aainstitutetext: Indian Institute of Technology Kanpur, Kanpur 208016, India.
bbinstitutetext: Centre de Physique Theorique, Ecole Polytechnique de Paris,
91128 Palaiseau Cedex, France.
ccinstitutetext: International Centre for Theoretical Sciences (ICTS-TIFR),
Bengaluru 560089, India.

# Non-Lorentzian Kač-Moody Algebras

Arjun Bagchi a Ritankar Chatterjee a, c Rishabh Kaushik a Amartya Saha a, c
and Debmalya Sarkar
(abagchi, ritankar<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We investigate two dimensional (2d) quantum field theories which exhibit Non-
Lorentzian Kač-Moody (NLKM) algebras as their underlying symmetry. Our
investigations encompass both 2d Galilean (speed of light $c\to\infty$) and
Carrollian ($c\to 0$) CFTs with additional number of infinite non-Abelian
currents, stemming from an isomorphism between the two algebras. We alternate
between an intrinsic and a limiting analysis. Our NLKM algebra is constructed
first through a contraction and then derived from an intrinsically Carrollian
perspective. We then go on to use the symmetries to derive a Non-Lorentzian
(NL) Sugawara construction and ultimately write down the NL equivalent of the
Knizhnik Zamolodchikov equations. All of these are also derived from
contractions, thus providing a robust cross-check of our analyses.

##  1 Introduction

Relativistic conformal field theory (CFT) is one of the most potent tools of
modern theoretical physics, with applications ranging from statistical
mechanics of phase transitions to quantum gravity through holography and
string theory. Especially powerful are methods of two dimensional (2d) CFTs
Belavin:1984vu  where symmetries enhance to two copies of the infinite
dimensional Virasoro algebra. The ideas and methods of 2d CFTs are of
particular importance to the success of string theory, where this arises as
residual symmetry on the string worldsheet after the fixing of conformal gauge
Friedan:1985ge .

Non-abelian current algebras arise on the string worldsheet when one considers
strings moving on arbitrary group manifolds Gepner:1986wi , generalising the
abelian versions which arise for strings propagating on flat backgrounds.
These Kač-Moody algebras give rise to the worldsheet 2d CFT by the Sugawara
construction. The construction of strings on arbitrary backgrounds is thus
intimately linked to these Kač-Moody algebras.

Kač-Moody (KM) algebras also arise when we think of 2d CFTs augmented by
additional symmetries Zamolodchikov:1985wn . For example, a CFT with
additional $U(1)$ global symmetry arises in a number of places, including the
study of black holes in AdS3 with charged $U(1)$ hair that are solutions to
Einstein-Chern Simons theory Das:2017vej . Virasoro with $U(1)$ KM symmetry
forms the chiral algebra of a large number of theories including
$\mathcal{N}=2$ superconformal field theories and theories with
$\mathcal{W}_{1+\infty}$ symmetry.

In this paper, we will be interested in the construction of Non-Lorentzian
(NL) versions of Kač-Moody algebras. Specifically, we are concerned with
Galilean and Carrollian CFTs in 2d with additional symmetry. Galilean and
Carrollian CFTs are obtained from their relativistic counterparts by a process
of contraction where the speed of light is taken to infinity (Galilean theory)
or zero (Carrollian theory). In two dimensions, the symmetry algebras turn out
to be isomorphic and it is in this 2d case we will focus our attention. We
will obtain the algebras of interest by a contraction and then construct
various properties of these algebras by methods that have no connections with
the limiting procedure and can be thought of as completely intrinsic analyses.
We also show that suitable singular limits also reproduce our intrinsic
answers.

### Possible applications

We have a variety of applications in mind for our algebraic explorations in
this work.

Holography of flat spacetimes

Following closely related observation in Bagchi:2010zz , it was shown in
Duval:2014uva  that $d$-dimensional Conformal Carrollian algebras are
isomorphic to asymptotic symmetry algebras of $(d+1)$ dimensional flat
spacetimes discovered first by Bondi, van der Burg, Metzner Bondi:1962px  and
Sachs Sachs:1962zza  and called BMS algebras. The Carroll CFTs can thus act as
putative duals to asypmtotically flat spacetimes Bagchi:2010zz ; Bagchi:2012cy
; Bagchi:2016bcd . Some important evidence for this duality has been provided
in e.g. Bagchi:2012xr ; Barnich:2012xq ; Barnich:2012aw ; Bagchi:2014iea ;
Bagchi:2015wna ; Hartong:2015usd ; Jiang:2017ecm . Of particular importance is
the recent result Bagchi:2022emh  that links 3d Carroll CFTs and scattering
amplitudes in 4d asymptotically flat spacetimes.

Our explorations in this paper are of direct importance in the context of 3d
flatspace. Here, the $U(1)$ version of NLKM symmetries are of interest for the
study of Flat Space Cosmological solutions Cornalba:2002fi  with $U(1)$ hair,
which are solutions of Einstein-Chern-Simons theory, like the AdS case. This
was addressed recently in Basu:2017aqn ; Bagchi:2022xug .

The above approach to holography in asymptotically flat spacetimes goes under
the name of Carrollian holography. There is an alternate formulation called
Celestial holography which posits that there is a 2d (relativistic) CFT that
computes $S$-matrix elements in 4d asymptotically flat spacetimes. This has
been instrumental in the uncovering of many new results in asymptotic
symmetries and scattering amplitudes in 4d. The interested reader is pointed
to the excellent reviews Strominger:2017zoo ; Pasterski:2021rjz ;
Raclariu:2021zjz . For connections between Celestial and Carrollian
holography, we point to Bagchi:2022emh ; Donnay:2022aba ; Donnay:2022wvx .

Interestingly, of late there have been studies of tree-level massless
scattering amplitudes which suggest that the asymptotic symmetries are far
richer than the extended BMS group. In Banerjee:2020zlg ; Banerjee:2020vnt ,
it was shown that there is an $SL(2)$ current algebra at level zero underlying
the symmetries of tree-level graviton amplitudes. More recently, massless
scattering amplitudes have revealed an infinite dimensional $w_{1+\infty}$
algebra Strominger:2021mtt .

If we assume that the field theory duals with these additional symmetries
would be related to a co-dimension one holographic description of 4d
flatspace, and that these theories should live on the whole of the null
boundary and not only on the celestial sphere, the structure emerging from the
above discussions should only be a part of the whole symmetries, very much
like the relation between the two copies of the Virasoro algebra that make up
the Celestial CFT and the whole (extended) BMS4 algebra. It is very likely
that the algebras of interest would then be the 3d versions of the Carrollian
Kac-Moody algebras that we discuss at length in this work.

Tensionless strings

The tensionless limit of strings, which is analogous to the massless limit of
point particles, is an important sector of string theory that remains
relatively less explored. In this limit, the string worldsheet becomes null
Schild:1976vq ; Isberg:1993av  and the 2d relativistic conformal symmetry that
arises on the tensile worldsheet is replaced by 2d Carrollian Conformal
symmetry in the tensionless theory Bagchi:2013bga ; Bagchi:2015nca ;
Bagchi:2020fpr . The study of tensionless strings on arbitrary group manifolds
would naturally incorporate the NLKM algebras we study in this paper. We
would, in future, attempt a construction of a Wess-Zumino-Witten model with
the NLKM algebras we discuss in this work.

It is of interest to mention that in Lindstrom:2003mg , it was argued that
tensionless strings appeared when the level of the affine algebra
corresponding to the WZW model of the strings propagating in a group manifold.
We believe that the intrinsic formulation of such strings should involve the
NLKM algebras we are studying here. A direction of future work is the
connection of these two ideas and the aim would be to show that the NLKM
algebras appear when the level of the relativistic affine algebras are dialled
to their critical value.

Other applications

There are, of course, other natural applications. The Galilean version of our
story is of relevance for understanding 2d Galilean CFTs with additional
symmetry which may appear in real life non-relativistic systems. These would
be of interest also in understanding non-relativistic strings in curved
Newton-Cartan backgrounds Harmark:2018cdl .

Interesting enough, Carrollian structures also arise condensed matter systems,
e.g. in the physics of flat bands that is of relevance in the context of
“magic” superconductivity in bi-layer graphene and also in fractional quantum
Hall systems Bagchi:2022eui . It is easy to envision condensed matter systems
with additional symmetry and hence our methods in this paper which lay the
foundation for systems with Carrollian (and Galilean) affine Lie algebras,
should have applicability in a wide number of condensed matter systems.
Finally, a $U(1)$ affine algebra was also found to emerge in studies of the
BMS scalar field theory in 2d Hao:2021urq . It would be of interest to figure
out if this is a feature of all free BMS field theories. Our methods outlined
in this paper should then be useful even for free BMS field theories.

### Outline of the paper

In section 2, we will start with a brief review of Carrollian and Galilean
CFTs in general dimensions, subsequently specializing the discussion to two
dimensions followed by a brief introduction to Non Lorentzian Kac Moody (NLKM)
algebras via contraction of relativistic Affine Lie Algebras. In section 3, we
present an intrinsic carrollian derivation to the NLKM Algebra. Then, we
formulate the non-Lorentzian version of the Sugawara construction for these
algebras in section 4 and verify its validity by showing its consistency with
the OPEs we obtained in section 3. In section 5, we work out the example of
tensionless strings on a flat background geometry which exhibits the $U(1)$
NLKM algebra as the current algebra. Section 6 contains the derivation of the
Non-Lorentzian analog of Knizhnik Zamolodchikov(KZ) equations using the OPE
definition of the primaries in section 3. And finally in section 7, we derive
NLKM Algebra, Sugawara Construction and the Non Lorentzian KZ equations
through contractions in detail. There are six appendices which collect the
details of various calculations that have been skipped in the main text for
the ease of readability.

##  2 Non-Lorentzian Kač-Moody algebra in 2d

In this section, we will begin our analysis by building the algebra we wish to
study in the remainder of the paper. The NLKM algebra will be defined as a
limit from the relativistic Virasoro KM algebra. We will see later how the
current part of the algebra can be used to generate the entire algebra by a NL
version of the Suwagara construction. We begin by reminding the reader of the
Galilean ($c\to\infty$) and the Carrollian ($c\to 0$) contractions of
relativistic CFTs in generic dimensions and the fact that this leads to
isomorphic algebras in $d=2$

###  2.1 Carrollian and Galilean CFTs

The Galilean $(c\to\infty)$ and Carrollian $(c\to 0)$ limits of the Poincare
algebra leads to two different contractions of the parent relativistic algebra
and two different algebras in the limit in general dimensions. The Poincare
algebra in $D$ dimension $ISO(D-1,1)$ is given by

$\displaystyle[P_{\mu},P_{\nu}]=0,\quad[M_{\mu\nu},P_{\rho}]=-2\eta_{\rho[\mu}P_{\nu]},\quad[M_{\mu\nu},M_{\rho\sigma}]=4\eta_{[\mu\rho}M_{\sigma]\nu]},$
(1)

where $\mu={0,1,2...,D-1}$, $P_{\mu}=-\partial_{\mu}$ are translation
generators and $M_{\mu\nu}=x_{\mu}\partial_{\nu}-x_{\nu}\partial_{\mu}$ are
Lorentz generators. Galilean limit is achieved by taking $c\to\infty$ limit,
alternatively by taking the contraction $t\to t$, $x^{i}\to\epsilon x^{i}$,
$\epsilon\to 0$ limit where $i\in\\{1,2,...,D-1\\}$ Bagchi:2009my ;
Bagchi:2009ca . Under this limit we see that

$\displaystyle
M_{0i}=t\partial_{i}+x_{i}\partial_{t}\to\frac{1}{\epsilon}t\partial_{i}+\epsilon
x_{i}\partial_{t}\quad\implies B_{i}=\lim_{\epsilon\to 0}\epsilon
M_{0i}=t\partial_{i}.$ (2)

The spatial rotation generators $M_{ij}$ remains same under this contraction.
Doing this contraction, we end up with the Galilei algebra, where all the non-
zero commutators are given by

$\displaystyle[M_{ij},M_{kl}]=4\delta_{i[k}M_{l]j]},\,[M_{ij},P_{k}]=-2\delta_{k[i}P_{j]},\,[M_{ij},B_{k}]=2\delta_{k[i}B_{j]},\,[B_{i},H]=-P_{i}.$
(3)

Carrollian limit is achieved by taking $c\to 0$ limit, alternatively by taking
the contraction $t\to\epsilon t$, $x^{i}\to x^{i}$, $\epsilon\to 0$. Under
this limit, we have

$\displaystyle M_{0i}=\epsilon t\partial_{i}+x_{i}\partial_{t}\to\epsilon
t\partial_{i}+\frac{1}{\epsilon}x_{i}\partial_{t}\quad\implies
B_{i}=\lim_{\epsilon\to 0}\epsilon M_{0i}=x_{i}\partial_{t},$ (4)

with $M_{ij}$ intact again. This gives us the Carroll algebra where the non-
zero commutators are given by

$\displaystyle[M_{ij},M_{kl}]=4\delta_{i[k}M_{l]j]},\,[M_{ij},P_{k}]=-2\delta_{k[i}P_{j]},\,[M_{ij},B_{k}]=-2\delta_{k[i}B_{j]}\,[P_{i},B_{j}]=\delta_{ij}H,$
(5)

where $H=-\partial_{t}$ is the Hamiltonian which has now become a central
element.

The tale of the two contractions is also true for the relativistic conformal
algebra. In relativistic Conformal symmetry group there are two additional
generators

$\displaystyle D=-x^{\mu}\partial_{\mu}\hskip
14.22636ptK_{\mu}=-(2x_{\mu}x^{\nu}\partial_{\nu}-x^{2}\partial_{\mu}),$ (6)

giving us the following additional commutators along with (1)

$\displaystyle[D,P_{\mu}]=P_{\mu},\,[D,K_{\mu}]=-K_{\mu},\,[K_{\mu},P_{\nu}]=2(\eta_{\mu\nu}D-M_{\mu\nu}),\,[K_{\rho},L_{\mu\nu}]=2\eta_{\rho[\mu}K_{\nu]}.$
(7)

Taking the non-relativistic limit this time, we will end up with the following
additional generators along with the generators of Galilei algebra

$\displaystyle D=-(x_{i}\partial_{i}+t\partial_{t})\hskip
14.22636ptK=K_{0}=-(2tx_{i}\partial_{i}+t^{2}\partial_{t})\hskip
14.22636ptK_{i}=t^{2}\partial_{i}.$ (8)

These generators, along with the generators of the Galilei algebra gives us
the Galilean Conformal Algebra (GCA) in $D$ dimensions Bagchi:2009my  where
non-zero commutators apart from (3) are given by

$\displaystyle[K,B_{i}]=K_{i},\hskip 14.22636pt[K,P_{i}]=2B_{i}\hskip
14.22636pt[M_{ij},K_{r}]=-2K_{[i}\delta_{j]r},$
$\displaystyle[H,K_{i}]=-2B_{i}\hskip 14.22636pt[D,K_{i}]=-2K_{i},\hskip
14.22636pt[D,P_{i}]=P_{i}$ $\displaystyle[D,H]=H,\hskip
14.22636pt[H,K]=-2D,\hskip 14.22636pt[D,K]=-K.$ (9)

When we take the Carrollian limit of the relativistic Conformal algebra, we
get the following new generators apart form the Carroll generators previously
encountered:

$\displaystyle D=-(x_{i}\partial_{i}+t\partial_{t})\hskip
14.22636ptK=K_{0}=-x_{i}x_{i}\partial_{t}\hskip
14.22636ptK_{j}=-2x_{j}(x_{i}\partial_{i}+t\partial_{t})+x_{i}x_{i}\partial_{j}.$
(10)

These additional generators give us the following non-vanishing commutators
along with those given in (5)

$\displaystyle[M_{ij},K_{k}]=\delta_{k[j}K_{i]}\hskip
14.22636pt[B_{i},K_{j}]=\delta_{ij}K,\hskip 14.22636pt[D,K]=-K$
$\displaystyle[K,P_{i}]=-2B_{i}\hskip
14.22636pt[K_{i},P_{j}]=-2D\delta_{ij}-2M_{ij},\hskip
14.22636pt[H,K_{i}]=2B_{i}$ $\displaystyle[D,H]=-H,\hskip
14.22636pt[D,P_{i}]=-P_{i}\hskip 14.22636pt[D,K_{i}]=K_{i}.$ (11)

(5) and (2.1) together form Carrollian Conformal Algebra (CCA). We see that in
general dimensions the two different contractions give us different algebras
that are not isomorphic to each other.

Let us now move to the interesting case of $d=2$. In 2d, the relativistic
conformal algebra becomes infinite dimensional and is given by two copies of
the Virasoro algebra:

$\displaystyle[\mathcal{L}_{m},\mathcal{L}_{n}]$
$\displaystyle=(m-n)\mathcal{L}_{m+n}+\frac{c}{12}(m^{3}-m)\delta_{m+n,0},$
(12) $\displaystyle[{\bar{\mathcal{L}}}_{m},{\bar{\mathcal{L}}}_{n}]$
$\displaystyle=(m-n){\bar{\mathcal{L}}}_{m+n}+\frac{\bar{c}}{12}(m^{3}-m)\delta_{m+n,0},$
(13) $\displaystyle[{\mathcal{L}}_{m},{\bar{\mathcal{L}}}_{n}]$
$\displaystyle=0.$

Here $c,\bar{c}$ are central charges of the Virasoro algebra (not to be
confused with the speed of light which we also had called $c$ earlier). The
Galilean contraction Bagchi:2009pe  of the above is given by

$L_{n}=\mathcal{L}_{n}+\bar{\mathcal{L}}_{n},\quad
M_{n}=\epsilon(\bar{\mathcal{L}}_{n}-\mathcal{L}_{n}),$ (14)

This contraction of the Virasoro algebra leads to

$\displaystyle[L_{m},L_{n}]$
$\displaystyle=(m-n)L_{m+n}+\frac{c_{L}}{12}(m^{3}-m)\delta_{n+m,0},$ (15a)
$\displaystyle[L_{m},M_{n}]$
$\displaystyle=(m-n)M_{m+n}+\frac{c_{M}}{12}(m^{3}-m)\delta_{n+m,0},$ (15b)
$\displaystyle[M_{m},M_{n}]$ $\displaystyle=0.$ (15c)

The way to see that this combination yields the non-relativistic limit, it is
instructive to write the generators of the Virasoro algebra in cylindrical
cooridinates

$\mathcal{L}_{n}=e^{in\omega}\partial_{\omega},\quad{\bar{\mathcal{L}}}_{n}=e^{in{\bar{\omega}}}\partial_{\bar{\omega}},\quad\omega,\bar{\omega}=\tau\pm\sigma,$
(16)

The limit (14) then translates to

$\sigma\to\epsilon\sigma,\,\tau\to\tau,\,\,\epsilon\to 0.$ (17)

which essentially means scaling velocities to be very small compared to 1 and
since we are doing all of this in units of speed of light $c=1$, this is
indeed the non-relativistic limit $c\to\infty$.

On the other hand, the Carrollian contraction of the Virasoro algebra is given
by

$L_{n}=\mathcal{L}_{n}-\bar{\mathcal{L}}_{-n},\quad
M_{n}=\epsilon(\mathcal{L}_{n}-\bar{\mathcal{L}}_{-n}).$ (18)

Again if we go back to the cylindrical coordinates, this limit translates to

$\sigma\to\sigma,\,\tau\to\epsilon\tau,\,\,\epsilon\to 0.$ (19)

The velocities are very large compared to 1 now and this means $v/c\to\infty$,
which equivalently translates to $c\to 0$. This is thus the Carrollian limit.
The surprising thing is that even this contraction yields the same algebra
(15). In order to avoid confusion with Galilean or Carrollian notions, we will
exclusively call this the BMS3 algebra. Galilei and Carroll contractions in
$d=2$ yield isomorphic algebras and this is down to the fact that there is
only one contracted and one uncontracted direction in each case. The algebra
does not differentiate between a contracted spatial and a contracted temporal
direction.

###  2.2 NL Affine Lie algebras

We start with the two copies of Virasoro Kac-Moody algebra whose holomorphic
part is,

$\displaystyle[\mathcal{L}_{m},\mathcal{L}_{n}]$
$\displaystyle=(m-n)\mathcal{L}_{m+n}+\frac{c}{12}(m^{3}-m)\delta_{m+n,0},\quad[\mathcal{L}_{m},j^{a}_{n}]=-nj^{a}_{m+n}$
$\displaystyle[j^{a}_{m},j^{b}_{n}]$
$\displaystyle=i\sum_{c=1}^{dim(g)}f^{abc}j^{c}_{m+n}+mk\delta_{m+n,0}\delta^{ab}.$
(20)

There is an equivalent anti-holomorphic part with $\bar{f}_{abc}$, $\bar{c}$
and $\bar{k}$ in place of $f_{abc}$, $c$ and $k$ respectively which are not
necessarily equal to their holomorphic counterparts. We are take
$f^{abc}\neq\bar{f}^{abc}$ for generality, but the dimensions of the two Lie
groups are the same.

We will take a contraction of the algebra as follows. We will work with the
following linear combinations of the relativistic KM generators:

$\displaystyle L_{n}=\mathcal{L}_{n}+\bar{\mathcal{L}}_{n},\quad
M_{n}=\epsilon(\bar{\mathcal{L}}_{n}-\mathcal{L}_{n}),\quad
J^{a}_{m}=j^{a}_{m}+\bar{j}^{a}_{m},\quad
K^{a}_{m}=\epsilon(\bar{j}^{a}_{m}-j^{a}_{m}).$ (21)

We will the consider the limit $\epsilon\to 0$. The contracted algebra is
given by:

$\displaystyle[L_{m},L_{n}]$
$\displaystyle=(m-n)L_{m+n}+\frac{c_{L}}{12}(m^{3}-m)\delta_{n+m,0},$ (22a)
$\displaystyle[L_{m},M_{n}]$
$\displaystyle=(m-n)M_{m+n}+\frac{c_{M}}{12}(m^{3}-m)\delta_{n+m,0},$ (22b)
$\displaystyle[L_{m},J^{a}_{n}]$ $\displaystyle=-nJ^{a}_{m+n}\ ,\
[L_{m},K^{a}_{n}]=-nK^{a}_{m+n}\ ,\ [M_{m},J^{a}_{n}]=-nK^{a}_{m+n}$ (22c)
$\displaystyle[J^{a}_{m},J^{b}_{n}]$
$\displaystyle=i\sum_{c=1}^{dim(g)}F^{abc}J^{c}_{m+n}+i\sum_{c=1}^{dim(g)}G^{abc}K^{c}_{m+n}+mk_{1}\delta^{ab}\delta_{m+n,0},$
(22d) $\displaystyle[J^{a}_{m},K^{b}_{n}]$
$\displaystyle=i\sum_{c=1}^{dim(g)}F^{abc}K^{c}_{m+n}+mk_{2}\delta^{ab}\delta_{m+n,0},$
(22e)

with rest of the commutators vanishing. We recognise the first two lines as
the familiar BMS3 algebra, equivalently the 2d Galilean or 2d Carrollian
Conformal Algebra. Throughout the paper, we will call this sub-algebra the BMS
algebra. In the above, the structure constants are related to their
relativistic counterparts by

$\displaystyle F^{abc}=\frac{1}{2}\left(f^{abc}+\bar{f}^{abc}\right)\ ,\
G^{abc}=\frac{1}{2\epsilon}\left(\bar{f}^{abc}-f^{abc}\right).$ (23)

while the central terms are given by

$c_{L}=c+\bar{c},\ \ c_{M}=\epsilon(\bar{c}-c),\,\ k_{1}=\bar{k}+k,\
k_{2}=\epsilon(\bar{k}-k).$ (24)

We will call the algebra (22) the 2d Non-Lorentzian Kač-Moody algebra. We can
also carry out the contraction ultrarelativistically for which we take the
following linear combinations of generators,

$\displaystyle L_{n}=\mathcal{L}_{n}-\bar{\mathcal{L}}_{-n},\quad
M_{n}=\epsilon(\mathcal{L}_{n}+\bar{\mathcal{L}}_{-n}),\quad
J^{a}_{n}=(j^{a}_{n}+\bar{j}^{a}_{-n}),\quad
K^{a}_{n}=\epsilon(j^{a}_{n}-\bar{j}^{a}_{-n}).$ (25)

Contracted algebra will be same as (22) with relations analogous to (23)
and(24) taking the following form:

$\displaystyle F^{abc}=\frac{1}{2}\bigg{(}f^{abc}+\bar{f}^{abc}\bigg{)}\ ,\
G^{abc}=\frac{1}{2\epsilon}\bigg{(}f^{abc}-\bar{f}^{abc}\bigg{)}$ (26)
$\displaystyle c_{L}=c-\bar{c},\quad c_{M}=\epsilon(c+\bar{c}),\quad
k_{1}=k-\bar{k},\quad k_{2}=\epsilon(k+\bar{k}).$ (27)

This (22) will be the algebra of interest for the rest of our paper.

Note that if we start with 2 identical Kac-Moody algebras for the holomorphic
and antiholomorphic sections, then after contraction we get

$f^{abc}=\bar{f}^{abc}\Rightarrow F^{abc}=f^{abc},\,G^{abc}=0.$ (28)

For most of our analysis, we will use this simplified algebra, but the results
can be easily generalised for the general algebra.

We should also clarify something about the new Lie algebra structure and our
notation. We started with (the affine version of) two copies of a Lie algebra
$g$ of dimension

$n=\text{dim}(g),$

constructed from the generators $\\{j^{a},a\in\\{1,2,\dots,n\\}\\}$ and
$\\{\bar{j}^{a},a\in\\{1,2,\dots,n\\}\\}$. Via contraction we obtain a (non-
semisimple) Lie algebra $\tilde{g}$ of dimension

$\tilde{n}={\text{dim}}(\tilde{g})=2\ {\text{dim}}(g)$

consisting of generators $\\{J^{a},K^{a},a\in\\{1,2,\dots,n\\}\\}$. So in all
our notation, the indices $a,b,c$ run from 1 to $n=\text{dim}(g)$ (not
$\tilde{n}$), and whenever we write $\text{dim}(g)$ (like in expressions of
central charge $c_{L}$ later in the paper), we mean the dimension of the
parent algebra $g$, which is actually half of the dimension of the new algebra
$\tilde{g}$. This is done for the tidiness of expressions.

Before we conclude this section, we would like to comment on the choice of
contraction of the currents to get to the NLKM algebra. Note that the chosen
linear combinations of the Kac Moody generators are motivated by the
Galilean/Carrollian contractions analogous to (17) and (19). The generators in
cylindrical coordinates look like,

$\displaystyle j^{a}_{n}=j^{a}\otimes
e^{in\omega},\quad\bar{j}^{a}_{n}=\bar{j}^{a}\otimes
e^{in\bar{\omega}},\quad\omega,\bar{\omega}=\tau\pm\sigma$ (29)

Galilean limit will correspond to (17) and Carrollian limit corresponds to
(19). In Galilean case, we have (upto linear order in $\epsilon$),

$\displaystyle j^{a}_{n}+\bar{j}^{a}_{n}=(j^{a}+\bar{j}^{a})\otimes
e^{in\tau}+in\sigma\epsilon(j^{a}-\bar{j}^{a})\otimes e^{in\tau},$ (30a)
$\displaystyle j^{a}_{n}-\bar{j}^{a}_{n}=(j^{a}-\bar{j}^{a})\otimes
e^{in\tau}+in\sigma\epsilon(j^{a}+\bar{j}^{a})\otimes e^{in\tau}.$ (30b)

We now have two choices, keeping the BMS contraction the same:

  * •

$J^{a}=j^{a}+\bar{j}^{a},\,\,K^{a}=\epsilon(\bar{j}^{a}-j^{a})\Rightarrow(\ref{chh14})$.

  * •

$J^{a}=\bar{j}^{a}-j^{a},K^{a}=\epsilon(\bar{j}^{a}+j^{a})\Rightarrow
J^{a}_{m}=\bar{j}^{a}_{m}-j^{a}_{m},\,\,K^{a}_{m}=\epsilon(\bar{j}^{a}_{m}+j^{a}_{m})$.

Both of these choices lead to relations (24) but (23) need to be altered for
the second choice to:

$\displaystyle F^{abc}=\frac{1}{2}(\bar{f}^{abc}-f^{abc}),\
G^{abc}=\frac{1}{2\epsilon}(f^{abc}+\bar{f}^{abc}).$ (31)

Similarly, in Carrollian case, we have two choices: (25) and

$J^{a}_{n}=j^{a}_{n}-j^{a}_{-n},\quad
K^{a}_{n}=\epsilon(j^{a}_{n}+j^{a}_{-n}).$

in which case (27) remains same but we have

$\displaystyle F^{abc}=\frac{1}{2}(f^{abc}-\bar{f}^{abc}),\quad
G^{abc}=\frac{1}{2\epsilon}(f^{abc}+\bar{f}^{abc})$ (32)

instead of (26). Since we will be considering the special case when
${f}^{abc}=\bar{f}^{abc}$, we will stick to the first choice in this paper.
For the intrinsic analysis, which only uses the algebra (22), of course these
choices do not matter. But when we derive results from the limit, in Sec. 7,
it is important to state everything works for the other contraction as well.
E.g. Sugawara construction through contraction (Sec 7.3) can be carried out
for the second choice if we specialize to the case $\bar{f}^{abc}=-f^{abc}$ in
case of Galilean contraction.

##  3 An intrinsic Carrollian derivation

Having derived the non-Lorentzian current algebra through a contraction, we
now go on to present an intrinsically Carrollian derivation of the same, where
we would not be alluding to a limiting procedure at all. This section heavily
borrows from the machinery detailed in Saha:2022gjw , some of the important
features of which are described in Appendix A. Although we will try and be
self consistent so that the section (with the help of the related appendix A)
stands on its own, for any details that we may have inadvertently skipped in
what follows, the reader is referred back to Saha:2022gjw .

###  3.1 An Infinity of Conserved Quantities

A 2D Carrollian CFT on the flat Carrollian background $(t,x)$ is invariant
under an infinite number of 2D Carrollian conformal (CC) transformations whose
infinitesimal versions are given as:

$\displaystyle x^{\prime}=x+\epsilon^{x}f(x)\hskip 7.11317pt,\hskip
7.11317ptt^{\prime}=t+\epsilon^{x}tf^{\prime}(x)+\epsilon^{t}g(x)$ (33)

As a consequence, the EM tensor components classically satisfy the following
conditions Saha:2022gjw :

$\displaystyle\partial_{\mu}T^{\mu}_{\hskip 4.2679pt\nu}=0\hskip
7.11317pt;\hskip 7.11317ptT^{x}_{\hskip 4.2679ptt}=0\hskip 7.11317pt;\hskip
7.11317ptT^{\mu}_{\hskip 4.2679pt\mu}=0$ $\displaystyle\Longrightarrow\hskip
7.11317pt$ $\displaystyle\partial_{t}T^{t}_{\hskip 4.2679ptt}=0\hskip
7.11317pt;\hskip 7.11317pt\partial_{t}T^{t}_{\hskip
4.2679ptx}=\partial_{x}T^{t}_{\hskip 4.2679ptt}.$ (34)

This allows for an infinite number of Noether currents. The two conserved
currents corresponding to the symmetry transformation (33) are noted below:

$\displaystyle j^{\mu}_{\hskip 4.2679ptt}=\left(j^{t}_{\hskip 4.2679ptt}\hskip
2.84526pt,\hskip 2.84526ptj^{x}_{\hskip
4.2679ptt}\right)=\left(g(x)T^{t}_{\hskip 4.2679ptt}\hskip 2.84526pt,\hskip
2.84526pt0\right),$ $\displaystyle j^{\mu}_{\hskip
4.2679ptx}=\left(j^{t}_{\hskip 4.2679ptx}\hskip 2.84526pt,\hskip
2.84526ptj^{x}_{\hskip 4.2679ptx}\right)=\left(f(x)T^{t}_{\hskip
4.2679ptx}+tf^{\prime}(x)T^{t}_{\hskip 4.2679ptt}\hskip 2.84526pt,\hskip
2.84526pt-f(x)T^{t}_{\hskip 4.2679ptt}\right).$ (35)

Now, we suppose that there are some other pairs of fields
$\\{\mathcal{J}^{a}_{x},\mathcal{J}^{a}_{t}\\}$ in the theory that obey the
following conditions analogous to (34):

$\displaystyle\partial_{t}\mathcal{J}^{a}_{t}(t,x)=0\hskip 7.11317pt;\hskip
7.11317pt\partial_{t}\mathcal{J}^{a}_{x}(t,x)=\partial_{x}\mathcal{J}^{a}_{t}(t,x)$
(36)

where $a$ is to be thought of as a ‘flavor’ index (but $t$ and $x$ in
subscript are not tensor indices). Using these fields, we can construct an
infinite number of conserved quantities:

$\displaystyle k^{a\mu}=\left(k^{at}\hskip 2.84526pt,\hskip
2.84526ptk^{ax}\right)=\left(g(x)\mathcal{J}^{a}_{t}\hskip 2.84526pt,\hskip
2.84526pt0\right),$ (37a) $\displaystyle j^{a\mu}=\left(j^{at}\hskip
2.84526pt,\hskip
2.84526ptj^{ax}\right)=\left(f(x)\mathcal{J}^{a}_{x}+tf^{\prime}(x)\mathcal{J}^{a}_{t}\hskip
2.84526pt,\hskip 2.84526pt-f(x)\mathcal{J}^{a}_{t}\right).$ (37b)

We shall regard these conserved quantities as the Noether currents associated
to some ‘internal’ symmetries of the action.

###  3.2 Current Ward Identities

We now consider an infinitesimal internal transformation of a (possibly multi-
component) field $\Phi(t,x)$:

$\displaystyle\Phi(t,x)\rightarrow\tilde{\Phi}(t,x)=\Phi(t,x)+\epsilon^{a}\left(\mathcal{F}_{a}\cdot\Phi\right)(t,x)$
(38)

where $\mathcal{F}_{a}\cdot\Phi$ denotes the functional changes of the (multi-
component) field $\Phi$ under infinitesimal transformations labeled by
$\epsilon^{a}$. So, the generator $G_{a}\Phi$ of this transformation is given
by:

$\displaystyle-i\epsilon^{a}G_{a}\Phi(t,x):=\tilde{\Phi}(t,x)-\Phi(t,x)=\epsilon^{a}\left(\mathcal{F}_{a}\cdot\Phi\right)(t,x)$
(39)

We shall now find the Ward identity corresponding to this internal
transformation which is assumed to be a symmetry of the 2D Carrollian CFT. For
this purpose, we analytically continue the real space variable
$x\in\mathbb{R}\cup\\{\infty\\}$ to the complex plane; thus, the Ward identity
reads Saha:2022gjw :

$\displaystyle\partial_{\mu}\langle j^{\mu}_{\hskip
4.2679pta}(t,{x})X\rangle\sim-i\sum\limits_{i=1}^{n}\text{
}\delta(t-{t_{i}})\left[\frac{-{(\mathcal{F}_{a})}_{i}\cdot\langle
X\rangle}{x-x_{i}}+\sum_{k\geq 2}\text{ }\frac{{\langle
Y^{(k)}_{a}\rangle}_{i}(\mathbf{x_{1}},...\mathbf{x_{n}})}{{(x-x_{i})}^{k}}\right]$
(40)

where the yet unknown correlators ${\langle Y^{(k)}_{a}\rangle}_{i}$ depend on
the transformation properties of the fields in the string-of-fields $X$ and
the transformation itself and $\sim$ denotes ‘modulo terms holomorphic in $x$
inside $[\ldots]$’. We also use the shorthand $\mathbf{x_{1}}=(t_{1},x_{1})$.
All the correlators are time-ordered.

Let us now assume that the symmetry transformation (38) is associated to the
following conserved current operator:

$\displaystyle k^{a\mu}=\left(\mathcal{J}^{a}_{t}\hskip 2.84526pt,\hskip
2.84526pt0\right).$ (41)

The corresponding Ward identity then is:

$\displaystyle\partial_{t}\langle\mathcal{J}^{a}_{t}(t,{x})X\rangle\sim-i\sum\limits_{i=1}^{n}\text{
}\delta(t-{t_{i}})\left[\frac{-{(\mathcal{F}_{a})}_{i}\cdot\langle
X\rangle}{x-x_{i}}+\sum_{k\geq 2}\text{ }\frac{{\langle
Y^{(k)}_{a}\rangle}_{i}(\mathbf{x_{1}},...\mathbf{x_{n}})}{{(x-x_{i})}^{k}}\right]$
$\displaystyle\Rightarrow\hskip 7.11317pt$
$\displaystyle\langle\mathcal{J}^{a}_{t}(t,{x})X\rangle=-i\sum\limits_{i=1}^{n}\text{
}\theta(t-{t_{i}})\left[\frac{-{(\mathcal{F}_{a})}_{i}\cdot\langle
X\rangle}{x-x_{i}}+\sum_{k\geq 2}\text{ }\frac{{\langle
Y^{(k)}_{a}\rangle}_{i}(\mathbf{x_{1}},...\mathbf{x_{n}})}{{(x-x_{i})}^{k}}\right]$
(42)

where, following Saha:2022gjw , the initial condition has been taken to be:

$\displaystyle\lim\limits_{t\rightarrow-\infty}\langle\mathcal{J}^{a}_{t}(t,{x})X\rangle=0$
(43)

and as in 2D relativistic CFT Zamolodchikov:1985wn ,
$\langle\mathcal{J}^{a}_{t}(t,{x})X\rangle$ is assumed to be finite whenever
$x\neq\\{x_{i}\\}$; this condition makes the holomorphic terms inside
$[\ldots]$ vanish in this Ward identity.

Similarly, if the conserved current:

$\displaystyle j^{a\mu}=\left(\mathcal{J}^{a}_{x}\hskip 2.84526pt,\hskip
2.84526pt-\mathcal{J}^{a}_{t}\right)$ (44)

is associated to another internal symmetry transformation:

$\displaystyle\Phi(t,x)\rightarrow\tilde{\Phi}(t,x)=\Phi(t,x)+\epsilon^{a}\left(\mathcal{G}_{a}\cdot\Phi\right)(t,x)$
(45)

the corresponding Ward identity is:

$\displaystyle\langle\left(\partial_{t}\mathcal{J}^{a}_{x}(t,{x})-\partial_{x}\mathcal{J}^{a}_{t}(t,x)\right)X\rangle\sim-i\sum\limits_{i=1}^{n}\text{
}\delta(t-{t_{i}})\left[\frac{-{(\mathcal{G}_{a})}_{i}\cdot\langle
X\rangle}{x-x_{i}}+\sum_{k\geq 2}\text{ }\frac{{\langle
Y^{(k)}_{a}\rangle}_{i}(\mathbf{x_{1}},...\mathbf{x_{n}})}{{(x-x_{i})}^{k}}\right]$
(46)

Assuming the initial condition:

$\displaystyle\lim\limits_{t\rightarrow-\infty}\langle\mathcal{J}^{a}_{x}(t,{x})X\rangle=0$
(47)

and the finite-ness property of $\langle\mathcal{J}^{a}_{x}(t,{x})X\rangle$
for $x\neq\\{x_{i}\\}$, this Ward identity together with (42) lead us to:

$\displaystyle\langle\mathcal{J}^{a}_{x}(t,{x})X\rangle=-i\sum\limits_{i=1}^{n}\text{
}\theta(t-{t_{i}})\left[\frac{-{(\mathcal{G}_{a})}_{i}\cdot\langle
X\rangle}{x-x_{i}}+\sum_{k\geq 2}\text{ }\frac{{\langle
Z^{(k)}_{a}\rangle}_{i}(\mathbf{x_{1}},...\mathbf{x_{n}})}{{(x-x_{i})}^{k}}\hskip
71.13188pt\right.$
$\displaystyle\left.-(t-t_{i})\left(\frac{-{(\mathcal{F}_{a})}_{i}\cdot\langle
X\rangle}{{(x-x_{i})}^{2}}+\sum_{k\geq 2}\text{ }k\frac{{\langle
Y^{(k)}_{a}\rangle}_{i}(\mathbf{x_{1}},...\mathbf{x_{n}})}{{(x-x_{i})}^{k+1}}\right)\right]$
(48)

Again, the correlators ${\langle Z^{(k)}_{a}\rangle}_{i}$ can not be
determined without knowing the explicit internal transformation properties of
the fields in $X$.

For future references, we note the $i\epsilon$-form Saha:2022gjw  of the Ward
identities (42) and (48) below with
$\Delta\tilde{x}_{p}:=x-x_{p}-i\epsilon(t-t_{p})$ 111We hope the reader does
not confuse the delta appearing in $\Delta\tilde{x}_{p}$ with the conformal
weight $\Delta$. The $\Delta$ appearing in the difference of coordinates would
always appear with a coordinate. :

$\displaystyle
i\langle\mathcal{J}^{a}_{t}(t,{x})X\rangle=\lim\limits_{\epsilon\rightarrow
0^{+}}\sum\limits_{i=1}^{n}\left[\frac{-{(\mathcal{F}_{a})}_{i}\cdot\langle
X\rangle}{\Delta\tilde{x}_{i}}+\sum_{k\geq 2}\text{ }\frac{{\langle
Y^{(k)}_{a}\rangle}_{i}}{{(\Delta\tilde{x}_{i})}^{k}}\right]$ (49)
$\displaystyle
i\langle\mathcal{J}^{a}_{x}(t,{x})X\rangle=\lim\limits_{\epsilon\rightarrow
0^{+}}\sum\limits_{i=1}^{n}\left[\frac{-{(\mathcal{G}_{a})}_{i}\cdot\langle
X\rangle}{\Delta\tilde{x}_{i}}+\sum_{k\geq 2}\frac{{\langle
Z^{(k)}_{a}\rangle}_{i}}{{(\Delta\tilde{x}_{i})}^{k}}\right.$
$\displaystyle\hskip
167.87125pt\left.-(t-t_{i})\left(\frac{-{(\mathcal{F}_{a})}_{i}\cdot\langle
X\rangle}{{(\Delta\tilde{x}_{i})}^{2}}+\sum_{k\geq 2}k\frac{{\langle
Y^{(k)}_{a}\rangle}_{i}}{{(\Delta\tilde{x}_{i})}^{k+1}}\right)\right]$ (50)

Thus, a general (possibly multi-component) 2D Carrollian conformal field
$\Phi(t,x)$ has the following OPEs (in the $i\epsilon$-form) with the current-
vector components
($\Delta\tilde{x}^{\prime}:=x^{\prime}-x-i\epsilon(t^{\prime}-t)$):

$\displaystyle
i\mathcal{J}^{a}_{t}(t^{\prime},x^{\prime})\Phi(t,x)\sim\lim\limits_{\epsilon\rightarrow
0^{+}}\left[\ldots+\frac{-(\mathcal{F}_{a}\cdot\Phi)(t,x)}{\Delta\tilde{x}^{\prime}}\right]$
(51) $\displaystyle
i\mathcal{J}^{a}_{x}(t^{\prime},x^{\prime})\Phi(t,x)\sim\lim\limits_{\epsilon\rightarrow
0^{+}}\left[\ldots+\frac{-(\mathcal{G}_{a}\cdot\Phi)(t,x)}{\Delta\tilde{x}^{\prime}}-(t^{\prime}-t)\left(\ldots+\frac{-(\mathcal{F}_{a}\cdot\Phi)(t,x)}{{(\Delta\tilde{x}^{\prime})}^{2}}\right)\right]$
(52)

where $\ldots$ represents higher order poles at $x^{\prime}=x$ and $\sim$
denotes ‘modulo terms holomorphic in $x^{\prime}$ that have vanishing VEVs’.

In this work, we shall only consider currents with scaling dimension
$\Delta=1$. Comparing the behavior of both sides of the OPEs (51) and (52)
under dilatation, we then infer that the scaling dimensions of the local
fields $\mathcal{F}_{a}\cdot\Phi$ and $\mathcal{G}_{a}\cdot\Phi$ must be same
as that of $\Phi$. Since this is true for any arbitrary local field $\Phi$, we
conclude that $\mathcal{F}_{a}\cdot\Phi$ and $\mathcal{G}_{a}\cdot\Phi$ must
be linear combinations of the components of the multi-component field $\Phi$
that ‘internally’ transforms under a bi-matrix representation of the global
current symmetry algebra; all of these components must have an equal scaling
dimension. Our goal is to find this global algebra and its infinite extension.

We now express $\mathcal{F}_{a}\cdot\Phi$ and $\mathcal{G}_{a}\cdot\Phi$ as
explicit linear combinations:

$\displaystyle\left(\mathcal{F}_{a}\cdot\Phi\right)^{i{i^{\prime}}}=\left(t_{K}^{a}\right)^{i}_{\hskip
2.84526ptj}\Phi^{j{i^{\prime}}}\hskip 7.11317pt;\hskip
7.11317pt\left(\mathcal{G}_{a}\cdot\Phi\right)^{i{i^{\prime}}}=\Phi^{i{j^{\prime}}}\left(t_{J}^{a}\right)^{\hskip
2.84526pt{i^{\prime}}}_{{j^{\prime}}}$ (53)

where $t^{a}_{J}$ and $t^{a}_{K}$ are just two matrices as of now. Later, we
shall relate them to the generators of the internal symmetry algebra.

###  3.3 Current-primary fields

In the operator formalism of a QFT, the conserved charge $Q_{A}$ is the
generator of an infinitesimal symmetry transformation on the space of the
quantum fields:

$\displaystyle\tilde{\Phi}(\mathbf{x})-{\Phi}(\mathbf{x})=-i\epsilon^{A}[Q_{A}\text{
},\text{ }{\Phi}(\mathbf{x})]$ (54)

In 2D Carroll CFT, the above generator equation for any conserved charge
operator $Q_{A}$ is related to the following contour integral prescription
involving an OPE Saha:2022gjw 222Section 5 of this reference contains a
derivation.:

$\displaystyle Q_{A}=\frac{1}{2\pi i}\oint\limits_{C_{u}}dx\text{
}j^{t}_{\hskip 4.2679ptA}(t,x)\text{\hskip 14.22636ptgenerates\hskip
14.22636pt}[Q_{A}\text{ },{\Phi}(t,x)]=\frac{1}{2\pi
i}\oint\limits_{x}dx^{\prime}\text{ }j^{t}_{\hskip
4.2679ptA}(t^{+},x^{\prime}){\Phi}(t,x)$ (55)

where $t^{+}>t$ and the counter-clockwise contour $C_{u}$ encloses the upper
half-plane along with the real line. The contour around $x$ must not enclose
any possible singularities of the vector field.

The conserved quantum charges $Q^{a}_{t}[g]$ and $Q^{a}_{x}[f]$ of the
respective currents (37a) and (37b) are thus given by:

$\displaystyle Q^{a}_{t}[g]=\frac{1}{2\pi i}\oint\limits_{C_{u}}dx\text{
}g(x)\mathcal{J}^{a}_{t}(t,x)$ (56) $\displaystyle Q^{a}_{x}[f]=\frac{1}{2\pi
i}\oint\limits_{C_{u}}dx\left[f(x)\mathcal{J}^{a}_{x}(t,x)+tf^{\prime}(x)\mathcal{J}^{a}_{t}(t,x)\right]$
(57)

The conserved charges of all flavors collectively induce the following
infinitesimal changes to a generic quantum field, as deduced from the OPEs
(51) and (52):

$\displaystyle-i\sum_{a}\epsilon^{a}\left[Q^{a}_{t}[g^{a}]\text{ },\text{
}{\Phi}(\mathbf{x})\right]$
$\displaystyle=-\frac{1}{2\pi}\sum_{a}\epsilon^{a}\oint\limits_{x}dx^{\prime}\text{
}g^{a}(x^{\prime})\mathcal{J}^{a}_{t}(t^{+},x^{\prime}){\Phi}(t,x)$
$\displaystyle=\sum_{a}\epsilon^{a}\left[g^{a}(x)\left(t^{a}_{K}\cdot{\Phi}\right)(t,x)+(\text{h.d.t.})\right]$
(58) $\displaystyle-i\sum_{a}\epsilon^{a}\left[Q^{a}_{x}[f^{a}]\text{ },\text{
}{\Phi}(\mathbf{x})\right]$
$\displaystyle=-\frac{1}{2\pi}\sum_{a}\epsilon^{a}\oint\limits_{x}dx^{\prime}\text{
}\left[f^{a}(x^{\prime})\mathcal{J}^{a}_{x}(t^{+},x^{\prime})+t^{+}f^{a\prime}(x^{\prime})\mathcal{J}^{a}_{t}(t^{+},x^{\prime})\right]{\Phi}(t,x)$
$\displaystyle=\sum_{a}\epsilon^{a}\left[f^{a}(x)\left({\Phi}\cdot
t^{a}_{J}\right)(t,x)+tf^{a\prime}(x)\left(t^{a}_{K}\cdot{\Phi}\right)(t,x)+(\text{h.d.t.})\right]$
(59)

where h.d.t. denotes terms necessarily containing derivatives (of order at
least 1) of $g^{a}(x)$ and $f^{a}(x)$.

For the currents (41) and (44), we simply have $f(x)=1=g(x)$. For any
arbitrary field $\Phi(t,x)$ this immediately leads to:

$\displaystyle-i\sum_{a}\epsilon^{a}\left[Q^{a}_{t}[1]\text{ },\text{
}{\Phi}(\mathbf{x})\right]=\sum_{a}\epsilon^{a}\left(t^{a}_{K}\cdot{\Phi}\right)(t,x)$
(60a) $\displaystyle-i\sum_{a}\epsilon^{a}\left[Q^{a}_{x}[1]\text{ },\text{
}{\Phi}(\mathbf{x})\right]=\sum_{a}\epsilon^{a}\left({\Phi}\cdot
t^{a}_{J}\right)(t,x)$ (60b)

Thus, the finite internal transformation that is generated by the charges
$\left\\{Q^{a}_{k}[1]\right\\}$ is:

$\displaystyle\Phi(t,x)\rightarrow\tilde{\Phi}(t,x)=e^{-i\sum_{a}\epsilon^{a}Q^{a}_{t}[1]}\Phi(t,x)e^{i\sum_{a}\epsilon^{a}Q^{a}_{t}[1]}=e^{\sum_{a}\epsilon^{a}t^{a}_{K}}\cdot\Phi(t,x)$
(61)

which is obtained by using the BCH lemma. Similarly, the charges
$\left\\{Q^{a}_{x}[1]\right\\}$ generate the following finite transformation:

$\displaystyle\Phi(t,x)\rightarrow\tilde{\Phi}(t,x)=\Phi(t,x)\cdot
e^{\sum_{a}\epsilon^{a}t^{a}_{J}}$ (62)

On the other hand, as derived from (58), an arbitrary field finitely
transforms under the action of the charges $\left\\{Q^{a}_{t}[g^{a}]\right\\}$
as:

$\displaystyle\Phi(t,x)\rightarrow\tilde{\Phi}(t,x)=e^{\sum_{a}\epsilon^{a}g^{a}(x)t^{a}_{K}}\cdot\Phi(t,x)+\text{
extra terms}$ (63)

while (59) leads to the following finite action of the charges
$\left\\{Q^{a}_{x}[f^{a}]\right\\}$:

$\displaystyle\Phi(t,x)\rightarrow\tilde{\Phi}(t,x)=e^{\sum_{a}\epsilon^{a}tf^{a\prime}(x)t^{a}_{K}}\cdot\Phi(t,x)\cdot
e^{\sum_{a}\epsilon^{a}f^{a}(x)t^{a}_{J}}+\text{ extra terms}$ (64)

In view of the above discussion, we emphasize that while a generic field
transforms covariantly under the action of the charges associated to the
conserved currents (41) and (44), that is not the case for the generic
conserved currents (37a) and (37b). A field that transforms covariantly (i.e.
for which the extra terms in (63) and (64) vanish) even under the action of
the charges of any currents of the form (37a) and (37b) is called a current-
primary field. Consequently, there is no h.d.t. in (58) and (59) appropriate
for a current-primary field. This enables us to completely specify the pole
structures of the current-primary OPEs for a primary field $\Phi(t,x)$:

$\displaystyle\mathcal{J}^{a}_{t}(t^{\prime},x^{\prime})\Phi(t,x)\sim\lim\limits_{\epsilon\rightarrow
0^{+}}i\frac{t^{a}_{K}\cdot\Phi(t,x)}{\Delta\tilde{x}^{\prime}}$ (65)
$\displaystyle\mathcal{J}^{a}_{x}(t^{\prime},x^{\prime})\Phi(t,x)\sim\lim\limits_{\epsilon\rightarrow
0^{+}}i\left[\frac{\Phi(t,x)\cdot
t^{a}_{J}}{\Delta\tilde{x}^{\prime}}-(t^{\prime}-t)\frac{t^{a}_{K}\cdot\Phi(t,x)}{{(\Delta\tilde{x}^{\prime})}^{2}}\right]$
(66)

that immediately imply the following Ward identities for a string $X$ of
primary fields:

$\displaystyle\langle\mathcal{J}^{a}_{t}(t,{x})X\rangle=\lim\limits_{\epsilon\rightarrow
0^{+}}i\sum\limits_{i=1}^{n}\frac{{(t^{a}_{K})}_{i}\cdot\langle
X\rangle}{\Delta\tilde{x}_{i}}$ (67)
$\displaystyle\langle\mathcal{J}^{a}_{x}(t,{x})X\rangle=\lim\limits_{\epsilon\rightarrow
0^{+}}i\sum\limits_{i=1}^{n}\left[\frac{\langle
X\rangle\cdot{(t^{a}_{J})}_{i}}{\Delta\tilde{x}_{i}}-(t-t_{i})\frac{{{(t^{a}_{K})}_{i}}\cdot\langle
X\rangle}{{(\Delta\tilde{x}_{i})}^{2}}\right]$ (68)

where ${(t^{a}_{J})}_{i}$ and ${(t^{a}_{K})}_{i}$ denotes transformation-
matrices appropriate for the $i$-th primary field in $X$.

###  3.4 Current-Current OPEs

To derive the current-current OPEs using the machinery just developed, we
shall assume that no field in the theory has negative scaling dimension with
the identity being the only field with $\Delta=0$.

Under these assumptions, the OPEs between the EM tensor components were
derived using only symmetry arguments in Saha:2022gjw ; the results are:

$\displaystyle T^{t}_{\hskip 4.2679ptt}({t^{\prime},x^{\prime}})T^{t}_{\hskip
4.2679ptt}(t,x)\sim 0$ $\displaystyle T^{t}_{\hskip
4.2679ptx}({t^{\prime},x^{\prime}})T^{t}_{\hskip
4.2679ptt}(t,x)\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-i\left[\frac{-i\frac{c_{M}}{2}}{{(\Delta\tilde{x}^{\prime})}^{4}}+\frac{2T^{t}_{\hskip
4.2679ptt}(t,x)}{{(\Delta\tilde{x}^{\prime})}^{2}}+\frac{{\partial_{x}}T^{t}_{\hskip
4.2679ptt}(t,x)}{\Delta\tilde{x}^{\prime}}\right]$ $\displaystyle
T^{t}_{\hskip 4.2679ptt}({t^{\prime},x^{\prime}})T^{t}_{\hskip
4.2679ptx}(t,x)\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-i\left[\frac{-i\frac{c_{M}}{2}}{{(\Delta\tilde{x}^{\prime})}^{4}}+\frac{2T^{t}_{\hskip
4.2679ptt}(t,x)}{{(\Delta\tilde{x}^{\prime})}^{2}}+\frac{{\partial_{t}}T^{t}_{\hskip
4.2679ptx}(t,x)}{\Delta\tilde{x}^{\prime}}\right]$ (69) $\displaystyle
T^{t}_{\hskip 4.2679ptx}({t^{\prime},x^{\prime}})T^{t}_{\hskip
4.2679ptx}(t,x)\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-i\left[\frac{-i\frac{c_{L}}{2}}{{(\Delta\tilde{x}^{\prime})}^{4}}+\frac{2T^{t}_{\hskip
4.2679ptx}(t,x)}{{(\Delta\tilde{x}^{\prime})}^{2}}+\frac{{\partial_{x}}T^{t}_{\hskip
4.2679ptx}(t,x)}{\Delta\tilde{x}^{\prime}}\right.$ $\displaystyle\left.\hskip
176.40707pt-({t^{\prime}}-{t})\left(\frac{-2ic_{M}}{{(\Delta\tilde{x}^{\prime})}^{5}}+\frac{4T^{t}_{\hskip
4.2679ptt}(t,x)}{{(\Delta\tilde{x}^{\prime})}^{3}}+\frac{{\partial_{t}}T^{t}_{\hskip
4.2679ptx}(t,x)}{{(\Delta\tilde{x}^{\prime})}^{2}}\right)\right].$

The constants $c_{L}$ and $c_{M}$ are the central charges of the 2D Carrollian
conformal QFT.

We shall use the same technique to find the current-current OPEs below.
Keeping in mind that here we are dealing with currents with scaling dimension
$\Delta=1$, below we note the most general allowed form of the
$\mathcal{J}_{t}-\mathcal{J}_{t}$ OPE compatible with the general OPE (51):

$\displaystyle\mathcal{J}^{a}_{t}(t^{\prime},x^{\prime})\mathcal{J}^{b}_{t}(t,x)\sim\lim\limits_{\epsilon\rightarrow
0^{+}}i\left[\frac{A^{ab}}{{\left(\Delta\tilde{x}^{\prime}\right)}^{2}}+\frac{\left(t^{a}_{K}\cdot\mathcal{J}_{t}\right)^{b}(t,x)}{\Delta\tilde{x}^{\prime}}\right]$
(70)

where $A^{ab}$ is a field proportional to identity so that it has vanishing
scaling dimension. Clearly,
$\left(t^{a}_{K}\cdot\mathcal{J}_{t}\right)^{b}(t,x)$ must have scaling
dimension $\Delta=1$. So, in a generic 2D CCFT,
$\left(t^{a}_{K}\cdot\mathcal{J}_{t}\right)^{b}$ must be a linear combination
of $\\{\mathcal{J}^{a}_{x},\mathcal{J}^{a}_{t}\\}$.

Correspondingly to the classical conservation equation
$\partial_{t}\mathcal{J}^{a}_{t}(t,x)=0$ , in the QFT we should have:

$\displaystyle\mathcal{J}^{a}_{t}(t^{\prime},x^{\prime})\partial_{t}\mathcal{J}^{b}_{t}(t,x)\sim
0\hskip 7.11317pt\Rightarrow\hskip
7.11317pt\partial_{t}\left(t^{a}_{K}\cdot\mathcal{J}_{t}\right)^{b}(t,x)=0$
(71)

which means that $\\{\mathcal{J}^{a}_{x}\\}$ can not contribute to the linear
combination $\left(t^{a}_{K}\cdot\mathcal{J}_{t}\right)^{b}$.

Since the currents have scaling dimension $\Delta=1$, they must satisfy the
bosonic333Since, as will be shown later, the currents are 2D CC primary fields
with integer scaling dimension, this statement is justified Saha:2022gjw .
exchange property. For the $\mathcal{J}^{a}_{t}(t,x)$ field, it is:

$\displaystyle\mathcal{J}^{a}_{t}(t^{\prime},x^{\prime})\mathcal{J}^{b}_{t}(t,x)=\mathcal{J}^{b}_{t}(t,x)\mathcal{J}^{a}_{t}(t^{\prime},x^{\prime})$
(72)

This condition implies the following restrictions:

$\displaystyle A^{ab}=A^{ba}\text{\hskip 9.95845pt and \hskip
9.95845pt}\left(t^{a}_{K}\cdot\mathcal{J}_{t}\right)^{b}=-\left(t^{b}_{K}\cdot\mathcal{J}_{t}\right)^{a}$
(73)

Looking at (52), we write the allowed form of a
$\mathcal{J}_{x}-\mathcal{J}_{t}$ OPE:

$\displaystyle\mathcal{J}^{a}_{x}(t^{\prime},x^{\prime})\mathcal{J}^{b}_{t}(t,x)\sim\lim\limits_{\epsilon\rightarrow
0^{+}}i\left[\frac{B^{ab}}{{\left(\Delta\tilde{x}^{\prime}\right)}^{2}}+\frac{\left(\mathcal{J}_{t}\cdot
t^{a}_{J}\right)^{b}(t,x)}{\Delta\tilde{x}^{\prime}}-(t^{\prime}-t)\left(\frac{2A^{ab}}{{\left(\Delta\tilde{x}^{\prime}\right)}^{3}}+\frac{\left(t^{a}_{K}\cdot\mathcal{J}_{t}\right)^{b}(t,x)}{{\left(\Delta\tilde{x}^{\prime}\right)}^{2}}\right)\right]$
(74)

where $B^{ab}$ are some constants. Now, we have the following restrictions:

$\displaystyle\mathcal{J}^{a}_{x}(t^{\prime},x^{\prime})\partial_{t}\mathcal{J}^{b}_{t}(t,x)\sim
0\hskip 11.38109pt\Longrightarrow\hskip 11.38109ptA^{ab}=0\text{\hskip
8.53581pt and \hskip
8.53581pt}\left(t^{a}_{K}\cdot\mathcal{J}_{t}\right)^{b}=0\text{\hskip
8.53581pt and \hskip 8.53581pt}\partial_{t}\left(\mathcal{J}_{t}\cdot
t^{a}_{J}\right)^{b}(t,x)=0$ (75)

Thus, again $\\{\mathcal{J}^{a}_{x}\\}$ do not contribute to the linear
combination $\left(\mathcal{J}_{t}\cdot t^{a}_{J}\right)^{b}$.

On the other hand, from (51), we get the following
$\mathcal{J}_{t}-\mathcal{J}_{x}$ OPE:

$\displaystyle\mathcal{J}^{b}_{t}(t^{\prime},x^{\prime})\mathcal{J}^{a}_{x}(t,x)\sim\lim\limits_{\epsilon\rightarrow
0^{+}}i\left[\frac{C^{ba}}{{\left(\Delta\tilde{x}^{\prime}\right)}^{2}}+\frac{\left(t^{b}_{K}\cdot\mathcal{J}_{x}\right)^{a}(t,x)}{\Delta\tilde{x}^{\prime}}\right]$
(76)

with $C^{ab}$ being constants. Using the following bosonic exchange property:

$\displaystyle\mathcal{J}^{b}_{t}(t^{\prime},x^{\prime})\mathcal{J}^{a}_{x}(t,x)=\mathcal{J}^{a}_{x}(t,x)\mathcal{J}^{b}_{t}(t^{\prime},x^{\prime})$

to compare the OPE (74) with (76), we get the following conditions:

$\displaystyle B^{ab}=C^{ba}\text{\hskip 9.95845pt and \hskip
9.95845pt}\left(\mathcal{J}_{t}\cdot
t^{a}_{J}\right)^{b}=-\left(t^{b}_{K}\cdot\mathcal{J}_{x}\right)^{a}$ (77)

which implies that $\\{\mathcal{J}^{a}_{x}\\}$ do not appear also in the
linear combination $\left(t^{b}_{K}\cdot\mathcal{J}_{x}\right)^{a}$.

Finally, we write the $\mathcal{J}_{x}-\mathcal{J}_{x}$ OPE in accordance with
the general form (52):

$\displaystyle\mathcal{J}^{a}_{x}(t^{\prime},x^{\prime})\mathcal{J}^{b}_{x}(t,x)\sim\lim\limits_{\epsilon\rightarrow
0^{+}}i\left[\frac{D^{ab}}{{\left(\Delta\tilde{x}^{\prime}\right)}^{2}}+\frac{\left(\mathcal{J}_{x}\cdot
t^{a}_{J}\right)^{b}(t,x)}{\Delta\tilde{x}^{\prime}}-(t^{\prime}-t)\left(\frac{2C^{ab}}{{\left(\Delta\tilde{x}^{\prime}\right)}^{3}}+\frac{\left(t^{a}_{K}\cdot\mathcal{J}_{x}\right)^{b}(t,x)}{{\left(\Delta\tilde{x}^{\prime}\right)}^{2}}\right)\right]$
(78)

from which, the bosonic exchange property:

$\displaystyle\mathcal{J}^{b}_{x}(t^{\prime},x^{\prime})\mathcal{J}^{a}_{x}(t,x)=\mathcal{J}^{a}_{x}(t,x)\mathcal{J}^{b}_{x}(t^{\prime},x^{\prime})$

leads to the following conditions:

$\displaystyle D^{ab}=D^{ba}\hskip 7.11317pt;\hskip 7.11317ptC^{ab}=C^{ba}$
(79) $\displaystyle\left(\mathcal{J}_{x}\cdot
t^{a}_{J}\right)^{b}=-\left(\mathcal{J}_{x}\cdot t^{b}_{J}\right)^{a}\hskip
7.11317pt;\hskip 7.11317pt$
$\displaystyle\left(t^{a}_{K}\cdot\mathcal{J}_{x}\right)^{b}=-\left(t^{b}_{K}\cdot\mathcal{J}_{x}\right)^{a}\hskip
7.11317pt;\hskip 7.11317pt\partial_{t}\left(\mathcal{J}_{x}\cdot
t^{a}_{J}\right)^{b}=\partial_{x}\left(t^{a}_{K}\cdot\mathcal{J}_{x}\right)^{b}$
(80)

It can be readily checked that these conditions are compatible with the
quantum versions (in the OPE language) of the classical conservation laws
(36).

We now explicitly write the allowed forms of the linear combinations appearing
in the above OPEs:

$\displaystyle\left(\mathcal{J}_{t}\cdot
t^{a}_{J}\right)^{b}=\left(t^{a}_{K}\cdot\mathcal{J}_{x}\right)^{b}=F^{abc}\mathcal{J}^{c}_{t}$
$\displaystyle\text{ with }\hskip 7.11317ptF^{abc}=-F^{bac},$
$\displaystyle\left(\mathcal{J}_{x}\cdot
t^{a}_{J}\right)^{b}=F^{abc}\mathcal{J}^{c}_{x}+G^{abc}\mathcal{J}^{c}_{t}$
$\displaystyle\text{ with }\hskip 7.11317ptG^{abc}=-G^{bac}.$ (81)

Thus, the final forms of the current-current OPEs are:

$\displaystyle\mathcal{J}^{a}_{t}(t^{\prime},x^{\prime})\mathcal{J}^{b}_{t}(t,x)\sim
0$
$\displaystyle\mathcal{J}^{a}_{x}(t^{\prime},x^{\prime})\mathcal{J}^{b}_{t}(t,x)\sim\lim\limits_{\epsilon\rightarrow
0^{+}}i\left[\frac{C^{ab}}{{\left(\Delta\tilde{x}^{\prime}\right)}^{2}}+\frac{F^{abc}\mathcal{J}^{c}_{t}(t,x)}{\Delta\tilde{x}^{\prime}}\right]$
$\displaystyle\mathcal{J}^{a}_{t}(t^{\prime},x^{\prime})\mathcal{J}^{b}_{x}(t,x)\sim\lim\limits_{\epsilon\rightarrow
0^{+}}i\left[\frac{C^{ab}}{{\left(\Delta\tilde{x}^{\prime}\right)}^{2}}+\frac{F^{abc}\mathcal{J}^{c}_{t}(t,x)}{\Delta\tilde{x}^{\prime}}\right]$
(82)
$\displaystyle\mathcal{J}^{a}_{x}(t^{\prime},x^{\prime})\mathcal{J}^{b}_{x}(t,x)\sim\lim\limits_{\epsilon\rightarrow
0^{+}}i\left[\frac{D^{ab}}{{\left(\Delta\tilde{x}^{\prime}\right)}^{2}}+\frac{\left(F^{abc}\mathcal{J}^{c}_{x}+G^{abc}\mathcal{J}^{c}_{t}\right)(t,x)}{\Delta\tilde{x}^{\prime}}-(t^{\prime}-t)\left(\frac{2C^{ab}}{{\left(\Delta\tilde{x}^{\prime}\right)}^{3}}+\frac{F^{abc}\mathcal{J}^{c}_{t}(t,x)}{{\left(\Delta\tilde{x}^{\prime}\right)}^{2}}\right)\right]$

These OPEs imply that the currents themselves are not current-primary fields
in general.

###  3.5 Global internal symmetry

All the correlation functions in the theory must be invariant under the global
internal transformations associated to which are the conserved currents (41)
and (44). This fact will put constraints on $C^{ab}$ and $D^{ab}$, as we will
now see.

We begin by noting the following 2-point correlators between the currents,
from the above OPEs:

$\displaystyle\left\langle\mathcal{J}^{a}_{t}(t^{\prime},x^{\prime})\mathcal{J}^{b}_{t}(t,x)\right\rangle=0$
$\displaystyle\left\langle\mathcal{J}^{a}_{t}(t^{\prime},x^{\prime})\mathcal{J}^{b}_{x}(t,x)\right\rangle=\left\langle\mathcal{J}^{a}_{x}(t^{\prime},x^{\prime})\mathcal{J}^{b}_{t}(t,x)\right\rangle=\lim\limits_{\epsilon\rightarrow
0^{+}}i\frac{C^{ab}}{{\left(\Delta\tilde{x}^{\prime}\right)}^{2}}$ (83)
$\displaystyle\left\langle\mathcal{J}^{a}_{x}(t^{\prime},x^{\prime})\mathcal{J}^{b}_{x}(t,x)\right\rangle=\lim\limits_{\epsilon\rightarrow
0^{+}}i\left[\frac{D^{ab}}{{\left(\Delta\tilde{x}^{\prime}\right)}^{2}}-(t^{\prime}-t)\frac{2C^{ab}}{{\left(\Delta\tilde{x}^{\prime}\right)}^{3}}\right]$

since the currents, with $\Delta=1$, must have vanishing VEVs.

Next, due to the global internal symmetry, an arbitrary $n$-point correlator
in the theory must satisfy:

$\displaystyle\sum_{i=1}^{n}\left\langle{\Phi_{1}}(t_{1},x_{1})\ldots\left({\Phi_{i}}\cdot
t^{a}_{J}\right)(t_{i},x_{i})\ldots{\Phi_{n}}(t_{n},x_{n})\right\rangle=0$
$\displaystyle\sum_{i=1}^{n}\left\langle{\Phi_{1}}(t_{1},x_{1})\ldots\left(t^{a}_{K}\cdot{\Phi_{i}}\right)(t_{i},x_{i})\ldots{\Phi_{n}}(t_{n},x_{n})\right\rangle=0$

Thus the 2-point current correlators explicitly satisfy:

$\displaystyle\left\langle\left(\mathcal{J}_{x}\cdot
t^{a}_{J}\right)^{b}(t_{1},x_{1})\mathcal{J}^{c}_{x}(t_{2},x_{2})\right\rangle+\left\langle\mathcal{J}^{b}_{x}(t_{1},x_{1})\left(\mathcal{J}_{x}\cdot
t^{a}_{J}\right)^{c}(t_{2},x_{2})\right\rangle=0$
$\displaystyle\Longrightarrow\hskip 7.11317pt$ $\displaystyle
F^{abd}D^{dc}+F^{acd}D^{db}+G^{abd}C^{dc}+G^{acd}C^{db}=0\hskip
14.22636pt\text{ and }\hskip 14.22636ptF^{abd}C^{dc}+F^{acd}C^{db}=0$ (84)

The analogues relations obtained from the invariance of the 3-point current
correlators are:

$\displaystyle F^{abe}F^{ecf}C^{fd}+F^{cae}F^{ebf}C^{fd}$
$\displaystyle+F^{ade}F^{bcf}C^{fe}=0$ $\displaystyle
F^{abe}F^{ecf}D^{fd}+F^{cae}F^{ebf}D^{fd}$
$\displaystyle+F^{ade}F^{bcf}D^{fe}+\left(F^{abe}G^{ecf}+G^{abe}F^{ecf}\right)C^{fd}$
$\displaystyle+\left(F^{cae}G^{ebf}+G^{cae}F^{ebf}\right)C^{fd}+\left(F^{ade}G^{bcf}+G^{ade}F^{bcf}\right)C^{fe}=0$
(85)

In what follows, we shall see that $\\{F^{abc}\\}$ and $\\{G^{abc}\\}$ must
also obey the following constraints arising as the Jacobi identity of the
infinite-dimensional Lie algebra of the current-modes, which we had previously
described in our initial algebraic description from the contraction in (22)
and also will derive independently later in this section (112):

$\displaystyle F^{abe}F^{ecd}+F^{cae}F^{ebd}+F^{bce}F^{ead}=0$ $\displaystyle
F^{abe}G^{ecd}+G^{abe}F^{ecd}+F^{cae}G^{ebd}+G^{cae}F^{ebd}+F^{bce}G^{ead}+G^{bce}F^{ead}=0$
(86)

No new constraint for $\\{F^{abc}\\}$ and $\\{G^{abc}\\}$ arises from the
global internal invariance of $n$-point current correlators for $n\geq 4$ .

Upon comparison, we notice that (85) reduces to (86) if we choose:

$\displaystyle D^{ab}=-ik_{1}\delta^{ab}\hskip 14.22636pt\text{and}\hskip
14.22636ptC^{ab}=-ik_{2}\delta^{ab}\hskip 7.11317pt\text{with}\hskip
7.11317ptk_{2}\neq 0$ (87)

In that case, $F^{abc}$ and $G^{abc}$ are anti-symmetric in all indices, as
seen from (84).

###  3.6 EM tensor-current OPEs

We now show under the assumption that no field in the theory has negative
scaling dimension with the identity field being the only one with $\Delta=0$ ,
that the currents $\mathcal{J}^{a}_{x}(t,x)$ and $\mathcal{J}^{a}_{t}(t,x)$
must transform as a rank-$\frac{1}{2}$ primary multiplet under 2D CC
transformations.

From Saha:2022gjw , we recall the OPEs of a general 2D CC (multi-component)
field $\Phi_{(l)}(t,x)$ having scaling dimension $\Delta$, Carrollian boost-
charge $\xi$ and boost rank $l$ with the EM tensor components:

$\displaystyle T^{t}_{\hskip
4.2679ptx}({t^{\prime},x^{\prime}})\Phi_{(l)}(t,x)\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-i\left[\ldots+\frac{{{\Delta}}\Phi_{(l)}(t,x)}{{(\Delta\tilde{x}^{\prime})}^{2}}+\frac{{\partial_{x}}\Phi_{(l)}(t,x)}{\Delta\tilde{x}^{\prime}}\right.$
$\displaystyle\left.\hskip
196.324pt-({t^{\prime}}-{t})\left(\ldots+\frac{2\left(\bm{\xi}\cdot\Phi_{(l)}\right)(t,x)}{{(\Delta\tilde{x}^{\prime})}^{3}}+\frac{{\partial_{t}}\Phi_{(l)}(t,x)}{{(\Delta\tilde{x}^{\prime})}^{2}}\right)\right]$
$\displaystyle T^{t}_{\hskip
4.2679ptt}(t^{\prime},x^{\prime})\Phi_{(l)}(t,x)\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-i\left[\ldots+\frac{\left(\bm{\xi}\cdot\Phi_{(l)}\right)(t,x)}{{(\Delta\tilde{x}^{\prime})}^{2}}+\frac{{\partial_{t}}\Phi_{(l)}(t,x)}{\Delta\tilde{x}^{\prime}}\right]$
(88)

The defining feature of 2D CC primary fields is the vanishing of the higher
order poles in the above OPEs that leads to:

$\displaystyle T^{t}_{\hskip
4.2679ptx}({t^{\prime},x^{\prime}})\Phi_{(l)}(t,x)\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-i\left[\frac{{{\Delta}}\Phi_{(l)}(t,x)}{{(\Delta\tilde{x}^{\prime})}^{2}}+\frac{{\partial_{x}}\Phi_{(l)}(t,x)}{\Delta\tilde{x}^{\prime}}\right.$
$\displaystyle\left.\hskip
196.324pt-({t^{\prime}}-{t})\left(\frac{2\left(\bm{\xi}\cdot\Phi_{(l)}\right)(t,x)}{{(\Delta\tilde{x}^{\prime})}^{3}}+\frac{{\partial_{t}}\Phi_{(l)}(t,x)}{{(\Delta\tilde{x}^{\prime})}^{2}}\right)\right]$
$\displaystyle T^{t}_{\hskip
4.2679ptt}(t^{\prime},x^{\prime})\Phi_{(l)}(t,x)\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-i\left[\frac{\left(\bm{\xi}\cdot\Phi_{(l)}\right)(t,x)}{{(\Delta\tilde{x}^{\prime})}^{2}}+\frac{{\partial_{t}}\Phi_{(l)}(t,x)}{\Delta\tilde{x}^{\prime}}\right]$
(89)

Now, along with the above assumption, the classical time-independence of the
field $\mathcal{J}^{a}_{t}$ restricts the general OPE (88) to the following
form:

$\displaystyle T^{t}_{\hskip
4.2679ptt}(t^{\prime},x^{\prime})\mathcal{J}^{a}_{t}(t,x)\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-i\left[\frac{A^{a}_{1}}{{(\Delta\tilde{x}^{\prime})}^{3}}+\frac{\left(\bm{\xi}\cdot\mathcal{J}^{a}_{t}\right)(t,x)}{{(\Delta\tilde{x}^{\prime})}^{2}}\right]$
(90)

where the field $A^{a}_{1}$ is proportional to the identity field. Similarly,
we write the most general allowed form of the following OPE from (88):

$\displaystyle T^{t}_{\hskip
4.2679ptx}(t^{\prime},x^{\prime})\mathcal{J}^{a}_{t}(t,x)\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-i\left[\frac{A^{a}_{2}}{{(\Delta\tilde{x}^{\prime})}^{3}}+\frac{\mathcal{J}^{a}_{t}(t,x)}{{(\Delta\tilde{x}^{\prime})}^{2}}+\frac{\partial_{x}\mathcal{J}^{a}_{t}(t,x)}{\Delta\tilde{x}^{\prime}}-(t^{\prime}-t)\left\\{\frac{3A^{a}_{1}}{{(\Delta\tilde{x}^{\prime})}^{4}}+\frac{2\left(\bm{\xi}\cdot\mathcal{J}^{a}_{t}\right)(t,x)}{{(\Delta\tilde{x}^{\prime})}^{3}}\right\\}\right]$
(91)

with $A^{a}_{2}$ being another constant. But, the quantum counterpart of the
conservation law (36) leads to:

$\displaystyle T^{t}_{\hskip
4.2679ptt}(t^{\prime},x^{\prime})\partial_{t}\mathcal{J}^{a}_{t}(t,x)\sim
0\text{ }\Longrightarrow\text{
}\partial_{t}\left(\bm{\xi}\cdot\mathcal{J}^{a}_{t}\right)(t,x)=0,$
$\displaystyle T^{t}_{\hskip
4.2679ptx}(t^{\prime},x^{\prime})\partial_{t}\mathcal{J}^{a}_{t}(t,x)\sim
0\text{ }\Longrightarrow\text{ }A^{a}_{1}=0\hskip 7.11317pt\text{ and }\hskip
7.11317pt\bm{\xi}\cdot\mathcal{J}^{a}_{t}=0.$ (92)

The OPEs for $\mathcal{J}^{a}_{x}(t,x)$ may have the most general form given
below:

$\displaystyle T^{t}_{\hskip
4.2679ptt}(t^{\prime},x^{\prime})\mathcal{J}^{a}_{x}(t,x)\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-i\left[\frac{A^{a}_{3}}{{(\Delta\tilde{x}^{\prime})}^{3}}+\frac{\left(\bm{\xi}\cdot\mathcal{J}^{a}_{x}\right)(t,x)}{{(\Delta\tilde{x}^{\prime})}^{2}}+\frac{\partial_{t}\mathcal{J}^{a}_{x}(t,x)}{\Delta\tilde{x}^{\prime}}\right]$
(93)

with the conservation law (36) forcing:

$\displaystyle\partial_{t}\left(\bm{\xi}\cdot\mathcal{J}^{a}_{x}\right)(t,x)=0.$
(94)

The remaining one OPE is given below:

$\displaystyle T^{t}_{\hskip
4.2679ptx}(t^{\prime},x^{\prime})\mathcal{J}^{a}_{x}(t,x)\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-i\left[\frac{A^{a}_{4}}{{(\Delta\tilde{x}^{\prime})}^{3}}+\frac{\mathcal{J}^{a}_{x}(t,x)}{{(\Delta\tilde{x}^{\prime})}^{2}}+\frac{\partial_{x}\mathcal{J}^{a}_{x}(t,x)}{\Delta\tilde{x}^{\prime}}\right.$
(95) $\displaystyle\left.\hskip
160.75806pt-(t^{\prime}-t)\left(\frac{3A^{a}_{3}}{{(\Delta\tilde{x}^{\prime})}^{4}}+\frac{2\left(\bm{\xi}\cdot\mathcal{J}^{a}_{x}\right)(t,x)}{{(\Delta\tilde{x}^{\prime})}^{3}}+\frac{\partial_{t}\mathcal{J}^{a}_{x}(t,x)}{{(\Delta\tilde{x}^{\prime})}^{2}}\right)\right]$

where $A^{a}_{3}$ and $A^{a}_{4}$ are constants. From the quantum version of
(36), one then obtains:

$\displaystyle T^{t}_{\hskip
4.2679ptx}(t^{\prime},x^{\prime})\left[\partial_{t}\mathcal{J}^{a}_{x}(t,x)-\partial_{x}\mathcal{J}^{a}_{t}(t,x)\right]\sim
0\hskip 14.22636pt\Longrightarrow\hskip
14.22636pt\bm{\xi}\cdot\mathcal{J}^{a}_{x}=\mathcal{J}^{a}_{t}\text{\hskip
9.95845pt and \hskip 9.95845pt}A^{a}_{2}=A^{a}_{3}$ (96)

i.e. the currents $\mathcal{J}^{a}_{x}(t,x)$ and $\mathcal{J}^{a}_{t}(t,x)$
transform under Carrollian boost as a rank-$\frac{1}{2}$ multiplet with boost
charge $\xi=1$.

The OPEs for the currents then are:

$\displaystyle T^{t}_{\hskip
4.2679ptt}(t^{\prime},x^{\prime})\mathcal{J}^{a}_{t}(t,x)\sim 0$
$\displaystyle T^{t}_{\hskip
4.2679ptx}(t^{\prime},x^{\prime})\mathcal{J}^{a}_{t}(t,x)\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-i\left[\frac{A^{a}_{3}}{{(\Delta\tilde{x}^{\prime})}^{3}}+\frac{\mathcal{J}^{a}_{t}(t,x)}{{(\Delta\tilde{x}^{\prime})}^{2}}+\frac{\partial_{x}\mathcal{J}^{a}_{t}(t,x)}{\Delta\tilde{x}^{\prime}}\right]$
$\displaystyle T^{t}_{\hskip
4.2679ptt}(t^{\prime},x^{\prime})\mathcal{J}^{a}_{x}(t,x)\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-i\left[\frac{A^{a}_{3}}{{(\Delta\tilde{x}^{\prime})}^{3}}+\frac{\mathcal{J}^{a}_{t}(t,x)}{{(\Delta\tilde{x}^{\prime})}^{2}}+\frac{\partial_{t}\mathcal{J}^{a}_{x}(t,x)}{\Delta\tilde{x}^{\prime}}\right]$
(97) $\displaystyle T^{t}_{\hskip
4.2679ptx}(t^{\prime},x^{\prime})\mathcal{J}^{a}_{x}(t,x)\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-i\left[\frac{A^{a}_{4}}{{(\Delta\tilde{x}^{\prime})}^{3}}+\frac{\mathcal{J}^{a}_{x}(t,x)}{{(\Delta\tilde{x}^{\prime})}^{2}}+\frac{\partial_{x}\mathcal{J}^{a}_{x}(t,x)}{\Delta\tilde{x}^{\prime}}\right.$
$\displaystyle\left.\hskip
160.75806pt-(t^{\prime}-t)\left(\frac{3A^{a}_{3}}{{(\Delta\tilde{x}^{\prime})}^{4}}+\frac{2\mathcal{J}^{a}_{t}(t,x)}{{(\Delta\tilde{x}^{\prime})}^{3}}+\frac{\partial_{t}\mathcal{J}^{a}_{x}(t,x)}{{(\Delta\tilde{x}^{\prime})}^{2}}\right)\right]$

On the other hand, applying the bosonic exchange property between the currents
and the EM tensor components, we obtain:

$\displaystyle\mathcal{J}^{a}_{t}(t^{\prime},x^{\prime})T^{t}_{\hskip
4.2679ptt}(t,x)\sim 0$
$\displaystyle\mathcal{J}^{a}_{t}(t^{\prime},x^{\prime})T^{t}_{\hskip
4.2679ptx}(t,x)\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-i\left[-\frac{A^{a}_{3}}{{(\Delta\tilde{x}^{\prime})}^{3}}+\frac{\mathcal{J}^{a}_{t}(t,x)}{{(\Delta\tilde{x}^{\prime})}^{2}}\right]$
$\displaystyle\mathcal{J}^{a}_{x}(t^{\prime},x^{\prime})T^{t}_{\hskip
4.2679ptt}(t,x)\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-i\left[-\frac{A^{a}_{3}}{{(\Delta\tilde{x}^{\prime})}^{3}}+\frac{\mathcal{J}^{a}_{t}(t,x)}{{(\Delta\tilde{x}^{\prime})}^{2}}\right]$
(98) $\displaystyle\mathcal{J}^{a}_{x}(t^{\prime},x^{\prime})T^{t}_{\hskip
4.2679ptx}(t,x)\sim\lim\limits_{\epsilon\rightarrow
0^{+}}-i\left[-\frac{A^{a}_{4}}{{(\Delta\tilde{x}^{\prime})}^{3}}+\frac{\mathcal{J}^{a}_{x}(t,x)}{{(\Delta\tilde{x}^{\prime})}^{2}}-(t^{\prime}-t)\left(-\frac{3A^{a}_{3}}{{(\Delta\tilde{x}^{\prime})}^{4}}+\frac{2\mathcal{J}^{a}_{t}(t,x)}{{(\Delta\tilde{x}^{\prime})}^{3}}\right)\right]$

Comparing these with (51) and (52), we immediately note the following:

$\displaystyle\left(t^{a}_{K}\cdot T^{t}_{\hskip
4.2679ptt}\right)(t,x)=\left(t^{a}_{K}\cdot T^{t}_{\hskip
4.2679ptx}\right)(t,x)=\left(T^{t}_{\hskip 4.2679ptt}\cdot
t^{a}_{J}\right)(t,x)=\left(T^{t}_{\hskip 4.2679ptx}\cdot
t^{a}_{J}\right)(t,x)=0$ (99)

which implies that the EM tensor components actually transform under the
singlet representation of the global internal symmetry algebra.

Next we look at the consequences of the global internal symmetry on the
2-point correlators between the currents and the EM tensor components to find
possible constraints on $A^{a}_{3}$ and $A^{a}_{4}$. It suffices to consider
the following:

$\displaystyle\left\langle\left(\mathcal{J}_{x}\cdot
t^{a}_{J}\right)^{b}(t_{1},x_{1})T^{t}_{\hskip
4.2679ptx}(t_{2},x_{2})\right\rangle+\left\langle\mathcal{J}^{b}_{x}(t_{1},x_{1})\left(T^{t}_{\hskip
4.2679ptx}\cdot t^{a}_{J}\right)(t_{2},x_{2})\right\rangle=0\hskip
7.11317pt\Longrightarrow\hskip 7.11317ptA^{a}_{3}=A^{a}_{4}=0$ (100)

Causing the vanishing of the poles of appropriate orders in the OPEs (97), we
finally conclude, comparing with (89), that:

the currents tranform as a 2D CC primary multiplet of rank-$\frac{1}{2}$ with
$\Delta=\xi=1$ .

Thus, the currents have the following infinitesimal 2D CC transformation
property under (33), as is obtained using the prescription (55) for the
corresponding conserved currents (35):

$\displaystyle-
iG_{x}\mathcal{J}^{a}_{t}(t,x)=-[f(x)\partial_{x}+f^{\prime}(x)]\mathcal{J}^{a}_{t}(t,x)\hskip
14.22636pt;\hskip 14.22636pt-iG_{t}\mathcal{J}^{a}_{t}(t,x)=0$ $\displaystyle-
iG_{t}\mathcal{J}^{a}_{x}(t,x)=-g(x)\partial_{t}\mathcal{J}^{a}_{x}(t,x)-g^{\prime}(x)\mathcal{J}^{a}_{t}(t,x)$
(101) $\displaystyle-
iG_{x}\mathcal{J}^{a}_{x}(t,x)=-[f(x)\partial_{x}+tf^{\prime}(x)\partial_{t}+f^{\prime}(x)]\mathcal{J}^{a}_{x}(t,x)-tf^{\prime\prime}(x)\mathcal{J}^{a}_{t}(t,x)$

As an aside, from (3.6) we note down below the infinitesimal internal
transformation properties of the EM tensor components, generated by the
conserved charges of the currents (37a) and (37b):

$\displaystyle-iG_{J}T^{t}_{\hskip
4.2679ptt}(t,x)=-f^{a\prime}(x)\mathcal{J}^{a}_{t}(t,x)\hskip
17.07164pt;\hskip 14.22636pt-iG_{K}T^{t}_{\hskip 4.2679ptt}(t,x)=0$ (102)
$\displaystyle-iG_{K}T^{t}_{\hskip
4.2679ptx}(t,x)=-g^{a{\prime}}(x)\mathcal{J}^{a}_{t}(t,x)\hskip
14.22636pt;\hskip 14.22636pt-iG_{J}T^{t}_{\hskip
4.2679ptx}(t,x)=-f^{a\prime}(x)\mathcal{J}^{a}_{x}(t,x)-tf^{a\prime\prime}(x)\mathcal{J}^{a}_{t}(t,x)$

###  3.7 The Algebra of Modes

The EM tensor components have the following mode-expansions:

$\displaystyle{T}^{t}_{\hskip
4.2679ptt}(t,x)=-i\sum_{n\in\mathbb{Z}}x^{-n-2}M_{n}\hskip 19.91692pt;\hskip
19.91692pt{T}^{t}_{\hskip
4.2679ptx}(t,x)=-i\sum_{n\in\mathbb{Z}}x^{-n-2}\text{
}[L_{n}-(n+2)\frac{t}{x}M_{n}]$ (103) $\displaystyle\Longrightarrow\hskip
7.11317pt$ $\displaystyle M_{n}=i\oint\limits_{0}\frac{dx}{2\pi i}\text{
}x^{n+1}\text{ }{T}^{t}_{\hskip 4.2679ptt}(t,x)\hskip 11.38109pt;\hskip
11.38109ptL_{n}=i\oint\limits_{0}\frac{dx}{2\pi i}\left[x^{n+1}{T}^{t}_{\hskip
4.2679ptx}(t,x)+(n+1)x^{n}t\text{ }{T}^{t}_{\hskip 4.2679ptt}(t,x)\right]$
(104)

In Saha:2022gjw , it was shown from the EM tensor OPEs (69) that the EM tensor
modes indeed generate the centrally extended BMS3 algebra:

$\displaystyle\left[M_{n}\hskip 2.84526pt,\hskip 2.84526ptM_{m}\right]=0$
$\displaystyle\left[L_{n}\hskip 2.84526pt,\hskip
2.84526ptM_{m}\right]=(n-m)M_{n+m}+\frac{c_{M}}{12}(n^{3}-n)\delta_{n+m,0}$
(105) $\displaystyle\left[L_{n}\hskip 2.84526pt,\hskip
2.84526ptL_{m}\right]=(n-m)L_{n+m}+\frac{c_{L}}{12}(n^{3}-n)\delta_{n+m,0}$

The infinitesimal 2D CC transformation properties of the currents are
expressed in the operator language Saha:2022gjw  using the prescription (55)
for the charges in (104):

$\displaystyle[L_{n}\text{
},\mathcal{J}^{a}_{x}(t,x)]=[x^{n+1}\partial_{x}+t(n+1)x^{n}\partial_{t}+(n+1)x^{n}]\mathcal{J}^{a}_{x}(t,x)+t(n+1)nx^{n-1}\mathcal{J}^{a}_{t}(t,x)$
$\displaystyle[M_{n}\text{
},\mathcal{J}^{a}_{x}(t,x)]=x^{n+1}\partial_{t}\mathcal{J}^{a}_{x}(t,x)+(n+1)x^{n}\mathcal{J}^{a}_{t}(t,x)$
(106) $\displaystyle[L_{n}\text{
},\mathcal{J}^{a}_{t}(t,x)]=x^{n+1}\partial_{x}\mathcal{J}^{a}_{t}(t,x)+(n+1)x^{n}\mathcal{J}^{a}_{t}(t,x)\hskip
14.22636pt;\hskip 14.22636pt[M_{n}\text{ },\mathcal{J}^{a}_{t}(t,x)]=0$

Next, the classical conservation laws (36) imply the following space-time
dependence of the fields $\\{\mathcal{J}^{a}_{t}\\}$ and
$\\{\mathcal{J}^{a}_{x}\\}$:

$\displaystyle\partial_{t}\mathcal{J}^{a}_{t}(t,x)=0\hskip
7.11317pt\Longrightarrow\hskip
7.11317pt\mathcal{J}^{a}_{t}(t,x)=\mathcal{J}^{a}_{t}(x)$
$\displaystyle\partial_{t}\mathcal{J}^{a}_{x}(t,x)=\partial_{x}\mathcal{J}^{a}_{t}(x)\hskip
7.11317pt\Longrightarrow\hskip
7.11317pt\mathcal{J}^{a}_{x}(t,x)=t\partial_{x}\mathcal{J}^{a}_{t}(x)+R^{a}(x)$

with $R^{a}(x)$ being arbitrary functions.

Guided by the above functional dependence and using the fact that the pair of
the fields $\mathcal{J}^{a}_{x}$ and $\mathcal{J}^{a}_{t}$ forms a 2D CC
primary rank-$\frac{1}{2}$ multiplet with scaling dimension $\Delta=1$, we
write down the mode-expansions following Saha:2022gjw :

$\displaystyle\mathcal{J}^{a}_{t}(x)=\sum_{n\in\mathbb{Z}}x^{-n-1}K^{a}_{n}\hskip
14.22636pt;\hskip
14.22636pt\mathcal{J}^{a}_{x}(t,x)=\sum_{n\in\mathbb{Z}}x^{-n-1}\left[J^{a}_{n}-(n+1)\frac{t}{x}K^{a}_{n}\right]$
(107) $\displaystyle K^{a}_{n}=\oint\limits_{C_{u}}\frac{dx}{2\pi i}\text{
}x^{n}\mathcal{J}^{a}_{t}(x)\hskip 15.649pt;\hskip
14.22636ptJ^{a}_{n}=\oint\limits_{C_{u}}\frac{dx}{2\pi
i}\left[x^{n}\mathcal{J}^{a}_{x}(t,x)+nx^{n-1}t\mathcal{J}^{a}_{t}(x)\right]$
(108)

where the counter-clockwise contour $C_{u}$ encloses the upper half-plane and
the real line.

Comparing (108) with (56) and (57), we immediately see that:

$J^{a}_{n}$ is the conserved charge of the current
$j^{a\mu}=\left(x^{n}\mathcal{J}^{a}_{x}+tnx^{n-1}\mathcal{J}^{a}_{t}\hskip
2.84526pt,\hskip 2.84526pt-x^{n}\mathcal{J}^{a}_{t}\right)$ $K^{a}_{n}$ is the
conserved charge of the current $k^{a\mu}=\left(x^{n}\mathcal{J}^{a}_{t}\hskip
2.84526pt,\hskip 2.84526pt0\right)$

Thus, using the prescription (55) on the OPEs (65) and (66) we reach the
definition of a current primary field ${\Phi}(t,x)$ in the operator formalism
(for any $n\in\mathbb{Z}$):

$\displaystyle[K^{a}_{n}\text{
},{\Phi}(t,x)]=ix^{n}\left(t^{a}_{K}\cdot{\Phi}\right)(t,x),$ (109a)
$\displaystyle[J^{a}_{n}\text{ },{\Phi}(t,x)]=i\left[x^{n}\left(\Phi\cdot
t^{a}_{J}\right)+tnx^{n-1}\left(t^{a}_{K}\cdot{\Phi}\right)\right](t,x)$
(109b)

For the EM tensor components, the analogues commutation relations are found
directly from (102):

$\displaystyle[K^{a}_{n}\text{ },T^{t}_{\hskip
4.2679ptt}(t,x)]=0,\,\,[J^{a}_{n}\text{ },T^{t}_{\hskip
4.2679ptt}(t,x)]=-inx^{n-1}\mathcal{J}^{a}_{t}(t,x),\,\,[K^{a}_{n}\text{
},T^{t}_{\hskip 4.2679ptx}(t,x)]=-inx^{n-1}\mathcal{J}^{a}_{t}(t,x),$
$\displaystyle[J^{a}_{n}\text{ },T^{t}_{\hskip
4.2679ptx}(t,x)]=-i\left[nx^{n-1}\mathcal{J}^{a}_{x}+tn(n-1)x^{n-2}\mathcal{J}^{a}_{t}\right](t,x)$
(110)

Substituting the current mode-expansion (107) and EM tensor mode expansion
(103) in these operator relations, we obtain the cross-commutation relations
between the modes of the EM tensor and the currents:

$\displaystyle[L_{n}\text{ },J^{a}_{m}]=-mJ^{a}_{m+n}\hskip 8.53581pt;\hskip
8.53581pt[M_{n}\text{ },J^{a}_{m}]=-mK^{a}_{m+n}=[L_{n}\text{
},K^{a}_{m}]\hskip 8.53581pt;\hskip 8.53581pt[M_{n}\text{ },K^{a}_{m}]=0$
(111)

Similarly, from the current-current OPEs (82), using the condition (87) and
the current mode-expansion (107), we reach the Lie algebra of the current
modes:

$\displaystyle[J^{a}_{n}\text{
},J^{b}_{m}]=iF^{abc}J^{c}_{n+m}+iG^{abc}K^{c}_{n+m}+nk_{1}\delta^{ab}\delta_{n+m,0}$
$\displaystyle[J^{a}_{n}\text{
},K^{b}_{m}]=iF^{abc}K^{c}_{n+m}+nk_{2}\delta^{ab}\delta_{n+m,0}\hskip
14.22636pt;\hskip 14.22636pt[K^{a}_{n}\text{ },K^{b}_{m}]=0$ (112)

The global internal symmetry is governed by the subalgebra of the zero-modes:

$\displaystyle[J^{a}_{0}\text{
},J^{b}_{0}]=iF^{abc}J^{c}_{0}+iG^{abc}K^{c}_{0}\hskip 14.22636pt;\hskip
14.22636pt[J^{a}_{0}\text{ },K^{b}_{0}]=iF^{abc}K^{c}_{0}\hskip
14.22636pt;\hskip 14.22636pt[K^{a}_{0}\text{ },K^{b}_{0}]=0$ (113)

This is precisely the algebra we have obtained for the NL currents in (22).

##  4 Sugawara Construction

As is well known, in 2d CFT, the Sugawara construction is employed to
construct (the modes of) the energy momentum tensor in terms of (the modes of)
the currents. In this section, here we will attempt to construct a NL version
of the Sugawara construction and express $L_{m}$ and $M_{m}$ in terms of
$J_{m}$ and $K_{m}$. We shall see that the $L_{m}$’s and $M_{m}$’s constructed
in such manner indeed satisfy the BMS algebra. Similar constructions have been
studied earlier in Rasmussen:2019zfu ; Ragoucy:2020elp .

###  4.1 Intrinsic construction from algebra

We begin here by assuming we have only the algebra of the NL currents, i.e.

$\displaystyle[J^{a}_{m},J^{b}_{n}]$
$\displaystyle=if^{abc}J^{c}_{m+n}+mk_{1}\delta^{ab}\delta_{m+n,0},$ (114a)
$\displaystyle[J^{a}_{m},K^{b}_{n}]$
$\displaystyle=if^{abc}K^{c}_{m+n}+mk_{2}\delta^{ab}\delta_{m+n,0},\quad[K^{a}_{m},K^{b}_{n}]=0.$
(114b)

In the above, the sum over the group indices e.g.
$f^{abc}J^{c}_{m+n}=\sum_{c=1}^{dim(\bar{g})}f^{abc}J^{c}_{m+n}$ is implied.
Comparing with (22), as stressed before, we work with the case where
$F^{abc}=f^{abc},G^{abc}=0$. Let us first consider the zero modes of $J$ and
$K$. Putting $n=m=0$ in (114), the algebra for the zero modes become

$\displaystyle[J^{a}_{0},J^{b}_{0}]=if^{abc}J^{c}_{0}\ ;\
\quad[J^{a}_{0},K^{b}_{0}]=if^{abc}K^{c}_{0}\ ;\ \quad[K^{a}_{0},K^{b}_{0}]=0$
(115)

Now let us look for quadratic Casimir operators for the above algebra. The
possible combinations are

$\displaystyle\sum_{a}J^{a}_{0}J^{a}_{0}\ ,\ \sum_{a}J^{a}_{0}K^{a}_{0}\ ,\
\sum_{a}K^{a}_{0}J^{a}_{0}\ ,\ \sum_{a}K^{a}_{0}K^{a}_{0}.$ (116)

Keeping in mind that Casimir operators must commute with all the generators,
we can exclude $\sum_{a}J^{a}_{0}J^{a}_{0}$ since it does not commute with
$K^{a}_{0}$

$\displaystyle\Big{[}\sum_{a}J^{a}_{0}J^{a}_{0},K^{b}_{0}\Big{]}=i\sum_{a,c}f^{abc}(J^{a}K^{c}+K^{c}J^{a})\neq
0$ (117)

All other combinations in (116) commute with all $J^{a}_{0}$’s and
$K^{a}_{0}$’s. Hence a generic Casimir operator constructed from the zero
modes of $J$ and $K$ will be a linear combination of the above three
combinations of $J^{a}_{0}$’s and $K^{a}_{0}$’s. We therefore want to
construct the zero level generator $L_{0},M_{0}$ from these combinations.

Since $L_{0}$ and $M_{0}$ are expected to be quadratic in terms of
$J^{a}_{n}$’s and $K^{a}_{n}$’s, we can write down the following generic
expression for them (The term $\sum JJ$ is excluded since we have already seen
that the term $\sum_{a}J^{a}_{0}J^{a}_{0}$ does not contribute to the zero
mode part of $L_{0}$ and $M_{0}$)

$\displaystyle L_{0}$
$\displaystyle=\alpha\sum_{a,l,n}J^{a}_{l}K^{a}_{n}+\beta\sum_{a,l,n}K^{a}_{l}J^{a}_{n}+\rho\sum_{a,l,n}K^{a}_{l}K^{a}_{n}$
(118a) $\displaystyle M_{0}$
$\displaystyle=\mu\sum_{a,l,n}J^{a}_{l}K^{a}_{n}+\nu\sum_{a,l,n}K^{a}_{l}J^{a}_{n}+\eta\sum_{a,l,n}K^{a}_{l}K^{a}_{n}$
(118b)

Now putting the conditions that

$[L_{0},J^{a}_{n}]=-nJ^{a}_{n},[M_{0},J^{a}_{n}]=-nK^{a}_{n}$

and looking at just the level of current generators on the RHS, we can see
that only $(l=-n)$ terms should contribute, so we obtain

$L_{0}=\alpha\sum_{a,n}J^{a}_{n}K^{a}_{-n}+\beta\sum_{a,n}K^{a}_{n}J^{a}_{-n}+\rho\sum_{a,n}K^{a}_{n}K^{a}_{-n}=\alpha
X_{0}+\beta Y_{0}+\rho Z_{0}$ (119a)
$M_{0}=\mu\sum_{a,l}J^{a}_{l}K^{a}_{-l}+\nu\sum_{a,l}K^{a}_{l}J^{a}_{-l}+\eta\sum_{a,l}K^{a}_{l}K^{a}_{-l}=\mu
X_{0}+\nu Y_{0}+\eta Z_{0}$ (119b)

Now take

$\begin{split}Y_{0}&=\sum_{a,l}K^{a}_{l}J^{a}_{-l}=\sum_{a,l}J^{a}_{-l}K^{a}_{l}+k_{2}\frac{\text{dim}g}{2}\sum_{l=-\infty}^{\infty}l=X_{0}+c,\end{split}$
(120)

where c is a (possibly infinite) constant. Since we always have the
independence of redefining BMS generators by constant shifts, we can define
level 0 BMS generators as

$L_{0}\equiv L^{\prime}_{0}=\frac{\alpha+\beta}{2}(X_{0}+Y_{0})+\rho
Z_{0},\quad M_{0}\equiv M^{\prime}_{0}=\frac{\mu+\nu}{2}(X_{0}+Y_{0})+\eta
Z_{0}.$ (121)

The normal ordering of the operators $L_{0},M_{0}$ is achieved by individual
normal ordering of $X_{0},Y_{0},Z_{0}$. We generalise the definition to the
other BMS generators as (with normal ordering)

$\displaystyle L_{m}$
$\displaystyle=\sum_{a=1}^{\text{dim}g}\left[\frac{\alpha+\beta}{2}\left\\{\sum_{l\leq-1}(J^{a}_{l}K^{a}_{m-l}+K^{a}_{l}J^{a}_{m-l})+\sum_{l>-1}(J^{a}_{m-l}K^{a}_{l}+K^{a}_{m-l}J^{a}_{l})\right\\}+\rho\sum_{l}K^{a}_{l}K^{a}_{m-l}\right]$
$\displaystyle M_{m}$
$\displaystyle=\sum_{a=1}^{\text{dim}g}\left[\frac{\mu+\nu}{2}\left\\{\sum_{l\leq-1}(J^{a}_{l}K^{a}_{m-l}+K^{a}_{l}J^{a}_{m-l})+\sum_{l>-1}(J^{a}_{m-l}K^{a}_{l}+K^{a}_{m-l}J^{a}_{l})\right\\}+\eta\sum_{l}K^{a}_{l}K^{a}_{m-l}\right]$
(122)

Using the form of the BMS generators in (4.1), and substituting in the BMS-
current cross commutators, i.e.

$[L_{m},J^{a}_{n}]=-nJ^{a}_{m+n}\ ;\ [L_{m},K^{a}_{n}]=-nK^{a}_{m+n}\ ;\
[M_{m},J^{a}_{n}]=-nK^{a}_{m+n},$ (123)

we obtain the following values for the coefficients

$\alpha+\beta=\frac{1}{k_{2}}\ ;\
\rho=-\frac{k_{1}+2C_{g}}{2k_{2}^{2}};\quad\mu+\nu=0\ ;\
\eta=\frac{1}{2k_{2}}.$ (124)

So we obtain the final form for our NL Sugawara construction

$\displaystyle L_{m}$
$\displaystyle=\frac{1}{2k_{2}}\sum_{a=1}^{dim(g)}\left\\{\sum_{l\leq-1}(J^{a}_{l}K^{a}_{m-l}+K^{a}_{l}J^{a}_{m-l})+\sum_{l>-1}(J^{a}_{m-l}K^{a}_{l}+K^{a}_{m-l}J^{a}_{l})-\frac{(k_{1}+2C_{g})}{k_{2}}\sum_{l}K^{a}_{l}K^{a}_{m-l}\right\\}$
$\displaystyle M_{m}$
$\displaystyle=\frac{1}{2k_{2}}\sum_{a=1}^{dim(g)}\sum_{l}K^{a}_{l}K^{a}_{m-l}$
(125)

As a check of the validity of the analysis, we compute the algebra of the $L$
and $M$ generators. These satisfy the following algebra (see Appendix B for
detailed calculations)

$\displaystyle[L_{m},L_{n}]=(m-n)L_{m+n}+\frac{dim(g)}{6}(m^{3}-m)\delta_{m+n,0}\
;\ [L_{m},M_{n}]=(m-n)M_{m+n}$ (126)

Hence, we see that by doing the NL Sugawara construction of NL currents, we
can find BMS algebra with

$c_{L}=2\text{dim}(g),\quad c_{M}=0.$ (127)

#### Non-zero $c_{M}$:

We can obtain a non-zero $c_{M}$ with a slight modification to the Sugawara
construction presented above. In case of Virasoro algebra with additional
symmetries, we can define new Virasoro generators Caroca:2017onr  as

$\tilde{\mathcal{L}}_{n}=\mathcal{L}^{S}_{n}+in\theta^{a}j^{a}_{n}+\frac{1}{2}k\theta^{2}\delta_{n,0}$
(128)

where $\mathcal{L}^{S}_{n}$ is the Virasoro generators obtained from Sugawara
construction and $\theta=\theta^{a}t^{a}$ is a vector belonging to the Lie
algebra with generators $t^{a}$. It can be showed that (see Appendix C) if
$\mathcal{L}_{n}$ satisfy Virasoro algebra with central charge $c$ then
$\tilde{\mathcal{L}}_{n}$ will satisfy Virasoro algebra with shifted central
charge $\tilde{c}$ where

$\tilde{c}=c+12k\theta^{2}.$ (129)

In case of NL Kac-Moody algebra, inspired by this, we introduce the following
redefinitions

$\displaystyle\tilde{L}_{n}$
$\displaystyle=L^{S}_{n}+in\theta^{a}J^{a}_{n}+\frac{1}{2}k_{1}\theta^{2}\delta_{n,0}$
$\displaystyle\tilde{M}_{n}$
$\displaystyle=M^{S}_{n}+in\theta^{a}K^{a}_{n}+\frac{1}{2}k_{2}\theta^{2}\delta_{n,0}$
(130)

where $L^{S}_{n}$ and $M^{S}_{n}$ are given by (4.1). This redefinition will
give us the following algebra (see Appendix C)

$\displaystyle[\tilde{L}_{m},\tilde{L}_{n}]$
$\displaystyle=(m-n)\tilde{L}_{m+n}+\frac{c_{L}}{12}(m^{3}-m)\delta_{n+m,0}$
$\displaystyle[\tilde{L}_{m},\tilde{M}_{n}]$
$\displaystyle=(m-n)\tilde{M}_{m+n}+\frac{c_{M}}{12}(m^{3}-m)\delta_{n+m,0}$
(131)

where the central charges are given by

$\displaystyle c_{L}$ $\displaystyle=2dim(g)+12k_{1}\theta^{2}$ $\displaystyle
c_{M}$ $\displaystyle=12k_{2}\theta^{2}$ (132)

Hence we can obtain the fully centrally extended BMS algebra.

###  4.2 Consistency with OPEs

We will recast the NL Sugawara construction in terms of the NL EM tensor that
we introduced in Sec. 3 instead of the generators of the BMS:
$\\{L_{n},M_{n}\\}$. $\\{L_{n},M_{n}\\}$ are of course modes of the NL EM
tensor. Hence the calculations in this subsection provide a sanity check for
our analysis making sure the various formulations are consistent with each
other.

The NL Sugawara construction gave us (4.1). We now normal order the products
to rewrite this as

$\begin{split}&L_{n}=\frac{1}{2k_{2}}\sum_{l,a}\Bigl{(}:J^{a}_{l}K^{a}_{n-l}:+:K^{a}_{l}J^{a}_{n-l}:-\frac{k_{1}+2C_{g}}{k_{2}}:K^{a}_{l}K^{a}_{n-l}:\Bigr{)}\\\
&M_{n}=\frac{1}{2k_{2}}\sum_{l,a}:K^{a}_{l}K^{a}_{n-l}:\end{split}$ (133)

Here, $:A_{l}:$ is a shorthand for normal ordered products. Now substituting
(133) in (103) and rearranging the terms, we get uv to xt

$\begin{split}&T_{u}(u,v)=\frac{1}{2k_{2}}\sum_{a}\left[\bigl{[}(\mathcal{J}^{a}_{u}\mathcal{J}^{a}_{v})(u,v)+(\mathcal{J}^{a}_{v}\mathcal{J}^{a}_{u})(u,v)\bigr{]}-\frac{k_{1}+2C_{g}}{k_{2}}(\mathcal{J}^{a}_{v}\mathcal{J}^{a}_{v})(v)\right]\\\
&T_{v}(v)=\frac{1}{2k_{2}}\sum_{a}(\mathcal{J}^{a}_{v}\mathcal{J}^{a}_{v})(v)\end{split}$
(134)

Here $(\dots)$ means normal ordered products of fields, which can be expressed
in terms of contour integrals. Here we are using $u,v$ coordinates instead of
$x,t$ of Sec. 3 because this construction applies to both Galilean and
Carrollian CFTs, because of the similarity between the 2 theories mentioned in
the introduction. So by substituting $u\to x,v\to t$ or $u\to t,v\to x$, we
can obtain the Galilean or Carrollian theory expressions, respectively.
So (134) gives the field expression for the NL Sugawara construction. For the
modified Sugawara construction as defined in (4.1), the EM tensor fields turn
out to be (by doing the mode expansion using the new generators)

$\begin{split}&\tilde{T}_{u}(u,v)=T_{u}(u,v)-i\theta^{a}\partial_{v}\mathcal{J}^{a}_{u}(u,v)-i\theta^{a}\frac{\mathcal{J}^{a}_{u}(u,v)}{v}-i\theta^{a}\frac{u\mathcal{J}^{a}_{v}(v)}{v^{2}}+\frac{1}{2}\frac{k_{1}\theta^{2}}{v^{2}}+\frac{uk_{2}\theta^{2}}{v^{3}}\\\
&\tilde{T}_{v}(v)=T_{v}(v)-i\theta^{a}\partial_{v}\mathcal{J}^{a}_{v}(v)-i\theta^{a}\frac{\mathcal{J}^{a}_{v}(v)}{v}+\frac{1}{2}\frac{k_{2}\theta^{2}}{v^{2}}\end{split}$
(135)

where, $T_{u}(u,v),T_{v}(u,v)$ refers to the quantities in (134), i.e. the
normal NL Sugawara construction expressions.

####  $T-\mathcal{J}$ OPE:

Now, for a consistency check of our analysis so far, we will rederive the OPEs
from the above definitions of the NL Sugawara construction. We begin with the
T-J OPEs. Taking the definitions as in (134) and contracting with the
currents, we get the following expressions (See Appendix D for detailed
calculation):

$\begin{split}&T_{v}(u_{1},v_{1})\mathcal{J}^{b}_{v}(u_{2},v_{2})\sim\,\text{regular}\\\
&T_{u}(u_{1},v_{1})\mathcal{J}^{b}_{v}(u_{2},v_{2})\sim\frac{\mathcal{J}^{b}_{v}(u_{2},v_{2})}{v_{12}^{2}}+\frac{\partial_{v}\mathcal{J}^{b}_{v}(u_{2},v_{2})}{v_{12}}+\dots\\\
&T_{v}(u_{1},v_{1})\mathcal{J}^{a}_{u}(u_{2},v_{2})\sim\frac{\mathcal{J}^{a}_{v}(u_{2},v_{2})}{v_{12}^{2}}+\frac{\partial_{v}\mathcal{J}^{a}_{v}(u_{2},v_{2})}{v_{12}}+\dots\\\
&T_{u}(u_{1},v_{1})\mathcal{J}^{a}_{u}(u_{2},v_{2})\sim\frac{\mathcal{J}^{a}_{u}(u_{2},v_{2})}{v_{12}^{2}}+\frac{\partial_{v}\mathcal{J}^{a}_{u}(u_{2},v_{2})}{v_{12}}+\frac{u_{12}}{v_{12}^{2}}\partial_{u}\mathcal{J}^{a}_{u}(u_{2},v_{2})\\\
&\qquad\qquad\qquad\qquad\qquad\qquad+\frac{2u_{12}}{v_{12}}\left(\frac{\mathcal{J}^{a}_{v}(u_{2},v_{2})}{v_{12}^{2}}+\frac{\partial_{v}\mathcal{J}^{a}_{v}(u_{2},v_{2})}{v_{12}}\right)+\dots\end{split}$
(136)

Which are equivalent to the $[L_{m},J^{a}_{n}]$ type commutation relations
given in (22). Clearly these relation satisfy the OPE relations (3.6) that the
currents and the EM tensors of a theory are supposed to satisfy.

####  $T-T$ OPE:

Next we use the definition of the Sugawara construction to compute the T-T
type OPEs. Doing the calculations (see Appendix D for detailed Calculations),
we get,

$\displaystyle T_{u}(u_{1},v_{1})T_{u}(u_{2},v_{2})\sim$
$\displaystyle\frac{2T_{u}(u_{2},v_{2})}{v_{12}^{2}}+\frac{\partial_{v}T_{u}(u_{2},v_{2})}{v_{12}}+\frac{u_{12}}{v_{12}^{2}}\partial_{u}T_{u}(u_{2},v_{2})$
$\displaystyle+\frac{2u_{12}}{v_{12}}\Bigl{(}\frac{2T_{v}(u_{2},v_{2})}{v_{12}^{2}}+\frac{\partial_{v}T_{v}(u_{2},v_{2})}{v_{12}}\Bigr{)}+\frac{dim(g)}{v_{12}^{4}}+\dots$
$\displaystyle T_{u}(u_{1},v_{1})T_{v}(u_{2},v_{2})\sim$
$\displaystyle\frac{2T_{v}(u_{2},v_{2})}{v_{12}^{2}}+\frac{\partial_{v}T_{v}(u_{2},v_{2})}{v_{12}}+\dots$
(137) $\displaystyle T_{v}(u_{1},v_{1})T_{v}(u_{2},v_{2})\sim$ regular

These results (4.2) match with the OPEs of the energy momentum tensors of a 2d
Carrollian or Galilean CFT, as given in (69). So, the NL Sugawara construction
really gives the EM tensors of a 2d NL CFT (proved both at algebra level and
now at field level). Also from the OPE expressions, we can verify the earlier
results of $c_{L}=2dim(g),c_{M}=0$ obtained earlier. Again if we start with
the modified Sugawara construction (135), we can obtain same OPE relations as
above, but with modified central charges
$c_{L}=2dim(g)+12k_{1}\theta^{2},c_{M}=12k_{2}\theta^{2}$. To see this, let’s
look at, for example, the $\tilde{T}_{u}(u_{1},v_{1})\tilde{T}_{v}(v_{2})$ OPE
and focus our attention to a specific term

$\tilde{T}_{u}(u_{1},v_{1})\tilde{T}_{v}(v_{2})\sim-\theta^{a}\theta^{b}\partial_{v}\mathcal{J}^{a}_{u}(u_{1},v_{1})\partial_{v}\mathcal{J}^{a}_{v}(v_{2})+\dots\sim-\theta^{2}\partial_{v_{1}}\partial_{v_{2}}\frac{k_{2}}{v_{12}^{2}}+\dots\sim\frac{6k_{2}\theta^{2}}{v_{12}^{4}}+\dots$

Matching the above expression with (69), we can see $c_{M}=12k_{2}\theta^{2}$.
Similarly we can obtain the other shifted central charge.

##  5 Tensionless String as NLKM

The tensionless limit of string theory is the limit that is diametrically
opposite to the usual point particle limit where known supergravity appears
from strings. This limit that explores the very strong gravity regime and,
when quantized, the very highly quantum and highly stringy regime of string
theory. In this section, we explore how this is connected to the NLKM
structures we have discussed in this paper. We will see that the currents
satisfying the $U(1)$ NLKM algebra come up intrinsically when we are looking
at tensionless strings propagating in flat spacetimes.

As is very well known, the action for a tensile relativistic string
propagating in a flat $d$-dimensional spacetime is given by the Polyakov
action:

$S_{\text{P}}=T\int
d^{2}\xi\,\sqrt{\gamma}\gamma^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\eta_{\mu\nu}.$
(138)

In order to take the tensionless limit, it is helpful to work with the phase
space action of string. One can then systematically take the limit and the
resulting action can be cast in a Polyakov like form, which we will call the
ILST action after the authors:

$S_{\text{ILST}}=\int
d^{2}\xi\,V^{\alpha}V^{\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\eta_{\mu\nu}.$
(139)

Here $X^{\mu}$ are the coordinates in the background flat space
$\eta_{\mu\nu}$ which are scalar fields on the worldsheet parametrized by
$\xi^{a}=\sigma,\tau$. The worldsheet metric $\gamma^{\alpha\beta}$
degenerates in the limit and the following replacement is made:

$T\sqrt{\gamma}\gamma^{\alpha\beta}\to V^{\alpha}V^{\beta},$ (140)

where $V^{\alpha}$ is a vector density. The action is invariant under
worldsheet diffeomorphisms and hence one needs to fix a gauge. It is helpful
to go to the equivalent of the conformal gauge

$V^{\alpha}=(v,0).$ (141)

There is some symmetry left over even after this gauge fixing. In the usual
tensile string, the residual symmetry gives two copies of the Virasoro algebra
and the appearance of a 2d CFT on the worldsheet is the central reason why we
understand string theory as well as we do. Now in the tensionless case, the
residual gauge symmetry that appears on the worldsheet is the BMS3 algebra.
This now dictates the theory of tensionless strings. The reason behind the
appearance of the BMS algebra is that the tensionless string in flat
spacetimes is actually a null string. This is the string equivalent of a
massless point particle which is constrained to travel on a null geodesic of
the ambient spacetime. The string sweeps out a worldsheet which is null and
since there are no mass terms, the resulting action has to have conformal
Carrollian symmetry in $d=2$ or equivalently BMS3.

Now, let us connect to our discussion in this paper. In the favourable
conformal gauge (141), the equations of motion and constraints arising from
(139) take the form

${\text{EOM:}}\quad\ddot{X}^{\mu}=0\ ;\
\quad{\text{Constraints:}}\quad\dot{X}^{2}=0\ ,\ \dot{X}.X^{\prime}=0.$ (142)

A convenient form of the solution is given by

$X^{\mu}(\sigma,\tau)=x^{\mu}+\sqrt{2c^{\prime}}\bigl{(}J^{\mu}_{0}\sigma+K^{\mu}_{0}\tau+i\sum_{n\neq
0}\frac{1}{n}(J^{\mu}_{n}-in\tau K^{\mu}_{n})e^{-in\sigma}\bigr{)}$ (143)

Here $\sigma,\tau$ are the coordinates on the cylinder, which are related to
the planar Non-Lorentzian coordinates as

$u=e^{-i\sigma}\ ;\ v=-\tau e^{-i\sigma}$ (144)

The periodic condition $X^{\mu}(\sigma+2\pi,\tau)=X^{\mu}(\sigma,\tau)$ forces
us to have $J^{\mu}_{0}=0$. From (143), we obtain

$\begin{split}J^{\mu}_{\tau}=\partial_{\sigma}X^{\mu}=\sqrt{2c^{\prime}}\sum_{n}(J^{\mu}_{n}-in\tau
K^{\mu}_{n})e^{-in\sigma},\quad
J^{\mu}_{\sigma}=-\partial_{\tau}X^{\mu}=-\sqrt{2c^{\prime}}\sum_{n}K^{\mu}_{n}e^{-in\sigma}\end{split}$
(145)

Clearly the currents in (145) satisfy similar relations as (36), i.e.

$\partial_{\tau}J^{\mu}_{\sigma}=0\ ;\
\partial_{\tau}J^{\mu}_{\tau}+\partial_{\sigma}J^{\mu}_{\sigma}=0.$ (146)

If we canonically quantise the system by demanding
$[X^{\mu}(\tau,\sigma),\Pi^{\nu}(\tau,\sigma^{\prime})]=i\delta(\sigma-\sigma^{\prime})\eta^{\mu\nu}$,
we obtain the following relations for the current modes

$[J^{\mu}_{m},J^{\nu}_{n}]=[K^{\mu}_{m},K^{\nu}_{n}]=0\ ;\
[J^{\mu}_{m},K^{\nu}_{n}]=2m\delta_{m+n,0}\eta^{\mu\nu}$ (147)

Clearly, the current modes for the same spacetime index $\mu$ follow a special
case of $U(1)$ NL Kac-Moody algebra (look at (22) for comparison), with
$k^{(\mu\nu)}_{1}=0,k^{(\mu\nu)}_{2}=2\eta^{\mu\nu}$.

Next if we look into the (classical) energy momentum tensor of this theory,
their mode expansion coefficients are given by

$L_{n}=\frac{1}{2}\sum_{m}J^{\mu}_{m}K^{\nu}_{n-m}\eta_{\mu\nu}\ ;\
M_{n}=\frac{1}{4}\sum_{m}K^{\mu}_{m}K^{\nu}_{n-m}\eta_{\mu\nu}.$ (148)

This matches classically with the relation (4.1) for the NL Sugawara
construction with

$k_{1}=0,\quad k_{2}=2.$ (149)

Note that $C_{g}=0$ for $U(1)$ algebra. The generators in (148) satisfy the
(centreless) BMS3 algebra, which is expected as the action (139) has BMS
symmetry as a residual gauge symmetry.

Thus we can see that the Non-Lorentzian $U(1)$ Kac-Moody algebra and the
associated NL Sugawara construction come up in the theory of tensionless or
null strings propagating on a flat background geometry. An outstanding
question is what happens when we look at the propagation of null strings on
arbitrary curved manifolds. These can be viewed as group manifolds and there
will be the NL equivalent of a Wess-Zumino-Witten model appearing for
tensionless strings in this context. We expect that the non-abelian NLKM
algebras explored in this work would naturally appear on these NL WZW models.
This is work in progress and we hope to report on this in the near future.

##  6 Non-Lorentzian Knizhnik Zamolodchikov Equations

In usual Lorentzian Kac-Moody algebras, the Knzhnik-Zamolodchikov equations
are the linear differential equations that are satisfied by the correlation
functions of these CFTs endowed with additional affine Lie symmetry. In this
section, we write down the non-Lorentzian analogues of these KZ equations
based on our construction of the NLKM algebra and its representations in this
paper.

In what follows, we will outline the steps to get to the NL Knizhnik
Zamolodchikov equations. The details of the calculation are presented in a
separate appendix E.

We begin with the OPE definition of BMS primary field (89) and obtain

$\begin{split}&\partial_{u}\Phi(u^{\prime},v^{\prime})=-\oint_{v^{\prime}}\frac{dv}{2\pi
i}\oint_{u^{\prime}}\frac{du}{2\pi
i}(u-u^{\prime})^{-1}T_{v}(u,v)\Phi(u^{\prime},v^{\prime})=-\oint_{v^{\prime}}\frac{dv}{2\pi
i}T_{v}(v)\Phi(u^{\prime},v^{\prime})\\\
&\partial_{v}\Phi(u^{\prime},v^{\prime})=\oint_{v^{\prime}}\frac{dv}{2\pi
i}\oint_{u^{\prime}}\frac{du}{2\pi
i}(u-u^{\prime})^{-1}T_{u}(u,v)\Phi(u^{\prime},v^{\prime})=\oint_{v^{\prime}}\frac{dv}{2\pi
i}T_{u}(u^{\prime},v)\Phi(u^{\prime},v^{\prime})\end{split}$ (150)

Next, we take the correlation function of a string of primary fields as shown
below

$\displaystyle\langle\partial_{u}\Phi(u,v)\Phi_{1}(u_{1},v_{1})\dots\Phi_{n}(u_{n},v_{n})\rangle$
$\displaystyle=\langle\partial_{u}\Phi(u,v)X(\\{u_{i},v_{i}\\})\rangle=\oint_{v}\frac{dv^{\prime}}{2\pi
i}\langle T_{v}(u^{\prime},v^{\prime})\Phi(u,v)X(\\{u_{i},v_{i}\\})\rangle$
$\displaystyle=\oint_{v}\frac{dv^{\prime}}{2\pi
i}\langle\frac{1}{2k_{2}}\Bigl{(}(\mathcal{J}^{a}_{v}\mathcal{J}^{a}_{v})(u^{\prime},v^{\prime})\Bigr{)}\Phi(u,v)X(\\{u_{i},v_{i}\\})\rangle$

and use relations (89, 150) to finally get the following expression (details
in Ap. E)

$\begin{split}&\langle\partial_{u}\Phi(u,v)X(\\{u_{i},v_{i}\\})\rangle=-\frac{1}{k_{2}}\Bigl{(}\sum_{j}\frac{t^{a}_{R,K}\otimes
t^{a}_{R_{j},K}}{(v-v_{j})}\Bigr{)}\langle\Phi(u,v)X(\\{u_{i},v_{i}\\})\rangle\end{split}$
(152)

which can be written as

$\Bigl{(}\partial_{u_{i}}+\frac{1}{k_{2}}\sum_{j\neq
i}\frac{t^{a}_{R_{i},K}\otimes
t^{a}_{R_{j},K}}{(v_{i}-v_{j})}\Bigr{)}\langle\Phi_{1}(u_{1},v_{1})\dots\Phi_{n}(u_{n},v_{n})\rangle=0.$
(153)

This is one of the Non-Lorentzian Knizhnik Zamolodchikov equations.

Similarly we can start with

$\begin{split}&\langle\partial_{v}\Phi(u,v)\Phi_{1}(u_{1},v_{1})\dots\Phi_{n}(u_{n},v_{n})\rangle=\langle\partial_{v}\Phi(u,v)X(\\{u_{i},v_{i}\\})\rangle=\oint_{v}\frac{dv^{\prime}}{2\pi
i}\langle T_{u}(u^{\prime},v^{\prime})\Phi(u,v)X(\\{u_{i},v_{i}\\})\rangle\\\
&=\oint_{v}\frac{dv^{\prime}}{2\pi
i}\langle\frac{1}{2k_{2}}\Bigl{(}(\mathcal{J}^{a}_{v}\mathcal{J}^{a}_{u})(u^{\prime},v^{\prime})+(\mathcal{J}^{a}_{u}\mathcal{J}^{a}_{v})(u^{\prime},v^{\prime})-\frac{k_{1}+2C_{g}}{k_{2}}(\mathcal{J}^{a}_{v}\mathcal{J}^{a}_{v})(u^{\prime},v^{\prime})\Bigr{)}\Phi(u,v)X(\\{u_{i},v_{i}\\})\rangle\end{split}$
(154)

From here, we can proceed in the same way as Appendix E to obtain the other
equation

$\begin{split}{\Bigl{(}\partial_{v_{i}}-\frac{1}{k_{2}}\sum_{j\neq
i}\frac{1}{(v_{i}-v_{j})}\sum_{a=1}^{dimg}\Bigl{[}\bigl{(}t^{a}_{R_{i},J}\otimes
t^{a}_{R_{j},K}+t^{a}_{R_{i},K}\otimes
t^{a}_{R_{j},J}\bigr{)}}&+{\bigl{(}\frac{u_{i}-u_{j}}{v_{i}-v_{j}}-\frac{k_{1}+2C_{g}}{k_{2}}\bigr{)}t^{a}_{R_{i},K}\otimes
t^{a}_{R_{j},K}\Bigr{]}\Bigr{)}}\\\
&{\langle\Phi_{R_{1}}(u_{1},v_{1})\Phi_{R_{2}}(u_{2},v_{2})...\Phi_{R_{n}}(u_{n},v_{n})\rangle=0}\end{split}$
(155)

This above equation (155) is the second of the Non-Lorentzian Knizhnik
Zamolodchikov equations. The solutions to these two equations (153), (155)
would give the correlation functions of the underlying NLKM theory. For usual
relativistic theories, the KZ equations are difficult to solve in general, but
one can do it for the four-point functions yielding a closed solution in terms
of hypergeometric functions. It would be instructive to check whether
something similar happens here. If one can find the solutions to the NLKM four
point functions, it would also be a nice exercise to check whether these can
be arrived at as a limit of the relativistic answers. This process of
attempting to generate the non-Lorentzian answers from the relativistic ones
is something we outline in detail in the section that follows.

##  7 NLKM from Contractions

In this section, we rederive various results we have obtained earlier in the
paper through a systematic limit on the algebraic structures obtained in the
relativistic set-up. Before moving to recovering the answers, we spend a bit
of time understanding which limit is appropriate for our purposes.

###  7.1 A brief detour to representations of BMS

We said in the introduction that there were two distinct contractions that
land us up on the BMS3 algebra starting from two copies of the Virasoro
algebra. One of them was a Carrollian or ultra-relativistic limit (18), where
there was a mixing of positive and negative Virasoro modes creating the BMS
generators, while the other was a Galilean or non-relativistic (14), where no
mixing took place. This mixing of modes is critical for the understanding of
the representations in the limit.

We begin with the Galilean contraction. In the parent relativistic CFT, the
theory is best described in terms of the highest weight representation. The
states of the theory are labelled by the zero modes:

${\mathcal{L}}_{0}|h,{\bar{h}}\rangle=h|h,{\bar{h}}\rangle,\quad\bar{{\mathcal{L}}}_{0}|h,{\bar{h}}\rangle={\bar{h}}|h,{\bar{h}}\rangle$
(156)

There is a class of states called primary states which are annihilated by all
positive modes:

${\mathcal{L}}_{n}|h,{\bar{h}}\rangle_{p}=0,\quad\bar{{\mathcal{L}}}_{n}|h,{\bar{h}}\rangle_{p}=0,\quad\forall
n>0.$ (157)

The Virasoro modules are built on these primary states by acting with raising
operators ${\mathcal{L}}_{-n}$. Now, looking back at the Galilean contraction
(14), we see that

${\mathcal{L}}_{n}=\frac{1}{2}\left(L_{n}+\frac{1}{\epsilon}M_{n}\right),\quad\bar{{\mathcal{L}}}_{n}=\frac{1}{2}\left(L_{n}-\frac{1}{\epsilon}M_{n}\right)$
(158)

The 2d CFT primary conditions then boil down to:

$L_{0}|\Delta,\xi\rangle=\Delta|\Delta,\xi\rangle,\,M_{0}|\Delta,\xi\rangle=\xi|\Delta,\xi\rangle;\quad
L_{n}|\Delta,\xi\rangle_{p}=0,\,M_{n}|\Delta,\xi\rangle_{p}=0,\,\forall n>0.$
(159)

In the above, the assumption is that the state $|h,{\bar{h}}\rangle$ goes to
the state $|\Delta,\xi\rangle$ in the limit. So we see that highest weight
states map to highest weight states in the Galilean contraction. This analysis
holds in a similar way when we consider the full NLKM algebra. We will be
using this for the analysis in the rest of the section.

But before we get there, let us point out why we are not using the Carroll
limit (18) for this purpose. In the Carroll contraction, we can read off

${\mathcal{L}}_{n}=\frac{1}{2}\left(L_{n}+\frac{1}{\epsilon}M_{n}\right),\quad\bar{{\mathcal{L}}}_{n}=\frac{1}{2}\left(-L_{-n}+\frac{1}{\epsilon}M_{-n}\right)$
(160)

The 2d CFT primary conditions in the Carroll limit become:

$M_{0}|M,s\rangle=M|M,s\rangle,\quad L_{0}|M,s\rangle=s|M,s\rangle,\quad
M_{n}|M,s\rangle=0,\quad\forall n\neq 0.$ (161)

In the above, the state $|h,{\bar{h}}\rangle$ in the Carroll limit becomes the
state $|M,s\rangle$. This is clearly not a highest weight state. The set of
these states form what is called the induced representation. The story is
again similar for the full NLKM. In the analysis that follows, we will not
focus on the induced representations. It would be of interest to consider them
in future work.

We should stress however, that the answers we have obtained in the intrinsic
Carrollian form earlier are for highest weight representations. The Galilean
limit would be an effective way of reproducing these answers with a flip of
space and time directions at the end of the analysis.

###  7.2 Contraction of the affine parameters

In this section we will establish the mode expansion of the NLKM currents
(108) from another approach, by doing a contraction of the original loop
extended algebra. Starting from the finite Lie algebra

$[j^{a},j^{b}]=if^{abc}j^{c}\ ;\
[\bar{j}^{a},\bar{j}^{b}]=if^{abc}\bar{j}^{c}$ (162)

we can define loop extended generators

$\displaystyle j^{a}_{n}=j^{a}\otimes z^{n}\ ;\
\bar{j}^{a}_{n}=\bar{j}^{a}\otimes\bar{z}^{n},$ (163)

which satisfy

$[j^{a}_{n},j^{b}_{m}]=if^{abc}j^{c}\otimes z^{n+m}=if^{abc}j^{c}_{n+m}.$
(164)

Loop extended algebra obtained above admits a central extension to give us the
current algebra in (12). Now, we can do a contraction of the affine parameter
as $z=t+\epsilon x$ and $\bar{z}=t-\epsilon x$ in the Galilean limit. We then
obtain (keeping upto first order in $\epsilon$),

$\displaystyle j^{a}_{n}$ $\displaystyle=j^{a}\otimes(t^{n}+n\epsilon
xt^{n-1}+...)\ ;\ \bar{j}^{a}_{n}=\bar{j}^{a}\otimes(t^{n}-n\epsilon
xt^{n-1}+...)$ (165)

Now we can define new contracted generators of the finite algebra as follows:

$\displaystyle J^{a}=j^{a}+\bar{j}^{a}\ ;\ K^{a}=\epsilon(\bar{j}^{a}-j^{a})$
(166)

Above definition can be extended to general modes of $J$ and $K$ using (165),

$\displaystyle J^{a}_{n}$ $\displaystyle=\lim_{\epsilon\rightarrow
0}(j^{a}_{n}+\bar{j}^{a}_{n})=J^{a}\otimes t^{n}-nK^{a}\otimes xt^{n-1}$
(167a) $\displaystyle K^{a}_{n}$ $\displaystyle=\lim_{\epsilon\rightarrow
0}\epsilon(\bar{j}^{a}_{n}-j^{a}_{n})=K^{a}\otimes t^{n}$ (167b)

In this way, we get the structure of the power series expansion of $J$ and $K$
in terms of $x$ and $t$, which is the Galilean analog of (108), or identical
upto the flip of temporal and spatial directions.

If we define primary field $\phi(x,t)$ as a representation of the finite
algebra, i.e. in terms of the action of zero modes of the generators $J$ and
$K$,

$\begin{split}[J^{a}_{0},\phi(x,t)]&\equiv[J^{a},\phi(x,t)]=t^{a}_{J}\phi(x,t)\\\
[K^{a}_{0},\phi(x,t)]&\equiv[K^{a},\phi(x,t)]=t^{a}_{K}\phi(x,t)\end{split}$
(168)

We can get the action of general modes using (167),

$\begin{split}[J^{a}_{n},\phi(x,t)]&=t^{a}_{J}\phi(x,t)\otimes
t^{n}-nt^{a}_{K}\phi(x,t)\otimes xt^{n-1}\equiv
t^{n}t^{a}_{J}\phi(x,t)-nxt^{n-1}t^{a}_{K}\phi(x,t)\\\
[K^{a}_{n},\phi(x,t)]&=t^{a}_{K}\phi(x,t)\otimes t^{n}\equiv
t^{n}t^{a}_{K}\phi(x,t)\end{split}$ (169)

This definition of a primary field agrees with our earlier result (109).

This action of $J^{a}_{n}$ and $K^{a}_{n}$ on primary fields also appears when
we draw motivation from Bagchi:2009ca . We can define the primary field as,

$\begin{split}[J^{a}_{0},\phi(0,0)]=t^{a}_{J}\phi(0,0)\ &;\
[K^{a}_{0},\phi(0,0)]=t^{a}_{k}\phi(0,0),\\\ [J^{a}_{n},\phi(0,0)]=0\ &;\
[K^{a}_{n},\phi(0,0)]=0\ \ \ \ \forall\ n>0.\end{split}$ (170)

For $n\geq 0$ and $U=e^{tL_{-1}-xM_{-1}}$,

$\displaystyle[J^{a}_{n},\phi(x,t)]$
$\displaystyle=[J^{a}_{n},U\phi(0)U^{-1}]=U[U^{-1}J^{a}_{n}U,\phi(0)]U^{-1}$
(171)

Consider,

$\displaystyle U^{-1}J^{a}_{n}U$
$\displaystyle=e^{-tL_{-1}+xM_{-1}}J^{a}_{n}e^{tL_{-1}-xM_{-1}}$
$\displaystyle=\sum_{k=0}^{n}\frac{t^{k}}{k!}\frac{n!}{(n-k)!}J^{a}_{n-k}-nx\sum_{k=0}^{n-1}\frac{t^{k}}{k!}\frac{(n-1)!}{(n-k-1)!}K^{a}_{n-k-1}$
(172)

where we have used the Baker-Campbell-Hausdorff(BCH) formula twice. Putting
this back in (171), we finally get,

$\begin{split}[J^{a}_{n},\phi(x,t)]&=U\left(\sum_{k=0}^{n}\frac{t^{k}}{k!}\frac{n!}{(n-k)!}[J^{a}_{n-k},\phi(0)]-nx\sum_{k=0}^{n-1}\frac{t^{k}}{k!}\frac{(n-1)!}{(n-k-1)!}[K^{a}_{n-k-1},\phi(0)]\right)U^{-1}\\\
&=(t^{n}t^{a}_{J}-nxt^{n-1}t^{a}_{K})\phi(x,t)\end{split}$ (173)

where only the terms corresponding to $k=n$ and $k=n-1$ survived respectively
in the first and the second sum because of (170). Similarly for $K^{a}_{n}$,
we get,

$\begin{split}[K^{a}_{n},\phi(x,t)]=U\left(\sum_{k=0}^{n}\frac{t^{k}}{k!}\frac{n!}{(n-k)!}[K^{a}_{n-k},\phi(0)]\right)U^{-1}=t^{n}t^{a}_{K}\phi(x,t)\end{split}$
(174)

These results verify (169) again from a different perspective.

###  7.3 Contracting the Sugawara construction

In section 4.1, we constructed the BMS generators by taking quadratic products
of the NL currents and ended up with (4.1). In this subsection, we shall
reproduce the same from the Galilean limit of Sugawara construction for CFT.
We start from the following expression for virasoro modes $\mathcal{L}_{m}$
and $\bar{\mathcal{L}}_{m}$ in relativistic sugawara construction.

$\displaystyle\mathcal{L}_{m}$
$\displaystyle=\gamma\sum_{a=1}^{dim(g)}(\sum_{l\leq-1}j^{a}_{l}j^{a}_{m-l}+\sum_{l>-1}j^{a}_{m-l}j^{a}_{l})$
(175a) $\displaystyle\bar{\mathcal{L}}_{m}$
$\displaystyle=\bar{\gamma}\sum_{a=1}^{dim(g)}(\sum_{l\leq-1}\bar{j}^{a}_{l}\bar{j}^{a}_{m-l}+\sum_{l>-1}\bar{j}^{a}_{m-l}\bar{j}^{a}_{l})$
(175b) $\displaystyle\text{ where }\gamma=\frac{1}{2(k+C_{g})},$
$\displaystyle\bar{\gamma}=\frac{1}{2(\bar{k}+C_{g})}\text{ and
}C_{g}=-\frac{1}{2dim(g)}\sum_{b,c}f^{bac}f^{bcd}$ (175c)

We can now take the Galilean limit by using the inverted version of relations
(21) and get the following form of Virasoro Generators by collecting the terms
with same order in $\epsilon$,

$\displaystyle\mathcal{L}_{m}=\frac{\gamma}{4}(A_{m}-\frac{B_{m}}{\epsilon}+\frac{C_{m}}{\epsilon^{2}})\
;\
\bar{\mathcal{L}}_{m}=\frac{\bar{\gamma}}{4}(A_{m}+\frac{B_{m}}{\epsilon}+\frac{C_{m}}{\epsilon^{2}})$
(176)

where,

$\displaystyle A_{m}$
$\displaystyle=\sum_{a=1}^{dim(g)}\sum_{l\leq-1}J^{a}_{l}J^{a}_{m-l}+\sum_{l>-1}J^{a}_{m-l}J^{a}_{l}$
(177a) $\displaystyle B_{m}$
$\displaystyle=\sum_{a=1}^{dim(g)}\\{\sum_{l\leq-1}(J^{a}_{l}K^{a}_{m-l}+K^{a}_{l}J^{a}_{m-l})+\sum_{l>-1}(J^{a}_{m-l}K^{a}_{l}+K^{a}_{m-l}J^{a}_{l})\\}$
(177b) $\displaystyle C_{m}$
$\displaystyle=\sum_{a=1}^{dim(g)}\sum_{l}K^{a}_{l}K^{a}_{m-l}$ (177c)

where we have employed the commutativity of $K$’s in order to write
$\frac{1}{\epsilon^{2}}$ term($C_{m}$) as a sum from $l=-\infty$ to $\infty$.
We can use (14) and the above relations to obtain:

$\displaystyle L_{m}$
$\displaystyle=\frac{1}{4}\Bigl{(}(\gamma+\bar{\gamma})A_{m}-\frac{(\gamma-\bar{\gamma})}{\epsilon}B_{m}+\frac{(\gamma+\bar{\gamma})}{\epsilon^{2}}C_{m}\Bigr{)}$
(178a) $\displaystyle M_{m}$
$\displaystyle=-\frac{1}{4}\Bigl{(}\epsilon(\gamma-\bar{\gamma})A_{m}-(\gamma+\bar{\gamma})B_{m}+\frac{(\gamma-\bar{\gamma})}{\epsilon}C_{m}\Bigr{)}$
(178b)

Inverting the relations (21), we can write $k$ and $\bar{k}$ in terms of
$k_{1}$ and $k_{2}$,

$\displaystyle k=\frac{1}{2}(k_{1}-\frac{k_{2}}{\epsilon})\ ;$ $\displaystyle\
\bar{k}=\frac{1}{2}(k_{1}+\frac{k_{2}}{\epsilon})$ (179)

Using the definitions of $\gamma$ and $\bar{\gamma}$ in (175) and using (179),
we can write,

$\displaystyle\lim_{\epsilon\to
0}\frac{\gamma+\bar{\gamma}}{\epsilon^{2}}=-\frac{2(k_{1}+2C_{g})}{k_{2}^{2}}\text{
and }\lim_{\epsilon\to
0}\frac{\gamma-\bar{\gamma}}{\epsilon}=-\frac{2}{k_{2}}$ (180)

Hence, in limit $\epsilon\rightarrow 0$ (178) can be written as,

$\displaystyle
L_{m}=\frac{1}{2k_{2}}\bigg{(}B_{m}-\frac{(k_{1}+2C_{g})}{k_{2}}C_{m}\bigg{)}\text{
and }M_{m}=\frac{1}{2k_{2}}C_{m}$ (181)

which agrees with (4.1).

As we have already seen in the previous section that (4.1) satisfies BMS
algebra with central charges $c_{L}=2dim(g)$ and $c_{M}=0$. This fact can also
be verified using the following definitions of central charges in the
relativistic Sugawara construction,

$\displaystyle c=\frac{kdim(g)}{k+C_{g}}\ ;\
\bar{c}=\frac{\bar{k}dim(g)}{\bar{k}+C_{g}}$ (182)

Using (182) and following similar steps as before using the (179), we can get
the following,

$\displaystyle c_{L}=\lim_{\epsilon\to 0}(c+\bar{c})$
$\displaystyle=\lim_{\epsilon\to
0}\bigg{\\{}dim(g)\bigg{(}\frac{\frac{1}{2}(k_{1}-\frac{k_{2}}{\epsilon})}{\frac{1}{2}(k_{1}-\frac{k_{2}}{\epsilon})+C_{g}}+\frac{\frac{1}{2}(k_{1}+\frac{k_{2}}{\epsilon})}{\frac{1}{2}(k_{1}+\frac{k_{2}}{\epsilon})+C_{g}}\bigg{)}\bigg{\\}}$
$\displaystyle\Rightarrow c_{L}=2dim(g)$ (183)

Similarly, we can get,

$\displaystyle c_{M}=\lim_{\epsilon\to 0}\epsilon(\bar{c}-c)=0$ (184)

These are the same values of central charges appeared in the $L$, $M$
commutation relations as we got in (126).

The modification we have done in order to get non-zero $c_{M}$ too can be
retrieved from contraction. For this, we can start from (128) and its
conjugate. Now if we take the limit in (21), we shall again retrieve the
redefined $L_{n}$s and $M_{n}$s as we have seen in (4.1), and the commutators
will be same as (4.1) with central charges (4.1).

###  7.4 NLKZ equations from Contraction

Finally, we show how to obtain the non-Lorentzian Knizhnik Zamolodchikov
equations from a limit of the ones for a 2d CFT with additional symmetry. Some
of the details of the analysis are contained in Appendix F. We start with the
original Knizhnik Zamolodchikov equations:

$\displaystyle\left(\partial_{w_{i}}-2\gamma\sum_{j\neq
i}\frac{\sum_{a}(t^{a}_{R_{i}})^{r_{i}}_{s_{i}}(t^{a}_{R_{j}})^{r_{j}}_{s_{j}}}{w_{i}-w_{j}}\right)\langle...\phi^{s_{i}}_{R_{i}}(w_{j})...\phi^{s_{j}}_{R_{j}}(w_{j})...\rangle=0$
(185a) $\displaystyle\left(\partial_{\bar{w}_{i}}-2\bar{\gamma}\sum_{j\neq
i}\frac{\sum_{a}(\bar{t}^{a}_{\bar{R}_{i}})^{\bar{r}_{i}}_{\bar{s}_{i}}(\bar{t}^{a}_{\bar{R}_{j}})^{\bar{r}_{j}}_{\bar{s}_{j}}}{\bar{w}_{i}-\bar{w}_{j}}\right)\langle...\bar{\phi}^{\bar{s}_{i}}_{\bar{R}_{i}}(\bar{w}_{j})...\bar{\phi}^{\bar{s}_{j}}_{\bar{R}_{j}}(\bar{w}_{j})...\rangle=0$
(185b)

where

$\displaystyle(t^{a}_{R_{i}})^{r_{i}}_{s_{i}}(t^{a}_{R_{j}})^{r_{j}}_{s_{j}}\langle...\phi^{s_{i}}_{R_{i}}(w_{j})...\phi^{s_{j}}_{R_{j}}(w_{j})...\rangle=((t^{a}_{R_{i}}\otimes
t^{a}_{R_{j}})\langle...\phi_{R_{i}}(w_{j})...\phi_{R_{j}}(w_{j})...\rangle)^{r_{i},r_{j}}$
(186)

Using the separability of the primary fields,
$\Phi_{R,\bar{R}}^{r,\bar{r}}(w,\bar{w})=\phi_{R}^{r}(w)\otimes^{\prime}\bar{\phi}_{\bar{R}}^{\bar{r}}(\bar{w})$,
we can write,

$\displaystyle\langle\Phi_{R_{1},\bar{R}_{1}}^{r_{1},\bar{r}_{1}}(w_{1},\bar{w}_{1})...\Phi_{R_{N},\bar{R}_{N}}^{r_{N},\bar{r}_{N}}(w_{N},\bar{w}_{N})\rangle=\langle\phi_{R_{1}}^{r_{1}}(w_{1})...\phi_{R_{N}}^{r_{N}}(w_{N})\rangle\langle\bar{\phi}_{\bar{R}_{1}}^{\bar{r}_{1}}(\bar{w}_{1})...\bar{\phi}_{\bar{R}_{N}}^{\bar{r}_{N}}(\bar{w}_{N})\rangle$
(187)

where the primed tensor product($\otimes^{\prime}$) ensures independent action
of operators on chiral and anti-chiral sectors whereas the unprimed tensor
product($\otimes$) ensures independent action on $i$th and $j$th insertion in
the $n$-point function.

Using the following linear combinations in limit $\epsilon\rightarrow 0$,

$\displaystyle(\ref{kz1})\times\langle\bar{\phi}_{\bar{R}_{1}}^{\bar{r}_{1}}(\bar{w}_{1})...\bar{\phi}_{\bar{R}_{N}}^{\bar{r}_{N}}(\bar{w}_{N})\rangle+(\ref{kz2})\times\langle\phi_{R_{1}}^{r_{1}}(w_{1})...\phi_{R_{N}}^{r_{N}}(w_{N})\rangle=0$
$\displaystyle\epsilon\\{(\ref{kz1})\times\langle\bar{\phi}_{\bar{R}_{1}}^{\bar{r}_{1}}(\bar{w}_{1})...\bar{\phi}_{\bar{R}_{N}}^{\bar{r}_{N}}(\bar{w}_{N})\rangle-(\ref{kz2})\times\langle\phi_{R_{1}}^{r_{1}}(w_{1})...\phi_{R_{N}}^{r_{N}}(w_{N})\rangle\\}=0$
(188)

We get (shown in details in Appendix F),

$\begin{split}&\bigg{[}\partial_{t_{i}}-\frac{1}{k_{2}}\sum_{j\neq
i}\bigg{\\{}\frac{\sum_{a}(t^{a}_{R_{i},J}\otimes
t^{a}_{R_{j},K}+t^{a}_{R_{i},K}\otimes
t^{a}_{R_{j},J})}{t_{ij}}+\bigg{(}\frac{x_{ij}}{t_{ij}^{2}}-\frac{(k_{1}+2C_{g})}{k_{2}t_{ij}}\bigg{)}\sum_{a}(t^{a}_{R_{i},K}\otimes
t^{a}_{R_{j},K})\bigg{\\}}\bigg{]}\\\ &\hskip
250.0pt\langle...\Phi_{R_{i},\bar{R}_{i}}(x_{i},t_{i})...\Phi_{R_{j},\bar{R_{j}}}(x_{j},t_{j})...\rangle=0\\\
&\bigg{[}\partial_{x_{i}}+\frac{1}{k_{2}}\sum_{j\neq
i}\frac{\sum_{a}(t^{a}_{R_{i},K}\otimes
t^{a}_{R_{j},K})}{t_{ij}}\bigg{]}\langle...\Phi_{R_{i},\bar{R}_{i}}(x_{i},t_{i})...\Phi_{R_{j},\bar{R}_{j}}(x_{j},t_{j})...\rangle=0\end{split}$
(189)

where,
$t^{a}_{R_{i},J}=t^{a}_{R_{i}}\otimes^{\prime}\bar{I}+I\otimes^{\prime}\bar{t}^{a}_{\bar{R}_{i}}$
and
$t^{a}_{R_{i},J}=\epsilon(I\otimes^{\prime}\bar{t}^{a}_{\bar{R}_{i}}-t^{a}_{R_{i}}\otimes^{\prime}\bar{I})$
in limit $\epsilon\rightarrow 0$. The above equations are same as what we got
form intrinsic analysis i.e. (153) and (155), with $t\to v$ and $x\to u$ as
its supposed to for a Galilean contracted result.

##  8 Conclusions

###  8.1 Summary

In this paper we have explored aspects of Non-Lorentzian CFTs with additional
Lie Algebraic symmetries. First we have reproduced the Non-Lorentzian Kač-
Moody algebra by taking singular limit from the Virasoro Kač-moody algebra.

After this we attempt to construct the same from intrinsic viewpoint without
any knowledge of the parent algebra. We see that 2d Carrollian Conformal
symmetry allows an infinite number of Noether currents. We then take a few
pairs of conserved currents (introducing flavour indices to distinguish them)
satisfying the conditions for EM tensor components in 2d Carrollian Conformal
symmetry. After this we derive the Ward identities associated with those
conserved currents. We also derive OPEs of a general 2d Carrollian Conformal
field with the current. After this we find the conserved charge operators
associated with these currents. The transformation generated by these charges
on a generic field is also derived. Here we first encounter the transformation
matrices which we encounter in relativistic CFT with Kac-Moody algebra. We
introduce current primary fields which turn out to be analogous to the
Virasoro Kač-Moody primary fields. After this the current current OPEs are
derived. While doing so, the structure constants emerge from the action of the
transformation matrices on the currents. After this, global internal symmetry
is applied on two point and three point correlation and it turns out that the
structure constants we have encountered while calculating the current current
OPEs satisfy Jacobi identity. After this we derive the OPE between the Energy
Momentum Tensor components and the current components. Using all these OPEs we
derive the algebra of the current modes and the Energy Momentum tensor modes.
The algebra transpire to be identical to the algebra obtained from limits of
Virasoro Kač-Moody algebra.

Later in the paper, we attempt to construct the EM tensor modes (which forms
the BMS algebra) from the current modes through Sugawara construction. Here we
see that the Sugawara construction only takes us to the BMS algebra with one
of the central charges to be zero. We needed another modification to the
Sugawara construction in order to get a fully centrally extended BMS algebra.
Using the expression of the EM tensor modes in terms of the current modes, we
calculate the OPEs of EM tensor fields with themselves and with currents and
see that the OPEs thus derived matches with the OPEs in the earlier section.

After this we have a brief look at the tensionless string. When we look at the
mode expansion of the coordinates (treating them as scalar fields), we see
that the modes satisfy a special case of Non Lorentzian $U(1)$ Kac-Moody
algebra. We also look at the expression of modes of the classical energy
momentum tensor in terms of these $U(1)$ modes and see that classically this
matches with the expression we have earlier derived for Non-Lorentzian
Sugawara construction. Hence we see that $U(1)$ NLKM currents are
intrinsically present in the tensionless string.

In section 6, we take a correlation function of a string of BMS current
primary fields. Using the OPE definition of the BMS primary field, we arrive
at the Non-Lorentzian version of the Knizhnik Zamolodchikov equations.
Finally, in section 7, we derive all the earlier results by taking limits from
the parent algebra.

###  8.2 Discussions and future directions

We mentioned in the introduction that our results in this paper lay the
groundwork for a large number of applications, most importantly to the
construction of a holographic dictionary for asymptotic flat spacetimes and
also the understanding of tensionless strings. We have built the underlying
algebraic structures in this paper which would be of importance to the quantum
field theories that are at the heart of these problems.

One of the most important and immediate next steps is to construct a Non-
Lorentzian Wess-Zumino-Witten model that realises these symmetries. This would
be central to the understanding of tensionless null strings moving on
arbitrarily curved manifolds. The work on this is currently underway.

The tensionless limit of string theory on arbitrary backgrounds is intimately
related to this and as mentioned in the introduction, we wish to revisit the
construction of Lindstrom:2003mg  in the light of our findings in this paper.
We believe that this limit would lead to null tensionless strings in AdS. This
is to be contrasted with tensionless strings in AdS considered in e.g.
Gaberdiel:2018rqv  and subsequent work in this direction, which are
tensionless but not null. A better understanding of the differences and
perhaps similarities between the two approaches would be important.

A generalisation of the methods outlined in this paper would be carried out
for 3d Carrollian and 3d Galilean theories. The structures are expected to
remain similar for the 3d Galilean theories, as the infinite dimensional
structure of the algebra without the extra currents remains intact when
generalised to higher dimensions. But for Carrollian theories, owing to the
fundamental difference between BMS3 and BMS4, where one copy of the Virasoro
in the 3d case gets enhanced to two Virasoros in the 4d case and
supertranslations develop two legs instead of one, the construction of the
quantum field theories with additional non-abelian currents would be more
involved.

### Acknowledgements

We thank Aritra Banerjee, Rudranil Basu and Niels Obers for interesting
discussions and comments on an initial version of the manuscript.

The work of AB is partially supported by a Swarnajayanti fellowship
(SB/SJF/2019-20/08) from the Science and Engineering Research Board (SERB)
India, the SERB grant (CRG/2020/002035), and a visiting professorship at École
Polytechnique Paris. AB also acknowledges the warm hospitality of the Niels
Bohr Institute, Copenhagen during later stages of this work. RC is supported
by the CSIR grant File No: 09/092(0991)/2018-EMR-I. AS is financially
supported by a PMRF fellowship, MHRD, India. RK acknowledges the support of
the Department of Atomic Energy, Government of India, under project number
RTI4001. DS would like to thank ICTS, Bengaluru for hospitality during the
course of this project.

## APPENDICES

##  Appendix A Carroll Multiplets

In this appendix, we review the construction of Carrollian boost multiplets in
two dimensions.

In two space-time dimensions, the Carrollian boost (CB) transformation is
defined as: $x\rightarrow x^{\prime}=x\text{ , }t\rightarrow t^{\prime}=t+vx$
; or equivalently, as:

$\displaystyle\begin{pmatrix}x\\\
t\end{pmatrix}\longrightarrow\begin{pmatrix}x^{\prime}\\\
t^{\prime}\end{pmatrix}=\left[\exp{\begin{pmatrix}0&0\\\
v&0\end{pmatrix}}\right]\begin{pmatrix}x\\\ t\end{pmatrix}\text{
}\Longleftrightarrow\text{
}x^{\mu}\rightarrow{x^{\prime}}^{\mu}={\left[e^{v\mathbf{B_{(2)}}}\right]}^{\mu}_{\hskip
5.69054pt\nu}\text{ }x^{\nu}$ (190)

with

$\displaystyle\mathbf{B_{(2)}}:=\begin{pmatrix}0&0\\\ 1&0\end{pmatrix}$ (191)

being the 2D representation of the CB generator $\mathbf{B}$ that is clearly
not diagonalizable.

Taking a cue from the Lorentz covariance of Lorentz tensors, it was postulated
in that a rank-$n$ Carrollian Cartesian tensor field $\Phi$ with ‘boost-
charge’ $\xi$ transforms under the Carrollian boost as:

$\displaystyle{\Phi}^{\mu_{1}...\mu_{n}}(t,x)\longrightarrow{\tilde{\Phi}}^{\mu_{1}...\mu_{n}}(t^{\prime},x^{\prime})={\left[e^{-\xi
v\mathbf{B_{(2)}}}\right]}^{\mu_{1}}_{\hskip
8.53581pt\nu_{1}}...{\left[e^{-\xi
v\mathbf{B_{(2)}}}\right]}^{\mu_{n}}_{\hskip
8.53581pt\nu_{n}}{\Phi}^{\nu_{1}...\nu_{n}}(t,x)$
$\displaystyle\Longleftrightarrow\qquad$
$\displaystyle{\mathbf{\Phi}}(t,x)\longrightarrow{{\mathbf{\tilde{\Phi}}}}(t^{\prime},x^{\prime})=\left[\bigotimes_{i=1}^{n}e^{-\xi
v\mathbf{B_{(2)}}}\right]{\mathbf{\Phi}}(t,x)=e^{-\xi
v\bigoplus\limits_{i=1}^{n}\mathbf{B_{(2)}}}{\mathbf{\Phi}}(t,x)$ (192)

where $\mu_{i},\nu_{i}$ are Carrollian space-time indices and for matrices,
the left index denotes row while the right one denotes column; repeated
indices are summed over and, in (192), indices are suppressed. It is to be
noted that the up/down appearance of a tensor-index does not matter; only the
left/right ordering is important.

Clearly, the Carrollian Cartesian tensors defined above are decomposible. So,
we now construct indecomposible Carrollian multiplets from these tensors. We
begin by recognizing that:

$\displaystyle\mathbf{B_{(2)}}=\mathbf{J}_{(l=\frac{1}{2})}^{\mathbf{-}}$
(193)

which is the lowering ladder operator in the $\text{SU}(2)$ spin-$\frac{1}{2}$
representation. Thus, $\bigoplus\limits_{i=1}^{n}\mathbf{B_{(2)}}$ in (192)
can be decomposed into indecomposable representations of
$\mathbf{J}^{\mathbf{-}}$ using the technique of ‘addition of $n$
spin-$\frac{1}{2}$ angular momenta’ in quantum mechanics, such that:

$\displaystyle\mathbf{B_{(d)}}\equiv\mathbf{J}_{(l=\frac{d-1}{2})}^{\mathbf{-}}$
(194)

It is evident that the representations $\mathbf{B_{(d)}}$ of the classical CB
generator are indecomposable since their only generalized eigenvalue is 0 and
it has geometric multiplicity 1. But, these representations are reducible for
$d\geq 2$.

A multi-component field transforming under the $d$-dimensional representation
of CB, $\mathbf{B_{(d)}}$, will be called a Carrollian multiplet of rank
$\frac{d-1}{2}$ with $d$ components, denoted by

$\displaystyle\Phi_{(l=\frac{d-1}{2})}^{m}\text{\hskip 7.11317pt with \hskip
7.11317pt}m=\frac{1-d}{2},\frac{3-d}{2},...,\frac{d-1}{2}$

By treating the $\mu=t$ index as spin-$\frac{1}{2}$ up-state and the $\mu=x$
index as spin-$\frac{1}{2}$ down-state, components $\Phi_{(l)}^{m}$ of a
Carrollian multiplet arise precisely as such linear combinations (with proper
Clebsch-Gordon coefficients) of the components of a Cartesian tensor of an
allowed rank $n$ that would appear while expanding the $|l,m\rangle$ states in
an allowed $|s_{1},s_{2},...,s_{n}\rangle$ basis (where $|s_{i}\rangle$ are
$\mathbf{J}_{(\frac{1}{2})}^{\mathbf{z}}$ eigenstates). So, as a linear
combination of the components of a rank-$n$ Cartesian tensor, one can obtain
multipltes of ranks: $0,1,2,...,\frac{n}{2}$ for even $n$ and
$\frac{1}{2},\frac{3}{2},\frac{5}{2},...,\frac{n}{2}$ for odd $n$. As an
example, we see how Carrollian multiplets of ranks $\frac{1}{2}$ and
$\frac{3}{2}$ are constructed from a rank-3 Cartesian tensor:

$\displaystyle\Phi_{(\frac{3}{2})}^{\frac{3}{2}}(t,x):=\Phi^{ttt}(t,x)\text{
}\text{ }\qquad\text{ }\text{
}\Phi_{(\frac{3}{2})}^{\frac{1}{2}}(t,x):=\frac{1}{\sqrt{3}}\left[\Phi^{ttx}+\Phi^{txt}+\Phi^{xtt}\right](t,x)$
$\displaystyle\Phi_{(\frac{3}{2})}^{-\frac{1}{2}}(t,x):=\frac{1}{\sqrt{3}}\left[\Phi^{txx}+\Phi^{xtx}+\Phi^{xxt}\right](t,x)\text{
}\text{ }\qquad\text{ }\text{
}\Phi_{(\frac{3}{2})}^{-\frac{3}{2}}(t,x):=\Phi^{xxx}(t,x)$
$\displaystyle\Phi_{(\frac{1}{2})}^{\frac{1}{2}}(t,x):=\frac{1}{\sqrt{a^{2}+b^{2}+c^{2}}}\left[a\Phi^{ttx}+b\Phi^{txt}+c\Phi^{xtt}\right](t,x)\text{
}\text{ }\qquad\text{with $a+b+c=0$}$
$\displaystyle\Phi_{(\frac{1}{2})}^{-\frac{1}{2}}(t,x):=\frac{1}{\sqrt{a^{2}+b^{2}+c^{2}}}\left[a\Phi^{xxt}+b\Phi^{xtx}+c\Phi^{txx}\right](t,x)$

(As the tuple $(a,b,c)$ in $\mathbb{R}^{3}$ lies on the plane $a+b+c=0$ which
is spanned by two basis vectors, two linearly independent rank-$\frac{1}{2}$
multiplets arise.)

A rank-$l$ Carrollian multiplet with boost-charge $\xi$ thus transforms under
the $2l+1$ dimensional representation of the CB as:

$\displaystyle\Phi_{(l)}^{m}(t,x)\longrightarrow\tilde{\Phi}_{(l)}^{m}(t^{\prime},x^{\prime})={\left[e^{-\xi
v\mathbf{J}_{(l)}^{\mathbf{-}}}\right]}^{m}_{\hskip
5.69054ptm^{\prime}}{\Phi}_{(l)}^{m^{\prime}}(t,x)$ (195)

After constructing the Carrollian multiplets from the Cartesian tensors as
demonstrated above, the components of the multiplets can always be redefined
such that:

$\displaystyle\text{in \eqref{x:2}, }\mathbf{J}_{(l)}^{\mathbf{-}}\text{ is
replaced by }\mathbf{M}_{(l)}:=\text{sub-diag }{(1,1,...,1)}_{2l+1}\text{ .}$

Hence, instead of the actual $\mathbf{J}_{(l)}^{\mathbf{-}}$ matrix, only the
indecomposable Jordan-block structure is important for defining the CB
transformation property of the Carrollian multiplets.

We conclude this appendix with the following observation. Since the finite
dimensional indecomposable representations of $\mathbf{B}$ are not symmetric
(or Hermitian), one can start with:

$\displaystyle\begin{pmatrix}t\\\
x\end{pmatrix}\longrightarrow\begin{pmatrix}t^{\prime}\\\
x^{\prime}\end{pmatrix}=\left[\exp{\begin{pmatrix}0&v\\\
0&0\end{pmatrix}}\right]\begin{pmatrix}t\\\ x\end{pmatrix}\text{
}\Longleftrightarrow\text{
}x^{\mu}\rightarrow{x^{\prime}}^{\mu}={\left[e^{v\mathbf{B^{\prime}_{(2)}}}\right]}^{\mu}_{\hskip
5.69054pt\nu}\text{ }x^{\nu}$

where

$\displaystyle\mathbf{B^{\prime}_{(2)}}:=\begin{pmatrix}0&1\\\
0&0\end{pmatrix}$

and follow the preceding argument to construct

$\displaystyle\mathbf{B^{\prime}_{(d)}}\equiv\mathbf{J}_{(l=\frac{d-1}{2})}^{\mathbf{+}}$

which is the $\text{SU}(2)$ raising ladder operator. But, as
$\mathbf{J}_{(l)}^{\mathbf{+}}={(\mathbf{J}_{(l)}^{\mathbf{-}})}^{\textbf{T}}$,
the raising and lowering operators’ representation matrices are related to
each other by the similarity transformation:

$\displaystyle S=\text{anti-diag }{(1,1,...,1)}_{2l+1}$

and consequently, $\mathbf{B}$ and $\mathbf{B^{\prime}}$ furnish two
equivalent representations of the CB generator.

##  Appendix B Calculation of Sugawara Construction Commutators

In this appendix, we provide the details of the calculation of commutators of
the NLKM algebra from the Sugawara construction.

### Calculating $[M_{m},J^{b}_{n}]$

$\displaystyle[M_{m},J^{b}_{n}]$
$\displaystyle=\frac{1}{2k_{2}}\sum_{a}\sum_{l}[K^{a}_{l}K^{a}_{m-l},J^{b}_{n}]\qquad\qquad\qquad\qquad\qquad\qquad\qquad\text{
(Using (\ref{chh4}))}$
$\displaystyle=\frac{1}{2k_{2}}\sum_{a}\sum_{l}\Bigl{(}[K^{a}_{l},J^{b}_{n}]K^{a}_{m-l}+K^{a}_{l}[K^{a}_{m-l},J^{b}_{n}]\Bigr{)}$
$\displaystyle=\frac{1}{2k_{2}}\sum_{a}\sum_{l}\bigg{\\{}(-i\sum_{c}f^{bac}K^{c}_{n+l}-nk_{2}\delta_{n+l,0}\delta^{ab})K^{a}_{m-l}$
$\displaystyle\qquad+K^{a}_{l}(-i\sum_{c}f^{bac}K^{c}_{m+n-l}-nk_{2}\delta_{m+n-l,0}\delta^{ab})\bigg{\\}}\text{
\qquad\qquad(Using (\ref{gkm}))}$
$\displaystyle=\frac{i}{2k_{2}}\sum_{a,c}\sum_{l}f^{abc}\Bigl{(}K^{c}_{n+l}K^{a}_{m-l}+K^{a}_{l}K^{c}_{m+n-l}\Bigr{)}-nK^{b}_{m+n}\qquad(\because-f^{bac}=f^{abc})$

where we have omitted the limits in summation over $a$ and $c$ and it is
understood that it runs over $1$ to $dim(g)$. Now,
$\sum_{l}K^{c}_{n+l}K^{a}_{m-l}=\sum_{l}K^{c}_{m+n-l}K^{a}_{l}$ simply by
translating $l$. Hence,

$\displaystyle[M_{m},J^{b}_{n}]=\frac{i}{k_{2}}\sum_{a,c}\sum_{l}f^{abc}K^{a}_{l}K^{c}_{m+n-l}-nK^{b}_{m+n}$
(196)

Now, again using antisymmetry property of $f^{abc}$

$\displaystyle\sum_{a,c}\sum_{l}f^{abc}K^{a}_{l}K^{c}_{m+n-l}$
$\displaystyle=\sum_{a,c}\sum_{l}f^{abc}K^{a}_{m+n-l}K^{c}_{l}$
$\displaystyle=\sum_{a,c}\sum_{l}f^{cba}K^{c}_{m+n-l}K^{a}_{l}$
$\displaystyle=-\sum_{a,c}\sum_{l}f^{abc}K^{a}_{l}K^{c}_{m+n-l}$
$\displaystyle\Rightarrow\sum_{a,c}\sum_{l}f^{abc}K^{a}_{l}K^{c}_{m+n-l}=0$
(197)

Hence, (196) can be written as,

$\displaystyle[M_{m},J^{b}_{n}]=-nK^{b}_{m+n}$ (198)

### Calculating $[L_{m},K^{b}_{n}]$

Again, using (4.1) and (22), we can get the following,

$\begin{split}[L_{m},K^{b}_{n}]&=\frac{1}{2k_{2}}\sum_{a}\bigg{[}\sum_{l\leq-1}([J^{a}_{l},K^{b}_{n}]K^{a}_{m-l}+K^{a}_{l}[J^{a}_{m-l},K^{b}_{n}])\\\
&\hskip
50.0pt+\sum_{l>-1}([J^{a}_{m-l},K^{b}_{n}]K^{a}_{l}+K^{a}_{m-l}[J^{a}_{l},K^{b}_{n}])\bigg{]}\\\
&=\frac{1}{2k_{2}}\sum_{a}\bigg{[}\sum_{l\leq-1}\bigg{\\{}i\sum_{c}f^{abc}(K^{c}_{l+n}K^{a}_{m-l}+K^{a}_{l}K^{c}_{m+n-l})+k_{2}lK^{a}_{m-l}\delta_{l+n,0}\delta^{ab}\\\
&\hskip
10.0pt+k_{2}(m-l)K^{a}_{l}\delta_{m+n-l,0}\delta^{ab}\bigg{\\}}\quad+\sum_{l>-1}\bigg{\\{}i\sum_{c}f^{abc}(K^{c}_{m+n-l}K^{a}_{l}+K^{a}_{m-l}K^{c}_{l+n})\\\
&\hskip
130.0pt+k_{2}(m-l)K^{a}_{l}\delta_{m+n-l,0}\delta^{ab}+k_{2}lK^{a}_{m-l}\delta_{l+n,0}\delta^{ab}\bigg{\\}}\bigg{]}\\\