ModMax meets GCA

Aritra<EMAIL_ADDRESS>Aditya
<EMAIL_ADDRESS>

a Okinawa Institute of Science and Technology,
1919-1 Tancha, Onna-son, Okinawa 904-0495, JAPAN

bDepartment of Physics, BITS-Pilani, K K Birla Goa Campus,
Zuarinagar, Goa-403726, INDIA

cSchool of Mathematics and Maxwell Institute for Mathematical Sciences,
University of Edinburgh, Peter Guthrie Tait Road,
Edinburgh EH9 3FD, UK

###### Abstract

A maximally symmetric non-linear extension of Maxwell’s theory in four
dimensions called ModMax has been recently introduced in the literature. This
theory preserves both electromagnetic duality and conformal invariance of the
linear theory. In this short paper, we introduce a Galilean cousin of the
ModMax theory, written in a covariant formalism, that is explicitly shown to
be invariant under Galilean Conformal Symmetries. We discuss the construction
of such a theory involving Galilean electromagnetic invariants, and show how
the classical structure of the theory is invariant under the action of
Galilean Conformal Algebra (GCA).

###### Contents

  1. 1 Introduction
  2. 2 ModMax as Conformal non-linear Electrodynamics
  3. 3 Covariant Galilean Conformal Electrodynamics
    1. 3.1 Galilean geometry
    2. 3.2 Galilean Isometries
    3. 3.3 Covariant Lagrangian and transformation laws
    4. 3.4 Electric and Magnetic invariants
  4. 4 Galilean Conformal ModMax-like Lagrangian
    1. 4.1 Symmetries of the Lagrangian
    2. 4.2 Equations of motion and Gauge invariance
    3. 4.3 Energy-momentum tensors
  5. 5 Discussions and conclusions

##  1 Introduction

Maxwell’s electrodynamics in $4d$ is special, in the sense it has two very
important symmetries, namely the four-dimensional conformal symmetry and
electric-magnetic duality. Maxwell theory is certainly the most successful and
well known Gauge Theory of $U(1)$ fields since its introduction one and half
century ago. Most notably Maxwell’s equations derived from this theory are
linear in the field strength $F_{\mu\nu}$. It has been widely accepted that a
generic quantum theory of Electrodynamics should have higher order corrections
to the linear terms in $F_{\mu\nu}$ in the Lagrangian, arising from loop
contributions [1], which reduces to the pure Maxwell term in the low energy
limit. This gave rise to the question of whether there exists other classical
Lagrangians for $U(1)$ gauge fields which are already non-linear in the field
strength and gives rise to Maxwell theory in an effective description. These
theories may capture new physics at different energy scales where the full
non-linear theory has to be taken care of, leading to corrections to known
results. Thus materialised the studies of Non-Linear Electrodynamics (NLED),
which has been going strong for more than a century already.

Notable examples of NLEDs include the famous Born-Infeld theory [2], a crucial
component of String Theory in the study of D-branes [3], that makes sure to
keep self energy of point particles finite. Other well studied examples
include actions involving various functionals of the field tensor and a nice
review for these constructions can be found in [4] and references therein. But
the main caveat lies in the problem that a generic NLED in $4d$ is not
conformally invariant, neither is it invariant under Hodge duality rotations.
So the question people have been asking for decades, reads, is it possible to
write down a Lorentz invariant non-linear theory of Electrodynamics that
preserves the symmetries of Maxwell Lagrangian? As far as conformal invariance
is concerned, it has been shown that as long as the lagrangian is a
homogeneous function of degree one of Maxwell Lorentz invariants, the theory
remains invariant. However, the requirement of electromagnetic duality
invariance takes a more involved form, as shown first by Bialynicki-Birula
[5]. Reconciling the two conditions seemed to be an involved problem for
decades, for example, Born-Infeld theory gives rise to manifestly duality
invariant equations, but it is not conformal invariant.

Only recently, the question we posed have been completely answered by Bandos,
Lechner, Sorokin, and Townsend [6] (see also [7]) who proposed a
simultaneously duality-invariant and conformal theory of $U(1)$ fields, that
reduces to Maxwell theory in a zero coupling limit. This theory has been
generically called Modified Maxwell theory or by the nickname “ModMax”. This
has generated considerable interest in last couple years and have been shown
to have many interesting properties as seen in the classical solutions [8],
hamiltonian formulation [9, 10], coupling to charged particles [11],
supersymmetric and other generalizations [12, 13, 14, 15] and connection to
black hole solutions [16, 17, 18, 19, 20]. It has also been shown that ModMax
theories can be generated by a $T\overline{T}$-like (or $\sqrt{T\overline{T}}$
like) deformation of Maxwell theories [21, 22, 23, 24], and a string theory
context for ModMax has been introduced in [25]. This list of course does not
do justice to the literature, and readers are directed to references and
citations of these papers. Certainly, the general symmetry structure of ModMax
is intriguing on its own and will be part of various studies in the near
future.

Our goal in this note, however, is to meander from the well traversed paths
and try something very new. We would like to focus on the conformal nature of
the ModMax theory, and would like to see what happens when one looks at these
symmetries going away from the relativistic situation. Specifically, we set
out trying to write an analogue Galilean Covariant nonlinear Lagrangian which
is invariant under the $4d$ Galilean Conformal Algebra (GCA). These symmetries
arise when the we take a non-relativistic (speed of light going to infinity)
limit on the $d$ dimensional conformal algebra. At each and every dimension,
this limit results in an infinite-dimensional Galilean conformal algebra [26],
in contrast to their relativistic cousins which are only infinite dimensional
in $2d$ [27]. Galilean electrodynamics has been studied for a long time,
starting as early as with Le Ballac and Levy-Leblond [28]. In recent years
theories of Galilean Electrodynamics have generated newfound interest due to
the larger and rich symmetry structures associated to it, and in [29] a
reformulation of Galilean Conformal Electrodynamics in various dimensions was
introduced via taking non-relativistic (NR) limits on the equations of motion
of the Maxwell theory, and has subsequently been developed in a bunch of works
[30, 31, 32, 33, 34, 35, 36, 37]. A caveat for taking such a limit is loss of
manifest electric-magnetic duality, as the physics splits in two sub-sectors,
where either the electric or the magnetic components of the gauge field
$A_{\mu}$ dominates. The other problem is, the procedure of taking NR limits
does not work accurately on the action formalism and hence a proper covariant
Galilean electrodynamic action is hard to write down.

The search for a covariant Galilean Electrodynamics theory needs to be
addressed by putting the gauge fields explicitly on a Non-Lorentzian manifold,
in this case a Newton-Cartan manifold. Newton-Cartan structures arise when we
take the speed of light to infinity and the usual Riemannian notion of a
manifold degenerates, paving the way for Galilean Relativity [38, 39, 30].
These manifolds in general have a fibre bundle structure that keeps temporal
and spatial diffeomorphisms separate from each other. An attempt to write down
a Galilean Covariant electrodynamics Lagrangian was recently made in [40], one
which simultaneously describes electric and magnetic dominated realms of the
theory. This Galilean Maxwell Lagrangian will be the building block of our
current work, and since that Lagrangian is manifestly invariant under the $4d$
GCA, we will try to introduce an explicit non-linear Covariant Lagrangian with
a ModMax-like form, consequently showing the GCA invariance for the same as
well. This will be the first instance of a nonlinear covariant Galilean
Conformal Electrodynamics (NLGCED) Lagrangian in the literature, to the best
of our knowledge.

The rest of this short paper is organised in the following way: in section 2
we will briefly review the structure of ModMax electrodynamics. In section 3
we will revisit Newton-Cartan structures and construction of a gauge field
Lagrangian on such a structure. Here we will slightly differ from the approach
of [40], and instead focus on the transformation of components of gauge fields
under Galilean Conformal symmetries. We will also discuss in detail the
structure of Galilean invariant field bilinear that will be important to our
construction. Then in section 4 we will go ahead and present our Galilean
ModMax-like Lagrangian and show its explicit invariance under GCA symmetries,
giving the generic form of ModMax Lagrangians (with square roots) an aura of
universality when it comes to conformal invariance both in and beyond
Lorentzian cases. In section 5 we will have further discussions and talk about
probable future extensions.

##  2 ModMax as Conformal non-linear Electrodynamics

As mentioned earlier, source-free ModMax theory in $4d$ Minkowski spacetime is
both conformally invariant and E-M duality invariant. In this section we will
briefly revisit some aspects of the ModMax Lagrangian, which will be later
crucial for the discussion of the analogous Galilean theory.

### Lagrangian and symmetries of the Relativistic ModMax

In $4d$, generic electrodynamics Lagrangians can only be functions of the
field strength tensor, and does not include the derivatives of those. For $4d$
Maxwell electrodynamics, there are just two Lorentz invariant quantities, i.e.

$S=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}=\frac{1}{2}\left(\bf{E}^{2}-\bf{B}^{2}\right),~{}~{}P=-\frac{1}{4}F^{\mu\nu}\tilde{F}_{\mu\nu}=\bf{E}\cdot\bf{B}.$
(1)

Here the first one is a Lorentz scalar and the second one is pseudo-scalar,
where the field strength and the Hodge dual field strength is defined as

$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu},~{}~{}~{}\tilde{F}_{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}F^{\rho\sigma},$
(2)

where $A^{\mu}$ is an $U(1)$ gauge field. A generic theory of Electrodynamics
in this dimension will be an analytic function of these Lorentz invariants and
will reduce to the pure Maxwell term in some (often weak field) limit.

As shown in [6], the unique one-parameter deformed family of Lorentz invariant
modifications of the Maxwell Lagrangian which stays conformal invariant and
produces E-M duality invariant equations of motion is the ModMax theory. The
Lagrangian density for this theory can be written down as

$\displaystyle\mathcal{L}_{MM}$ $\displaystyle=$
$\displaystyle\cosh\gamma~{}S+\sinh\gamma~{}\sqrt{S^{2}+P^{2}}$
$\displaystyle=$
$\displaystyle-\frac{\text{cosh}\gamma}{4}\big{[}F^{\mu\nu}F_{\mu\nu}\big{]}+\frac{\text{sinh}\gamma}{4}\sqrt{(F^{\mu\nu}F_{\mu\nu})^{2}+(\tilde{F}^{\mu\nu}F_{\mu\nu})^{2}}.$

Here $\gamma$ is the dimensionless parameter that controls the deformation,
and at $\gamma=0$ the theory reduces down to pure Maxwell, with $\gamma>0$
having a well defined solution space. The conditions posed by causality and
unitarity demand the coupling constant to be non-negative $\gamma\geq 0$ [6].
The square root term can be thought of as a deformation to Maxwell theory, and
in general is proportional to $\left(\bf{E}^{2}+\bf{B}^{2}\right)$, i.e. the
Hamiltonian of the pure Maxwell theory. The term under square root can also be
rewritten as $\mathcal{F}^{\mu\nu}\mathcal{\bar{F}}_{\mu\nu}$, where
$\mathcal{F}^{\mu\nu}=F^{\mu\nu}+i\tilde{F}^{\mu\nu}$, and the bar denotes
complex conjugation. This structure of the deformation turns out to be crucial
to show invariance of the equations of motion under conventional
Electromagnetic duality as well.

The unique nature of the ModMax Lagrangian makes sure of conformal invariance
as it is a homogeneous function of degree one under scale transformations
[10]. One can see this symmetry via explicitly computing the stress-energy
tensors of this theory, which turn out to be traceless and explicitly
proportional to their Maxwell cousins.

### Equations of motion

The equations of motion of the Lagrangian (2) is given by

$\displaystyle\partial_{\mu}\Bigg{[}(\text{cosh}\gamma)F^{\mu\nu}-\text{sinh}\gamma\Bigg{(}\frac{(F^{\alpha\beta}F_{\alpha\beta})F^{\mu\nu}+(\tilde{F}^{\alpha\beta}F_{\alpha\beta})\tilde{F}^{\mu\nu}}{\sqrt{(F^{\alpha\beta}F_{\alpha\beta})^{2}+(\tilde{F}^{\alpha\beta}F_{\alpha\beta})^{2}}}\Bigg{)}\Bigg{]}=0,$
(4)

which again boils down to the standard equation $dF=0$ when we put $\gamma=0$.
One can note that these equations of motion for generic $\gamma$ are ill-
defined for solutions having null electromagnetic fields, i.e.
$FF=F\tilde{F}=0$, like in the case of Electromagnetic waves. It has been
shown that these pathologies can be cured if one instead works in the
Hamiltonian formalism [6].

The duality invariance associated to these equations, although not so
important for our discussion here, is given by the so called Galliard-Zumino
duality conditions [41] for non-linear Electrodynamics,

$\tilde{F}^{\mu\nu}F_{\mu\nu}=\tilde{E}^{\mu\nu}E_{\mu\nu},$ (5)

which replaces the usual rotations of field strengths well-known for the
source free Maxwell equations. Here the excitation field strength is given by
$E_{\mu\nu}=\frac{\partial\mathcal{L}_{MM}}{\partial F^{\mu\nu}}$ and the
hodge dual of that is defined in the usual way. Notice that $E^{\mu\nu}$ turns
out to be just $F^{\mu\nu}$ for a pure Maxwell theory. So it is clear that
equations of motion deduced from general non-linear theories would demand a
symmetry under $U(1)$ rotations of $E^{\mu\nu}$ and $\tilde{F}^{\mu\nu}$.
333One can however write down an action principle for generic non-linear
electrodynamics that also manifestly shows electromagnetic duality symmetry
beyond the equations of motion. See [15] for such a democratic formulation.
One can notice that our equation of motion (4) can be rewritten using the
tensor $E^{\mu\nu}$ in (5) as

$\partial_{\mu}E^{\mu\nu}=0,$ (6)

which makes sure this and the usual Bianchi identity $d\tilde{F}=0$ are a set
of duality invariant non-linear equations of motion for the ModMax theory.

##  3 Covariant Galilean Conformal Electrodynamics

In this section, we will discuss the formalism associated to a Covariant
formulation of Galilean Electrodynamics. As we have introduced before, it is
essential to start with a geometric formulation of tensors on a Newton-Cartan
manifold and put gauge fields explicitly on it to understand the true nature
of Galilean conformal symmetries.

###  3.1 Galilean geometry

In Galilean sense, usual Riemannian metrics are of no use since they are
degenerate and can’t be used to raise or lower indices on objects. At the
limit of $c\to\infty$ the Poincaré group is replaced by the Galilei group, and
the kinematical structure of the group allows one to define a manifold, called
the Newton-Cartan manifold. The main ingredients of an intrinsically Galilean
(or Newton-Cartan) manifold is the degenerate spatial metric $h^{\mu\nu}$ and
a choice of null time direction $\tau_{\mu}$ that gives rise to another two-
index object $\tau_{\mu\nu}=-\tau_{\mu}\tau_{\nu}$ [42, 43, 44, 33]. In $4d$,
the simplest choice to represent these tensors are:

$\displaystyle h^{\mu\nu}=\begin{bmatrix}[r]0&\phantom{-}0\\\
0&\phantom{-}1_{3\times 3}\\\
\end{bmatrix},\quad\tau_{\mu\nu}=\begin{bmatrix}[r]-1&\phantom{-}0\\\
0&\phantom{-}0_{3\times 3}\\\ \end{bmatrix}.$ (7)

These two non-invertible Galilean tensors are used to define contravariant and
covariant Galilean vectors and the nowhere vanishing time-like vector field
$\tau_{\mu}$ is given by,

$\displaystyle\tau_{\mu}=\begin{bmatrix}[r]1&0&0&0\end{bmatrix}~{}~{}~{}~{}\tau_{\mu}\tau^{\mu}=1.$
(8)

These two geometric ingredients are orthogonal in the sense
$h^{\mu\nu}\tau_{\nu}=0$. Both these objects remain invariant under Galilean
boosts and rotation. We can remind the reader that time is absolute in
Galilean relativity, which is inherent in these invariant structures. For the
$(h,\tau)$ duo, which defines a gauge choice (i.e. the form of the tensors in
(7)) for a Newton-Cartan spacetime, the respective covariant and contravariant
objects are given by projective inverses of $(h^{\mu\nu},\tau_{\nu})$, and are
given by,

$\displaystyle h_{\mu\nu}=\begin{bmatrix}[r]a&\phantom{-}b_{i}\\\
b_{i}&1_{3\times
3}\end{bmatrix},\quad\tau^{\mu\nu}=\tau^{\mu}\tau^{\nu}\;\;\text{where}\;\tau^{\mu}=(1,c_{1},c_{2},c_{3}).$
(9)

These expressions follow from the definition of the Galilean tensors using
projective inverse definitions given by $\tau^{\mu}\tau_{\mu}=1$ and by
$h^{\mu\alpha}h_{\alpha\beta}h^{\beta\nu}=h^{\mu\nu}$. One should note that
these projective inverses are not generally Galilean invariants for all
choices of the constants $(a,b_{i},c_{i})$.

These tensors are crucial in defining Galilean objects in the theory, i.e. a
covariant vector $\tilde{K}_{\mu}$ will be defined from the knowledge of a
contravariant vector $K^{\mu}$ via the operation
$\tilde{K}_{\mu}=\tau_{\mu\nu}K^{\nu}$, and an opposite operation
$(K_{\mu}\to\tilde{K}^{\mu})$ will be done via $h^{\mu\nu}$. Contrary to
relativistic tensors, these operations are not invertible since temporal and
spatial components are split from each other due to the structure of
$(h^{\mu\nu},\tau_{\nu})$. Similarly, covariant derivatives are defined as
$\partial_{\mu}=(\partial_{t},\partial_{x},\partial_{y},\partial_{z})$, while
the associated contravariant object reads
$\partial^{\mu}=h^{\mu\nu}\partial_{\nu}=(0,\partial_{x},\partial_{y},\partial_{z})$.
We will be using these Galilean derivatives throughout the rest of this work,
and they are not to be confused with usual derivatives used in section 2 for
example.

###  3.2 Galilean Isometries

As mentioned in the introduction, GCA is a Galilean or Non-Relativistic
contraction of the relativistic conformal algebra. Equivalently to the
intrinsic description in the last section, Galilean Conformal objects can be
realised by taking the following contraction of coordinates on associated
conformal theories [26]:

$\displaystyle x_{i}\rightarrow\epsilon x_{i},t\rightarrow
t,\epsilon\rightarrow 0.$ (10)

The above turns out to be equivalent to taking a $c\to\infty$ scaling.
Remember that in $4d$ the conformal algebra is a finite dimensional algebra,
and hence to start with we only get the finite part of the GCA (fGCA) when the
above mentioned contraction is acted upon. This finite algebra is generated by
rotations ($J_{ij}$), spacetime translations ($H$ and $P_{i}$), boosts
($B_{i}$), scaling ($D$) and special conformal transformations ($K$ and
$K_{i}$). The vector fields associated to these generators are given by:

$\displaystyle J_{ij}=-(x_{i}\partial_{j}-x_{j}\partial_{i}),\qquad
P_{i}=\partial_{i},\qquad H=-\partial_{t},\qquad B_{i}=t\partial_{i}$ (11)
$\displaystyle D=-(t\partial_{t}+x^{i}\partial_{i}),\qquad
K=-(2tx^{i}\partial_{i}+t^{2}\partial_{t}),\qquad K_{i}=t^{2}\partial_{i}.$
(12)

The $i,j$ indices all correspond to purely spatial components in the above.
Consider an extension of the generators in an $n$ dependent form

$L^{(n)}=-(n+1)t^{n}x^{i}\partial_{i}-t^{n+1}\partial_{t},\qquad
M_{i}^{(n)}=t^{n+1}\partial_{i},$ (13)

where for $n=0,\pm 1$, the generators $L^{(n)},M^{(n)}_{i}$ denotes the set of
finite GCA generators

$\displaystyle
L^{(-1,0,1)}=\\{H,D,K\\},~{}~{}M_{i}^{(-1,0,1)}=\\{P_{i},B_{i},K_{i}\\},$ (14)

but in principle any value of $n$ is admissible, hence giving rise to
generators spanning an infinite dimensional vector space. The rotation
generators could also be given an infinite dimensional lift as follows:

$J_{ij}^{(n)}=-t^{n}(x_{i}\partial_{j}-x_{j}\partial_{i}).$ (15)

Armed with these new generators the full infinite dimensional extended GCA can
be written in the following form:

$\displaystyle\left[L^{(n)},L^{(m)}\right]=(n-m)L^{(n+m)},~{}\left[L^{(n)},M_{i}^{(m)}\right]=(n-m)M_{i}^{(n+m)},\left[M_{i}^{(n)},M_{j}^{(m)}\right]=0,$
$\displaystyle\left[L^{(n)},J_{ij}^{(m)}\right]=-mJ_{ij}^{(n+m)},~{}\left[J_{ij}^{(n)},M_{r}^{(m)}\right]=-(M_{i}^{(n+m)}\delta_{jr}-M_{j}^{(n+m)}\delta_{ir}),$
$\displaystyle\left[J_{ij}^{(n)},J_{rs}^{(m)}\right]=\delta_{is}J_{rj}^{(n+m)}+\delta_{jr}J_{si}^{(n+m)}+\delta_{ir}J_{js}^{(n+m)}+\delta_{js}J_{ir}^{(n+m)}.$
(16)

In the rest of the paper, we will be talking about theories which are
manifestly invariant under transformations induced by this algebra.

###  3.3 Covariant Lagrangian and transformation laws

Conventionally, Galilean electrodynamics, and more specifically the conformal
cousin of it, has been studied in the literature from an equation of motion
point of view [28, 29]. This hinges on the fact that there can be two distinct
limits of relativistic Electrodynamics, known as the Electric and Magnetic
ones, that may correspond to a theory of Galilean Electrodynamics. In the
electric limit, the time-like components of the gauge field $A^{\mu}$ dominate
(i.e. $E_{i}\gg B_{i}$), while in the magnetic case the space-like components
of the same dominate (i.e. $B_{i}\gg E_{i}$). For a theory with sources, one
could take the same limits on the currents to write electric and magnetic
equations of motion. To this effect, in [40] a composite albeit covariant
Lagrangian was introduced, which consistently reproduces both electric and
magnetic equations of motion.

The source free action for the Galilean Covariant Lagrangian can be written
as:

$\displaystyle
S(a_{\mu},a^{\mu},\partial_{\mu}a_{\nu},\partial^{\mu}a^{\nu})=\int
d^{3}x\,dt\,\Big{[}-\frac{1}{4}f_{\mu\nu}f^{\mu\nu}\Big{]},$ (17)

where the contravariant and covariant field strengths are given in terms of
Galilean gauge field $a^{\mu}$ and $a_{\mu}$. These fields here are distinct
objects due to the structure of Galilean tensors, and the respective field
strengths read:

$f^{\mu\nu}=(\partial^{\mu}a^{\nu}-\partial^{\nu}a^{\mu}),~{}~{}f_{\mu\nu}=(\partial_{\mu}a_{\nu}-\partial_{\nu}a_{\mu}).$
(18)

So there are two distinct equations of motion, obtained by varying the above
action w.r.t. one of the two kinds of gauge fields, which do not depend on
each other. One can see while variation w.r.t. $a^{\mu}$ gives rise to,

$\partial_{\nu}f^{\mu\nu}=0,~{}$ (19)

which are the equations of motion in the Electric limit, on the other hand,
variation w.r.t $a_{\mu}$ leads one to

$\partial^{\nu}f_{\mu\nu}=0,$ (20)

generating the magnetic equations of motion 444In component form, the Electric
and Magnetic equations can be written as
$\displaystyle\partial_{i}\partial^{i}a^{0}=0,~{}\partial^{j}(\partial_{t}a^{0}+\partial_{i}a^{i})=(\partial_{i}\partial^{i})a^{j}~{}~{}~{}\text{(Electric)}$
(21)
$\displaystyle\partial_{t}\partial_{i}a_{i}=\partial_{i}\partial^{i}a_{0},~{}\partial^{j}\partial_{i}a_{i}=\partial_{i}\partial^{i}a_{j}~{}~{}~{}\text{(Magnetic)}.$
(see [29] for details). So evidently, the contravariant gauge fields are
responsible for the electric limit of the theory, while covariant ones are
responsible for the magnetic limit of the theory555Covariant gauge fields are
defined by $a^{\mu}\tau_{\mu}=0$ and contravariant ones are defined by
$a_{\mu}h^{\mu\nu}=0$, these belong to invariant vector spaces under the
action of the Galilean group..

We will now write down the transformation of both kinds of the gauge fields
under GCA666To get a better understanding of the representation theory, we
urge the reader to look at [29, 40]., which will be important in the two
separate limits. The covariant formalism used here ensures that the theory is
invariant under Galilean boosts, rotations and translations. For gauge fields
in magnetic limit, the transformation laws for the covariant fields under
rotations, Galilean boost, scale transformations and the special conformal
transformations (SCT) take the following form: Rotations:

$\displaystyle\delta_{J}a_{0}=(x^{i}\partial^{j}-x^{j}\partial^{i})a_{0},~{}~{}\delta_{J}a_{k}=(x^{i}\partial^{j}-x^{j}\partial^{i})a_{k}+(\delta^{i}_{k}a_{j}-\delta^{j}_{k}a_{i}).$
(22)

Galilean boosts:

$\displaystyle\delta_{B_{m}}a_{0}=-t\partial_{m}a_{0}-a_{m},~{}~{}\delta_{B_{m}}a_{i}=-t\partial_{m}a_{i}.$
(23)

Scale transformations:

$\displaystyle\delta_{D}(a_{0},a_{i})=(t\partial_{t}+x^{l}\partial_{l}+1)(a_{0},a_{i}),$
(24)

SCT:

$\displaystyle\delta_{K}a_{0}=(t^{2}\partial_{t}+2tx^{l}\partial_{l}+2t)a_{0}+2x^{l}a_{l},~{}~{}\delta_{K}a_{i}=(t^{2}\partial_{t}+2tx^{l}\partial_{l}+2t)a_{i},$
(25a)
$\displaystyle\delta_{K_{m}}a_{0}=-t^{2}\partial_{m}a_{0}-2ta_{m},~{}~{}\delta_{K_{m}}a_{i}=-t^{2}\partial_{m}a_{i}.$
(25b)

Finally, we will write the variation of gauge fields in magnetic limit under
infinite dimensional GCA generators $(L^{(n)},M^{(n)}_{m})$. They are given by

$\displaystyle\delta_{L^{(n)}}a_{0}=(t^{n+1}\partial_{t}+(n+1)t^{n}x^{l}\partial_{l}+(n+1)t^{n})a_{0}+n(n+1)t^{n-1}x^{l}a_{l},$
(26a)
$\displaystyle\delta_{L^{(n)}}a_{i}=(t^{n+1}\partial_{t}+(n+1)t^{n}x^{l}\partial_{l}+(n+1)t^{n})a_{i},$
(26b)
$\displaystyle\delta_{M^{(n)}_{m}}a_{0}=-t^{n+1}\partial_{m}a_{0}-(n+1)t^{n}a_{m},$
(26c) $\displaystyle\delta_{M^{(n)}_{m}}a_{i}=-t^{n+1}\partial_{m}a_{i}.$
(26d)

Similarly, in the electric limit, we have transformation laws for
contravariant gauge fields:
Rotations:

$\displaystyle\delta_{J}a^{0}=(x^{i}\partial^{j}-x^{j}\partial^{i})a^{0},~{}~{}\delta_{J}a^{k}=(x^{i}\partial^{j}-x^{j}\partial^{i})a^{k}+(\delta^{ki}a^{j}-\delta^{kj}a^{i}).$
(27)

Galilean boosts:

$\displaystyle\delta_{B_{m}}a^{0}=-t\partial_{m}a^{0},~{}~{}\delta_{B_{m}}a^{i}=-t\partial_{m}a^{i}+\delta^{i}_{m}a^{0}.$
(28)

Scale transformations:

$\displaystyle\delta_{D}(a^{0},a^{i})=(t\partial_{t}+x^{l}\partial_{l}+1)(a^{0},a^{i}).$
(29)

SCT:

$\displaystyle\delta_{K}a^{0}=(t^{2}\partial_{t}+2tx^{l}\partial_{l}+2t)a^{0},~{}~{}\delta_{K}a^{i}=(t^{2}\partial_{t}+2tx^{l}\partial_{l}+2t)a^{i}-2x^{i}a^{0},$
(30a)
$\displaystyle\delta_{K_{m}}a^{0}=-t^{2}\partial_{m}a^{0},~{}~{}\delta_{K_{m}}a^{i}=-t^{2}\partial_{m}a^{i}+2t\delta^{i}_{m}a^{0}.$
(30b)

Under $(L^{(n)},M^{(n)}_{m})$, the transformation laws are given by

$\displaystyle\delta_{L^{(n)}}a^{0}=(t^{n+1}\partial_{t}+(n+1)t^{n}x^{l}\partial_{l}+(n+1)t^{n})a^{0},$
(31a)
$\displaystyle\delta_{L^{(n)}}a^{i}=(t^{n+1}\partial_{t}+(n+1)t^{n}x^{l}\partial_{l}+(n+1)t^{n})a^{i}-n(n+1)t^{n-1}x^{i}a^{0},$
(31b) $\displaystyle\delta_{M^{(n)}_{m}}a^{0}=-t^{n+1}\partial_{m}a^{0},$
(31c)
$\displaystyle\delta_{M^{(n)}_{m}}a^{i}=-t^{n+1}\partial_{m}a^{i}+(n+1)t^{n}\delta^{i}_{m}a^{0}.$
(31d)

One can easily see the sheer asymmetry between the transformations of
covariant and contravariant objects in this case, and of course the same shows
up between temporal and spatial components. Using these above transformations,
one could deduce the relevant transformation laws for the electric and
magnetic field strengths as well, and explicitly check the invariance of (17)
under the same.

###  3.4 Electric and Magnetic invariants

We have seen earlier that a relativistic ModMax Lagrangian depends on both
Lorentz invariants in Electrodynamics. Hence for the purpose of this work,
defining Electromagnetic invariants under Galilean transformations are very
important. Now for example, $f^{\mu\nu}$ is clearly an Electric object, since
the gauge fields are contravariant here, similarly $f_{\mu\nu}$ is a magnetic
object for similar reasons. To mark their properties, we call them
$f^{\mu\nu}_{(E)}$ and $f_{\mu\nu}^{(M)}$ from now and hereon.

The obvious invariant quantity is the covariant Lagrangian for Galilean
Maxwell theory, which is a composite of Electric and Magnetic objects,

$\mathcal{L}=-\frac{1}{4}f^{\mu\nu}_{(E)}f_{\mu\nu}^{(M)},$ (32)

i.e. a “true” Lagrangian is one composed of both Electric and Magnetic tensors
with contracted indices, and gives the right Electric or Magnetic EOM when
varied w.r.t one or the other gauge fields. But this is not the Lagrangian
when one takes the relativistic Lagrangian and performs an Electric or a
Magnetic limit. In that case, both field strength components of the Lagrangian
change, i.e. we get either of

$\mathcal{L}^{(E)}=-\frac{1}{4}f^{\mu\nu}_{(E)}f_{\mu\nu}^{(E)},~{}~{}\mathcal{L}^{(M)}=-\frac{1}{4}f^{\mu\nu}_{(M)}f_{\mu\nu}^{(M)},$
(33)

which are only useful in one or the other limits. Here, the inverse field
strengths $f_{\mu\nu}^{(E)}$ and $f^{\mu\nu}_{(M)}$ are not electric or
magnetic objects respectively, not at least by contra/co-variance of the
associated gauge field. But they are electric inverse of the electric field
strength and magnetic inverse of the magnetic field strength, in the same vein
as defining the projective inverses for our Galilean tensors. To connect to
the notation of [40], these are actually defined by the tilde conjugation,
which acts via a Galilean contraction of gauge fields. For example, in the
Electric case,

$\displaystyle
f_{\mu\nu}^{(E)}=\tilde{f}_{\mu\nu}=-f^{\alpha\beta}T_{\mu\alpha\beta\nu}=(\partial_{\mu}\tilde{a}_{\nu}-\partial_{\nu}\tilde{a}_{\mu}),~{}\tilde{a}_{\mu}=a^{\nu}\tau_{\mu\nu},$
(34)

Where the tensor $T$ is defined as a combination of $\tau$ and $h$ to achieve
this,

$\displaystyle
T_{\alpha\beta\mu\nu}:=4\tau_{[\alpha}h_{\beta][\mu}\tau_{\nu]}=(\tau_{\alpha}h_{\beta\mu}\tau_{\nu}-\tau_{\beta}h_{\alpha\mu}\tau_{\nu}-\tau_{\alpha}h_{\beta\nu}\tau_{\mu}+\tau_{\beta}h_{\alpha\nu}\tau_{\mu}).$
(35)

It is easy to see that $T_{\alpha\beta\mu\nu}$ is symmetric under exchange of
$\alpha$ and $\nu$ and of $\beta$ and $\mu$ i.e.
$T_{\alpha\beta\mu\nu}=T_{\nu\mu\beta\alpha}$. Notice that
$T_{\alpha\beta\mu\nu}$ is antisymmetric if we exchange $\alpha$ with $\beta$
or $\mu$ with $\nu$:

$T_{\alpha\beta\mu\nu}=-T_{\beta\alpha\mu\nu}=-T_{\alpha\beta\nu\mu}=T_{\beta\alpha\nu\mu}.$
(36)

From these it also follows that
$T_{\alpha\beta\mu\nu}-T_{\alpha\mu\beta\nu}=T_{\alpha\nu\mu\beta}$. Similarly
the magnetic inverse is given by contraction with only $h$’s,

$f^{\mu\nu}_{(M)}=\tilde{f}^{\mu\nu}=h^{\alpha\mu}f_{\alpha\beta}h^{\beta\nu}=(\partial^{\mu}\tilde{a}^{\nu}-\partial^{\nu}\tilde{a}^{\mu}),~{}~{}\tilde{a}^{\mu}=h^{\mu\nu}a_{\nu}.$
(37)

Note, one of these field strengths are dualised by $\tau_{\mu\nu}$ and the
other by $h^{\mu\nu}$, thereby giving them the notion of an Electric
(temporal) or a Magnetic (spatial) contraction. Hence the Lagrangians (33)
$\mathcal{L}^{(E/M)}$ are not “true” Lagrangians, but limits of relativistic
Lagrangians in the respective regimes where only notion of Electric terms or
Magnetic terms survive. By definition, $\mathcal{L}^{(E/M)}$ both are Galilean
invariants as one can explicitly show, and we will go ahead to define an
additional GCA invariant other than that of $\mathcal{L}$ in (32):

$\mathcal{M}=\frac{1}{\sqrt{2}}\sqrt{\mathcal{L}^{(E)~{}2}+\mathcal{L}^{(M)~{}2}}.$
(38)

This particular quantity will be crucial in our later discussions.

Although we have defined a notion of “dual” field strengths for Galilean
theories using the tilde conjugation, a real Hodge dual in this case is ill-
defined as the associated metric degenerates. Evidently, the notion of EM
duality is lost as the two regimes aren’t simply connected in a Galilean
theory. To the best of our knowledge, the notion of Hodge duals on a Newton-
Cartan manifold is not discussed in the literature as well. However, we can
always go ahead and define the Hodge-dual-like field strength tensor for the
Galilean case in accordance with its relativistic counterpart: $\star
f_{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}f^{\rho\sigma}$, assuming the
definition of Levi-Civita will not change under NR limits. In this sense it
relates true electric and magnetic representations on either sides of the
equality

$\star
f_{\mu\nu}^{(M)}=\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}f^{\rho\sigma}_{(E)}~{}~{}\text{and}~{}~{}\star
f^{\mu\nu}_{(E)}=\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}f_{\rho\sigma}^{(M)}.$
(39)

As we discussed before, a true contracted Galilean object will be a
combination of purely Electric and Magnetic tensors. And based on the
definitions we provided earlier, we can show that

$\mathcal{L}^{*(E)}=-\frac{1}{4}f^{\mu\nu}_{(E)}\star
f_{\mu\nu}^{(E)}~{}~{}\text{or}~{}~{}\mathcal{L}^{*(M)}=-\frac{1}{4}f_{\mu\nu}^{(M)}\star
f^{\mu\nu}_{(M)},$ (40)

are both invariant under the GCA transformations as well. Here we have gone
further to define the Electric and Magnetic inverses of the Hodge dual tensor:

$\star
f_{\mu\nu}^{(E)}=\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}f^{\rho\sigma}_{(M)}=\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}\tilde{f}^{\rho\sigma},~{}~{}~{}\star
f^{\mu\nu}_{(M)}=\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}f_{\rho\sigma}^{(E)}=\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}\tilde{f}_{\rho\sigma}.$
(41)

We should again remind the reader, the objects defined under this star
conjugation are not real Galilean objects, and are only defined in an ad-hoc
basis. We will comment on these invariants later in the paper.

##  4 Galilean Conformal ModMax-like Lagrangian

###  4.1 Symmetries of the Lagrangian

Let us now come to the crux of this paper, that is to construct a non-linear
ModMax-like, albeit GCA invariant, Galilean electrodynamics theory. As in the
case of the Relativistic ModMax theory, we can define a Lagrangian in terms of
the two GCA invariants $\mathcal{L}$ and $\mathcal{M}$ we described in the
last section (see (32) and (38)), with the same general distinctive structure:

$\displaystyle\mathcal{L}_{GMM}$ $\displaystyle=$
$\displaystyle-\frac{\cosh\gamma}{4}~{}\mathcal{L}+\frac{\sinh\gamma}{4}~{}\sqrt{\mathcal{L}^{2}+\mathcal{M}^{2}}$
$\displaystyle=$
$\displaystyle-\frac{\text{cosh}\gamma}{4}\big{[}f^{\mu\nu}f_{\mu\nu}\big{]}+\frac{\text{sinh}\gamma}{4}\sqrt{(f^{\mu\nu}f_{\mu\nu})^{2}+\frac{1}{2}(\tilde{f}^{\mu\nu}f_{\mu\nu})^{2}+\frac{1}{2}(\tilde{f}_{\mu\nu}f^{\mu\nu})^{2}}.$

This simply becomes (17) when we choose $\gamma=0$. In component form, the
Lagrangian can be written down as

$\displaystyle\mathcal{L}_{GMM}=-\frac{\text{cosh}\gamma}{4}E+\frac{\text{sinh}\gamma}{4}\sqrt{C},$
(43)

where written in component form, the quantities read:
$C=[E^{2}+\frac{1}{2}(\tilde{f}^{ij}f_{ij})^{2}+\frac{1}{2}(2\tilde{f}_{i0}(\partial^{i}a^{0}))^{2}]$
and $E=[2f_{i0}(\partial^{i}a^{0})+f^{ij}f_{ij}]$. Let us now move on to the
invariance of this Lagrangian (43) under GCA. The Lagrangian is trivially
invariant under translations and rotations. We will only show the invariance
under the boost, scale transformations and SCT. The change of the Lagrangian
under action of the Boost generator is given by

$\displaystyle\delta_{B}\mathcal{L}_{GMM}=\frac{\text{cosh}\gamma}{4}\partial_{m}[t(2f_{i0}(\partial^{i}a^{0})+f^{ij}f_{ij})]-\frac{\text{sinh}\gamma}{4}\frac{1}{2\sqrt{C}}\partial_{m}(tC)=\partial_{m}(-t\mathcal{L}_{GMM}).$
(44)

We will next check the variation under scale transformation. It is given by

$\displaystyle\delta_{D}\mathcal{L}_{GMM}=\frac{\text{cosh}\gamma}{4}[(t\partial_{t}+x^{m}\partial_{m}+4)E]+\frac{\text{sinh}\gamma}{4}[(t\partial_{t}+x^{m}\partial_{m}+4)\sqrt{C}]$
$\displaystyle\hskip
48.36958pt=(t\partial_{t}+x^{m}\partial_{m}+4)\mathcal{L}_{GMM}=\partial_{t}(t\mathcal{L}_{GMM})+\partial_{m}(x^{m}\mathcal{L}_{GMM}).$
(45)

Finally looking at the change of (43) under Special Conformal Transformations
$K$, we get

$\displaystyle\delta_{K}\mathcal{L}_{GMM}=(t^{2}\partial_{t}+2tx^{m}\partial_{m}+8t)\Big{[}-\frac{\text{cosh}\gamma}{4}E+\frac{\text{sinh}\gamma}{4}\sqrt{C}\Big{]}$
$\displaystyle\hskip
48.36958pt=\partial_{t}(t^{2}\mathcal{L}_{GMM})+\partial_{m}(2tx^{m}\mathcal{L}_{GMM}).$
(46)

We have looked at the invariance of (43) under finite part of GCA. Next step
will be to move on to the infinite extension of GCA. Under the generators
$M^{(n)}_{m}$, we get the variation:

$\displaystyle\delta_{M}\mathcal{L}_{GMM}=\frac{\text{cosh}\gamma}{4}\partial_{m}(t^{n+1}E)-\frac{\text{sinh}\gamma}{4}(t^{n+1}\partial_{m}\sqrt{C})=\partial_{m}(-t^{n+1}\mathcal{L}_{GMM}).$
(47)

Similarly, under $L^{(n)}$, we have,

$\displaystyle\delta_{L}\mathcal{L}_{GMM}=(t^{n+1}\partial_{t}+(n+1)t^{n}x^{m}\partial_{m}+4(n+1)t^{n})\Big{[}-\frac{\text{cosh}\gamma}{4}E+\frac{\text{sinh}\gamma}{4}\sqrt{C}\Big{]}$
$\displaystyle\hskip
48.36958pt=\partial_{t}(t^{n+1}\mathcal{L}_{GMM})+(n+1)\partial_{m}(t^{n}x^{m}\mathcal{L}_{GMM}).$
(48)

In all of these cases, the transformations change the Lagrangian via a total
derivative term, and hence we see that the theory is fully invariant under
extended part of GCA.

Now few comments are in order at this point. Clearly the structure of (4.1),
like the relativistic counterpart, depends on the use of electromagnetic
invariants, which occur directly in the Lagrangian. In the relativistic case,
choice of these invariants are straightforward, however it is evidently not so
simple in the Galilean counterpart, as we have discussed before. It actually
turns out that instead of $\mathcal{M}$ one could choose some other Galilean
invariant as mentioned in 53. Consequently one may go ahead and write down a
test Lagrangian of the form:

$\mathcal{\bar{L}}_{GMM}=-\frac{\text{cosh}\gamma}{4}[f^{\mu\nu}_{(E)}f_{\mu\nu}^{(M)}]+\frac{\text{sinh}\gamma}{4}\sqrt{(f^{\mu\nu}_{(E)}f_{\mu\nu}^{(M)})^{2}+(f^{\mu\nu}_{(E)}\star
f_{\mu\nu}^{(E)})^{2}}$ (49)

or an equivalent one with the $f\star f$ term replaced by the magnetic
version. It can be explicitly shown using methods we discussed in this section
that the above Lagrangian is invariant under GCA symmetries as well. However
as we mentioned earlier, these $f\star f$ terms in the Galilean case are
defined in an ad-hoc way, hence we do not delve into detail discussions on
them. For us, (4.1) will be the master Lagrangian to follow through.

###  4.2 Equations of motion and Gauge invariance

The equations of motion from (4.1) comes out to be twofold, as in the case of
its Maxwellian cousin. Varying the action w.r.t. the contravariant fields
$a^{\mu}$, we get the Electric-like equation of motion:

$\partial_{\mu}\Bigg{[}(\text{cosh}\gamma)f^{\mu\nu}-\text{sinh}\gamma\Bigg{(}\frac{(f^{\alpha\beta}f_{\alpha\beta})f^{\mu\nu}+(\tilde{f}^{\alpha\beta}f_{\alpha\beta})\tilde{f}^{\mu\nu}}{\sqrt{(f^{\alpha\beta}f_{\alpha\beta})^{2}+\frac{1}{2}(\tilde{f}^{\alpha\beta}f_{\alpha\beta})^{2}+\frac{1}{2}(\tilde{f}_{\alpha\beta}f^{\alpha\beta})^{2}}}\Bigg{)}\Bigg{]}=0,$
(50)

while varying w.r.t. covariant gauge fields $a_{\mu}$ gives us the Magnetic-
like equation of motion

$\partial^{\mu}\Bigg{[}(\text{cosh}\gamma)f_{\mu\nu}-\text{sinh}\gamma\Bigg{(}\frac{(f^{\alpha\beta}f_{\alpha\beta})f_{\mu\nu}+(\tilde{f}_{\alpha\beta}f^{\alpha\beta})\tilde{f}_{\mu\nu}}{\sqrt{(f^{\alpha\beta}f_{\alpha\beta})^{2}+\frac{1}{2}(\tilde{f}^{\alpha\beta}f_{\alpha\beta})^{2}+\frac{1}{2}(\tilde{f}_{\alpha\beta}f^{\alpha\beta})^{2}}}\Bigg{)}\Bigg{]}=0.$
(51)

Observe that both equations reduce down to the Galilean Maxwell Electric and
Magnetic equations of motion when we put in $\gamma=0$. Also these are in the
same footing as the equations of motion from Relativistic version of the
ModMax theory. One can look at these equations and note that the electric and
the magnetic sectors are interchanged via the exchange of field tensors
$f^{\mu\nu}\leftrightarrow f_{\mu\nu}$ (which is equivalent to doing $E_{i}\to
B_{i}$ and $B_{i}\to-E_{i}$ in the relativistic case). One can think of this
as a Galilean remnant of the original electromagnetic duality.

We will now look at the gauge transformations (GT) for this theory. The gauge
transformations for $a^{\mu}$ and $a_{\mu}$, that keep the structure of
Lagrangian unchanged, are given by

$\displaystyle a^{\mu}\rightarrow a^{\mu}+\partial^{\mu}\Lambda_{1}\implies
f^{\mu\nu}\rightarrow f^{\mu\nu}$ (52a) $\displaystyle a_{\mu}\rightarrow
a_{\mu}+\partial_{\mu}\Lambda_{2}\implies f_{\mu\nu}\rightarrow f_{\mu\nu}$
(52b)

whereas $(\Lambda_{1,2})$ are two different gauge potentials corresponding to
symmetries in either limit. 777One would use an analogue of the Lorenz gauge
in either limit i.e. $\partial_{\mu}a^{\mu}=0$ or $\partial^{\mu}a_{\mu}=0$,
and this would imply the gauge potentials satisfy a Laplace equation
$\nabla^{2}\Lambda_{1,2}=0$. Similarly, the transformation of the conjugate
field strenghts $\tilde{f}^{\mu\nu}$ and $\tilde{f}_{\mu\nu}$ under (52) are
given by

$\displaystyle\delta_{GT}\tilde{f}^{\alpha\beta}=h^{\mu\alpha}(\delta_{GT}f_{\mu\nu})h^{\nu\beta}=0,~{}~{}\delta_{GT}\tilde{f}_{\alpha\beta}=-(\delta_{GT}f^{\mu\nu})T_{\alpha\mu\nu\beta}=0.$
(53)

We this conclude that the Lagrangian and equations of motion are invariant
under gauge transformations (52).

###  4.3 Energy-momentum tensors

Let us now write down the energy-momentum (EM) tensors in the Electric and
Magnetic limit of Galilean ModMax theory (4.1). We will use the Noether charge
methodology followed in [40] to deduce purely electric or magnetic stress
tensors in either of those limits. In the Electric limit, it is given by

${T_{E}}^{\mu}_{\;\;\nu}=\Big{(}f^{\mu\alpha}\tilde{f}_{\alpha\nu}+\frac{1}{4}\delta^{\mu}_{\nu}f^{\alpha\beta}\tilde{f}_{\alpha\beta}\Big{)}\Big{(}\cosh\gamma-\sinh\gamma\frac{\mathcal{L}^{(E)}}{\sqrt{\mathcal{L}^{2}+\mathcal{M}^{2}}}\Big{)},$
(54)

whereas in the Magnetic limit, the stress tensor reads

${T_{M}}^{\mu}_{\;\;\nu}=\Big{(}\tilde{f}^{\mu\alpha}f_{\alpha\nu}+\frac{1}{4}\delta^{\mu}_{\nu}\tilde{f}^{\alpha\beta}f_{\alpha\beta}\Big{)}\Big{(}\cosh\gamma-\sinh\gamma\frac{\mathcal{L}^{(M)}}{\sqrt{\mathcal{L}^{2}+\mathcal{M}^{2}}}\Big{)}.$
(55)

This is again reminiscent of the relativistic ModMax case, as our stress
tensors in either limits are explicitly proportional to the Galilean Maxwell
ones of [40], with a multiplied contracted term. As we know for generic
Galilean Conformal theories, the stress tensor needs to be traceless i.e.
${T}^{\mu}_{\;\;\mu}=0$ and the condition on the component $T^{0}_{~{}i}=0$
has to be satisfied, since there is no momentum flux in non-relativistic
theories [45]. In case of the Electric limit, it is easy to check these
conditions:

$\displaystyle{T_{E}}^{\mu}_{\;\;\mu}=\Big{(}-f^{\alpha\mu}\tilde{f}_{\alpha\mu}+f^{\alpha\beta}\tilde{f}_{\alpha\beta}\Big{)}\Big{(}\cosh\gamma-\sinh\gamma\frac{\mathcal{L}^{(E)}}{\sqrt{\mathcal{L}^{2}+\mathcal{M}^{2}}}\Big{)}=0,$
(56) $\displaystyle{T_{E}}^{0}_{~{}i}=\Big{(}f^{0\alpha}\tilde{f}_{\alpha
i}\Big{)}\Big{(}\cosh\gamma-\sinh\gamma\frac{\mathcal{L}^{(E)}}{\sqrt{\mathcal{L}^{2}+\mathcal{M}^{2}}}\Big{)}=0.$
(57)

Similarly, in the Magnetic limit, the stress tensor satisfies

$\displaystyle{T_{M}}^{\mu}_{\;\;\mu}=0,~{}{T_{M}}^{0}_{~{}i}=0.$ (58)

This shows both electric and magnetic sectors of our non-linear theory is
explicitly Galilean invariant as well.

##  5 Discussions and conclusions

In this short paper, we described a nonlinear Galilean Covariant Lagrangian
that is invariant under Galilean Conformal symmetries by construction.
Interestingly, the Lagrangian was written in the same vein of the ModMax
Lagrangian and hence reaffirms the conformal nature of the ModMax construction
beyond the relativistic case. We focussed on the invariants of the Galilean
Maxwell theory and used them as building blocks to build our Lagrangian, with
an explicit proof of invariance under GCA transformations. We also discussed
the Galilean equations of motion and stress tensors, both in the Electric and
Magnetic limits of the theory. The nonlinear equations in all of the cases
reduce to the known Galilean equations in the $\gamma=0$ limit.

As we mentioned earlier, our calculation here introduces the first example of
a nonlinear Galilean covariant electrodynamics theory. It remains to explore
whether the usual NLED physics, like classical solutions, carry forward to
these sort of Galilean theories as well. As we mentioned earlier, it is useful
to go into the Hamiltonian formalism for the relativistic ModMax theory in
order to make sense of null solutions. It would be interesting to explore the
canonical structure of the Galilean theory as well along this route, starting
from (17) and proceeding to find the same for the total Galilean ModMax
theory. Another very straightforward extension would be to discuss Galilean
superconformal extension of this theory in the vein of [13]. Super-GCA
algebras have already been described well in the literature [46, 47, 48] and
one would hope to find a nonlinear realisation of that as well in the super-
ModMax-like theory. We hope to come back to these issues in future work.

## Acknowledgements

The authors would like to thank Arjun Bagchi and Rudranil Basu for useful
discussions and comments on the manuscript. The work of ArB is supported by
the Quantum Gravity Unit of the Okinawa Institute of Science and Technology
Graduate University (OIST). AM acknowledges support of the Department of
Physics at BITS Goa through the grant SERB CRG/2020/002035 and School of
Mathematics at University of Edinburgh where part of this work was conducted.
ArB acknowledges kind hospitality from IIT Kanpur and Kyoto University during
the course of this work.

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