aainstitutetext: Indian Institute of Technology Kanpur, Kalyanpur, Kanpur
208016. INDIA.
bbinstitutetext: Department of Physics, BITS-Pilani, K K Birla Goa Campus,
Zuarinagar, Goa-403726. INDIA
ccinstitutetext: Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-
Institut, 14476, Golm, Germany.

# Galilean Gauge Theories from Null Reductions

Arjun Bagchi b Rudranil Basu a Minhajul Islam a Kedar S. Kolekar b,c Aditya
Mehra
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The procedure of null reduction provides a concrete way of constructing field
theories with Galilean invariance. We use this to examine Galilean gauge
theories, viz. Galilean electrodynamics and Yang-Mills theories in spacetime
dimensions 3 and 4. Different non-relativistic conformal symmetries arise in
these contexts: Schrödinger symmetry in $d=3$ and Galilean conformal symmetry
in $d=4$. A canonical analysis further reveals that the symmetries enhance to
their infinite dimensional versions in phase space and pick up central
extensions. In addition, for the Abelian theory, we discuss non-relativistic
electro-magnetic duality in $d=3$ and its difference with the $d=4$ version.
We also mention some quantum aspects for both Abelian and non-Abelian
theories.

##  1 Introduction

Our present understanding of Nature rests heavily on the framework of quantum
field theory. For the theory of elementary particles, naturally this comes
hand in hand with relativistic physics. So the formulation of quantum field
theories, specifically gauge theories, are intimately connected with Lorentz
or Poincaré symmetry. For various applications, it is necessary to look beyond
relativistic theories. For effective descriptions of systems where typical
velocities are much lower than the speed of light, the Galilean algebra
replaces the Poincaré algebra as the symmetry of interest. This happens, e.g.
in many real-life condensed matter systems, (non-relativistic) hydrodynamics.
For systems such as these, quantum field theoretic descriptions naturally to
incorporate Galilean symmetries instead of Lorentz or Poincaré symmetry.
Although this seems like an age-old problem, interesting progress in the field
has been achieved only recently, coinciding with an effort in understanding
the Holographic Principle away from its usual setting in relativistic Anti-de
Sitter (AdS) spacetimes Son:2008ye ; Balasubramanian:2008dm ; Kachru:2008yh ;
Bagchi:2009my ; 2009 . In this paper, we shall be interested in the
construction of Galilean gauge theories. Initial works on Galilean gauge
theories date back to the construction of Galilean electrodynamics by Le
Ballac and Levy-Leblond LBLL  in the 1970s. The more recent papers in this
direction are Bagchi:2014ysa ; Duval:2014uoa ; Bagchi:2015qcw ;
VandenBleeken:2015rzu ; Bergshoeff:2015sic ; Festuccia:2016caf ;
Banerjee:2019axy ; Mehra:2021sfx .

Relativistic quantum field theories are defined on fixed (pseudo-) Riemannian
manifolds with a definite metric structure. When we move to Galilean field
theories, the background metric on the manifold becomes degenerate as the
speed of light goes to infinity and the (pseudo-) Riemannian structure is
replaced by a so-called Newton-Cartan structure which is a quadruple
$(\mathcal{G},h,\tau,\Gamma)$ Duval:1993pe ; Duval:2009vt . $\mathcal{G}$ is a
$(d+1)$ dimensional manifold where we can choose a set of coordinates
$(t,x^{i})$. $h$ is a contravariant spatial metric $h^{\mu\nu}$ and
$\tau=\tau_{a}dx^{a}$ a non-vanishing one-form called the clock form. $\Gamma$
is torsion-free linear connection. A flat NC structure is given by a vanishing
$\Gamma$. The Galilean algebra is defined as the Lie algebra of vector fields
$\xi=\xi^{a}\partial_{a}$ satisfying

$\mathcal{L}_{\xi}h^{\mu\nu}=0,\quad\mathcal{L}_{\xi}\tau_{a}=0,\quad\tau_{a}h^{ab}=0.$
(1)

The algebra generated is the same as the one obtained by an Inönü-Wigner
contraction of the Poincaré algebra.

The absence of a non-degenerate metric makes the writing of an action a
difficult proposition. Some of the recent literature, include some of our
earlier work in this direction Bagchi:2014ysa ; Bagchi:2015qcw ;
Bagchi:2017yvj , thus concentrated on the aspects of symmetries arising from
the equations of motion (EOM) which were arrived at by a limiting procedure
starting from a relativistic quantum field theory. Particular attention was
given to the conformal versions of these Galilean QFTs, or Galilean Conformal
Field Theories. It was shown that Galilean gauge theories in four dimensions,
viz. Galilean electrodynamics and Galilean Yang-Mills theories had EOM that
exhibited invariance under the infinite dimensional Galilean Conformal
Algebra. These infinite enhancements remained when one coupled the gauge
theories to massless matter (scalars or fermions). We will review some of this
briefly later in the paper. But our focus in this current paper is the
construction of explicit actions for these gauge theories. We do this by the
process of null reduction Duval:1984cj ; 1991 ; 1995 ; Santos:2004pq  that has
been previously employed for generating non-relativistic theories from
relativistic theories in one higher dimension.

In particular, we focus on Galilean gauge theories in dimensions three and
four. We start with relativistic gauge theories in $d=4,5$ and null reduce to
generate actions for Galilean electrodynamics and Galilean Yang-Mills theories
in $d=3,4$. For the abelian theories, this process yields results that are
already present in the literature, reproducing the so-called Magnetic sector
of GED in $d=4$ Bagchi:2014ysa ; Festuccia:2016caf  and a known action in
$d=3$ Chapman:2020vtn . The actions of Galilean YM are again straight-forward
generalisations of their abelian counterparts. Interesting results appear when
we delve deeper into the symmetries of these actions and their corresponding
EOM. We find that for the gauge theories non-relativistic conformal algebras,
the Galilean Conformal Algebra in $d=4$ and the Schrödinger algebra in $d=3$
are realised at the level of the action. These are actually partially enhanced
to their infinite dimensional algebras in the action itself. At the level of
the EOM, there is further enhancement of symmetries. While the appearance of
the infinite dimensional GCA is not unexpected given earlier works, the
appearance of the Schrödinger-Virasoro algebra in $d=3$ is new. Further
surprises await us when we look at the canonical analysis later in the paper.
Here we find that the entire infinite dimensional algebras are realised as
symmetries in phase space. There are also central extensions which we find.

Along the way, we also investigate aspects of electromagnetic duality for
Galilean Electromagnetism. While in $d=4$ the duality has been known to
exchange electric and magnetic theories, we find that in $d=3$ the theory
itself has an in-built electromagnetic duality. This is inherited from the
null reduction of the higher dimensional relativistic theory. We end our
analysis with a few quantum aspects of Galilean Yang-Mills theory. A detailed
exposition of this will be carried out in later work.

##### Outline:

The paper is organised as follows. We begin in Sec 2 with a review of non-
relativistic conformal symmetries, specifically focussing on the Galilean
conformal algebra and then the Schrödinger algebra first discussing the finite
dimensional algebras and then going onto their infinite dimensional
extensions. We briefly comment about their geometric realisations. We end the
section with a quick review of the procedure of null reduction.

In Sec. 3, we address Abelian Galilean gauge theories. We begin with some
review material about Galilean electrodynamics in $d=4$ and then proceed to
find actions by null reductions in $d=4$ and $d=3$. Symmetries arising from
the actions and the corresponding EOM are discussed. While most of this is
review material, we discover a realisation of the infinite dimensional
Schrödinger-Virasoro algebra in $d=3$. We then discuss electromagnetic duality
for the $d=3$ Galilean theory and its differences in the $d=4$ case. Much of
the $d=3$ analysis is based on null reductions.

Sec. 4 addresses non-Abelian theories. The section is structured similar to
the Abelian one, but we provide more details of the symmetry analyses of the
actions and EOM. Unlike the Abelian case, we find that our EOMs don’t reduce
to earlier work in $d=4$ when some fields are turned off. Interestingly,
though we again find an enhancement of symmetries in $d=4$, as well as in
$d=3$.

We perform a canonical analysis of symmetries in Sec. 5 and find that the
infinite dimensional Virasoro sub-algebras are non-trivially realised in phase
space for the Yang-Mills theories in both dimensions. In addition, there are
state-dependent central extensions to the Virasoro algebra. In Sec. 6, we
briefly discuss quantum mechanical aspects and write down the Feynman rules
for Galilean Yang-Mills theories. We conclude with a summary of our results
and a list of future directions in Sec. 7. An appendix contains some aspects
of GCA representation theory.

##  2 Non-relativistic Conformal Symmetries

Different versions of conformal symmetry have been discussed in the non-
relativistic setting Taylor:2008tg . We will focus our attention on two of
these, viz. Galilean Conformal symmetry Bagchi:2009my  and Schrödinger
symmetry Hagen:1972pd ; Niederer:1972zz ; Henkel:1993sg ; Son:2008ye .
Galilean conformal symmetry is arrived at by an Inonu-Wigner contraction of
relativistic conformal symmetry, much the same way as one derives the Galilean
algebra from the Poincaré algebra. This is the symmetry for a massless or
gapless non-relativistic system. Schrödinger symmetry on the other hand is the
group of symmetries of the free Schrödinger equation.

In order to set notation, let us remind ourselves of the relativistic
conformal generators in $d$ dimensional Minkowski spacetime. The conformal
group is constructed from translations [$P_{\mu}$], Lorentz transformations
$[J_{\mu\nu}]$, dilatation [$D$], and special conformal transformations (SCT)
[$K_{\mu}$]. In vector field notation, they are given by

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

where $\mu=(0,i=1,...,d-1)$. The commutation relations satisfied by these
generators form the conformal algebra, which is isomorphic to $so(d,2)$.

###  2.1 Galilean Conformal Symmetries

As stated earlier, the Galilean Conformal Algebra is obtained from the
relativistic conformal algebra through an Inönü-Wigner contraction. We will
perform the non-relativistic limit at the spacetime level and in units where
$c=1$. In this limit, the velocities are small as compared to the speed of
light. The systematic procedure to take the limit is as follows

$\displaystyle x_{i}\rightarrow\epsilon x_{i},\,t\rightarrow t\quad\text{along
with}~{}~{}\epsilon\rightarrow 0.$ (3)

The scaling in (3) is equivalent to $c\rightarrow\infty$ scaling. Contracting
(2) in this way we get

$\displaystyle H=-\partial_{t},\quad
D=-(t\partial_{t}+x^{i}\partial_{i}),\quad
K=-(t^{2}\partial_{t}+2tx^{i}\partial_{i}),$ (4a) $\displaystyle
P_{i}=\partial_{i},\quad B_{i}=t\partial_{i},\quad K_{i}=t^{2}\partial_{i},$
(4b) $\displaystyle J_{ij}=(x_{i}\partial_{j}-x_{j}\partial_{i}).$ (4c)

where $(H,D,K)\rightarrow$ {Hamiltonian, dilatation and temporal SCT} and
$(P_{i},B_{i},K_{i})\rightarrow$ {translation, boost and spatial SCT}.
Finally, $J_{ij}$ are the generators of spatial rotations. The algebra of
these generators are given by

$\displaystyle[J_{ij},J_{kl}]=(\delta_{ik}J_{jl}-\delta_{jk}J_{il}-\delta_{il}J_{jk}+\delta_{jl}J_{ik}),\quad[H,B_{i}]=P_{i},$
$\displaystyle[P_{i},B_{j}]=0,\quad[J_{ij},X_{k}]=(\delta_{ik}X_{j}-\delta_{jk}X_{i}),\quad[B_{i},B_{j}]=0,$
$\displaystyle[D,H]=H,\quad[D,P_{i}]=P_{i},\quad[D,K]=-K,\quad[D,K_{i}]=-K_{i},$
$\displaystyle[K,H]=2D,\quad[K,P_{i}]=2B_{i},\quad[B_{i},K]=-K_{i},$
$\displaystyle[B_{i},K_{j}]=0,\quad[K_{i},P_{j}]=0,\quad[K_{i},H]=2B_{i},$ (5)

where $X_{i}=(P_{i},B_{i},K_{i})$. This is a finite dimensional Galilean
conformal algebra (fGCA). The generators can also be written in a suggestive
form for $n=0,\pm 1$ given as

$\displaystyle
L^{(n)}=-t^{n+1}\partial_{t}-(n+1)t^{n}x^{i}\partial_{i},\quad(L^{(-1,0,1)}\rightarrow
H,D,K)$ (6a) $\displaystyle
M_{i}^{(n)}=t^{n+1}\partial_{i},\quad\quad(M^{(-1,0,1)}_{i}\rightarrow
P_{i},B_{i},K_{i})$ (6b)

The fGCA can be written down as

$\displaystyle\big{[}L^{(n)},L^{(m)}\big{]}=(n-m)L^{(n+m)},~{}\big{[}L^{(n)},M_{i}^{(m)}\big{]}=(n-m)M_{i}^{(n+m)},~{}\big{[}M^{(n)}_{i},M^{(m)}_{j}\big{]}=0,$
$\displaystyle\big{[}L^{(n)},J_{ij}\big{]}=0,~{}\big{[}J_{ij},M^{(n)}_{k}\big{]}=M^{(n)}_{j}\delta_{ik}-M^{(n)}_{i}\delta_{jk}.$
(7)

One sees from above that the algebra continues to hold for any integer value
of $n$. Surprisingly the algebra is extended to an infinite-dimensional
algebra which we name I-GCA or simply GCA from now on. The algebra above can
admit central terms in the usual Virasoro subalgebra. There is a further
central extension which the $[L,M]$ commutator gets in the special case of
$d=2$. In the systems we study later in the paper, we will find that in their
realisation in phase space, the symmetries develop a Virasoro central
extension.

The action of infinite extension of GCA on the fields at general space-time
points is given by Bagchi:2009ca ; Bagchi:2014ysa ; Bagchi:2017yvj :

$\displaystyle\big{[}L^{(n)},\varphi(t,x)\big{]}=\big{(}t^{n+1}\partial_{t}+(n+1)t^{n}x^{k}\partial_{k}+(n+1)t^{n}\Delta\big{)}\varphi(t,x)$
$\displaystyle\hskip
76.82234pt-n(n+1)t^{n-1}x^{k}U\big{[}M_{k}^{(0)},\varphi(0,0)\big{]}U^{-1},$
(8a)
$\displaystyle\big{[}M^{(n)}_{k},\varphi(t,x)\big{]}=-t^{n+1}\partial_{k}\varphi(t,x)+(n+1)t^{n}U\big{[}M_{k}^{(0)},\varphi(0,0)\big{]}U^{-1}.$
(8b)

In the above, $\varphi(t,x)$ represents a field of generic spin. For the
purpose of this paper, we shall only be interested in scalars and gauge fields
and hence the transformation laws read:

$\big{[}M_{k}^{(0)},\varphi_{A}(0,0)\big{]}=a\phi_{k}\delta_{A0}+s\delta_{At}a_{k}+r\delta_{Ai}\phi\delta_{ik},$
(9)

where the index $i$ is not summed over and $\varphi_{0}(0,0)=\phi$,
$\varphi_{t}(0,0)=a_{t}$, $\varphi_{i}(0,0)=a_{i}$
$(\phi\rightarrow\text{scalar field},\\{a_{t},a_{i}\\}\rightarrow\text{gauge
fields})$. The values of constants $(a,s,r)$ are determined by demanding
inputs from the dynamics. We will clarify later in the paper how the values of
these constants get fixed for the theory under consideration (see _e.g._
Sec[4.3]). We will use (8) to check the symmetries of the system we are
dealing with in this paper. For the detailed analysis of the actions of GCA
generators on fields, the reader is directed to appendix A.

###  2.2 Schrödinger Symmetries

We now revisit another non-relativistic algebra that has less symmetries in
comparison to GCA. The Schrödinger group Hagen:1972pd ; Niederer:1972zz ,
$Sch(d)$ is a symmetry group of the free Schrödinger equation in $d$
dimensions. This symmetry has been discussed in the context of fermions at
unitarity Nishida:2007pj . Some recent literature on the Schrödinger algebra
includes Hellerman:2021qzz ; Karananas:2021bqw ; Pellizzani:2021hzx ;
Hellerman:2020eff ; Kravec:2018qnu ; Favrod:2018xov ; Pal:2018idc .

The Schrödinger algebra is generated by the generators of centrally extended
Galilean algebra with two additional generators; the dilatation $\tilde{D}$
and a special conformal transformation $\tilde{K}$. These generators are given
by

$\displaystyle P_{i}=\partial_{i},\quad H=\partial_{t},\quad
G_{i}=t\partial_{i}+x_{i}M,\quad
J_{ij}=(x_{i}\partial_{j}-x_{j}\partial_{i}),$
$\displaystyle\tilde{D}=2t\partial_{t}+x^{i}\partial_{i},\quad\tilde{K}={2}t^{2}\partial_{t}+2tx^{i}\partial_{i}+x^{2}M.$
(10)

The non-zero commutations relations of the Schrödinger algebra are

$\displaystyle[H,\tilde{D}]=2H,\quad[P_{i},\tilde{D}]=P_{i},\quad[\tilde{D},G_{i}]=G_{i},\quad[\tilde{D},\tilde{K}]=2\tilde{K},$
$\displaystyle[P_{i},G_{j}]=-\delta_{ij}M,\quad[H,\tilde{K}]=\tilde{D},\quad[H,G_{i}]=-P_{i},\quad[G_{i},\tilde{K}]=-G_{i},$
$\displaystyle[J_{ij},P_{k}]=\delta_{ik}P_{j}-\delta_{jk}P_{i},\quad[J_{ij},G_{k}]=\delta_{ik}G_{j}-\delta_{jk}G_{i},$
$\displaystyle[J_{ij},J_{kl}]=(\delta_{ik}J_{jl}+\delta_{jl}J_{ik}-\delta_{il}J_{jk}-\delta_{jk}J_{il}).$
(11)

Here $G_{i}$ is the Schrödinger boost, and $M$ is a central element, that can
be viewed as the mass or the particle number of the system. The number of
generators of the group, including the central element $M$, in $d=4$ is 13,
which is to be contrasted with the finite GCA, which has 15. The difference
comes about in the number of special conformal generators, which is a vector-
worth in the GCA compared to a single one in the Schrödinger algebra.

We now briefly discuss the representation of the Schrödinger algebra following
Nishida:2007pj . A primary operator $\varphi(0,0)$ at $(t=0,\ x^{i}=0)$ of
scaling dimension $\Delta_{\varphi}$ is defined by

$[\tilde{D},\varphi(0,0)]=\Delta_{\varphi}\varphi(0,0);\quad[G_{i},\varphi(0,0)]=0,\quad[\tilde{K},\varphi(0,0)]=0.$
(12)

Then the tower of operators built by repeated action of $P_{i}$ and $H$ on a
primary operator forms an irreducible representation of the Schrödinger
algebra. In this way, we can see that all local operators can be divided into
irreducible representations built on different primary operators. For a
primary operator at an arbitrary spacetime point $(t,x^{i})$, the generators
$G_{i}$ and $\tilde{K}$ act as

$\displaystyle\big{[}G_{i},\varphi(t,x)\big{]}=(t\partial_{i}+m_{\varphi}x_{i})\varphi(t,x),$
$\displaystyle\big{[}\tilde{K},\varphi(t,x)\big{]}=[(2t^{2}\partial_{t}+2tx^{i}\partial_{i}+t\Delta_{\varphi})+x^{2}m_{\varphi}]\varphi(t,x),$
(13)

where $m_{\varphi}$ is the eigenvalue of $M$ on $\varphi$. For any local
operator at an arbitrary spacetime point, the generators $P_{i}$, $H$ and
$\tilde{D}$ act as

$\displaystyle\big{[}P_{i},\varphi(t,x)\big{]}=\partial_{i}\varphi(t,x),~{}\big{[}H,\varphi(t,x)\big{]}=\partial_{t}\varphi(t,x),$
$\displaystyle\big{[}\tilde{D},\varphi(t,x)\big{]}=(2t\partial_{t}+x^{i}\partial_{i}+\Delta_{\varphi})\varphi(t,x).$
(14)

The boost and dilatation generators act on the fields of our interest
$\varphi=(\phi\rightarrow\text{scalar field},\\\
\\{a_{t},a_{i}\\}\rightarrow\text{gauge fields})$ as

$\displaystyle\delta_{G_{k}}\phi=-t\partial_{k}\phi,~{}\delta_{G_{k}}a_{t}=-(t\partial_{k}a_{t}+a_{k}),~{}\delta_{G_{k}}a_{i}=-(t\partial_{k}a_{i}-\delta_{ik}\phi),$
(15a)
$\displaystyle\delta_{\tilde{D}}\varphi=(2t\partial_{t}+x^{k}\partial_{k}+\Delta_{\varphi})\varphi.$
(15b)

The generator of the special conformal transformation $\tilde{K}$ acts on
$\varphi$ as

$\displaystyle\delta_{{\tilde{K}}}a_{t}=(2t^{2}\partial_{t}a_{t}+2tx^{k}\partial_{k}a_{t}+4ta_{t}+2x^{k}a_{k}),$
(16a)
$\displaystyle\delta_{\tilde{K}}a_{i}=(2t^{2}\partial_{t}a_{i}+2tx^{k}\partial_{k}a_{i}+2ta_{i}-2x_{i}\phi),$
(16b)
$\displaystyle\delta_{\tilde{K}}\phi=(2t^{2}\partial_{t}\phi+2tx^{k}\partial_{k}\phi).$
(16c)

Since we would be interested in gauge fields in the bulk of the paper, we
would be dealing with massless systems and hence will restrict ourselves to
the zero eigenvalue subsector of $M$ $:m_{\varphi}=0$.

Although the Schrödinger algebra as reviewed above is rather well known,
somewhat less studied is the interesting observation that even this algebra
admits an infinite extension Henkel:1993sg ; Alishahiha:2009nm . To get this
infinite extension, we first rewrite the generators as

$\displaystyle
Z^{(n)}=2t^{n+1}\partial_{t}+(n+1)t^{n}x^{i}\partial_{i}+\frac{1}{2}n(n+1)t^{n-1}x^{2}M,$
$\displaystyle
Y_{i}^{(m)}=t^{m+\frac{1}{2}}\partial_{i}+\big{(}m+\frac{1}{2}\big{)}t^{m-\frac{1}{2}}x_{i}M,$
$\displaystyle T^{(n)}=t^{n}M,$ (17)

where $n=(-1,0,1)$, $m=(-\frac{1}{2},\frac{1}{2})$ and
$(Z^{(-1,0,1)}\rightarrow H,\tilde{D},\tilde{K})$,
$(Y^{(-\frac{1}{2},\frac{1}{2})}_{i}\rightarrow P_{i},G_{i})$. To distinguish
from the GCA case, we use the notation $Z^{(n)}$ instead of the usual
$L^{(n)}$ notation for the Schrödinger case. In terms of these redefined
generators the Schrödinger algebra becomes

$\displaystyle[Z^{(n)},Z^{(m)}]=(n-m)Z^{(n+m)},\quad[Y_{i}^{(n)},Y_{j}^{(m)}]=(n-m)\delta_{ij}T^{(n+m)},$
$\displaystyle[Z^{(n)},Y_{i}^{(m)}]=(\frac{n}{2}-m)Y_{i}^{(n+m)},\quad[Z^{(n)},T^{(m)}]=-mT^{(n+m)}.$
(18)

We see that the above algebra is satisfied if we extend the values of $n$, $m$
to $n\in\mathbb{Z}$, $m\in\mathbb{Z}+\frac{1}{2}$, thus giving us an infinite
extension of the Schrödinger algebra, which in literature is referred to as
the Schrödinger-Virasoro algebra. The Virasoro sub-algebra again admits a
central extension, which we will encounter in our analysis later in the paper.
The action of infinite generators $Z^{(n)}$ on our fields of interest
$\varphi=(\phi,a_{t},a_{i})$ can be written down as

$\displaystyle\delta_{Z^{(n)}}\phi=2t^{n+1}\partial_{t}\phi+(n+1)t^{n}(x^{k}\partial_{k}+\Delta_{\phi})\phi,$
$\displaystyle\delta_{Z^{(n)}}a_{i}=2t^{n+1}\partial_{t}a_{i}+(n+1)t^{n}(x^{k}\partial_{k}+\Delta_{a_{i}})a_{i}-n(n+1)t^{n-1}x_{i}\phi,$
$\displaystyle\delta_{Z^{(n)}}a_{t}=2t^{n+1}\partial_{t}{a}_{t}+(n+1)t^{n}(x^{k}\partial_{k}+\Delta_{a_{t}})a_{t}+n(n+1)t^{n-1}x^{k}a_{k}.$
(19)

Similarly the action of $Y^{(n)}_{k}$ generators on the fields $\varphi$ is

$\displaystyle\delta_{Y^{(m)}_{k}}\phi=t^{m+\frac{1}{2}}\,\partial_{k}\phi,~{}\delta_{Y^{(m)}_{k}}a_{i}=t^{m+\frac{1}{2}}\partial_{k}a_{i}-\big{(}m+\frac{1}{2}\big{)}t^{m-\frac{1}{2}}\phi,$
$\displaystyle\delta_{Y^{(m)}_{k}}a_{t}=t^{m+\frac{1}{2}}\partial_{k}a_{t}+\big{(}m+\frac{1}{2}\big{)}t^{m-\frac{1}{2}}a_{k}.$
(20)

These transformations above for the Abelian fields $(\phi,a_{t},a_{i})$ (and
their generalizations to non-Abelian fields) will be useful to show the
invariance of Galilean versions of electrodynamics (and Yang-Mills) under
Schrödinger algebra in $3$ dimensions.

###  2.3 Geometrical realization of non-relativistic conformal symmetries

In the introduction, we described the Galilean algebra as the isometry algebra
of a flat Newton-Cartan manifold (1). The non-relativistic conformal algebras
that we have discussed above from an algebraic perspective can also be look at
geometrically. In terms of the previously discussed Newton-Cartan structures,
these symmetries arise as conformal isometries Duval:2009vt ; Duval:2014lpa .

Generically, the infinite dimensional conformal isometries of the flat NC
spacetime are generated by vector fields $\xi^{a}$ which satisfy the non-
relativistic conformal Killing equations

$\mathcal{L}_{\xi}h^{\mu\nu}=\lambda
h^{\mu\nu},\quad\mathcal{L}_{\xi}\tau_{a}=\mu\tau_{a};\quad\lambda+N\mu=0.$
(21)

Solving these equations, the vectors fields $\xi^{a}$ span the _conformal
Galilei algebra of level $N$_, denoted by
$\mathfrak{cgal}_{N}(\mathcal{G},h,\tau)$. The level $N$ of the algebra is
related to the dynamical exponent $z$, which characterizes the unequal scaling
of space and time under dilatation, as $z=\frac{2}{N}$.

For $N=2$, _i.e._ $z=1$ the algebra
$\mathfrak{cgal}_{2}(\mathcal{G},h,\tau)=\mathfrak{gca}(d+1)$ is the infinite
dimensional Galilean conformal algebra. For $N=1$, _i.e._ $z=2$ we get the
infinite dimensional Schrödinger-Virasoro algebra
$\mathfrak{cgal}_{1}(\mathcal{G},h,\tau)=\mathbf{sv}(d+1)$.

Further demanding that the non-relativistic conformal transformations preserve
the form of the geodesic equation _i.e._ the projective structures associated
with the flat connection $\Gamma=0$, the algebra reduces to finite dimensional
Schrödinger algebra
$\mathfrak{cgal}_{1}(\mathcal{G},h,\tau,\Gamma)=\mathfrak{sch}(d+1)$ for
$z=2$, and finite dimensional GCA for $z=1$. We refer the reader to
Duval:2009vt  for further detials.

###  2.4 Null Reduction

Before moving onto the main body of the paper, let us briefly review the
process of null reduction which will be our primary tool in constructing non-
relativistic actions. Null reduction is a method to obtain a $d$-dimensional
Galilean invariant theory from a $(d+1)$-dimensional relativistic theory
Duval:1984cj ; 1991 ; 1995 ; Santos:2004pq . Below we elucidate this by
applying it to a scalar field theory.

To begin with, consider the metric of a $(d+1)$-dimensional Minkowski
spacetime

$\displaystyle
ds^{2}=\eta_{\mathcal{B}\mathcal{C}}dx^{\mathcal{B}}dx^{\mathcal{C}}=-(dx^{0})^{2}+(dx^{1})^{2}+\delta_{AB}dx^{A}dx^{B},$
(22)

where $\mathcal{B},\mathcal{C}=(0,1,...,d)$ and $A,B=(2,\dots,d)$. We will
make a change of coordinates to the usual light-cone coordinates given below

$\displaystyle u=\frac{1}{\sqrt{2}}(x^{0}-x^{1}),\quad
t=\frac{1}{\sqrt{2}}(x^{0}+x^{1}).$ (23)

In lightcone coordinates, the metric becomes

$\displaystyle
ds^{2}=\eta_{\tilde{\mu}\tilde{\nu}}dx^{\tilde{\mu}}dx^{\tilde{\nu}}=2du\,dt+\delta_{AB}dx^{A}dx^{B},$
(24)

where $\tilde{\mu}=(u,t,2,3,...)$. We will now look how to implement this
technique on a field theory. We will first start with an action of Lorentz
invariant theory on the coordinate system (22). We then reduce the action of
that theory along the $u$ null direction of the coordinate system (24). Here,
the components of $(d+1)$-dimensional generic fields reduce to different
fields in $d$-dimensional Galilean invariant theory, _e.g._ a gauge field
reduces as
$\mathcal{A}_{\tilde{\mu}}=(\mathcal{A}_{u},\mathcal{A}_{t},\mathcal{A}_{i})$.
The fields $\mathcal{A}$ are taken to be independent of the coordinate $u$.
The $t$ coordinate becomes time in $d$-dimensional Galilean theory and $(A,B)$
becomes the spatial coordinates $i,j=(1,2,3,...)$.

Let us discuss this technique through an example of a scalar field theory.
Consider a relativistic scalar field theory in $5$-dimensional Minkowski
spacetime, described by the action

$\displaystyle\mathcal{S}=\int
dt\,d^{4}x\big{(}\frac{1}{2}\eta^{\mathcal{B}\mathcal{C}}\partial_{\mathcal{B}}\phi\partial_{\mathcal{C}}\phi\big{)}=\int
dt\,d^{4}x\big{(}-\frac{1}{2}(\partial_{0}\phi)^{2}+\frac{1}{2}(\partial_{1}\phi)^{2}+\frac{1}{2}(\partial_{i}\phi)^{2}\big{)},$
(25)

where $\mathcal{B}=[0,1,i=(2,3,4)]$. We will now express (25) in coordinate
system (24) so that we can perform a null reduction. The action becomes

$\displaystyle\mathcal{S}=\int
d^{5}x\big{(}\frac{1}{2}\eta^{\tilde{\mu}\tilde{\nu}}\partial_{\tilde{\mu}}\phi\partial_{\tilde{\nu}}\phi\big{)}$
$\displaystyle\hskip 23.33147pt=\int
dt\,du\,d^{3}x\big{(}\frac{1}{2}\eta^{tu}\partial_{t}\phi\partial_{u}\phi+\frac{1}{2}\eta^{ut}\partial_{u}\phi\partial_{t}\phi+\frac{1}{2}\eta^{ij}\partial_{i}\phi\partial_{j}\phi\big{)}.$
(26)

With $\phi(t,x^{i})$ independent of the null coordinate $u$, we perform the
null reduction on (2.4) along $u$-direction and get the action as

$\mathcal{S}_{G}=\int
dt\,d^{3}x\big{(}\frac{1}{2}\eta^{ij}\partial_{i}\Phi\partial_{j}\Phi\big{)}$
(27)

where the subscript $G$ denotes Galilean and $\phi(t,x^{i})$ has reduced to
$\Phi(t,x^{i})$. This resultant action (27) is invariant under GCA in $d=4$
dimensions.

##  3 Abelian Galilean Gauge theory

In this section, we will address Galilean electrodynamics in dimensions $d=4$
and then $d=3$. We will construct actions by null reducing relativistic
electrodynamics in one higher dimension and investigate the symmetries
associate with the action and the corresponding EOM. Much of this is review
material, although the discovery of the (partial) realisation of the infinite
dimensional Schrödinger Virasoro in the $d=3$ case is new. We then address the
question of electromagnetic duality in these theories, focussing on the $d=3$
case and point out differences to the one found earlier in the $d=4$ case.

###  3.1 Galilean electrodynamics in $d=4$

Le Bellac and Lévy-Leblond first introduced Galilean electrodynamics (GED) in
LBLL  and wrote it down in the language of electric ($\vec{E}$) and magnetic
($\vec{B}$) fields. In (Bagchi:2014ysa, ), we (a subset of the current
authors) discussed the Galilean electrodynamics in terms of potential
formulation, scaling the scalar and vector potentials in addition to the
scaling of coordinates (3). The non-relativistic scaling of the $4$-vector
potential was considered in two different ways:

$\displaystyle a_{t}\rightarrow a_{t},\,\,a_{i}\rightarrow\epsilon
a_{i}~{}~{}\text{and}~{}~{}a_{t}\rightarrow\epsilon a_{t},\,\,a_{i}\rightarrow
a_{i}.$ (28)

The first limit is known as the Electric limit whereas the second limit is
called as Magnetic limit. The EOM of these two sectors (in absence of sources)
are as follows:

Electric limit:
$\displaystyle\partial^{i}\partial_{i}a_{t}=0,\quad\partial^{j}\partial_{j}a_{i}-\partial_{i}\partial_{j}a^{j}+\partial_{t}\partial_{i}a_{t}=0;$
(29a) Magnetic limit:
$\displaystyle\partial^{j}\partial_{j}a_{i}-\partial_{i}\partial_{j}a^{j}=0,\quad\partial^{i}\partial_{i}a_{t}-\partial_{i}\partial_{t}a^{i}=0.$
(29b)

These equations can also be written in terms of $\vec{E}$ and $\vec{B}$. They
become

Electric limit:
$\displaystyle\vec{\nabla}\cdot\vec{E}=0,\quad\vec{\nabla}\cdot\vec{B}=0,\quad\vec{\nabla}\times\vec{E}=0,\quad\vec{\nabla}\times\vec{B}=\frac{\partial\vec{E}}{\partial
t}$ (30a) Magnetic limit:
$\displaystyle\vec{\nabla}\cdot\vec{E}=0,\quad\vec{\nabla}\cdot\vec{B}=0,\quad\vec{\nabla}\times\vec{E}=-\frac{\partial\vec{B}}{\partial
t},\quad\vec{\nabla}\times\vec{B}=0.$ (30b)

By looking at (30), we notice that under the exchange of electric and magnetic
fields, i.e. under the following transformation

$\displaystyle\vec{E}\longrightarrow\vec{B},\quad\vec{B}\longrightarrow-\vec{E},$
(31)

the EOM in the electric and the magnetic sectors get interchanged. This is
referred to as the electric-magnetic duality between “electric” and “magnetic”
sectors of Galilean electrodynamics Duval:2014uoa .

We will now look at the invariance of EOM (29) of Galilean electrodynamics
under infinite-dimensional Galilean conformal symmetry. For that, we have to
use the details of representation theory (look at Appendix A). The
transformation of equations of Electric limit under the generators
$(L^{(n)},M^{(n)}_{k})$ of GCA are given by

$\displaystyle[L^{(n)},\partial^{i}\partial_{i}a_{t}]=0,~{}[L^{(n)},\partial^{j}\partial_{j}a_{i}-\partial_{i}\partial^{j}a_{j}+\partial_{t}\partial_{i}a_{t}]=-\frac{1}{2}\,n(n+1)\,t^{n-1}(d-4)\partial_{i}a_{t},$
$\displaystyle[M^{(n)}_{k},\partial^{i}\partial_{i}a_{t}]=0,~{}[M^{(n)}_{k},\partial^{j}\partial_{j}a_{i}-\partial_{i}\partial^{j}a_{j}+\partial_{t}\partial_{i}a_{t}]=0.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(32)

We see that for $d=4$, the equations are invariant under the full infinite
dimensional GCA. Similarly for Magnetic case, we have

$\displaystyle[L^{(n)},\partial^{j}\partial_{j}a_{i}-\partial_{i}\partial^{j}a_{j}]=0,~{}[L^{(n)},\partial^{i}\partial_{i}a_{t}-\partial_{t}\partial^{i}a_{i}]=-\frac{1}{2}n(n+1)(d-4)t^{n-1}\partial^{i}a_{i},$
$\displaystyle[M^{(n)}_{k},\partial^{j}\partial_{j}a_{i}-\partial_{i}\partial^{j}a_{j}]=0,~{}[M^{(n)}_{k},\partial^{i}\partial_{i}a_{t}-\partial_{t}\partial^{i}a_{i}]=0.$
(33)

From (3.1) and (3.1), we see that Galilean electrodynamics is invariant under
GCA in $d=4$ case. For further explanations, the reader is directed to
Bagchi:2014ysa .

###  3.2 An action for Galilean electrodynamics in $d=4$

In this section, as a warm-up to our analysis in Yang Mills theory, we review
the construction of the action for Galilean electrodynamics by null reducing
electrodynamics (ED) in one higher dimension Festuccia:2016caf ;
Chapman:2020vtn . For that, let us consider the 5-dimensional Maxwell action

$\displaystyle\mathcal{S}_{ED}=\int
d^{5}x\bigg{(}-\frac{1}{4}\eta^{\tilde{\mu}\tilde{\rho}}\eta^{\tilde{\nu}\tilde{\sigma}}F_{\tilde{\mu}\tilde{\nu}}F_{\tilde{\rho}\tilde{\sigma}}\bigg{)},$
(34)

in the coordinate system (24). The field strength is given by
$F_{\tilde{\mu}\tilde{\nu}}=(\partial_{\tilde{\mu}}A_{\tilde{\nu}}-\partial_{\tilde{\nu}}A_{\tilde{\mu}})$.
We take the vector potential $A_{\tilde{\mu}}$ to be independent of the null
coordinate ‘$u$’ and decompose the components of $A_{\tilde{\mu}}$ as

$A_{u}=\phi,\ A_{t}=a_{t},\ A_{i}=a_{i}.$ (35)

Then performing null reduction, we get an action for Galilean electrodynamics
in $4$-dimensions as

$\displaystyle\mathcal{S}_{GED}=\int
dt\,d^{3}x\bigg{(}-\frac{1}{4}W^{ij}W_{ij}+E^{i}\partial_{i}\phi+\frac{1}{2}(\partial_{t}\phi)^{2}\bigg{)},$
(36)

where $W_{ij}=(\partial_{i}a_{j}-\partial_{j}a_{i})$ and
$E_{i}=(\partial_{t}a_{i}-\partial_{i}a_{t})$. The associated EOM are given by

$\displaystyle\partial^{2}_{t}\phi+\partial_{i}E_{i}=0,\quad\partial^{i}\partial_{i}\phi=0,\quad\partial^{j}W_{ji}-\partial_{t}\partial_{i}\phi=0.$
(37)

We see that the action (36) is invariant under GCA (8) in $d=4$. We further
see that setting $\phi=0$, equations (37) reduce to

$\displaystyle\partial^{j}W_{ji}=0,\quad\partial_{i}E_{i}=0.$ (38)

which are the EOM in the magnetic limit of GED (29b) Bagchi:2014ysa . Thus we
see that from a null reduction, we only get the magnetic sector of GED.

###  3.3 Galilean electrodynamics in $d=3$ dimensions

We will study Galilean electrodynamics in $3$-dimensions obtained by null
reduction of electrodynamics in $4$-dimensions. We see that the action looks
identical to the case in four dimensions (36) with the difference being in the
values of spatial indices $i,j$. However the symmetries are different: GED in
$3$-dimensions is invariant under Schrödinger algebra Chapman:2020vtn .

We will now look at the invariance of the GED action under the Schrödinger
symmetries. Under boost transformations, using the transformations of the
fields (15a), the action transforms as

$\delta_{G_{k}}\mathcal{S}_{GED}=\int
dt\,d^{2}x\,\delta_{G_{k}}\mathcal{L}_{GED}=\int
dtd^{2}x\,\partial_{k}\Big{[}-t\Big{(}\frac{1}{2}\partial_{t}\phi\partial_{t}\phi+E^{i}\partial_{i}\phi-\frac{1}{4}W^{ij}W_{ij}\Big{)}\Big{]},$
(39)

where
$\mathcal{L}_{GED}=\Big{(}\frac{1}{2}\partial_{t}\phi\partial_{t}\phi+E^{i}\partial_{i}\phi-\frac{1}{4}W^{ij}W_{ij}\Big{)}$
is the Lagrangian density. Under dilatation, the fields transform as given in
(15b) with scaling dimensions $(\Delta_{\phi},\Delta_{a_{t}},\Delta_{a_{t}})$,
to be determined. We see that using these transformations of the fields, the
action transforms to a total derivative

$\delta_{\tilde{D}}\mathcal{S}_{GED}=\int
dt\,d^{2}x\,\delta_{\tilde{D}}\mathcal{L}_{GED}=\int
d^{2}x\Big{[}\partial_{t}(2\,t\mathcal{L}_{GED})+\partial_{k}(x^{k}\mathcal{L}_{GED})\Big{]}$
(40)

if the scaling dimensions of the fields are taken to be

$\Delta_{\phi}=0,\,\Delta_{a_{i}}=1,\,\Delta_{a_{t}}=2.$ (41)

Under special conformal transformations, using (16) the action changes as

$\displaystyle\delta_{\tilde{K}}\mathcal{S}_{GED}=\int
dtd^{2}x\Big{[}\partial_{t}(2t^{2}\mathcal{L}_{GED})+\partial_{k}(2tx^{k}\mathcal{L}_{GED})\Big{]}.$
(42)

We see that the action transforms to a total derivatives under boosts,
dilatation and the the special conformal transformation. Thus, the action is
invariant under the Schrödinger symmetry in $d=3$ dimensions.

Now we will show that the GED action is invariant under the infinite
symmetries generated by $Y_{i}^{(n)}$ and the global symmetries generated by
$Z^{(n)}$ for $n=-1,0,1$. To see this, we use the transformations of the
fields (2.2), (2.2) under $Y_{i}^{(n)}$ and $Z^{(n)}$, and get the changes in
the action as

$\displaystyle\delta_{Z^{(n)}}\mathcal{S}_{GED}=\int
dtd^{2}x\Big{[}\partial_{t}\Big{(}2t^{n+1}\mathcal{L}_{GED}\Big{)}+\partial_{k}\Big{(}\,(n+1)t^{n}x^{k}\mathcal{L}_{GED}\Big{)}$
$\displaystyle\hskip 113.81102pt-n(n+1)(n-1)\,t^{n-2}(\phi)^{2}\Big{]},$
$\displaystyle\delta_{Y^{(n)}_{k}}\mathcal{S}_{GED}=\int
dt\,d^{2}x\,\partial_{k}\Big{[}-\Big{(}t^{n+\frac{1}{2}}\,\mathcal{L}_{GED}-\frac{1}{2}(n-\frac{1}{2})(n+\frac{1}{2})t^{n-\frac{3}{2}}(\phi)^{2}\Big{)}\Big{]}.$
(43)

We see that $\delta_{Y_{k}^{(n)}}\mathcal{S}_{GED}$ is a total derivative for
all $n$, and $\delta_{Z^{(n)}}\mathcal{S}_{GED}$ is a total derivative only
for $n=(-1,0,1)$, implying invariance under $Z^{(-1,0,1)}\equiv(H,D,K)$.

The next thing we are interested in is looking at the symmetries of the EOM
(37) of the theory. The analysis here is very similar to what we will present
later on in the Yang-Mills case. Instead of repeating ourselves, we point the
reader to Sec 4.4 and all the results would hold by putting the structure
constants to zero. The upshot is that the EOM are invariant under the finite
Schrödinger algebra as well as the infinite dimensional boosts, as in the case
of the action. However, like in the $d=4$ GED case, when the field $\phi$ is
turned off, the EOM exhibit an extended symmetry and remain invariant under
the full infinite Schrödinger-Virasoro algebra.

###  3.4 Electromagnetic duality

Earlier in this section, we encountered the electric-magnetic duality in 4$d$
GED which exchanges the electric and magnetic sectors (31). Then it is natural
to ask if there is some notion of similar duality in 3$d$ GED obtained by null
reduction. Given the result in 4$d$ GED, this is not obvious since we get only
the magnetic sector from null reduction. As we will see in this section, there
are electromagnetic type dualities in GED in $3$-dimensions, and these are of
different kind than the electric-magnetic duality in $4$-dimensions. We will
heavily use null reduction in the following.

####  3.4.1 Free theory

The electromagnetic duality in relativistic electrodynamics in $4$-dimensions
is the invariance of Maxwell’s equations $\partial_{\mu}F^{\mu\nu}=0$ and
Bianchi identity $\partial_{\mu}\tilde{F}^{\mu\nu}=0$ under the duality
transformation

$F^{\mu\nu}\rightarrow
F^{\prime\mu\nu}=\tilde{F}^{\mu\nu},\quad\tilde{F}^{\mu\nu}\rightarrow\tilde{F}^{\prime\mu\nu}=F^{\mu\nu}.$
(44)

Here the dual of the field strength is defined, in our conventions, as

$\tilde{F}^{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma},$
(45)

where $\epsilon^{\mu\nu\rho\sigma}$ is the totally antisymmetric tensor with
$\epsilon^{0123}=1$.

To get electromagnetic duality in GED in $3$-dimensions, let us perform null
reduction using the reduction ansatz (24), (35). The field strength reduces to
field strength like variables in the Galilean theory as

$\displaystyle W_{t}=F_{ut}=-\partial_{t}\phi,\quad
W_{i}=F_{ui}=-\partial_{i}\phi,\quad
W_{it}=F_{it}=\partial_{i}a_{t}-\partial_{t}a_{i},$ $\displaystyle
W_{ij}=F_{ij}=\partial_{i}a_{j}-\partial_{j}a_{i},$ (46)

and the dual field strength reduces to dual field strength like variables in
the Galilean theory as

$\displaystyle\tilde{W}^{t}=\tilde{F}^{ut}=\frac{1}{2}\epsilon^{tij}W_{ij},\quad\tilde{W}^{i}=\tilde{F}^{ui}=-\epsilon^{tij}W_{tj},\quad\tilde{W}^{it}=\tilde{F}^{it}=-\epsilon^{tij}W_{j},$
$\displaystyle\tilde{W}^{ij}=\tilde{F}^{ij}=\epsilon^{tij}W_{t}.$ (47)

Here $\epsilon^{tij}$ is totally antisymmetric with $\epsilon^{t23}=-1$ and it
is related to the totally antisymmetric tensor in $4$-dimensions as
$\epsilon^{utij}=\epsilon^{tij}$. Then the Galilean EOM can be written in
terms of the field strength like variables as

$\displaystyle\delta^{ij}\partial_{i}W_{j}=0,\quad\delta^{jk}\partial_{j}W_{ki}+\partial_{t}W_{i}=0,\quad\partial_{t}W_{t}+\delta^{ij}\partial_{i}W_{jt}=0.$
(48)

The null reduction of the Bianchi identity
$\partial_{\mu}\tilde{F}^{\mu\nu}=0$ gives Bianchi identities in the Galilean
theory as

$\displaystyle\partial_{t}\tilde{W}^{t}+\partial_{i}\tilde{W}^{i}=0,\quad\partial_{i}\tilde{W}^{it}=0,\quad\partial_{t}\tilde{W}^{ti}+\partial_{k}\tilde{W}^{ki}=0.$
(49)

Now to do null reduction of the relativistic electromagnetic duality
transformation (44), let us first express it in terms of field strength
$F_{\mu\nu}$ and dual field strength $\tilde{F}^{\mu\nu}$ with proper index
structure:

$\displaystyle
F^{\prime}_{\alpha\beta}=\eta_{\alpha\mu}\eta_{\beta\nu}\tilde{F}^{\mu\nu},\quad\tilde{F}^{\prime\mu\nu}=\eta^{\mu\alpha}\eta^{\beta\nu}F_{\alpha\beta}.$
(50)

Null reducing these expressions, we get the duality transformations in
Galilean theory, $W\rightarrow W^{\prime}=\tilde{W}$,
$\tilde{W}\rightarrow\tilde{W}^{\prime}=W$ as

$\displaystyle W_{t}\rightarrow W^{\prime}_{t}=-\tilde{W}^{t},\quad
W_{i}\rightarrow W^{\prime}_{i}=\delta_{ij}\tilde{W}^{tj},\quad
W_{ti}\rightarrow W^{\prime}_{ti}=\delta_{ij}\tilde{W}^{j},$ $\displaystyle
W_{ij}\rightarrow W^{\prime}_{ij}=\delta_{ik}\delta_{jl}\tilde{W}^{kl},$
$\displaystyle\tilde{W}^{t}\rightarrow\tilde{W}^{\prime
t}=-W_{t},\quad\tilde{W}^{i}\rightarrow\tilde{W}^{\prime
i}=\delta^{ij}W_{tj},\quad\tilde{W}^{ti}\rightarrow\tilde{W}^{\prime
ti}=\delta^{ij}W_{j},$
$\displaystyle\tilde{W}^{ij}\rightarrow\tilde{W}^{\prime
ij}=\delta^{ik}\delta^{jl}W_{kl}.$ (51)

We see that under the transformations (51), the Galilean EOM (48) for
$(W_{t},W_{i},W_{ti},W_{ij})$ become Galilean Bianchi identities (49) for
$(\tilde{W}^{\prime t},\tilde{W}^{\prime i},\tilde{W}^{\prime
ti},\tilde{W}^{\prime ij})$ and the Galilean Bianchi identities for
$(\tilde{W}^{t},\tilde{W}^{i},\tilde{W}^{ti},\tilde{W}^{ij})$ become Galilean
EOM for $(W^{\prime}_{t},W^{\prime}_{i},W^{\prime}_{ti},W^{\prime}_{ij})$.
Thus the field equations (48) and (49) are invariant under the transformations
(51).

####  3.4.2 Interacting theory

We have a generalization of the electromagnetic duality to $SL(2,\mathbb{R})$
duality in a relativistic interacting theory of a complex scalar $Z(x)$ and
the electromagnetic field $A_{\mu}(x)$ in $4$-dimensions Freedman:2012zz .
This theory is described by the Lagrangian density

$\mathcal{L}_{ED}=-\frac{1}{4}(ImZ)F_{\mu\nu}F^{\mu\nu}-\frac{1}{8}(ReZ)\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma},$
(52)

and the EOM

$\partial_{\mu}\big{[}(ImZ)F^{\mu\nu}+(ReZ)\tilde{F}^{\mu\nu}\big{]}=0,$ (53)

along with the Bianchi identity. These EOM and Bianchi identity are invariant
under $SL(2,\mathbb{R})$ duality transformations

$\displaystyle\mathcal{S}=\bigg{(}\begin{matrix}d&c\\\
b&a\end{matrix}\bigg{)},\quad\quad ad-bc=1,$ (54)

with the fields transforming as

$\displaystyle F_{\mu\nu}\rightarrow
F^{\prime}_{\mu\nu}=\big{[}d+\frac{c}{2}(Z+\bar{Z})\big{]}F_{\mu\nu}+\frac{ic}{2}(Z-\bar{Z})\tilde{F}^{\alpha\beta}\eta_{\alpha\mu}\eta_{\beta\nu},$
$\displaystyle\tilde{F}^{\mu\nu}\rightarrow\tilde{F}^{\prime\mu\nu}=\big{[}d+\frac{c}{2}(Z+\bar{Z})\big{]}\tilde{F}^{\mu\nu}-\frac{ic}{2}(Z-\bar{Z}){F}_{\alpha\beta}\eta^{\alpha\mu}\eta^{\beta\nu},$
(55)

and

$Z^{\prime}=\frac{aZ+b}{cZ+d}.$ (56)

We note that this duality is a symmetry of the EOM and the Bianchi identity
and not of the Lagrangian (52), as can be checked by a straightforward
calculation Freedman:2012zz .

Performing null reduction of (52), we get the Lagrangian describing a theory
of complex scalar and GED in $3$-dimensions:

$\displaystyle\mathcal{L_{M+S}}$ $\displaystyle=$
$\displaystyle(\text{Im}Z)\big{[}\frac{1}{4}W^{ij}W_{ij}+W^{i}W_{ti}+\frac{1}{2}W_{t}W_{t}\big{]}$
(57)
$\displaystyle-\frac{(\text{Re}Z)}{2}\big{[}\tilde{W}^{t}W_{t}+\tilde{W}^{i}W_{i}+\tilde{W}^{it}W_{it}+\frac{1}{2}\tilde{W}^{ij}W_{ij}\big{]}.$

The null reduction of (53) gives the EOM

$\displaystyle\partial_{t}\big{[}i(Z-\bar{Z})W_{t}+(Z+\bar{Z})\tilde{W}^{t}\big{]}+\partial_{i}\big{[}i(Z-\bar{Z})\delta^{ij}W_{jt}+(Z+\bar{Z})\tilde{W}^{i}\big{]}=0,$
$\displaystyle\partial_{i}\big{[}i(Z-\bar{Z})\delta^{ij}W_{j}+(Z+\bar{Z})\tilde{W}^{it}\big{]}=0,$
(58)
$\displaystyle\partial_{t}\big{[}i(Z-\bar{Z})\delta^{ij}W_{j}-(Z+\bar{Z})\tilde{W}^{ti}\big{]}+\partial_{k}\big{[}i(Z-\bar{Z})\delta^{kl}\delta^{ij}W_{lj}-(Z+\bar{Z})\tilde{W}^{ki}\big{]}=0.$

The Bianchi identities are same as given in (49). Now we perform null
reduction on the action of $SL(2,\mathbb{R})$ transformations on relativistic
fields (55). This null reduction, using (46), (47), gives the action of
$SL(2,\mathbb{R})$ transformations on the fields in the Galilean theory as

$\displaystyle W_{t}\ \rightarrow\
W^{\prime}_{t}=\big{[}d+\frac{c}{2}(Z+\bar{Z})\big{]}W_{t}-\frac{ic}{2}(Z-\bar{Z})\tilde{W}^{t},$
$\displaystyle W_{i}\ \rightarrow\
W^{\prime}_{i}=\big{[}d+\frac{c}{2}(Z+\bar{Z})\big{]}W_{i}+\frac{ic}{2}(Z-\bar{Z})\tilde{W}^{tj}\delta_{ji},$
$\displaystyle W_{it}\ \rightarrow\
W^{\prime}_{it}=\big{[}d+\frac{c}{2}(Z+\bar{Z})\big{]}W_{it}-\frac{ic}{2}(Z-\bar{Z})\tilde{W}^{j}\delta_{ji},$
$\displaystyle W_{ij}\ \rightarrow\
W^{\prime}_{ij}=\big{[}d+\frac{c}{2}(Z+\bar{Z})\big{]}W_{ij}+\frac{ic}{2}(Z-\bar{Z})\tilde{W}^{kl}\delta_{ki}\delta_{lj},$
(59)

and

$\displaystyle\tilde{W}^{t}\ \rightarrow\ \tilde{W}^{\prime
t}=\big{[}d+\frac{c}{2}(Z+\bar{Z})\big{]}\tilde{W}^{t}+\frac{ic}{2}(Z-\bar{Z})W_{t},$
$\displaystyle\tilde{W}^{i}\ \rightarrow\ \tilde{W}^{\prime
i}=\big{[}d+\frac{c}{2}(Z+\bar{Z})\big{]}\tilde{W}^{i}-\frac{ic}{2}(Z-\bar{Z})W_{tj}\delta{ji},$
$\displaystyle\tilde{W}^{it}\ \rightarrow\ \tilde{W}^{\prime
it}=\big{[}d+\frac{c}{2}(Z+\bar{Z})\big{]}\tilde{W}^{it}+\frac{ic}{2}(Z-\bar{Z})W_{j}\delta^{ji},$
$\displaystyle\tilde{W}^{ij}\ \rightarrow\ \tilde{W}^{\prime
ij}=\big{[}d+\frac{c}{2}(Z+\bar{Z})\big{]}\tilde{W}^{ij}-\frac{ic}{2}(Z-\bar{Z})W_{kl}\delta^{ki}\delta^{lj}.$
(60)

To see the invariance of the field equations, we can first replace fields
$(W,\tilde{W})$ by $(W^{\prime},\tilde{W}^{\prime})$ in (58),(49), _i.e._
assume that $(W^{\prime},\tilde{W}^{\prime})$ satisfy the field equations and
the Bianchi identities. Then using the transformations (59), (60) and $ad-
bc=1$, we can show that the fields $(W,\tilde{W})$ satisfy equations (58) and
(49).

The takeaway point from this analysis is that electromagnetic duality in
Galilean electrodynamics is very different in four and three dimensions. While
in four dimensions, there is an exchange between the electric and magnetic
sectors, which are in effect two different theories, in three dimensions, the
duality exists already in one sector, which reduces to the magnetic sector by
putting an extra field to zero. A priori, this may seem like a surprise, as
the actions of both the 4d and the 3d theories are almost identical and
obtained in the same way from the higher dimensional relativistic theory. The
answer to the difference is of course that the original 5d relativistic theory
from which the 4d action is derived, is not invariant under electromagnetic
duality, whereas the 4d relativistic theory is. The duality structure of the
3d Galilean theory is directly inherited from the 4d relativistic theory
through null reductions.

##  4 Non-Abelian Galilean Gauge theory

In this section, we move on to the non-Abelian Galilean gauge theories. Before
doing so, in order to set notation, let us very briefly recall the
relativistic theory and the symmetries associated with it.

It is well known that classical relativistic Yang-Mills theory is invariant
under conformal symmetry in four spacetime dimensions. The Yang-Mills action
in $(d+1)$-dimensions is:

$\displaystyle\mathcal{S}_{YM}=\int d^{d+1}x\,\mathcal{L}_{YM}=\int
d^{d+1}x\,\Big{(}-\frac{1}{4}F^{\tilde{\mu}\tilde{\nu}a}F_{\tilde{\mu}\tilde{\nu}}^{a}\Big{)},$
(61)

and the EOM

$\displaystyle\partial_{\tilde{\mu}}F^{\tilde{\mu}\tilde{\nu}a}+gf^{abc}A_{\tilde{\mu}}^{b}F^{\tilde{\mu}\tilde{\nu}c}=0,$
(62)

where $a=1,2,...,N^{2}-1$. The non-abelian field strength is defined as
$F_{\tilde{\mu}\tilde{\nu}}^{a}=\partial_{\tilde{\mu}}A_{\tilde{\nu}}^{a}-\partial_{\tilde{\nu}}A_{\tilde{\mu}}^{a}+gf^{abc}A_{\tilde{\mu}}^{b}A_{\tilde{\nu}}^{c}$.
Here, $A^{a}_{\tilde{\mu}}$ is the gauge field and $f^{abc}$ is the structure
constant of the underlying gauge group. The action (61) and equations (62) are
trivially invariant under Poincaré and scale transformations in all
dimensions.

$\displaystyle\delta_{K_{\tilde{\sigma}}}\big{(}\partial_{\tilde{\mu}}F^{\tilde{\mu}\tilde{\nu}a}+gf^{abc}A_{\tilde{\mu}}^{b}F^{\tilde{\mu}\tilde{\nu}c}\big{)}=\big{(}d-3\big{)}\big{[}F_{\tilde{\sigma}\tilde{\nu}}^{a}+\big{(}\partial_{\tilde{\sigma}}A^{a}_{\tilde{\nu}}-\eta_{\tilde{\sigma}\tilde{\nu}}\partial_{\tilde{\rho}}A^{\tilde{\rho}a}\big{)}+gf^{abc}\bigl{\\{}2A^{b}_{\tilde{\sigma}}A_{\tilde{\nu}}^{c}$
$\displaystyle\hskip
199.16928pt-\eta_{\tilde{\sigma}\tilde{\nu}}A^{\tilde{\rho}b}A_{\tilde{\rho}}^{c}+\eta_{\tilde{\sigma}\tilde{\rho}}x^{\tilde{\rho}}\big{(}\partial^{\tilde{\rho}}(A^{b}_{\tilde{\rho}}A^{c}_{\tilde{\nu}})$
$\displaystyle\hskip
199.16928pt+A^{\tilde{\rho}b}F_{\tilde{\rho}\tilde{\nu}}^{c}+gf^{cde}A^{\tilde{\rho}b}A^{d}_{\tilde{\rho}}A^{e}_{\tilde{\nu}}\big{)}\bigr{\\}}\big{]}.$
(63)

We see that the EOM are invariant under SCT in $d+1=4$ spacetime dimensions.
So relativistic Yang-Mills theory is invariant under the entire conformal
group at the classical level in four spacetime dimensions.

###  4.1 Galilean Yang-Mills Theory

We begin our investigation of non-Abelian gauge theories with a brief review
of existing literature. In Bagchi:2015qcw , a subset of us looked at the
generalised $SU(N)$ Galilean field theory. Below we review the simplest case,
i.e. the $SU(2)$ theory. The first non-trivial generalisation is the existence
of skewed limits when one deviates from a $U(1)$ theory. It is due to the
presence of three different gauge fields that leads to four limits instead of
two in the $U(1)$ case. The scaling of gauge fields
($A_{\mu}^{a}=a_{t}^{a},a_{i}^{a}$) are defined as

Electric limit: $\displaystyle a^{a}_{t}\rightarrow
a^{a}_{t},\,\,a^{a}_{i}\rightarrow\epsilon a^{a}_{i},$ (64a) Magnetic limit:
$\displaystyle a^{a}_{t}\rightarrow\epsilon a^{a}_{t},\,\,a^{a}_{i}\rightarrow
a^{a}_{i},$ (64b) EEM limit: $\displaystyle a^{1,2}_{t}\rightarrow
a^{1,2}_{t},\,\,a^{1,2}_{i}\rightarrow\epsilon
a^{1,2}_{i},~{}a^{3}_{t}\rightarrow\epsilon a^{3}_{t},\,\,a^{3}_{i}\rightarrow
a^{3}_{i},$ (64c) EMM limit: $\displaystyle a^{1}_{t}\rightarrow
a^{1}_{t},\,\,a^{1}_{i}\rightarrow\epsilon
a^{1}_{i},~{}a^{2,3}_{t}\rightarrow\epsilon
a^{2,3}_{t},\,\,a^{2,3}_{i}\rightarrow a^{2,3}_{i}.$ (64d)

Here, $a=(1,2,3)$ and $E$ denotes for electric and $M$ for magnetic case. When
we apply these four sets of scaling on the Non-abelian EOM (62), we will get
the vanilla limits (Electric and Magnetic limits) as well as the skewed limits
(EEM and EMM limits). We will now write down the EOM for each sector. They are
given by

  * •

Electric limit:

$\displaystyle\partial_{i}\partial_{i}a_{i}^{a}-\partial_{j}\partial_{i}a_{j}^{a}+\partial_{t}\partial_{i}a_{t}^{a}+g\epsilon^{abc}a_{t}^{b}\partial_{i}a_{t}^{c}=0,\quad\partial_{i}\partial_{i}A_{t}^{a}=0.$
(65)

  * •

Magnetic limit:

$\partial^{i}\partial_{i}a^{a}_{t}-\partial^{i}\partial_{t}a^{a}_{i}=0,\quad\partial^{j}\partial_{j}a^{a}_{i}-\partial^{j}\partial_{i}a^{a}_{j}=0.$
(66)

  * •

EEM limit:

$\displaystyle\partial^{i}\partial_{i}a_{t}^{1,2}=0,\quad\quad\partial^{i}\partial_{i}a_{j}^{1,2}-\partial^{i}\partial_{j}a_{i}^{1,2}+\partial_{t}\partial_{j}a_{t}^{1,2}=0,$
(67a)
$\displaystyle\partial^{i}\partial_{i}a_{t}^{3}-\partial^{i}\partial_{t}a_{i}^{3}=0,\quad\partial^{i}(\partial_{i}a_{j}^{3}-\partial_{j}a_{i}^{3})=0.$
(67b)

  * •

EMM limit:

$\displaystyle\partial^{i}(\partial_{i}a_{j}^{1}-\partial_{j}a_{i}^{1})+\partial_{t}\partial_{j}a_{t}^{1}+g\partial^{i}(a^{2}_{i}a^{3}_{j}-a^{3}_{i}a^{2}_{j})$
$\displaystyle+ga^{2}_{i}(\partial_{i}a^{3}_{j}-\partial_{j}a^{3}_{i})+ga^{3}_{i}(\partial_{j}a^{2}_{i}-\partial_{i}a^{2}_{j})$
$\displaystyle=$ $\displaystyle 0,$ (68a)
$\displaystyle\partial^{i}\partial_{i}a^{1}_{t}=0,\quad\partial^{i}(\partial_{i}a_{j}^{2,3}-\partial_{j}a_{i}^{2,3})$
$\displaystyle=$ $\displaystyle 0,$ (68b)
$\displaystyle\partial^{i}\partial_{i}a_{t}^{3}-\partial^{i}\partial_{t}a_{i}^{3}-2ga_{i}^{2}\partial^{i}a_{t}^{1}-gA_{t}^{1}\partial^{i}a_{i}^{2}$
$\displaystyle=$ $\displaystyle 0,$ (68c)
$\displaystyle\partial^{i}\partial_{i}a_{t}^{2}-\partial^{i}\partial_{t}a_{i}^{2}+2ga_{i}^{3}\partial^{i}a_{t}^{1}+ga_{t}^{1}\partial^{i}a_{i}^{3}$
$\displaystyle=$ $\displaystyle 0.$ (68d)

For $SU(2)$ theory, the structure constant $f^{abc}\equiv\varepsilon^{abc}$
with $\varepsilon^{abc}=\pm 1$ for different permutations. In Bagchi:2015qcw ,
the invariance of EOM of these four limits of Galilean Yang-Mills was seen
under infinite-dimensional GCA.

Below we will obtain a different set of EOM from the ones above by looking at
a null reduction of relativistic Yang-Mills in $d=4$. Needless to say, turning
the coupling $g=0$, and setting an extra field to zero, like in the
electrodynamics case, we will recover the Magnetic sector EOM above (68b). The
magnetic sector in the above is the same as the $U(1)$ case, with some added
gauge indices. So given the abelian null-reduced Galilean theory reproduces
the Magnetic equations, it is a foregone conclusion that turning off the gauge
coupling (and setting the extra field to zero) would reduce the EOM to (68b).
What is unexpected is that the EOM from the action we derive does not fit into
any one we derived using the limiting procedure. The infinite dimensional
symmetries, however, would still emerge from this new set of equations.

###  4.2 Null Reduction of Yang-Mills theory

We will construct the action for Galilean Yang-Mills theory by using the null
reduction procedure. We will follow the same technique that was applied to
scalar field theory and Maxwell theory to get their Galilean counterparts. We
write the Lagrangian density of Yang-Mills theory in $(d+1)$ dimensions (61)
in null coordinates (24):

$\displaystyle\mathcal{L}_{YM}=-\frac{1}{4}\eta^{\tilde{\mu}\tilde{\rho}}\eta^{\tilde{\nu}\tilde{\sigma}}F_{\tilde{\mu}\tilde{\nu}}^{a}F_{\tilde{\rho}\tilde{\sigma}}^{a}=-\frac{1}{4}\Big{[}2F^{a}_{ut}F^{a}_{tu}+F^{ija}F^{a}_{ij}+4F^{a}_{ui}F^{ia}_{t}\Big{]},$
(69)

and perform null reduction along the null direction parametrized by the
coordinate $u$. We take the gauge field to be independent of the
$u$-coordinate, _i.e._ $\partial_{u}A_{\tilde{\mu}}^{a}=0$, and decompose its
components as

$\displaystyle A_{u}^{a}=\phi^{a},\quad A_{t}^{a}=a_{t}^{a},\quad
A_{i}^{a}=a_{i}^{a}.$ (70)

Then the null reduction gives the Lagrangian density in $d$ spacetime
dimensions as

$\displaystyle\mathcal{L}_{GYM}=\Big{[}\frac{1}{2}(\partial_{t}\phi^{a}-gf^{abc}\phi^{b}a_{t}^{c})(\partial_{t}\phi^{a}-gf^{ade}\phi^{d}a_{t}^{e})-\frac{1}{4}(\partial^{i}a^{j}-\partial^{j}a^{i}+gf^{ade}a^{id}a^{je})$
$\displaystyle(\partial_{i}a_{j}-\partial_{j}a_{i}+gf^{abc}a_{i}^{b}a_{j}^{c})+(\partial_{i}\phi^{a}-gf^{abc}\phi^{b}a_{i}^{c})(\partial_{t}a^{ia}-\partial^{i}a_{t}^{a}+gf^{abc}a_{t}^{b}{a}^{ic})\Big{]},$
(71)

where $GYM$ stands for Galilean Yang-Mills. It can also be written in a
compact form given by

$\displaystyle\mathcal{L}_{GYM}=\frac{1}{2}D_{t}\phi^{a}D_{t}\phi^{a}+D_{i}\phi^{a}E^{ia}-\frac{1}{4}W^{ija}W_{ij}^{a},$
(72)

where $D_{t}$, $D_{i}$ are gauge-covariant derivatives and $E^{ia}$,
$W^{a}_{ij}$ are field strength variables defined as

$\displaystyle
D_{t}\phi^{a}=\partial_{t}\phi^{a}-gf^{abc}\phi^{b}a_{t}^{c},~{}D_{i}\phi^{a}=\partial_{i}\phi^{a}-gf^{abc}\phi^{b}a_{i}^{c},$
(73a) $\displaystyle
E^{ia}=\partial_{t}a^{ia}-\partial^{i}a_{t}^{a}+gf^{abc}a_{t}^{b}{a}^{ic},~{}W_{ij}^{a}=\partial_{i}a_{j}-\partial_{j}a_{i}+gf^{abc}a_{i}^{b}a_{j}^{c}.$
(73b)

The EOM for the Lagrangian (72) are given by

$\displaystyle D_{t}D_{t}\phi^{a}+D_{i}E^{ia}=0,$ (74a) $\displaystyle
D_{i}D_{i}\phi^{a}+gf^{abc}\phi^{b}D_{t}\phi^{c}=0,$ (74b) $\displaystyle
D_{t}D_{i}\phi^{a}-D_{j}W_{ji}^{a}-gf^{abc}\phi^{b}E^{ic}=0.$ (74c)

We can also find these equations by doing the procedure of null reduction on
relativistic equations.111The Lagrangian was also introduced in Gomis:2020fui
to derive the EOM for Galilean Yang-Mills theory with $U(N)$ gauge group
obtained as an effective theory from non-relativistic open string theory.

If we put $\phi^{a}=0$, the equations become

$\displaystyle D_{i}E^{ia}=0,~{}~{}D_{j}W_{ji}^{a}=0.$ (75)

###  4.3 Symmetries of Galilean Yang-Mills theory in $d=4$

We will explicitly look at the symmetries of the Galilean Yang-Mills at the
level of Lagrangian and EOM in $d=4$. We will use the action of GCA to find
the symmetries of (72) and (74). In the representation theory, we have some
undefined constants given as $(\Delta,a,r,s)$. They depend on the fields of
the theory under consideration. The scaling weight ($\Delta$) for a particular
field gets fixed when we impose invariance of the Lagrangian under scale
transformation. Similarly, the constants $(a,r,s)$ get fixed by comparing the
results from (8b) with $n=0$ and taking the non-relativistic limit on boost
transformations for a particular field. For our case, the values of these
constants are given by

$\displaystyle\Delta_{\phi}=\Delta_{a_{i}}=\Delta_{a_{t}}=1,~{}a=0,r=-1,s=1.$
(76)

In the coming section, we will use these values of the constants to find the
invariance of the Lagrangian and EOM under GCA.

####  4.3.1 Gauge invariance of Action

Before looking at spacetime symmetries of the Lagrangian, we will first
discuss in details the gauge transformations. We will evaluate non-
relativistic gauge transformation by doing a null reduction on relativistic
transformations. We will begin with relativistic gauge transformations given
by

$\displaystyle A_{\mu}^{a}\rightarrow
A_{\mu}^{a}+\frac{1}{g}\partial_{\mu}\alpha^{a}+f^{abc}A^{b}_{\mu}\alpha^{c}.$
(77)

Performing a null reduction along $u$ direction and using (70), the final
result comes out to be

$\displaystyle a_{i}^{a}\rightarrow
a_{i}^{a}+\frac{1}{g}\partial_{i}\alpha^{a}+f^{abc}a_{i}^{b}\alpha^{c},$ (78a)
$\displaystyle a_{t}^{a}\rightarrow
a_{t}^{a}+\frac{1}{g}\partial_{t}\alpha^{a}+f^{abc}a_{t}^{b}\alpha^{c},$ (78b)
$\displaystyle\phi^{a}\rightarrow\phi^{a}+f^{abc}\phi^{b}\alpha^{c}.$ (78c)

Here, the field $\phi^{a}$ now transforms as a scalar field in adjoint
representation. Let us see how the Lagrangian changes under these
transformations. The quantities (73) transform as

$\displaystyle D_{t}\phi^{a}\rightarrow
D_{t}\phi^{a}+f^{abc}\alpha^{c}D_{t}\phi^{b},~{}~{}D_{i}\phi^{a}\rightarrow
D_{i}\phi^{a}+f^{abc}\alpha^{c}D_{i}\phi^{b},$ (79a) $\displaystyle
E^{ia}\rightarrow E^{ia}+f^{abc}\alpha^{c}E^{ib},~{}~{}W_{ij}^{a}\rightarrow
W_{ij}^{a}+f^{abc}\alpha^{c}W_{ij}^{b}.$ (79b)

Using (79), the Lagrangian density change as $\delta\mathcal{L}_{GYM}=0$. It
tell us that the action is invariant under the gauge transformations (78).

####  4.3.2 Spacetime symmetries at the level of Action

We will first look into the symmetries of the action and then at the level of
equations. The action and EOM are trivially invariant under translations
$(H,P_{i})$ and rotations $(J_{ij})$. We will only show the invariance under
boost $(B_{i})$, scale transformation $(D)$ and SCT $(K,K_{i})$ and then will
move on to exhibit the invariance under infinite extension
$(L^{(n)},M^{(n)}_{i})$ of GCA.

Boost transformations: For our case, fields transform under boosts as

$\displaystyle\delta_{B_{k}}\phi=-(t\partial_{k}\phi),~{}\delta_{B_{k}}a_{t}^{a}=-(t\partial_{k}a_{t}^{a}+a_{k}^{a}),~{}\delta_{B_{k}}a_{i}^{a}=-(t\partial_{k}a_{i}^{a}-\delta_{ik}\phi^{a}),$
(80)

We can find these transformations by looking at (8b) with $n=0$ and using the
values of the constants as given in (76). To compare these transformation to
the ones we have in (8b) with $n=0$, we have used the relation
$\delta_{\epsilon}\varphi=[\epsilon Q,\varphi(t,x)]$, where $Q$ is the
generator of the infinitesimal symmetry transformations acting on a generic
field $\varphi$ and $\epsilon$ is the symmetry parameter. The action (72)
transforms as

$\displaystyle\delta_{B_{k}}S_{GYM}=\int
dt\,d^{3}x\,\Big{(}\delta_{B_{k}}\mathcal{L}_{GYM}\Big{)}$
$\displaystyle\hskip 17.07182pt=\int
dt\,d^{3}x\,\partial_{k}\Big{[}-t\Big{(}\frac{1}{2}D_{t}\phi^{a}D_{t}\phi^{a}+E^{ia}D_{i}\phi^{a}-\frac{1}{4}W^{ija}W_{ij}^{a}\Big{)}\Big{]}$
(81)

We see that the action comes out to be invariant under boost transformation
(80).

Scale transformation: We will see how dilatation $(D)$ affects the Lagrangian.
The field transformations under dilatation is given by

$\displaystyle\delta_{D}\Phi^{a}=\,(t\partial_{t}+x^{k}\partial_{k}+1)\Phi^{a},$
(82)

where $\Phi^{a}\equiv(\phi^{a},a_{i}^{a},a_{t}^{a})$. The action changes as

$\displaystyle\delta_{D}S_{GYM}=\int
dtd^{3}x\Big{[}\partial_{t}(\,t\mathcal{L}_{GYM})+\partial_{k}(\,x^{k}\mathcal{L}_{GYM})\Big{]}.$
(83)

The action is invariant under dilatation.

$K_{i}$ transformations: The fields $\Phi^{a}=(\phi^{a},a_{i}^{a},a_{t}^{a})$
transforms under $K_{i}$ as

$\displaystyle\delta_{K_{i}}\phi^{a}=-(t^{2}\partial_{i}\phi^{a}),~{}\delta_{K_{i}}a_{t}^{a}=-(t^{2}\partial_{i}a_{t}^{a}+2ta_{i}^{a}),~{}\delta_{K_{i}}a_{j}^{a}=-(t^{2}\partial_{i}a_{j}^{a}-2t\delta_{ij}\phi^{a}).$
(84)

Using these transformations of the fields, we get the change in the action

$\displaystyle\delta_{K_{i}}S_{GYM}=\int
dt\,d^{3}x\,\Big{(}\delta_{K_{i}}\mathcal{L}_{GYM}\Big{)}$
$\displaystyle\hskip 17.07182pt=\int
dt\,d^{3}x\,\partial_{l}\Big{[}-t^{2}\Big{(}\frac{1}{2}D_{t}\phi^{a}D_{t}\phi^{a}+E^{ia}D_{i}\phi^{a}-\frac{1}{4}W^{ija}W_{ij}^{a}\Big{)}+(\phi^{a})^{2}\Big{]}$
$\displaystyle\hskip 17.07182pt=\int
dt\,d^{3}x\,\partial_{l}\Big{[}-t^{2}\,\mathcal{L}_{GYM}+(\phi^{a})^{2}\Big{]}.$
(85)

Again, the action is invariant under $K_{i}$ transformation.

$K$ transformation: Under $K$, the fields transforms as

$\displaystyle\delta_{K}a_{t}^{a}=(t^{2}\partial_{t}a_{t}^{a}+2tx^{k}\partial_{k}a_{t}^{a}+2ta_{t}^{a}+2x^{k}a_{k}^{a}),$
(86a)
$\displaystyle\delta_{K}a_{i}^{a}=(t^{2}\partial_{t}a_{i}^{a}+2tx^{k}\partial_{k}a_{i}^{a}+2ta_{i}^{a}-2x_{i}\phi^{a}),$
(86b)
$\displaystyle\delta_{K}\phi^{a}=(t^{2}\partial_{t}\phi^{a}+2tx^{k}\partial_{k}\phi^{a}+2t\phi^{a}).$
(86c)

We have used (76) in (8) with $n=1$ to get these variations in fields. Using
these transformations in action, we get

$\displaystyle\delta_{K}S_{GYM}=\int
dt\,d^{3}x\,\Big{[}\partial_{t}\Big{\\{}t^{2}\Big{(}\frac{1}{2}D_{t}\phi^{a}D_{t}\phi^{a}+E^{ia}D_{i}\phi^{a}-\frac{1}{4}W^{ija}W_{ij}^{a}\Big{)}+(\phi^{a})^{2}\Big{\\}}$
$\displaystyle\hskip
113.81102pt+\partial_{k}\Big{\\{}2tx^{k}\Big{(}\frac{1}{2}D_{t}\phi^{a}D_{t}\phi^{a}+E^{ia}D_{i}\phi^{a}-\frac{1}{4}W^{ija}W_{ij}^{a}\Big{)}\Big{\\}}\Big{]}$
$\displaystyle\hskip 42.67912pt=\int
dt\,d^{3}x\,\Big{[}\partial_{t}\Big{(}t^{2}\mathcal{L}_{GYM}+(\phi^{a})^{2}\Big{)}+\partial_{k}\Big{(}2tx^{k}\mathcal{L}_{GYM}\Big{)}\Big{]}.$
(87)

The action comes out to be invariant under $K$ transformation.

We have seen that the action is invariant under finite GCA. We will now move
on to the infinite extension of GCA. We will see how the action changes and
get the required invariance under these transformations (8).

$M_{k}^{(n)}$ transformations: The fields under $M^{(n)}_{k}$ transform as

$\displaystyle\delta_{M^{(n)}_{k}}a_{t}^{a}=-(t^{n+1}\partial_{k}a_{t}^{a}+(n+1)t^{n}a_{k}^{a}),~{}~{}\delta_{M^{(n)}_{k}}\phi^{a}=-(t^{n+1}\partial_{k}\phi^{a}),$
(88a)
$\displaystyle\delta_{M^{(n)}_{k}}a_{i}^{a}=-(t^{n+1}\partial_{k}a_{i}^{a}-(n+1)t^{n}\delta_{ik}\phi^{a}).$
(88b)

If we take $n=0,\pm 1$ in (88), we get back the global transformations. The
action changes as

$\displaystyle\delta_{M^{(n)}_{k}}S_{GYM}=\int
dt\,d^{3}x\,\partial_{k}\Big{[}-\,t^{n+1}\Big{(}\frac{1}{2}D_{t}\phi^{a}D_{t}\phi^{a}+E^{ia}D_{i}\phi^{a}-\frac{1}{4}W^{ija}W_{ij}^{a}\Big{)}$
$\displaystyle\hskip
199.16928pt+\frac{1}{2}n(n+1)t^{n-1}(\phi^{a})^{2}\Big{]}$
$\displaystyle\hskip 22.76228pt=\int
dt\,d^{3}x\,\partial_{k}\Big{[}-\Big{(}t^{n+1}\,\mathcal{L}_{GYM}-\frac{1}{2}n(n+1)t^{n-1}(\phi^{a})^{2}\Big{)}\Big{]}.$
(89)

The action becomes invariant under $M^{(n)}_{k}$ transformations.

$L^{(n)}$ transformations: The fields under $L^{(n)}$ transforms as

$\displaystyle\delta_{L^{(n)}}a_{t}^{a}=\Big{(}t^{n+1}\partial_{t}a_{t}^{a}+(n+1)t^{n}x^{k}\partial_{k}a_{t}^{a}+(n+1)t^{n}a_{t}^{a}+n(n+1)t^{n-1}x^{k}a_{k}^{a}\Big{)},$
(90a)
$\displaystyle\delta_{L^{(n)}}a_{i}^{a}=\Big{(}t^{n+1}\partial_{t}a_{i}^{a}+(n+1)t^{n}x^{k}\partial_{k}a_{i}^{a}+(n+1)t^{n}a_{i}^{a}-n(n+1)t^{n-1}x_{i}\phi^{a}\Big{)},$
(90b)
$\displaystyle\delta_{L^{(n)}}\phi^{a}=\Big{(}t^{n+1}\partial_{t}\phi^{a}+(n+1)t^{n}x^{k}\partial_{k}\phi^{a}+(n+1)t^{n}\phi^{a}\Big{)}.$
(90c)

Under these transformations the action changes as

$\displaystyle\delta_{L^{(n)}}S_{GYM}=\int
dtd^{3}x\Big{[}\partial_{t}\Big{(}\Big{\\{}t^{n+1}\mathcal{L}_{GYM}+\frac{1}{2}n(n+1)t^{n-1}(\phi^{a})^{2}\Big{\\}}\Big{)}$
$\displaystyle\hskip
28.45274pt+\partial_{k}\Big{(}\,(n+1)t^{n}x^{k}\mathcal{L}_{GYM}\Big{)}-n(n+1)(n-1)\,t^{n-2}(\phi^{a})^{2}\Big{]}.$
(91)

The action is not invariant under $L^{(n)}$ transformations for all $n$, but
invariant only under the global part, _i.e._ $L^{(-1,0,1)}\equiv(H,D,K)$.

In conclusion, the action is invariant under $M^{(n)}_{k}$ but not invariant
under $L^{(n)}$ for all $n$. It comes out to be invariant under the global
part of $L^{(-1,0,1)}\equiv(H,D,K)$.

####  4.3.3 Spacetime symmetries at the level of EOM

We will now see the invariance at the level of EOM. Here, we will only show
the changes of equations under $L^{(n)},M^{(n)}_{k}$. We can find the global
part by taking the value of $n=0\pm 1$.

$M_{k}^{(n)}$ transformations: We will use (88) to find the invariance of
equations (74) under $M_{k}^{(n)}$. We get the result as

$\displaystyle\delta_{M^{(n)}_{k}}\eqref{EOM-1}=-t^{n+1}\partial_{k}\eqref{EOM-1}-(n+1)t^{n}\eqref{EOM3}=0,$
(92a)
$\displaystyle\delta_{M^{(n)}_{k}}\eqref{EOM3}=-t^{n+1}\partial_{k}\eqref{EOM3}-(n+1)t^{n}\delta_{ik}\eqref{EOM2}=0,$
(92b)
$\displaystyle\delta_{M^{(n)}_{k}}\eqref{EOM2}=-t^{n+1}\partial_{k}\eqref{EOM2}=0.$
(92c)

Here, 74a, 74b, etc means that we are considering only the LHS of (74).

$L^{(n)}$ transformations: Similarly, using (90) on (74), we get

$\displaystyle\delta_{L^{(n)}}\eqref{EOM-1}=(t^{n+1}\partial_{t}+(n+1)t^{n}x^{k}\partial_{k}+3(n+1)t^{n})\eqref{EOM-1}$
$\displaystyle\hskip
62.59596pt+n(n+1)t^{n-1}x^{k}\eqref{EOM3}+n(n+1)(n-1)t^{n-2}x^{k}D_{k}\phi^{a}$
$\displaystyle\hskip 170.71652pt+n(n+1)(n-1)t^{n-2}\phi^{a},$ (93a)
$\displaystyle\delta_{L^{(n)}}\eqref{EOM2}=(t^{n+1}\partial_{t}+(n+1)t^{n}x^{k}\partial_{k}+3(n+1)t^{n})\eqref{EOM2}=0,$
(93b)
$\displaystyle\delta_{L^{(n)}}\eqref{EOM3}=(t^{n+1}\partial_{t}+(n+1)t^{n}x^{k}\partial_{k}+3(n+1)t^{n})\eqref{EOM3}$
$\displaystyle\hskip 142.26378pt+n(n+1)t^{n-1}x^{k}\eqref{EOM2}=0.$ (93c)

We see that the (74b, 74c) are invariant under $L^{(n)}$ for all $n$. However,
(74a) is invariant only under $L^{(-1,0,1)}$.

In conclusion, the equations are completely invariant under $M^{(n)}_{k}$ but
not invariant under $L^{(n)}$ for all $n$. They come out to be invariant under
the global part of $L^{(-1,0,1)}\equiv(H,D,K)$.

Symmetries of equations when $\phi^{a}=0$: When we take $\phi^{a}=0$ in (74),
we get the equations given in (75). We will now look for the symmetries of
these equations under infinite extension of GCA. To find the fields variation
for this case, we have to put $\phi^{a}=0$ in the transformations (88, 90).
The invariance of (75) under $L^{(n)}$ is given by

$\displaystyle\delta_{L^{(n)}}(D_{i}E^{ia})=\Big{[}t^{n+1}\partial_{t}+(n+1)t^{n}x^{k}\partial_{k}+3(n+1)t^{n}\Big{]}(D_{i}E^{ia})$
$\displaystyle\hskip 162.18062pt+n(n+1)t^{n-1}x^{k}(D^{i}W_{ki}^{a})=0,$ (94a)
$\displaystyle\delta_{L^{(n)}}(D^{i}W_{ij}^{a})=\Big{[}t^{n+1}\partial_{t}+(n+1)t^{n}x^{k}\partial_{k}+3(n+1)t^{n}\Big{]}(D^{i}W_{ij}^{a})=0.$
(94b)

Similarly, under $M^{(n)}_{k}$

$\displaystyle\delta_{M^{(n)}_{k}}(D_{i}E^{ia})=0,~{}~{}\delta_{M^{(n)}_{k}}(D^{i}W_{ij}^{a})=0.$
(95)

These equations are invariant under full infinite dimensional GCA.

So, in conclusion, the symmetry analysis of Galilean Yang-Mills in $d=4$
obtained by null reducing the relativistic Yang-Mills action in $d=5$ gives us

  * •

At the level of action: Invariance under finite GCA + infinite dimensional
boosts.

  * •

At the level of EOM: Invariance under finite GCA + infinite dimensional
boosts.

  * •

At the level of EOM, with fields turned off: Invariance under infinite GCA.

An interesting point to re-emphasise is that the EOM even with the fields
$\phi^{a}$ turned off are different from the ones obtained by taking limits as
was described earlier in the section. It is possible that one needs to
consider scalings of the gauge coupling $g$ in order to obtain these
results222A similar phenomenon was observed when constructing the action of
Carrollian scalar electrodynamics in Bagchi:2019clu ..

###  4.4 Spacetime symmetries of Galilean Yang-Mills theory in $d=3$

We will now focus on symmetries of the Galilean Yang-Mills action in three
dimensions. We will show that the Galilean Yang-Mills action is invariant
under Schrödinger symmetries. The symmetry analysis for this theory under the
Galilean transformations $(H,P_{i},G_{i},J_{ij})$ is very similar to the $d=4$
case, as discussed in the previous section. The following straightforward yet
crucial analysis reveals the infinite number of $Y^{(n)}$ generators to be
symmetries of theory.

####  4.4.1 Symmetries at the level of action

Under scale transformation (15b), the action transforms as

$\delta_{\tilde{D}}S_{GYM}=\int
dt\,d^{2}x\,\Big{(}\delta_{\tilde{D}}\mathcal{L}_{GYM}\Big{)}=\int
dtd^{2}x\big{[}\partial_{t}(2t\mathcal{L}_{GYM})+\partial_{k}(x^{k}\mathcal{L}_{GYM})\big{]},$
(96)

which is a total derivative, implying the invariance of the action under
dilatation. Under the special conformal transformation (16), the action
changes as

$\delta_{\tilde{K}}S=\int
dtd^{2}x\big{[}\partial_{t}(2t^{2}\mathcal{L}_{GYM})+\partial_{k}(2tx^{k}\mathcal{L}_{GYM})\big{]}.$
(97)

Thus the action is invariant under special conformal transformation in $d=3$.
We will now move on to the invariance of the action under the infinite
extension of Schrödinger algebra. Under $Z^{(n)}$ and $Y_{k}^{(n)}$
transformations, the fields transform as

$\displaystyle\delta_{Z^{(n)}}\phi^{a}=2t^{n+1}\partial_{t}\phi^{a}+(n+1)t^{n}(x^{k}\partial_{k}+\Delta_{\phi})\phi^{a},$
$\displaystyle\delta_{Z^{(n)}}a_{i}^{a}=2t^{n+1}\partial_{t}a_{i}^{a}+(n+1)t^{n}(x^{k}\partial_{k}+\Delta_{a_{i}})a_{i}^{a}-n(n+1)t^{n-1}x_{i}\phi^{a},$
$\displaystyle\delta_{Z^{(n)}}a_{t}^{a}=2t^{n+1}\partial_{t}{a}_{t}^{a}+(n+1)t^{n}(x^{k}\partial_{k}+\Delta_{a_{t}})a_{t}^{a}+n(n+1)t^{n-1}x^{k}a_{k}^{a}.$
(98)

Similarly for $Y$, we have

$\displaystyle\delta_{Y^{(m)}_{k}}\phi^{a}=t^{m+\frac{1}{2}}\,\partial_{k}\phi^{a},~{}\delta_{Y^{(m)}_{k}}a_{i}^{a}=t^{m+\frac{1}{2}}\partial_{k}a_{i}^{a}-\big{(}m+\frac{1}{2}\big{)}t^{m-\frac{1}{2}}\phi^{a},$
$\displaystyle\delta_{Y^{(m)}_{k}}a_{t}^{a}=t^{m+\frac{1}{2}}\partial_{k}a_{t}^{a}+\big{(}m+\frac{1}{2}\big{)}t^{m-\frac{1}{2}}a_{k}^{a}.$
(99)

Under $Z$ and $Y$ transformations above, the changes in the action are

$\displaystyle\delta_{Z^{(n)}}S_{GYM}=\int
dtd^{2}x\Big{[}\partial_{t}\Big{(}2t^{n+1}\mathcal{L}_{GYM}\Big{)}+\partial_{k}\Big{(}\,(n+1)t^{n}x^{k}\mathcal{L}_{GYM}\Big{)}$
$\displaystyle\hskip 85.35826pt-n(n+1)(n-1)\,t^{n-2}(\phi^{a})^{2}\Big{]},$
(100) $\displaystyle\delta_{Y^{(n)}_{k}}S_{GYM}=\int
dt\,d^{2}x\,\partial_{k}\Big{[}-\Big{(}t^{n+\frac{1}{2}}\,\mathcal{L}_{GYM}-\frac{1}{2}(n-\frac{1}{2})(n+\frac{1}{2})t^{n-\frac{3}{2}}(\phi^{a})^{2}\Big{)}\Big{]}.$
(101)

In conclusion, the action comes out to be invariant under
$Z^{(-1,0,1)}\equiv(H,D,K)$ and completely invariant under $Y^{(n)}_{k}$
transformations.

####  4.4.2 Symmetries at the level of EOM

We will focus on the invariance of the EOM under the infinite dimensional
Schrödinger algebra.

$Y_{k}^{(n)}$ transformations: The invariance of EOM are given as below

$\displaystyle\delta_{Y^{(n)}_{k}}\eqref{EOM-1}=-t^{n+\frac{1}{2}}\partial_{k}\eqref{EOM-1}-(n+\frac{1}{2})t^{n-\frac{1}{2}}\eqref{EOM3}=0,$
(102a)
$\displaystyle\delta_{Y^{(n)}_{k}}\eqref{EOM3}=-t^{n+\frac{1}{2}}\partial_{k}\eqref{EOM3}-(n+\frac{1}{2})t^{n-\frac{1}{2}}\delta_{ik}\eqref{EOM2}=0,$
(102b)
$\displaystyle\delta_{Y^{(n)}_{k}}\eqref{EOM2}=-t^{n+\frac{1}{2}}\partial_{k}\eqref{EOM2}=0.$
(102c)

$Z^{(n)}$ transformations: Under (4.4.1), the equatons (74) transforms as

$\displaystyle\delta_{Z^{(n)}}\eqref{EOM-1}=(2t^{n+1}\partial_{t}+(n+1)t^{n}x^{k}\partial_{k}+4(n+1)t^{n})\eqref{EOM-1}$
$\displaystyle\hskip
62.59596pt+n(n+1)t^{n-1}x^{k}\eqref{EOM3}-2n(n+1)(n-1)t^{n-2}\phi^{a},$ (103a)
$\displaystyle\delta_{Z^{(n)}}\eqref{EOM2}=(2t^{n+1}\partial_{t}+(n+1)t^{n}x^{k}\partial_{k}+2(n+1)t^{n})\eqref{EOM2}=0,$
(103b)
$\displaystyle\delta_{Z^{(n)}}\eqref{EOM3}=(2t^{n+1}\partial_{t}+(n+1)t^{n}x^{k}\partial_{k}+3(n+1)t^{n})\eqref{EOM3}$
$\displaystyle\hskip 142.26378pt-n(n+1)t^{n-1}x^{k}\eqref{EOM2}=0.$ (103c)

We see that the equations (74) are invariant under $Z^{(-1,0,1)}$ and
$Y_{k}^{(n)}$ for all $n$. Now for the case with $\phi^{a}=0$. The invariance
of (75) under $Z^{(n)}$ is given by

$\displaystyle\delta_{Z^{(n)}}(D_{i}E^{ia})=\Big{[}2t^{n+1}\partial_{t}+(n+1)t^{n}x^{k}\partial_{k}+4(n+1)t^{n}\Big{]}(D_{i}E^{ia})$
$\displaystyle\hskip 162.18062pt-n(n+1)t^{n-1}x^{k}(D^{i}W_{ki}^{a})=0,$
(104a)
$\displaystyle\delta_{Z^{(n)}}(D^{i}W_{ij}^{a})=\Big{[}2t^{n+1}\partial_{t}+(n+1)t^{n}x^{k}\partial_{k}+2(n+1)t^{n}\Big{]}(D^{i}W_{ij}^{a})=0.$
(104b)

Similarly, under $Y^{(n)}_{k}$

$\displaystyle\delta_{Y^{(n)}_{k}}(D_{i}E^{ia})=0,~{}~{}\delta_{Y^{(n)}_{k}}(D^{i}W_{ij}^{a})=0.$
(105)

We see that the equations (75) are invariant under the infinite extension of
Schr$\ddot{o}$dinger symmetry.

Again to summarise, the symmetry analysis of Galilean Yang-Mills in $d=3$
obtained by null reducing the relativistic Yang-Mills action in $d=4$ gives us

  * •

At the level of action: Invariance under finite Schrödinger algebra + infinite
dimensional boosts.

  * •

At the level of EOM: Invariance under finite Schrödinger algebra + infinite
dimensional boosts.

  * •

At the level of EOM, with field turned off: Invariance under infinite
Schrödinger-Virasoro algebra.

###  4.5 Scale transformation with generic $z$

Dynamical exponent $z$ is defined as the relative scaling of space and time
under the Dilatation operator:

$D:\quad t\to\lambda^{z}t,\quad x\to\lambda x.$ (106)

For the case of the Schrödinger algebra $z=2$ and, rather interestingly, for
the GCA $z=1$. Usually, $z=1$ indicates a relativistic theory, but GCFTs are
examples of non-relativistic QFTs with $z=1$.

We now discuss whether scale transformations with generic dynamical exponent
$z$ can be a symmetry at the level of Lagrangian (72) in $d$ spacetime
dimensions. To understand it, we first write down the scale transformation of
different fields $\Phi^{a}=(\phi^{a},\,a_{t}^{a},\,a_{i}^{a})$, given by

$\delta_{\bar{D}}\Phi^{a}=-(zt\partial_{t}+x^{k}\partial_{k}+\Delta_{\Phi})\Phi^{a}.$
(107)

We will now look at the transformation of different covariant terms (73) under
(107):

$\displaystyle\delta_{\bar{D}}(D_{t}\phi^{a})=\delta_{\bar{D}}(\partial_{t}\phi^{a}-gf^{abc}\phi^{b}a_{t}^{c})$
$\displaystyle=-(zt\partial_{t}+x^{k}\partial_{k})D_{t}\phi^{a}-(\Delta_{\phi}+z)\partial_{t}\phi^{a}+(\Delta_{\phi}+\Delta_{a_{t}})gf^{abc}\phi^{b}a_{t}^{c}$
(108)

If we demand that $(D_{t}\phi^{a})$ transforms as a primary, we find

$\Delta_{\phi}+z=\Delta_{\phi}+\Delta_{a_{t}}\Rightarrow\Delta_{a_{t}}=z.$
(109)

Similarly, demanding the same for $D_{i}\phi^{a},W_{ij}^{a},E^{ia}$, we get

$\Delta_{\phi}+1=\Delta_{\phi}+\Delta_{a_{i}}\implies\Delta_{a_{i}}=1$ (110)

Finally, we will now look at the change in the action. The remaining terms are
given by

$\displaystyle\delta_{\tilde{D}}S_{GYM}=\int
dt\,d^{d-1}x\,\Big{\\{}\Big{[}\frac{d-1-z}{2}-\Delta_{\phi}\Big{]}(D_{t}\phi^{a})^{2}-(\Delta_{\phi}+\Delta_{a_{i}}-d+2)D_{i}\phi^{a}E^{ia}$
$\displaystyle\hskip
170.71652pt-\Big{[}\frac{z+(d-1)}{4}-\frac{\Delta_{a_{i}}+1}{2}\Big{]}W^{ija}W_{ij}^{a}\Big{\\}}.$
(111)

The conditions we get from above are

$\displaystyle\Delta_{a_{i}}=1,~{}\Delta_{a_{t}}=z,~{}d-1-z=2\Delta_{\phi},~{}\Delta_{\phi}=d-3.$
(112)

So, from these equations we arrive at

$z+d=5.$ (113)

The significance of this equation for $d>5$ is not clear to us. This condition
should be satisfied to get the dilatation invariance of the Lagrangian (72).
Clearly, we get $z=1$ when $d=4$ and $z=2$ when $d=3$, which are both in
agreement with our analysis in the paper so far.

##  5 Noether charges and phase space analysis

The analysis above clearly reflects that the infinite dimensional Abelian
ideal along with 3 generators of the $SL(2)$ subalgebra for both the GCA and
the Schrodinger algebra, respectively for $d=4$ and $d=3$ are symmetries of
the theory. In the following we will investigate the dynamical realization of
these symmetries in terms of Noether charges. However, for both the cases, we
also noticed that the infinite dimensional enhancement of the $SL(2)$ part,
i.e. the Witt algebra generates on-shell symmetries, at the level of equations
of motion, when a particular field is turned off. This is a reflection of the
fact that these are weak symmetries in the sense elaborated in Beisert:2017pnr
; Beisert:2018zxs ; Bagchi:2019clu ; Banerjee:2020qjj . To investigate the
roles of these later ones, we look at their realizations on phase space, where
they generate Hamiltonian vector fields.

###  5.1 Galilean Yang-Mills theory in $d=4$

We will now examine the dynamical realization of the infinite dimensional GCA
on the space of fields satisfying the EOM. The Lagrangian is invariant under
$M^{(n)}_{i}$ and finite part of $L^{(n)}$, obviously giving rise to conserved
quantities. In spite of the fact that $L^{(n)},\,\forall n\in\mathbb{Z}$ are
not symmetry generators of the theory (72), we will see that they are the
symmetries of the phase space.

In order to construct the conserved quantities, we make use of the pre-
symplectic potential on the space of solutions, defined from the Lagrangian
via:

$\displaystyle\delta L=\int
d^{d-1}x\>\partial_{t}\underbrace{\Theta(\delta)}_{\mbox{(pre)-symplectic
potential}}~{}~{}:\text{on-shell}$ (114)

for a generic variation $\delta\varphi$. The pre-symplectic potential arising
from (72) is

$\displaystyle\Theta(\delta)=D_{t}\phi^{a}(\delta\phi^{a})+D_{i}\phi^{a}(\delta
a^{ia}).$ (115)

Now, for a specific symmetry transformation
$\varphi\to\varphi+\delta_{\epsilon}\varphi$:

$\displaystyle\delta_{\epsilon}L=\int
d^{d-1}x\>\partial_{t}\beta(\delta_{\epsilon})~{}~{}:\text{off-shell,}$ (116)

for some function $\beta$ in field space. Hence, comparing (114) and (116), we
deduce that on-shell, the Noether charges $Q_{\epsilon}$ are given by :

$\displaystyle Q_{\epsilon}=\int
d^{d-1}x\left(\Theta(\delta_{\epsilon})-\beta(\delta_{\epsilon})\right)$ (117)

The Noether charges for GCA in $d=4$ can now be calculated using this
technique. We will focus on the Noether charges for $L^{(-1,0,1)}=H,D,K$.

$\displaystyle Q_{H}=\int d^{3}x\,\Gamma\equiv H,$ (118) $\displaystyle
Q_{D}=\int
d^{3}x\,\Big{[}t\Gamma+(x^{k}\partial_{k}\phi^{a}+\phi^{a})D_{t}\phi^{a}+(x^{k}\partial_{k}a_{i}^{a}+a_{i}^{a})D_{i}\phi^{a}\Big{]},$
(119) $\displaystyle Q_{K}=\int
d^{3}x\,\Big{[}t^{2}\Gamma+2t(x^{k}\partial_{k}\phi^{a}+\phi^{a})D_{t}\phi^{a}+(2tx^{k}\partial_{k}a_{i}^{a}+2ta_{i}^{a}-2x_{i}\phi^{a})D_{i}\phi^{a}+(\phi^{a})^{2}\Big{]},$
(120)

where

$\Gamma=\frac{1}{2}D_{t}\phi^{a}D_{t}\phi^{a}+D_{i}\phi^{a}D_{i}a_{t}^{a}+\frac{1}{4}W^{ija}W_{ij}^{a}+gf^{abc}\phi^{b}a_{t}^{c}D_{t}\phi^{a}$

and $H$ is the Hamiltonian of our system. Interestingly, as in the Abelian
case Banerjee:2019axy  the charges corresponding to the $M^{(n)}_{k}$
transformations vanish for all $n$:

$Q_{M}=\int
d^{3}x\,\left(\Theta(\delta_{M^{(n)}_{k}})-\beta(\delta_{M^{(n)}_{k}})\right)=\int
d^{3}x\,\left(D_{t}\phi^{a}(\delta_{M^{(n)}_{k}}\phi^{a})+D_{i}\phi^{a}(\delta_{M^{(n)}_{k}}a^{ia})\right)=0.$
(121)

### Pre-symplectic structure

In order to inspect the action of the $L^{(n)}$ generators on the phase space,
we start with the pre-symplectic structure for Galilean Yang-Mills theory
given by the exterior derivative of the pre-symplectic potential on the space
of fields:

$\displaystyle\Omega(\delta_{1},\delta_{2})=\int
d^{3}x\,\left(\delta_{2}\Theta(\delta_{1})-\delta_{1}\Theta(\delta_{2})\right)$
$\displaystyle=\int
d^{3}x\,[D_{t}\delta_{2}\phi^{a}\delta_{1}\phi^{a}+D_{i}\delta_{2}\phi^{a}\delta_{1}a^{ia}-D_{t}\delta_{1}\phi^{a}\delta_{2}\phi^{a}-D_{i}\delta_{1}\phi^{a}\delta_{2}a^{ia}$
$\displaystyle\hskip 54.06006pt-
gf^{abc}(\phi^{b}\delta_{2}a_{t}^{c}\delta_{1}\phi^{a}+\phi^{b}\delta_{2}a_{i}^{c}\delta_{1}a^{ia}-\phi^{b}\delta_{1}a_{t}^{c}\delta_{2}\phi^{a}-\phi^{b}\delta_{1}a_{i}^{c}\delta_{2}a^{ia})].$
(122)

As expected in any gauge theory, the gauge transformation (78) gives a
degenerate direction in space of solutions:

$\displaystyle\Omega(\delta_{\alpha^{a}},\delta)=0$ (123)

for any arbitrary gauge parameter $\alpha^{a}$.

$M_{k}^{(n)}$ and $L^{(n)}$ transformations: Consistent with the finding that
conserved quantities associated with the $M_{k}^{(n)}$ transformations (88)
are all 0, we observe that their corresponding Hamiltonian functions are
trivially zero as well:

$\displaystyle\Omega(\delta_{M_{k}^{(n)}},\delta)=0.$ (124)

Curiously, $L^{(n)}$ transformations for all $n$ are Hamiltonian vector
fields, giving rise to non-trivial generators of canonical transformations,
despite the fact that only the $n=0,\pm 1$ modes are symmetries of the theory:

$\displaystyle\Omega(\delta_{L^{(n)}},\delta)=\delta\int
d^{3}x\,\Big{[}t^{n+1}\Big{\\{}\frac{1}{2}D_{t}\phi^{a}D_{t}\phi^{a}+\frac{1}{4}W^{ija}W_{ija}\Big{\\}}-(n+1)t^{n}\Big{\\{}2(a^{ia}D_{i}\phi^{a}+\phi^{a}D_{t}\phi^{a})$
$\displaystyle\hskip
71.13188pt+x^{k}(\phi^{a}\partial_{k}D_{t}\phi^{a}+a_{i}^{a}\partial_{k}D^{i}\phi^{a})\Big{\\}}+n(n+1)t^{n-1}(\phi^{a})^{2}\Big{]}=\delta
Q[f(t)].$ (125)

Algebra of the Hamiltonian functions: One of the most important questions in
the phase space analysis of GCA in context of Galilean Yang Mills theory is
whether the Witt sub-algebra of GCA generated by the Hamiltonian vector fields
$\delta_{L^{(n)}}$ gets centrally extended or not at the level of the
corresponding Hamiltonian functions. The answer turns out to be affirmative:

$\displaystyle\Omega(\delta_{L^{(n)}},\delta_{L^{(m)}})=(n-m)Q[L^{(n+m)}]+K_{L^{(n)},L^{(m)}}$
(126)

where $K_{L^{(n)},L^{(m)}}=\big{[}n(n^{2}-1)-m(m^{2}-1)\big{]}t^{n+m-1}\int
d^{3}x(\phi^{a})^{2}$ is a state dependent central extension, routinely
observed in asymptotic symmetries of gravitational theories Barnich:2011mi
and satisfies the 2 co-cycle condition:

$K_{[L^{(n)},L^{(m)}],L^{(r)}}+K_{[L^{(m)},L^{(r)}],L^{(n)}}+K_{[L^{(r)},L^{(n)}],L^{(m)}}=0.$
(127)

The same feature was observed in the in the Abelian case Banerjee:2019axy  as
well. The realization of the commutation relation
$[L^{n},M^{(m)}_{k}]=(n-m)M^{(n+m)}_{k}$ is trivial as the Hamiltonian
functions corresponding to $M^{(m)}_{k}$ are all zero.

###  5.2 Galilean Yang-Mills theory in $d=3$

Since the Lagrangian in $d=3$ has the same form as that of $d=4$, the
construction for Noether charges follow exactly the same route. Before going
into the details of some of the expressions for charges and the algebra, it is
important to point out that everything that we construct in this sub-section
also holds for the $U(1)$ case, just by turning off the structure constants.
So our conclusions for this sub-section equally applies to Galilean
Electrodynamics in $d=3$.

Conserved quantities corresponding to the $Z^{(n)},\,(n=0,\pm 1)$ symmetry
generators (4.4.1) of the Schrödinger algebra are as follows.

$\displaystyle Q_{H}=\int d^{2}x\,\mathcal{H}\equiv\text{Hamiltonian},$ (128)
$\displaystyle Q_{\tilde{D}}=\int
d^{2}x\,\Big{[}t\mathcal{H}+x^{k}\partial_{k}\phi^{a}D_{t}\phi^{a}+(x^{k}\partial_{k}a_{i}^{a}+a_{i}^{a})D_{i}\phi^{a}\Big{]},$
(129) $\displaystyle Q_{\tilde{K}}=\int
d^{2}x\,\Big{[}t^{2}\mathcal{H}+2tx^{k}\partial_{k}\phi^{a}D_{t}\phi^{a}+(2tx^{k}\partial_{k}a_{i}^{a}+2ta_{i}^{a}-2x_{i}\phi^{a})D_{i}\phi^{a}\Big{]},$
(130)

where
$\mathcal{H}=2\Big{[}\frac{1}{2}D_{t}\phi^{a}D_{t}\phi^{a}+D_{i}\phi^{a}D_{i}a_{t}^{a}+\frac{1}{4}W^{ija}W_{ij}^{a}+gf^{abc}\phi^{b}a_{t}^{c}D_{t}\phi^{a}\Big{]}$.
However the charges corresponding to the $Y^{(n)}_{k}$ transformations vanish
trivially for all $n$, analogous to the role of the $M^{(n)}$ generators in
the 4 dimensional counterpart:

$Q_{Y}=\int
d^{3}x\,\left(\Theta(\delta_{Y^{(n)}_{k}})-\beta(\delta_{Y^{(n)}_{k}})\right)=\int
d^{3}x\,\left(D_{t}\phi^{a}(\delta_{Y^{(n)}_{k}}\phi^{a})+D_{i}\phi^{a}(\delta_{Y^{(n)}_{k}}a^{ia})\right)=0.$
(131)

#### Hamiltonian functions

As expected from the symmetry analysis above, we notice that the
transformations $\delta_{Y_{k}^{(n)}}$ are Hamiltonian vector fields on the
space of solutions, but with vanishing dynamical generators:

$\Omega({\delta_{Y_{k}^{(n)}},\delta})=0.$ (132)

On the other hand, all the infinite modes of the $Z^{(n)}$ generators give
rise integrable Hamiltonian functions on the space of solutions. Substituting
these expressions in (5), we have

$\displaystyle\Omega(\delta_{Z^{(n)}},\delta)=\delta\int
d^{2}x\,\Big{[}2t^{n+1}\Big{\\{}\frac{1}{2}D_{t}\phi^{a}D_{t}\phi^{a}+\frac{1}{4}W^{ija}W_{ija}\Big{\\}}-(n+1)t^{n}\Big{\\{}(a^{ia}D_{i}\phi^{a}+2\phi^{a}D_{t}\phi^{a})$
$\displaystyle\hskip
79.6678pt+x^{k}(\phi^{a}\partial_{k}D_{t}\phi^{a}+a_{i}^{a}\partial_{k}D^{i}\phi^{a})\Big{\\}}+(n+1)nt^{n-1}(\phi^{a})^{2}\Big{]}=\delta
Q[Z_{n}].$ (133)

For the phase space realization of the Schrödinger algebra (2.2), we readily
notice that the central mass term drops off, since the Hamiltonian functions
corresponding to $Y$ generators themselves vanish. For the very same reason,
the brackets $[Z^{(n)},Y_{i}^{(m)}]$ trivialize at the level of Hamiltonian
functions:

$\Omega(\delta_{Z^{(n)}},\delta_{Y_{i}^{(m)}})=0.$ (134)

However the Witt subalgebra of the Schrödinger algebra generated by $Z^{(n)}$
gets centrally extended on the space of solutions:

$\Omega(\delta_{Z^{(n)}},\delta_{Z^{(m)}})=(n-m)Q[Z^{(n+m)}]+2\big{[}n(n^{2}-1)-m(m^{2}-1)\big{]}t^{n+m-1}\int
d^{3}x(\phi^{a})^{2}.$ (135)

As in the Witt subalgebra of GCA, the second term in (135) is also a state
dependent central term satisfying the 2 co-cycle condition (127). So indeed,
even for Galilean gauge theories in 3d (both electrodynamics and Yang-Mills),
the entire infinite dimensional Virasoro sub-algebra of the Schrödinger-
Virasoro algebra is realised in phase space, along with the central
extensions. This is rather remarkable given that this same sub-algebra did not
turn out to be symmetries of the action or the subsequent EOM.

##  6 Quantum Aspects: Propagators and vertices

In earlier sections, we focused on classical aspects of Galilean
electrodynamics and Galilean Yang-Mills theories. Now we would to like to look
at the quantization of these theories. In this paper, we take the initial
steps to our goal.

A detailed study of the quantum theory of GED coupled to a scalar field in
$3$-dimensions was done in Chapman:2020vtn . For our purposes, where the
additional scalar field is absent, we only review here the expression for the
propagators of the gauge fields of GED:

$\displaystyle\langle\mathcal{A}_{I}(k)\mathcal{A}_{J}(-k)\rangle\equiv
D_{IJ}(k)=-\frac{i}{\vec{k}^{2}}\begin{pmatrix}0&\quad 1&\quad 0\\\
1&\quad-(1-\xi)\frac{\omega^{2}}{\vec{k}^{2}}&\quad(1-\xi)\frac{\omega
k_{j}}{\vec{k}^{2}}\\\ 0&\quad(1-\xi)\frac{\omega
k_{i}}{\vec{k}^{2}}&\quad\delta_{ij}-(1-\xi)\frac{k^{i}k^{j}}{\vec{k}^{2}}\end{pmatrix},$
(136)

where $k=(\omega,\vec{k})$, $\mathcal{A}_{I}=(\phi,a_{t},a_{i})$ and
$i=\\{1,2\\}$. (Performing a similar analysis in $4$-dimensions, we get the
same expression (136) for gauge field propagators in $4$-dimensions with
$i=\\{1,2,3\\}$.) This will be useful when we try to find the propagators for
the Galilean Yang-Mills theory.

Figure 1: GED propagator $\langle\mathcal{A}_{I}\mathcal{A}_{J}\rangle$.

The construction of the full quantum theory of Galilean Yang-Mills theory is
quite non-trivial. In this section, we begin by obtaining tree level
propagators and vertices, and write Feynman rules for Galilean Yang Mills
theory. We hope to return to more details of the full quantum mechanical
theory in future work.

Let us first recall the total gauge-fixed Lagrangian of the relativistic Yang-
Mills theory including the gauge fixing and ghost terms:

$\displaystyle\mathcal{L}=-\frac{1}{4}F^{\tilde{\mu}\tilde{\nu}a}F^{a}_{\tilde{\mu}\tilde{\nu}}-\frac{1}{2\xi}\big{(}\partial^{\tilde{\mu}}A_{\tilde{\mu}}\big{)}^{2}-\partial^{\tilde{\mu}}\bar{c}^{a}D_{\tilde{\mu}}c^{a},$
(137)

where $\xi$ is the gauge fixing parameter, $(c^{a},\bar{c}^{a})$ are ghost
fields and $D_{\mu}c^{a}=\partial_{\mu}c^{a}-gf^{abc}A_{\mu}^{b}c^{c}$.
Performing null reduction along $u$-direction, we get the total Lagrangian
density, including the gauge fixing term and the ghost term, for Galilean
Yang-Mills theory in four dimensions

$\displaystyle\tilde{\mathcal{L}}_{GYM}$ $\displaystyle=$
$\displaystyle\frac{1}{2}D_{t}\phi^{a}D_{t}\phi^{a}+D_{i}\phi^{a}E^{ia}-\frac{1}{4}W^{ija}W_{ij}^{a}$
(138)
$\displaystyle-\frac{1}{2\xi}(\partial_{t}\phi^{a}+\partial^{i}a_{i}^{a})^{2}-gf^{abc}\partial_{t}\bar{c}^{a}\phi^{b}c^{c}-\delta^{ij}\partial_{i}\bar{c}^{a}(D_{j}c)^{a}.$

From this Lagrangian density, we have the kinetic terms as

$\displaystyle\mathcal{L}_{kin}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\partial_{t}\phi^{a}\partial_{t}\phi^{a}+\delta^{ij}\partial_{i}\phi^{a}(\partial_{t}a^{a}_{j}-\partial_{j}a_{t}^{a})-\frac{1}{2}\delta^{ik}\delta^{jl}(\partial_{i}a_{j}^{a}\partial_{k}a_{l}^{a}-\partial_{i}a_{j}^{a}\partial_{l}a_{k}^{a})$
(139)
$\displaystyle-\frac{1}{2\xi}(\partial_{t}\phi^{a}\partial_{t}\phi^{a}+2\partial_{t}\phi^{a}\partial^{i}a_{i}^{a}+\delta^{ij}\delta^{kl}\partial_{i}a_{j}^{a}\partial_{k}a_{l}^{a})-\delta^{ij}\partial_{i}\bar{c}^{a}\partial_{j}c^{a}.$

In order to get propagators from this kinetic part of the Lagrangian, let us
first introduce Fourier transformation to momentum space

$\Phi^{a}(t,\vec{x})=\int\frac{d\omega}{2\pi}\frac{d^{3}\vec{k}}{(2\pi)^{3}}e^{-i\omega
t}e^{i\vec{k}\cdot\vec{x}}\tilde{\Phi}^{a}(\omega,\vec{k}),$ (140)

where $\Phi^{a}=(\phi^{a},a_{t}^{a},a_{i}^{a},c^{a},\bar{c}^{a})$, and delta
functions

$\int\frac{d\omega}{2\pi}e^{-i\omega
t}=\delta(\omega),\quad\int\frac{d^{3}\vec{x}}{(2\pi)^{3}}e^{i\vec{k}\cdot\vec{x}}=\delta^{(3)}(\vec{k}).$
(141)

We also introduce the notation, $k=(\omega,\vec{k})$ and
$\mathcal{A}_{I}^{a}=(\phi^{a},a_{t}^{a},a_{i}^{a})$. Taking Fourier
transformation and using delta functions, the kinetic part of the action
becomes

$\mathcal{S}_{kin}=\int\frac{d\omega
d^{3}\vec{k}}{(2\pi)^{4}}\Big{(}\frac{1}{2}\mathcal{A}_{I}^{a}(k)d^{IJab}\mathcal{A}_{J}^{b}(-k)+\bar{c}^{a}(k)\big{(}-\vec{k}^{2}\big{)}c^{a}(-k)\Big{)},$
(142)

where

$\displaystyle
d^{IJab}(k)=\begin{pmatrix}\Big{(}1-\frac{1}{\xi}\Big{)}\omega^{2}&\quad-\vec{k}^{2}&\quad-\Big{(}1-\frac{1}{\xi}\Big{)}\omega
k^{j}\\\ -\vec{k}^{2}&\quad 0&\quad 0\\\ -\Big{(}1-\frac{1}{\xi}\Big{)}\omega
k^{i}&\quad
0&\quad-\delta^{ij}\vec{k}^{2}+\Big{(}1-\frac{1}{\xi}\Big{)}k^{i}k^{j}\end{pmatrix}.$
(143)

Then from the inverse of $d^{IJab}$, we get the propagators for the fields
$\mathcal{A}_{I}^{a}$ as

$\displaystyle\langle A_{I}^{a}(k)A_{J}^{b}(-k)\rangle\equiv
D^{ab}_{IJ}(k)=-\frac{i\delta^{ab}}{\vec{k}^{2}}\begin{pmatrix}0&\quad 1&\quad
0\\\ 1&\quad-(1-\xi)\frac{\omega^{2}}{\vec{k}^{2}}&\quad(1-\xi)\frac{\omega
k_{j}}{\vec{k}^{2}}\\\ 0&\quad(1-\xi)\frac{\omega
k_{i}}{\vec{k}^{2}}&\quad\delta_{ij}-(1-\xi)\frac{k^{i}k^{j}}{\vec{k}^{2}}\end{pmatrix}.$
(144)

The inverse of the coefficient of $\bar{c}c$ in (142) gives the propagator for
ghost fields as

$\displaystyle\langle\bar{c}^{a}(k)c^{b}(-k)\rangle=\frac{i\delta^{ab}}{\vec{k}^{2}}.$
(145)

We see that the denominators of propagators above depend only on spatial
momentum and are independent of frequency implying instantaneous propagation
of field excitations. In other words, there are no local degrees of freedom in
the theory. This is consistent with the fact that in spite of presence of the
infinite number of $M{(n)}$ symmetry generators, they have no non-trivial
dynamical realization.

(a) Gauge field propagator
$\langle\mathcal{A}^{a}_{I}\mathcal{A}^{b}_{J}\rangle$

(b) Ghost propagator

Figure 2: Two-point Propagators

In order to obtain vertices, let us write the interaction part of the
Lagrangian density (138):

$\displaystyle\mathcal{L}_{int}$ $\displaystyle=$ $\displaystyle
gf^{abc}\Big{(}\partial_{t}\phi^{a}a_{t}^{b}\phi^{c}+\delta^{ij}\partial_{i}\phi^{a}a_{t}^{b}a_{j}^{c}-\delta^{ij}a_{i}^{b}\phi^{c}\partial_{j}a_{t}^{a}+\delta^{ij}a_{i}^{b}\phi^{c}\partial_{t}a_{j}^{a}$
(146)
$\displaystyle-\delta^{ik}\delta^{jl}\partial_{i}a_{j}^{a}a_{k}^{b}a_{l}^{c}-\partial_{t}\bar{c}^{a}\phi^{b}c^{c}-\delta^{ij}\partial_{i}\bar{c}^{a}a_{j}^{b}c^{c}\Big{)}$
$\displaystyle+g^{2}f^{abc}f^{ade}\Big{(}\frac{1}{2}a_{t}^{b}\phi^{c}a_{t}^{d}\phi^{e}+\phi^{c}a_{t}^{d}a_{i}^{b}a_{j}^{e}\delta^{ij}-\frac{1}{4}\delta^{ik}\delta^{jl}a_{i}^{b}a_{j}^{c}a_{k}^{d}a_{l}^{e}\Big{)}.$

Transforming to momentum space and using delta functions, we can write the $3$
field interaction terms in the action as

$\displaystyle\mathcal{S}^{(3)}_{int}$ $\displaystyle=$
$\displaystyle\int\frac{1}{(2\pi)^{12}}{\displaystyle{\prod_{i=1}^{3}d\omega_{i}d^{3}\vec{k}_{i}}}\,(2\pi)^{4}\delta(\omega_{1}+\omega_{2}+\omega_{3})\delta^{(3)}(\vec{k}_{1}+\vec{k}_{2}+\vec{k}_{3})gf^{abc}\times$
(147)
$\displaystyle\quad\Big{[}\frac{i}{2}(\omega_{1}-\omega_{2})\phi^{a}(k_{1})\phi^{b}(k_{2})a_{t}^{c}(k_{3})+i\delta^{ij}(k_{1}-k_{2})_{i}\phi^{a}(k_{1})a_{t}^{b}(k_{2})a_{j}^{c}(k_{3})$
$\displaystyle\quad+\frac{i}{2}(\omega_{3}-\omega_{2})\phi^{a}(k_{1})a_{i}^{b}(k_{2})a_{j}^{c}(k_{3})\delta^{ij}$
$\displaystyle\quad+\frac{i}{6}\big{(}(k_{1}-k_{2})_{i}\delta^{il}\delta^{jk}+(k_{2}-k_{3})_{i}\delta^{ij}\delta^{lk}+(k_{3}-k_{1})_{i}\delta^{ik}\delta^{jl}\big{)}a_{j}^{a}(k_{1})a_{k}^{b}(k_{2})a_{l}^{c}(k_{3})$
$\displaystyle\quad-i\omega_{2}\phi^{a}(k_{1})\bar{c}^{b}(k_{2})c^{c}(k_{3})+i\delta^{ij}k_{2j}a_{i}^{a}(k_{1})\bar{c}^{b}(k_{2})c^{c}(k_{3})\Big{]},$

where
${\displaystyle\prod_{i=1}^{n}d\omega_{i}d^{3}\vec{k}_{i}}=d\omega_{1}d^{3}\vec{k}_{1}...d\omega_{n}d^{3}\vec{k}_{n}$.
From this expression for the action in momentum space, we can write the
$3$-point vertices as

$\displaystyle V_{3\,\phi\phi
a_{t}}^{abc}=-gf^{abc}(\omega_{1}-\omega_{2}),\quad V_{3\,\phi
a_{t}a_{i}}^{abc\,i}=-gf^{abc}(k_{1}-k_{2})^{i},\quad V_{3\,\phi
a_{i}a_{j}}^{abc\,ij}=gf^{abc}(\omega_{2}-\omega_{3})\delta^{ij},$
$\displaystyle
V_{3\,a_{i}a_{j}a_{k}}^{abc\,ijk}=-gf^{abc}\big{(}(k_{1}-k_{2})^{k}\delta^{ij}+(k_{2}-k_{3})^{i}\delta^{jk}+(k_{3}-k_{1})^{j}\delta^{ik}\big{)},$
$\displaystyle V_{3\,\phi\bar{c}c}^{abc}=gf^{abc}\omega_{2},\quad
V_{3\,a_{i}\bar{c}c}^{abc}=-gf^{abc}k_{2}^{i}.$ (148)

Similarly, transforming the $4$ field interaction terms in $\mathcal{S}_{int}$
to momentum space, we get

$\displaystyle\mathcal{S}^{(4)}_{int}$ $\displaystyle=$
$\displaystyle\int\frac{1}{(2\pi)^{16}}{\displaystyle{\prod_{i=1}^{4}d\omega_{i}d^{3}\vec{k}_{i}}}\,(2\pi)^{4}\delta(\omega_{1}+\omega_{2}+\omega_{3}+\omega_{4})\delta^{(3)}(\vec{k}_{1}+\vec{k}_{2}+\vec{k}_{3}+\vec{k}_{4})\times$
(149) $\displaystyle\quad
g^{2}\Big{[}\frac{1}{4}\big{(}f^{abd}f^{ace}+f^{abe}f^{acd}\big{)}\phi^{b}(k_{1})\phi^{c}(k_{2})a_{t}^{d}(k_{3})a_{t}^{e}(k_{4})$
$\displaystyle\qquad-\big{(}f^{abd}f^{ace}+f^{abe}f^{acd}\big{)}\phi^{b}(k_{1})a_{t}^{c}(k_{2})a_{i}^{d}(k_{3})a_{j}^{e}(k_{4})\delta^{ij}$
$\displaystyle\qquad-\frac{1}{24}\big{(}f^{abc}f^{ade}(\delta^{ik}\delta^{jl}-\delta^{il}\delta^{jk})+f^{abd}f^{ace}(\delta^{ij}\delta^{kl}-\delta^{il}\delta^{jk})$
$\displaystyle\qquad+f^{abe}f^{acd}(\delta^{ij}\delta^{kl}-\delta^{ik}\delta^{jl})\big{)}a_{i}^{b}(k_{1})a_{j}^{c}(k_{2})a_{k}^{d}(k_{3})a_{l}^{e}(k_{4})\Big{]},$

from which we can read of the $4$-point vertices

$\displaystyle V_{4\,\phi\phi
a_{t}a_{t}}^{bcde}=ig^{2}\big{(}f^{abd}f^{ace}+f^{abe}f^{acd}\big{)},$
$\displaystyle V_{4\,\phi
a_{t}a_{i}a_{j}}^{bcde\,ij}=-ig^{2}\big{(}f^{abd}f^{ace}+f^{abe}f^{acd}\big{)}\delta^{ij},$
$\displaystyle
V_{4\,a_{i}a_{j}a_{k}a_{l}}^{bcde\,ijkl}=-4ig^{2}\Big{(}f^{abc}f^{ade}(\delta^{ik}\delta^{jl}-\delta^{il}\delta^{jk})+f^{abd}f^{ace}(\delta^{ij}\delta^{kl}-\delta^{il}\delta^{jk})$
$\displaystyle\hskip
99.58464pt+f^{abe}f^{acd}(\delta^{ij}\delta^{kl}-\delta^{ik}\delta^{jl})\Big{)}.$
(150)

(a) $V_{3\,\phi\phi a_{t}}^{abc}$

(b) $V_{3\,\phi a_{t}a_{i}}^{abc\,i}$

(c) $V_{3\,\phi a_{i}a_{j}}^{abc\,ij}$

(d) $V_{3\,a_{i}a_{j}a_{k}}^{abc\,ijk}$

(e) $V_{3\,\phi\bar{c}c}^{abc}$

(f) $V_{3\,a_{i}\bar{c}c}^{abc}$

Figure 3: Three-point Vertices

(a) $V_{4\,\phi\phi a_{t}a_{t}}^{bcde}$

(b) $V_{4\,\phi a_{t}a_{i}a_{j}}^{bcde\,ij}$

(c) $V_{4\,a_{i}a_{j}a_{k}a_{l}}^{bcde\,ijkl}$

Figure 4: Four-point Vertices

In future work, in order to make the theory contain non-trivial degrees of
freedom, we would be coupling matter degrees of freedom to the above
Lagrangian. The Feynman rules we have derived above would be useful for
computing quantum mechanical processes in the non-abelian Galilean gauge
theories, now coupled with matter.

##  7  Conclusions and Future Directions

### Summary

In this paper, we have investigated various properties of Galilean gauge
theories. We have arrived at actions for abelian and non-Abelian Galilean QFTs
by a process of null reduction from higher dimensional relativistic QFTs. We
principally investigated the symmetry structures associated with these gauge
theories and found that there are symmetry enhancements at various levels.

For three dimensional theories (both abelian and non-abelian), the gauge
theories exhibit invariance under the Schrödinger group. The Schrödinger
algebra admits an infinite extension giving the so-called Virasoro-Schrödinger
algebra. At the level of the action of these Galilean gauge theories, we found
that the symmetries enhance from the finite version to a subset of the
infinite algebra. At the level of the EOM, we found even more enhancements.
Finally, in our analysis of the pre-symplectic structure, the symmetries in
phase space exhibited the full Virasoro-Schroödinger algebra, together with a
state-dependent central extension.

The story in four spacetime dimensions proceeded along similar lines, now with
the Schrödinger algebra replaced by the finite Galilean Conformal Algebra and
the Virasoro-Schrödinger algebra replaced by the infinite version of the GCA.

Apart from this, as an aside, we also considered electro-magnetic duality in
the $d=3$ abelian theory and found that unlike the $d=4$ case where the
duality transformation exchanged electric and magnetic theories, the field
equations of the theory we considered were invariant under the duality. This
invariance was inherited from relativistic duality symmetric theory via null
reduction. The $d=4$ case, even though also obtainable from null reductions,
does not inherit any duality symmetry since the relativistic $d=5$ theory
itself is not duality invariant.

We concluded our analysis with a sneak-peek into the quantum-mechanical regime
of these gauge theories and wrote down the propagators and vertices for the
theories. The full-blown quantum analysis is left for future work.

### Looking ahead

A lot of recent attention has been devoted to Carrollian theories, where the
speed of light, instead of going to infinity like in Galilean theories, goes
to zero LevyLeblond . In these peculiar theories, the lightcones thus close up
and a priori, these theories seem like non-sensical ones which would be of no
use. Surprisingly however, these Carrollian theories have been shown to
encompass a great deal of interesting physics. These Carrollian structures
show up whenever one attempts to write down a QFT on a null surface
Bagchi:2019clu ; Ciambelli:2019lap  and hence is vitally important for
understanding holographic duals to asymptotically flat spacetimes living on
$\mathscr{I}^{\pm}$ Bagchi:2010zz ; Bagchi:2012cy ; Barnich:2012aw ;
Bagchi:2012xr ; Barnich:2012xq ; Bagchi:2014iea ; Hartong:2015usd ;
Jiang:2017ecm ; Ciambelli:2018wre . These are also important for any generic
null surface and hence crucial for theories defined on black hole event
horizons Donnay:2019jiz . It has been shown that 2d Carrollian CFTs (the
symmetries of which are actually isomorphic to 2d Galilean CFTs Bagchi:2010zz
) replace usual relativistic CFTs on the worldsheet of strings in the
tensionless or null limit Bagchi:2013bga ; Duval:2014uva ; Bagchi:2015nca ;
Bagchi:2019cay ; Bagchi:2020ats ; Bagchi:2021ban . For higher dimensional
objects like branes, higher dimensional Carrollian CFTs should be important.
Of late, Carrollian theories have been found to be of importance in
cosmological scenarios including dark energy deBoer:2021jej , and in the
theory of fractons in condensed matter physics Bidussi:2021nmp . Given such a
wide array of applicability, field theories with Carrollian symmetry,
especially Carrollian gauge theories are going to play a pivotal role going
forward. Analysis of abelian and non-abelian Carrollian gauge theories have
been carried out using EOM Bagchi:2019xfx . It is however of greater
importance to construct actions of quantum field theories with Carroll
symmetry. Some work in this direction are Bagchi:2019clu ; Banerjee:2020qjj ;
Hao:2021urq ; Chen:2021xkw . We would like to generalise these constructions
to non-abelian theories and study their quantum mechanical structure.

Returning to the Galilean arena, we obviously wish to investigate the quantum
mechanical structure of the gauge theories we have developed. This is work in
progress. We wish to generalise the constructions in this paper to
Supersymmetric QFTs. The process of null reduction would be useful in this
case as well. We plan to return to these and other related questions in the
near future.

## Acknowledgments

AB is partially supported by a Swarnajayanti fellowship of the Department of
Science and Technology, India and by the following grants from the Science and
Engineering Research Board: SB/SJF/2019-20/08, MTR/2017/000740,
CGR/2020/002035. RB is supported by the grants SRG/2020/001037 and
CRG/2020/002035 from the Science and Engineering Research Board. MI would like
to thank Supratim Das Bakshi for useful discussions. AM would like to thank
Alexander von Humboldt Foundation for a Humboldt Postdoctoral Research
Fellowship which funded his tenure at AEI Potsdam.

##  Appendix A Scale-Spin Representation of GCA

The GCA is given as

$\displaystyle\big{[}L^{(n)},L^{(m)}\big{]}=(n-m)L^{(n+m)},~{}\big{[}L^{(n)},M_{i}^{(m)}\big{]}=(n-m)M_{i}^{(n+m)},~{}\big{[}M^{(n)}_{i},M^{(m)}_{j}\big{]}=0,$
$\displaystyle\big{[}L^{(n)},J_{ij}\big{]}=0,~{}\big{[}J_{ij},M^{(n)}_{k}\big{]}=M^{(n)}_{j}\delta_{ik}-M^{(n)}_{i}\delta_{jk}$
(151)

where
$L^{(-1,0,1)}\rightarrow(H,D,K),~{}M^{(-1,0,1)}_{i}\rightarrow(P_{i},B_{i},K_{i})$
respectively. We will now construct the representation theory for this
algebra. We will begin by labelling the primary states 333The primary states
in the non-relativistic setting are defined as  $\displaystyle
L^{n}|\varphi\rangle_{p}=M_{i}^{n}|\varphi\rangle_{p}=0,\quad\forall~{}n>0.$
(152) under dilatation ($L^{(0)}$) and rotation ($J_{ij}$) as Bagchi:2009ca ;
Bagchi:2014ysa ; Bagchi:2017yvj :

$\displaystyle
L^{(0)}|\varphi\rangle_{p}=\Delta|\varphi\rangle_{p},~{}J_{ij}|\varphi\rangle_{p}=\Sigma_{ij}|\varphi\rangle_{p}.$
(153)

where $\Delta$ is the scaling weight and $\Sigma_{ij}$ denotes the action of
rotation in the particular representation of $SO(d-1)$. The labeling of the
states can be performed in this manner because both $D$ and $J_{ij}$ commutes
(A) and we can use eigenvalues of both the operators simultaneously. We will
write down the action of the finite part of GCA on the primary operators as

$\displaystyle\big{[}L^{(-1)},\varphi(t,x)\big{]}=\partial_{t}\varphi(t,x),\quad\big{[}M_{i}^{(-1)},\varphi(t,x)\big{]}=-\partial_{i}\varphi(t,x)$
(154a)
$\displaystyle\big{[}J_{ij},\varphi(0,0)\big{]}=\Sigma_{ij}\varphi(0,0),\quad\big{[}L^{0},\varphi(0,0)\big{]}=\Delta\varphi(0,0)$
(154b)

where we have used the state-operator correspondence (it relates the primary
state and the vacuum as $|\varphi\rangle_{p}=\varphi(0,0)|0\rangle$) in the
intermediate steps. Next step will be to calculate the action of $L^{(0)}$ and
$J_{ij}$ on $\varphi$ at an arbitrary spacetime point $(t,x)$. For that, we
will use

$\displaystyle\varphi(t,x)=U\varphi(0,0)U^{-1}\quad\text{with}\quad
U=e^{tL^{(-1)}-x^{k}M_{k}^{(-1)}}$ (155)

along with Baker-Campbell-Hausdrorff (BCH) formula and the commutation
relations of GCA. The final result becomes

$\displaystyle\big{[}J_{ij},\varphi(t,x)\big{]}=\big{(}x_{i}\partial_{j}-x_{j}\partial_{i}\big{)}\varphi(t,x)+\Sigma_{ij}\varphi(t,x)$
(156a)
$\displaystyle\big{[}L^{(0)},\varphi({t,x})\big{]}=\big{(}t\partial_{t}+x^{k}\partial_{k}+\Delta\big{)}\varphi(t,x)$
(156b)

We will now look at the action of boost and SCT on the primaries. We know from
(A) that boost and rotation generators do not commute. Its direct consequences
will be that we cannot find the action of boost as simple as (156). We have to
think of another method to evaluate boost action on the primaries. One such
way is to use the Jacobi identity given below:

$\displaystyle\big{[}J_{ij},\big{[}B_{k},\varphi(0,0)\big{]}\big{]}=\big{[}B_{k},\Sigma_{ij}\varphi(0,0)\big{]}+\delta_{ki}\big{[}B_{j},\varphi(0,0)\big{]}-\delta_{jk}\big{[}B_{i},\varphi(0,0)\big{]}$
(157)

By solving (157), the action of boost on $\varphi(0,0)$ can be calculated for
different theory. For our case where the field content is given by
$(\phi\rightarrow\text{scalar field},\\{a_{t},a_{i}\\}\rightarrow\text{gauge
fields})$, we get

$\displaystyle\big{[}B_{k},\varphi(0,0)\big{]}=a\phi_{k}+sa_{k}+r\phi\delta_{ik}$
(158)

where $\varphi(0,0)=\\{\phi,a_{t},a_{i}\\}$. The values of constants $(a,s,r)$
are determined by demanding inputs from the dynamics. The action of boost on
$\varphi(t,x)$ at finite spacetime points is given by

$\displaystyle\big{[}B_{k},\varphi(t,x)\big{]}=-t\partial_{k}\varphi(t,x)+U\big{[}B_{k},\varphi(0,0)\big{]}U^{-1}$
(159)

Once we have (158), the action of SCT can be calculated easily. They are

$\displaystyle\big{[}K,\varphi(t,x)\big{]}=\big{(}t^{2}\partial_{t}+2tx^{k}\partial_{k}+2t\Delta\big{)}\varphi(t,x)-2x^{k}U\big{[}B_{k},\varphi(0,0)\big{]}U^{-1},$
(160a)
$\displaystyle\big{[}K_{k},\varphi(t,x)\big{]}=-t^{2}\partial_{k}\varphi(t,x)+2tU\big{[}B_{k},\varphi(0,0)\big{]}U^{-1}.$
(160b)

Similarly, the action of infinite dimensional generators ($L^{(n)}$ and
$M^{(n)}_{i}$) on $\varphi(t,x)$ at arbitrary spacetime point can be found.
They are given by

$\displaystyle\big{[}L^{(n)},\varphi(t,x)\big{]}=\big{(}t^{n+1}\partial_{t}+(n+1)t^{n}x^{k}\partial_{k}+(n+1)t^{n}\Delta\big{)}\varphi(t,x)$
$\displaystyle\hskip
76.82234pt-n(n+1)t^{n-1}x^{k}U\big{[}M_{k}^{0},\varphi(0,0)\big{]}U^{-1},$
(161a)
$\displaystyle\big{[}M^{(n)}_{k},\varphi(t,x)\big{]}=-t^{n+1}\partial_{k}\varphi(t,x)+(n+1)t^{n}U\big{[}M_{k}^{0},\varphi(0,0)\big{]}U^{-1}.$
(161b)

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