# On self-dual Carrollian conformal nonlinear electrodynamics

Bin Chen1,2,3, Jue Hou4, Haowei Sun1

###### Abstract

In this work, we study the duality symmetry group of Carrollian (nonlinear)
electrodynamics and propose a family of Carrollian ModMax theories, which are
invariant under Carrollian $\text{SO}(2)$ electromagnetic (EM) duality
transformations and conformal transformation. We define the Carrollian
$\text{SO}(2)$ EM transformations, with the help of Hodge duality in
Carrollian geometry, then we rederive the Gaillard-Zumino consistency
condition for EM duality of Carrollian nonlinear electrodynamics. Together
with the traceless condition for the energy-momentum tensor, we are able to
determine the Lagrangian of the Carrollian ModMax theories among pure
electrodynamics. We furthermore study their behaviors under the
$\sqrt{T\bar{T}}$ deformation flow, and show that these theories deform to
each other and may reach two endpoints under the flow, with one of the
endpoint being the Carrollian Maxwell theory. As a byproduct, we construct a
family of two-dimensional Carrollian ModMax-like multiple scalar theories,
which are closed under the $\sqrt{T\bar{T}}$ flow and may flow to a BMS free
multi-scalar model.

1School of Physics, Peking University, No.5 Yiheyuan Rd, Beijing 100871, P. R.
China
2Collaborative Innovation Center of Quantum Matter, No.5 Yiheyuan Rd, Beijing
100871, P. R. China
3Center for High Energy Physics, Peking University, No.5 Yiheyuan Rd, Beijing
100871, P. R. China
4School of physics & Shing-Tung Yau Center, Southeast University, Nanjing
211189,
P. R. China

###### Contents

  1. 1 Introduction
  2. 2 Basics of Carrollian symmetry
    1. 2.1 Carrollian symmetry and Carrollian conformal symmetry
    2. 2.2 Carrollian geometry
  3. 3 Non-Riemannian Hodge duality
    1. 3.1 Non-Riemannian Levi-Civita tensors
    2. 3.2 Carrollian Hodge-$*$ and electromagnetic duality
  4. 4 Generalized duality symmetry of Carrollian nonlinear electrodynamics
    1. 4.1 Gaillard-Zumino condition for Carrollian NEDs
    2. 4.2 Generalized duality symmetry of (electric-type) Carrollian Maxwell electrodynamics
    3. 4.3 Carrollian ModMax theories
  5. 5 Carrollian $\sqrt{T\bar{T}}$ deformation flows
    1. 5.1 Carrollian ModMax under generalized $\sqrt{T\bar{T}}$ deformation
    2. 5.2 $\sqrt{T\bar{T}}$ deformation of $2D$ Carrollian modified scalar theories
  6. 6 Discussions
  7. A Criterion of Carrollian conformal symmetry
    1. A.1 Carrollian Bessel-Hagen criterion
    2. A.2 Energy-momentum tensor of Carrollian NED

##  1 Introduction

Electromagnetic (EM) duality is one of the most important features of Maxwell
electrodynamics. It reveals the invariance of Maxwell’s equations under the
transformation between electric fields and magnetic fields. The idea of EM
duality was further developed to the Montonen–Olive duality [1] and the
S-duality in field theory and string theory[2], which relates a strongly
coupled theory to a weakly coupled one. Although generally a duality relates
two different theories, Maxwell theory in vacuum is said to be self-dual as
the EM duality maps it to itself. Moreover, Larmor [3] and Rainich [4] noticed
that the EM discrete duality transformation of Maxwell equations can be
generalized to an one-parameter $\text{U}(1)$ rotation
$(\mathbf{E}+i\mathbf{B})\to e^{-i\theta}(\mathbf{E}+i\mathbf{B})$. Such kind
of generalized duality symmetry is also shared by many other nonlinear
electrodynamics (NEDs) [5], among which one notable example is the famous
Born-Infeld theory. Most generally, as studied by Gaillard and Zumino in [6,
7, 8, 9], the duality symmetry group $\text{U}(1)$ (or $\text{SO}(2)$) can be
extended to other subgroups of the general linear group
$\text{GL}(2,\mathbb{R})$ in nonlinear electrodynamics interacting with
matters, or subgroups of $\text{GL}(2n,\mathbb{R})$ in interacting theories of
$n$ gauge fields. Among other cases, duality symmetries also present in
supersymmetric field theories[10, 11] and supergravity theories [12, 13, 14].

Apart from the duality rotation symmetry, the Maxwell theory is also conformal
invariant. The existence of duality rotation invariant NEDs inspires naturally
the question: is there any NED that preserves all the symmetries of the
Maxwell theory? Historically, people gave a partially negative answer to this
question and proved that the only possible Lorentz conformal invariant
equations from the Lagrangians is the Maxwell electrodynamics, if the
Lagrangian is a real analytic function of electromagnetic fields [15, 16].
Recently, it was found that the Modified Maxwell (ModMax) theory [16, 17]
presented a counter-example, in which case the Lagrangian and the Hamiltonian
are not analytic functions. The ModMax theory is an one-parameter family of
maximally symmetric nonlinear extensions of the Maxwell theory, in the sense
that it is the unique Lorentz invariant deformation of the Maxwell theory
which is both conformal invariant and duality rotation invariant. The
Lagrangian of ModMax is of the form

$\mathcal{L}_{\gamma}(\mathcal{S},\mathcal{P})=-\frac{1}{2}\cosh{\gamma}\mathcal{S}+\frac{1}{2}\sinh{\gamma}\sqrt{\mathcal{S}^{2}+\mathcal{P}^{2}};\qquad\mathcal{S}\equiv\frac{1}{2}F_{\mu\nu}F^{\mu\nu},\mathcal{P}\equiv\frac{1}{2}F_{\mu\nu}(*F)^{\mu\nu},$
(1.1)

where $\gamma$ is a real parameter. The free Maxwell electrodynamics can be
viewed as the $\gamma=0$ ModMax, and at small $\gamma$ ModMax can be
considered as an alternative to Maxwell theory, with some physical
distinguishable implications such as vacuum birefringence [16]. Interestingly,
the ModMax and its $2D$ cousin can be obtained from a $T\bar{T}$-like
deformations of Maxwell electrodynamics [18, 19, 20, 21, 22]. The
supersymmetric extension of ModMax was considered in [23, 24, 23] and the
higher-form extension was discussed in [25]. Other recent works about ModMax
include [26, 27, 28, 29, 30, 31].

It is notable that in most of the discussions on EM duality symmetries, the
theories are supposed to be Lorentz invariant. However, the Lorentz invariance
is not the only option. Actually, before the discovery of ModMax, it has been
noticed in [32] that there is a deep link between duality rotation invariance
and the Poincaré symmetry, and remarkably the zero signature contraction of
the Poincaré group called Carrollian group is also allowed to have duality
invariant electrodynamics. The Carrollian symmetry group was first found by
Lévy-Leblond in 1965[33] (and independently by Sen Gupta [34]) by studying the
ultra-relativistic ($c\to 0$) contraction of the Poincaré group. The
Carrollian physics appear in various fields and has attracted much attention
especially in the past decade. Some of the most recent researches include [35,
36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]. As far as we
know, except [32] all other discussions about Carrollian NEDs [52, 53] or
Carrollian electrodynamics did not touch the duality symmetry of the theory.
It is demanding to investigate the duality symmetry group of Carrollian
(nonlinear) electrodynamics and the existence of a Carrollian version of
ModMax.

In this work, we try to study the EM duality in $4D$ Carrollian
electromagnetic theories. As first pointed out in [54], the Carrollian Maxwell
theories are no longer invariant under the ultra-relativistic limit of the EM
transformations. This is essentially because that the Hodge-$\ast$ operator is
no longer well-defined for degenerate metrics. In order to discuss the EM
duality in Carrollian geometry, we have to define Carrollian Hodge duality
appropriately. We will show that the electric-type Carrollian Maxwell theory
is self-dual under a modified discrete EM duality, but it is not invariant
under a continuous $\text{SO}(2)$ EM duality. By studying the Gaillard-Zumino
condition for Carrollian nonlinear electrodynamics, we find a family of
Carrollian ModMax theories, which is Carrollian $\text{SO}(2)$ EM duality
invariant as well as conformal invariant. The Lagrangian of Carrollian ModMax
theories is

$\mathcal{L}_{\gamma}(\mathcal{S},\mathcal{P})=\frac{1}{4}\left(e^{\gamma}\mathcal{S}\mp
e^{-\gamma}\frac{\mathcal{P}^{2}}{\mathcal{S}}\right);\qquad\mathcal{S}\equiv\frac{1}{2}F_{\mu\nu}F^{\mu\nu}=m^{\mu\rho}\gamma^{\nu\sigma}F_{\mu\nu}F_{\rho\sigma},\hskip
8.61108pt\mathcal{P}\equiv\frac{1}{2}F_{\mu\nu}\widetilde{F}^{\mu\nu}.$ (1.2)

It is remarkable that the Carrollian $\text{SO}(2)$ EM duality is not the
ultra-relativisitc limit of the usual $\text{SO}(2)$ EM duality.

Furthermore, we investigate the flow of the $4D$ Carrollian ModMax theories
under $\sqrt{T\bar{T}}$ deformation. In the relativistic case, the ModMax
theories can be generated from the Maxwell theory by the $\sqrt{T\bar{T}}$
flow. However, we show a different picture in the Carrollian case: under
$\sqrt{T\bar{T}}$ deformation, the Carrollian ModMax theories are closed in
the sense that they are related to each other by the flow, and they can flow
to two endpoints, one of them being the electric-type Carrollian Maxwell
theory, and the theories at the endpoints are the fixed points of the flow.
The last fact suggests that the Carrollian ModMax theories cannot be generated
from the Carrollian Maxwell theory, in accord with the fact that they have
different duality symmetries.

As a byproduct of our study, we construct 2D Carrollian ModMax-like $N$-scalar
theories and discuss their behaviors under the $\sqrt{T\bar{T}}$ flow. In the
$N=2$ case, the Lagrangian of the theories can be obtained by dimensional
reduction from $4D$ Carrollian ModMax theories. We generalize the Lagrangian
to $N$-scalar and show that they are closed under $\sqrt{T\bar{T}}$
deformation. Similar to the Carrollian ModMax theories, these scalar theories
flow to two fixed endpoints, with one of them being the BMS free $N$-scalar
theory, and the flow at the endpoints is not invertible.

The remaining parts of this paper are organized as follows. In Section 2, we
briefly review the basics of Carrollian (conformal) symmetry and Carrollian
geometry. In Section 3, we bring the ideas from [55] to define Carrollian
Levi-Civita tensors and the Carrollian Hodge-$*$ operator, and use it to
discuss the discrete Hodge duality in the Carrollian Maxwell theory. In
Section 4, we generalize the discussion of continuous generalized duality
symmetry to Carrollian NEDs. We find out the maximal duality group of the
electric-type Carrollian Maxwell theory, and solve a one-parameter family of
Carrollian NEDs - the Carrollian ModMax theories, which are both conformal and
SO(2)-duality invariant. We also find the electric-type Carrollian Maxwell
theory as one of the limits of the Carrollian ModMax theories. In Section 5,
we investigate the behavior of $4D$ Carrollian ModMax theories under the
$\sqrt{T\bar{T}}$ deformation flow, and generalize the study to a family of
$2D$ modified BMS multiple-scalar model. The $4D$ Carrollian ModMax provides
another example to show that the duality-invariant NEDs are generally related
to $\sqrt{T\bar{T}}$-like flows [56]. We end with some discussions in Section
6. We pack some technical details into an appendix.

#### Conventions:

We use the Greek alphabets $\mu,\nu,\cdots$ as the spacetime indices, and use
the lower-case Latin alphabets $i,j,\cdots$ as spatial indices only. We use
the upper-case alphabets $A,B,\cdots$ as internal indices in the tetrad
formalism. Besides, we use $a,b$ as the indices for $2D$ spacetime. When
speaking of $D$-dimensional spacetime, we are talking about spacetime with a
total dimension $D$. For Lorentzian spacetime, we select the almost positive
signature $(-+++)$.

##  2 Basics of Carrollian symmetry

We begin with a brief introduction to basic concepts about Carrollian symmetry
and its conformal extension. Our focus is mainly on the $4d$ Carrollian
electrodynamics, so we only discuss the global part of Carrollian symmetry and
Carrollian conformal symmetry. However, it is worth noting that the global
Carrollian conformal algebras in $d=2,3$ have extensions that are isomorphic
to the infinite-dimensional BMSd+1 algebra [57]. One may also refer to [48]
for further details about representations of higher dimensional Carrollian
algebras, and [54] for more details about the dual concepts of Newton-Cartan
geometry and Carrollian geometry.

###  2.1 Carrollian symmetry and Carrollian conformal symmetry

As the most famous non-trivial examples of kinematical algebras classified by
Bacry and Lévy-Leblond[58], both the Galilean and the Carrollian algebra
emerge as the Inönü–Wigner contractions of the Poincaré group, by taking the
$c\to\infty$ (Galilean/non-relativistic) and the $c\to 0$ (Carrollian/ultra-
relativistic) limit respectively. After taking the Carrollian limit, the
Lorentzian boosts become the Carrollian boosts:

$\left\\{\begin{aligned} t&\to t-\mathbf{b}\cdot\mathbf{x},\\\
\mathbf{x}&\to\mathbf{x}.\\\ \end{aligned}\right.$ (2.1)

At the same time, other parts of Poincaré group remain the same, so the
Carrollian algebra is made up of the Carrollian boosts $B_{i}$, the spatial
rotations $J_{ij}$ and the spacetime translations $P_{\mu}$.

Conformal field theories (CFTs) are the theories with relativistic conformal
symmetry and play a central role in modern theoretical physics. The Carrollian
conformal symmetry naturally arises from the $c\to 0$ limit of relativistic
conformal symmetry [59], and its symmetry algebra is called Carrollian
conformal algebra (CCA), which is generated by the generators of the
Carrollian algebra $\\{P_{\mu},J_{ij},B_{i}\\}$ plus the scaling generator $D$
and the Carrollian special conformal transformations (SCTs) $K_{\mu}$. The
generators in CCA correspond to the Carrollian conformal Killing vectors in
Table 1, and their commutation relations are:

$\displaystyle[D,P_{\mu}]=P_{\mu},~{}~{}[D,K_{\mu}]=-K_{\mu},~{}~{}[D,B_{i}]=[D,J_{ij}]=0,$
(2.2)
$\displaystyle[J_{ij},G_{k}]=\delta_{ik}G_{j}-\delta_{jk}G_{i},~{}~{}G\in\\{P,K,B\\},~{}~{}$
$\displaystyle[J_{ij},P_{0}]=[J_{ij},K_{0}]=0,$
$\displaystyle[J_{ij},J_{kl}]=\delta_{ik}J_{jl}-\delta_{il}J_{jk}+\delta_{jl}J_{ik}-\delta_{jk}J_{il},$
$\displaystyle[B_{i},P_{j}]=\delta_{ij}P_{0},~{}~{}[B_{i},K_{j}]=\delta_{ij}K_{0},~{}~{}[B_{i},B_{j}]=[B_{i},P_{0}]=[B_{i},K_{0}]=0,$
$\displaystyle[K_{0},P_{0}]=0,~{}~{}[K_{0},P_{i}]=-2B_{i},~{}~{}[K_{i},P_{0}]=2B_{i},~{}~{}[K_{i},P_{j}]=2\delta_{ij}D+2J_{ij}.$

Here $\mu=0,1,\dots,d-1$, and $i,j=1,\dots,d-1$.

Table 1: The generators of the CCA and the Carrollian conformal Killing vector fields. Generators |  Vector fields $\mathbf{v}=v^{\mu}\partial_{\mu}$
---|---
$D$ |  $\mathbf{d}=t\partial_{t}+x^{i}\partial_{i}$
$P_{\mu}$ |  $\mathbf{p}_{\mu}=\left(\partial_{t}~{},~{}\vec{\partial}\right)$
$K_{\mu}$ |  $\mathbf{k}_{\mu}=\left(-\vec{x}^{2}\partial_{t},2\vec{x}x_{\mu}\partial^{\mu}-\vec{x}^{2}\vec{\partial}\right)$
$B_{i}$ |  $\mathbf{b}_{i}=x^{i}\partial_{t}$
$J_{ij}$ |  $\mathbf{j}_{ij}=x^{i}\partial_{j}-x^{j}\partial_{i}$

Recently, different methods have been proposed to construct the field theories
with Carrollian symmetry and Carrollian conformal symmetry. One can proceed by
taking the Carrollian limit [59, 60, 61, 62], or performing null reduction
from Bargmann field theories [49], or using Galilean theories as seed theories
[63, 47]. Each of these methods allows us to find two different sectors (3.4)
of Carrollian $\text{U}(1)$ gauge theory, which are referred to as the
electric sector and the magnetic sector respectively, and they are important
examples of Carrollian conformal field theories.

###  2.2 Carrollian geometry

The relativistic field theory is defined on a Minkowski spacetime, and
Einstein’s general relativity is most conveniently defined in terms of
(pseudo-)Riemann geometry. It is natural to ask what kind of geometries
underlying the Carrollian field theory and the Carrollian gravity are. Over
the years, the concept of Carrollian geometry has been well established, both
intrinsically [54, 64, 65, 66] and by considering the Carrollian limit or
small $c$ expansion of the Lorentzian geometry [67, 68].

The Carrollian geometry is described in a manner dual to the Newton-Cartan
geometry which is used for non-relativistic gravity. A (weak) Carrollian
structure is given by a triplet $(\mathcal{C},h_{\mu\nu},n^{\mu})$ where
$h_{\mu\nu}$ is a degenerate metric and the temporal vector $n^{\mu}$ is its
kernel. Just like the Lorentz group is the isometry group of the flat
Minkowski spacetime, the Carrollian group is the isometry group of the flat
Carrollian structure with

$h_{\mu\nu}=\delta_{ij}dx^{i}dx^{j},\ n^{\mu}=\partial_{0}.$ (2.3)

Moreover, the Carrollian conformal group is the conformal group of the flat
Carrollian structure, whose definition is

$\varphi^{*}h=\Omega^{2}h,\qquad\varphi_{*}n=\Omega^{-2/k}n.$ (2.4)

Here $\varphi$ is the spacetime transformation associated to a group element
and $\Omega(x)$ is the conformal factor. Usually, the Carrollian conformal
symmetry has $k=2$ to fix the scaling in the $n^{\mu}$ direction to be the
same as the ones in other directions.

We will denote $(n^{\mu},h_{\mu\nu})$ as the Carrollian structure, and
$(\tau_{\mu},\gamma^{\mu\nu})$ for its inverse. The meaning of the inverse is
that they satisfy the following relations:

$\displaystyle n^{\mu}h_{\mu\nu}=\tau_{\mu}\gamma^{\mu\nu}=0,\quad
n^{\mu}\tau_{\mu}=1,$ (2.5) $\displaystyle
n^{\mu}\tau_{\nu}+\gamma^{\mu\rho}h_{\rho\nu}=\delta^{\mu}_{\ \nu}.$

Since only the Carrollian structure is relevant to the physics, any Carrollian
physical quantity needs to be invariant under a gauge symmetry generated by
$b_{\mu}$ called the local boost transformation, acting on
$(\tau_{\mu},\gamma^{\mu\nu})$ as:

$\tau_{\mu}\to\tau_{\mu}+b_{\mu},\quad\gamma^{\mu\nu}\to\gamma^{\mu\nu}+2b^{(\mu}n^{\nu)}\qquad\text{where}\qquad
n^{\mu}b_{\mu}=0,\quad b^{\mu}=\gamma^{\mu\nu}b_{\nu}.$ (2.6)

Notice that $(n^{\mu},h_{\mu\nu})$ does not have a unique inverse. Instead, if
$(\tau_{\mu},\gamma^{\mu\nu})$ is an inverse of $(n^{\mu},h_{\mu\nu})$, any
$(\tau^{\prime}_{\mu},\gamma^{\prime\mu\nu})$ related to it by (2.6) is also
an inverse.

One can also describe the Carrollian geometry in the tetrad formalism. In this
formalism, the Carrollian structure and its inverse can be reorganized into
the tetrad $\mathcal{E}^{A}_{\mu}=(\tau_{\mu},e^{i}_{\mu})$ and its inverse,
the cotetrad $\mathcal{E}^{\mu}_{A}=(n^{\mu},e_{i}^{\mu})$,

$\mathcal{E}^{A}_{\nu}\mathcal{E}_{A}^{\mu}=\delta^{\mu}_{\
\nu},\quad\mathcal{E}^{A}_{\mu}\mathcal{E}_{B}^{\mu}=\delta^{A}_{\ B}.$ (2.7)

The local Carrollian symmetry acts on the tetrads by

$\mathcal{E}^{A}_{\mu}\to C^{A}_{\ B}\mathcal{E}^{B}_{\mu},\hskip
12.91663pt\mathcal{E}^{\mu}_{A}\to(C^{-1})^{\ B}_{A}\mathcal{E}^{\mu}_{B},$
(2.8)

where

$C=\begin{pmatrix}1&\mathbf{b}^{\intercal}\\\ 0&\mathbf{R}\in\text{SO}(d-1)\\\
\end{pmatrix}\in\text{Carr}_{0}(d)$ (2.9)

is the $(d\crossproduct d)$ representation matrix of an element in the
isometry subgroup $\text{Carr}_{0}(d)$ of the $d$-dimensional Carrollian
group, which satisfies $\det C=1$. Consequently, the determinant of the tetrad
matrix $\det(\mathcal{E}^{A}_{\mu})$ is invariant under the local Carrollian
symmetry. Last, the Carrollian metric is recovered by
$h_{\mu\nu}=e^{i}_{\mu}e^{i}_{\nu}$. Different from that the local Lorentz
symmetry acts trivially on the Lorentz metric, the local Carrollian boost
symmetry acts non-trivially on the Carrollian structure. It can be easily
checked that this transformation is consistent with the above local boost
transformation (2.6), which provides another origin of the local boost
symmetry. From now on, we speak of Carrollian invariance by requiring both
Carrollian spacetime transformation invariance and local boost invariance.

##  3 Non-Riemannian Hodge duality

In four-dimensional ($4D$) relativistic Maxwell theory, the equations of
motion are Maxwell’s equations

$\displaystyle\left\\{\begin{aligned}
\nabla\cdot\mathbf{E}&=0,&\nabla\times\mathbf{B}-\frac{1}{c^{2}}\frac{\partial\mathbf{E}}{\partial
t}&=0,&\\\
\nabla\cdot\mathbf{B}&=0,&\nabla\times\mathbf{E}+\frac{\partial\mathbf{B}}{\partial
t}&=0.&\end{aligned}\right.\quad\text{(Maxwell)}$ (3.1)

The well-known electromagnetic duality exchanges the electric and magnetic
fields

$\mathbf{E}\to c\mathbf{B},\qquad c\mathbf{B}\to-\mathbf{E},$ (3.2)

and is a symmetry of Maxwell’s equations. This duality map can be
interpretated as the map between the covariant field strength 2-form
$F_{\mu\nu}=2\partial_{[\mu}A_{\nu]}$ and its “Hodge-$*$” dual:

$F_{\mu\nu}\to(*F)_{\mu\nu},\qquad(*F)_{\mu\nu}\to-F_{\mu\nu}.$ (3.3)

The Carrollian cousins of Maxwell theory, commonly referred to as Carrollian
electrodynamics or Carrollian Maxwell theories, have been studied extensively
[54, 49, 59, 60, 61, 63, 47, 62, 69]. It turns out that there are two types of
Carrollian Maxwell’s equations, the electric-type and the magnetic-type,

$\displaystyle\left.\left\\{\begin{aligned}
\nabla\cdot\mathbf{E}_{e}&=0,&\frac{\partial\mathbf{E}_{e}}{\partial
t}&=0,&\\\
\nabla\cdot\mathbf{B}_{e}&=0,&\nabla\times\mathbf{E}_{e}+\frac{\partial\mathbf{B}_{e}}{\partial
t}&=0.&\end{aligned}\right.\text{(elec.)}\quad\right|\left\\{\begin{aligned}
\nabla\cdot\mathbf{E}_{m}&=0,&\nabla\times\mathbf{B}_{m}-\frac{\partial\mathbf{E}_{m}}{\partial
t}&=0,&\\\
\nabla\cdot\mathbf{B}_{m}&=0,&\frac{\partial\mathbf{B}_{m}}{\partial
t}&=0.&\end{aligned}\right.\text{(mag.)}$ (3.4)

These two sectors arise either as two different limits of Maxwell’s equations
under suitable rescaling of $c$,

$\displaystyle\mathbf{E}_{e}$ $\displaystyle=$ $\displaystyle\mathbf{E},$
$\displaystyle\qquad\mathbf{B}_{e}$ $\displaystyle=$ $\displaystyle\mathbf{B}$
, (3.5) $\displaystyle\mathbf{E}_{m}$ $\displaystyle=$ $\displaystyle
c^{-1}\mathbf{E},$ $\displaystyle\qquad\mathbf{B}_{m}$ $\displaystyle=$
$\displaystyle c\mathbf{B}$ ,

or as the equations of motions of the theories obtained by doing null
reduction from two different Bargmann $\text{U}(1)$ gauge theories[49].

As first pointed out in [54], the Carrollian electrodynamics are no longer
invariant under the ultra-relativistic limit (3.5) of the (discrete) electric-
magnetic duality transformation (3.2):

$\mathbf{E}_{e}\to\mathbf{B}_{m},\qquad\mathbf{B}_{e}\to-\mathbf{E}_{m}.$
(3.6)

Instead, this transformation intertwines the electric-type equations with the
magnetic-type ones in (3.4).

The breaking of electric-magnetic duality is essentially the consequence of
the fact that the standard Hodge-$*$ operator is no longer well-defined for
degenerate metrics. Nevertheless, one may generalize the definition of
Hodge-$*$ operator in Riemann geometry to both Galilean and Carrollian
spacetimes[55]. We will review the definition of the Carrollian Hodge-$\ast$
operator, and moreover show that with the help of the generalized Carrollian
Hodge-$*$ operator, even though the electric-type111For magnetic-type theory,
we can still find a similar duality transformation which preserves the
magnetic-type Carrollian Maxwell’s equations. However, it will be much more
subtle to treat it in a fully Carrollian covariant manner compared to the
electric-type theory. As can be seen in [49], the Lagrangian description of
the magnetic-type theory is more complicated because some additional auxiliary
fields should be introduced. So in this sense, the magnetic-type theory is not
a “pure electrodynamics”, whose Lagrangian $\mathcal{L}(F)$ depends only on
$F_{\mu\nu}$. See more in the discussion part. Carrollian electrodynamics is
not strictly “self-dual” under the duality transformation rule given by the
modified Hodge-$*$ operator

$F_{\mu\nu}\to(*F)_{\mu\nu},\qquad(*F)_{\mu\nu}\to(*^{2}F)_{\mu\nu}=0,$ (3.7)

the theory can be invariant under a modified duality transformation

$F_{\mu\nu}\to F_{\mu\nu}+(*F)_{\mu\nu},\qquad(*F)_{\mu\nu}\to(*F)_{\mu\nu}.$
(3.8)

###  3.1 Non-Riemannian Levi-Civita tensors

We first review the classical theory of the Hodge-$*$ operator on a
(pseudo-)Riemann manifold. The Hodge-$*$ operator is a linear map defined on
the space of differential forms $\Omega^{*}(\mathcal{M})$ of a
(pseudo-)Riemann manifold $\mathcal{M}$ which is endowed with a non-degenerate
metric $g_{\mu\nu}$ and its inverse $g^{\mu\nu}$. Conventionally, we use $|g|$
for the determinant of $g$ and $\norm{g}$ for the absolute value of the
determinant. On a $d$-dimensional manifold, the Hodge-$*$ operator maps
$k$-forms to $(d-k)$-forms. Practically, the Levi-Civita tensor
$\varepsilon_{\mu_{1}\cdots\mu_{d}}$ (or lifted to
$\varepsilon^{\mu_{1}\cdots\mu_{d}}$ by $g^{\mu\nu}$) is used to explicitly
write down the Hodge dual of a $k$-form $A_{\nu_{1}\cdots\nu_{p}}$:

$\displaystyle\varepsilon_{\mu_{1}\cdots\mu_{d}}\equiv\sqrt{\norm{g}}\widetilde{\varepsilon}_{\mu_{1}\cdots\mu_{d}},\quad\text{where}\quad\widetilde{\varepsilon}_{\mu_{1}\cdots\mu_{d}}=\widetilde{\varepsilon}^{\mu_{1}\cdots\mu_{d}}=\left\\{\begin{aligned}
+&1&(\mu_{1}\cdots\mu_{d})&\text{ is an even permutation}\\\
-&1&(\mu_{1}\cdots\mu_{d})&\text{ is an odd permutation}\\\
&0&\text{otherwise}\end{aligned}\right.,$ (3.9)
$\displaystyle\varepsilon^{\mu_{1}\cdots\mu_{d}}=\ g^{\mu_{1}\nu_{1}}\cdots
g^{\mu_{d}\nu_{d}}\
\varepsilon_{\nu_{1}\cdots\nu_{d}}=(-1)^{s}\sqrt{\norm{g}}^{-1}\widetilde{\varepsilon}_{\mu_{1}\cdots\mu_{d}},\qquad\text{$s$
is the signature of $g$},$
$\displaystyle(*A)_{\mu_{1}\cdots\mu_{d-p}}\equiv\frac{1}{p!}\
g^{\rho_{1}\nu_{1}}\cdots g^{\rho_{p}\nu_{p}}\
\varepsilon_{\mu_{1}\cdots\mu_{d-p}\rho_{1}\cdots\rho_{p}}A_{\nu_{1}\cdots\nu_{p}}.$

Geometrically, the Levi-Civita tensor defines an orientation that is
compatible with the (pseudo-)Riemann metric. Although a non-degenerate metric
$g_{\mu\nu}$ is necessary for the standard definition of the Hodge dual of a
differential form, the concept of Levi-Civita tensor can be generalized to the
situations where the spacetime is only an oriented manifold, not necessarily a
(pseudo-)Riemann manifold. It could be the covariant orientation form (also
named volume form) of the manifold compatible with other kinds of geometric
data such as the Carrollian structure.

Fortunately, there exist two Carrollian covariant and locally boost invariant
tensors in Carrollian spacetime [55], one being lower indexed and the other
being upper indexed, which can be used as the Carrollian Levi-Civita tensors.
Formally, one of these Carrollian Levi-Civita tensors for Carrollian structure
$(n^{\mu},h_{\mu\nu})$ is

$\varepsilon_{\mu_{1}\cdots\mu_{d}}\equiv\sqrt{\norm{g_{\tau}}}\widetilde{\varepsilon}_{\mu_{1}\cdots\mu_{d}}=\sqrt{(-1)^{s}|g_{\tau}|}\widetilde{\varepsilon}_{\mu_{1}\cdots\mu_{d}},\qquad(g_{\tau})_{\mu\nu}\equiv(-1)^{s}\tau_{\mu}\tau_{\nu}+h_{\mu\nu}.$
(3.10)

It is the same as the standard Levi-Civita tensor of a non-degenerate metric
$g_{\tau}$. The metric $g_{\tau}$ depends on the selection of
$(\tau_{\mu},\gamma^{\mu\nu})$ so it is not locally boost invariant, but the
determinant of $g_{\tau}$ is independent of this choice and is locally boost
invariant. This can be most easily seen in the tetrad formalism introduced in
Section 2.2 because the determinant $|g_{\tau}|$ is the square of the tetrad
determinant $\det(\mathcal{E}^{A}_{\mu})$. The signature $(-1)^{s}$ of the
Carrollian manifold contributes only a sign in the determinant and does not
have an exact meaning as in the Lorentz case, and can be chosen among $\pm 1$
arbitrarily. Though $(-1)^{s}=-1$ is more convenient to compare with the
ultra-relativistic limit of Lorentz theory, in this paper we use $(-1)^{s}=1$
for simplicity.

By considering the dual of a volume form, we have the following $n$-vector,
which is also Carrollian covariant and locally boost invariant,

$\varepsilon^{\mu_{1}\cdots\mu_{d}}\equiv(-1)^{s}\sqrt{\norm{g_{\gamma}}}^{-1}\widetilde{\varepsilon}^{\mu_{1}\cdots\mu_{d}},\qquad(g_{\gamma})^{\mu\nu}\equiv(-1)^{s}n^{\mu}n^{\nu}+\gamma^{\mu\nu}.$
(3.11)

We also call it the Carrollian Levi-Civita tensor. The relation between this
tensor and the volume form (3.10) is

$\varepsilon^{\mu_{1}\cdots\mu_{d}}=\ g_{\gamma}^{\mu_{1}\nu_{1}}\cdots
g_{\gamma}^{\mu_{d}\nu_{d}}\
\varepsilon_{\nu_{1}\cdots\nu_{d}},\qquad\varepsilon_{\mu_{1}\cdots\mu_{d}}=\
(g_{\tau})_{\mu_{1}\nu_{1}}\cdots(g_{\tau})_{\mu_{d}\nu_{d}}\
\varepsilon^{\nu_{1}\cdots\nu_{d}}.$ (3.12)

###  3.2 Carrollian Hodge-$*$ and electromagnetic duality

With the help of the Carrollian Levi-Civita tensor, one may formally define
the Carrollian Hodge-$*$ operator as

$\displaystyle(*A)_{\mu_{1}\cdots\mu_{d-p}}$ $\displaystyle\equiv\frac{1}{p!}\
g_{\gamma}^{\rho_{1}\nu_{1}}\cdots g_{\gamma}^{\rho_{p}\nu_{p}}\
\varepsilon_{\mu_{1}\cdots\mu_{d-p}\rho_{1}\cdots\rho_{p}}A_{\nu_{1}\cdots\nu_{p}}$
(3.13) $\displaystyle=\frac{1}{p!}\
(g_{\tau})_{\mu_{1}\nu_{1}}\cdots(g_{\tau})_{\mu_{d-p}\nu_{d-p}}\
\varepsilon^{\nu_{1}\cdots\nu_{d-p}\rho_{1}\cdots\rho_{p}}A_{\rho_{1}\cdots\rho_{p}}.$

However, this is not a good definition because $(g_{\tau})_{\mu\nu},\
g_{\gamma}^{\mu\nu}$ enters so it depends on the choice of
$(\tau_{\mu},\gamma^{\mu\nu})$. To make the definition locally boost
invariant, we can replace $(g_{\tau})_{\mu\nu},\ g_{\gamma}^{\mu\nu}$ with the
locally boost invariant quantities $h_{\mu\nu},\
m^{\mu\nu}\equiv(-1)^{s}n^{\mu}n^{\nu}$, and the two lines in (3.13) no longer
equal. Due to the antisymmetric nature of the Levi-Civita tensor, the
contraction of $\varepsilon_{\mu_{1}\cdots\mu_{d}}$ with more than two
$m^{\mu\nu}$-s vanishes. Thus the more interesting choice is to use
$\varepsilon^{\mu_{1}\cdots\mu_{d}}$ and $h_{\mu\nu}$ to define

$(*A)_{\mu_{1}\cdots\mu_{d-p}}\equiv\frac{1}{p!}\ h_{\mu_{1}\nu_{1}}\cdots
h_{\mu_{d-p}\nu_{d-p}}\
\varepsilon^{\nu_{1}\cdots\nu_{d-p}\rho_{1}\cdots\rho_{p}}A_{\rho_{1}\cdots\rho_{p}}.$
(3.14)

We will use this as our definition of the Carrollian Hodge-$*$ operator, which
is one of the two Hodge stars defined in [55].

Different from the (anti-)idempotent property $*^{2}=(-1)^{s}$ of standard
Hodge-$*$ operator, the Carrollian Hodge-$*$ operator defined above is
nilpotent, $*^{2}=0$. Then in electromagnetic duality if we take
$F_{\mu\nu}\to(*F)_{\mu\nu}$, for self-consistence we are only allowed to
choose $(*F)_{\mu\nu}\to 0$ instead of $(*F)_{\mu\nu}\to(-1)^{s}F_{\mu\nu}$.
Using $F_{\mu\nu}=E_{e}^{i}dx^{i}\wedge
dt+\frac{1}{2}\varepsilon_{ijk}B_{e}^{i}dx^{j}\wedge dx^{k}$, the $4D$
electric-type Carrollian Maxwell’s equations in (3.4) can be rewritten in
terms of the Carrollian Hodge-$*$ and exterior derivative $d$ as

$d*F=0,\qquad dF=0.$ (3.15)

It is easy to check that the equations degenerate under the electromagnetic
duality (3.7) and are not strictly self-dual. However, upon adding an extra
$F_{\mu\nu}$ into the transformation such that $F_{\mu\nu}\to
F_{\mu\nu}+(*F)_{\mu\nu}$, we have $(*F)_{\mu\nu}\to(*F)_{\mu\nu}$, as shown
in (3.8), the full electric-type Carrollian Maxwell’s equations are self-dual
under this transformation.

##  4 Generalized duality symmetry of Carrollian nonlinear electrodynamics

In Section 3 we discuss the discrete duality symmetry of Carrollian
electrodynamics. For relativistic Maxwell theory, it is well-known that it is
conformal invariant as well and the discrete duality symmetry can be
generalized to a continuous $\text{SO}(2)$ EM duality symmetry:

$\mathbf{E}\to\cos{\theta}\ \mathbf{E}+\sin{\theta}\ c\mathbf{B},\qquad
c\mathbf{B}\to\cos{\theta}\ c\mathbf{B}-\sin{\theta}\ \mathbf{E}.$ (4.1)

Actually, the Maxwell theory is not the unique theory with $\text{SO}(2)$ EM
duality symmetry. The duality rotation symmetry has been found in many other
nonlinear electromagnetic theories (NEDs) such as the famous Born-Infeld
theory. A few years ago[16, 17], a one-parameter family of NEDs was discovered
to share almost the same symmetries as the Maxwell theory. This so-called
ModMax (Modified Maxwell) theory has a Lagrangian

$\mathcal{L}_{\gamma}(\mathcal{S},\mathcal{P})=-\frac{1}{2}\cosh{\gamma}\mathcal{S}+\frac{1}{2}\sinh{\gamma}\sqrt{\mathcal{S}^{2}+\mathcal{P}^{2}};\qquad\mathcal{S}\equiv\frac{1}{2}F_{\mu\nu}F^{\mu\nu},~{}~{}\mathcal{P}\equiv\frac{1}{2}F_{\mu\nu}(*F)^{\mu\nu},$
(4.2)

where $\gamma\in\mathbb{R}$ is a real free parameter. Together with the
Bialynicki-Birula theory, they are the only NED theories with both conformal
symmetry and an $\text{SO}(2)$ EM duality symmetry. Later, the intriguing
relation between the relativistic ModMax theories in various dimensions and
the $\sqrt{T\bar{T}}$ flow was exploited in [19, 20, 22, 21, 18].

In this section, we first derive the Carrollian Gaillard-Zumino condition for
generalized duality symmetry of NEDs by following the original relativistic
formalism in [6]. Then making use of this condition, we investigate the
generalized duality symmetry of electric-type Carrollian Maxwell theory.
Furthermore, though electric-type Carrollian Maxwell theory lacks
$\text{SO}(2)$ EM duality symmetry, we discover a family of Carrollian
conformal NED theories that are $\text{SO}(2)$ EM duality invariant. This new
family of theories will be referred to as Carrollian ModMax theories. We show
that the electric-type Carrollian Maxwell theory can be obtained as the
limiting theories of this new family. We would like to point out that the
Carrollian ModMax theories we get here are NOT the ultra-relativistic limit of
the Lorentz ModMax theories. Actually, taking the Carrollian limit directly,
as the Galilean limit of ModMax taken in [70], breaks up the $\text{SO}(2)$ EM
duality symmetry. For a review of generalized duality symmetries and their
applications, one may refer to [71]. In [72], an overview of general
properties of nonlinear electrodynamics, in particular, the ModMax
electrodynamics, is given.

###  4.1 Gaillard-Zumino condition for Carrollian NEDs

It is illuminating to start from the covariant formalism of NED [71]. In this
formalism, one may introduce the concept of dual field strength222In some
literature the definition here is replaced by $\widetilde{G}^{\mu\nu}\equiv
2\frac{\partial}{\partial F_{\mu\nu}}\mathcal{L}(F)$, because they use
$\frac{\partial{F_{\mu\nu}}}{\partial{F_{\rho\sigma}}}=\delta^{\mu}_{\rho}\delta^{\nu}_{\sigma}$
rather than
$\frac{\partial{F_{\mu\nu}}}{\partial{F_{\rho\sigma}}}=\delta^{\mu}_{\rho}\delta^{\nu}_{\sigma}-\delta^{\mu}_{\sigma}\delta^{\nu}_{\rho}$.
We will avoid this trick and differentiate by independent components of the
antisymmetric tensors.

$\widetilde{G}^{\mu\nu}\equiv\frac{\partial}{\partial
F_{\mu\nu}}\mathcal{L}(F)$ (4.3)

and the $\sim$-operation on antisymmetric tensors $X$

$\widetilde{X}^{\mu_{1}\cdots\mu_{d-p}}\equiv\frac{1}{p!}\varepsilon^{\mu_{1}\cdots\mu_{d-p}\nu_{1}\cdots\nu_{p}}X_{\nu_{1}\cdots\nu_{p}},\qquad\widetilde{X}_{\mu_{1}\cdots\mu_{d-p}}\equiv\frac{1}{p!}\varepsilon_{\mu_{1}\cdots\mu_{d-p}\nu_{1}\cdots\nu_{p}}X^{\nu_{1}\cdots\nu_{p}},$
(4.4)

which plays a central role in the Carrollian treatment of generalized duality.
These two definitions depend only on the Levi-Civita tensors and apply to the
Lorentzian case as well. Different from the Hodge-$*$ operator which maps a
$p$-form to an $(d-p)$-form, the $\sim$-operator maps differential forms to
antisymmetric polyvectors and vice versa, and this operation does not lose any
information because $\widetilde{\widetilde{X}}=(-1)^{s}X$.

We say “pure electrodynamics” when the Lagrangian of the electromagnetic
theory $\mathcal{L}(F)$ depends on $F_{\mu\nu}$ only. The equations of motion
obtained from the Lagrangian $\mathcal{L}(F)$ and the Bianchi identity are
then

$\partial_{\mu}\widetilde{G}^{\mu\nu}=0,\qquad\partial_{\mu}\widetilde{F}^{\mu\nu}=0.$
(4.5)

It is not difficult to observe that $F$ and $G$ are on equal footing. The most
general form of infinitesimal transformations of generalized duality symmetry
for pure electrodynamics is usually written as

$\delta\begin{pmatrix}{F}\\\ {G}\end{pmatrix}=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\begin{pmatrix}{F}\\\ {G}\end{pmatrix}.$ (4.6)

The convention is unambiguous for relativistic theory, but to discuss
Carrollian theories we should strictly distinguish the positions of indices.
By using the $\sim$-operator we can define
$G_{\mu\nu}\equiv(-1)^{s}\widetilde{\widetilde{G}^{\mu\nu}}$ from the dual
field strength $\widetilde{G}^{\mu\nu}$. Then (4.6) implies the following
infinitesimal duality transformations of $(F,G)$ and
$(\widetilde{F},\widetilde{G})$:

$\delta\begin{pmatrix}{F_{\mu\nu}}\\\
{G_{\mu\nu}}\end{pmatrix}=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\begin{pmatrix}{F_{\mu\nu}}\\\
{G_{\mu\nu}}\end{pmatrix},\qquad\delta\begin{pmatrix}{\widetilde{F}^{\mu\nu}}\\\
{\widetilde{G}^{\mu\nu}}\end{pmatrix}=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\begin{pmatrix}{\widetilde{F}^{\mu\nu}}\\\
{\widetilde{G}^{\mu\nu}}\end{pmatrix}.$ (4.7)

In relativistic Maxwell theory, since the Lagrangian is
$\mathcal{L}(F)=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$, the dual field strength is
$\widetilde{G}^{\mu\nu}=-F^{\mu\nu}$ and the Hodge dual of field strength is
$(*F)_{\mu\nu}=G_{\mu\nu}$. The $\text{SO}(2)$ EM duality is generated by
(4.7) with $a=d=0,\quad b=-c$.

The Gaillard-Zumino condition of generalized duality symmetry is just the
requirement of self-consistence in the definition of duality transformation
because $G$ depends on $F$ through the constitutive equation (4.3). It can be
expressed as

$\widetilde{G}^{\mu\nu}+\delta\widetilde{G}^{\mu\nu}=\frac{\partial\mathcal{L}(F+\delta
F)}{\partial(F_{\mu\nu}+\delta F_{\mu\nu})}.$ (4.8)

Now we have

R.H.S. $\displaystyle=\frac{1}{2}\frac{\partial\mathcal{L}(F+\delta
F)}{\partial F_{\rho\sigma}}\cdot\frac{\partial
F_{\rho\sigma}}{\partial(F_{\mu\nu}+\delta F_{\mu\nu})}$ (4.9)
$\displaystyle=\frac{\partial\mathcal{L}(F+\delta F)}{\partial
F_{\mu\nu}}-a\frac{\partial\mathcal{L}(F)}{\partial
F_{\mu\nu}}-\frac{1}{2}b\widetilde{G}^{\rho\sigma}\frac{\partial
G_{\rho\sigma}}{\partial F_{\mu\nu}},$

and also

$\text{L.H.S.}=\frac{\partial\mathcal{L}(F)}{\partial
F_{\mu\nu}}+c\widetilde{F}^{\mu\nu}+d\widetilde{G}^{\mu\nu}.$ (4.10)

Comparing the two sides and using

$\displaystyle\frac{\partial}{\partial
F_{\mu\nu}}\left(F_{\rho\sigma}\widetilde{F}^{\rho\sigma}\right)=4\widetilde{F}^{\mu\nu},\qquad\frac{\partial}{\partial
F_{\mu\nu}}\left(G_{\rho\sigma}\widetilde{G}^{\rho\sigma}\right)=2\widetilde{G}^{\rho\sigma}\frac{\partial
G_{\rho\sigma}}{\partial F_{\mu\nu}},$ (4.11)
$\displaystyle\mathcal{L}(F+\delta
F)-\mathcal{L}(F)=\frac{1}{2}\frac{\partial\mathcal{L}}{\partial
F_{\mu\nu}}\delta
F_{\mu\nu}=\frac{1}{2}\widetilde{G}^{\mu\nu}(aF_{\mu\nu}+bG_{\mu\nu}),$

we get the Gaillard-Zumino consistency condition

$\frac{b}{4}G_{\mu\nu}\widetilde{G}^{\mu\nu}-\frac{c}{4}F_{\mu\nu}\widetilde{F}^{\mu\nu}=(a+d)\mathcal{L}(F)-\frac{a}{2}F_{\mu\nu}\widetilde{G}^{\mu\nu}+\text{const}.$
(4.12)

We can see that the establishment of this condition only depends on the
$\sim$-operation so that it applies not only to relativistic electromagnetism
but also to Carrollian theories. Specifically, when we take $a=d=0,\quad b=-c$
and make the assertion333For the usual relativistic NED theories this
assertion is achieved [5] by requiring that the Lagrangian tends to the usual
Maxwell Lagrangian in the weak-field limit $F\to 0$. But since the Lagrangian
of the ModMax theory is not analytic in $F$, it seems that we can not take the
weak-field limit in general case. However, even in the situations that the
weak-field limit is not well-defined, we are allowed to apply this requirement
if one can find any path along which $F\to 0$ and the Lagrangian reduces to
the Maxwell Lagrangian. For example, in the ModMax and our Carrollian ModMax
theories, we can consider the zero magnetic field configurations $F_{ij}=0$,
under which the Lagrangians become analytic and have a weak-field limit, which
leads to the conclusion that the constant in (4.12) is zero. that the constant
in (4.12) vanishes, we arrive at the Gaillard-Zumino condition for
$\text{SO}(2)$ EM duality invariance:

$G_{\mu\nu}\widetilde{G}^{\mu\nu}+F_{\mu\nu}\widetilde{F}^{\mu\nu}=0.$ (4.13)

It is easy to check that the Lagrangian of electric-type Carrollian Maxwell
electrodynamics $\mathcal{L}=\frac{1}{2}F_{0i}^{2}$ does not satisfy the
condition (4.13).

###  4.2 Generalized duality symmetry of (electric-type) Carrollian Maxwell
electrodynamics

In the previous section we have found discrete duality transformation of
Carrollian Maxwell electrodynamics, it is natural to ask whether this theory
possesses generalized duality. It has been shown that $\text{SO}(2)$ EM
duality is not the symmetry of this theory, so we need to solve for the
general form of Gaillard-Zumino condition (4.12). The Lagrangian
$\mathcal{L}=\frac{1}{2}F_{0i}^{2}$ tells that
$\widetilde{G}^{0i}=-\widetilde{G}^{i0}=F_{0i}$, and other components of dual
field strength vanish. Consequently, $G_{\mu\nu}\widetilde{G}^{\mu\nu}=0$ and
$\mathcal{L}(F=0)=0,\ \widetilde{G}(F=0)=0$. Substituting into (4.12) we have

$-\frac{c}{4}F_{\mu\nu}\widetilde{F}^{\mu\nu}=\frac{a-d}{2}F_{0i}^{2}.$ (4.14)

This condition holds if and only if $c=0$ and $a=d$. These constraints
restrict the duality transformations to the dilation given by $b=c=0,\ a=d$
and a “shift” $a=c=d=0$. The duality group is isomorphic to
$\mathbb{R}_{>0}\times\mathbb{R}$.

It can be proved that the dilation is equivalent to the scaling symmetry when
the spacetime dimension is $d=4$, both for relativistic theories and
Carrollian theories. The reason is that the energy-momentum tensor of
$\mathcal{L}(F)$ can be expressed as

$T^{\mu}_{\
\nu}=-\left(\widetilde{G}^{\mu\rho}F_{\rho\nu}+\delta^{\mu}_{\nu}\mathcal{L}\right).$
(4.15)

Without virial currents, the conservation equation of the current
corresponding to the scaling symmetry requires the energy-momentum tensor to
be traceless. Using the fact that the Gaillard-Zumino condition of dilation
$b=c=0,\ a=d$ is

$2a\mathcal{L}=\frac{a}{2}F_{\mu\nu}\widetilde{G}^{\mu\nu},$ (4.16)

then it is exactly equivalent to the traceless condition

$T^{\mu}_{\ \mu}=\widetilde{G}^{\mu\nu}F_{\mu\nu}-4\mathcal{L}=0.$ (4.17)

Moreover, the traceless condition also implies the whole Carrollian conformal
symmetry, as we prove in Appendix A.

On the other hand, the $\mathbb{R}$-duality shift symmetry gives the
transformation in electric and magnetic fields

$\mathbf{E}_{e}\to\mathbf{E}_{e},\qquad\mathbf{B}_{e}\to\mathbf{B}_{e}+\xi\mathbf{E}_{e}.$
(4.18)

The electric-type Carrollian Maxwell’s equations are invariant under these
duality shift symmetries, and the duality transformation in (3.8) is the
transformation (4.18) with $\xi=1$.

###  4.3 Carrollian ModMax theories

The ModMax theories is a family of conformal $\text{SO}(2)$ EM duality
invariant NED theories. They were first discovered in [16], and were rederived
by using a much simpler method in [17]. In this subsection, we are going to
construct Carrollian ModMax theories, which are conformal invariant and
$\text{SO}(2)$ EM duality invariant.

From now on we use $F^{\mu\nu}$ for
$F^{\mu\nu}\equiv(m^{\mu\rho}\gamma^{\nu\sigma}+m^{\nu\rho}\gamma^{\mu\sigma})F_{\rho\sigma}$,
and this notation will not cause any confusion because the $\sim$-operation of
$F^{\mu\nu}$ satisfies
$\widetilde{F}_{\mu\nu}=\widetilde{F}^{\rho\sigma}h_{\rho\mu}h_{\sigma\nu}$.
Moreover, we introduce the Carrollian invariant quantities as the building
blocks of the Lagrangian,

$\displaystyle\mathcal{S}\equiv\frac{1}{2}F_{\mu\nu}F^{\mu\nu}=m^{\mu\rho}\gamma^{\nu\sigma}F_{\mu\nu}F_{\rho\sigma},$
$\displaystyle\mathcal{P}\equiv\frac{1}{2}F_{\mu\nu}\widetilde{F}^{\mu\nu},$
(4.19)

and introduce the notations

$\displaystyle\mathcal{L}_{\mathcal{S}}\equiv\frac{\partial\mathcal{L}(\mathcal{S},\mathcal{P})}{\partial\mathcal{S}},$
$\displaystyle\mathcal{L}_{\mathcal{P}}\equiv\frac{\partial\mathcal{L}(\mathcal{S},\mathcal{P})}{\partial\mathcal{P}}.$
(4.20)

Notice that the Carrollian $\mathcal{P}$ is opposite to the Lorentz
$\mathcal{P}$ due to our signature choice. For relativistic pure
electrodynamics, all of the Lorentz invariant Lagrangians can be expressed as
a function of relativistic $\mathcal{S},\ \mathcal{P}$ defined in (4.2), which
correspond to the Maxwell Lagrangian and the $\theta$-term respectively. When
it comes to the Carrollian symmetry, finding all Carrollian invariant
quantities becomes more complicated because of the existence of magnetic-type
quantities. However, if we exclude them by restricting ourselves to pure
electrodynamics and focus on the electric-type theory, the Carrollian
invariant NED Lagrangians $\mathcal{L}(\mathcal{S},\mathcal{P})$ are assumed
to be functions of Carrollian $\mathcal{S},\ \mathcal{P}$. Here $\mathcal{S}$
is proportional to the Lagrangian $\mathcal{L}=\frac{1}{2}\mathcal{S}$ of
electric-type Carrollian Maxwell electrodynamics, which can be checked to be
locally boost invariant, and the topological term $\mathcal{P}$ is the same as
in the relativistic case.

By direct computation, we find the dual field strength

$\widetilde{G}^{\mu\nu}=2\left(\mathcal{L}_{\mathcal{S}}F^{\mu\nu}+\mathcal{L}_{\mathcal{P}}\widetilde{F}^{\mu\nu}\right),$
(4.21)

and the Gaillard-Zumino conditions of dilation (or equivalently the scaling
symmetry) and $\text{SO}(2)$ EM duality

$\left\\{\begin{aligned} &F_{\mu\nu}\widetilde{G}^{\mu\nu}=4\mathcal{L},\\\
&F_{\mu\nu}\widetilde{F}^{\mu\nu}+G_{\mu\nu}\widetilde{G}^{\mu\nu}=0,\end{aligned}\right.$
(4.22)

which look like the equations found in [17]. However, recalling that in
Carrollian theory $F^{\mu\nu}\widetilde{F}_{\mu\nu}=0$, which is not equal to
$F_{\mu\nu}\widetilde{F}^{\mu\nu}$, we find that the equations (4.22) can be
expressed in terms of $\mathcal{S},\ \mathcal{P}$

$\left\\{\begin{aligned}
&\mathcal{S}\mathcal{L}_{\mathcal{S}}+\mathcal{P}\mathcal{L}_{\mathcal{P}}=\mathcal{L},\\\
&4\mathcal{L}_{\mathcal{P}}(2\mathcal{S}\mathcal{L}_{\mathcal{S}}+\mathcal{P}\mathcal{L}_{\mathcal{P}})+\mathcal{P}=0.\end{aligned}\right.$
(4.23)

If we impose the reality condition on the Lagrangian444The reality condition
restricts the Lagrangian to be purely real or imaginary, and it causes the
$\mp$-sign to appear in the Lagrangian (4.24). Strictly speaking, the
Lagrangian with the plus sign is a solution to (4.23) only if an extra overall
$i$ factor is included. The Lagrangians with two signs can be converted into
each other through $\gamma\to\gamma+i\frac{\pi}{2}$. At the same time, the
Wick rotation $x^{0}\to ix^{0}$ also exchanges this sign, and the overall
imaginary unit factor can be interpreted as the consequence of Wick rotation.,
the only solution to this partial differential equations is

$\mathcal{L}_{\gamma}(\mathcal{S},\mathcal{P})=\frac{1}{4}\left(e^{\gamma}\mathcal{S}\mp
e^{-\gamma}\frac{\mathcal{P}^{2}}{\mathcal{S}}\right),$ (4.24)

where $\gamma$ is a free real parameter, as in ModMax. This one-parameter
family of theories provides the only Carrollian conformal and $\text{SO}(2)$
EM duality invariant NEDs. Obviously, the Lagrangians (4.24) cannot be
realized as the ultra-relativistic limit of Lorentz ModMax Lagrangian

$\mathcal{L}_{\gamma}(\mathcal{S},\mathcal{P})=-\frac{1}{2}\cosh{\gamma}\mathcal{S}+\frac{1}{2}\sinh{\gamma}\sqrt{\mathcal{S}^{2}+\mathcal{P}^{2}}.$
(4.25)

Up to a renormalization, the Carrollian ModMax family (4.24) reaches two
different limits as $\gamma\to\pm\infty$

$\mathcal{L}_{+\infty}(\mathcal{S},\mathcal{P})\sim\mathcal{S},\qquad\mathcal{L}_{-\infty}(\mathcal{S},\mathcal{P})\sim\frac{\mathcal{P}^{2}}{\mathcal{S}}.$
(4.26)

The $\mathcal{L}_{+\infty}$ limit is just the Lagrangian of electric-type
Carrollian Maxwell electrodynamics, while $\mathcal{L}_{-\infty}$ is a novel
Carrollian NED. Using the generalized Gaillard-Zumino condition, the
$\mathcal{L}_{-\infty}$ shares the same duality symmetries as the electric-
type Carrollian Maxwell electrodynamics, whose duality group is
$\mathbb{R}_{>0}\times\mathbb{R}$. The duality group of Carrollian ModMax
theories is $\mathbb{R}_{>0}\times\text{SO}(2)$, which is similar to the one
of relativistic ModMax and Maxwell theories. Again, using the Gaillard-Zumino
condition of $c=0,\ a=d$ one can check that $\mathcal{L}_{-\infty}$ is the
only (pure) Carrollian NED that shares the same
$(\mathbb{R}_{>0}\times\mathbb{R})$-duality symmetry with electric-type
Carrollian Maxwell theory.

##  5 Carrollian $\sqrt{T\bar{T}}$ deformation flows

The $T\bar{T}$-like deformations have raised broad interest in various areas
of theoretical physics because they provide important examples of solvable
deformations that preserve integrability and duality symmetry. Most generally,
the $T\bar{T}$ deformation is generated by an irrelevant operator whose
expression in dimension $d$ is [21, 73]:

$O^{\lambda}_{T^{2}}=\frac{1}{d}\left(T^{\mu}_{\ \nu}T^{\nu}_{\
\mu}-rT^{\mu}_{\ \mu}T^{\nu}_{\ \nu}\right),\qquad r\in\mathbb{R}.$ (5.1)

The parameter $r$ can be chosen differently for different purposes [21, 74],
but it has no effect if the energy-momentum tensor is traceless. The
$\sqrt{T\bar{T}}$ deformation is a deformation generated by the classically
marginal operator $O^{\lambda}_{\sqrt{T^{2}}}$ [75, 21, 22], which is the
square root of (5.1),

$O^{\gamma}_{\sqrt{T^{2}}}=\sqrt{\frac{1}{d}\left(T^{\mu}_{\ \nu}T^{\nu}_{\
\mu}-rT^{\mu}_{\ \mu}T^{\nu}_{\ \nu}\right)}.$ (5.2)

As we have mentioned, relativistic ModMax theories are closely related to the
Maxwell theory through the $\sqrt{T\bar{T}}$ flow [19, 20]. In this section,
we will investigate this relationship in the context of the Carrollian ModMax
theories. We find that under the $\sqrt{T\bar{T}}$ flow, the Carrollian ModMax
theories reach two fixed points, one of which is the electric-type Carrollian
Maxwell theory, and the flow is non-invertible at these endpoint theories.
This makes the story quite different from the relativistic $\sqrt{T\bar{T}}$
flow of the ModMax theories in which the $\sqrt{T\bar{T}}$ flow can generate
the ModMax theories from the Maxwell theory. The Carrollian ModMax theories
provide a concrete and novel example of $\sqrt{T\bar{T}}$ deformation of non-
Lorentzian theories.

Furthermore, inspired by $4D$ ModMax theories, a few research groups proposed
independently in [22, 21, 18] that a one-parameter family of $2D$ ModMax-like
theories of multiple free scalars could be generated by the $2D$
$\sqrt{T\bar{T}}$ deformation flow from a free multi-scalar theory. We will
show that the $2D$-dimensional reduction of $4D$ Carrollian ModMax theories
and its generalization lead to a family of modified $2D$ multiple BMS-scalar
theories[76], which are also closed under the $\sqrt{T\bar{T}}$ flow.

###  5.1 Carrollian ModMax under generalized $\sqrt{T\bar{T}}$ deformation

The relativistic ModMax theories (4.25) are known to be generated from the
Maxwell theory by the $\sqrt{T\bar{T}}$ flow perturbatively in $d=4$, in the
sense that

$\frac{\partial\mathcal{L}^{\text{ModMax}}_{\gamma}}{\partial\gamma}=O^{\gamma}_{\sqrt{T^{2}}},\qquad\mathcal{L}_{\gamma}^{\text{ModMax}}=\mathcal{L}^{\text{Maxwell}}+\int
O^{\gamma}_{\sqrt{T^{2}}}d\gamma.$ (5.3)

To convince this, we just need to substitute the ModMax Lagrangians (4.2) and
the expression of $\sqrt{T\bar{T}}$ operator (5.2) into the above equation and
verify that it is a solution. Now we look at the Carrollian ModMax theories
$\mathcal{L}^{\text{CMM}}_{\gamma}$ in (4.24):

$\mathcal{L}^{\text{CMM}}_{\gamma}(\mathcal{S},\mathcal{P})=\frac{1}{4}\left(e^{\gamma}\mathcal{S}\mp
e^{-\gamma}\frac{\mathcal{P}^{2}}{\mathcal{S}}\right).$ (5.4)

The energy-momentum tensor of a general relativistic nonlinear electrodynamics
is [77]

$T^{\mu}_{\
\nu}=-\left(\widetilde{G}^{\mu\rho}F_{\rho\nu}+\delta^{\mu}_{\nu}\mathcal{L}\right).$
(5.5)

This expression also works for Carrollian nonlinear electrodynamics, and we
prove it in Appendix A.2. We use ${(T_{0})}^{\mu}_{\ \nu}$ as the energy-
momentum tensor of the electric-type Carrollian Maxwell theory
$\mathcal{L}^{\text{eCMax}}=\frac{1}{2}\mathcal{S}$,

${(T_{0})}^{\mu}_{\
\nu}=-\left(F^{\mu\rho}F_{\rho\nu}+\frac{1}{4}F^{\rho\sigma}F_{\rho\sigma}\delta^{\mu}_{\nu}\right).$
(5.6)

The energy-momentum tensors of the Carrollian ModMax theories are

${(T_{\gamma})}^{\mu}_{\ \nu}=\frac{1}{2}{(T_{0})}^{\mu}_{\
\nu}\left(e^{\gamma}\pm
e^{-\gamma}\frac{\mathcal{P}^{2}}{\mathcal{S}^{2}}\right).$ (5.7)

Then the $T\bar{T}$ operators of electric-type Carrollian Maxwell theory and
Carrollian ModMax theories are respectively

$\displaystyle O_{T_{0}^{2}}$ $\displaystyle=\frac{1}{4}\mathcal{S}^{2},$
(5.8) $\displaystyle O^{\gamma}_{T^{2}}$
$\displaystyle=\frac{1}{16}\mathcal{S}^{2}\left(e^{\gamma}\pm
e^{-\gamma}\frac{\mathcal{P}^{2}}{\mathcal{S}^{2}}\right)^{2}=\left(\frac{e^{\gamma}\mathcal{S}^{2}\pm
e^{-\gamma}\mathcal{P}^{2}}{4\mathcal{S}}\right)^{2}.$

Therefore, we have

$\frac{\partial\mathcal{L}_{\gamma}^{\text{CMM}}}{\partial\gamma}=\frac{1}{4}\left(e^{\gamma}\mathcal{S}\pm
e^{-\gamma}\frac{\mathcal{P}^{2}}{\mathcal{S}}\right)=O^{\gamma}_{\sqrt{T^{2}}}.$
(5.9)

We observe that the family of Carrollian ModMax theories (5.4) deform among
themselves under $\sqrt{T\bar{T}}$ flow. As the parameter $\gamma$ tends to
infinity, the endpoints of this flow, we find the Lagrangian of electric-type
Carrollian Maxwell theory $\mathcal{L}\sim\mathcal{S}$ and the other
$\mathbb{R}$-duality shift invariant Carrollian conformal NED
$\mathcal{L}\sim\frac{\mathcal{P}^{2}}{\mathcal{S}}$. However, we cannot
deform the endpoint theories via the $\sqrt{T\bar{T}}$ flow to get the
Carrollian ModMax theories, because the flow is non-invertible at these two
points. The endpoint theories are the fixed points of the $\sqrt{T\bar{T}}$
flow, because

$\displaystyle O^{\gamma}_{\sqrt{T^{2}}}$
$\displaystyle=e^{\gamma}\frac{\mathcal{S}}{2}$ for
$\displaystyle\mathcal{L}=e^{\gamma}\frac{\mathcal{S}}{2},$ (5.10)
$\displaystyle O^{\gamma}_{\sqrt{T^{2}}}$
$\displaystyle=e^{-\gamma}\frac{\mathcal{P}^{2}}{2\mathcal{S}}$ for
$\displaystyle\mathcal{L}=-e^{-\gamma}\frac{\mathcal{P}^{2}}{2\mathcal{S}}.$

This is reasonable because $\sqrt{T\bar{T}}$ flow preserves the duality
symmetry while the endpoint theories have different duality groups from
general Carrollian ModMax theories. The relationship between the endpoint
theories and Carrollian ModMax theories is illustrated in the following
diagram, and the red points contain an infinite renormalization factor:

$\leavevmode\hbox to155.89pt{\vbox
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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0.0pt\hfil&\hfil\hskip 37.26387pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\pgfsys<EMAIL_ADDRESS>(5.11)

###  5.2 $\sqrt{T\bar{T}}$ deformation of $2D$ Carrollian modified scalar
theories

In the last subsection, we discuss the behavior of $4D$ Carrollian ModMax
theories under the $\sqrt{T\bar{T}}$ flow, now we show that the story can be
generalized to $2D$, and we construct a Carrollian ModMax-like extension of
$2D$ BMS free scalar theory. To be more specific, we consider $2D$ scalar
field theories, where the scalar has internal degrees of freedom and can be
taken as $N$ copies of bosonic scalar fields, or in other words we consider
$2D$ $\text{O}(N)$ free scalar theory. We reproduce part of the relativistic
picture that there is a link between the $2D$ ModMax-like modified scalar
theories and the $4D$ ModMax theories [19, 18, 21]. For the Carrollian cases,
we construct $2D$ $\sqrt{T\bar{T}}$-closed scalar theories from 4D Carrollian
ModMax theories, by first doing a dimensional reduction to obtain $N=2$ scalar
theories and then generalizing to the case that $N$ takes any positive integer
value. Different from relativistic cases, the $\sqrt{T\bar{T}}$ flow of the
$2D$ Carrollian ModMax-like modified scalar theories have two fixed points as
in (5.11), where the flow is again non-invertible.

In two dimensions, the $\sqrt{T\bar{T}}$ flow is generated by the operator
(5.2) with $d=2$ and $r=1$, and the $T\bar{T}$ operator (5.1) equals the
opposite of the determinant of $2D$ stress tensor $T^{a}_{\ b}$:

$(O^{\gamma}_{\sqrt{T^{2}}})=\sqrt{\frac{1}{2}\left(T^{a}_{\ b}T^{b}_{\
a}-T^{a}_{\ a}T^{b}_{\ b}\right)}=\sqrt{-\det(T^{a}_{\ b})}.$ (5.12)

We begin with a short review of the relativistic $2D$ ModMax-like scalar
theories. As the relativistic ModMax theories are generated from the Maxwell
theory by the $\sqrt{T\bar{T}}$ flow perturbatively, the relativistic $2D$
ModMax-like scalar theories are generated by the $2D$ $\sqrt{T\bar{T}}$ flow
from a free $N$ bosonic scalar theory, which could be taken as the seed
theory. The Lagrangian of the free $N$-scalar theory is given by

$\mathcal{L}_{0}^{N,\text{L}}=-\frac{1}{2}\sum_{i=1}^{N}\partial_{a}\phi^{i}\partial^{a}\phi^{i}.$
(5.13)

From now on $i$ is always summed over the internal indices. The stress tensor
of this theory is obtained as

$T^{a}_{\
b}=\frac{\partial\mathcal{L}}{\partial\partial_{a}\phi^{i}}\partial_{b}\phi^{i}-\delta^{a}_{\
b}\mathcal{L}.$ (5.14)

Under the $\sqrt{T\bar{T}}$ deformation, one can get the deformed Lagrangian:

$\mathcal{L}_{\gamma}^{N,\text{L}}=-\frac{\cosh{\gamma}}{2}\partial_{a}\phi^{i}\partial^{a}\phi^{i}+\frac{\sinh{\gamma}}{2}\sqrt{2\partial_{a}\phi^{i}\partial^{b}\phi^{i}\partial_{b}\phi^{j}\partial^{a}\phi^{j}-(\partial_{a}\phi^{i}\partial^{a}\phi^{i})^{2}}.$
(5.15)

The deformed Lagrangian reduces to the free massless $N$-scalar theory at
$\gamma=0$ after a rescaling. When $N=1$, the Lagrangian
$\mathcal{L}_{\gamma}^{N=1,\text{L}}=e^{\gamma}\mathcal{L}_{0}^{N=1,\text{L}}$
is just a rescaling of the free theory Lagrangian, so the deformation is
trivial. When $N>1$, due to the fact that

$\partial_{a}\phi^{i}\partial^{b}\phi^{i}\partial_{b}\phi^{j}\partial^{a}\phi^{j}\neq(\partial_{a}\phi^{i}\partial^{a}\phi^{i})^{2},$

the Lagrangian can not be rescaled to $\mathcal{L}_{0}^{N,\text{L}}$, so the
deformation is nontrivial.

The deformed Lagrangian (5.15) can be understood from the perspective of
dimensional reduction from 4D ModMax. Recalling that the Lagrangian of
Lorentzian ModMax theories is given by:

$\mathcal{L}_{\gamma}(\mathcal{S},\mathcal{P})=-\frac{1}{2}\cosh{\gamma}\mathcal{S}+\frac{1}{2}\sinh{\gamma}\sqrt{\mathcal{S}^{2}+\mathcal{P}^{2}};\qquad\mathcal{S}\equiv\frac{1}{2}F_{\mu\nu}F^{\mu\nu},\mathcal{P}\equiv\frac{1}{2}F_{\mu\nu}(*F)^{\mu\nu}.$
(5.16)

Inspired by the connection between particular solutions relating to scattering
of plane waves of the $4D$ Born-Infeld theory and the solutions of the $2D$
Nambu-Goto theory in static gauge with $(N+2)$-dimensional target space for
$N=2$ [21], one can perform a dimensional reduction of the Lorentzian $4D$
ModMax theories to two dimensions. This is achieved by identifying
$\phi^{1}=A_{2},\ \phi^{2}=A_{3}$ and specifying the following field
configurations,

$\displaystyle A_{\mu}=A_{\mu}(x^{0},x^{1}),$ (5.17) $\displaystyle
F_{01}=\partial_{0}A_{1}-\partial_{1}A_{0}=0.$

Then $F_{\mu\nu}$ has only four independent non-vanishing components:

$F_{02}=\partial_{0}\phi^{1},\quad F_{03}=\partial_{0}\phi^{2},\quad
F_{12}=\partial_{1}\phi^{1},\quad F_{13}=\partial_{1}\phi^{2},$ (5.18)

and the Lorentzian invariants $\mathcal{S}$ and $\mathcal{P}$ reduce to

$\displaystyle\mathcal{S}=\frac{1}{2}F_{\mu\nu}F^{\mu\nu}=\sum_{i=1}^{2}\partial_{a}\phi^{i}\partial^{a}\phi^{i},$
(5.19)
$\displaystyle\mathcal{P}=\frac{1}{2}F_{\mu\nu}(*F)^{\mu\nu}=-2(\partial_{1}\phi^{1}\partial_{0}\phi^{2}-\partial_{0}\phi^{1}\partial_{1}\phi^{2}),\qquad(\varepsilon^{0123}=-1).$

Plugging them into (5.16), the reduced theories have the Lagrangian (5.15)
with $N=2$. One can further generalize to $N$-scalar case with the Lagrangian
(5.15). One may also introduce the following notion to write this Lagrangian
more compactly

$\displaystyle(\Phi_{N})_{ab}\equiv\sum_{i=1}^{N}\partial_{a}\phi^{i}\partial_{b}\phi^{i},\qquad(a,b\in\\{0,1\\}),$
(5.20)
$\displaystyle\mathcal{L}_{\gamma}^{N,\text{L}}=-\frac{\cosh{\gamma}}{2}(\Phi_{N})_{a}^{\
a}+\frac{\sinh{\gamma}}{2}\sqrt{2(\Phi_{N})_{ab}(\Phi_{N})^{ab}-\left((\Phi_{N})_{a}^{\
a}\right)^{2}},$

It can be directly checked that $\mathcal{L}_{\gamma}^{N,\text{L}}$ satisfies
the $\sqrt{T\bar{T}}$ flow equation

$\displaystyle\frac{\partial\mathcal{L}_{\gamma}^{N,\text{L}}}{\partial\gamma}=(O^{\gamma}_{\sqrt{T^{2}}})^{N,\text{L}}.$
(5.21)

In analogy with the Lorentzian dimensional reduction, we can finally explore
the $\sqrt{T\bar{T}}$ deformation of Carrollian scalar theories from $4D$ seed
theories. The $4D$ Carrollian ModMax Lagrangian is given by (4.24):

$\mathcal{L}_{\gamma}(\mathcal{S},\mathcal{P})=\frac{1}{4}\left(e^{\gamma}\mathcal{S}\mp
e^{-\gamma}\frac{\mathcal{P}^{2}}{\mathcal{S}}\right);\qquad\mathcal{S}\equiv\frac{1}{2}F_{\mu\nu}F^{\mu\nu}=m^{\mu\rho}\gamma^{\nu\sigma}F_{\mu\nu}F_{\rho\sigma},~{}~{}\mathcal{P}\equiv\frac{1}{2}F_{\mu\nu}\widetilde{F}^{\mu\nu}.$
(5.22)

According to (5.17) we have

$\mathcal{S}=\sum_{i=1}^{2}\partial_{0}\phi^{i}\partial_{0}\phi^{i},\hskip
12.91663pt\mathcal{P}=2(\partial_{1}\phi^{1}\partial_{0}\phi^{2}-\partial_{0}\phi^{1}\partial_{1}\phi^{2}),\qquad(\varepsilon^{0123}=+1).$
(5.23)

Plugging the above relations into (4.24), we get:

$\displaystyle\mathcal{L}_{\gamma}^{N=2,\text{C}}$
$\displaystyle=\frac{1}{4}e^{\gamma}\partial_{0}\phi^{i}\partial_{0}\phi^{i}\mp
e^{-\gamma}\frac{(\partial_{1}\phi^{1}\partial_{0}\phi^{2}-\partial_{0}\phi^{1}\partial_{1}\phi^{2})^{2}}{\partial_{0}\phi^{i}\partial_{0}\phi^{i}}$
(5.24) $\displaystyle=\frac{1}{4}e^{\gamma}(\Phi_{2})_{00}\pm
e^{-\gamma}\frac{\det(\Phi_{2})}{(\Phi_{2})_{00}}.$

It is easy to check that

$\frac{\partial\mathcal{L}_{\gamma}^{N=2,\text{C}}}{\partial\gamma}=(O^{\gamma}_{\sqrt{T^{2}}})^{N=2,\text{C}}.$
(5.25)

We can then generalize the result to obtain the Carrollian
$\sqrt{T\bar{T}}$-deformed Lagrangian of $N$-scalars,

$\mathcal{L}_{\gamma}^{N,C}=\frac{1}{4}e^{\gamma}(\Phi_{N})_{00}\pm
e^{-\gamma}\frac{\det(\Phi_{N})}{(\Phi_{N})_{00}}.$ (5.26)

The $N=1$ theory is trivial because $\det\Phi_{1}=0$. For $N\geq 2$, this
Lagrangian is nontrivial because as we can see from (5.24) that
$\det\Phi_{2}=(\partial_{1}\phi^{1}\partial_{0}\phi^{2}-\partial_{0}\phi^{1}\partial_{1}\phi^{2})^{2}\neq
0$, and it satisfies

$\frac{\partial\mathcal{L}_{\gamma}^{N,C}}{\partial\gamma}=(O^{\gamma}_{\sqrt{T^{2}}})^{N,C}.$
(5.27)

However, just as in the discussion of the $\sqrt{T\bar{T}}$ flow of the $4d$
Carrollian ModMax theories in the last subsection, we cannot generate the
whole flow from the Carrollian Maxwell theory as seed theory because it lies
on a fixed point of the $\sqrt{T\bar{T}}$ flow. After dimensional reduction
from $4D$ Carrollian Maxwell theory, we find the $2D$ BMS $N$-free scalar
theory, which is invariant under $\sqrt{T\bar{T}}$ flow [40],

$\mathcal{L}_{0}^{N,\text{C}}=\frac{1}{2}\sum_{i=1}^{N}\partial_{0}\phi^{i}\partial_{0}\phi^{i},\hskip
12.91663pt\mathcal{L}_{\gamma}^{N,\text{C}}=e^{\gamma}\mathcal{L}_{0}^{N,\text{C}}.$
(5.28)

This theory cannot be taken as a seed theory to generate other theories on the
flow.

In short, the Lorentzian $\sqrt{T\bar{T}}$ flow and the Carrollian
$\sqrt{T\bar{T}}$ flow are again very different in $2D$. For the Lorentzian
case, we can get the deformed Lagrangian by using the free scalar theory as
the seed theory. However, for the Carrollian case, the BMS free scalar theory
$\mathcal{L}^{N,C}=\frac{1}{2}{\left(\Phi_{N}\right)_{00}}$ and the scalar
theory with
$\mathcal{L}^{N,C}=\frac{\det\left(\Phi_{N}\right)}{\left(\Phi_{N}\right)_{00}}$
are fixed points at $\gamma=\pm\infty$ of the $\sqrt{T\bar{T}}$ deformation
flow. We can generate the Carrollian $\sqrt{T\bar{T}}$-deformed Lagrangian
from any theory living on the flow except the endpoint theories. We show this
$\sqrt{T\bar{T}}$ flow in the following diagram:

$\leavevmode\hbox to241.64pt{\vbox
to53.9pt{\pgfpicture\makeatletter\hbox{\hskip
120.81879pt\lower-26.95131pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{${\mathcal{L}\sim\frac{\det\left(\Phi_{N}\right)}{\left(\Phi_{N}\right)_{00}}}$}
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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0.0pt\hfil&\hfil\hskip 49.56328pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys<EMAIL_ADDRESS>(5.29)

##  6 Discussions

In this paper, we constructed a one-parameter family of Carrollian ModMax
theories and studied their properties under the Carrollian electromagnetic
duality transformations and the $\sqrt{T\bar{T}}$ flow. Firstly, We noticed
that the Carrollian EM duality transformation could be defined with the help
of the non-Riemannian Hodge duality. Secondly, we used the Carrollian
Gaillard-Zumino consistency condition for Carrollian NEDs to show that the
Carrollian Maxwell theory is invariant under dilation and shift, which is
different from the well-known transformation originating from the Carrollian
limit of the EM duality. Furthermore, we showed that if we focused on pure
electrodynamics, the Carrollian ModMax theories are the only Carrollian
conformal and $\text{SO}(2)$ EM duality invariant NEDs. Finally we discussed
the $\sqrt{T\bar{T}}$ flow of Carrollian theories. We showed that the
Carrollian ModMax theories could flow to the Carrollian Maxwell theory by the
$\sqrt{T\bar{T}}$ deformation. However, the Carrollian Maxwell theory is the
fixed point under the $\sqrt{T\bar{T}}$ flow, and cannot lead to Carrollian
ModMax theories. In other words, the $\sqrt{T\bar{T}}$ deformation is not
invertible at the endpoints of the flow. This is very different from the usual
picture between the Lorentzian ModMax theories and Maxwell theory, which are
connected by the $\sqrt{T\bar{T}}$ flow. Moreover, we investigated the
$\sqrt{T\bar{T}}$ flow of a family of $2D$ Carrollian ModMax-like $N$-scalar
models, whose construction is inspired by the dimensional reduction of the
$4D$ Carrollian ModMax theories, and we found that they are driven to the $2D$
BMS free $N$-scalar model by the flow.

There are still several important issues to discuss. First of all, although
our discussion is based on Carrollian symmetry and Carrollian NEDs, we can
generalize our discussion to the Galilean NEDs without any difficulty. The
Galilean analog of Carrollian structure is the Newton-Cartan structure which
is given by the pair $(\gamma^{\mu\nu},\tau_{\mu})$. Here $\gamma^{\mu\nu}$ is
also degenerate and $\tau_{\mu}$ is a one-form belonging to its kernel.
Covariant Newton-Cartan theories should be invariant under the Galilean
cousins of Carrollian local boosts - the Milne boosts parameterized by
$b^{\mu}$ that act on the inverse $(h_{\mu\nu},n^{\mu})$ of Newton-Cartan
structure as

$n^{\mu}\to n^{\mu}+b^{\mu},\quad h_{\mu\nu}\to
h_{\mu\nu}+2b_{(\mu}\tau_{\nu)}\qquad\text{where}\qquad\tau_{\mu}b^{\mu}=0,\quad
b_{\mu}=h_{\mu\nu}b^{\nu}.$ (6.1)

One can similarly define the Levi-Civita tensors through the determinant of

$(g_{\tau})_{\mu\nu}\equiv(-1)^{s}\tau_{\mu}\tau_{\nu}+h_{\mu\nu},$ (6.2)

which is invariant for a given Newton-Cartan structure, and discuss the
Galilean Hodge duality and the continuous generalized duality symmetries for
Galilean NEDs. The only quadratic gauge and Galilean invariant electromagnetic
theory has the Lagrangian555This Lagrangian however is non-dynamical and not
the Lagrangian of Galilean electrodynamics (GED) [78, 79, 80], which includes
an extra scalar field. Different from the Carrollian case, the two Galilean
limits of Maxwell’s equations both have no simple Lagrangian descriptions,
instead, the GED Lagrangian is more commonly considered, which is closely
related to the magnetic Galilean limit of Maxwell’s equations. Our discussion
will need more adjustments to deal with the GED theory.  [81]:

$\mathcal{L}=\frac{1}{2}F_{ij}F^{ij}.$ (6.3)

We can take $F^{\mu\nu}\equiv\gamma^{\mu\rho}\gamma^{\nu\sigma}F_{\rho\sigma}$
and find the Galilean invariant quantities

$\mathcal{S}=\frac{1}{2}F_{\mu\nu}F^{\mu\nu}=\frac{1}{2}\gamma^{\mu\rho}\gamma^{\nu\sigma}F_{\mu\nu}F_{\rho\sigma},\qquad\mathcal{P}=\frac{1}{2}F_{\mu\nu}\widetilde{F}^{\mu\nu}.$
(6.4)

General Galilean NEDs without extra fields are given by
$\mathcal{L}(\mathcal{S},\mathcal{P})$, and their duality symmetries are also
determined by the Gaillard-Zumino condition (4.12). By solving it, the
generalized duality group of Galilean electromagnetic theory (6.3) is again
$\mathbb{R}_{>0}\times\mathbb{R}$, consisting of the dilation and Galilean
shift. The Lagrangian of Galilean NEDs with generalized duality group
$\mathbb{R}_{>0}\times\text{SO}(2)$ is

$\mathcal{L}^{\text{GMM}}_{\gamma}(\mathcal{S},\mathcal{P})=\frac{1}{4}\left(e^{\gamma}\mathcal{S}\mp
e^{-\gamma}\frac{\mathcal{P}^{2}}{\mathcal{S}}\right).$ (6.5)

These theories are the only Galilean NEDs that are Galilean conformal and
$\text{SO}(2)$ EM duality invariant. It can be observed that they have the
same expression as Carrollian ModMax theories if we replace the Carrollian
invariant quantity $\mathcal{S}$ with its Galilean version. The quantity
$\mathcal{P}$ remains the same. Moreover, they are also closed under
$\sqrt{T\bar{T}}$ deformation and have (6.3) as one of the endpoints under the
flow. The dimensional reduction to two dimensions as in Section 5.2 gives us
the same result for the Galilean ModMax theories, with only the indices $0$
being replaced by $1$. It reflects the fact that the $2D$ Galilean conformal
algebra (GCA) is isomorphic to the $2D$ Carrollian conformal algebra (CCA).

Secondly, the original form of the Carrollian ModMax-like scalar theories
(5.26) is hard to be quantized because of the appearance of the denominator.
The general Carrollian ModMax-like scalar theories can be linearized
classically with the help of a Langangian-multiplier field $\eta$ and a world-
sheet metric $h_{ab}$ as

$\displaystyle\mathcal{L}$ $\displaystyle\sim\frac{1}{4}e^{\gamma}\Phi_{00}\pm
4e^{-\gamma}\left(\eta^{2}\Phi_{00}-\eta\sqrt{-\det(\Phi)}\right),$ (6.6)
$\displaystyle\sim\left(\frac{1}{4}e^{\gamma}\pm
4e^{-\gamma}\eta^{2}\right)\Phi_{00}\mp
2e^{-\gamma}\eta\left(-h\right)^{\frac{1}{2}}h^{ab}\Phi_{ab}.\qquad a,b\ =\
0,1$

It will be interesting to explore the connection between this linearized form
and the Polyakov world-sheet action of the conventional string. Besides, we
have discussed the $\sqrt{T\bar{T}}$ deformation of the Carrollian ModMax-like
scalar theories. It would be interesting to discuss the $\sqrt{T\bar{T}}$
deformation of BMS fermionic fields [82, 83, 84].

In the end, we want to comment on the magnetic-type Carrollian Maxwell theory.
Although the magnetic-type theory is not the focus of this paper, the
magnetic-type Carrollian Maxwell’s equations also acquire a duality
transformation as

$\mathbf{E}_{m}\to\mathbf{E}_{m}+\mathbf{B}_{m},\qquad\mathbf{B}_{m}\to\mathbf{B}_{m}.$
(6.7)

To put them in a covariant way needs some work. The magnetic-type Lagrangian
is666Here, for simplicity, we have omitted a decoupled $\Pi_{0}^{2}$-term that
would be necessary to self-consistently realize manifest off-shell Carrollian
invariance on the Lagrangian. It is a consequence of the requirement that
$[\delta_{B_{i}},\delta_{B_{j}}]=\delta_{[B_{i},B_{j}]}=0$. [49]:

$\displaystyle\mathcal{L}[A_{i},A_{0};\Pi_{i}]=\Pi_{i}(\partial_{i}A_{0}-\partial_{0}A_{i})+\frac{1}{4}(\partial_{i}A_{j}-\partial_{j}A_{i})^{2}.$
(6.8)

The magnetic-type electric fields $\mathbf{E}_{m}$ should no longer be
interpreted as part of the electromagnetic tensor
$F_{\mu\nu}=2\partial_{[\mu}A_{\nu]}$, but to be identified with the
multiplier fields $\Pi_{i}$. And there is an additional constraint
$\iota_{\hat{n}}F=0$ to put the temporal part of $F_{\mu\nu}$ to zero, where
$\iota_{\hat{n}}$ is the inner derivative on the differential form given by
$\hat{n}$. This makes finding the covariant expression of the magnetic-type
duality transformation a more subtle task. To address this concern, the
correct treatment is to encode the magnetic-type electric and magnetic fields
$\mathbf{E}_{m},\mathbf{B}_{m}$ into the dual field strengths, which is
defined as $\widetilde{G}^{\mu\nu}\equiv\frac{\partial}{\partial
F_{\mu\nu}}\mathcal{L}(F;\Pi)$. Now the magnetic-type electric and magnetic
fields satisfy

$\widetilde{G}^{\mu\nu}=\Pi_{i}\partial_{i}\wedge\partial_{t}+F_{ij}\partial_{i}\wedge\partial_{j}=E_{m}^{i}\partial_{i}\wedge\partial_{t}+\frac{1}{2}\varepsilon_{ijk}B_{m}^{i}\partial_{j}\wedge\partial_{k}.$
(6.9)

Next define
$G_{\mu\nu}\equiv\frac{1}{2}\varepsilon_{\mu\nu\rho\sigma}\widetilde{G}^{\rho\sigma}$,
and one can check that using the Carrollian Hodge-$*$, the magnetic-type
Carrollian Maxwell’s equations can be recast into the forms

$d*G=0,\qquad dG=0,\qquad(\iota_{\hat{n}}F=0).$ (6.10)

These equations are invariant under the following duality transformations

$G\to G+*G,\qquad*G\to*G,\qquad(F\to F),$ (6.11)

which gives (6.7). Recalling that there is a swapping duality between the
electric-type and magnetic-type Carrollian Maxwell theories, this magnetic-
type discrete duality can be viewed as the combination of the swapping duality
and the electric-type discrete self-duality. To discuss the generalized
duality symmetry of magnetic-type theory in detail, it is necessary to study
the Gaillard-Zumino condition for interacting Carrollian theories [6], since
we have not only the electromagnetic fields. Besides, it is also important to
discuss the sourced Carrollian electromagnetic theories to establish the full
picture of Carrollian electromagnetic duality. We leave these issues to future
work.

## Acknowledgments

We are grateful to Roberto Tateo for some helpful discussions. We would like
to thank Yu-fan Zheng for discussions about Carrollian electromagnetic duality
and Yunfeng Jiang, Miao He and Yi-Jun He for discussions about
$\sqrt{T\bar{T}}$. We also thank Zezhou Hu, Yunsong Wei for other valuable
discussions. This research is supported by NSFC Grant No. 11735001, 12275004.
Jue Hou is also supported by the Jiangsu Funding Program for Excellent
Postdoctoral Talent.

##  Appendix A A Criterion of Carrollian conformal symmetry

In this appendix, we discuss the Carrollian version of the Bessel-Hagen
criterion for conformal field theory (CFT) to show that on the flat background
the scale invariant Carrollian NED is also Carrollian conformal invariant if
there exists a proper traceless energy-momentum tensor. We also exhibit how to
obtain the energy-momentum tensor (4.15) of Carrollian NED without a non-
degenerate Riemann metric.

###  A.1 Carrollian Bessel-Hagen criterion

The Bessel-Hagen criterion [85] tells us that in relativistic field theory,
the existence of a traceless symmetric energy-momentum tensor

$T^{\mu}_{\ \mu}=0,\qquad(T^{\mu\nu}=T^{\nu\mu})$ (A.1)

implies the whole conformal symmetry. However, the energy-momentum tensor of a
Carrollian theory cannot be arbitrarily raised or lowered and cannot be made
symmetric. It remains to figure out whether the condition $T^{\mu}_{\ \mu}=0$
implies the whole Carrollian conformal symmetry. We aim to prove that a
Bessel-Hagen-like criterion for Carrollian theory.

The relativistic symmetric energy-momentum tensor $T^{\mu\nu}$ has an
definition through the variation of metric $g_{\mu\nu}$ by

$\delta S=\int d^{n}x\
\delta\left(\sqrt{||g||}\mathcal{L}(\text{matter};g)\right)=\int\sqrt{||g||}d^{n}x\
\frac{1}{2}T^{\mu\nu}\delta g_{\mu\nu}.$ (A.2)

For infinitesimal transformation generated by a vector $\xi^{\mu}$, the metric
changes by

$\delta_{\hat{\xi}}g_{\mu\nu}=(L_{\hat{\xi}}{\hat{g}})_{\mu\nu}=2\nabla_{(\mu}\xi_{\nu)},$
(A.3)

so the action is invariant under all conformal transformations satisfying
$(L_{\xi}g)_{\mu\nu}=\lambda g_{\mu\nu}$,

$\delta S=\int\sqrt{-g}d^{n}x\ T^{\mu\nu}\lambda
g_{\mu\nu}=\lambda\int\sqrt{-g}d^{n}x\ T^{\mu}_{\ \mu}=0,$ (A.4)

provided that the energy-momentum tensor is traceless.

However, we need to be careful when dealing with the indices for Carrollian
field theories. Without raising or lowering any indices, the energy-momentum
tensor refers to such a tensor that for any Killing vector $\xi^{\mu}$, the
action varies as

$\delta S=\int\mathcal{E}d^{n}x\ T^{\mu}_{\ \nu}\nabla_{\mu}\xi^{\nu}.$ (A.5)

Here we use $\mathcal{E}$ for $\sqrt{||g_{\tau}||}$ in the Carrollian volume
form (3.10). The $T^{\mu}_{\ \nu}$ can be chosen to be the canonical energy-
momentum tensor in the Noether process, but it is not symmetric. Another way
to derive it is by relating it to the variation of the pair
$(n^{\mu},h_{\mu\nu})$. Some authors prefer to do the variation of the
Carrollian “metric” with regard to pair $(\tau_{\mu},h_{\mu\nu})$ or
$(n^{\mu},\gamma^{\mu\nu})$ [47], but we do the variation using exactly the
Carrollian structure data $(n^{\mu},h_{\mu\nu})$

$\delta S=\int d^{n}x\
\delta\left(\mathcal{E}\mathcal{L}(\text{matter};n^{\mu},h_{\mu\nu})\right)=\int\mathcal{E}d^{n}x\
\left(-{\mathcal{T}_{n}}_{\mu}\delta
n^{\mu}+\frac{1}{2}{\mathcal{T}_{h}}^{\mu\nu}\delta h_{\mu\nu}\right).$ (A.6)

such that

$T^{\mu}_{\
\nu}\equiv-\left({\mathcal{T}_{n}}_{\nu}n^{\mu}+{\mathcal{T}_{h}}^{\mu\rho}h_{\rho\nu}\right).$
(A.7)

The ${\mathcal{T}_{h}}^{\mu\nu}$ is symmetric and determined up to an
arbitrary term proportional to $n^{\mu}n^{\nu}$. It should be pointed out here
that the variation by $n_{\mu}$ and $h_{\mu\nu}$ are not independent because
$n^{\mu}h_{\mu\nu}=0$, so ${\mathcal{T}_{n}}_{\mu}$ and
${\mathcal{T}_{h}}^{\mu\nu}$ are not uniquely determined. Actually we have
$h_{\mu\nu}\delta n^{\mu}+n^{\mu}\delta h_{\mu\nu}=0$, so the following
transformation gives equivalent energy-momentum tensors for any $a^{\mu}$:

$\left\\{\begin{aligned}
&{\mathcal{T}_{n}}_{\mu}\to{\mathcal{T}_{n}}_{\mu}+a^{\mu}h_{\mu\nu},\\\
&{\mathcal{T}_{h}}^{\mu\nu}\to{\mathcal{T}_{h}}^{\mu\nu}+2n^{(\mu}a^{\nu)}.\\\
\end{aligned}\right.$ (A.8)

Then for Carrollian conformal Killing vector $\xi^{\mu}$, we have

$(L_{\hat{\xi}}{\hat{n}})^{\mu}=-\alpha
n^{\mu},\qquad(L_{\hat{\xi}}{\hat{h}})_{\mu\nu}=2\alpha h_{\mu\nu}.$ (A.9)

And we also have

$(L_{\hat{\xi}}{\hat{n}})^{\mu}=\xi^{\nu}\nabla_{\nu}n^{\mu}-n^{\nu}\nabla_{\nu}\xi^{\mu},\qquad(L_{\hat{\xi}}{\hat{h}})_{\mu\nu}=2\nabla_{(\mu}\xi_{\nu)}.$
(A.10)

Here $\nabla_{\mu}$ is any torsionless connection adapted to Carrollian
structure $(n^{\mu},h_{\mu\nu})$ with vanishing extrinsic curvature
$K_{\mu\nu}=\frac{1}{2}(L_{\hat{n}}{\hat{h}})_{\mu\nu}$,

$\nabla_{\rho}n^{\mu}=\nabla_{\rho}h_{\mu\nu}=0.$ (A.11)

Here $\xi_{\mu}$ is lowered by $h_{\mu\nu}$. Then the variation of the action
is

$\displaystyle\delta_{\xi}S$ $\displaystyle=\int\mathcal{E}d^{n}x\
\left({\mathcal{T}_{n}}_{\nu}(L_{\hat{\xi}}{\hat{n}})^{\nu}+\frac{1}{2}{\mathcal{T}_{h}}^{\mu\nu}(L_{\hat{\xi}}{\hat{h}})_{\mu\nu}\right).$
(A.12) $\displaystyle=\int\mathcal{E}d^{n}x\
\left(\left({\mathcal{T}_{n}}_{\nu}n^{\mu}+{\mathcal{T}_{h}}^{\mu\rho}h_{\rho\nu}\right)\nabla_{\mu}\xi^{\nu}+{\mathcal{T}_{n}}_{\nu}\xi^{\mu}\nabla_{\mu}n^{\nu}\right)$
$\displaystyle=\int\mathcal{E}d^{n}x\ T^{\mu}_{\ \nu}\nabla_{\mu}\xi^{\nu}+0.$

In this way one can prove that the tensor defined in (A.7) satisfies the
definition of an energy-momentum tensor. Actually, the energy-momentum tensor
(4.15),(5.5) can be given in this way.

We require the background Carrollian metric to have vanishing extrinsic
curvature because only in this case the torsionless adapted connection
$\nabla_{\mu}$ exist. The flat Carrollian metric trivially satisfies this
requirement, as well as any magnetic-type Carrollian gravity solutions [68].
When the extrinsic curvature $K_{\mu\nu}$ does not vanish, we can only find
torsional connections such that (A.10) will involve $K_{\mu\nu}$. Such
torsional connections invalidate the arguments here.

Now substituting (A.9) into (A.6) we get for Carrollian conformal Killing
vectors,

$\displaystyle\delta_{\xi}S$ $\displaystyle=\alpha\int\mathcal{E}d^{n}x\
\left({\mathcal{T}_{n}}_{\mu}n^{\mu}+{\mathcal{T}_{h}}^{\mu\nu}h_{\mu\nu}\right).$
(A.13) $\displaystyle=\alpha\int\mathcal{E}d^{n}x\ T^{\mu}_{\ \mu}.$

In this way, we find the Bessel-Hagen-like criterion of Carrollian conformal
invariance that a Carrollian field theory is Carrollian conformal invariant if
the energy-momentum tensor given by variation of metric data (A.7) is
traceless $T^{\mu}_{\ \mu}=0$.

###  A.2 Energy-momentum tensor of Carrollian NED

In this section, we show how to use the method in the last section to find the
energy-momentum tensor of scale invariant Carrollian NED as (4.15). We should
do the variation of the Lagrangian of Carrollian NED according to the
Carrollian structure. We need to use the following identities:

$\displaystyle\delta\mathcal{E}$
$\displaystyle=\frac{1}{2}\mathcal{E}{g_{\gamma}}^{\mu\nu}\delta{(g_{\tau})}_{\mu\nu}=\frac{1}{2}\mathcal{E}\left(2n^{\mu}\delta\tau_{\mu}+\gamma^{\mu\nu}\delta
h_{\mu\nu}\right)$ (A.14)
$\displaystyle\simeq\mathcal{E}\left(-\tau_{\mu}\delta
n^{\mu}+\frac{1}{2}\gamma^{\mu\nu}\delta h_{\mu\nu}\right),$
$\displaystyle\delta{\mathcal{S}}$
$\displaystyle=\delta\left(m^{\mu\rho}\gamma^{\nu\sigma}F_{\mu\nu}F_{\rho\sigma}\right)=\frac{1}{2}\delta\left(h_{\mu\rho}h_{\nu\sigma}\widetilde{F}^{\mu\nu}\widetilde{F}^{\rho\sigma}\right)$
$\displaystyle\simeq-2\mathcal{S}\left(-\tau_{\mu}\delta
n^{\mu}+\frac{1}{2}\gamma^{\mu\nu}\delta
h_{\mu\nu}\right)+\left(h_{\nu\sigma}\widetilde{F}^{\mu\nu}\widetilde{F}^{\rho\sigma}\right)\delta
h_{\mu\rho},$ $\displaystyle\delta(\mathcal{E}\mathcal{P})$
$\displaystyle=\mathcal{P}\delta\mathcal{E}+\mathcal{E}\delta\mathcal{P}=0,\qquad\text{($\mathcal{P}$
is topological)}.$

Here $\simeq$ means equal up to the term proportional to $n^{\mu}n^{\nu}$.

Moreover, for general $\mathcal{L}=\mathcal{L}(\mathcal{S},\mathcal{P})$,
noticing that

$\mathcal{P}\delta^{\mu}_{\
\nu}=-2\widetilde{F}^{\mu\rho}F_{\rho\nu},\qquad\mathcal{S}\delta^{\mu}_{\
\nu}=-F^{\mu\rho}F_{\rho\nu}-\widetilde{F}^{\mu\rho}\widetilde{F}_{\rho\nu},$
(A.15)

we have

$\delta S=\int d^{4}x\
\mathcal{L}\delta\mathcal{E}+\mathcal{E}\mathcal{L}_{\mathcal{S}}\delta\mathcal{S}-\mathcal{P}\mathcal{L}_{\mathcal{P}}\delta\mathcal{E}.$
(A.16)

So the components of the energy-momentum tensor are

$\displaystyle{\mathcal{T}_{n}}_{\mu}$
$\displaystyle=(2\mathcal{S}\mathcal{L}_{\mathcal{S}}-\mathcal{L}+\mathcal{P}\mathcal{L}_{\mathcal{P}})\tau_{\mu},$
(A.17) $\displaystyle{\mathcal{T}_{h}}^{\mu\nu}$
$\displaystyle=-(2\mathcal{S}\mathcal{L}_{\mathcal{S}}-\mathcal{L}+\mathcal{P}\mathcal{L}_{\mathcal{P}})\gamma^{\mu\nu}+2\mathcal{L}_{\mathcal{S}}h_{\rho\sigma}\widetilde{F}^{\mu\rho}\widetilde{F}^{\nu\sigma}.$
$\displaystyle-T^{\mu}_{\ \nu}$
$\displaystyle=-(2\mathcal{S}\mathcal{L}_{\mathcal{S}}-\mathcal{L}+\mathcal{P}\mathcal{L}_{\mathcal{P}})\left(\gamma^{\mu\rho}h_{\rho\nu}+n^{\mu}\tau_{\nu}\right)+2\mathcal{L}_{\mathcal{S}}\widetilde{F}^{\mu\rho}\widetilde{F}_{\nu\rho}$
$\displaystyle=\mathcal{L}\delta^{\mu}_{\
\nu}-\left(2\mathcal{S}\mathcal{L}_{\mathcal{S}}+\mathcal{P}\mathcal{L}_{\mathcal{P}}\right)\delta^{\mu}_{\
\nu}-2\mathcal{L}_{\mathcal{S}}\widetilde{F}^{\mu\rho}\widetilde{F}_{\rho\nu}$
$\displaystyle=\mathcal{L}\delta^{\mu}_{\
\nu}+\left(2\mathcal{L}_{\mathcal{S}}F^{\mu\rho}+2\mathcal{L}_{\mathcal{P}}\widetilde{F}^{\mu\rho}\right)F_{\rho\nu}$
$\displaystyle=\widetilde{G}^{\mu\rho}F_{\rho\nu}+\mathcal{L}\delta^{\mu}_{\
\nu}.$

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