aainstitutetext: Indian Institute of Technology Kanpur, Kalyanpur, Kanpur
208016. INDIA.
bbinstitutetext: Department of Mathematical Sciences, Durham University,
Stockton Road, DH1 3LE, Durham, United Kingdom.
ccinstitutetext: Department of Physics, Indian Institute of Technology (Indian
School of Mines) Dhanbad, Jharkhand 826004, India.
ddinstitutetext: School of Basic and Applied Sciences, JSPM University, Gate
No. 720, Wagholi, Pune 412207, India.

# 3d Carrollian Chern-Simons theory & 2d Yang-Mills

Arjun Bagchi b Arthur Lipstein c Mangesh Mandlik d Aditya Mehra
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

With the goal of building a concrete co-dimension one holographically dual
field theory for four dimensional asymptotically flat spacetimes (4d AFS) as a
limit of AdS4/CFT3, we begin an investigation of 3d Chern-Simons matter (CSM)
theories in the Carroll regime. We perform a Carroll (speed of light $c\to 0$)
expansion of the relativistic Chern-Simons action coupled to a massless scalar
and obtain Carrollian CSM theories, which we show are invariant under the
infinite dimensional 3d conformal Carroll or 4d Bondi-van der Burg-Metzner-
Sachs (BMS4) symmetries, thus making them putative duals for 4d AFS.
Concentrating on the leading-order electric Carroll CSM theory, we perform a
null reduction of the 3d theory. Null reduction is a procedure to obtain non-
relativistic theories from a higher dimensional relativistic theory.
Curiously, null reduction of a Carrollian theory yields a relativistic lower-
dimensional theory. We work with $SU(N)\times SU(M)$ CS theory coupled to bi-
fundamental matter and show that when $N=M$, we obtain (rather surprisingly) a
2d Euclidean Yang-Mills theory after null reduction. We also comment on the
reduction when $N\neq M$ and possible connections of the null-reduced Carroll
theory to a candidate 2d Celestial CFT.

##  1 Introduction

The Holographic Principle tHooft:1993dmi ; Susskind:1994vu  has been our
preferred path in attempts to understand the quantum nature of gravity in
recent years. After the initial ideas originating from the area law of black
hole entropy, holography has become almost synonymous with its formulation in
Anti de Sitter (AdS) spacetimes through the celebrated AdS/CFT correspondence
Maldacena:1997re . The Maldacena conjecture gave us the first concrete dual
pair involving type IIB superstring theory on AdS${}_{5}\times\mathbbm{S}^{5}$
in the bulk and $\mathcal{N}=4$ SU$(N)$ Supersymmetric Yang-Mills theory on
the four-dimensional (4d) flat boundary of AdS5. This is sometimes called the
AdS5/CFT4 correspondence to distinguish it from similar correspondences in
other dimensions.

The more recent AdS4/CFT3 correspondence connects type IIA superstring theory
on AdS${}_{4}\times\mathbbm{CP}^{3}$ with ABJM theory Aharony:2008ug , which
is a $\mathcal{N}=6$ Superconformal Chern-Simons matter theory with a gauge
group $U(N)\times U(N)$ living on the 3d boundary of AdS4. When the string
coupling becomes large, type IIA superstring theory goes over to M-theory and
hence for generic values of parameters, ABJM theory is dual to M-theory on
AdS${}_{4}\times\mathbbm{S}^{7}/\mathbbm{Z}_{k}$. There is also a
generalisation of ABJM theory to unequal gauge groups $U(M)\times U(N)$
Aharony:2008gk . For more details on the subject, the reader is pointed to the
review Bagger:2012jb .

Of late, there is a renewed interest in the formulation of holography beyond
its original home in AdS, specifically to asymptotically flat spacetimes
(AFS). The case of 4d asymptotically flat space in the bulk is of particular
interest because of its obvious connection to the real world. There has been a
wealth of new connections in the infra-red established between seemingly
unrelated corners of asymptotic symmetries, soft theorems and memory effects
Strominger:2017zoo . Questions of holography in this context have followed in
a natural way.

There are now two principle routes to flat holography, viz. Celestial and
Carrollian holography. Celestial holography is the proposal that the
holographic dual to 4d AFS is a 2d relativistic CFT which lives on the
celestial sphere at null infinity. This makes use of the fact that the bulk
Lorentz group acts as global conformal transformations on the celestial
sphere. The reader is pointed to the recent reviews Strominger:2017zoo ;
Pasterski:2021raf ; Pasterski:2021rjz ; Raclariu:2021zjz  and the references
within. Carrollian holography, on the other hand, proposes a co-dimension one
hologram in terms of a 3d Carrollian CFT. A Carrollian theory can be obtained
from a relativistic one by sending the speed of light $c$ to zero LBLL ;
SenGupta:1966qer  and these are naturally defined on null surfaces. In
contrast with Celestial holography, the Carrollian version takes into account
the whole Poincare group which now acts as global Carrollian conformal
transformations on the whole of the null boundary, crucially keeping track of
the null direction. An incomplete set of references on Carrollian holography
is Bagchi:2016bcd ; Bagchi:2022emh ; Donnay:2022aba ; Donnay:2022wvx ;
Bagchi:2023fbj ; Salzer:2023jqv ; Saha:2023hsl ; Nguyen:2023vfz ;
Bagchi:2023cen ; Mason:2023mti ; Alday:2024yyj ; Ruzziconi:2024zkr  and older
work in this direction, especially in the context of lower dimensions include
Bagchi:2010eg ; Bagchi:2012cy ; Barnich:2012aw ; Bagchi:2012yk ; Bagchi:2012xr
; Barnich:2012xq ; Barnich:2012rz ; Bagchi:2014iea ; Hartong:2015usd .

The approaches to flat holography have been principally bottom up, with
Celestial holography relying on bulk physics to learn about the features of
the dual 2d CFT, and Carrollian holography mainly adopting a similar approach.
However see some recent attempts at top-down approaches involving twistor
theory Costello:2022wso ; Costello:2022jpg . It is natural to attempt to build
a theory of flat holography by taking a systematic limit of AdS/CFT
Susskind:1998vk ; Polchinski:1999ry ; Giddings:1999jq  and some recent
attempts in this direction include Ball:2019atb ; Casali:2022fro ;
Bagchi:2023fbj ; Bagchi:2023cen ; Alday:2024yyj . We will be interested in
this line of inquiry and will focus on 4d AFS. The large radius limit of AdS
induces a Carrollian limit in the boundary CFT Bagchi:2012cy . With this in
mind, our aim is to build the Carrollian equivalent of the ABJM model to
connect this to the flat version of the AdS4/CFT3 correspondence. In this
paper, we take the first steps towards this broader goal. We construct the
Carrollian limit of Chern-Simons (CS) matter theories in $d=3$.

It is by now well known that Carrollian limits come in two varieties called
the electric and magnetic limits. Given the action of a relativistic quantum
field theory, one can systematically expand out the relevant dynamic fields in
powers of the speed of light (this expansion is called the Carroll or
$c$-expansion, where $c$ is the speed of light) and the leading term in this
action is what goes under the name of the Electric theory. This is, by
construction, invariant under Carroll symmetries. The Carrollian electric
theories exhibit ultralocal correlation functions containing spatial delta-
functions. Such correlators of Carrollian CFTs can be mapped to S-matrix
elements in the bulk 4d asymptotically flat spacetimes by the so-called
modified Mellin transformation Bagchi:2022emh ; Banerjee:2018gce ;
Banerjee:2019prz . Electric Carrollian CFTs are thus prototypical of holograms
of flat spacetime. In our paper, we will mostly be interested in Electric
Carrollian theories. Magnetic Carrollian theories arise out of the next-to-
leading order (NLO) terms in the above mentioned $c$-expansion. The NLO term
by itself is not Carroll boost invariant and in order to restore Carrollian
symmetries, one needs to put in appropriate Lagrange multipliers. We will
briefly look at Magnetic Carrollian CSM theories in two appendices at the end
of the paper.

One of the important differences between holography in AdS4 and 4d AFS is the
symmetry structure at the boundary. A usual recipe for holography is to
consider the asymptotic symmetry group (ASG) as the symmetry that dictates the
dual field theory. The ASG is the group of allowed diffeomorphisms for a given
set of boundary conditions modded out by the trivial diffeomorphisms. For many
cases, as with AdS4, the ASG is simply the isometry group of the background
i.e. SO(3,2). In 4d AFS, however, the ASG at its null boundary enhances from
the usual Poincare group ISO(3,1) and becomes the infinite dimensional 4d
Bondi-van der Burg-Metzner-Sachs (BMS4) group Bondi:1962px ; Sachs:1962zza .
The 3d dual field theory is hence supposed to inherit this infinite
dimensional asymptotic BMS4 symmetry from the bulk Bagchi:2016bcd . Although
this process is non-trivial from the point of view of the Carrollian limit of
the CS-matter theory, we will show later in the paper that the 3d Carrollian
field theory that we obtain in the limit does admit this infinite dimensional
symmetry structure. For the uninitiated, this may seem like a magic trick
since the original theory only had finite dimensional symmetries. BMS
symmetries are isomorphic to conformal Carroll symmetries which are conformal
isometries of the background null structure Henneaux:1979vn ; Duval:2014uoa ;
Duval:2014uva  and hence the degeneration of the background Lorentzian
structure to form the Carrollian structure gives rise to these infinite
symmetries. 111The expectation that the theories obtained in the Carrollian
limit would lead to infinite dimensional symmetries in generic dimension was
shown to be true at the level of equations of motion for a wide variety of
theories in Bagchi:2019xfx . We elaborate on this later in the paper.

The main surprise in our paper comes in the next part of our analysis. In this
work, we are develop a specific 3d Carrollian CFT as a putative dual to a
gravitational theory in 4d AFS. As we mentioned above, there is also the
Celestial approach which proposes a 2d dual relativistic CFT. The 2d Celestial
CFT does not depend on the null direction and lives only on the celestial
sphere. In an attempt to obtain a 2d Celestial CFT from a 3d Carrollian one,
we propose to reduce the 3d Carrollian theory along the null direction. The
null reduction of the non-Abelian Carrollian CS matter theory interestingly
leads to a 2d Euclidean Yang-Mills theory. The choice of matter here is
crucial. We find that only bifundamental matter leads to 2d non-abelian Yang-
Mills, while fundamental matter leads to 2d electrodynamics. The Carroll limit
of the bosonic version of the ABJM theory with SU(N) $\times$ SU(N) gauge
group will lead us to SU(N) Yang-Mills theory in 2d. We also comment on the
more general SU(N) $\times$ SU(M) theory. We may expect the null-reduced
theory to represent a 2d Celestial CFTs, but a priori Yang-Mills theory in
$d=2$ is not conformally invariant. We argue that the theory one gets from the
limit inherits scale invariance, and hence full conformal invariance in $d=2$,
through the process of null reduction.

An outline of the rest of the paper is the following. We take a quick tour of
Carrollian and Conformal Carrollian symmetries in Sec. 2. Here we also touch
upon aspects of representation theory we would need later in the paper. We
focus on Abelian Chern-Simons matter theories in Sec. 3 and explain the
$c$-expansion and obtain the Electric and Magnetic Carroll CSM theories. We
discuss the emergence of infinite dimensional conformal Carroll symmetries of
the Electric theory in the main text while the symmetry structure of the
Magnetic sector is discussed in Appendix A. We then give some details of the
null reduction of Carroll CSM theories and obtain 2d electrodynamics starting
from the electric theory. The magnetic theory is discussed in Appendix B. Sec.
4 contains the generalisation to non-Abelian CSM theories, its Carrollian
construction and the details of the null reduced theory which now becomes a 2d
SU(N) Yang-Mills if we begin with bi-fundamental matter in CS theory with
gauge group SU(N) $\times$ SU(N). We also outline the construction for the
general SU(N) $\times$ SU(M) theory and discuss how the null-reduced theory
shows an emergent 2d conformal symmetry making it a candidate 2d Celestial
CFT. We conclude with various remarks.

##  2 Carroll and Conformal Carroll Symmetries

Carroll symmetry, first introduced by Levy-Leblond LBLL  and Sengupta
SenGupta:1966qer , has become very important of late with emerging
applications in a wide variety of physical scenarios, starting from condensed
matter Bidussi:2021nmp ; Bagchi:2022eui  and ultra-relativistic fluids
Bagchi:2023ysc ; Bagchi:2023rwd  to gravitational physics Donnay:2019jiz ;
deBoer:2021jej  and string theory Bagchi:2013bga ; Bagchi:2015nca ;
Bagchi:2020fpr ; Bagchi:2023cfp . These symmetries arise naturally on null
surfaces and hence are found on the event horizons of generic black holes and
also at the asymptotic null boundary of flat spacetimes, where the symmetries
enhances to their conformal version. The latter is where we would be
interested in for our explorations in this paper. In order to set up our
calculations in the coming sections, below we give a quick summary of Carroll
and conformal Carroll symmetry first from an algebraic and then from a
geometric point of view.

###  2.1 Algebraic and Geometric preliminaries

The Carroll algebra is an Inönü-Wigner contraction of the relativistic
Poincare algebra where one takes the speed of light to zero ($c\to 0$). The
conformal Carroll can be obtained by a similar contraction of the relativistic
conformal algebra. Starting with the differential representation of the
relativistic conformal algebra:

$\displaystyle J_{\mu\nu}=x_{\mu}\partial_{\nu}-x_{\mu}\partial_{\nu},\quad
P_{\mu}=\partial_{\mu},\quad D=x^{\mu}\partial_{\mu},\quad
K_{\mu}=2x_{\mu}x^{\nu}\partial_{\nu}-x^{\nu}x_{\nu}\partial_{\mu}$ (1)

one can take the $c\to 0$ limit by sending $t\to\epsilon t,\,x_{i}\to x_{i}$
to get the set of generators for the conformal Carroll algebra:

$\displaystyle H=\partial_{t},\quad P_{i}=\partial_{i},\quad
J_{ij}=x_{i}\partial_{j}-x_{j}\partial_{i},\quad B_{i}=x_{i}\partial_{t}$ (2a)
$\displaystyle D=t\partial_{t}+x^{i}\partial_{i},\quad
K=x^{i}x_{i}\partial_{t},\quad
K_{j}=2x_{j}(t\partial_{t}+x^{i}\partial_{i})-(x^{i}x_{i})\partial_{j}.$ (2b)

The non-zero commutation relations of these above generators that form the
conformal Carrollian algebra are:

$\displaystyle[J_{ij},B_{k}]=\delta_{k[j}B_{i]},~{}[J_{ij},P_{k}]=\delta_{k[j}P_{i]},~{}[J_{ij},K_{k}]=\delta_{k[j}K_{i]},~{}[B_{i},P_{j}]=-\delta_{ij}H,$
$\displaystyle[B_{i},K_{j}]=\delta_{ij}K,~{}[D,K]=K,~{}[K,P_{i}]=-2B_{i},~{}[K_{i},P_{j}]=-2D\delta_{ij}-2J_{ij},$
$\displaystyle[H,K_{i}]=2B_{i},~{}[D,H]=-H,~{}[D,P_{i}]=-P_{i},~{}[D,K_{i}]=K_{i}.$
(3)

The sub-algebra $\\{J_{ij},B_{i},P_{i},H\\}$ forms the Carroll algebra. We
will now focus on (2+1)-dimensions. Let us recombine the above generators as

$\displaystyle
L_{0}=\frac{1}{2}(D+iJ_{xy}),~{}~{}L_{-1}=-\frac{1}{2}(P_{x}-iP_{y}),~{}~{}L_{1}=\frac{1}{2}(K_{x}+iK_{y}),$
(4a)
$\displaystyle\bar{L}_{0}=\frac{1}{2}(D-iJ_{xy}),~{}~{}\bar{L}_{-1}=-\frac{1}{2}(P_{x}+iP_{y}),~{}~{}\bar{L}_{1}=\frac{1}{2}(K_{x}-iK_{y}),$
(4b) $\displaystyle
M_{00}=P_{0},~{}~{}M_{01}=B_{x}-iB_{y},~{}~{}M_{10}=B_{x}+iB_{y},~{}~{}M_{11}=K_{0}.$
(4c)

Using the differential representation of the conformal Carroll algebra and the
definitions (4), we obtain a suggestive form for the Conformal Carroll
generators:

$\displaystyle
L_{n}=z^{n+1}\partial_{z}+\frac{1}{2}(n+1){z^{n}}t\partial_{t},~{}\bar{L}_{n}=\bar{z}^{n+1}\partial_{\bar{z}}+\frac{1}{2}(n+1)\bar{z}^{n}t\partial_{t},~{}M_{nm}=z^{n}\bar{z}^{m}\partial_{t}.\quad$
(5)

where $z=x+iy$ and $\bar{z}=x-iy$. The conformal Carroll algebra now takes the
form

$\displaystyle[L_{n},L_{m}]=(n-m)L_{n+m},~{}~{}[\bar{L}_{n},\bar{L}_{m}]=(n-m)\bar{L}_{n+m},$
(6a)
$\displaystyle~{}[L_{n},M_{rs}]=\Big{(}\frac{n+1}{2}-r\Big{)}M_{(n+r)s},~{}~{}[\bar{L}_{n},M_{rs}]=\Big{(}\frac{n+1}{2}-s\Big{)}M_{r(n+s)}.$
(6b) $\displaystyle~{}[M_{rs},M_{pq}]=0.$ (6c)

where $n=-1,0,1$ and $r,s=0,1$. If we now extend the generators (5) for
arbitrary integer $n,r,s$, the algebra above (6) is infinite dimensional. This
algebra is isomorphic to the four dimensional Bondi-van der Burg-Metzner-Sachs
algebra (BMS4) which is the asymptotic symmetry algebra of asymptotically flat
4d spacetimes at the null boundary Bondi:1962px ; Sachs:1962zza .

We now give a geometric account of these symmetries Henneaux:1979vn ;
Duval:2014uoa ; Duval:2014uva . In flat space, it is very evident the Carroll
limit makes the Minkowski metric degenerate. The metric with covariant indices
becomes:

$\eta_{\mu\nu}=\begin{pmatrix}-c^{2}&0&0\\\ 0&1&0\\\
0&0&1\end{pmatrix},\quad\eta_{\mu\nu}\xrightarrow{c\to
0}h_{\mu\nu}=\begin{pmatrix}0&0&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix},$ (7)

while the contravariant version takes the following form

$\eta^{\mu\nu}=\begin{pmatrix}-\frac{1}{c^{2}}&0&0\\\ 0&1&0\\\
0&0&1\end{pmatrix},\quad-c^{2}\eta^{\mu\nu}\xrightarrow{c\to
0}\Theta^{\mu\nu}=\begin{pmatrix}1&0&0\\\ 0&0&0\\\
0&0&0\end{pmatrix}=\theta^{\mu}\theta^{\nu},\,\,\text{where}\,\,\theta^{\mu}=\begin{pmatrix}1\\\
0\\\ 0\end{pmatrix}.$ (8)

It is clear from the above that we have

$h_{\mu\nu}\theta^{\nu}=0$ (9)

One can generalise this structure to define general Carrollian manifolds with
the pair $(h_{\mu\nu},\theta^{\mu})$. Formally, a Carroll manifold is defined
as a $d$-dimensional manifold endowed with a degenerate symmetric positive
covariant tensor field $h_{\mu\nu}$ and nowhere vanishing vector field
$\theta$ which generates the kernel of $h$. This is a “weak” Carrollian
structure as opposed to a “strong" structure which also requires the existence
of a symmetric affine connection compatible with both $h$ and $\theta$.

The Carroll algebra is obtained as the isometry of a flat Carroll manifold

$\mathcal{L}_{\zeta}\theta^{\mu}=0,\quad\mathcal{L}_{\zeta}h_{\mu\nu}=0.$ (10)

Here $\mathcal{L}_{\zeta}$ represents a Lie derivative along the vector field
$\zeta$. This actually leads to an infinite dimensional algebra, which reduces
to the finite dimensional Carroll algebra we obtained above in the limit when
we restrict to linear functions. We shall mostly be interested in the
conformal structures on these manifolds. The conformal isometry is generated
by

$\mathcal{L}_{\zeta}\theta^{\mu}=\lambda\theta^{\mu},\quad\mathcal{L}_{\zeta}h_{\mu\nu}=-2\lambda
h_{\mu\nu}.$ (11)

Here $\lambda$ is the conformal factor222In general, one could choose
different conformal factors $\lambda_{1}$ and $\lambda_{2}$ for $\theta$ and
$h$ and this would lead to the so-called $N$-conformal Carroll algebras, where
$N=-\lambda_{2}/\lambda_{1}$ and this is related to the anisotropy factor
$z=N/2$ which dictates the relative scaling of space and time under
dilatations. From the point of view of holography of asymptotically flat
spacetimes, where the bulk is a 4d relativistic spacetime, we are interested
in 3d field theories that have uniform scaling of space and time, $z=1$ and
the above choice is valid.. For flat Carroll backgrounds, the solution to the
conformal isometry equations above is given by:

$\displaystyle\xi=\Big{(}\alpha(x^{i})+\frac{t}{2}\partial_{i}f^{i}(x^{j})\Big{)}\partial_{t}+f^{i}(x^{j})\partial_{i}.$
(12)

Here $x^{i}$ denotes the $(d-1)$ spatial directions of the $d$-dimensional
Carroll manifold. $\alpha(x^{i})$ are arbitrary functions of these spatial
coordinates and parametrise supertranslations. $f^{i}(x^{j})$ also satisfy
conformal killing equations on the spatial slice. We are interested in the
case $d=3$ and hence here $f^{i}(x^{j})$ are restricted to be to be
holomorphic/anti-holomorphic functions, i.e. $f\equiv f(z)$ and
$\bar{f}\equiv\bar{f}(\bar{z})$. It is clear from the above that we can define
the generators of the algebra of Carrollian conformal isometry as follows

$L(f)=f(z)\partial_{z}+\frac{t}{2}\partial_{z}f(z)\,\partial_{t},\quad
L(\bar{f})=\bar{f}({\bar{z}})\partial_{\bar{z}}+\frac{t}{2}\partial_{\bar{z}}\bar{f}({\bar{z}})\partial_{t},\quad
M(\alpha)=\alpha(z,{\bar{z}})\partial_{t}.$ (13)

If we break this up into modes

$\displaystyle f(z)$ $\displaystyle=$
$\displaystyle\sum_{n}a_{n}z^{n+1},\quad\bar{f}({\bar{z}})=\sum_{n}{\bar{a}}_{n}{\bar{z}}^{n+1},\quad\alpha(z,{\bar{z}})=\sum_{r,s}b_{r,s}z^{r}{\bar{z}}^{s}$
(14) $\displaystyle L(f)$ $\displaystyle=$
$\displaystyle\sum_{n}a_{n}L_{n},\quad
L(\bar{f})=\sum_{n}{\bar{a}}_{n}\bar{L}_{n},\quad
M(\alpha)=\sum_{r,s}b_{r,s}M_{r,s}$ (15)

it is straight-forward to check that the generators are the same as (5) and
obey the infinite dimensional BMS4 algebra.

###  2.2 Aspects of representation theory

In this subsection, we briefly recall aspects of representations of Carrollian
CFTs. The construction of the representations of Carrollian CFTs is similar to
the relativistic conformal case. Our construction here would be important to
understand the symmetries of the specific Carrollian field theories, i.e. the
Carroll CSM theories we will focus on later in the paper.

Let us consider how conformal Carrollian symmetry acts on a generic field
$\Phi$ which can be looked upon as a multiplet of different fields $\phi_{i}$:

$\Phi=\begin{pmatrix}\phi_{1}\\\ \vdots\\\ \phi_{n}\end{pmatrix}.$ (16)

We first focus on the little group that keeps the origin ($t=0,x_{i}=0$)
invariant. This is the subgroup generated by the rotations, Carroll boosts,
dilatations, and Carroll SCTs. The action of the generators on $\Phi$ is given
by

$\displaystyle[J_{ij},\Phi(0)]=\mathcal{S}_{ij}\Phi(0),~{}~{}[B_{i},\Phi(0)]=\mathcal{B}_{i}\Phi(0),~{}~{}[D,\Phi(0)]=\Delta\Phi(0),$
(17a)
$\displaystyle~{}[K_{i},\Phi(0)]=k_{i}\Phi(0),~{}~{}[K,\Phi(0)]=k\Phi(0).$
(17b)

The little group generators form a matrix representation at the origin. We can
set $k$ and $k_{i}$ to zero as a consequence of the algebra of these
generators, and this is very similar to the usual relativistic CFT analysis.
The representations of the whole conformal Carroll algebra are induced from
this. The transformations of the fields under the action of the different
generators of the algebra at arbitrary points are given by using the
translation generators on the generators to move them to act on the field at
that point

${\mathcal{O}}(t,x_{i})=e^{iHt}e^{iP_{i}x^{i}}{\mathcal{O}}(0)e^{-iHt}e^{-iP_{i}x^{i}},$
(18)

where ${\mathcal{O}}$ represents a generic operator, and in this case a member
of the generators, and by repeated use of the Baker-Campbell-Hausdorff
formula. This yields:

$\displaystyle[J_{ij},\Phi(t,x_{i})]=(x_{i}\partial_{j}-x_{j}\partial_{i}+\mathcal{S}_{ij})\Phi(0),$
(19a)
$\displaystyle[B_{i},\Phi(t,x_{i})]=(x_{i}\partial_{t}+\mathcal{B}_{i})\Phi(0),$
(19b)
$\displaystyle[D,\Phi(t,x_{i})]=(t\partial_{t}+x^{i}\partial_{i}+\Delta)\Phi(0),$
(19c)
$\displaystyle[K,\Phi(t,x_{i})]=(x^{2}\partial_{t}+2x_{i}\mathcal{B}_{i})\Phi(0),$
(19d)
$\displaystyle[K_{i},\Phi(t,x_{i})]=(2x_{i}\Delta-2x_{j}\mathcal{S}_{ij}+2t\mathcal{B}_{i}+2tx_{i}\partial_{t}+2x_{i}x_{j}\partial_{j}-x^{2}\partial_{i})\Phi(0).$
(19e)

In the Carrollian CFTs, we label fields by their dilatation weight $\Delta$
and consider various spins $\mathcal{S}_{ij}$. The non-trivial features are
encoded in the boost matrices $\mathcal{B}_{i}$, as we will see below.

### Action of infinite dimensional generators on fields in 3d

We now focus on $d=3$ and discuss aspects of the representations of the
infinite dimensional algebra. We define primaries of the whole infinite
dimensional conformal Carroll algebra. All fields are labelled under $L_{0}$
and $\bar{L}_{0}$

$[L_{0},\Phi]=h\Phi,\quad[\bar{L}_{0},\Phi]={\bar{h}}\Phi$ (20)

Here $h+{\bar{h}}=\Delta$ and $h-\bar{h}=\mathcal{S}$. Drawing analogies with
usual CFT, we define Carrollian primaries are those for which the weights
cannot be lowered further:

$[L_{n},\Phi]=0,\quad[\bar{L}_{n},\Phi]=0,[M_{r,s},\Phi]=0,\quad\forall
n,r,s>0$ (21)

Quasi-primaries are primaries with respect to the global Poincare sub-algebra.
In $d=3$, the algebra of the spin matrices related to Carroll boosts
$(\mathcal{B}_{x},\mathcal{B}_{y})$ and rotations $\mathcal{S}$ is given by

$[\mathcal{S},\mathcal{B}_{x}]=-\mathcal{B}_{y},~{}~{}[\mathcal{S},\mathcal{B}_{y}]=\mathcal{B}_{x},~{}~{}[\mathcal{B}_{x},\mathcal{B}_{y}]=0.$
(22)

The commuting nature of the boosts makes it possible to have different boost
labels for a particular spin.

#### Spin 0 case:

We first look at the scalar representation. This is simply obtained by setting

$\mathcal{S}=\mathcal{B}_{x}=\mathcal{B}_{y}=0.$ (23)

With the input above, we can write down the transformation of the primaries
$\Phi(t,z,\bar{z})\equiv\phi(t,z,\bar{z})$ for the whole infinite dimensional
algebra:

$\displaystyle[M_{nm},\phi(t,z,\bar{z})]=z^{n}\bar{z}^{m}\partial_{t}\phi(t,z,\bar{z}),$
(24a)
$\displaystyle[L_{n},\phi(t,z,\bar{z})]=\frac{1}{2}[(z^{n}(n+1)(\Delta_{\phi}+t\partial_{t})+2z^{n+1}\partial_{z})]\phi(t,z,\bar{z}),$
(24b)
$\displaystyle[\bar{L}_{n},\phi(t,z,\bar{z})]=\frac{1}{2}[(\bar{z}^{n}(n+1)(\Delta_{\phi}+t\partial_{t})+2\bar{z}^{n+1}\partial_{\bar{z}})]\phi(t,z,\bar{z}).$
(24c)

Here, $\Delta_{\phi}$ denotes the scaling weight of field $\phi$. The
subscripts $(n,m)>0$. This is again done by translating the generator to
$(x,t)$ by (18) and using the BCH formula. We also invoke (21).

#### Spin 1 case:

The spin 1 representation of rotation means that we have

$\mathcal{S}_{ij}=\begin{bmatrix}0&0&0\\\ 0&0&-1\\\ 0&1&0\end{bmatrix}.$ (25)

We now have options for our boost generators consistent with (22). One of this
is the trivial representation:

$\mathcal{B}_{x}=\mathcal{B}_{y}=0.$ (26)

The non-Lorentzian nature of the algebra means that one can have more than one
representation for the boost generators corresponding to a particular spin. We
will be interested in the non-trivial representation:

$\displaystyle\mathcal{B}_{x}=\begin{bmatrix}0&0&0\\\ 1&0&0\\\
0&0&0\end{bmatrix},~{}\mathcal{B}_{y}=\begin{bmatrix}0&0&0\\\ 0&0&0\\\
1&0&0\end{bmatrix}.$ (27)

These non-trivial boost matrices described are non-diagonalisable that means
the components of spinning primaries would mix under boost transformations. We
will work in a basis where the spin 1 $\Phi$ field is given by:

$\Phi(t,z,\bar{z})=\begin{pmatrix}a_{t}(t,z,{\bar{z}})\\\
a_{z}(t,z,{\bar{z}})\\\ a_{\bar{z}}(t,z,{\bar{z}})\end{pmatrix}$ (28)

where $a_{z}=(a_{x}-ia_{y})$ and $a_{\bar{z}}=(a_{x}+ia_{y})$. The action of
supertranslation on the different components is given by:

$\displaystyle[M_{nm},a_{t}]=z^{n}\bar{z}^{m}\partial_{t}a_{t},$ (29a)
$\displaystyle[M_{nm},a_{z}]=z^{n}\bar{z}^{m}\partial_{t}a_{z}+2nz^{n-1}\bar{z}^{m}a_{t},$
(29b)
$\displaystyle[M_{nm},a_{\bar{z}}]=z^{n}\bar{z}^{m}\partial_{t}a_{\bar{z}}+2mz^{n}\bar{z}^{m-1}a_{t}.$
(29c)

Notice that the Jordan block structure of the boosts mean that $a_{t}$ is
present in the transformation of $a_{z},a_{\bar{z}}$ but only transforms into
itself under supertranslations. Similarly, the action of superrotations on the
components are given by

$\displaystyle[L_{n},a_{t}]=\left[z^{n+1}\partial_{z}+\frac{1}{2}z^{n}(n+1)(\Delta+t\partial_{t})\right]a_{t},$
(30a)
$\displaystyle[L_{n},a_{z}]=\left[z^{n+1}\partial_{z}+\frac{1}{2}z^{n}(n+1)(\Delta+1+t\partial_{t})\right]a_{z}+2tn(n+1)a_{t}z^{n-1},$
(30b)
$\displaystyle[L_{n},a_{\bar{z}}]=\left[z^{n+1}\partial_{z}+\frac{1}{2}z^{n}(n+1)(\Delta-1+t\partial_{t})\right]a_{\bar{z}}.$
(30c)

and for the anti-holomorphic counterpart:

$\displaystyle[\bar{L}_{n},a_{t}]=\left[{\bar{z}}^{n+1}\partial_{\bar{z}}+\frac{1}{2}{\bar{z}}^{n}(n+1)(\Delta+t\partial_{t})\right]a_{t},,$
(31a)
$\displaystyle[\bar{L}_{n},a_{z}]=\left[{\bar{z}}^{n+1}\partial_{\bar{z}}+\frac{1}{2}{\bar{z}}^{n}(n+1)(\Delta-1+t\partial_{t})\right]a_{z},$
(31b)
$\displaystyle[\bar{L}_{n},a_{\bar{z}}]=\left[{\bar{z}}^{n+1}\partial_{\bar{z}}+\frac{1}{2}{\bar{z}}^{n}(n+1)(\Delta+1+t\partial_{t})\right]a_{\bar{z}}+2tn(n+1)a_{t}\bar{z}^{n-1},$
(31c)

Notice that the different components have different scaling dimensions.
Comparing with (20), we see that

$h_{a_{t}}=\bar{h}_{a_{t}}=\frac{\Delta}{2};\quad
h_{a_{z}}=\frac{\Delta+1}{2},\bar{h}_{a_{z}}=\frac{\Delta-1}{2};\quad
h_{a_{\bar{z}}}=\frac{\Delta-1}{2},\bar{h}_{a_{\bar{z}}}=\frac{\Delta+1}{2}.$
(32)

One can similarly build higher spin representations. There will be more
choices for non-trivial boost matrices as one increases the spin, with all the
lower spin boost matrices showing up. For the purposes of this paper, we will
be concerned with spin-0 and spin-1 cases.

##  3 Abelian Chern-Simons coupled to scalar matter

Our goal in this paper is to construct Carrollian versions of Chern-Simons
matter theories. We will focus on 3 dimensions. In this section, we give an
overview of the basic construction of these theories from the point of view of
an expansion in powers of the speed of light $c$ following deBoer:2021jej  and
demonstrate the technique for the Abelian CS theory before we move onto the
more interesting non-Abelian case in the next section. This section provides a
warm-up for the more involved case to be discussed later. We will also comment
on the symmetries of the actions derived and the most interesting point, the
dimensional reduction of the Carrollian CSM theory.

We begin with the well-known relativistic $U(1)$ Chern-Simons theory and
couple this to a scalar. This theory is described by the action:

$S=\int
dtd^{2}x\,\left\\{\frac{k}{4\pi}\epsilon^{\mu\nu\rho}A_{\mu}\partial_{\nu}A_{\rho}+(D_{\mu}\phi)^{*}(D^{\mu}\phi)\right\\},$
(33)

where $\mu=0,1,2$, $k$ is the level of the Chern-Simons term and $D_{\mu}$ is
the gauge covariant derivative: $D_{\mu}=\partial_{\mu}-ieA_{\mu}$. We note
that under a general coordinate transformation, the gauge field transforms as:

$\delta
A_{\mu}=\xi^{\nu}\partial_{\nu}A_{\mu}+\partial_{\mu}\xi^{\nu}A_{\nu}.$ (34)

We would be interested in splitting up spatial and temporal components in
order to consider the Carroll limit. The restriction of the above general
coordinate transformation to a Lorentz transformation is

$\xi^{0}=\frac{\beta_{i}x^{i}}{c},\quad\xi^{i}=\frac{x^{0}}{c}\beta^{i}=t\beta^{i}$
(35)

The gauge field $A_{\mu}$ and real scalar field transforms under Lorentz
boosts as

$\displaystyle\delta
A_{\mu}=ct\beta^{i}\partial_{i}A_{\mu}+\frac{1}{c}\beta_{i}x^{i}\partial_{t}A_{\mu}+\bar{\delta}A_{\mu},~{}~{}\text{where}~{}\bar{\delta}A_{0}=\beta^{i}A_{i},\,\bar{\delta}A_{i}=\beta_{i}A_{0},$
(36a)
$\displaystyle\delta\phi=ct\beta^{i}\partial_{i}\phi+\frac{1}{c}\beta_{i}x^{i}\partial_{t}\phi.$
(36b)

###  3.1 Carrollian expansion

We will now construct the Carrollian version of the CS theory coupled to a
scalar field. We will use an expansion of all fields in a power series in
$c^{2}$ deBoer:2021jej . The leading term would become what is known as the
Electric Carroll theory, while the sub-leading term, with appropriate
modifications, becomes the Magnetic theory. The fields in our theory are
expanded as:

$A_{t}=\sum^{\infty}_{n=0}c^{\lambda}c^{2n}a^{(n)}_{t},~{}A_{i}=\sum^{\infty}_{n=0}c^{\lambda}c^{2n}a^{(n)}_{i},~{}\phi=\sum^{\infty}_{n=0}c^{\gamma}c^{2n}\phi^{(n)}.$
(37)

where we use $A_{t}=cA_{0}$. We find the transformation rules of the fields at
a generic level $(n)$ by considering the expansion of the relativistic fields
again. Let us specifically look at the transformation under boosts. We define

$\beta_{i}=cb_{i},$ (38)

where $b_{i}$ is the Carroll boost parameter. The fields then transforms as

$\displaystyle\delta
a_{t}^{(n)}=b_{i}x^{i}\partial_{t}a_{t}^{(n)}+tb^{i}\partial_{i}a_{t}^{(n-1)}+b^{i}a_{i}^{(n-1)},$
(39a) $\displaystyle\delta
a_{i}^{(n)}=b_{j}x^{j}\partial_{t}a_{i}^{(n)}+b^{j}t\partial_{j}a_{i}^{(n-1)}+b^{i}a_{t}^{(n)}.$
(39b)

where for $n=0$, the transformations are included using $a_{\mu}^{(-1)}=0$. It
is straight-forward to see that the leading $n=0$ transformations are
identical to what we had derived earlier from the representations of the
conformal Carroll algebra in (29). In conclusion, the set
$(a^{(0)}_{t},a^{(0)}_{i})$ acts like a vector field with respect to Carroll
transformations. These rules are also applicable for the scalar field. The
resultant higher modes in the expansion transforms into each other under these
boosts.

###  3.2 Electric and Magnetic Actions

We will now study the expansion of Chern Simon theory coupled to scalar field.
The action (33) with explicit $c$ factors is given by

$S=\int
dtd^{2}x\,\frac{k}{4\pi}\Big{[}\epsilon^{\mu\nu\rho}A_{\mu}\partial_{\nu}A_{\rho}\Big{]}-\frac{1}{c^{2}}(D_{t}\phi)^{*}(D_{t}\phi)+(D_{i}\phi)^{*}(D_{i}\phi)=\int
dtd^{2}x\,\mathcal{L}.$ (40)

We will plug (37) into (40) and extract the leading and subleading pieces. We
will take $\lambda=\gamma-1$ since we wish to keep the Chern-Simons term at
leading order. Interestingly, we get two distinct theories corresponding to
$\lambda=0$ and $\lambda\neq 0$. For $\lambda\neq 0$, it is a straight-forward
exercise to check that the interaction terms between the gauge fields and
scalars (in the covariant derivative) disappear. We will thus focus on the
$\lambda=0$ sector alone.

The leading order Lagrangian, which we will call the Electric Carroll
Lagrangian is given by:

$\mathcal{L}_{0}=\frac{k}{4\pi}\epsilon^{\mu\nu\rho}a_{\mu}^{(0)}\partial_{\nu}a_{\rho}^{(0)}-(D_{t}\phi)^{(0)*}(D_{t}\phi)^{(0)},$
(41)

where we have used the abbreviation
$(D_{t}\phi)^{(0)}=(\partial_{t}-iea^{(0)}_{t})\phi^{(0)}$. We will see below
that this Lagrangian has Carroll and indeed (infinite) conformal Carroll
symmetries.

The next-to-leading order (NLO) Lagrangian is given by:

$\displaystyle\mathcal{L}_{1}=\frac{k}{4\pi}\epsilon^{\mu\nu\rho}\Big{(}a_{\mu}^{(1)}\partial_{\nu}a_{\rho}^{(0)}+a_{\mu}^{(0)}\partial_{\nu}a_{\rho}^{(1)}\Big{)}-(D_{t}\phi)^{(1)*}(D_{t}\phi)^{(0)}-(D_{t}\phi)^{(0)*}(D_{t}\phi)^{(1)}$
$\displaystyle+(D_{i}\phi)^{(0)*}(D_{i}\phi)^{(0)},$ (42)

where we have defined

$(D_{i}\phi)^{(0)}=D^{(0)}_{i}\phi^{(0)}=(\partial_{i}-iea^{(0)}_{i})\phi^{(0)},\quad(D_{t}\phi)^{(1)}=\partial_{t}\phi^{(1)}-iea^{(0)}_{t}\phi^{(1)}-iea_{t}^{(1)}\phi^{(0)}$
(43)

This Lagrangian is not Carroll invariant, specifically it is not Carroll boost
invariant. In order to rectify this, we modify it by adding Lagrange
multipliers $(\chi_{\mu},\xi)$ to make it Carroll boost invariant. We re-write
the Lagrangian after adding Lagrange multipliers, to get:

$\displaystyle\mathcal{L}_{1}=$
$\displaystyle\frac{k}{4\pi}\epsilon^{\mu\nu\rho}\Big{(}\tilde{\chi}_{\mu}\partial_{\nu}a_{\rho}^{(0)}+a_{\mu}^{(0)}\partial_{\nu}\tilde{\chi}_{\rho}\Big{)}-(\tilde{\xi}^{*}+ie\tilde{\chi}_{t}\phi^{*(0)})(D_{t}\phi)^{(0)}+(D_{t}\phi)^{(0)*}(\tilde{\xi}-ie\tilde{\chi}_{t}\phi^{(0)})$
(44) $\displaystyle\qquad+(D_{i}\phi)^{(0)*}(D_{i}\phi)^{(0)},$

where we have redefined $\tilde{\xi}=(D_{t}^{(0)}{\phi}^{(1)}+\xi)$ and
$\tilde{\chi}_{\mu}=(a_{\mu}^{(1)}+\chi_{\mu})$. As we elaborate in Appendix
A, by ascribing certain transformation properties to the Lagrange multipliers,
the above Lagrangian can be made Carroll invariant. In conclusion, the
Carrollian Chern-Simons matter theories that we would be interested in have
the following Lagrangians:

Electric:
$\displaystyle\quad\mathcal{L}_{e}=\frac{k}{4\pi}\epsilon^{\mu\nu\rho}a_{\mu}\partial_{\nu}a_{\rho}-(D_{t}\phi)^{*}(D_{t}\phi),$
(45) Magnetic:
$\displaystyle\quad\mathcal{L}_{m}=\frac{k}{4\pi}\epsilon^{\mu\nu\rho}\Big{(}{\chi}_{\mu}\partial_{\nu}a_{\rho}+a_{\mu}\partial_{\nu}{\chi}_{\rho}\Big{)}-\Big{[}J_{t}^{*}(D_{t}\phi)+(D_{t}\phi)^{*}J_{t}\Big{]}+(D_{i}\phi)^{*}(D_{i}\phi)$

Here we have $D_{t}=\partial_{t}-iea_{t},D_{i}=\partial_{i}-iea_{i}$,
$J_{t}\equiv\xi-ie\chi_{t}\phi$, $a_{t}^{(0)}\equiv a_{t}$ and
$\tilde{\chi}\equiv\chi$ and so on. We have dropped all superscripts on the
fields.

###  3.3 Symmetries for the electric action

We now briefly delve into the symmetries of the electric action (45). The
transformation of the vector fields $\Phi=(a_{t},a_{z},a_{\bar{z}})$ under the
conformal Carroll algebra are given by the equations (29), (30) and (31). The
transformation of the scalar $\phi$ is given by (24). The dilation weights of
the different components of the vector field are given by

$\Delta_{a_{\mu}}=1\Rightarrow h_{a_{t}}=\bar{h}_{a_{t}}=\frac{1}{2};\quad
h_{a_{z}}=1,\,\bar{h}_{a_{z}}=0;\quad
h_{a_{\bar{z}}}=0,\,\bar{h}_{a_{\bar{z}}}=1.$ (47)

For the scalar we have

$\Delta_{\phi}=\frac{1}{2}.$ (48)

These can be deduced from the relativistic theory in the same way as we
constructed the change under the boosts. The dilatation weights don’t change
under the limit $c\to$ since the dilatation generator
$D=t\partial_{t}+x^{i}\partial_{i}$ does not change under contraction.

We can now explicitly check for the invariance of the Lagrangian (45) under
supertranslations the action of which on the fields are given by (29) and
(24). This yields

$\delta_{M}\mathcal{L}_{e}=\partial_{t}[z^{n}\bar{z}^{m}\mathcal{L}_{e}].$
(49)

So we see that the electric action is invariant under infinite dimensional
supertranslations. Similarly, the action of the “holomorphic” superrotations
are given by (30) and (24). This gives

$\delta_{L}\mathcal{L}_{e}=\partial_{t}\Big{[}\frac{1}{2}z^{n}(n+1)t\mathcal{L}_{e}\Big{]}+\partial_{z}\Big{[}z^{n+1}\mathcal{L}_{e}\Big{]}.$
(50)

In the above, we have explicitly used the weights (47) and (48). We thus have
only total derivative terms under the variation of the electric Lagrangian and
hence the action is invariant under the infinite dimensional superrotations as
well.

Let us put in perspective what we have found. The relativistic Chern-Simons
action in 3d coupled to massless scalar matter is conformally invariant, but
this symmetry is finite dimensional. We have taken a Carrollian expansion on
this action and considered the leading electric Carroll action. This action is
now invariant under an infinite dimensional symmetry, viz. the 3d conformal
Carrollian or BMS4 algebra. This theory is a potential model of a field
theoretic dual to a gravitational theory in 4d asymptotically flat spacetimes.

One can also look at the symmetries of the magnetic action (LABEL:car-mag) and
conclude the emergence of infinite dimensional symmetries there. We give the
details of this in Appendix A.

###  3.4 Null reduction of Carrollian theory

One of the objectives of our work is to relate the two approaches to
holography in asymptotically flat spacetimes, the Carroll and the Celestial.
As indicated in the introduction, Carrollian holography proposes a co-
dimension one dual to 4d asymptotically flat spacetimes living on the entire
null boundary, while Celestial holography advocates a co-dimension two dual
that resides on the celestial sphere. The 3d Carrollian field theory is
defined on the null line $\mathbbm{R}_{u}$ as well as the sphere
$\mathbbm{S}^{2}$ at $\mathscr{I}^{\pm}$. It is thus natural to ask what
happens if we reduce the 3d theory along the null direction and this is what
we will do below.

Before proceeding, it is important to remind the reader that when one does a
null reduction of a relativistic theory in $(d+1)$ dimensions, one ends up
with a Galilean theory in $d$ dimensions. In order to null reduce, the
relativistic theory is written in lightcone coordinates
$x^{\pm}=\frac{1}{\sqrt{2}}(x^{0}\pm x^{d})$ and then the derivative along
$x^{+}$ is set to zero: $\partial_{+}=0$. For the purposes of this quick
comment, let us focus on 4d theories. In terms of the metric, in the lightcone
coordinates in four dimensions, we have

$\eta_{4\times 4}=\left[\begin{array}[]{cccc}0&1&0&0\\\
\cline{2-4}\cr\lx@intercol\hfil
1\hfil\lx@intercol\vrule\lx@intercol&{0}&0&\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol\\\ \lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&0&1&\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol\\\ \lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&0&0&\lx@intercol\hfil
1\hfil\lx@intercol\vrule\lx@intercol\\\
\cline{2-4}\cr\end{array}\right]=\begin{pmatrix}0&\tau_{3\times 1}\\\
\tau_{1\times 3}&\,\,h_{3\times 3}\end{pmatrix}$ (51)

The null reduction focuses on the lower $3\times 3$ block. This is a
degenerate metric giving rise to a 3d Galilean structure. Now let us attempt
the same on a 4d Carrollian theory. We know that here we already have a
degenerate metric:

$g_{4\times 4}=\left[\begin{array}[]{cccc}0&0&0&0\\\
\cline{2-4}\cr\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&{1}&0&\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol\\\ \lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&0&1&\lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol\\\ \lx@intercol\hfil
0\hfil\lx@intercol\vrule\lx@intercol&0&0&\lx@intercol\hfil
1\hfil\lx@intercol\vrule\lx@intercol\\\
\cline{2-4}\cr\end{array}\right]=\begin{pmatrix}0&0_{3\times 1}\\\ 0_{1\times
3}&\,\,\delta_{3\times 3}\end{pmatrix}$ (52)

The null reduction again will focus on the lower $3\times 3$ block, but now in
contrast to the relativistic case, we have a 3d Euclidean non-degenerate
metric. We might expect that a null reduction of a Carrollian theory thus
would generate a Euclidean theory in one lower dimension333There has been
recent work relating lower dimensional non-Lorentzian theories (both Galilean
and Carrollian) to relativistic theories in lightcone coordinates in one
higher dimension from a geometric perspective Bagchi:2024epw . It would be of
interest to see if something similar can be attempted for higher dimensional
Carroll theories and lower dimensional relativistic ones.. This expectation is
borne out by our analyses in this paper.

Armed with this intuition, we will now Kaluza Klein reduce the Carrollian
theory along the null or $t$-direction. Splitting the space and time indices,
we see that the electric Lagrangian is given by

$\mathcal{L}_{e}=\frac{k}{4\pi}\epsilon^{txy}[a_{t}f_{xy}-a_{x}(\partial_{t}a_{y}-\partial_{y}a_{t})+a_{y}(\partial_{t}a_{x}-\partial_{x}a_{t})]-(D_{t}\phi)^{*}(D_{t}\phi).$
(53)

The process of null reduction, as just mentioned above, means that any
derivative in $t$-direction is set to zero. Doing this we get:

$\mathcal{L}_{null-
red}=\frac{k}{4\pi}\epsilon^{txy}[a_{t}f_{xy}+a_{x}\partial_{y}a_{t}-a_{y}\partial_{x}a_{t}]-e^{2}a_{t}^{2}\phi^{*}\phi=\frac{k}{2\pi}\epsilon^{txy}a_{t}f_{xy}-e^{2}a_{t}^{2}\phi^{*}\phi,$
(54)

where we have dropped a total derivative in the intermediate steps. Our aim
now is to integrate out the $a_{t}$ field. The equation of motion of $a_{t}$
is given by:

$\frac{k}{4\pi}\epsilon^{txy}f_{xy}=e^{2}a_{t}\phi^{*}\phi.$ (55)

Completing square(s), (54) can be written as

$\displaystyle\mathcal{L}_{null-
red}=-e^{2}\phi^{*}\phi\left(a_{t}-\frac{k}{4\pi
e^{2}}\frac{f_{xy}}{\phi^{*}\phi}\right)^{2}+\left(\frac{k}{4\pi
e}\right)^{2}\frac{f_{xy}^{2}}{\phi^{*}\phi}$ (56)

Classically, the $a_{t}$ equation of motion suggests that the bracket of the
first term vanishes. In the path integral language, the bracket gives a
gaussian integral in shifted $a_{t}$, which just yields a determinant. In
either case, we are left with only the second term after integrating out
$a_{t}$. So we find that

$\mathcal{L}_{null-red}=\left(\frac{k}{4\pi
e}\right)^{2}\frac{f_{xy}^{2}}{\phi^{*}\phi}$ (57)

This is a 2D Euclidean pure Maxwell theory with coupling
$\frac{1}{g^{2}}=\left(\frac{k}{4\pi e|\phi|}\right)^{2}$, provided $|\phi|$
acquires a vacuum expectation value by some mechanism.

The magnetic Carroll theory can also be null reduced and we provide some
details in Appendix B. This is more involved and we will not be concerned with
this in the main body of the paper.

In conclusion, we have shown that starting with a 3d relativistic Abelian
Chern-Simons theory coupled to scalar matter, one can do a Carroll expansion
in powers of the speed of light to obtain two Carroll Chern-Simons matter
theories in 3d, which exhibit infinite dimensional Conformal Carroll symmetry.
Now, null reducing the electric Carroll CS theory and integrating out the
$a_{t}$ field, we have ended up in a 2d Euclidean Maxwell theory. This section
provides a warm-up for the more involved non-Abelian case we would be
addressing in the coming sections. It is rather curious that one can end up
with a lower dimensional Maxwell theory from Chern-Simons theory in this way.

We started out this sub-section saying that we wanted to relate 3d Conformal
Carroll theories to 2d Celestial CFTs via null reductions. We have obtained a
2d Euclidean Maxwell theory. Now Maxwell theory is only classically
conformally invariant in $d=4$. So a priori, it is not clear at all that we
have ended up with a relativistic CFT in $d=2$. We will however argue that
this is the case when we move to the details of the non-Abelian theory in the
coming sections.

##  4 Bifundamental CSM Theories

We will now consider a non-abelian generalisation of our construction in the
previous section, viz. a Chern-Simons matter theory with bifundamental matter
and gauge group $SU(N)\times SU(M)$. Such theories famously arise in the
context of AdS4/CFT3 duality Aharony:2008ug . Note that the ABJM theory has
$U(N)\times U(N)$ gauge group, the $U(1)\times U(1)$ part of which can be
gauge-fixed to a discrete subgroup. We will avoid this subtlety by working
with special unitary groups. For simplicity, we will also neglect fermions and
scalar potential terms. First we will take the Carrollian limit to obtain a
Chern-Simons matter theory with Carrollian conformal symmetry (or BMS4
symmetry), which can be thought of as living at null infinity of Minkowski
space, providing a toy model for flat space holography. Then we will perform
dimensional reduction along the null direction to obtain a relativistic two-
dimensional theory. It is notable if start with a relativistic theory and
apply a Carrollian limit followed by a null reduction, we end up with a
relativistic theory in one lower dimension. In a sense, we can think of the
non-relativistic limit encoded by the null reduction as cancelling out the
ultra-relativistic limit encoded by the Carrollian limit. We expect this to be
a more general phenomenon. Moreover, we will show that the resulting theory
has relativistic 2d conformal symmetry and may therefore be a celestial CFT.

We will show below that upon giving the scalar fields a vacuum expectation
value, the null-reduced 3d theory becomes a Euclidean 2d Yang-Mills theory. To
our knowledge, such a connection between 3d CSM and 2d YM (YM) theory has not
previously been observed. In particular, if $M\leq N$ the gauge group of 2d YM
will be $SU(M)$. From this we see that having fundamental matter in 3d (which
corresponds to having $M=1$) will lead to an abelian theory in 2d even if the
3d theory has a non-abelian gauge group. Hence, it is crucial to have
bifundamental matter in 3d in order to get an interacting theory in 2d. It is
intriguing that the necessity of bifundamental matter was previously
discovered using completely different reasoning in the context of AdS/CFT
Schwarz:2004yj ; Bagger:2007vi ; Aharony:2008ug . This suggests that if a
concrete realisation of flat space holography exists, it should indeed arise
from taking the flat space limit of AdS/CFT.

###  4.1 Carrollian bifundamental CSM

We begin by considering the relativistic CS theory coupled to bifundamental
scalar matter:

$\displaystyle S=\int dt\,dx\,dy$
$\displaystyle\left\\{\frac{ik_{M}}{8\pi}e^{\mu\nu\rho}\operatorname{Tr}_{N}\left(A_{\mu}\partial_{\nu}A_{\rho}+\frac{2i}{3}A_{\mu}A_{\nu}A_{\rho}\right)\right.$
$\displaystyle+\frac{ik_{M}}{8\pi}\epsilon^{\mu\nu\rho}\operatorname{Tr}_{M}\left(B_{\mu}\partial_{\nu}B_{\rho}+\frac{2i}{3}B_{\mu}B_{\nu}B_{\rho}\right)$
$\displaystyle\left.+\operatorname{Tr}_{M}\left[\left(D_{\mu}\phi\right)^{\dagger}\left(D_{\mu}\phi\right)\right]\right\\}$
(58)

where $\phi$ is a scalar field in the in $(N,\bar{M})$ representation of
$\operatorname{SU}(N)\times\operatorname{SU}(M)$, $A_{\mu}$ and $B_{\mu}$ are
$SU(N)$ and $SU(M)$ gauge fields, respectively, and

$D_{\mu}\phi=\partial_{\mu}\phi-iA_{\mu}\phi+i\phi B_{\mu}.$ (59)

We will choose the Chern-Simons levels to be $k_{N}=-k_{M}=k$. We will see
later that this choice gives 2d YM theory after taking the Carrollian limit
followed by dimensional reduction. It is also the choice which appears in the
ABJ(M) theory Aharony:2008ug ; Aharony:2008gk .

We employ the same Carroll expansion (37), but now for both gauge fields
$A_{\mu}$ and $B_{\mu}$, along with the scalar $\phi$. This results in a
generalisation of the Abelian Carroll actions we wrote down earlier. We will
focus solely on the leading electric action in this case (but the magnetic
case can be similarly obtained). The Carrollian electric non-Abelian CSM
action is given by

$\displaystyle S_{e}=\int dtdxdy$
$\displaystyle\left\\{\frac{ik}{8\pi}\epsilon^{\mu\nu\rho}\operatorname{Tr}_{N}\left(a_{\mu}\partial_{\nu}a_{\rho}+\frac{2i}{3}a_{\mu}a_{\nu}a_{\rho}\right)\right.$
$\displaystyle-\frac{ik}{8\pi}\epsilon^{\mu\nu\rho}\operatorname{Tr}_{M}\left(b_{\mu}\partial_{\nu}b_{\rho}+\frac{2i}{3}b_{\mu}b_{\nu}b_{\rho}\right)$
$\displaystyle\left.-\operatorname{Tr}_{M}\left[\left(D_{t}\phi\right)^{\dagger}\cdot
D_{t}\phi\right]\right\\},$ (60)

where we have

$D_{t}\phi=\partial_{t}\phi-ia_{t}\phi+i\phi b_{t}.$ (61)

This theory can be shown to have infinite dimensional Carrollian conformal
symmetry, like its Abelian counterpart, and can be thought of as a CFT living
in null boundary of Minkowski space, presumably dual to some gravitational
theory in the bulk.

###  4.2 Dimensional Reduction and emergence of 2d Yang-Mills

In continuation of the construction in the Abelian case, we will now
dimensionally reduce along the null direction, $t$. We remind the reader again
that this is a null reduction, which normally gives a lower-dimensional non-
relativistic theory when applied to a relativistic theory. However applied to
a Carrollian theory, this yields a relativistic Euclidean theory, so in a
sense the non-relativistic nature of the null reduction counters the ultra-
relativistic nature of the Carroll theory leading to a relativistic theory at
the end of the process. When applied to our Carrollian CSM, the lower
dimensional theory is again relativistic. We will show that is contains 2d
Yang-Mills theory and enjoys 2d relativistic conformal symmetry, and can
therefore be interpreted as a celestial CFT.

To perform the dimensional reduction, simply take $\partial_{t}\rightarrow 0$.
After doing so, we obtain

$\displaystyle S_{2d}=\int dxdy$
$\displaystyle\left\\{\frac{ik}{4\pi}\operatorname{Tr}_{N}\left(aF_{xy}\right)-\frac{ik}{4\pi}\operatorname{Tr}_{M}\left(b\tilde{F}_{xy}\right)\right.$
$\displaystyle\left.+\operatorname{Tr}_{M}\left[(a\phi-\phi
b)^{\dagger}(a\phi-\phi b)\right]\right\\},$ (62)

where $a=a_{t}$, $b=b_{t}$, and

$\displaystyle F_{xy}=$
$\displaystyle\partial_{x}a_{y}-\partial_{y}a_{x}+i\left[a_{x},a_{y}\right],$
(63a) $\displaystyle\tilde{F}_{xy}=$
$\displaystyle\partial_{x}b_{y}-\partial_{y}b_{x}+i\left[b_{x},b_{y}\right].$
(63b)

We will now integrate out $a,b$. To simplify the analysis and the physical
interpretation of the result, we will give $\phi$ a vacuum expectation value
(vev). The simplest case is $N=M$. In this case we obtain $SU(N)$ YM. For
$M<N$, we get $SU(M)$ YM plus additional terms whose physical interpretation
we will discuss later.

#### Case 1: $M=N$.

Let us first consider $N=M$. In this case we can set
$\phi=v\mathbbm{1}_{N\times N}$ giving

$\displaystyle S_{2d}=\int dxdy$
$\displaystyle\left\\{\frac{ik}{8\pi}\operatorname{Tr}_{N}\left[a_{+}\left(F_{xy}-\tilde{F}_{xy}\right)\right]\right.$
$\displaystyle\left.+\frac{ik}{8\pi}\operatorname{Tr}_{N}\left[a_{-}\left(F_{xy}+\tilde{F}_{xy}\right)\right]+v^{2}\operatorname{Tr}_{N}\left(a_{-}^{2}\right)\right\\},$
(64)

where $a_{\pm}=a+b$. We then find the following equations of motion:

$\displaystyle a_{+}\text{ eom: }F_{xy}=\tilde{F}_{xy}$ (65a) $\displaystyle
a_{-}\text{ eom: }a_{-}=-\frac{ik}{8\pi v^{2}}F_{xy}.$ (65b)

Plugging these back into the action then gives

$S_{2d}=\frac{1}{g_{yM}^{2}}\int
dxdy\operatorname{Tr}_{N}\left(F_{xy}^{2}\right),\,\,\,g_{\text{YM}}^{2}=\frac{64\pi^{2}v^{2}}{k^{2}}.$
(66)

#### Case 2: $M<N$.

We now focus on the more complicated case $M<N$. In this case, we may set

$\phi=v\binom{\mathbbm{1}_{M\times M}}{0_{(N-M)\times M}}.$ (67)

It is also convenient to split the gauge fields and field strengths into
blocks as follows:

$a=\left(\begin{array}[]{ll}a_{M\times M}&a_{M\times(N-M)}^{\prime\dagger}\\\
a_{(N-M)\times
M}^{\prime}&a_{(N-M)\times(N-M)}^{\prime\prime}\end{array}\right),\,\,\,b=b_{M\times
M},$ (68) $F_{xy}=\left(\begin{array}[]{ll}F_{xy}^{M\times
M}&F_{xy}^{\prime\dagger M\times(N-M)}\\\ F_{xy}^{\prime(N-M)\times
M}&F_{xy}^{\prime\prime(N-M)\times(N-M)}\end{array}\right),\,\,\,\tilde{F}_{xy}=\tilde{F}_{xy}^{M\times
M}.$ (69)

After doing so, we find that

$\displaystyle S_{2d}=\int dxdy$
$\displaystyle\left\\{\frac{ik}{8\pi}\operatorname{Tr}_{M}\left[a_{+}\left(F_{xy}-\tilde{F}_{xy}\right)^{M\times
M}\right]\right.+\frac{ik}{8\pi}\operatorname{Tr}_{M}\left[a_{-}\left(F_{xy}+\tilde{F}_{xy}\right)^{M\times
M}\right]$
$\displaystyle+\frac{ik}{4\pi}\left[\operatorname{Tr}_{N-M}\left(a^{\prime\prime}F_{xy}^{\prime\prime}\right)+\operatorname{Tr}_{N-M}\left(a^{\prime}F_{xy}^{\prime\dagger}\right)+\operatorname{Tr}_{M}\left(a^{\prime\dagger}F_{xy}^{\prime}\right)\right]$
$\displaystyle\left.+v^{2}\left[\operatorname{Tr}_{M}a_{-}^{2}+\operatorname{Tr}_{M}\left(a^{\prime\dagger}a^{\prime}\right)\right]\right\\},$
(70)

where $a_{\pm}=\left(a\pm b\right)_{M\times M}$. We then find the following
equations of motion:

$\displaystyle a_{+}\text{ eom: }F_{xy}^{M\times M}=\tilde{F}_{xy}^{M\times
M}$ (71a) $\displaystyle a_{-}\text{ eom:
}a_{-}=-\frac{ik}{8\pi^{2}v^{2}}F_{xy}^{M\times M}$ (71b) $\displaystyle
a^{\prime}\text{ eom: }a^{\prime}=-\frac{ik}{4\pi v^{2}}F^{\prime}_{xy}$ (71c)
$\displaystyle a^{\prime\prime}\text{ eom: }F_{xy}^{\prime\prime}=0.$ (71d)

Plugging these back into the action finally gives

$S_{2d}=\frac{1}{g_{\mathrm{YM}}^{2}}\int
dxdy\left[\operatorname{Tr}_{M}\left(F_{xy}^{M\times
M}\right)^{2}+4\operatorname{Tr}_{M}\left(F_{xy}^{\prime\dagger}F_{xy}^{\prime}\right)\right],\quad\text{where}\,\,g_{\mathrm{YM}}^{2}=\frac{64\pi^{2}v^{2}}{k^{2}}.$
(72)

Note that the first term describes 2d $SU(M)$ YM, while the second term
involves the field strength $F^{\prime}_{xy}$ which is an $M\times(N-M)$ non-
hermitian matrix. The physical interpretation of the second term is unclear in
general, but when $M=1$, $F^{\prime}_{xy}$ is an $(N-1)$-component vector,
i.e. $F_{xy}^{\prime}=\left(F_{xy}^{(1)},\ldots,F_{xy}^{(N-1)}\right)$ and the
second term reduces to a sum over $(N-1)$ abelian non-Hermitian fields:

$\operatorname{Tr}_{M}\left(F_{xy}^{\prime+}F_{xy}^{\prime}\right)=\sum_{\alpha=1}^{N-1}\left|F_{xy}^{(\alpha)}\right|^{2}.$
(73)

Note that $M=1$ corresponds to having fundamental matter coupled to $SU(N)$
Chern-Simons theory in the original 3d theory but after dimensional reduction
we end up with an Abelian theory if even the original theory was non-Abelian.

From our findings above, we clearly see that having bifundamental matter in
three dimensions is required in order to have an interacting theory after
dimensional reduction. Interestingly, the same conclusion was reached from a
very different perspective when constructing a consistent example of the
AdS4/CFT3 correspondence. We believe that this is not a coincidence.

###  4.3 Hints of 2d relativistic conformal symmetry

In this sub-section, we will indicate how the 2d theory in (62) exhibits an
emergent conformal symmetry arising from dimensional reduction. To motivate
this, first recall the vector representation of the 3d Carrollian conformal
group (5), which we re-write here for ease of reading:

$\displaystyle L_{n}=z^{n+1}\partial_{z}+\frac{1}{2}(n+1)z^{n}t\partial_{t},$
(74a) $\displaystyle
L_{n}=\bar{z}^{n+1}\partial_{\bar{z}}+\frac{1}{2}(n+1)\bar{z}^{n}t\partial_{t},$
(74b) $\displaystyle M_{n,s}=z^{r}\bar{z}^{s}\partial_{t}$ (74c)

Here the first two lines represents the superrotations which close to two
copies of Virasoro algebra, but are in an unusual 3d representation with
$(t,z,{\bar{z}})$ and the third line represents the generators of angle-
dependent supertranslations along the null direction $t$. Dimensional
reduction along the null direction sets the $t$ derivatives to zero, i.e.
$\partial_{t}\equiv 0$ and we are left with

$L_{n}=z^{n+1}\partial_{z},\quad L_{n}=\bar{z}^{n+1}\partial_{z}.$ (75)

These are the usual representation of the generators of the two copies of the
Virasoro algebra in $d=2$. We thus expect the 2d theory, which is a null-
reduced 3d Carrollian CFT, to have 2d relativistic conformal symmetry.

Let us now understand how the 2d Yang-Mills theory can have an emergent scale
invariance. Looking at the first line in (62), we see that $a,b$ must have
scaling dimension zero since the strengths $F_{xy}$ and $\tilde{F}_{xy}$ have
scaling dimension two. Applying this to the second line in (62) then implies
that $\phi$ has scaling dimension one. After giving $\phi$ a vev and
integrating out $a,b$ we see that the resulting 2d YM theory is also scale-
invariant since $g^{2}_{\mathrm{YM}}$ has scaling dimension two. In summary,
we find that

$\Delta_{a}=\Delta_{b}=0,\,\,\,\Delta_{\phi}=1.$ (76)

The crucial point here is that the fields that are to be integrated out from
the 3d theory $a_{t}=a$ and $b_{t}=b$ have changed scaling dimensions from
what we started out with, as has the field which acquires a vev, i.e. $\phi$.
Since $a,b$ are scalars in the 2d picture, it is natural to set the scaling
dimension of $a=b=0$.

Although we don’t claim to understand the process of null reduction at the
level of the representation theory completely, let us attempt some more
explanations. We wish to figure out how the 2d conformal representations
appear naturally from the 3d Carroll representations under this process. In
particular, the transformation of the fields $a_{t},a_{i}$ and $\phi$ in the
3d action before the null reduction are given by Eqs. (29)–(31) and (24). The
process of null reduction would change the dilation weights of $a_{t}$ and
$\phi$. In particular, due to the different scaling dimensions for $a_{t}$ and
$a_{i}$, the 3d Carroll boosts do not mix these components of the spin-one
field into each other. So these objects under Carroll boosts would transform
in the trivial representation (26) instead of the non-trivial one (27) for the
spin-one multiplet. In particular, the transformation of each field would be
according to (24) and doing the null reduction by setting the $t$-derivatives
here to zero gives us a natural 2d conformal transformation:

$\displaystyle[L_{n},\Phi(z,{\bar{z}})]=\left[z^{n+1}\partial_{z}+z^{n}(n+1)h\right]\Phi(z,\bar{z}),$
(77a)
$\displaystyle[\bar{L}_{n},\Phi(z,\bar{z})]=\left[{\bar{z}}^{n+1}\partial_{\bar{z}}+{\bar{z}}^{n}(n+1)\bar{h}\right]\Phi(z,\bar{z}),$
(77b)

where $\Phi(z,{\bar{z}})=(a_{z},a_{\bar{z}})$. The weights of the fields are
give by

$h_{a_{z}}=1,~{}\bar{h}_{a_{z}}=0~{};\quad
h_{a_{\bar{z}}}=0,~{}\bar{h}_{a_{\bar{z}}}=1.$ (78)

These follow directly from (32) since $\Delta_{a_{z}}=\Delta_{a_{\bar{z}}}=1$,
which does not change with the dimensional reduction. The above transformation
can also obtained from Eqs. (30) and (31) by setting $\partial_{t}\equiv 0$
and $a_{t}\equiv 0$. It is now straightforward to show that the theory in (62)
enjoys 2d conformal symmetry.

It is interesting to note that we obtain a theory with relativistic conformal
symmetry by performing a null reduction of a theory with Carrollian conformal
symmetry. We believe that this mechanism is not special to Carrollian CSM
theory, and should hold for any theory which arises from taking the Carrollian
limit of a relativistic theory essentially because the non-relativistic limit
encoded by the null reduction cancels out the ultrarelativstic limit of the
Carrollian limit.

##  5 Conclusions

###  5.1 Summary

Motivated by the ABJM construction of a concrete dual to AdS4 spacetimes in
terms of 3d CSM theories, in this paper we laid out the basic construction of
a holographic dual to 4d AFS in terms of a 3d Carrollian CSM theory. We
arrived at the Carrollian theories by considering a $c$-expansion of the
fields in the relativistic theory and showed that the leading Electric Carroll
CSM theory has an infinite dimensional BMS4 symmetry. This makes the theory a
candidate for a field theory dual to 4d AFS, since it inherits the asymptotic
symmetries of the bulk gravitational theory. In Appendix A, we discuss aspects
of the sub-leading magnetic theory, which also exhibits similar symmetry
structures.

We then performed a null reduction of the 3d Carrollian theories. Reducing
along the null direction, we ended up with 2d (Euclidean) relativistic
theories. The theory we reduced to depended very crucially on the matter
content of the parent theory. We considered bi-fundamental matter and non-
Abelian relativistic CS theories and then the process of first taking the
Carroll limit followed by a null reduction landed us on a 2d Yang-Mills theory
with $SU(N)$ gauge symmetry, if we started out with two equal gauge groups
$SU(N)\times SU(N)$. For the $SU(N)\times SU(M)$ case ($N>M$), the results
were more involved, with a 2d SU(M) YM theory with additional interactions.
For fundamental matter, the theory reduced to 2d electrodynamics. This rather
surprising connection between 3d CSM theories and 2d YM theories, to the best
of our knowledge, is completely novel and could be the tip of the iceberg of a
deep connection between 3d-2d theories via this curious ultrarelativistic-
nonrelativistic reduction.

We ultimately provided some hints as to how the 2d YM theory we obtained has
an emergent 2d relativistic conformal symmetry and thus may provide a bridge
between 3d Carrollian CFTs and 2d Celestial CFTs. We provide more comments
below.

###  5.2 Discussions and future directions

Our work raises several tantalising questions and below we discuss some of
them.

  * $\star$

Relating Carroll and Celestial CFTs through null reductions.
As described in the introduction, in recent years, there has been a major
theoretical effort to formulate flat space holography in terms of a 2d CFT
living on the sphere at null infinity, known as the Celestial CFT
Strominger:2017zoo ; Pasterski:2021raf . Given that the 2d theory we obtain by
performing a null reduction of a 3d Carrollian CFT has 2d conformal symmmetry,
we believe that this theory may provide a concrete relisation of a celestial
CFT, or at least be closely related to one. Let us suggest a speculative
holographic argument which lends support to this claim. First recall the
formula for a bulk-to-boundary propagator for a field dual to a scalar
operator with dimension $\Delta$ in a Carrollian CFT Bagchi:2023fbj :

$\tilde{G}_{\triangle}=\frac{1}{(t+q\cdot x)^{\Delta}},$ (79)

where $\vec{q}$ is a null vector which can be interpreted as the momentum of a
massless particle in 4d Minkowski space. This propagator was derived by
writing the AdS4 propagator in 5d embedding corrdinates and taking the flat
space limit. If we restrict our attention to one edge of null infinity
parametrised by $0<u<\infty$ and impose appropriate boundary conditions, we
can extract the zero mode of the operators along this interval of the boundary
by simply performing an integral over $u$ as follows:

$\displaystyle\int_{0}^{\infty}duG_{\Delta}$
$\displaystyle=\int_{0}^{\infty}\frac{du}{(u+q\cdot
x)^{\Delta}}=\left.\frac{1}{1-\Delta}\frac{1}{(u+q\cdot
x)^{\Delta-1}}\right|_{0}^{\infty}$ (80)
$\displaystyle=\frac{1}{\Delta-1}\frac{1}{(q\cdot
x)^{\Delta-1}},\,\,\,\Delta\neq 1.$ (81)

We recognise the second line as the bulk-to-boundary propagator in AdS3 which
can be derived from the Mellin transform of a plane wave in 4d Minkowski space
Cheung:2016iub . More generally, performing this Mellin transform maps
scattering amplitudes to Celestial correlators Pasterski:2017kqt . Hence,
dimensional reduction maps a Carrollian CFT operator with scaling dimension
$\Delta$ to a celestial CFT operator with scaling dimension $\Delta-1$.

There have been other similar suggestions for relating 3d Carrollian and 2d
Celestial CFTs (see e.g. Donnay:2022wvx ). We hope to follow up on this,
specifically in the context of 3d CSM theories we have discussed above. It
would also be interesting to explore if there is any relation between 2d YM
and other recent proposals for Celestial CFTs Costello:2023hmi ;
Stieberger:2023fju ; Melton:2024gyu .

  * $\star$

Limits and reductions
We have performed a Carroll limit followed by a null reduction on the 3d
relativistic CSM theories to end up with 2d Yang-Mills theories. It would be
intriguing to figure out what happens if one does the opposite, i.e. null-
reduce the 3d relativistic theory and perform a Carroll limit on the resulting
theory and to generalise this story to other spacetime dimensions. We hope to
report on this in the near future.

  * $\star$

Computing correlation functions
Given a concrete proposal for a Carrollian CFT, it would be of great interest
to compute its correlation functions in order to probe the dynamics of the
bulk theory. For this puropse, it would be useful to adapt the Feynman rules
recently derived for Carrollian YM theories in Islam:2023rnc  to Carrollian
CSM theories. A natural target to derive from the boundary perspective would
be tree-level Einstein gravity amplitudes, which were recently mapped to
Carrollian correlators in Bagchi:2023cen ; Mason:2023mti . In general, we
expect boundary correlators to produce amplitudes of Einstein gravity plus an
infinite tower of higher derivative corrections which arise from the low
energy expansion of a UV finite theory of quantum gravity such as string
theory. While reproducing bulk locality at four-points may require performing
a non-perturbative calculation in the boundary Maldacena:2015iua , we should
already be able to get some insight into the bulk dynamics by computing three-
point functions. Indeed, conformal Ward identities imply that three-point
stress tensor correlators in relativistic CFT’s must be a linear combination
of two different structures which correspond to two-derivative and six-
derivative gravitational interactions in the bulk Osborn:1993cr ;
Bzowski:2017poo ; Farrow:2018yni , so one expects to have a similar statement
for 3d Carrollian CFT’s.

  * $\star$

Supersymmetrization
One of the most important directions is to generalise our discussion to
include supersymmetry and in particular figure out what the Carroll limit of
3d $\mathcal{N}=6$ Supersymmetric CS theory is so that we can actually focus
on the flat limit of the AdS4/CFT3 correspondence. Supersymmetric versions of
Carrollian theories in dimensions higher than two have been addressed in
Bagchi:2022owq . It would be of interest to use these algebraic structures in
the construction of an explicit supersymmetric CSM model. Understanding the
analogue of this limit for type IIA string theory on AdS${}_{4}\times$ CP3 is
also an important project, but one may have to work a lot harder for a full
string theoretic understanding of the bulk.

We hope to come back to these, and other questions of interest, very soon.

### Acknowledgements

We thank Rudranil Basu, Prateksh Dhivakar, Sudipta Dutta, Romain Ruzziconi,
and Akshay Yelleshpur Srikant for useful discussions. AB is partially
supported by a Swarnajayanti Fellowship from the Science and Engineering
Research Board (SERB) under grant SB/SJF/2019-20/08 and also by SERB grant
CRG/2022/006165. AB thanks the participants and organisers of the workshop
“Carrollian Physics and Holography” organised at the Erwin Schrödinger
Institute (ESI), University of Vienna, for interesting discussions, and the
ESI for hospitality during the visit. AL is supported by an STFC Consolidated
Grant ST/T000708/1.

## APPENDICES

##  Appendix A Symmetries of Magnetic limit

In this appendix, we will look into the magnetic limit and the symmetries of
the action. The action in the magnetic limit is given by

$\displaystyle\mathcal{L}_{mag}=\frac{k}{4\pi}\,\Big{[}\epsilon^{tij}\Big{(}\chi_{t}\partial_{i}a_{j}-\chi_{i}\partial_{t}a_{j}+\chi_{i}\partial_{j}a_{t}+a_{t}\partial_{i}\chi_{j}-a_{i}\partial_{t}\chi_{j}+a_{i}\partial_{j}\chi_{t}\Big{)}\Big{]}$
$\displaystyle-\Big{[}J_{t}^{*}(D_{t}\phi)+(D_{t}\phi)^{*}J_{t}\Big{]}+(D_{i}\phi)^{*}(D_{i}\phi),$
(82)

where we have $D_{t}=(\partial_{t}-iea_{t}),D_{i}=(\partial_{i}-iea_{i})$,
$J_{t}\equiv(\xi-ie\chi_{t}\phi)$, $a_{t}^{(0)}\equiv a_{t}$ and
$\tilde{\chi}\equiv\chi$ and so on. The equations of motion are

$\displaystyle\frac{k}{2\pi}\epsilon^{tij}\tilde{f}_{jt}+ie[\phi^{*}(D_{i}\phi)-\phi(D_{i}\phi)^{*}]=0,~{}\frac{k}{4\pi}\epsilon^{tij}\tilde{f}_{ij}+ie[\phi
J_{t}^{*}-\phi^{*}J_{t}]=0,$ (83)
$\displaystyle\frac{k}{4\pi}\epsilon^{tij}f_{ij}-ie[\phi^{*}(D_{t}\phi)-\phi(D_{t}\phi)^{*}]=0,~{}\frac{k}{2\pi}\epsilon^{tij}f_{ti}=0,$
(84) $\displaystyle
D_{t}(J_{t})-ie\chi_{t}(D_{t}\phi)-D_{i}D_{i}\phi=0,~{}D_{t}\phi=0.$ (85)

where $\tilde{f}_{ab}=(\partial_{a}\chi_{b}-\partial_{b}\chi_{a})$ and
$a=(t,i)$. We will now look at the symmetries of the Lagrangian (A).

#### Boost transformation:

The transformations of the various fields in the Lagrangian under the action
of Carroll boosts is given by

$\displaystyle[B_{i},a_{t}(x^{i},t)]=x_{i}\partial_{t}a_{t},\quad[B_{i},a_{j}(x^{i},t)]=x_{i}\partial_{t}a_{j}+\delta_{ij}a_{t}$
(86)
$\displaystyle~{}[B_{i},\chi_{t}(x^{i},t)]=x_{i}\partial_{t}\chi_{t},\quad[B_{i},\chi_{j}(x^{i},t)]=x_{i}\partial_{t}\chi_{j}+\delta_{ij}\chi_{t},$
(87)
$\displaystyle~{}[B_{i},\phi(x^{i},t)]=x_{i}\partial_{t}\phi,\quad[B_{i},\xi(x^{i},t)]=x_{i}\partial_{t}\xi+(D_{i}\phi).$
(88)

The boost transformations of the Lagrange multipliers $(\chi_{a},\xi)$ are
chosen in a manner so as to make sure that the action is invariant under
Carroll boosts. Below we see this explicitly. The variation of Lagrangian
under boost transformation is given by

$\displaystyle\delta_{B}\mathcal{L}_{mag}=x_{l}\partial_{t}\mathcal{L}_{mag}=\partial_{t}[x_{l}\mathcal{L}_{mag}].$
(89)

The magnetic action thus is invariant under Carroll boosts.

#### Scale transformation:

The transformation of the fields under dilatations is given by:

$\displaystyle[D,\Phi(x^{i},t)]=(t\partial_{t}+x^{i}\partial_{i}+\Delta_{\Phi})\Phi,$
(90)

where $\Phi\equiv(a_{t},a_{i},\phi,\chi,\xi)$ and $\Delta_{\Phi}$ denotes each
fields respected scaling weight. Using it to understand the variation of the
Lagrangian, we get

$\displaystyle\delta_{D}\mathcal{L}_{mag}=\partial_{l}[x^{l}\mathcal{L}_{mag}]+\partial_{t}[t\mathcal{L}_{mag}]+(2\Delta_{\phi}-1)[(D_{i}\phi)^{*}(D_{i}\phi)]$
$\displaystyle+(\Delta_{\chi}-1)\frac{k}{4\pi}\,\Big{[}\epsilon^{tij}\Big{(}\chi_{t}\partial_{i}a_{j}-\chi_{i}\partial_{t}a_{j}+\chi_{\mu}\partial_{\nu}a_{\rho}+a_{t}\partial_{i}\chi_{j}-a_{i}\partial_{t}\chi_{j}+a_{i}\partial_{j}\chi_{t}\Big{)}\Big{]}$
$\displaystyle-\Big{(}\Delta_{\phi}-\frac{1}{2}\Big{)}\Big{[}(\xi^{*}+ie\chi_{t}\phi^{*})(D_{t}\phi)+(D_{t}\phi)^{*}(\xi-
ie\chi_{t}\phi)\Big{]}$
$\displaystyle-\Big{(}\Delta_{\xi}-\frac{3}{2}\Big{)}\Big{[}\xi^{*}(D_{t}\phi)+(D_{t}\phi)^{*}\xi\Big{]}-\Big{(}\Delta_{\chi}+\Delta_{\phi}-\frac{3}{2}\Big{)}ie\chi_{t}\Big{[}\phi^{*}(D_{t}\phi)-(D_{t}\phi)^{*}\phi\Big{]}$
(91)

We have already taken $\Delta=1$ for the gauge field $(a_{t},a_{i})$ in the
intermediate steps. All extra terms vanishes when we take

$\Big{[}\Delta=1,\Delta_{\chi}=1,\Delta_{\xi}=\frac{3}{2},\Delta_{\phi}=\frac{1}{2}\Big{]}.$
(92)

Finally the result becomes

$\delta_{D}\mathcal{L}_{mag}=\partial_{l}[x^{l}\mathcal{L}_{mag}]+\partial_{t}[t\mathcal{L}_{mag}].$
(93)

The magnetic action is thus invariant under scale transformation given the
scaling dimensions of the fields (92).

#### Supertranslation transformation:

We will now look into the supertranslations and the invariance of the magnetic
limit. The transformations of the fields under supertranslation is given by

  * •

For the scalar $\phi$: (24).

  * •

For the vector field $\vec{a}=(a_{t},a_{z},a_{\bar{z}})$, and Lagrange
multiplier $\vec{\chi}=(\chi_{t},\chi_{z},\chi_{\bar{z}})$: (29).

  * •

For the Lagrange multiplier $\xi$:

$[M_{nm},\xi]=z^{n}\bar{z}^{m}\partial_{t}\xi+nz^{n-1}\bar{z}^{m}D_{z}\phi+mz^{n}\bar{z}^{m-1}D_{\bar{z}}\phi.$
(94)

The variation of (A) under supertranslation comes out to be

$\displaystyle\delta_{M}\mathcal{L}_{mag}=\partial_{t}[z^{n}\bar{z}^{m}\mathcal{L}_{mag}]$
(95)

The Magnetic Carrollian CSM theory thus has infinite dimensional
supertranslation invariance.

#### Superrotations transformation:

We now move on to superrotations. The transformations of the fields under
superrotations are given by:

  * •

For the scalar $\phi$: (24).

  * •

For the vector field $\vec{a}=(a_{t},a_{z},a_{\bar{z}})$, and the vector
Lagrange multiplier $\vec{\chi}=(\chi_{t},\chi_{z},\chi_{\bar{z}})$: (30) and
(31).

  * •

For the Lagrange multiplier $\xi$:

$[L_{n},\xi]=\frac{1}{2}[(z^{n}(n+1)(\Delta_{\xi}+t\partial_{t})+2z^{n+1}\partial_{z})\xi+tn(n+1)(D_{z}\phi)z^{n-1}].$
(96)

Using the above, the variation under superrotations of (A) comes out to be

$\displaystyle\delta_{L}\mathcal{L}_{mag}=\partial_{t}\Big{[}\frac{1}{2}z^{n}(n+1)t\mathcal{L}_{mag}\Big{]}+\partial_{z}\Big{[}z^{n+1}\mathcal{L}_{mag}\Big{]}.$
(97)

We thus see that the magnetic action is invariant under infinite dimensional
superrotations. The magnetic Carrollian CSM action thus has all the infinite
dimensional symmetries of the extended BMS4 algebra.

##  Appendix B Null reduction of magnetic theory

In this appendix, we provide some details of the null reduction of the
magnetic Carrollian CSM theory. For simplicity, we focus on the Abelian case.
The Lagrangian is given by

$\displaystyle\mathcal{L}_{mag}=\frac{k}{4\pi}\,\Big{[}\epsilon^{txy}\Big{(}\chi_{t}f_{xy}-\chi_{x}f_{ty}+\chi_{y}f_{tx}+a_{t}\tilde{f}_{xy}-a_{x}\tilde{f}_{ty}+a_{y}\tilde{f}_{tx}\Big{)}\Big{]}$
$\displaystyle-\Big{[}J_{t}^{*}(D_{t}\phi)+(D_{t}\phi)^{*}J_{t}\Big{]}+(D_{x}\phi)^{*}(D_{x}\phi)+(D_{y}\phi)^{*}(D_{y}\phi).$
(98)

In order to null reduce the theory, we set the derivatives $\partial_{t}\equiv
0$. The action of the reduced theory thus becomes

$\displaystyle\mathcal{L}_{mag}=\frac{k}{4\pi}\,\Big{[}\epsilon^{txy}\Big{(}\chi_{t}f_{xy}+\chi_{x}\partial_{y}a_{t}-\chi_{y}\partial_{x}a_{t}+a_{t}\tilde{f}_{xy}+a_{x}\partial_{y}\chi_{t}-a_{y}\partial_{x}\chi_{t}\Big{)}\Big{]}$
$\displaystyle+iea_{t}\Big{[}J_{t}^{*}\phi-\phi^{*}J_{t}\Big{]}+(D_{x}\phi)^{*}(D_{x}\phi)+(D_{y}\phi)^{*}(D_{y}\phi),$
(99)

taking the total derivatives, we get

$\displaystyle\mathcal{L}_{mag}=\frac{k}{2\pi}\,\Big{[}\epsilon^{txy}\Big{(}\chi_{t}f_{xy}+a_{t}\tilde{f}_{xy}\Big{)}\Big{]}+iea_{t}\Big{[}J_{t}^{*}\phi-\phi^{*}J_{t}\Big{]}+(D_{x}\phi)^{*}(D_{x}\phi)+(D_{y}\phi)^{*}(D_{y}\phi).$

The equations of motion for $a_{t}$ and $\chi_{t}$ are given by

$\displaystyle\frac{k}{2\pi}\epsilon^{txy}\tilde{f}_{xy}=-ie[J^{*}_{t}\phi-
J_{t}\phi^{*}],~{}\frac{k}{2\pi}\epsilon^{txy}f_{xy}-2e^{2}a_{t}\phi^{*}\phi=0.$
(100)

Looking above, we get two cases. Let us discuss each case in details. First
one is integrating out $a_{t}$ and auxiliary fields and second one is if $\xi$
is not integrated out.

### Integrating out $a_{t}$ and auxiliary fields

The magnetic action for an Abelian Chern-Simons field minimally coupled to a
scalar, after taking away $\partial_{t}$ is

$\displaystyle\mathcal{L}_{mag}=\frac{\kappa}{2\pi}\,\Big{[}\epsilon^{txy}\Big{(}\chi_{t}f_{xy}+a_{t}\tilde{f}_{xy}\Big{)}\Big{]}+iea_{t}\Big{[}J_{t}^{*}\phi-\phi^{*}J_{t}\Big{]}+(D_{x}\phi)^{*}(D_{x}\phi)+(D_{y}\phi)^{*}(D_{y}\phi)$

Substituting the definition $J_{t}=\xi-i\chi_{t}\phi$, we get

$\mathcal{L}_{mag}=a_{t}\left(ie\phi\xi^{*}-ie\phi^{*}\xi-2e^{2}\phi^{*}\phi\chi_{t}\right)+\frac{\kappa}{2\pi}\left(f_{xy}\chi_{t}+\tilde{f}_{xy}a_{t}\right)+|D_{i}\phi|^{2}$
(101)

The goal is to integrate out $a_{t}$, $\chi_{t}$ and $\xi$. A slight
generalization of the well known fact that a product can be written as a
difference of two squares helps us write the “quadratic" terms in (101) as

$a_{t}\left(ie\phi\xi^{*}-ie\phi^{*}\xi-2e^{2}\phi^{*}\phi\chi_{t}\right)=|\left(a_{t}+\frac{ie\phi^{*}}{2}\xi-\frac{e^{2}\phi^{*}\phi}{2}\chi_{t}\right)|^{2}-|\left(a_{t}-\frac{ie\phi^{*}}{2}\xi+\frac{e^{2}\phi^{*}\phi}{2}\chi_{t}\right)|^{2}$

Let’s define

$V_{1}=a_{t}+\frac{ie\phi^{*}}{2}\xi-\frac{e^{2}\phi^{*}\phi}{2}\chi_{t},~{}~{}~{}~{}V_{2}=a_{t}-\frac{ie\phi^{*}}{2}\xi+\frac{e^{2}\phi^{*}\phi}{2}\chi_{t}$

So the quadratic piece is $V_{1}^{*}V_{1}-V_{2}^{*}V_{2}$. Now let’s look at
the linear piece. $a_{t}$ is simply $\frac{V_{1}+V_{2}}{2}$, while
$\chi_{t}=\frac{V_{2}-V_{1}}{e^{2}\phi^{*}\phi}+\frac{i}{e\phi}\xi$. So we can
now write (101) as

$\begin{split}\mathcal{L}_{mag}=&~{}V_{1}^{*}V_{1}-V_{2}^{*}V_{2}+\frac{\kappa}{4\pi}\left[\left(\frac{\tilde{f}_{xy}}{2}-\frac{f_{xy}}{e^{2}\phi^{*}\phi}\right)(V_{1}+V_{1}^{*})+\left(\frac{\tilde{f}_{xy}}{2}+\frac{f_{xy}}{e^{2}\phi^{*}\phi}\right)(V_{2}+V_{2}^{*})\right]\\\
&+\frac{i\kappa f_{xy}}{4\pi e\phi}\xi-\frac{i\kappa f_{xy}}{4\pi
e\phi^{*}}\xi^{*}+|D_{i}\phi|^{2}.\end{split}$ (102)

This is quadratic in $V_{1}$ and $V_{2}$, but only linear in $\xi$, which acts
as a Lagrange multiplier that imposes the constraint $f_{xy}=0$. Immediately
using this constraint, (102) further simplifies to

$\begin{split}\mathcal{L}_{mag}=&~{}V_{1}^{*}V_{1}-V_{2}^{*}V_{2}+\frac{\kappa\tilde{f}_{xy}}{8\pi}(V_{1}+V_{1}^{*}+V_{2}+V_{2}^{*})+|D_{i}\phi|^{2},\\\
=&\left(V_{1}+\frac{\kappa\tilde{f}_{xy}}{8\pi}\right)^{*}\left(V_{1}+\frac{\kappa\tilde{f}_{xy}}{8\pi}\right)-\left(V_{2}-\frac{\kappa\tilde{f}_{xy}}{8\pi}\right)^{*}\left(V_{2}-\frac{\kappa\tilde{f}_{xy}}{8\pi}\right)+|D_{i}\phi|^{2},\end{split}$
(103)

where we have added and subtracted the same term to complete both perfect
squares. Integrating out these perfect squares, we are left with

$\mathcal{L}_{1}=|D_{i}\phi|^{2}$ (104)

with the constraint $f_{xy}=0$.

### If $\xi$ is not integrated out

If we keep $\xi$ as a field in the reduced theory, We go back to (102). Now we
don’t impose $f_{xy}=0$ since $\xi$ is no longer a Lagrange multiplier. We can
again complete the squares involving $V_{1}$ and $V_{2}$, and doing that we
get

$\begin{split}\mathcal{L}_{mag}=&~{}\left(V_{1}+\frac{\kappa\tilde{f}_{xy}}{8\pi}-\frac{\kappa
f_{xy}}{4\pi
e^{2}\phi^{*}\phi}\right)^{*}\left(V_{1}+\frac{\kappa\tilde{f}_{xy}}{8\pi}-\frac{\kappa
f_{xy}}{4\pi
e^{2}\phi^{*}\phi}\right)-\left(\frac{\kappa}{4\pi}\right)^{2}\left(\frac{\tilde{f}_{xy}}{2}-\frac{f_{xy}}{e^{2}\phi^{*}\phi}\right)^{2}\\\
&-\left(V_{2}-\frac{\kappa\tilde{f}_{xy}}{8\pi}-\frac{\kappa f_{xy}}{4\pi
e^{2}\phi^{*}\phi}\right)^{*}\left(V_{2}-\frac{\kappa\tilde{f}_{xy}}{8\pi}-\frac{\kappa
f_{xy}}{4\pi
e^{2}\phi^{*}\phi}\right)+\left(\frac{\kappa}{4\pi}\right)^{2}\left(\frac{\tilde{f}_{xy}}{2}+\frac{f_{xy}}{e^{2}\phi^{*}\phi}\right)^{2}\\\
&+\frac{i\kappa f_{xy}}{4\pi e\phi}\xi-\frac{i\kappa f_{xy}}{4\pi
e\phi^{*}}\xi^{*}+|D_{i}\phi|^{2}.\end{split}$ (105)

Integrating out the perfect squares we get

$\mathcal{L}_{mag}=2\left(\frac{\kappa}{4\pi
e}\right)^{2}\frac{f_{xy}\tilde{f}_{xy}}{\phi^{*}\phi}+\frac{i\kappa
f_{xy}}{4\pi e\phi}\xi-\frac{i\kappa f_{xy}}{4\pi
e\phi^{*}}\xi^{*}+|D_{i}\phi|^{2}.$ (106)

If we change our minds now and integrate out $\xi$, it sets $f_{xy}=0$ giving
back the action (104).

At this juncture, it is not clear to us what the null reduction of the
magnetic CSM theory is hinting at. We hope to come back to this in more detail
in the near future.

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