{x}=1,\quad\tau_{t}=1-\tau_{x},\quad\tau_{x}\sim\text{arbitrary},$
$\displaystyle e_{t}=-1,\quad e_{x}=1,\quad e^{t}=-\tau_{x},\quad
e^{x}=1-\tau_{x}.$ (44)

The vanishing of the spacetime connection (41), $\Gamma^{\rho}_{\mu\nu}=0$
gives $\Omega_{\mu}=\partial_{\mu}\tau_{x}$, given in terms of the arbitrary
unfixed vielbein $\tau_{x}$ as before. This form of the boost spin-connection
again leads to vanishing of boost curvature
$R_{\mu\nu}(C)=2\partial_{[\mu}\Omega_{\nu]}=0$, and hence vanishing Riemann
tensor $\mathcal{R}_{\mu\nu\sigma}^{\quad\
\rho}=\tau^{\rho}e_{\sigma}R_{\mu\nu}(C)=0$. All of this makes sure this new
choice still belongs to the flat Carroll class.

With all the components of the geometry at our disposal, we will start looking
at field theories living on this $2d$ flat Carroll spacetime. In what follows,
we will mostly be working with the standard flat Carroll choice of
$h=\text{diag}(0,1)$, unless otherwise specified.

##  3 BMS invariant actions for a $2d$ scalar field

$2d$ scalar field theories invariant under the BMS3 algebra (1) have
previously been studied at length in connection to null string theories
Bagchi:2013bga ; Bagchi:2015nca ; Bagchi:2020fpr  and more recently as field
theory in Hao:2021urq . These in general rely on null or Carrollian limits
taken from conformal scalar field and/or conformal gauge fixed string
theories. On general grounds, it is expected that considering (quantum) field
theories on an inherently Carroll manifold would give rise to a wider class of
theories than those appearing in the Carroll limit. With the Carroll covariant
formalism at our behest, we can now make this more robust by exploiting the
isomorphism of conformal Carroll groups and BMS groups. We will focus on
scalar field theories.

To start with, we construct independent classes of Carroll covariant actions
for a free scalar field $\Phi(x,t)$ on a 2d Carroll spacetime that are
manifestly invariant under Carroll-diffeomorphisms (35) and BMS-Weyl
transformations (36). Using the composite formalism discussed in the last
section, we find three distinct ones:

$\displaystyle S_{00}=\int
dtdx\,e\,e^{\mu}_{0}e^{\nu}_{0}\,\partial_{\mu}\Phi\partial_{\nu}\Phi,$ (45a)
$\displaystyle S_{01}=\int
dtdx\,e\,e^{\mu}_{0}e^{\nu}_{1}\,\partial_{\mu}\Phi\partial_{\nu}\Phi,$ (45b)
$\displaystyle S_{11}=\int
dtdx\,e\,e^{\mu}_{1}e^{\nu}_{1}\,\partial_{\mu}\Phi\partial_{\nu}\Phi.$ (45c)

The Carroll diffeomorphism invariance of each of the actions can be seen from
a straight-forward calculation. To see the BMS-Weyl invariance, we note that
the scalar field has a BMS-Weyl weight zero, _i.e._ it doesn’t transform under
the BMS-Weyl transformation. Introducing arbitrary constants 222It is fine to
do so in two dimensions, however there may be more than what meets the eye for
higher dimensional constructions. See sec. 3.2 for a discussion. $W^{AB}$ that
are symmetric in $A,B$ ($A,B,\dots=0,1$), we can collectively write the three
Carroll covariant actions into a single Carroll covariant action:

$S=W^{00}S_{00}+2W^{01}S_{01}+W^{11}S_{11}=W^{AB}\int
dtdx\,e\,e^{\mu}_{A}e^{\nu}_{B}\,\partial_{\mu}\Phi\partial_{\nu}\Phi.$ (46)

We would like to emphasize that $W^{AB}$ are just arbitrary constants
introduced for our convenience to write the unified Carroll covariant action
(46). More precisely, $W^{AB}$ should not be thought of as the flat metric on
the tangent space, since the tangent space is degenerate as well and the
correct (flat) geometric data describing the tangent space is $\tau^{A}=(1,0)$
and the spatial metric $h_{AB}=\text{diag}(0,1)$. In the rest of the section,
we will explicitly discuss the symmetry invariance for all these three
component actions.

###  3.1 BMS invariance on flat Carroll background

Defined on the flat Carroll background (42), we will call the three actions we
encountered in (45) “timelike”, “mixed-derivative” and “spacelike” actions
respectively,

$\displaystyle S_{00}\ \rightarrow\ S_{t}=\int dtdx(\partial_{t}\Phi)^{2},$
$\displaystyle S_{01}\ \rightarrow\ S_{m}=\int
dtdx\big{(}\partial_{x}\Phi\partial_{t}\Phi+e^{t}_{1}(\partial_{t}\Phi)^{2}\big{)},$
$\displaystyle S_{11}\ \rightarrow\ S_{sp}=\int
dtdx(\partial_{x}\Phi+e^{t}_{1}\partial_{t}\Phi)^{2}.$ (47)

The names of course reflect their spacetime structures. Once we choose the
background geometry to be flat Carroll spacetime, these actions (47) are no
longer invariant under Carroll-diffeomorphisms and BMS-Weyl transformations,
as expected. However they are invariant under a residual set of
transformations. To find these residual symmetries, we gauge fix the
degenerate metric $h_{\mu\nu}$ and the vector field $\zeta^{\mu}$ to their
flat spacetime values (39) such that they are invariant under a combined
action of a diffeomorphism and a BMS-Weyl transformation.

An infinitesimal diffeomorphism is given by

$x^{\mu}\rightarrow x^{\mu}+\xi^{\mu},$

and we choose the BMS-Weyl factor $\Omega(t,x)$ be small. Then the gauge-
fixing of $h_{\mu\nu}$ and $\zeta^{\mu}$ to flat spacetime values leads to the
conformal Killing equations

$\mathcal{L}_{\xi}h_{\mu\nu}=2\Omega
h_{\mu\nu},\qquad\mathcal{L}_{\xi}\zeta^{\mu}=-\Omega\zeta^{\mu},$ (48)

where $\mathcal{L}$ is the Lie derivative of these fields under the
diffeomorphism. Solving these equations for the flat Carroll spacetime (39),
we can constrain the components of the vector field $\xi$,

$\partial_{t}\xi^{t}=\partial_{x}\xi^{x},\quad\partial_{t}\xi^{x}=0,\quad\Omega=\partial_{x}\xi^{x}.$
(49)

We can recognize these as BMS3 transformations, which give rise to the
solutions for the Killing fields

$\xi^{t}=f^{\prime}(x)t+g(x),~{}~{}\xi^{x}=f(x),$ (50)

where $f^{\prime}(x)=\partial_{x}f$. Under these conformal Carroll
transformations, the inverse zweibeins transform as

$e^{\prime\mu}_{A}(x^{\prime})e^{\Omega}=\frac{\partial
x^{\prime\mu}}{\partial x^{\nu}}e^{\nu}_{A}(x).$ (51)

From this we can now explicitly check that $e^{t}_{0}$, $e^{x}_{1}$ and
$e^{x}_{0}$ are all invariant under these conformal Carroll transformations.
Thus we see that gauge fixing $h_{\mu\nu}$ and $\zeta^{\mu}$ amounts to gauge-
fixing $e^{t}_{0}$, $e^{x}_{1}$ and $e^{x}_{0}$, while leaving $e^{t}_{1}$
arbitrary, as found earlier. Starting from the transformation law for the
unfixed zweibein $e^{t}_{1}$ in the form

$e^{\prime t}_{1}(x^{\prime})=e^{-\Omega(x)}\frac{\partial
t^{\prime}}{\partial x^{\nu}}e^{\nu}_{1}(x),$ (52)

we get the corresponding transformation law for small $\Omega$ and $\xi$

$\displaystyle e^{\prime
t}_{1}(x^{\prime})=e^{t}_{1}+e^{\nu}_{1}\partial_{\nu}\xi^{t}-\Omega
e^{t}_{1}=e^{t}_{1}+\partial_{x}\xi^{t}.$ (53)

Using this result, we can go ahead to find the conformal Carroll
transformation (49) of derivatives of the scalar field:

$\displaystyle\partial^{\prime}_{\mu}\Phi^{\prime}(x^{\prime})=\partial_{\mu}\Phi(x)-\partial_{\mu}\xi^{\nu}\partial_{\nu}\Phi(x),$
$\displaystyle i.e.\quad$
$\displaystyle\partial_{t^{\prime}}\Phi^{\prime}(x^{\prime})=\partial_{t}\Phi(1-\partial_{t}\xi^{t}),\qquad\partial_{x^{\prime}}\Phi^{\prime}(x^{\prime})=\partial_{x}\Phi(1-\partial_{x}\xi^{x})-\partial_{x}\xi^{t}\partial_{t}\Phi.$
(54)

Using (53) and (54), we now show that the each of the action in (47) is
invariant under BMS3 _i.e._ the set of conformal Carroll transformations (49).
Below we give detailed expressions below for all three classes of actions.

(1) Timelike action :

$\displaystyle S^{\prime}_{t}$ $\displaystyle=$ $\displaystyle\int
dt^{\prime}dx^{\prime}(\partial_{t^{\prime}}\Phi^{\prime})^{2}$ (55)
$\displaystyle=$ $\displaystyle\int
dtdx(1+2\partial_{t}\xi^{t})(1-\partial_{t}\xi^{t})^{2}(\partial_{t}\Phi)^{2}$
$\displaystyle=$ $\displaystyle\int dtdx(\partial_{t}\Phi)^{2}=S_{t}.$

(2) Mixed-derivative action :

$\displaystyle S^{\prime}_{m}$ $\displaystyle=$ $\displaystyle\int
dt^{\prime}dx^{\prime}\Big{(}\partial_{x^{\prime}}\Phi^{\prime}\partial_{t^{\prime}}\Phi^{\prime}+e^{\prime
t}_{1}(\partial_{t^{\prime}}\Phi^{\prime})^{2}\Big{)}$ (56) $\displaystyle=$
$\displaystyle\int
dtdx(1+2\partial_{t}\xi^{t})\Big{(}(\partial_{x}\Phi(1-\partial_{x}\xi^{x})-\partial_{x}\xi^{t}\partial_{t}\Phi)(1-\partial_{t}\xi^{t})\partial_{t}\Phi$
$\displaystyle\qquad+(e^{t}_{1}+\partial_{x}\xi^{t})(1-2\partial_{t}\xi^{t})(\partial_{t}\Phi)^{2}\Big{)}$
$\displaystyle=$ $\displaystyle\int
dtdx\Big{(}\partial_{x}\Phi\partial_{t}\Phi+e^{t}_{1}(\partial_{t}\Phi)^{2}\Big{)}=S_{m}.$

(3) Spacelike action :

$\displaystyle S^{\prime}_{sp}$ $\displaystyle=$ $\displaystyle\int
dt^{\prime}dx^{\prime}\Big{(}(\partial_{x^{\prime}}\Phi^{\prime})^{2}+2e^{\prime
t}_{1}\partial_{x^{\prime}}\Phi^{\prime}\partial_{t^{\prime}}\Phi^{\prime}+(e^{\prime
t}_{1})^{2}(\partial_{t^{\prime}}\Phi^{\prime})^{2}\Big{)}$ (57)
$\displaystyle=$ $\displaystyle\int
dtdx(1+2\partial_{t}\xi^{t})\Big{(}(\partial_{x}\Phi)^{2}(1-2\partial_{x}\xi^{x})-2\partial_{x}\xi^{t}\partial_{x}\Phi\partial_{t}\Phi$
$\displaystyle\qquad+2(e^{t}_{1}+\partial_{x}\xi^{t})(\partial_{x}\Phi(1-\partial_{x}\xi^{x})-\partial_{x}\xi^{t}\partial_{t}\Phi)(1-\partial_{t}\xi^{t})\partial_{t}\Phi$
$\displaystyle\qquad+((e^{t}_{1})^{2}+2e^{t}_{1}\partial_{x}\xi^{t})(1-2\partial_{t}\xi^{t})(\partial_{t}\Phi)^{2}\Big{)}$
$\displaystyle=$ $\displaystyle\int
dtdx\Big{(}(\partial_{x}\Phi)^{2}+2e^{t}_{1}\partial_{x}\Phi\partial_{t}\Phi+(e^{t}_{1})^{2}(\partial_{t}\Phi)^{2}\Big{)}=S_{sp}.$

With the explicit invariance under BMS3 transformations shown, some comments
are in order here. As the well versed reader can spot, the “timelike” action
in our notion is indeed the single scalar field version of the null string
action Isberg:1993av , and also the action discussed in Hao:2021urq . However
the other two actions are somewhat unique as the unfixed vielbein directly
appears in them and is absolutely crucial in making them invariant under the
transformations in question. In this sense, these are new classes of BMS3
invariant actions, where unlike relativistic theories, fixing background
fields doesn’t suffice in fixing all ingredients of the geometry.

###  3.2 Invariance under tangent space transformations

We also need our actions to be invariant under tangent space Carroll boosts.
The invariance of $e^{\mu}_{0}$ under tangent space transformations (37),
along with $e^{x}_{0}=0$ implies that the action $S_{00}$ is automatically
invariant under tangent space transformations.

However the inverse zweibein $e^{\mu}_{1}$ is not invariant under local
transformations (37). So for the actions $S_{01}$ and $S_{11}$ to be
invariant, we require that the derivatives of the scalar field transform as

$\delta_{\lambda}(\partial_{t}\Phi)=0,\quad\delta_{\lambda}(\partial_{x}\Phi)=\lambda_{x}\partial_{t}\Phi$
(58)

under local transformations, where $\lambda_{x}=e^{1}_{x}\lambda_{1}$.
Naively, at a first glance, it may seem odd that the derivatives of the scalar
field, which are spacetime quantities, transform non-trivially under tangent
space transformations. Indeed, in the frame formalism, the ordinary
derivatives do not transform under local transformations. However we remind
the reader that in the composite formalism, the ordinary derivatives do
transform under local transformations, as we have seen in (20). This can be
interpreted to mean that in the composite formalism, where we have only
Carroll diffeomorphisms in the spacetime, the ordinary derivatives are not
tensors with respect to Carroll diffeomorphisms. The correctly defined tensors
in the Carrollian sense are

$\hat{\partial}_{t}\Phi=\tau^{\mu}\partial_{\mu}\Phi,\quad\hat{\partial}_{x}\Phi=(\partial_{x}-\tau_{x}\tau^{\mu}\partial_{\mu})\Phi.$
(59)

Under Carroll diffeomorphisms $t^{\prime}=t^{\prime}(t,x)$,
$x^{\prime}=x^{\prime}(x)$, we see that $\hat{\partial}_{t}\Phi$ and
$\hat{\partial}_{x}\Phi$ transform as Carroll scalar and Carroll vector,
respectively, with respect to the base space, _i.e._

$(\hat{\partial}_{t}\Phi)^{\prime}=\hat{\partial}_{t}\Phi,\quad(\hat{\partial}_{x}\Phi)^{\prime}=\frac{\partial
x}{\partial x^{\prime}}\hat{\partial}_{x}\Phi.$ (60)

Then, as should be the case, we see that the Carroll covariant derivatives
above, being true tensors in Carrollian sense in our formalism, are invariant
under tangent space transformations provided the ordinary derivatives
transform as given in (58). Note that for the flat Carroll choice of
$\tau^{\mu}=(1,0)$, the covariant derivative $\hat{\partial}_{t}\Phi$ is
simply $\partial_{t}\Phi$.

We can now use the Carroll covariant derivatives defined in (59) to rewrite
the three actions $S_{00}$, $S_{01}$ and $S_{11}$ such that these exhibit both
Carroll diffeomorphism invariance and local Carroll boost invariance
manifestly. This is done by using the completeness relations
$\tau_{\mu}\tau^{\mu}=1$, $\tau_{\mu}e^{\mu}=0$ along with $\tau^{x}=0$ in
(45) to get

$\displaystyle S_{00}=\int
dtdx\,e\,\hat{\partial}_{t}\Phi\hat{\partial}_{t}\Phi,$ $\displaystyle
S_{01}=\int dtdx\,e\,\hat{\partial}_{t}\Phi e^{x}_{1}\hat{\partial}_{x}\Phi,$
$\displaystyle S_{11}=\int
dtdx\,e\,h^{xx}\hat{\partial}_{x}\Phi\hat{\partial}_{x}\Phi.$ (61)

Here $e^{x}_{1}$ is a Carroll vector and $h^{xx}=e^{x}_{1}e^{x}_{1}$ is a
Carroll tensor. Under Carroll diffeomorphisms $t^{\prime}=t^{\prime}(t,x)$,
$x^{\prime}=x^{\prime}(x)$, these transform as

$e^{\prime x}_{1}=\frac{\partial x^{\prime}}{\partial x}e^{x}_{1},\quad
h^{\prime xx}=\frac{\partial x^{\prime}}{\partial x}\frac{\partial
x^{\prime}}{\partial x}h^{xx},$ (62)

thus making the actions $S_{01}$ and $S_{11}$ in (3.2) manifestly covariant
under Carroll diffeomorphisms. Under local Carroll boosts parametrized by
$\lambda_{1}$, we see from (37), (38) (using $\tau^{x}=0$) that

$\delta_{\lambda_{1}}e^{x}_{1}=0,\quad\delta_{\lambda_{1}}h^{xx}=0,$ (63)

thus making the actions $S_{01}$ and $S_{11}$ in (3.2) manifestly invariant
under local Carroll boosts.

We would like to mention that, though rewriting in terms Carroll covariant
derivatives brings out the symmetries manifestly, we will continue to use the
vielbeins and the ordinary derivatives in conjunction in the rest of the
paper, for consistency of the notation. While doing so, we will keep in mind
the appropriate transformation properties of the vielbeins and the ordinary
derivatives under Carroll diffeomorphisms and local Carroll boosts, as given
in the preceding discussions.

As an aside, this also leads us to the question of tangent space invariance in
higher dimensions. Let us consider a naive generalization of our actions to
$(d+1)$ dimensions:

$\displaystyle S$ $\displaystyle=$ $\displaystyle W^{00}\int
dtd^{d}x\,e\,e^{\mu}_{0}e^{\nu}_{0}\,\partial_{\mu}\Phi\partial_{\nu}\Phi+2W^{0a}\int
dtd^{d}x\,e\,e^{\mu}_{0}e^{\nu}_{a}\,\partial_{\mu}\Phi\partial_{\nu}\Phi$
(64) $\displaystyle\quad+W^{ab}\int
dtd^{d}x\,e\,e^{\mu}_{a}e^{\nu}_{b}\,\partial_{\mu}\Phi\partial_{\nu}\Phi$
$\displaystyle\equiv$ $\displaystyle W^{00}S_{00}+2W^{0a}S_{0a}+W^{ab}S_{ab}.$

In $(d+1)$ dimensions, we have both local Carroll boost and local rotations in
the tangent space. From (14), we see that $e^{\mu}_{0}$ is invariant under
local Carroll boost and local rotations. Thus we get that the action $S_{00}$
is invariant under tangent space transformations in higher dimensions as well,
with $W^{00}$ an arbitrary constant. Requiring local Carroll boost invariance
of $S_{ab}$ we get the generalization of the transformations for the
derivatives of the scalar field (58) to

$\delta_{\lambda}(\partial_{t}\Phi)=0,\quad\delta_{\lambda}(\partial_{i}\Phi)=\lambda_{i}\partial_{t}\Phi;\quad\lambda_{i}=\lambda_{a}e^{a}_{i}.$
(65)

This generalization to higher dimensions can be seen as coming from the
invariance under tangent space transformations of general Carroll-covariant
derivatives333Using the identification (31), we note that these Carroll
covariant derivatives were also introduced in Ciambelli:2018ojf  in an
alternate formalism in terms of the metric data $(\omega,b_{i},a_{ij})$.

$\hat{\partial}_{t}\Phi=\tau^{\mu}\partial_{\mu}\Phi,\quad\hat{\partial}_{i}\Phi=(\partial_{i}-\tau_{i}\tau^{\mu}\partial_{\mu})\Phi,$
(66)

which again transform as Carroll scalar and Carroll vector, respectively,
under Carroll diffeomorphisms:

$(\hat{\partial}_{t}\Phi)^{\prime}=\hat{\partial}_{t}\Phi,\quad(\hat{\partial}_{i}\Phi)^{\prime}=\frac{\partial
x^{j}}{\partial x^{\prime i}}\hat{\partial}_{j}\Phi.$ (67)

From (14) and (15), we see that though the vielbein $e^{\mu}_{a}$ transforms
under local rotations, $h^{\mu\nu}=e^{\mu}_{a}e^{\nu}_{b}\delta^{ab}$ is
invariant under local rotations. Thus identifying $W^{ab}=\delta^{ab}$, we see
that the action

$W^{ab}S_{ab}=\int
dtd^{d}x\,e\,h^{\mu\nu}\,\partial_{\mu}\Phi\partial_{\nu}\Phi$ (68)

i.e. the higher dimensional analog of the spacelike action, is also invariant
under tangent space transformations.

Now let us turn to the remaining action $S_{0a}$. Using (65), we see that
$S_{0a}$ is local boost invariant. However $S_{0a}$ is not local rotation
invariant due to non-trivial transformation of $e^{\mu}_{a}$. The only way to
have local rotation invariance is to take $W^{0a}$ as a function
$W^{a}(t,x^{i})$ such that it transforms as a vector in the tangent space
under local rotations but remains invariant under local boosts, _i.e._ $\delta
W^{a}=\lambda^{a}_{\ b}W^{b}$. Then the action in question becomes:

$W^{0a}S_{0a}\rightarrow\int
dtd^{d}x\,e\,W^{a}e^{\mu}_{0}e^{\nu}_{a}\,\partial_{\mu}\Phi\partial_{\nu}\Phi.$
(69)

We conclude that we can write $S_{01}$ as a ‘minimally’ coupled action for a
scalar field on a Carroll background in $2$-dimensions. However, we cannot
write $S_{0a}$ as a meaningful action in higher dimensions without including
the vector field $W^{a}(t,x^{i})$ in the action. In this sense, our BMS3
invariant mixed-derivative action in two dimensions is rather intriguing and
unique.

###  3.3 Comments on fixing $e^{t}_{1}$

What now remains is to ponder over the unfixed vielbeins in our theory. From a
purely geometric perspective, _i.e._ for pure Carroll gravity, $e^{t}_{1}$ is
completely arbitrary for the flat Carroll manifold. For _e.g._ , choosing any
value for $e^{t}_{1}$ doesn’t affect the curvatures; those are still
vanishing. However when matter is coupled to the flat Carroll spacetime, the
complete arbitrariness of $e^{t}_{1}$ is lost. We saw that for unfixed
$e^{t}_{1}$, its non-trivial transformation is essential to ensure BMS
invariance of the mixed-derivative and the spacelike actions for the scalar
field. However upon fixing $e^{t}_{1}$ to a constant value, the resultant
mixed-derivative and spacelike actions are not BMS invariant in general.
Further, $e^{t}_{1}=0$ turns out to be special compared to other non-zero
constant values, where we see certain differences in the symmetries and
dynamics of the mixed-derivative and spacelike theories. We will elaborate on
this in later sections.

For now, let us focus on the reduced symmetries of the scalar field actions
after fixing $e^{t}_{1}$ to constant. The reduced set of transformations which
keep these resultant actions invariant are those that do not change the fixed
value of $e^{t}_{1}$. From (53), these reduced transformations should satisfy
$\partial_{x}\xi^{t}=0$. Solving this extra condition along with (49), we get
that the solution for $\xi$ turns out to be

$\xi^{\mu}=\lambda x^{\mu}+\kappa^{\mu},$ (70)

where $\lambda$ and $\kappa^{\mu}$ are constants describing scaling and
translations respectively. Notice that with this choice, the boost spin
connection $\Omega_{\mu}^{1}=0$, effectively eliminating all gauge connections
on the manifold.

To see that the resultant actions have only translations and scaling as the
reduced symmetries, let us look at the extra terms generated in the variation
of $S_{m}$ and $S_{sp}$ under BMS transformations (49):

$\delta_{\xi}S_{m}\propto\int
dtdx\,\partial_{x}\xi^{t}~{}\dot{\Phi}^{2},\quad\delta_{\xi}S_{sp}\propto\int
dtdx\,\partial_{x}\xi^{t}~{}\dot{\Phi}(\Phi^{\prime}+k\dot{\Phi}).$ (71)

With no further constraints on $\Phi$, the invariance of the actions $S_{m}$
and $S_{sp}$ obviously requires $\partial_{x}\xi^{t}=0$.

Thus we see that for a constant fixed $e^{t}_{1}$, the residual symmetries of
$S_{m}$ and $S_{sp}$ are reduced from BMS3 to only scaling and translations.
Interestingly, as we will argue later, the BMS3 invariance for the $S_{sp}$
case can be restored by going to the correct frame defined by the right
$\tau^{\mu}$. More intriguingly, following Henneaux:2021yzg , we see that in
the mixed action, $e^{t}_{1}$ can be thought of as having an additional gauge
symmetry. If we take this potentially useful gauge symmetry into
consideration, this allows us to fix $e^{t}_{1}$ to constant values without
actually imposing the extra condition $\partial_{x}\xi^{t}=0$ on the Killing
field $\xi$. This effective compensation, although unclear from physical
perspective but assuming it makes sense physically, may enhance the residual
symmetry of the gauge fixed mixed-derivative action. See sec. 6.1 for more
details on this gauge symmetry.

###  3.4 Stress tensors

The next section onwards we will be turning our attention to the residual
symmetries of our three classes of actions with fixed vielbiens, and it is
imperative to start with the constraint structure dictated by the
corresponding stress tensors. In relativistic $2d$ CFTs, a straightforward
consequence of conformal invariance is vanishing of the trace of the stress
tensor classically. Analogously in BMS3 invariant theories, the BMS-Weyl
symmetry also leads to vanishing of the trace of the stress tensor. As one of
our primary goals is to discuss these actions in connection with the
tensionless strings, to keep the context clear, from here onwards we choose to
work on (Carrollian) cylinder parametrized by $(\tau,\sigma)$ as coordinates.
Here $\tau$ is the (null) time coordinate and $\sigma$ is the spatial
coordinate with periodicity $\sigma\sim\sigma+2\pi$. For the combined action
(46), the stress tensor is defined as a variation with respect to the
zweibeins

$T^{\mu}_{\ \ \nu}=\frac{e^{\mu}_{A}}{2e}\frac{\delta S}{\delta e^{\nu}_{A}}$
(72)

which explicitly gives

$\displaystyle T^{\mu}_{\ \ \nu}$ $\displaystyle=$ $\displaystyle
W^{AB}\Big{(}e^{\mu}_{A}e^{\lambda}_{B}\partial_{\lambda}\Phi\partial_{\nu}\Phi-\frac{\delta^{\mu}_{\nu}}{2}e^{\rho}_{A}e^{\lambda}_{B}\partial_{\rho}\Phi\partial_{\lambda}\Phi\Big{)}.$
(73)

On the flat Carroll background (42) now taken to be the flat Carrollian
cylinder, the coefficients of $W^{AB}$ in (73) give stress tensors for the
three actions in (47) as follows.

$1$. Coefficient of $W^{00}$ gives stress tensor for the timelike action :

$\displaystyle T^{\tau}_{\ \tau}=-T^{\sigma}_{\
\sigma}=\frac{1}{2}(\partial_{\tau}\Phi)^{2},\quad T^{\tau}_{\
\sigma}=\partial_{\tau}\Phi\partial_{\sigma}\Phi,\quad T^{\sigma}_{\ \tau}=0.$
(74)

$2$. Coefficient of $2W^{01}$ gives stress tensor for the mixed derivative
action :

$\displaystyle T^{\tau}_{\ \tau}=-T^{\sigma}_{\
\sigma}=\frac{1}{2}e^{\tau}_{1}(\partial_{\tau}\Phi)^{2},\quad T^{\tau}_{\
\sigma}=\frac{1}{2}(\partial_{\sigma}\Phi)^{2}+e^{\tau}_{1}\partial_{\sigma}\Phi\partial_{\tau}\Phi,\quad
T^{\sigma}_{\ \tau}=\frac{1}{2}(\partial_{\tau}\Phi)^{2}.$ (75)

$3$. Coefficient of $W^{11}$ gives stress tensor for the spacelike action :

$\displaystyle T^{\tau}_{\ \tau}=-T^{\sigma}_{\
\sigma}=\frac{1}{2}\left(-(\partial_{\sigma}\Phi)^{2}+(e^{\tau}_{1})^{2}(\partial_{\tau}\Phi)^{2}\right),$
$\displaystyle T^{\tau}_{\
\sigma}=e^{\tau}_{1}\partial_{\sigma}\Phi(\partial_{\sigma}\Phi+e^{\tau}_{1}\partial_{\tau}\Phi),\qquad
T^{\sigma}_{\
\tau}=\partial_{\tau}\Phi(\partial_{\sigma}\Phi+e^{\tau}_{1}\partial_{\tau}\Phi).$
(76)

Keeping in with the spirit of Carollian structures, we can see the Energy-
Momentum tensors are not symmetric in spacetime indices to begin with. But
also we would like to remind the reader that for a Carroll boost invariant
theory one would require $T^{\sigma}_{~{}\tau}=0$ on shell,444We would like to
note that using the equation of motion for $e^{\tau}_{1}$, _i.e._
$\partial_{\tau}\Phi(\partial_{\sigma}\Phi+e^{\tau}_{1}\partial_{\tau}\Phi)=0$,
the component $T^{\sigma}_{\ \tau}$ in (76) vanishes. see, _e.g._
deBoer:2021jej  for a nice discussion on this.555In contrast for a Galilean
boost invariant theory one would demand $T^{\tau}_{~{}\sigma}=0$. In the rest
of the paper, we will be using these stress tensors to study the residual
symmetries of our three classes of actions. As described before, these
symmetries will emerge in the later two cases when we gauge fix $e^{\tau}_{1}$
to a constant value.

##  4 Warming up: Residual symmetries of the timelike action

We will start our discussion with the simplest example, the $2d$ Conformal
Carroll model for free scalar field Hao:2021urq , which is our timelike action
$S_{00}$ in (47). On a flat Carrollian cylinder parametrized by coordinates
$(\tau,\sigma)$, the action simply reads

$S_{t}=\int d\tau d\sigma(\partial_{\tau}\Phi)^{2}.$ (77)

Note that this action can be interpreted as direct Carrollian limit $(c\to
0~{}\text{or}~{}\sigma\to\sigma,\tau\to\epsilon\tau,~{}\epsilon\to 0)$ on a
$2d$ scalar CFT action. The equation of motion in the cylindrical coordinates
is:

$\partial_{\tau}^{2}\Phi=0.$ (78)

Assuming periodic boundary conditions, the general solution in terms of mode
expansions can be written as,

$\Phi(\tau,\sigma)=\phi_{0}+A_{0}\sigma+B_{0}\tau+\sum_{n\neq
0}\frac{i}{n}(A_{n}-in\tau B_{n})e^{-in\sigma},$ (79)

where periodicity along $\sigma$ requires $A_{0}=0$, giving the periodic
solution

$\Phi(\tau,\sigma)=\phi_{0}+B_{0}\tau+\sum_{n\neq 0}\frac{i}{n}(A_{n}-in\tau
B_{n})e^{-in\sigma}.$ (80)

The mode expansion for the conjugate momentum $\Pi=\partial_{\tau}\Phi$ is

$\Pi(\tau,\sigma)=\sum_{n}B_{n}e^{-in\sigma}.$ (81)

It is straightforward to directly check that this action (77) is manifestly
invariant under BMS transformations (50).

As mentioned earlier, the timelike action (77) represents a single scalar
field version of the gauge-fixed tensionless string action Bagchi:2013bga ;
Bagchi:2015nca ; Bagchi:2020fpr . Thus, from a string theory perspective, we
can interpret this action as describing a Carroll string on a flat Carroll
worldsheet geometry, which is null by construction. This gauge-fixing of the
worldsheet geometry leads to the constraints $T^{\mu}_{\ \nu}=0$. From the
expression of the stress tensor (74), these constraints can be written as,

$T^{\mu}_{\ \nu}=0\implies T_{1}\equiv T^{\tau}_{\ \sigma}=0,~{}~{}T_{2}\equiv
T^{\tau}_{\ \tau}=0.$ (82)

Written in terms of oscillator modes from (79), these constraints become

$\displaystyle(\partial_{\tau}\Phi)^{2}=\sum_{n,m}B_{-m}B_{n+m}e^{-in\sigma}=\sum_{n}M_{n}e^{-in\sigma}=0,$
$\displaystyle\partial_{\tau}\Phi\partial_{\sigma}\Phi=\sum_{n,m}(A_{-m}-in\tau
B_{-m})B_{n+m}e^{-in\sigma}=\sum_{n}(L_{n}-in\tau M_{n})e^{-in\sigma}=0,$ (83)

where the modes of the stress tensor are bilinears of the oscillators,

$L_{n}=\frac{1}{2}\sum_{m}A_{-m}B_{m+n},\quad
M_{n}=\frac{1}{2}\sum_{m}B_{-m}B_{m+n}.$ (84)

We find the algebra of oscillators by imposing the canonical Poisson’s
brackets between $\Pi(\tau,\sigma)$ and $\Phi(\tau,\sigma)$. For Carrollian
theories, we employ the equal time Poisson’s brackets

$\displaystyle\\{\Phi(\tau,\sigma),\Pi(\tau,\sigma^{\prime})\\}_{PB}=\delta(\sigma-\sigma^{\prime}),$
$\displaystyle\\{\Phi(\tau,\sigma),\Phi(\tau,\sigma^{\prime})\\}_{PB}=\\{\Pi(\tau,\sigma),\Pi(\tau,\sigma^{\prime})\\}_{PB}=0.$
(85)

It is easy to show they imply the algebra

$\\{A_{n},B_{m}\\}_{PB}=-2in\delta_{n+m,0}\quad\\{A_{n},A_{m}\\}_{PB}=0,\quad\\{B_{n},B_{m}\\}_{PB}=0.$
(86)

Bear in mind, this isn’t the usual oscillator algebra of scalar field modes,
instead they look like those belonging to a particle theory. Using the
canonical Poisson brackets, the algebra of generators $L_{n}$ and $M_{n}$
turns out to be

$\\{L_{n},L_{m}\\}_{PB}=-i(n-m)L_{n+m},\quad\\{L_{n},M_{m}\\}_{PB}=-i(n-m)M_{n+m},\quad\\{M_{n},M_{m}\\}_{PB}=0.$
(87)

This is the classical part of the BMS3 algebra, which is indeed the residual
symmetry algebra for this action. We can identify $L_{n}$ and $M_{n}$ as
superrotation and supertranslation generators respectively.

A lot of discussions have appeared in the literature related to the symmetries
of this action, including quantum versions thereof, especially in relation to
worldsheet symmetries of null string theories. Here we won’t go into the
details, stopping only at the classical symmetry analysis. Readers are
directed to Bagchi:2019cay ; Bagchi:2020fpr ; Bagchi:2020ats ; Bagchi:2021rfw
; Bagchi:2021ban  for further reading.

Complementing the preceeding analysis, where we studied the timelike action
(77) from a string theory perspective, we can independently study the same
action as a field theory describing a scalar field minimally coupled to a flat
Carroll spacetime, _e.g._ as discussed in Hao:2021urq . Let us briefly comment
on this point of view, again taking the background to be a flat Carroll
cylinder. Then the equation of motion and the solution are as given in
(78)-(81). We can vary the action (77) under the infinitesimal BMS
transformations (50) and use equations of motion to get the conserved Noether
currents

$j^{\tau}=\frac{1}{2}(\partial_{\tau}\Phi)^{2}\xi^{\tau}+\partial_{\tau}\Phi\partial_{\sigma}\Phi\xi^{\sigma},\quad
j^{\sigma}=(\partial_{\tau}\Phi)^{2}\xi^{\sigma}.$ (88)

We integrate the temporal component $j^{\tau}$ over the $\sigma$-cirlce, and
use the mode expansion (80) along with the Fourier expansions
$f(\sigma)=\sum_{n}a_{n}e^{in\sigma}$, $g(\sigma)=\sum_{n}b_{n}e^{in\sigma}$
in $\xi^{\tau}$, $\xi^{\sigma}$ 666The killing vectors on a cylinder are
$\xi^{\tau}=f^{\prime}(\sigma)\tau+g(\sigma),~{}~{}\xi^{\sigma}=f(\sigma)$. to
get the conserved charge

$Q=\int d\sigma j^{\tau}=\sum_{n}a_{n}L_{n}+b_{n}M_{n}.$ (89)

Then looking at (87), we can conclude that the above conserved charge
generates BMS3 algebra, which are the symmetries of our timelike action (77).
With this vanilla example out of our way, we can then move onto more involved
cousins of this action in the next sections.

##  5 Symmetries of the Spacelike action

In this section, we investigate the action $S_{11}$ in (45) from a string
theory perspective with the geometrical fields being dynamical. As in the
usual string theory, we gauge-fix the worldsheet geometry to flat Carroll
cylinder $\tau^{\mu}=(1,0)$, $h_{\mu\nu}=diag(0,1)$. This reduces $S_{11}$ to
the spacelike action given in (47), now in cylindrical coordinates
$(\tau,\sigma)$:

$S_{sp}=\int d\tau
d\sigma(\partial_{\sigma}\Phi+e^{\tau}_{1}\partial_{\tau}\Phi)^{2}.$ (90)

As we saw earlier, this action is BMS3 invariant for arbitrary
$e^{\tau}_{1}(\tau,\sigma)$, which changes in a particular way under BMS
transformations. Thus, a priori, we are not allowed to arbitrarily fix
$e^{\tau}_{1}$ to a constant. Nonetheless, we find that for specific constant
values of $e^{\tau}_{1}$, the spacelike action has connections with some known
interesting actions in the literature, as we elaborate below. In particular,
the value $e^{\tau}_{1}=1$ turns out to be of particular importance, for which
the action takes the form

$S_{sp}=\int d\tau
d\sigma(\partial_{\sigma}\Phi+\partial_{\tau}\Phi)^{2}=2\int d\tau
d\sigma(\partial_{+}\Phi)^{2},$ (91)

where $\sigma^{\pm}=\tau\pm\sigma$. Note the difference with the usual $2d$
CFT action: the holomorphic derivative in the action just drops off here. This
particular class of gauge fixed actions has appeared in many contexts in
physics, especially in the study of chiral scalars Sonnenschein:1988ug  and
more recently in Ambitwistor string theories Casali:2016atr ; Casali:2017zkz ,
where the action customarily corresponds to the choice $e^{\tau}_{1}=-1$
instead.

The components of the stress tensor with this choice take the form

$T^{\tau}_{\ \tau}=-T^{\sigma}_{\
\sigma}=\frac{1}{2}(\dot{\Phi}+\Phi^{\prime})(\dot{\Phi}-\Phi^{\prime}),\quad
T^{\tau}_{\ \sigma}=(\dot{\Phi}+\Phi^{\prime})\Phi^{\prime},\quad
T^{\sigma}_{\ \tau}=(\dot{\Phi}+\Phi^{\prime})\dot{\Phi},$ (92)

which can be obtained by putting $e^{\tau}_{1}=1$ in (76). Now let us write
the stress tensor in terms of the momentum conjugate to $\Phi$:

$P=\frac{\delta\mathcal{L}}{\delta\dot{\Phi}}=2(\dot{\Phi}+\Phi^{\prime})=\partial_{+}\Phi.$
(93)

Then the components of stress tensor in terms of $P$ are

$T^{\tau}_{\ \tau}=-T^{\sigma}_{\
\sigma}=\frac{P^{2}}{8}-\frac{P\Phi^{\prime}}{2},\quad T^{\tau}_{\
\sigma}=\frac{P\Phi^{\prime}}{2},\quad T^{\sigma}_{\
\tau}=\frac{P^{2}}{4}-\frac{P\Phi^{\prime}}{2}.$ (94)

This stress tensor is traceless and conserved but clearly $T^{\sigma}_{\
\tau}\neq 0$, which is a telltale sign of Carroll boost invariance being
broken. In keeping with the string theory description, we now have to impose
the constraints

$T^{\mu}_{\ \nu}=0\quad\implies\quad P\Phi^{\prime}=0,\quad P^{2}=0.$ (95)

###  5.1 Symmetries in “lightcone” coordinates

Although the gauge-fixed spacelike action does not have full Carroll
symmetries anymore, we can go to lightcone coordinates to restore Carroll
boost invariance in one of the lightcone directions. This is also evident in
the chiral structure of the spacelike Lagrangian. Taking a cue from
Ambitwistor string literature Casali:2016atr ; Casali:2017zkz  we write the
action in lightcone coordinates on a Carrollian cylinder using,

$\displaystyle\sigma^{\pm}=\tau\pm\sigma,\quad\partial_{\pm}=\frac{1}{2}(\partial_{\tau}\pm\partial_{\sigma}).$
(96)

Now the gauge-fixed spacelike action becomes

$S=2\int d\sigma^{+}d\sigma^{-}(\partial_{+}\Phi)^{2}.$ (97)

We see that this action is manifeslty invariant under the BMS3 transformations
in lightcone coordinates:

$\sigma^{\pm}\rightarrow\sigma^{\pm}+\xi^{\pm};\qquad\xi^{+}=\sigma^{+}\partial_{-}f(\sigma^{-})+g(\sigma^{-}),\quad\xi^{-}=f(\sigma^{-}).$
(98)

However note that this isn’t equivalent to having BMS3 in $(\tau,\sigma)$
coordinates as the Killing equations are drastically different,

$\displaystyle\partial_{+}\xi^{+}=\partial_{-}\xi^{-}\quad\implies\quad\partial_{\tau}\xi^{\sigma}+\partial_{\sigma}\xi^{\tau}=0,$
$\displaystyle\partial_{+}\xi^{-}=0\quad\implies\quad\partial_{\tau}\xi^{\tau}-\partial_{\sigma}\xi^{\sigma}-\partial_{\tau}\xi^{\sigma}+\partial_{\sigma}\xi^{\tau}=0.$
(99)

We can write the conformal Killing equations as
$\partial_{\tau}\xi^{\sigma}=-\partial_{\sigma}\xi^{\tau},\quad\partial_{\tau}\xi^{\tau}-\partial_{\sigma}\xi^{\sigma}=2\partial_{\tau}\xi^{\sigma},$
which are not equivalent to BMS transformations in the original coordinates
(49). This happens because the frame defined by (96) is Carroll inequivalent
to the usual $(\tau,\sigma)$ frame we have been working on till now.

An alternative way to look at this action is to consider it to be the
“timelike” action corresponding to the separate Carroll frame (i.e. choice of
$h_{\mu\nu}$ and $\tau^{\mu}$) we mentioned in (43). In this case the new
“timelike” action in cylindrical coordinates is:

$\tilde{S}_{t}\sim\int d\tau
d\sigma\,e\,\tau^{\mu}\tau^{\nu}\,\partial_{\mu}\Phi\partial_{\nu}\Phi=\int
d\tau d\sigma(\partial_{+}\Phi)^{2},$ (100)

since $\tau^{\mu}=(1,1)$ is the new null vector in this case. This fits in
perfectly with what we have already found, i.e. the new action is invariant
under BMS3 transformations, albeit in the $\sigma^{\pm}$ direction. The (BMS)
conformal Killing equations $\mathcal{L}_{\xi}h_{\mu\nu}=-2\lambda
h_{\mu\nu},~{}\mathcal{L}_{\xi}\tau^{\mu}=\lambda\tau^{\mu}$ explicitly take
the form (99) for this case.777This choice of the null vector also relates to
a cousin of the “Ambitwistor” gauge of null string theories, details of which
will be discussed later.  Also notice that for the spacelike action with any
constant $e^{\tau}_{1}=k,~{}k>0$ can be re-interpreted as the timelike action
in a Carroll frame with $\tau^{\mu}=(1,\frac{1}{k})$, and hence can be shown
to be BMS3 invariant in a judiciously chosen frame where the Carroll boost
invariance is automatically restored 888 For more details on this choice of
frame one can refer to the appendix A.. Note here, in these cases the action
could be rewritten, upto a constant, using the timelike covariant derivative
(59),

$\tilde{S}_{t}\sim\int d\tau d\sigma(\hat{\partial}_{\tau}\Phi)^{2},$ (101)

which guarantees invariance under BMS3 transformations since
$\hat{\partial}_{t}\Phi$ always transforms as a Carroll scalar. However with
the choice of $e^{\tau}_{1}=0$, this argument does not work as the null vector
then has divergent components. In fact one can show with $e^{\tau}_{1}=0$ we
get a purely Galilean covariant action. This is also supported by the fact
that with this choice, the stress tensor component $T^{\tau}_{~{}\sigma}$
clearly vanishes (76), implying an underlying Galilean takeover. However in
two dimensions, Galilean and Carrollian conformal algebras are classically
isomorphic to each other, so even at this singular point, the form of the
symmetry algebra remains unchanged.

Coming back to the lightcone case, using the transformation $\Sigma^{\mu}_{\
\nu}(\sigma^{+},\sigma^{-})=\frac{\partial\sigma^{\mu}}{\partial
x^{\alpha}}\frac{\partial x^{\beta}}{\partial\sigma^{\nu}}T^{\alpha}_{\
\beta}$, the components of the stress tensor (92) can be written in lightcone
coordinates as

$\Sigma^{+}_{\ +}=-\Sigma^{-}_{\ -}=2(\partial_{+}\Phi)^{2},\quad\Sigma^{+}_{\
-}=4\partial_{+}\Phi\partial_{-}\Phi,\quad\Sigma^{-}_{\ +}=0.$ (102)

One can clearly see that $\Sigma^{-}_{\ +}$ vanishes here, signalling once
again the restoration of Carroll boost invariance in these coordinates. In
lightcone coordinates, the conjugate momentum is $P=4\partial_{+}\Phi$, which
dictates the form of stress tensors as follows:

$\Sigma^{+}_{\ +}=-\Sigma^{-}_{\ -}=\frac{P^{2}}{8},\quad\Sigma^{+}_{\
-}=\frac{P^{2}}{4}-P\Phi^{\prime},\quad\Sigma^{-}_{\ +}=0.$ (103)

From (103), we see that imposing $\Sigma^{\mu}_{\ \nu}=0$ gives usual “null
string” constraints

$P\Phi^{\prime}=0,\quad P^{2}=0.$ (104)

However from (102), we get the constraints

$\partial_{+}\Phi\partial_{-}\Phi=0,\quad(\partial_{+}\Phi)^{2}=0,\quad\emph{i.e.}\quad
P\partial_{-}\Phi=0,\quad P^{2}=0,$ (105)

respectively. The constraint $P\Phi^{\prime}=0$ here appears to be just a
combination of $P\partial_{-}\Phi=0$ and $P^{2}=0$, hence isn’t an independent
object.

###  5.2 Mode expansion and charges

In lightcone coordinates, the Euler-Lagrange equation of motion coming from
(97) reads

$\partial^{2}_{+}\Phi=0,$ (106)

and the Hamilton’s equations are $P=4\partial_{+}\Phi$, $\partial_{+}P=0$. The
general solution for $\Phi$ associated to these equations is,

$\Phi(\sigma^{+},\sigma^{-})=A(\sigma^{-})+\frac{\sigma^{+}}{4}P(\sigma^{-}),$
(107)

where the arbitrary functions have mode expansions

$A(\sigma^{-})=\sum_{n}A_{n}e^{-in\sigma^{-}},\quad
P(\sigma^{-})=\sum_{n}P_{n}e^{-in\sigma^{-}}.$ (108)

Note that this solution is not periodic in $\sigma$ (or in
$\sigma^{\pm}\rightarrow\sigma^{\pm}\pm 2\pi$). A similar problem for the
modes was noticed in Casali:2016atr , and was remedied by introducing the weak
periodicity conditions

$P(\sigma^{-}+2\pi)=P(\sigma^{-}),~{}~{}A(\sigma^{-}+2\pi)\approx
A(\sigma^{-})-\frac{\pi}{2}P(\sigma^{-}),$ (109)

which imposes a condition on the $A_{n}$ modes as

$A_{n}\approx A_{n}-\frac{\pi}{2}P_{n}.$ (110)

This identification clearly does not hamper the canonical Poisson brackets
imposed on the system that reads,

$\\{A_{n},P_{m}\\}=\delta_{n+m,0},\quad\\{A_{n},A_{m}\\}=0,\quad\\{P_{n},P_{m}\\}=0.$
(111)

But this is a subtle problem related to the actual Ambitwistor gauge that
cannot be addressed in the second order equations of motion, so the confusion
regarding the interpretation of the non-periodic solution persists. We will
however take these equations at the face value and proceed for now, and a
detailed discussion will be given in section 7. The constraints coming from
the lightcone stress tensor are

$\displaystyle\Sigma^{+}_{\ +}\sim P^{2}$ $\displaystyle=$
$\displaystyle\sum_{n,m}P_{-m}P_{n+m}e^{-in\sigma^{-}}\sim\sum_{n}M_{n}e^{-in\sigma^{-}},$
(112) $\displaystyle\Sigma^{+}_{\ -}=P\partial_{-}\Phi$ $\displaystyle=$
$\displaystyle\sum_{n,m}\Big{(}imA_{-m}P_{n+m}-\frac{in\sigma^{+}}{8}P_{-m}P_{n+m}\Big{)}e^{-in\sigma^{-}}$
(113) $\displaystyle=$
$\displaystyle\sum_{n}(L_{n}-in\sigma^{+}M_{n})e^{-in\sigma^{-}}.$

We see that even with the non-periodic solution, we can write the constraint
$P\partial_{-}\Phi$ in the desired form of BMS3 stress tensor. The above
constraints again generate the classical part of the BMS3 algebra (87). We
also notice the modes of the second constraint $(L_{n}-in\sigma^{+}M_{n})$
generate conformal transformations along the $\sigma^{-}$, bringing our
argument throughout this section to a full circle.

##  6 The Mixed-derivative theory

In this section, we concentrate on one of the main findings of this work, the
covariant mixed-derivative action $S_{m}$ in (47). This one will be
technically different from the other two cases we have discussed in the
preceding sections. We will see the fate of the symmetries when we fix the
unspecified vielbein $e^{\tau}_{1}$ in this case. As discussed before, fixing
$e^{\tau}_{1}$ reduces the conformal Carroll symmetries down to scaling and
translations only. However as we also mentioned earlier, a gauge symmetry of
$e^{\tau}_{1}$ can potentially compensate this effect and bring back boost
invariance to the system.

###  6.1 Gauge symmetry of $e^{\tau}_{1}$

In sec. 3, we saw that the mixed derivative action, given by

$\ S_{m}=\int d\tau
d\sigma\Big{(}\partial_{\sigma}\Phi\partial_{\tau}\Phi+e^{\tau}_{1}(\partial_{\tau}\Phi)^{2}\Big{)}$
(114)

with arbitrary $e^{\tau}_{1}(\tau,\sigma)$ is manifestly BMS3 invariant. Now
to see the aforementioned gauge symmetries associated to the unfixed zweibein
$e^{\tau}_{1}$, we need to recast $S_{m}$ into first order form. The conjugate
momentum to $\Phi$ in this case is,

$\Pi=\frac{\delta\mathcal{L}_{m}}{\delta\dot{\Phi}}=2e^{\tau}_{1}\dot{\Phi}+\Phi^{\prime}$
(115)

where $\dot{\Phi}=\partial_{\tau}\Phi$ and
$\Phi^{\prime}=\partial_{\sigma}\Phi$. The Hamiltonian is:

$\mathcal{H}_{m}=\Pi\dot{\Phi}-\mathcal{L}_{m}=e^{\tau}_{1}\dot{\Phi}^{2}=\frac{(\Pi-\Phi^{\prime})^{2}}{4e^{\tau}_{1}}.$
(116)

We can write the first order action using the Hamiltonian as

$S^{H}_{m}=\int
d^{2}\sigma\Big{(}\Pi\dot{\Phi}-\frac{(\Pi-\Phi^{\prime})^{2}}{4e^{\tau}_{1}}\Big{)}.$
(117)

Since $e^{\tau}_{1}$ is arbitrary in this action, we redefine it as
$\theta=\frac{1}{e^{\tau}_{1}}$. Then the Hamiltonian action takes the form,

$S^{H}_{m}=\int
d^{2}\sigma\Big{(}\Pi\dot{\Phi}-\frac{\theta}{4}(\Pi-\Phi^{\prime})^{2}\Big{)}.$
(118)

Now we remind ourselves, generically the Hamiltonian action for gauge theories
is written as

$I_{H}=\int d^{2}\sigma\big{(}\Pi\dot{\Phi}-H_{0}+\lambda^{a}\phi_{a}\big{)},$
(119)

where $H_{0}$ is the canonical Hamiltonian and $\lambda^{a}$ are Lagrange
multipliers that imposes the constraints $\phi^{a}=0$. For our case, $\theta$
is the only Lagrange multiplier and the corresponding constraint is given by
$\phi=\frac{1}{4}(\Pi-\Phi^{\prime})^{2}=0$. The canonical Hamiltonian $H_{0}$
is also equal to zero in this case. Using the Poisson brackets

$\displaystyle[\Phi(\tau,\sigma),\Pi(\tau,\sigma^{\prime})]=\delta(\sigma-\sigma^{\prime}),\quad[\Phi(\tau,\sigma),\Phi(\tau,\sigma^{\prime})]=[\Pi(\tau,\sigma),\Pi(\tau,\sigma^{\prime})]=0,$
(120)

it can be verified that the constraint is indeed a first class constraint and
hence will generate gauge symmetries that keep the action invariant. For an
arbitrary gauge function $\epsilon(\tau,\sigma)$, the field $\Phi$ transforms
as999Using the expression (115) for the conjugate momentum, we can write the
gauge transformation (121) of the scalar field as
$\delta_{\epsilon}\Phi=2\epsilon e^{\tau}_{1}\partial_{\tau}\Phi$.

$\delta_{\epsilon}\Phi=\int
d^{2}\sigma^{\prime}\epsilon(\tau,\sigma^{\prime})\big{[}\Phi(\tau,\sigma),\phi(\tau,\sigma^{\prime})\big{]}=\epsilon(\tau,\sigma)(\Pi-\Phi^{\prime})$
(121)

and the canonical conjugate $\Pi$ will transform under the gauge symmetry as,

$\delta_{\epsilon}\Pi=\int
d^{2}\sigma^{\prime}\epsilon(\tau,\sigma^{\prime})\big{[}\Pi(\tau,\sigma),\phi(\tau,\sigma^{\prime})\big{]}=-2\big{(}\epsilon(\tau,\sigma)(\Pi-\Phi^{\prime})\big{)}^{\prime}.$
(122)

Then we have the gauge transformations of various terms in the action
$S_{m}^{H}$ as

$\displaystyle\delta_{\epsilon}(\Pi\dot{\Phi})=-\epsilon\dot{\phi}+B=\dot{\epsilon}\phi+B_{1},$
$\displaystyle\delta_{\epsilon}(\Pi-\Phi^{\prime})^{2}=4\big{(}(\epsilon\phi)^{\prime}+\epsilon^{\prime}\phi\big{)}\
\implies\
-\frac{1}{4}\theta\delta_{\epsilon}\phi=-\theta\epsilon^{\prime}\phi-\epsilon\phi\theta^{\prime}+B_{2},$
(123)

where $B$, $B_{1}$, $B_{2}$ are total derivative terms. Using these
expressions, we have the transformation of the total Lagrangian as the
following,

$\displaystyle\delta_{\epsilon}\mathcal{L}^{H}_{m}$ $\displaystyle=$
$\displaystyle\delta_{\epsilon}(\Pi\dot{\Phi})-\frac{1}{4}\theta\delta_{\epsilon}\phi-\frac{1}{4}\delta_{\epsilon}\theta\phi$
(124) $\displaystyle=$
$\displaystyle(\dot{\epsilon}-\theta\epsilon^{\prime}-\epsilon\theta^{\prime})\phi-\frac{1}{4}\delta_{\epsilon}\theta\phi+\text{boundary
terms}.$

In order for the action to be gauge invariant, the Lagrange multiplier
$\theta$ has to transform as

$\delta_{\epsilon}\theta=4(\dot{\epsilon}-\theta\epsilon^{\prime}-\epsilon\theta^{\prime}).$
(125)

Then using $e^{\tau}_{1}=\frac{1}{\theta}$, the gauge transformation of
$e^{\tau}_{1}$ that keeps the action $S_{m}$ in (114) gauge-invariant is

$\delta_{\epsilon}e^{\tau}_{1}=-(e^{\tau}_{1})^{2}\dot{\epsilon}+e^{\tau}_{1}\epsilon^{\prime}-\epsilon(e^{\tau}_{1})^{\prime}.$
(126)

Towards a way to gauge fix $e^{\tau}_{1}=1$:

From (53), we see that under BMS3 transformations (49), $e^{\tau}_{1}$ changes
as

$\delta_{\xi}e^{\tau}_{1}=-\xi^{\nu}\partial_{\nu}e^{\tau}_{1}+\partial_{\sigma}\xi^{\tau}.$

That was the reason the action lost invariance under the full BMS3 group when
$e^{\tau}_{1}$ assumes a fixed value. Now, following a similar logic presented
in Henneaux:2021yzg , we could possibly use the gauge symmetries mentioned
above and restore BMS invariance by simultaneously making a suitable gauge
transformation that will preserve the condition $e^{\tau}_{1}=1$, i.e,

$(\delta_{\xi}+\delta_{\epsilon})e^{\tau}_{1}=0\ \implies\
(\xi^{\tau})^{\prime}+\epsilon^{\prime}-\dot{\epsilon}=0.$ (127)

From conformal Killing equations on a flat Carroll background (49)-(50), we
have $\xi^{\tau}=\tau f^{\prime}(\sigma)+g(\sigma)$. Hence the above equation
can be solved generically if we choose

$\epsilon(\tau,\sigma)=-\big{(}f^{\prime}(\sigma)\tau+g(\sigma)+f(\sigma)+k\big{)}.$
(128)

Here $f$, $g$ are arbitrary functions of $\sigma$ and $k$ is a constant,
signifying we can always choose an $\epsilon$ of this form. Then we can
potentially argue that the gauge-fixed action

$S_{m}=\int d\tau d\sigma(\dot{\Phi}^{2}+\dot{\Phi}\Phi^{\prime})$ (129)

would be invariant under the BMS transformation, if we simultaneously make a
compensating gauge transformation, _i.e._
$(\delta_{\xi}+\delta_{\epsilon})S_{m}=0$101010The idea of compensating a
diffeomorphism with a gauge transformation works mathematically to give rise
to a symmetry enhancement, but the physical implication of such a composition
of two physically distinct transformations is not entirely clear to us..
Remember, appearance of this gauge symmetry does not depend on the constant
value of $e^{\tau}_{1}$ being chosen, and the resulting physics should always
be independent of this value.

With the offshell symmetries of the mixed-derivative action taken care of, we
now put the theory onshell, in what follows. While doing so, we fix
$e^{\tau}_{1}=1$ without using the (compensating) gauge symmetry discussed
above, which leads to breaking of symmetries to only translations and scaling,
as mentioned before. However we will see that, onshell, there is an
enhancement in symmetries to one copy of the Virasoro algebra. This is an
interesting feature since it shows similarities with the Floreanini-Jackiw
(FJ) chiral scalar field theory Floreanini:1987as , and thus leads us to
interpret the action (129) as a time-space inverted version of the FJ action.
However we would also like to note that this equivalence between the mixed-
derivative and the FJ theories only occurs onshell. Offshell, the FJ action is
a relativistic theory unlike the mixed-derivative action, which by
construction is a Carrollian theory. Close cousins of this chiral action have
appeared in myriad of physical systems, most recently in the arena of near
horizon dynamics of scalar fields Grumiller:2019tyl .

###  6.2 Onshell Symmetries

In the previous discussions we have seen that upon fixing $e^{\tau}_{1}$ to a
constant value, without using the compensating gauge transformation introduced
in sec. 6.1, the action loses the boost invariance and consequently the
infinite BMS extensions. However in this section, we show that these
symmetries can be partially lifted to one copy of Virasoro when we go onshell.

The Euler Lagrange equation of motion from the variation of the gauge-fixed
action (129) is

$\partial^{2}_{\tau}\Phi+\partial_{\tau}\partial_{\sigma}\Phi=0.$ (130)

As the covariant action has Carrollian diffeomorphism and Weyl invariance, we
are interested in residual conformal Carrollian transformations. The conformal
Killing equations on flat Carroll background are given by (49). Since the Weyl
weight for the scalars in $(1+1)$ dimensions is zero, the transformations of
$\Phi$ under these conformal Carroll transformations would be

$\delta_{\xi}\Phi=-\xi^{\rho}\partial_{\rho}\Phi$ (131)

Now under these transformations satisfying the conformal Killing equations,
different terms in the Lagrangian will transform as follows. The $\dot{\Phi}$
term transforms as

$\displaystyle\delta_{\xi}[(\partial_{\tau}\Phi)^{2}]$ $\displaystyle=$
$\displaystyle-2\partial_{\tau}\Phi\delta_{\xi}(\partial_{\tau}\Phi)$
$\displaystyle=$
$\displaystyle-2\partial_{\tau}\Phi[(\partial_{\tau}\xi^{\tau})\partial_{\tau}\Phi+\xi^{\tau}\partial_{\tau}^{2}\Phi+\xi^{\sigma}\partial_{\tau}\partial_{\sigma}\Phi]$
$\displaystyle=$
$\displaystyle-\partial_{\tau}[\xi^{\tau}(\partial_{\tau}\Phi)^{2}]-\partial_{\sigma}[\xi^{\sigma}(\partial_{\tau}\Phi)^{2}]$

and the cross derivative term transforms as

$\displaystyle\delta_{\xi}(\partial_{\tau}\Phi\partial_{\sigma}\Phi)=-\partial_{\tau}[\xi^{\tau}(\partial_{\tau}\Phi\partial_{\sigma}\Phi)]-\partial_{\sigma}[\xi^{\sigma}(\partial_{\tau}\Phi\partial_{\sigma}\Phi)]-\partial_{\sigma}\xi^{\tau}(\partial_{\tau}\Phi)^{2}.$
(133)

The last term in the above equation seems to be problematic, but it also turns
out to be a boundary term if the equations of motion for $\Phi$ is used:

$\partial_{\sigma}\xi^{\tau}(\partial_{\tau}\Phi)^{2}=(\partial_{\tau}+\partial_{\sigma})[(\xi^{\tau}-\xi^{\sigma})(\partial_{\tau}\Phi)^{2}]$
(134)

Hence we have the total variation of the Lagrangian

$\delta_{\xi}\mathcal{L}=-\partial_{\rho}\Lambda^{\rho},$ (135)

where we have

$\displaystyle\Lambda^{\tau}=\xi^{\tau}[2(\partial_{\tau}\Phi)^{2}+\partial_{\tau}\Phi\partial_{\sigma}\Phi]-\xi^{\sigma}(\partial_{\tau}\Phi)^{2},\quad\Lambda^{\sigma}=[\xi^{\tau}(\partial_{\tau}\Phi)^{2}+\xi^{\sigma}\partial_{\tau}\Phi\partial_{\sigma}\Phi].$
(136)

Using this expression for $\delta_{\xi}\mathcal{L}$, we compute the onshell
expression for the Noether current:

$J^{\alpha}=\frac{\partial\mathcal{L}}{\partial(\partial_{\alpha}\Phi)}\delta_{\xi}\Phi-\Lambda^{\alpha},$
(137)

whose components are

$J^{\tau}=\xi^{\sigma}(\partial_{\tau}\Phi+\partial_{\sigma}\Phi)^{2},\quad
J^{\sigma}=0.$ (138)

This current is conserved onshell and the charges associated with it can be
computed by integrating $J^{\tau}$ over the spatial slice:

$Q_{\xi}=\int
d\sigma[\xi^{\sigma}(\partial_{\tau}\Phi+\partial_{\sigma}\Phi)^{2}].$ (139)

One noteworthy point here is that the dependence on $\xi^{\tau}$ is entirely
cancelled out in the expression of charges. This indicates that all the
supertranslation charges ($M_{n}$) vanish onshell, leaving only the
superrotation charges ($L_{n}$) non-trivial. These superrotation charges
satisfy one chiral copy of Virasoro algebra. This means the symmetries of the
theory are enhanced to chiral Virasoro onshell .

Same conclusion can be achieved by improving the stress tensor of the theory
as well. The conserved currents associated with BMS transformations can be
constructed using stress tensor components as

$J^{\alpha}=T^{\alpha}_{\ \beta}\xi^{\beta}.$ (140)

Upon using the conservation of stress tensor and the tracelessness condition,
we get from the above equation

$\partial_{\alpha}J^{\alpha}=(\partial_{\sigma}\xi^{\tau})T^{\sigma}_{\
\tau}.$ (141)

We see that the conservation of Noether current requires us to have a stress
tensor with $T^{\sigma}_{\ \tau}=0$. In quantum theory this condition on the
stress tensor can be accounted for Carrollian boost Ward identity. For our
case, the components of the stress tensor (75) for $e^{\tau}_{1}=1$ are

$\displaystyle T^{\tau}_{\ \tau}=-T^{\sigma}_{\
\sigma}=\frac{1}{2}(\partial_{\tau}\Phi)^{2},\quad T^{\tau}_{\
\sigma}=\frac{1}{2}(\partial_{\sigma}\Phi)^{2}+\partial_{\sigma}\Phi\partial_{\tau}\Phi,\quad
T^{\sigma}_{\ \tau}=\frac{1}{2}(\partial_{\tau}\Phi)^{2}.$ (142)

This stress tensor is traceless and conserved but $T^{\sigma}_{\ \tau}\neq 0$.
This again implies boost non-invariance but it is possible to improve the
stress tensor and set $T^{\sigma}_{\ \tau}=0$ by using equations of motion.

In order to do this improvement we add $B^{\alpha}_{\ \beta}$ to
$T^{\alpha}_{\ \beta}$ such that $B^{\tau}_{\ \tau}=-B^{\sigma}_{\
\sigma}\equiv f_{1}(\tau,\sigma)$, $B^{\tau}_{\ \sigma}\equiv
f_{2}(\tau,\sigma)$ and $B^{\sigma}_{\ \tau}=-(\partial_{\tau}\Phi)^{2}$. The
conservation equations $\partial_{\alpha}B^{\alpha}_{\ \beta}=0$, since
$T^{\alpha}_{\ \beta}$ is conserved, become

$\partial_{\tau}f_{1}=\partial_{\tau}\Phi\partial_{\tau}\partial_{\sigma}\Phi,\quad\partial_{\tau}f_{2}=\partial_{\sigma}f_{1}.$
(143)

A general (not using double derivatives acting on $\Phi$) solution of the
first equation above is

$f_{1}=\frac{1}{2}[\lambda(\partial_{\tau}\Phi+\partial_{\sigma}\Phi)^{2}-(\partial_{\tau}\Phi)^{2}],$
(144)

where the $\lambda$-term, $\lambda$ being an arbitrary constant parameter,
corresponds to the equation of motion in the conservation equations. With this
solution for $f_{1}$, the second conservation equation becomes

$\partial_{\tau}f_{2}=\lambda(\partial_{\tau}\Phi+\partial_{\sigma}\Phi)(\partial_{\tau}\partial_{\sigma}\Phi+\partial^{2}_{\sigma}\Phi)-\partial_{\tau}\Phi\partial_{\tau}\partial_{\sigma}\Phi.$
(145)

For simplicity and without loss of generality, we choose $\lambda=0$ and get

$f_{2}=-\frac{1}{2}(\partial_{\tau}\Phi)^{2}.$ (146)

This solution for $(f_{1},f_{2})$ gives us the improved stress tensor
$\Sigma^{\mu}_{\ \nu}=T^{\mu}_{\ \nu}+B^{\mu}_{\ \nu}$, whose components are

$T_{1}\equiv\Sigma^{\tau}_{\
\sigma}=\frac{1}{2}(\partial_{\tau}\Phi+\partial_{\sigma}\Phi)^{2},\quad
T_{2}\equiv\Sigma^{\tau}_{\ \tau}=-\Sigma^{\sigma}_{\
\sigma}=0,\quad\Sigma^{\sigma}_{\ \tau}=0.$ (147)

This meets the criteria for constructing conserved currents and matches with
what we have seen above from Noether’s procedure.

We also note that there is another way to obtain the improved stress tensor
directly from the variation of the action by making use of the degenerate
nature of the spacetime. In particular, owing to the constraint
$e^{\mu}_{1}\tau_{\mu}=0$, we project out the pure timelike component while
taking variation with respect to $e^{\mu}_{1}$, _i.e._

$\frac{\delta}{\delta
e^{\nu}_{1}}e^{\mu}_{1}\partial_{\mu}\Phi=\partial_{\nu}\Phi-\tau_{\nu}\tau^{\mu}\partial_{\mu}\Phi.$
(148)

Using this modified variation with respect to $e^{\mu}_{1}$ in the formula
(72) for the stress tensor and putting the flat values of vielbeins (42) along
with $e^{\tau}_{1}=1$ in the resulting expressions, we directly get (147).
Naively, though, it may seem that this method is different than improving the
stress tensor. However, a closer look reveals that the two procedures are
indeed equivalent. To see this, we note that while improving the stress tensor
to make $T^{\sigma}_{\ \tau}$ vanishing, we are effectively subtracting out
$(\partial_{\tau}\Phi)^{2}$ (recall that $B^{\sigma}_{\
\tau}=-(\partial_{\tau}\Phi)^{2}$). In the second procedure using the modified
variation (148), projecting out the timelike component effectively amounts to
imposing the equation of motion for $e^{\tau}_{1}$, _i.e._
$(\partial_{\tau}\Phi)^{2}=0$. Then putting $(\partial_{\tau}\Phi)^{2}=0$ in
(142) gives the improved stress tensor (147). Thus the two procedures are
equivalent and give the desired improved stress tensor.

###  6.3 Mode expansions and symmetry algebra

We start with the equation of motion,

$\partial^{2}_{\tau}\Phi+\partial_{\tau}\partial_{\sigma}\Phi=0.$ (149)

The most general solution ansatz subject to periodic boundary conditions would
be

$\Phi(\tau,\sigma)=\sum_{n}\Phi_{n}(\tau)e^{-in\sigma},$ (150)

where the modes $\Phi_{n}(\tau)$ satisfy

$\partial^{2}_{\tau}{\Phi_{n}(\tau)}-in\partial_{\tau}{\Phi_{n}(\tau)}=0.$
(151)

These equations are generically solved by

$\Phi_{0}(\tau)=\phi_{0}+B_{0}\tau,\quad\Phi_{n}(\tau)=\frac{i}{n}(A_{n}-B_{n}e^{in\tau})\
\forall\ n\neq 0,$ (152)

which gives the mode expansion for $\Phi$ as

$\Phi(\tau,\sigma)=\phi_{0}+B_{0}\tau+i\sum_{n\neq
0}\frac{1}{n}(A_{n}-B_{n}e^{in\tau})e^{-in\sigma}.$ (153)

Note that the general solution to (149) can be written as,

$\Phi(\tau,\sigma)=\phi_{0}+\frac{(A_{0}+B_{0})}{2}\tau+\frac{(A_{0}-B_{0})}{2}\sigma+i\sum_{n\neq
0}\frac{1}{n}(A_{n}-B_{n}e^{in\tau})e^{-in\sigma}.$

Imposing periodicity in $\sigma$ into the above implies $A_{0}=B_{0}$, giving
the periodic solution (153).

#### Poisson’s brackets and charge algebra:

The canonical momenta conjugate to $\Phi$ is

$\Pi=\frac{\delta\mathcal{L}}{\delta\dot{\Phi\textbullet}}=2\dot{\Phi}+\Phi^{\prime},$
(154)

where $\dot{\Phi}$ and $\Phi^{\prime}$ denote $\tau$ and $\sigma$ derivatives
of $\Phi$ respectively. In terms of the oscillators

$\displaystyle\dot{\Phi}(\tau,\sigma)=B_{0}+\sum_{n\neq
0}B_{n}e^{in(\tau-\sigma)}=\sum_{n}B_{n}e^{in(\tau-\sigma)},$
$\displaystyle\Phi^{\prime}(\tau,\sigma)=\sum_{n\neq
0}(A_{n}-B_{n}e^{in\tau})e^{-in\sigma}=\sum_{n}(A_{n}-B_{n}e^{in\tau})e^{-in\sigma},$
(155)

where we have used $B_{0}=A_{0}$ in the second equality in $\Phi^{\prime}$
above. Then the expression for the conjugate field momentum becomes

$\Pi(\tau,\sigma)=\sum_{n}(A_{n}+B_{n}e^{in\tau})e^{-in\sigma}.$ (156)

We find the algebra of oscillators by imposing the canonical Poisson’s
brackets between $\Pi(\tau,\sigma)$ and $\Phi(\tau,\sigma)$. For Carrollian
theories, the equal time Poisson’s brackets (85) imply the algebra for
oscillators:

$\\{A_{n},A_{m}\\}_{PB}=-in\delta_{n+m,0}\quad\\{B_{n},B_{m}\\}_{PB}=in\delta_{n+m,0},\quad\\{A_{n},B_{m}\\}_{PB}=0.$
(157)

Using this oscillator algebra, it is straightforward to show that the charges
satisfy one copy of Virasoro algebra. To see this, recall that the solutions
of the conformal Killing equations (49) on flat Carrollian backgrounds is
given by

$\xi^{\tau}=f^{\prime}(\sigma)\tau+g(\sigma)\quad,\quad\xi^{\sigma}=f(\sigma),$
(158)

where $f$ and $g$ are functions of $\sigma$ only. Taking
$f(\sigma)=\sum_{n}a_{n}e^{in\sigma}$, the charge in (139) gives

$\displaystyle Q_{\xi}=\sum_{n}a_{n}L_{n};\quad L_{n}=\int
d\sigma(\dot{\Phi}+\Phi^{\prime})^{2}e^{in\sigma}$ (159)

Using the mode expansion, we get an expression for $L_{n}$ in terms of
oscillators as

$L_{n}=\frac{1}{2}\sum_{p,q}\int d\sigma
A_{p}A_{q}e^{-i(p+q-n)\sigma}=\frac{1}{2}\sum_{p}A_{p}A_{n-p}.$ (160)

Alternately, we can also find expressions for $L_{n}$ and $M_{n}$ from the
components of the improved stress tensor (147). From $T_{2}=0$, using the
expression $T_{2}=\sum_{n}M_{n}e^{-in\sigma}$, we get $M_{n}=0$. On the other
hand,

$T_{1}=\frac{1}{2}(\partial_{\sigma}\Phi+\partial_{\tau}\Phi)^{2}=\sum_{m}\Big{(}\frac{1}{2}\sum_{n}A_{-n}A_{n+m}\Big{)}e^{-im\sigma}.$
(161)

Comparing this with the expression $T_{1}=\sum_{n}(L_{n}-in\tau
M_{n})e^{-in\sigma}$, along with $M_{n}=0$, one finds
$L_{m}=\frac{1}{2}\sum_{n}A_{-n}A_{n+m}$. Then using the oscillator algebra
(157), we find that these generators satisfy

$\\{L_{n},L_{m}\\}_{PB}=-i(n-m)L_{n+m},$ (162)

which is one copy of the Virasoro algebra, thus providing a confirmation of
what we argued from the Noether prescription as well.

###  6.4 Application: Flatspace Chiral Gravity

To wrap up this section, let us comment on a possible intriguing connection of
our mixed derivative theory with holography in asymptotically flat spacetimes.

As we mentioned in the introduction, the BMS3 algebra is the asymptotic
symmetry algebra of flat spacetimes at the null boundary. The details of the
gravitational theory with asymptotically flat boundary conditions is encoded
in the two central terms $c_{L}$ and $c_{M}$. For Einstein gravity, these turn
out to be $c_{L}=0,c_{M}=3/G$, where $G$ is the Newton’s constant in $3d$.

If one wishes to add a non-zero central term, an easy way of doing this is to
turn on a Gravitational Chern-Simons (GCS) term in the bulk theory, which now
is given by the action111111Below the subscript EH means Einstein-Hilbert.:

$S_{\text{bulk}}=S_{\text{EH}}+S_{\text{GCS}}=\frac{1}{16\pi G}\int
d^{3}x\sqrt{-g}\ R+\frac{1}{32\pi G\mu}\int
d^{3}x\sqrt{-g}\left(\Gamma\partial\Gamma+\frac{2}{3}\Gamma^{3}\right).$ (163)

In the above, the indices on the connections have been suppressed
($\Gamma^{a}_{\ bc}\equiv\Gamma$). This theory goes under the name of
Topologically Massive Gravity (TMG) Deser:1982vy , and in the AdS context this
has been used to provide evidence for a holographic theory of chiral gravity
Li:2008dq  and logarithmic gravity Grumiller:2008qz , when the parameter $\mu$
is tuned to a particular critical value. The central charges, in the case of
asymptotically flat boundary conditions, become

$c^{\text{(TMG)}}_{L}=\frac{3}{\mu G},\quad c^{\text{(TMG)}}_{M}=\frac{3}{G}.$
(164)

One can now take a particular interesting double scaling limit on this theory
such that

$\mu=\epsilon\to 0,\,G=\frac{1}{8k\epsilon}\to\infty,\,\text{with}\quad\mu
G=\frac{1}{8k}.$ (165)

This limit send the Einstein-Hilbert term in the bulk action to zero and one
is only left with the GCS term. This theory is called (conformal) Chern-Simons
Gravity.

$S_{\text{CSG}}=\frac{k}{4\pi}\int
d^{3}x\sqrt{-g}\left(\Gamma\partial\Gamma+\frac{2}{3}\Gamma^{3}\right).$ (166)

The central charge of the BMS3 that is obtained as the asymptotic symmetries
of this theory has

$c^{\text{(CSG)}}_{L}=24k,\quad c^{\text{(CSG)}}_{M}=0.$ (167)

It can further be shown that all the $M_{n}$ charges of the theory vanish
onshell. The symmetries are thus reduced from the BMS3 to a single copy of the
Virasoro algebra. This is a feature of $3d$ BMS invariant field theories with
$c_{M}=0$ and can be shown by an analysis of null vectors.121212Some evidence
to the contrary have been reported recently in Hao:2021urq . This depends
crucially on the Jordan-block structures that arise in more general BMS
invariant theories. It is possible that like the AdS story relating Chiral
gravity and Log gravity Maloney:2009ck , that Flatspace Chiral gravity exists
as a sector (singlet sector) within a more general theory. The dual theory to
CSG with asymptotically flat boundary conditions is thus a chiral $2d$ CFT
governed by the symmetries of a single Virasoro algebra. This has been called
Flatspace Chiral Gravity (F$\chi$G) Bagchi:2012yk . See Bagchi:2018ryy  for a
supersymmetric version.

Our discussions in this section have been focused on the mixed-derivative
theory, which was defined on a null surface, but had a chiral Virasoro
symmetry on-shell. We saw the truncation of symmetries from BMS3 to Virasoro
in this explicit example. It should be clear from the above discussion of
F$\chi$G that the mixed derivative theory has features that resemble a
possible dual field theory to Chern-Simons Gravity with flat boundary
conditions. The theory lives on a null manifold, as is expected for the dual
to CSG which should live on $\mathscr{I}^{\pm}$. And crucially, the symmetries
of the theory reduce from BMS to a single copy of the Virasoro algebra.

##  7 Null string theories and Carrollian actions

Tensionless or null string theories have been a well-studied example of a
Carrollian limit of a relativistic theory, in this case that from the
relativistic string worldsheet. The main ingredient in this theory has been
the ILST action Isberg:1993av  which replaces the well known Polyakov action
on a Carrollian worldsheet,

$S_{\text{ILST}}=\int d^{2}\sigma
V^{\mu}V^{\nu}\partial_{\mu}\Phi\partial_{\nu}\Phi,$ (168)

where $V^{\mu}$’s are vector densities under worldsheet diffeomorphisms and
the equations of motion for them imply that the worldsheet metric has
degenerated. Here we have suppressed the spacetime indices on the scalar field
$\Phi$, and we assume a $D$ dimensional flat target space geometry. The above
action can be thought of as intrinsic worldsheet action for null strings, or
equivalently a UR limit of the tensile string theory Bagchi:2015nca . In this
section, we will revisit this action in light of our discussions throughout
the bulk of this paper, and offer more insight into certain classes of actions
which appeared in previous sections.

###  7.1 Mapping ILST and zweibein formulations from UR limit

As one can understand, taking a $T\to 0$ limit on a tensile sigma model action
isn’t well defined as the string tension appears as the coupling constant.
Then one has to go to the Hamiltonian formulation and introduce auxiliary
fields to impose string constraints. In the formulation of ILST, the
worldsheet metric of a relativistic string (see Isberg:1993av  for more
details) is parametrized by these two Lagrange multipliers $(\lambda,\rho)$ as

$g^{\mu\nu}=\begin{pmatrix}-1&\rho\\\ \rho&\
-\rho^{2}+4\lambda^{2}T^{2}\end{pmatrix};\quad\det(g^{\mu\nu})=-4\lambda^{2}T^{2}.$
(169)

In the tensionless limit $T\rightarrow 0$, we see that the worldsheet
(inverse) metric degenerates with $\det(g^{\mu\nu})=0$. For comparison with
the zweibein formulation in previous sections, where we have
$\det(g_{\mu\nu})=0$, we use the conformal invariance of the Polyakov action
and perform a conformal transformation on (169):

$g_{\mu\nu}\rightarrow
G_{\mu\nu}=\frac{T^{2}}{4}g_{\mu\nu}=\frac{1}{16\lambda^{2}}\begin{pmatrix}\rho^{2}-4\lambda^{2}T^{2}&\rho\\\
\rho&1\end{pmatrix};\quad\det(G_{\mu\nu})=-\frac{T^{2}}{64\lambda^{2}},$ (170)

for which we have $\det(G_{\mu\nu})=0$ as $T\rightarrow 0$. One should note
here that in general the conformal transformation we made stops making sense
at $T=0$, and as a consequence the object $G_{\mu\nu}$ does not really exist
in the ILST formalism. However, continuing with the above, we can expand the
metric density around $T=0$ as

$-\frac{T}{2}\sqrt{-G}G^{\mu\nu}=\frac{1}{4\lambda}\begin{pmatrix}1&-\rho\\\
-\rho&\rho^{2}\end{pmatrix}-T^{2}\begin{pmatrix}0&0\\\
0&\lambda\end{pmatrix}.$ (171)

In the frame formulation, Carroll geometry is described by zweibeins
$e^{0}_{\mu}$, $e^{1}_{\mu}$ which are obtained from an ultrarelativistic
limit of the relativistic zweibeins as

$E^{0}_{\mu}=\epsilon e^{0}_{\mu},\quad E^{1}_{\mu}=e^{1}_{\mu},\quad
E=\epsilon e;\quad\epsilon\rightarrow 0.$ (172)

Then expanding $G^{\mu\nu}=\eta^{AB}E^{\mu}_{A}E^{\nu}_{B}$, with
$\sqrt{-G}=\sqrt{-\det(G_{\mu\nu})}=E$ and $\eta_{AB}=diag(-1,1)$, in
$\epsilon$ we can write the metric density in terms of zweibeins as

$\displaystyle-\epsilon\sqrt{-G}G^{\mu\nu}=ee^{\mu}_{0}e^{\nu}_{0}-\epsilon^{2}ee^{\mu}_{1}e^{\nu}_{1}.$
(173)

In the strict tensionless limit ($\epsilon=0$, or $T=0$), we can write the
degenerate metric density, _i.e._ the first term in (171) in terms of a vector
density $V^{\mu}$:

$\lim_{T\rightarrow 0}-\frac{T}{2}\sqrt{-G}G^{\mu\nu}=V^{\mu}V^{\nu};\quad
V^{\mu}=\frac{1}{2\sqrt{\lambda}}(1,-\rho).$ (174)

In the UR limit, the first term in (173) gives the degenerate metric density
in terms of zweibeins:

$\lim_{\epsilon\rightarrow
0}-\epsilon\sqrt{-G}G^{\mu\nu}=ee^{\mu}_{0}e^{\nu}_{0}.$ (175)

Identifying $\frac{T}{2}=\epsilon$,131313On the flat (Minkowski) worldsheet
described by $4\lambda=1$, $\rho=0$ giving
$ds^{2}=-d\tau^{2}+\frac{4}{T^{2}}d\sigma^{2}$, $ds^{2}=0$ gives
$|\frac{d\sigma}{d\tau}|$=$\frac{T}{2}\sim$ speed of light on the worldsheet.
we see that the tensionless limit is equivalent to the UR limit. Then
comparing (174) and (175), we get

$V^{\mu}=\sqrt{e}e^{\mu}_{0}.$ (176)

So that the ILST action explicitly becomes our timelike action, and no analog
of spacelike or mixed-derivative action can be found in this formalism.
Further, one can naively compare the subleading terms in (171) and (173), to
get

$e(e^{\tau}_{1})^{2}=0,\quad ee^{\tau}_{1}e^{\sigma}_{1}=0,\quad
e(e^{\sigma}_{1})^{2}=4\lambda,$ (177)

so that we seem to conclude that $e^{\tau}_{1}=0$ for Carroll geometry
described in ILST formulation. But considering $G_{\mu\nu}|_{T=0}$ as the
degenerate metric $h_{\mu\nu}$ in the Carrollian sense, we can see this
conclusion may not be right as one cannot constrain $e^{\tau}_{1}$ from this
structure. In Appendix A, we try to extend the ILST formalism beyond only the
timelike action using this choice of $h_{\mu\nu}$. This is also important
since $(\lambda,\rho)$ in general could be arbitrary functions of the
coordinates, and the conditions of flat Carroll geometry should impose
constraints on their structure. It does turn out that Carrollian structures
impose that the Lagrange multipliers have to be constants, however an unfixed
vielbien can freely exist in the theory as before.

###  7.2 Different gauges

Learning from the null string with degenerate worldsheet metric, _i.e._
$\det(g_{\mu\nu})=0$, we now try to understand some of the actions we have
discussed in this work. We would restrict ourselves for only one scalar for
the time being and start with the theory described by the action Isberg:1993av
, which in general reads

$S=\int d^{2}\sigma V^{\mu}V^{\nu}\partial_{\mu}\Phi\partial_{\nu}\Phi=\int
d^{2}\sigma\frac{1}{4\lambda}\big{(}\dot{\Phi}-\rho\Phi^{\prime}\big{)}^{2},$
(178)

but here $V^{\mu}=\frac{1}{2\sqrt{\lambda}}(1,-\rho)$ and $\lambda$, $\rho$
are generically non-zero constant Lagrange multipliers imposing the
constraints

$P^{2}=0,\quad P\partial_{\sigma}\Phi=0$ (179)

respectively. The stress tensor is given by varying the above action with
respect to $V^{\mu}$:

$T^{\mu}_{\
\nu}=V^{\mu}V^{\alpha}\partial_{\alpha}\Phi\partial_{\nu}\Phi-\frac{\delta^{\mu}_{\nu}}{2}V^{\alpha}V^{\beta}\partial_{\alpha}\Phi\partial_{\beta}\Phi,$
(180)

whose components are

$\displaystyle T^{\tau}_{\ \tau}=-T^{\sigma}_{\
\sigma}=\frac{1}{8\lambda}(\dot{\Phi}-\rho\Phi^{\prime})(\dot{\Phi}+\rho\Phi^{\prime}),$
$\displaystyle T^{\tau}_{\
\sigma}=\frac{1}{4\lambda}(\dot{\Phi}-\rho\Phi^{\prime})\Phi^{\prime},\quad
T^{\sigma}_{\
\tau}=\frac{1}{4\lambda}(\dot{\Phi}-\rho\Phi^{\prime})(-\rho\dot{\Phi}).$
(181)

Using the expression for the conjugate momentum,
$P=\frac{1}{2\lambda}(\dot{\Phi}-\rho\Phi^{\prime})$, we can write the
constraints (179) as

$\frac{1}{4\lambda^{2}}(\dot{\Phi}-\rho\Phi^{\prime})^{2}=0,\quad\frac{1}{2\lambda}(\dot{\Phi}-\rho\Phi^{\prime})\Phi^{\prime}=0.$
(182)

Equivalently, in first order formulation, the null string action, including
the constraints, is written as

$S=\int d^{2}\sigma(P\dot{\Phi}-\lambda P^{2}-\rho P\Phi^{\prime}),$ (183)

with Hamiltonian equations of motions having the compact form

$V^{\mu}\partial_{\mu}\Phi=\lambda P,~{}~{}V^{\mu}\partial_{\mu}P=0.$ (184)

The first order formalism is more interesting to use here as we can deal with
a choice where $\lambda=0$. Note here, due to structure of $V^{\mu}$ (174) and
equivalently that of $\tau^{\mu}$, the $\lambda=0$ dynamics cannot be
reflected into the second order equations of motion and constraints. Let us
discuss some explicitly chosen values of $(\lambda,\rho)$ to elaborate on
this.

#### 1) Static gauge: $\rho=0$, $\lambda=0$

The first order equations of motion here are

$\dot{\Phi}=0,~{}~{}\dot{P}=0,$ (185)

i.e. the solutions are static, there is no time evolution of the null string,

$\Phi(\tau,\sigma)=\Phi_{0}(\sigma),~{}~{}P(\tau,\sigma)=P_{0}(\sigma).$ (186)

Using a periodic boundary condition in $\sigma$, we can expand the fields into
modes

$\Phi_{0}(\sigma)=\sum_{n}A_{n}e^{-in\sigma},~{}~{}P_{0}(\sigma)=\sum_{n}P_{n}e^{-in\sigma}.$
(187)

And the Poisson brackets of modes read

$\\{A_{n},P_{m}\\}=\delta_{n+m,0},\quad\\{A_{n},A_{m}\\}=0,\quad\\{P_{n},P_{m}\\}=0.$
(188)

Then ignoring constant factors, one could write the constraints (179) as

$P\Phi^{\prime}\sim\sum_{n}L_{n}e^{-in\sigma}=0,~{}~{}~{}P^{2}\sim\sum_{n}M_{n}e^{-in\sigma}=0,$
(189)

where $L_{n}$ and $M_{n}$ again satisfy the classical part of the BMS3
algebra.

#### 2) “Timelike” gauge: $\rho=0$, $4\lambda=1$

From (178) we can see the second order action in this case is explicitly our
timelike action

$S=\int d\tau d\sigma(\partial_{\tau}\Phi)^{2}.$ (190)

The equations of motion in this case are

$\dot{\Phi}=\frac{P}{4},~{}~{}\dot{P}=0,$ (191)

so that the solution with periodic boundary conditons has a linear dependence
on time,

$\Phi(\tau,\sigma)=\Phi_{0}(\sigma)+\frac{\tau}{4}P_{0}(\sigma).$ (192)

One can again perform a mode expansion as before and compare this with the
solutions for our timelike action (80). This makes the structures of null
string constraints evident:

$P\Phi^{\prime}\sim\sum_{n}(L_{n}-in\tau
M_{n})e^{-in\sigma}=0,~{}~{}~{}P^{2}\sim\sum_{n}M_{n}e^{-in\sigma}=0.$ (193)

The generation of BMS3 algebra is guaranteed from implementation of these
constraints as well, due to the following intriguing automorphism of the
algebra:

$L_{n}^{\prime}\to L_{n}-in\tau M_{n},$ (194)

which keeps the bracket structure invariant.

#### 3) Ambitwistor gauge: $\rho=-1$, $\lambda=0$

In second order formulation, the ambitwistor gauge is singular due to the
presence of $\frac{1}{\lambda}$ in the action (178). However in the first
order formulation with the action (183), the ambitwistor gauge is well defined
Casali:2016atr . We should note herein lies the difference between our
spacelike action with $e^{\tau}_{1}=1$ (91) and pure Ambitwistor strings. In
the Ambitwistor gauge, the Hamiltonian action (183) becomes

$S=\int d^{2}\sigma(P\dot{\Phi}+P\Phi^{\prime})=\int d^{2}\sigma
P\partial_{+}\Phi.$ (195)

For $\lambda=0$, equations of motion become

$\partial_{+}P=0,\quad\partial_{+}\Phi=0;$ (196)

giving rise to solutions and mode expansions of the form

$\Phi=A(\sigma^{-})=\sum_{n}A_{n}e^{-in\sigma^{-}},~{}~{}P=P(\sigma^{-})=\sum_{n}P_{n}e^{-in\sigma^{-}}.$
(197)

One could see these solutions do not suffer from a periodicity issue like we
had in section (5.2) as there are no linear terms present in the mode
expansion. Like our static gauge case, the constraints are simple

$P\Phi^{\prime}\sim\sum_{n}L_{n}e^{-in\sigma^{-}}=0,~{}~{}~{}P^{2}\sim\sum_{n}M_{n}e^{-in\sigma^{-}}=0.$
(198)

It goes without saying the these still generate the BMS3 algebra.

#### 4) “Spacelike” gauge: $\rho=-1$, $4\lambda=1$

This is explicitly the case we have discussed in section 5. In this gauge, the
second order action (178) becomes

$S=\int d^{2}\sigma(\dot{\Phi}+\Phi^{\prime})^{2}=4\int
d\sigma^{+}d\sigma^{-}(\partial_{+}\Phi)^{2}.$ (199)

The Hamilton’s equation of motion and their solution turns out to be

$\partial_{+}P=0,\quad\partial_{+}\Phi=\lambda
P;\qquad\Phi=A(\sigma^{-})+\lambda\sigma^{+}P(\sigma^{-}),$ (200)

which clearly lacks periodicity in $\sigma$, as we encountered before. The
conjugate momentum is $P=2(\dot{\Phi}+\Phi^{\prime})=4\partial_{+}\Phi$ and
the constraints (182) become

$(\dot{\Phi}+\Phi^{\prime})^{2}=0,\quad(\dot{\Phi}+\Phi^{\prime})\Phi^{\prime}=0.$
(201)

The components of the stress tensor in first order formulation are

$\displaystyle T^{\tau}_{\ \tau}=-T^{\sigma}_{\
\sigma}=\frac{P^{2}}{8}-\frac{P\Phi^{\prime}}{2},\quad T^{\tau}_{\
\sigma}=\frac{1}{2}P\Phi^{\prime},\quad T^{\sigma}_{\
\tau}=\frac{P^{2}}{4}-\frac{P\Phi^{\prime}}{2}.$ (202)

We see that the action and the stress tensor above for the null string in the
gauge $\rho=-1$, $4\lambda=1$ match exactly with those of the spacelike theory
in (94). As we saw in that case, the symmetry structures are more evident in
the “lightcone” coordinates, where it still generates a BMS3 algebra.

The “take home” message from this section is that the ILST formalism can
generate certain gauge fixed versions of the geometrically obtained Carroll
Conformal actions in the preceding sections. Specifically those are the purely
timelike action and the spacelike action with a fixed $e_{1}^{\tau}=1$.
However, there is no clear description for any cousin of the mixed derivative
action in this formalism, making our discussion of the same even more unique.
See Appendix B for more discussion on this issue. The description of the mixed
action as a null string theory, or a close relative thereof, thus remains an
open question. We make a few speculative remarks about this at the end of our
discussions section, which is up next.

##  8 Discussions and conclusions

Summary

In this paper, we have analyzed various classical properties of Carroll
covariant actions for a massless scalar field in $2d$ Carroll spacetimes.
Using the zweibeins $(\tau^{\mu},e^{\mu})$ describing the Carroll geometry to
contract with $\partial_{\mu}\Phi$, we constructed three actions: the timelike
action, the spacelike action and the mixed-derivative action. On a generic
Carroll spacetime, each of these actions is Carroll diffeomorphism covariant
and BMS-Weyl invariant. We saw that in our formalism, fixing the background to
be flat Carroll spacetime leaves the component $e^{\tau}$ arbitrary. From a
purely geometric perspective, this unfixed zweibein $e^{\tau}$ is truely
redundant and leaving it arbitrary or choosing any constant value doesn’t
affect geometric quantities. However, on coupling the scalar field to the flat
Carroll spacetime, $e^{\tau}$ behaves non-trivially. In particular, its
arbitrariness ensure BMS3 invariance of all three actions for the scalar field
on the flat Carroll spacetime. But evidently, on taking $e^{\tau}$ to be
constant, the resultant spacelike and mixed-derivative actions have reduced
set of (offshell) residual symmetries, _i.e._ only translations and scaling.
This leads us to treat the spacelike and mixed-derivative actions differently.

For the spacelike action with $e^{\tau}$ fixed to a constant value, we found
that we recover the offshell BMS3 symmetries by transforming to lightcone
coordinates. We would like to emphasize that this is not inconsistent with the
earlier statement about residual symmetries being reduced to only translations
and scaling. To clarify, this reduction of residual symmetries happens in the
standard (flat) Carroll frame parametrized by ($\tau,\sigma$) coordinates,
whereas the BMS3 symmetries are recovered in lightcone coordinates which
parametrize an inequivalent (flat) Carroll frame (see sec. 2.3 for more
details).

Our novel mixed-derivative theory turned out to have quite a rich structure.
Offshell, the mixed-derivative action can be argued to possess a gauge
symmetry under gauge transformations of $e^{\tau}$. This gauge symmetry could
be interpreted to have a compensatory effect that allows us to gauge-fix
$e^{\tau}$ to a constant value without sacrificing the BMS symmetries.
Onshell, the mixed-derivative theory shows enhancement in the symmetries to
one copy of the Virasoro algebra.

To close our discussion, we made contact with existing literature on
Tensionless strings and described how it fits as a subset of the classes of
BMS invariant actions we have discussed in this work. Thus this work
encompassed examples of BMS as both global and gauge symmetries in physical
systems.

Future directions

This paper has been a small first step along a very interesting thoroughfare,
and we certainly have miles to go ahead. Since we focussed solely on the $2d$
case here, the immediate extension that comes to mind is to discuss analogous
actions in higher dimensions. A nice discussion has already appeared in
Gupta:2020dtl , but the corresponding analogue of our mixed derivative action
remains absent in that formalism.

Another point of interest would be to geometrically understand the
significance of the unfixed vielbien $e^{\tau}$. Although we could show there
is a gauge redundancy of that quantity in our formalism, the actual nature of
the freedom associated to it on a null manifold is subtle. Especially in the
case of $3d$ conformal Carroll actions, which should be important in studying
duals of $4d$ flat space gravity, this gauge redundancy may hold some more
surprises in store. Similar questions actually would survive if we want to
interpret our two dimensional actions with an unfixed $e^{\tau}$ as valid
string theories. We plan to come back to these problems in an upcoming work.

A very natural follow-up work would be to try and quantize the theories we
explored in this paper. Although for the timelike case, a large amount of
literature exist on quantum structures Bagchi:2020fpr ; Bagchi:2021rfw , and
the vacuum structure associated to Ambitwistor string is also very well
explored Casali:2016atr ; Casali:2017zkz ; Bagchi:2020fpr , the quantization