aainstitutetext: Indian Institute of Technology Kanpur, Kalyanpur, Kanpur
208016. INDIAbbinstitutetext: Birla Institute of Technology and Science,
Pilani Campus, Pilani, Rajasthan 333031, INDIA. ccinstitutetext: Asia Pacific
Center for Theoretical Physics, POSTECH, Pohang 37673, KOREA.ddinstitutetext:
Center for Geometry and Physics, Institute for Basic Science (IBS), Pohang
37673, KOREA.

# Beyond Wilson? Carroll from current deformations

Arjun Bagchi b Aritra Banerjee a Saikat Mondal c Debangshu Mukherjee d and
Hisayoshi Muraki
abagchi<EMAIL_ADDRESS>saikatmd@iitk.ac.in,debangshu.mukherjee<EMAIL_ADDRESS>
###### Abstract

At extreme energies, both low and high, the spacetime symmetries of
relativistic quantum field theories (QFTs) are expected to change with
Galilean symmetries emerging in the very low energy domain and, as we will
argue, Carrollian symmetries appearing at very high energies. The formulation
of Wilsonian renormalisation group seems inadequate for handling these changes
of the underlying Poincare symmetry of QFTs and it seems unlikely that these
drastic changes can be seen within the realms of relativistic QFT. We show
that contrary to this expectation, changes in the spacetime algebra occurs at
the very edges of parameter space. In particular, we focus on the very high
energy sector and show how bilinears of $U(1)$ currents added to a two
dimensional (massless) scalar field theory deform the relativistic spacetime
conformal algebra to conformal Carroll as the effective coupling of the
deformation is dialed to infinity. We demonstrate this using both a symmetric
and an antisymmetric current-current deformation for theories with multiple
scalar fields. These two operators generate distinct kinds of quantum flows in
the coupling space, the symmetric driven by Bogoliubov transformations and the
antisymmetric by spectral flows, both leading to Carrollian CFTs at the end of
the flow.

##  1 Introduction

The modern understanding of quantum field theory is to a large part based on
the wonderful picture of renormalisation of Wilson RevModPhys.47.773 , where
high energy modes are systematically integrated out in order to reach an
effective low energy theory. Given a quantum field theory, this process of
continually integrating out modes takes the theory from an arbitrary point on
parameter space to an infra-red (IR) fixed point, governed by a conformal
field theory (CFT). There is also an ultra-violet (UV) fixed point governed by
a UV CFT.

Deformations of a quantum field theory can be relevant, marginal and
irrelevant. CFTs are fixed points in these renormalisation group (RG) flows
and the parameter space may contain CFTs other than UV and IR CFTs. Relevant
deformations move one along RG trajectories from the UV CFT to the IR CFT
through intermediate points which are a series of quantum field theories.
Marginal deformations (the exactly marginal ones) govern flows along a line of
CFTs while irrelevant deformations have the promise of running up RG flows at
the cost of locality in the resulting QFTs. These are notoriously difficult to
track and work on irrelevant deformations has been limited before the recent
advances in a specific class of deformations called the $T{\bar{T}}$
deformations. See Jiang:2019epa  for a review and a comprehensive list of
papers on the subject. When there are other conserved charges in the theories,
one can also have other solvable deformations e.g. $J{\bar{T}}$ deformations
Guica:2017lia .

The space of all QFTs can be constructed as RG flows away from these fixed
points governed by CFTs. The very ambitious programme of trying to classify
all quantum field theories can be recast into the question of classifying all
CFTs, which is perhaps a bit more tractable. One of the major goals of the
conformal bootstrap programme 111See, for example, Poland:2018epd  for an
exposition to the subject and detailed list of references. is to attempt
answering this question.

###  1.1 Possible breakdown of Wilsonian RG at very low energies

The formulation of Wilsonian RG for relativistic quantum field theories does
not allow changes in the spacetime algebra. However, it is very easy to
envision a scenario where this should run into trouble. If we are interested
in low energies and real world phenomena in e.g. condensed matter physics, the
non-relativistic approximation is often a very good approximation. So,
formulated generally, a renormalisation group flow should be able to reduce a
theory which is a relativistic QFT to a Galilean QFT if we are sufficiently
low in energies. The framework of RG that we use seems to be incapable to
understanding changes in the spacetime algebra that are required in order to
go from the relativistic to the non-relativistic world 222Wilson’s original
formulation RevModPhys.47.773  of course was done in the context of non-
relativistic field theories. But even here the theory cannot accommodate
changes in the spacetime algebra. Moving up the RG flow to a spacetime algebra
which is Poincare instead of Galilean cannot be handled.. The various attempts
at non-relativistic effective field theories in the literature suffer from
this inherent and basic problem. These theories are not Galilean invariant
theories, but Poincare invariant theories masquerading as non-relativistic
field theories.

This paper aims to initiate a programme devoted to the understanding of the
very low and the very high energy sectors of relativistic theories where the
spacetime algebras themselves get transformed. We have discussed above how one
can move down the RG flow lines from higher to lower energies via relevant
deformations and when we go to sufficiently low energies we should end up in
effect in the non-relativistic world where all the relativistic degrees of
freedom have been integrated out and the theory should become Galilean
invariant. We also stressed that this is possibly not achievable in the usual
Wilsonian framework, since there is no notion of a change in spacetime algebra
in this regard. We now discuss the ultra-high energy regime which would be the
focus of the current paper.

###  1.2 Emergence of Carroll at very high energies

Figure 1: Emergent symmetries in relativistic QFTs

As mentioned above, moving up the RG flow is fraught with danger, and
irrelevant operators, which would trigger such flows, also necessitate the
addition of more and more counter terms to the Lagrangian of the QFT in
question to render it finite. A loss of locality is a natural and problematic
consequence. Now let us think of the very high energy theory instead of the
very low energy one where we argued that Galilean invariance should naturally
appear. The Galilean algebra in $d$ dimensions arises as an Inönü-Wigner
contraction of the Poincare algebra when the speed of light $c\to\infty$, and
the non-zero commutators read:

$\displaystyle[J_{ij},B_{k}]$
$\displaystyle=\delta_{k[j}B_{i]},~{}~{}~{}[J_{ij},\,P_{k}]=\delta_{k[j}P_{i]},~{}~{}[H,B_{i}]=P_{i}.$
(1)

where $[J_{ij},J_{kl}]$ gives the usual spatial rotation algebra in one less
dimension, $P_{i}$ are spatial momenta and $B_{i}$ are the boosts, with $H$
being the Hamiltonian. As a major point of difference from the Poincare
algebra, notice here that Galilean boosts commute

$[B_{i},B_{j}]=0$ (2)

as opposed to Lorentzian boosts which close to rotations
$[J_{0i},J_{0j}]=J_{ij}$.

There exists a diametrically opposite limit of the Poincare algebra where the
speed of light, instead of going to infinity, goes to zero. This apparently
non-sensical $c\to 0$ limit of the Poincare algebra has been named the Carroll
algebra LBLL ; NDS , after Lewis Carroll, the creator of Alice in Wonderland
3179 , where the commutators of interest (except spatial rotations, which obey
the same algebra as in the Lorentz or Galilean case) are:

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

Carroll boosts again commute. But we see a reversal of roles between the
spatial momenta and the Hamiltonian in this algebra compared to the Galilean
algebra above. We claim that the very high energy sector of a relativistic
quantum field theory is governed by the Carroll algebra, in a manner similar
to the expectation that the Galilean algebra dictates the very low energy
sector. We now justify our rather radical claim.

The very high energy theory may be thought of, in the spirit of Wilson, as a
theory where all the low energy modes are inconsequential. It is conceivable
that such a theory may be reached from a lower energy theory by systematically
throwing away or “integrating out” all modes except the very high energy ones.
In a sense we are envisioning this extreme high energy theory as being built
out of a sector of the original theory, very much like the very low energy
theory. This very high energy theory should be limited to excitations that
travel extremely fast and as a first approximation, can be thought of only
containing modes which travel at the speed of light. This in a sense means
that the theory should contain only excitations moving on null hypersurfaces
in the spacetime on which the theory was originally defined. Now, it is known
that Carroll symmetries appear on generic null hypersurfaces and hence it can
be expected that the very high energy sector is governed by Carrollian
symmetries.

We now offer more evidence that shows our claim may not be as radical as may
have been assumed at first. We will do this by showing the emergence of
Carrollian symmetries in various ultra-high energy contexts. But before we go
into specific examples, let us offer some general explanations. The $c\to 0$
limit can be viewed as a limit where

$t\to\varepsilon t,\,x^{i}\to x^{i},\quad\varepsilon\to 0,$ (4)

where $\varepsilon$ is a dimensionless parameter which plays the role of the
speed of light. Now the contraction of the time direction essentially means
that the energies become much larger than the spatial momenta, i.e.

${}E\to E/\varepsilon,\,p^{i}\to p^{i}$ (5)

So, the Carroll limit naturally takes the theory to high energies. Below we
exemplify this in various contexts and show how Carrollian structures arise in
the ultra-high energy sectors of various theories.

###  1.3 Carroll and extreme high energies: Examples

We now give specific examples of how Carrollian symmetries arise in the
extreme high energy limits of various theories of interest.

  * $\square$

Tensionless strings: Non-interacting string theory is most conveniently
described in terms of the Polyakov action:

${}S_{P}=\frac{1}{2\pi\alpha^{\prime}}\int
d^{2}\xi\sqrt{\gamma}\gamma^{ab}\partial_{a}X^{\mu}\partial_{b}X^{\nu}\eta_{\mu\nu}$
(6)

In the above, $\gamma^{ab}$ is the metric on the worldsheet, while
$\eta_{\mu\nu}$ is the flat background metric indicating that the string moves
in a flat spacetime geometry. The theory has one tuneable parameter
$\alpha^{\prime}$, the length of the fundamental string. The Polyakov action
has diffeomorphism and Weyl invariance which need gauge fixing. In the so-
called conformal gauge, where the worldsheet metric is fixed to be flat, there
is still a residual gauge symmetry (diffeomorphisms that can be undone by a
Weyl transformation) which leads to conformal invariance of the worldsheet
theory. This helps organise the spectrum and indeed the quantum theory of the
tensile relativistic string.

String theory reduces to Einstein gravity (supersymmetric version thereof)
when the length of the string goes to zero, i.e. $\alpha^{\prime}\to 0$. This
is the low-energy limit of string theory. The opposite limit, the ultra-high
energy limit is where $\alpha^{\prime}\to\infty$. Since the tension is the
inverse of the string length, this is also the tensionless limit, where the
string becomes long and floppy. In this limit the worldsheet metric
degenerates and the action can be written as Isberg:1993av

${}S_{\text{ILST}}=\int
d^{2}\xi\,V^{a}V^{b}\partial_{a}X^{\mu}\partial_{b}X^{\nu}\eta_{\mu\nu}$ (7)

where $V^{a}$’s are vector densities and $V^{a}V^{b}$ replaces
$\sqrt{\gamma}\gamma^{ab}$ as the tension vanishes. Carrollian structures thus
appear on the worldsheet. The residual symmetries in the analogue of the
conformal gauge change from the two copies of the Virasoro to the 2d conformal
Carroll algebra Bagchi:2013bga ; Bagchi:2015nca ; Bagchi:2020fpr  indicating
this change of worldsheet geometry.

The very high energy regime of string theory thus indicates the emergence of
Carrollian structures, in this case on the worldsheet of the string, which
becomes tensionless in this sector.

  * $\square$

Ultra-relativistic fluids: The soup called Quark-Gluon-Plasma (QGP) formed
immediately after the Big Bang is effectively modelled by fluids flowing at
velocities near the speed of light. One of the simplest and most useful models
in this scenario was put forward by Bjorken Bjorken:1982qr  and this
effectively two dimensional model goes under the name of Bjorken flow. It has
been recently shown that this model admits Carrollian symmetry bjorken1 . A
generalisation of this model to include dependences in the transverse
directions is the so-called Gubser flow Gubser:2010ze . It was shown even more
recently that Carroll hydrodynamics can effectively describe even Gubser flows
bjorken2 . In general, Carrollian hydrodynamics should be able to capture all
ultra-relativistic fluids flowing at velocities comparable to the speed of
light. The very high energy sector of fluid dynamics hence is effectively
captured by a Carroll sector.

  * $\square$

Flat space holography: Holography beyond AdS/CFT and in particular flat space
holography has got a lot of attention of late and progress has been achieved
in two slightly differing approaches called Celestial holography (see
Strominger:2017zoo ; Pasterski:2021rjz ; Raclariu:2021zjz  for excellent
reviews) and Carrollian holography. See Bagchi:2010zz ; Bagchi:2012cy ;
Bagchi:2012xr ; Barnich:2012xq ; Bagchi:2015wna ; Bagchi:2014iea ;
Bagchi:2016bcd  for earlier work in the 3d bulk/2d boundary context. More
recent developments for 4d bulk and 3d boundary have followed from
Bagchi:2022emh ; Donnay:2022aba . The main difference between the two is that
while the Celestial version advocates a co-dimension two hologram in terms of
a relativistic conformal field theory living on the celestial sphere on the
null boundary of flat space, the Carroll formulation keeps track of the null
direction at $\mathscr{I}^{\pm}$ and hence puts forward a co-dimension one
dual. As the name suggests, Carrollian holography posits that 4d
asymptotically flat spacetime has a holographic dual which is a 3d Carrollian
CFT. See Bagchi:2023cen  for a discussion on the points of difference between
the approaches.

Now, flat spacetimes can be obtained by an infinite radius limit of AdS. This
limit can also be thought of as focusing on length scales very small compared
to the radius of AdS deep inside the bulk where the effect of curvature would
not be felt. In terms of the boundary theory, this region corresponds to the
deep UV region of the CFT. Hence flat holography can be thought of as a very
high energy sector of usual AdS/CFT. Now, it has been recently made precise
that this limit on AdS/CFT gives rise to the Carrollian hologram described
above Bagchi:2023fbj ; Bagchi:2023cen . This was done by showing how various
AdS Witten diagrams in the limit led to Carrollian correlation functions.
Again, the emergence of Carrollian structures, now in the very high energy
sector of AdS/CFT is indicative of the fact that Carrollian symmetry may in
general be associated to the deep UV of a generic quantum field theory and
this symmetry may be emergent in the same way as Galilean symmetries emerge in
the very low energy sector of any relativistic set up.

Having offered several examples and hopefully having convinced the reader that
the initial claim of the emergence of Carrollian symmetry in the ultra-high
energy sector of any relativistic theory is not unnatural, we make a final
comment before outlining how we will go about showing this in the context of
RG flows in QFTs in this paper. The operators we naturally associate with
flows from the IR to the UV are irrelevant operators and we have remarked
earlier that perturbation of a QFT by generic irrelevant operators would lead
to a loss of locality in the theory. It seems from our discussion above that
somewhere in the extreme UV, the relativistic theory morphs into a Carrollian
theory. One would expect locality (in the sense that it is defined in the
relativistic regime) to be destroyed in this Carrollian theory since the most
“natural” way of reaching this regime would be through irrelevant
deformations. The loss of locality (defined in the relativistic sense) is
manifest in the Carroll theory since lightcones completely close up in this
regime. If anything, the theory becomes ultra-local.

###  1.4 What is in this paper?

Having set the stage in the previous subsections, our aim in this paper is to
show how Carrollian symmetry indeed appears in quantum field theories through
renormalisation group flows. While an important future goal is to show that
this happens generically, in the present work we will be very specific and
will focus on two dimensional relativistic conformal field theories and
perturbation of 2d CFTs by marginal operators. We will show that when we dial
some effective parameter controlling the size of the perturbation all the way
to infinity so as to reach the edge of the parameter space, we indeed find the
emergence of Carrollian structures. In this case, the marginal deformations
which generate continuous lines of flows from one to another CFT end up in a
2d Carrollian CFT at the end of this process.

##### Outline of the paper:

Sec.(2) contains a summary of our results and this should be useful for
readers who wish to just get the gist of the paper without ploughing through
the technicalities. In particular, Table (1) contains the most vital
information which we would go on to show in detail in the later sections. The
two following sections contains the main details of the paper. Sec.(3) deals
with the symmetric current-current deformations. We first address the
classical aspects and then move into the quantum mechanical aspects later in
the section. Sec.(4) addresses the antisymmetric deformation and again we
delve into both classical and quantum mechanical details of the deformed
theory. The two deformations are shown to act very differently, but as
advertised above, they both lead to theories with Carrollian conformal
symmetry at the end of the flow. We end with comments and future directions in
Sec.(5). A number of appendices at the end contain calculations omitted in the
main text and supplementary discussions.

##  2 Flowing to Carroll: the set up

###  2.1 Summary of results

In this brief section, we will outline our strategy for the rest of the paper.
The theory we will concentrate on is the free massless relativistic scalar
theory in $d=2$. This is of course a relativistic conformal field theory. We
will, in general, consider $D$ such scalar fields:

$\mathcal{L}=\frac{1}{2}\sum_{i=1}^{D}\left(\left(\partial_{\tau}\phi_{i}\right)^{2}-\left(\partial_{\sigma}\phi_{i}\right)^{2}\right).$
(8)

which gives rise to the equations of motion for each field:

$\left(\partial_{\tau}^{2}-\partial_{\sigma}^{2}\right)\phi_{i}=0.$ (9)

We will perturb this theory in two different ways, both of them will involve
adding bilinears of $U(1)$ Kac-Moody currents.

  * $\star$

Symmetric current-current deformation: We first look at deformations of the
parent Lagrangian by a symmetric current-current deformation:

$\widetilde{\mathcal{L}}^{\alpha}=\mathcal{L}+\frac{\alpha}{2}\sum_{i}\left({{J}}^{2}_{i}+\overline{{J}}^{2}_{i}\right).$
(10)

Here $J,\overline{{J}}$ are conserved currents:

${J}_{i}=\frac{1}{\sqrt{2}}(\dot{\phi}_{i}+\phi_{i}^{\prime}),\quad\overline{{J}}_{i}=\frac{1}{\sqrt{2}}(\dot{\phi}_{i}-\phi_{i}^{\prime}),$
(11)

where $\dot{\phi}_{i}=\partial_{\tau}{\phi}_{i}$ and
${\phi}^{\prime}_{i}=\partial_{\sigma}{\phi}_{i}$ are the temporal and spatial
derivatives, and $\alpha$ is a continuous parameter. The currents obey the
conservation laws $\bar{\partial}J_{i}=\partial\overline{J}_{i}=0$ with
$\partial,\bar{\partial}=\partial_{\tau}\pm\partial_{\sigma}$. More
explicitly, the deformed Lagrangian reads:

$\widetilde{\mathcal{L}}^{\alpha}=\frac{1}{2}\sum_{i=1}^{D}\left((1+\alpha)\left(\partial_{\tau}\phi_{i}\right)^{2}-(1-\alpha)\left(\partial_{\sigma}\phi_{i}\right)^{2}\right).$
(12)

The deformation scales the time derivatives up in the action and scales down
space derivatives, or vice versa, depending on the chosen sign of $\alpha$.
When $\alpha\neq\pm 1$, one can rescale temporal and spatial coordinates and
“restore” conformal symmetry. $\alpha=\pm 1$ acts as boundaries of the
parameter space, at which one of the derivatives in the action is lost. We
will show that $\alpha=1$ is the Carroll point (and $\alpha=-1$ the Galilean
point) and hitting this leads to a change of space-time symmetry. Both from
the classical and quantum perspective, this is the exact limit where Carroll
algebra takes over.

The unequal scaling of spacetime coordinates as manifest in action (12) leads
to a modification of the effective speed of light in the theory. Using this
handle over the propagation velocity of modes, we will tune the effective
speed of light to zero, where only fast-moving excitations are important. In
Bagchi:2022nvj , similar conclusions were reached via a “boost” transformation
on the (anti)holomorphic currents, parameterized by $\alpha$. We will show
that this effectively leads to a Bogoliubov transformation on the modes of the
theory, and different values of $\alpha$ define different physical vacua on
which a line of CFTs are defined, albeit with more and more faster moving
excitations.

Property | Symmetric | Antisymmetric
---|---|---
Deformed Lagrangian | $\widetilde{\mathcal{L}}^{\alpha}=\mathcal{L}+\frac{\alpha}{2}\left({{J}}^{2}_{i}+\overline{{J}}^{2}_{i}\right)$ | $\widetilde{\mathcal{L}}^{g}=\mathcal{L}+g\epsilon_{ab}J^{a}\bar{J}^{b}$
Deformed Hamiltonian | $\widetilde{\mathcal{H}}^{\alpha}=\mathcal{H}\left[\pi(\alpha),\phi\right]+\alpha J_{i}\overline{J}_{i}$ | $\widetilde{\mathcal{H}}^{g}=\mathcal{H}\left[\pi(g),\phi\right]+g\epsilon_{ab}J^{a}\bar{J}^{b}$
End-point of flow | $\alpha\to\pm 1$ | $g\to\infty$
Symmetry at end-point | BMS3 | BMS3
Speed of light | Changes continuously to $0$ | Abruptly $\to 0$ @ end-point
Vacuum | Continuous change | Sudden jump @ end-point
Central charges | Continuous change | Sudden jump @ end-point
Nature of flow | Bogoliubov transformation driven | Spectral flow driven
Table 1: Properties of deformations.

  * $\star$

Antisymmetric current-current deformation: We will then focus on current-
current deformations of the form $\epsilon_{ab}J^{a}\bar{J}^{b}$. These are
exactly marginal operators, and have been discussed in the literature in
connection to string theory Apolo:2019zai . A salient feature of such
deformations is that these mix spacetime indices on the fields, as opposed to
the symmetric $J\bar{J}$ case where spacetime indices just decouple from each
other.

We will restrict ourselves to the two dimensional Lagrangian density of two
bosonic fields $\phi_{1,2}$ deformed by a $J^{1}\wedge\bar{J}^{2}$ operator
for simplicity. Generalisation to $2D$ bosons is immediate. Here again the
currents $J^{i}=\partial_{z}\phi^{i}$ and
$\overline{J}^{i}=\partial_{\bar{z}}\phi^{i}$ are the (anti-)holomorphic
$U(1)$ currents of the theory. The deformed Lagrangian, as discussed in
Guica:2021fkv , is

$\mathcal{\tilde{L}}^{(g)}=\frac{1}{1+g^{2}}\left[\frac{1}{2}\sum_{i=1,2}\left(\left(\partial_{\tau}\phi_{i}\right)^{2}-\left(\partial_{\sigma}\phi_{i}\right)^{2}\right)-g\left(\partial_{\tau}\phi_{1}\partial_{\sigma}\phi_{2}-\partial_{\sigma}\phi_{1}\partial_{\tau}\phi_{2}\right)\right].$
(13)

The action of this deformation is very different from the symmetric case. For
all finite values of the coupling $g$, the theory is a relativistic CFT, as
the perturbation term can be considered a boundary contribution. This
$g$-deformed line of CFTs however continue to have the same effective speeds
of light, as the equation of motion for propagating modes remain the same. The
change of spacetime algebra only becomes apparent when this boundary term
becomes unmistakably large at $g\to\infty$. We will show how a non-invertible
change in boundary conditions for the fields in this limit implies zero
velocity of mode propagation, thereby bringing about an abrupt change in
spacetime algebra.

Quantization of the theory needs careful consideration of the boundary term
contribution. There is no change in physical vacuum of the theory at finite
$g$. However the gradual take-over by higher and higher energy modes is
achieved by a process called the “spectral flow” of energy eigenstates. The
bare conformal dimensions of undeformed fields pick up $g$ dependent
contributions as the strength of the perturbation increases.

For finite $g$, these deformed states continue to transform under Virasoso-
Kac-Moody symmetries, as the (momentum dependent) spectral flow parameter
commutes with all the generators. Starting with the highest weight
representations of the undeformed Virasoro algebra, the CFT Hilbert space
reorganizes itself at the strict $g\to\infty$ limit into the spectrum of the
Carrollian conformal algebra.

From our brief summary above, it should be clear to the reader that although
the two deformations take us from the same relativistic theory to a theory
with Carrollian symmetries, the energy flows upwards from the relativistic
regime as triggered by different operators have very different effects on the
theories, as elucidated in Table (1) as well. This is the key idea that we
will delve into in our work.

###  2.2 Carrollian limits and the Dirac-Schwinger condition

Before we begin our investigations of the edges of parameter spaces of QFTs,
we first need to acquaint ourselves with some properties of Carrollian
theories and the limits which lead up to these theories. As discussed in the
introduction, the Carroll algebra arises out of a $c\to 0$ limit of the
Poincare algebra and is given by (3). We are interested in quantum field
theories exhibiting Carrollian (and Conformal Carrollian) symmetries (see
Bagchi:2016bcd ; Bagchi:2019clu ; Bagchi:2019xfx ; Hao:2021urq ; Basu:2018dub
; Saha:2022gjw ; Banerjee:2023jpi ; deBoer:2023fnj  for a non-exhaustive
list). Following the algebra, an obvious way to get to Carrollian QFTs is to
dial $c\to 0$ in particular relativistic QFTs. Below we switch to the
Hamiltonian framework to elucidate some properties of this limit.

To understand how Carrollian dynamics occurs in field theories, we now revisit
Carrollian limits from the point of view of the Dirac-Schwinger condition of
classical field theory. We will be mostly following the notations introduced
in Henneaux:2021yzg  in what follows. Let us take the example of a massless
relativistic scalar field on flat D-dimensional Minkowski space, and reinstate
the $c$ factors so that we can clearly analyze the effect of taking $c\to 0$
limit. Here the Lagrangian is given by:

$\mathcal{L}=\frac{1}{2c^{2}}\left(\dot{\phi}^{2}-c^{2}(\nabla_{k}\phi)^{2}\right).$
(14)

As discussed before, Carrollian limit corresponds to taking $c\to 0$ limit on
the theory. So naively, the limiting theory would be the one where the
temporal derivative term leads over the spatial derivative, and the resulting
theory becomes invariant under Carroll boosts. However, one could also have a
subtle alternative limit, where spatial gradients survive333Note that in the
opposite limit $c\to\infty$, just the spatial derivative part of the action
survives, and becomes invariant under Galilean boosts instead, as discussed
earlier. This is subtly different from this alternative Carroll limit with
spatial gradients.. These sectors will also be evident in the current work,
albeit we will mostly use the convention $c=1$ and the Carroll limit will be
imposed via dialing of some auxiliary parameters as promised in the previous
subsection. A nice way to understand these two Carroll limits is to consider
the Hamiltonian formalism, where energy (Hamiltonian) density $\mathcal{H}$
and momentum density $\mathcal{P}_{k}$ form the local generators of the
Carroll symmetry algebra. The Hamiltonian density corresponding to the scalar
field is:

$\mathcal{H}=\frac{1}{2}\left(c^{2}\pi^{2}+(\nabla_{k}\phi)^{2}\right),~{}~{}\dot{\phi}=\frac{\partial\mathcal{H}}{\partial\pi}$
(15)

and the invariant action reads

$S=\int dt~{}d^{d}x\left(\pi\dot{\phi}-\mathcal{H}\right).$ (16)

For a relativistic scalar field, the Poisson bracket of energy densities at
different spatial points generate the Dirac-Schwinger condition:

$\left\\{\mathcal{H}(x),\mathcal{H}\left(x^{\prime}\right)\right\\}_{\mathrm{PB}}=2\mathcal{P}^{k}(x)\partial_{k}\delta\left(x-x^{\prime}\right)+\left(\partial_{k}\mathcal{P}^{k}(x^{\prime})\right)\delta\left(x-x^{\prime}\right),$
(17)

where one uses the canonical commutation relations

$\\{\phi(x),\pi(x^{\prime})\\}_{\rm PB}=-\\{\pi(x),\phi(x^{\prime})\\}_{\rm
PB}=\delta(x-x^{\prime}),$ (18)

and the momentum flux density is given by
$\mathcal{P}_{k}=c^{2}\pi\nabla_{k}\phi$.

The hallmark of Carroll dynamics is the vanishing of the Poisson bracket
between Hamiltonian densities, since the Carrollian momentum flux density is
identically zero. So for us, the rule of thumb to construct Carroll invariant
systems would be to look for the ones having

$\\{\mathcal{H}(x),\mathcal{H}(x^{\prime})\\}_{\rm PB}=0.$ (19)

Consequently one can generalise this statement to the quantum case where the
commutator vanishes. 444This condition can also be dubbed as the “flat band”
condition as used by condensed matter physicists, and a striking connection to
Carroll physics to such systems was shown in Bagchi:2022eui .  The simplest
example of such a system corresponds to taking $c\to 0$ directly in (14), as
mentioned beforehand, where only the time derivative part of the action
remains. This version of Carrollian limit was dubbed as the “Magnetic” limit
since $\mathcal{H}$ just contains the spatial gradient, and the momenta cannot
be expressed in terms of the derivative of fields. On the other hand one could
canonically rescale the fields and momenta using $\phi\to c\phi$ and
$\pi\to\frac{1}{c}\pi$, and then take $c\to 0$, leading to
$\mathcal{H}\sim\pi^{2}$. This other limit is called the “Electric” Carroll
limit, and in this case equations of motion simplify to $\pi=\dot{\phi}$ and
$\dot{\pi}=0$. The nomenclature is sort of opposite in spirit to similar
Carroll limits taken from the Lagrangian perspective that we started out with.
See deBoer:2021jej  for more details on Carroll expansions of scalar fields
and related boost invariance .

In both kind of limits (19) however is sacrosanct. This brings forth the
possibility that for a multi-field system one might have a myriad of ways of
taking Carroll limits, and even a mixture of electric and magnetic Carroll
fields may be admissible. This could lead to very complicated phase space
structures, reducing it down to a combination of a subset over volume forms.
In the rest of this manuscript, we will be talking about deformation of scalar
field theories via current bilinears, that exactly achieve different kinds of
Carroll limits on the system as their coupling saturates some relevant bound.
Upon quantization this would mean the resulting spectrum approaches the
Carrollian one via different paths in the space of couplings. We would be
consistently switching between Lagrangian and Hamiltonian languages to get
these points through to the reader.

##  3 The symmetric deformation

We begin our detailed study of deformations with the symmetric current-current
deformation of the scalar theory:

$\widetilde{\mathcal{L}}^{\alpha}=\mathcal{L}_{\text{free}}+\mathcal{O}_{\text{def}}=\frac{1}{2}\sum_{i=1}^{D}\left(\left(\partial_{\tau}\phi_{i}\right)^{2}-\left(\partial_{\sigma}\phi_{i}\right)^{2}\right)+\frac{\alpha}{2}\sum_{i}\left({{J}}^{2}_{i}+\overline{{J}}^{2}_{i}\right).$
(20)

where as stated before
${J}_{i}=\frac{1}{\sqrt{2}}(\dot{\phi}_{i}+\phi_{i}^{\prime}),\,\overline{{J}}_{i}=\frac{1}{\sqrt{2}}(\dot{\phi}_{i}-\phi_{i}^{\prime})$
are conserved currents.

###  3.1 Classical considerations

We will find it convenient to move to the Hamiltonian formalism, especially
for the purposes of quantizing the theory. We first set up some notation.

####  3.1.1 Setting the stage

The canonical commutation relation in this case reads

$\\{\phi_{i}(\sigma),\pi_{j}(\sigma^{\prime})\\}_{\rm
PB}=-\\{\pi_{i}(\sigma),\phi_{j}(\sigma^{\prime})\\}_{\rm
PB}=\delta_{ij}\delta(\sigma-\sigma^{\prime}),$ (21)

with all other brackets vanishing. Here $\pi_{i}$ are the conjugate momenta
for the fields $\phi_{i}$, given by

$\pi_{i}=\frac{\partial\mathcal{L}_{\text{free}}}{\partial\dot{\phi}_{i}}=\dot{\phi}_{i}.$
(22)

These relations imply the algebra among the currents

$\displaystyle\\{J_{i}(\sigma),J_{j}(\sigma^{\prime})\\}_{\rm
PB}=\delta_{ij}\partial_{\sigma}\delta(\sigma-\sigma^{\prime}),\quad\\{\overline{J}_{i}(\sigma),\overline{J}_{j}(\sigma^{\prime})\\}_{\rm
PB}=-\delta_{ij}\partial_{\sigma}\delta(\sigma-\sigma^{\prime}),$ (23)

with $J_{i}$ and $\overline{J}_{j}$ Poisson commuting. The (anti)holomorphic
stress tensors for the whole system are (classically) obtained via Sugawara
construction from the currents

$T=\frac{1}{2}\sum_{i}{J}_{i}^{2},\qquad\quad\overline{T}=\frac{1}{2}\sum_{i}\overline{J}_{i}^{2},$
(24)

which satisfy the following algebra

$\displaystyle\left\\{T(\sigma),T\left(\sigma^{\prime}\right)\right\\}_{\mathrm{PB}}$
$\displaystyle=2T(\sigma)\partial_{\sigma}\delta\left(\sigma-\sigma^{\prime}\right)+\left(\partial_{\sigma}T(\sigma)\right)\delta\left(\sigma-\sigma^{\prime}\right),$
(25a)
$\displaystyle\left\\{\overline{T}(\sigma),\overline{T}\left(\sigma^{\prime}\right)\right\\}_{\mathrm{PB}}$
$\displaystyle=-2\overline{T}(\sigma)\partial_{\sigma}\delta\left(\sigma-\sigma^{\prime}\right)-\left(\partial_{\sigma}\overline{T}(\sigma)\right)\delta\left(\sigma-\sigma^{\prime}\right),$
(25b)
$\displaystyle\left\\{T(\sigma),\overline{T}\left(\sigma^{\prime}\right)\right\\}_{\mathrm{PB}}$
$\displaystyle=0.$ (25c)

As a consequence, when we expand the stress tensors in terms of Fourier modes:

$T(\sigma)=\sum_{m}\mathcal{L}_{m}\,e^{-im\sigma}\qquad\quad\overline{T}(\sigma)=\sum_{m}\overline{\mathcal{L}}_{m}\,e^{+im\sigma},$
(26)

the generators $\mathcal{L}$’s correspond to two copies of (the classical part
of) the Virasoro algebra. The above Poisson brackets between stress tensors
can then be rewritten in terms of these Fourier modes:

$\\{\mathcal{L}_{n},\mathcal{L}_{m}\\}_{\text{PB}}=-i(n-m)\mathcal{L}_{n+m},\qquad\\{\overline{\mathcal{L}}_{n},\overline{\mathcal{L}}_{m}\\}_{\text{PB}}=-i(n-m)\overline{\mathcal{L}}_{n+m},$
(27)

which are the well known form of centerless Virasoro or Witt algebra. In the
quantum case, one will also have central charges $c,\bar{c}$ appearing in the
commutators of these generators.

####  3.1.2 BMS at the end of parameter space

With our notations in place, we now show how the spacetime algebra would
change at the ends of the flow generated by this deformation. Let us now focus
on the conjugate momenta to the fields as calculated from (20):

$\widetilde{\pi}^{\alpha}_{i}=\frac{\partial\widetilde{\mathcal{L}}^{\alpha}}{\partial\dot{\phi}_{i}}=(1+\alpha)\dot{\phi}_{i}$
(28)

Thus the Hamiltonian density can be written as

$\widetilde{\mathcal{H}}^{\alpha}=\sum_{i=1}^{D}\widetilde{\pi}_{i}\dot{\phi}_{i}-\widetilde{\mathcal{L}}^{\alpha}=\frac{\kappa}{2}\sum_{i=1}^{D}\left(\epsilon\,\widetilde{\pi}_{i}^{2}+\frac{1}{\epsilon}\left(\partial_{\sigma}\phi_{i}\right)^{2}\right)$
(29)

where we have defined new parameters

$\kappa=\sqrt{\frac{1-\alpha}{1+\alpha}},\qquad\epsilon=\frac{1}{\sqrt{1-\alpha^{2}}}.$
(30)

These would be of interest in what follows. Now, the deformed Hamiltonian (29)
written in terms of the undeformed one is

$\widetilde{\mathcal{H}}^{\alpha}=\mathcal{H}_{\text{free}}+2\alpha\sum_{i}{{J}}_{i}\overline{{J}}_{i}=T+\overline{T}+2\alpha\sum_{i}{{J}}_{i}\overline{{J}}_{i}.$
(31)

where $\mathcal{H}_{\text{free}}$ is just the Hamiltonian of $D$ free massless
relativistic scalars and $T$ and $\overline{T}$ are stress tensors defined
above. Along with this, we also define

$\mathcal{J}=T-\overline{T}.$ (32)

Following Bagchi:2022nvj , we now calculate the algebra of
$\widetilde{\mathcal{H}}^{\alpha}$ and $\mathcal{J}$ using the current algebra
as mentioned in the last section. This leads us to the following algebra:

$\displaystyle\left\\{\mathcal{J}(\sigma),\mathcal{J}\left(\sigma^{\prime}\right)\right\\}_{\mathrm{PB}}=2\mathcal{J}(\sigma)\partial_{\sigma}\delta\left(\sigma-\sigma^{\prime}\right)+\left(\partial_{\sigma}\mathcal{J}(\sigma)\right)\delta\left(\sigma-\sigma^{\prime}\right)$
(33)
$\displaystyle\left\\{\mathcal{J}(\sigma),\widetilde{\mathcal{H}}^{\alpha}\left(\sigma^{\prime}\right)\right\\}_{\mathrm{PB}}=2\widetilde{\mathcal{H}}^{\alpha}(\sigma)\partial_{\sigma}\delta\left(\sigma-\sigma^{\prime}\right)+\left(\partial_{\sigma}\widetilde{\mathcal{H}}^{\alpha}(\sigma)\right)\delta\left(\sigma-\sigma^{\prime}\right)$
$\displaystyle\left\\{\widetilde{\mathcal{H}}^{\alpha}(\sigma),\widetilde{\mathcal{H}}^{\alpha}\left(\sigma^{\prime}\right)\right\\}_{\mathrm{PB}}=\left(1-{\alpha}^{2}\right)\left(2\mathcal{J}(\sigma)\partial_{\sigma}\delta\left(\sigma-\sigma^{\prime}\right)+\left(\partial_{\sigma}\mathcal{J}(\sigma)\right)\delta\left(\sigma-\sigma^{\prime}\right)\right).$

Written in terms of modes

$\mathcal{J}(\sigma)=\sum_{m}L_{m}\,e^{-im\sigma},\qquad\widetilde{\mathcal{H}}^{\alpha}(\sigma)=\sum_{m}M^{\alpha}_{m}\,e^{-im\sigma},$
(34)

we have the algebra

$\displaystyle\\{L_{n},L_{m}\\}_{\text{PB}}$
$\displaystyle=-i(n-m)L_{n+m},\quad\\{L_{n},M^{\alpha}_{m}\\}_{\text{PB}}=-i(n-m)M^{\alpha}_{n+m},$
(35) $\displaystyle\\{M^{\alpha}_{n},M^{\alpha}_{m}\\}_{\text{PB}}$
$\displaystyle=\left(1-{\alpha}^{2}\right)\Big{[}-i(n-m)L_{n+m}\Big{]}.$ (36)

Here $\alpha=0$ is the usual conformal algebra in two dimensions. For
$-1<\alpha<1$, we can always rescale the $\widetilde{\mathcal{H}}^{\alpha}$ by
a constant factor and redefine the above as the usual conformal algebra.

The interesting thing happens at the edge of the parameter space, given by
$\alpha=\pm 1$. The above procedure of rescaling fails at these end points and
the last bracket $\\{\mathcal{H},\mathcal{H}\\}$ vanishes. The spacetime
algebra deforms from the two copies of the Virasoro algebra to (the classical
part of) the BMS3 algebra Bondi:1 ; Sachs:1962zza 555The last condition is
nothing but smooth vanishing of the Dirac-Schwinger condition, as described
earlier. This can also be interpreted as a $SL(2,\mathbb{R})$ transformation
on the Kac-Moody currents that degenerates in certain limits.. The BMS3
algebra is isomorphic to the Conformal Carrollian Algebra in $d=2$ (CCA2) and
hence we see the emergence of Carrollian symmetry at the very end of parameter
space. We will see below that $\alpha=1$ is the Carroll point where the
effective speed of light goes to zero. 666It is imperative to point out here
that the generators of the BMS3 algebra ($L,M$) can be constructed by an
Inönü-Wigner contraction of two copies of undeformed virasoro generators as in
(27):
$L_{n}=\mathcal{L}_{n}-\overline{\mathcal{L}}_{-n},~{}~{}~{}M_{n}=\varepsilon\left(\mathcal{L}_{n}+\overline{\mathcal{L}}_{-n}\right),~{}~{}\varepsilon\to
0,$ (37) which is the consequence of the ultrarelativistic limit (4). Despite
using same notations, the above contraction of $M$ is certainly different than
what (31) implies at the level of modes.

There is also the other end $\alpha=-1$ where the speed goes to infinity and
this is the Galilean point. The emergence of the same algebra both deep in the
UV and deep in the IR of the original relativistic CFT points to a curious
feature of $d=2$ where the Carrollian and Galilean algebras, as well as their
conformal extensions are isomorphic Bagchi:2010zz . We will deal mainly with
the Carrollian very high energy theory in this paper and leave the details of
the Galilean low energy theory for future work.

##### Dialling the speed of light.

We now wish to identify the parameters we introduced to the speed of light in
the system and thereby connect to Carrollian symmetries in a more intuitive
fashion. Since the canonical relations are set by

$\\{\widetilde{\pi}_{i}(\sigma),\phi^{\prime}_{j}(\sigma^{\prime})\\}_{\rm
PB}=\\{\phi^{\prime}_{i}(\sigma),\widetilde{\pi}_{j}(\sigma^{\prime})\\}_{\rm
PB}=\delta_{ij}\partial_{\sigma}\delta(\sigma-\sigma^{\prime})$ (38)

and otherwise vanishes, the time evolution of the system is given by the
following Hamiltonian equations of motion:

$\displaystyle\dot{\phi}_{i}(\sigma)$
$\displaystyle=\\{\phi_{i}(\sigma),\widetilde{H}\\}_{\rm
PB}\qquad\qquad\left(\widetilde{H}=\int
d\sigma^{\prime}\,\widetilde{\mathcal{H}}(\sigma^{\prime})\right)$
$\displaystyle=\int
d\sigma^{\prime}\,\sum_{j=1}^{D}\frac{\widetilde{\pi}_{j}(\sigma^{\prime})}{1+\alpha}\\{\phi_{i}(\sigma),\widetilde{\pi}_{j}(\sigma^{\prime})\\}_{\rm
PB}$ $\displaystyle=\int
d\sigma^{\prime}\,\sum_{j=1}^{D}\frac{\widetilde{\pi}_{j}(\sigma^{\prime})}{1+\alpha}\delta_{ij}\delta(\sigma-\sigma^{\prime})=\frac{\widetilde{\pi}_{i}(\sigma)}{1+\alpha}$
(39)

which is nothing but reproducing the relation (28). In a similar manner,

$\dot{\widetilde{\pi}}_{i}(\sigma)=\\{\widetilde{\pi}_{i}(\sigma),\widetilde{H}\\}_{\rm
PB}=(1-\alpha)\phi^{\prime\prime}_{i}(\sigma).$ (40)

Hence the scalar fields in the deformed theory obey the wave equation with an
“effective speed of light” $\kappa$,

$\ddot{\phi}_{i}-\kappa^{2}{\phi}^{\prime\prime}_{i}=0,$ (41)

where $\kappa$ is as given in (30). Hence the BMS points, where $\alpha=\pm
1$, correspond to speed of wave propagation either going to zero (for
$\alpha=1$) or to infinity (for $\alpha=-1$). At $\kappa=0$, the Hamiltonian
density in (29) Poisson commutes with itself, signifying an emergence of
Carrollian symmetries. 777The reader would argue that this is also true at
$\kappa\to\infty$ and since the Hamiltonian Poisson commutes with itself here
as well, this point also leads to Carrollian and not Galilean symmetries. The
issue is a bit more subtle however. The Carroll and Galilei (conformal)
algebras may be isomorphic, but the identification of generators is non-
trivial and the temporal and spatial directions are flipped in the
identification. The Carroll Hamiltonian is no longer the Galilean Hamiltonian,
but identified the Galilean momentum. Hence what Poisson commutes is the
Galilean momenta and not the Hamiltonian. See Appendix (B) for more details on
this issue. Note that the relative sizes between the momentum and spatial
derivative terms in this case is controlled by the effective parameter
$\epsilon$ (29), which runs off to infinity.

###  3.2 The quantum perspective

Having spent some time trying to put in place the classical aspects of the
effects of the symmetric current-current deformation on free massless scalar
theory, we now turn our attention to the quantum aspects. To begin with, we
will revisit the previous analysis of Bagchi:2022nvj  of rewriting the
deformation in terms of a field rotation of underlying currents. We will then
use this insight to make statements about first the deformation of the vacuum
of the theory and then move on to study the deformation of the spectrum.

####  3.2.1 Deformation as Bogoliubov transformation

An important observation in Bagchi:2022nvj  was that the current-current
deformation in question (31) can be generated by a $SL(2,\mathbb{R})$ rotation
of the $U(1)$ Kac-Moody currents associated to the stress tensor of the
conformal fields. Writing in a matrix form, this transformation at the level
of currents looks like:

$\displaystyle\begin{pmatrix}J^{\mu}(\tilde{\alpha})\\\
\bar{J}^{\mu}(\tilde{\alpha})\end{pmatrix}=\begin{pmatrix}\cosh\theta(\tilde{\alpha})\,\,&\,\,\sinh\theta(\tilde{\alpha})\\\
\sinh\theta(\tilde{\alpha})\,\,&\,\,\cosh\theta(\tilde{\alpha})\end{pmatrix}\begin{pmatrix}J^{\mu}(0)\\\
\bar{J}^{\mu}(0)\end{pmatrix}$ (42)

Here $\tilde{\alpha}$ is a boost parameter in the two-dimensional space
spanned by (anti-)holomorphic coordinates $z,\bar{z}$. Using the properties of
2d boosts in this space, yields the simple solution

$\cosh\theta(\tilde{\alpha})=\frac{1}{\sqrt{1-\tilde{\alpha}^{2}}},~{}~{}\sinh\theta(\tilde{\alpha})=\frac{\tilde{\alpha}}{\sqrt{1-\tilde{\alpha}^{2}}}.$
(43)

With this transformation between deformed and undeformed bases, we can now
define the combinations of current bilinears:

$\displaystyle\mathcal{J}(\tilde{\alpha};\sigma)$ $\displaystyle=$
$\displaystyle\sum_{\mu}\,\big{(}J^{\mu}(\tilde{\alpha})J^{\mu}(\tilde{\alpha})-\bar{J}^{\mu}(\tilde{\alpha})\bar{J}^{\mu}(\tilde{\alpha})\big{)}=T(\sigma)-\overline{T}(\sigma)$
(44a) $\displaystyle\mathcal{H}(\tilde{\alpha};\sigma)$ $\displaystyle=$
$\displaystyle\frac{1}{(\cosh^{2}\theta(\tilde{\alpha})+\sinh^{2}\theta(\tilde{\alpha}))}\sum_{\mu}\,\big{(}J^{\mu}(\tilde{\alpha})J^{\mu}(\tilde{\alpha})+\bar{J}^{\mu}(\tilde{\alpha})\bar{J}^{\mu}(\tilde{\alpha})\big{)}$
(44b) $\displaystyle=$ $\displaystyle
T(\sigma)+\overline{T}(\sigma)+2\alpha\sum_{\mu}J^{\mu}(0)\bar{J}^{\mu}(0),$

where in the very last line we have re-defined the parameter as888Note that
this redefinition makes sure that $\tilde{\alpha}=\pm 1\implies\alpha=\pm 1$,
consequently $\tanh\theta=\pm 1$.

$\alpha=\frac{2\cosh\theta\sinh\theta}{\cosh^{2}\theta+\sinh^{2}\theta}.$ (45)

This is in sync with our discussions earlier around (31), this $\mathcal{H}$
turns out to be our deformed Hamiltonian as expected. We can see with the new
parameterisation with $\alpha$, the stress tensor combinations remain the
explicit CFT ones at $\theta=0~{}(\alpha=0)$ and flow over to those of
Carrollian algebra at $\theta\to\infty~{}(\alpha\to 1)$. In what follows, we
will continue to mark all our deformed objects with an $\alpha$ label, by
dropping the tilde notations, to distinguish them from the undeformed ones.

Now let us define Fourier modes associated to the $U(1)$ currents $J,\bar{J}$
as following:

${}J=\sum_{n}j_{n}e^{-in\sigma},~{}~{}\bar{J}=\sum_{n}\bar{j}_{n}e^{+in\sigma}.$
(46)

Note that we are working with level one currents $k=\bar{k}=1$. When we
quantize, the algebra of currents lead one to the following commutators for
modes,

${}[j_{n},j_{m}]=n\delta_{m+n,0},~{}~{}[\bar{j}_{n},\bar{j}_{m}]=n\delta_{m+n,0}.$
(47)

Consequently, (42) can be identified as a Bogoliubov transformation between
deformed and undeformed modes:

$\displaystyle
j_{n}(\alpha)=\cosh\theta~{}j_{n}(0)+\sinh\theta~{}\bar{j}_{-n}(0);\quad\bar{j}_{n}(\alpha)=\cosh\theta~{}\bar{j}_{n}(0)+\sinh\theta~{}j_{-n}(0).$
(48)

The above relations keeps the canonical commutation relations invariant. We
can then follow the usual Sugawara method to construct stress tensors out of
bilinears of modes, and that in turn gives rise to the stress tensor modes
from (44):

$\displaystyle\mathcal{J}_{n}(\alpha;\sigma)$ $\displaystyle=$
$\displaystyle\sum_{a,m}\left(j_{-m}^{a}j_{m+n}^{a}-\overline{j}_{-m}^{a}\overline{j}_{m-n}^{a}\right),$
(49a) $\displaystyle\mathcal{H}_{n}(\alpha;\sigma)$ $\displaystyle=$
$\displaystyle\sum_{m}\left(j_{-m}^{a}j_{m+n}^{a}+\overline{j}_{-m}^{a}\overline{j}_{m-n}^{a}+2\alpha
j_{-m}^{a}\overline{j}_{-m-n}^{a}\right),$ (49b)

where the current modes are the undeformed ones
$j_{n}^{a}=j_{n}^{a}(\alpha=0)$. The algebra of the modes again explicitly
leads to the BMS3 algebra when $\alpha=1$, as we have discussed in the
classical case. Note that this upped bound on $\alpha$ means the Bogoliubov
coefficients diverge in these limits. However, since these modes are related
via continuous Bogoliubov transformations, the spectrum also should smoothly
interpolate to the BMS spectrum. 999This structure has been particularly well
explored within the recent tensionless string literature, see for example
Bagchi:2015nca ; Bagchi:2019cay ; Bagchi:2020fpr ; Bagchi:2020ats . In
Bagchi:2021ban  it was shown that this oscillator structures help one
understand these theories as those on infinitely accelerated string
worldsheets, most noticeably near a black hole horizon.

####  3.2.2 Deformation of the vacuum

Let us now concentrate on the deformed quantum theory. The highest weight
vacuum of the undeformed field theory, as given by

$j_{n}(0)|0\rangle_{0}=0=\bar{j}_{n}(0)|0\rangle_{0}$ (50)

has now changed due to the Bogoliubov transformed oscillators, and the
deformed vacuum condition reads:

$\displaystyle\left(j_{n}(0)+\tanh\theta~{}\bar{j}_{-n}(0)\right)|0\rangle_{\alpha}=0;\quad\left(\bar{j}_{n}(0)+\tanh\theta~{}j_{-n}(0)\right)|0\rangle_{\alpha}=0.$
(51)

This means one can write the deformed vacuum as a two-mode squeezed state on
the undeformed vacuum:

$|0\rangle_{\alpha}={\mathcal{N}}\prod_{n=1}^{\infty}\exp\left[\,\frac{1}{n}\tanh\theta~{}j_{-n}\cdot\overline{j}_{-n}\right]|0\rangle_{0}$
(52)

In the above, ${\mathcal{N}}$ is a suitable normalisation factor. At the
critical points $\tanh\theta\to\pm 1$ the above state becomes a boundary
state, either Dirichlet or Neumann based on the sign of the parameter.
101010This class of smoothly interpolating vacua has been considered in the
tensionless string literature Bagchi:2020fpr .

####  3.2.3 Deformed central charges

We go forward and elaborate on the spectrum of the deformed theory now. Note
that the Bogoliubov transformation of the oscillators lead to a set of new
Virasoro generators where the undeformed holomorphic and anti-holomorphic ones
mix together:

$\displaystyle\mathcal{L}_{n}(\alpha)=\cosh^{2}{\theta}\mathcal{L}_{n}(0)+\sinh^{2}{\theta}\bar{\mathcal{L}}_{-n}(0)+\frac{1}{2}\cosh{\theta}\sinh{\theta}\sum_{p}(j_{n-p}\tilde{j}_{-p}+j_{p}\tilde{j}_{-n+p}),$
$\displaystyle\bar{\mathcal{L}}_{n}(\alpha)=\cosh^{2}{\theta}\bar{\mathcal{L}}_{n}(0)+\sinh^{2}{\theta}{\mathcal{L}}_{-n}(0)+\frac{1}{2}\cosh{\theta}\sinh{\theta}\sum_{p}(\tilde{j}_{n-p}{j}_{-p}+\tilde{j}_{p}{j}_{-n+p}),$

where we have used the standard form of bilinears of oscillators coming from
the Sugawara construction and performed a basis transformation. Clearly, the
interpolating symmetry generator modes of the stress tensors as defined in
(44) can be written as combination of these deformed generators,

$\displaystyle\mathcal{J}_{n}(\alpha)=\mathcal{L}_{n}(\alpha)-\bar{\mathcal{L}}_{-n}(\alpha),\quad\mathcal{H}_{n}(\alpha)=\frac{1}{(\cosh^{2}\theta+\sinh^{2}\theta)}\bigg{(}\mathcal{L}_{n}(\alpha)+\bar{\mathcal{L}}_{-n}(\alpha)\bigg{)},$
(53)

so that we have
$\mathcal{J}(\alpha;\sigma)=\sum_{n}\mathcal{J}_{n}(\alpha)e^{-in\sigma}$ etc.
These modes follow the centrally extended interpolating algebra

$\displaystyle[\mathcal{J}_{n}(\alpha),\mathcal{J}_{m}(\alpha)]=(n-m)\mathcal{J}_{n+m}(\alpha)+\frac{c_{L}}{12}\,n(n^{2}-1)\delta_{n+m,0}$
(54a)
$\displaystyle[\mathcal{J}_{n}(\alpha),\mathcal{H}_{m}(\alpha)]=(n-m)\mathcal{H}_{n+m}(\alpha)+\frac{c_{M}}{12}\,n(n^{2}-1)\delta_{n+m,0}$
(54b)
$\displaystyle[\mathcal{H}_{n}(\alpha),\mathcal{H}_{m}(\alpha)]=\frac{1}{(\cosh^{2}{\theta}+\sinh^{2}{\theta})^{2}}\big{[}(n-m)\mathcal{J}_{n+m}(\alpha)+\frac{c_{L}}{12}\,n(n^{2}-1)\delta_{n+m}\big{]}.$
(54c)

Comparing these to a pair of undeformed Virasoro algebras with central charges
$c,\bar{c}$, we can easily read off the deformed central terms:

$c_{L}=c-\bar{c}\,,\qquad
c_{M}=(\cosh^{2}{\theta}+\sinh^{2}{\theta})^{-1}(c+\bar{c}).$ (55)

Observe that what we have got here is an effective contraction of the central
extensions as $\theta\to\infty$. Near this extreme value, the scaling factor
$(\cosh^{2}{\theta}+\sinh^{2}{\theta})^{-1}$ gradually goes to zero, making
sure that we have the centrally extended BMS3 algebra.

####  3.2.4 Deformed spectrum

To calculate the deformed spectrum, we consider the scalars compactified over
$D$ circles of radius $R_{i}$. For the undeformed case, the periodic boundary
condition was applied on the fields

$\phi_{i}(\tau,\sigma+1)\equiv\phi_{i}(\tau,\sigma)+m_{i}R_{i},$ (56)

where $m_{i}$ are the winding numbers and we have rescaled the periodicity for
simplicity. This, in conjunction with the relevant equations of motion, leads
to the following mode expansion of the fields

$\phi_{i}(\tau,\sigma)=\phi_{i}^{(0)}+\frac{n_{i}}{R_{i}}\tau+m_{i}R_{i}\sigma+\frac{i}{\sqrt{2}}\sum_{n\neq
0}\frac{1}{n}\left\\{j^{i}_{n}e^{-2\pi{in(\tau+\sigma)}}+\bar{j}^{i}_{n}e^{-2\pi{in(\tau-\sigma)}}\right\\}.$
(57)

The Kaluza-Klein (KK) momentum $(n_{i})$ and winding numbers are related to
zero modes of the undeformed field as

$\displaystyle m^{i}=j^{i}_{0}-\bar{j}^{i}_{0}\,,\quad
n^{i}=j^{i}_{0}+\bar{j}^{i}_{0}.$ (58)

Since we still have oscillatory modes in the deformed theory, as made clear by
the equations of motion, one can also impose similar periodic boundary
conditions in the basis:

$\phi_{i}(\tau,\sigma+1)\equiv\phi_{i}(\tau,\sigma)+\bar{m}_{i}\bar{R}_{i},$
(59)

with corresponding new winding numbers $\bar{m}_{i}$ (and KK momenta
$\bar{n}_{i}$) defined on circles of radii $\bar{R}_{i}$. Reminding ourselves
that in the deformed theory the scalar field still follows a wave equation
(41) with rescaled speed of light, this leads to the following mode expansion
of the fields in terms of oscillators $j(\alpha)$:

$\phi_{i}(\tau,\sigma;\alpha)=\phi_{i}^{(0)}+\frac{\bar{n}_{i}}{\bar{R}_{i}}\tau+\bar{m}_{i}\bar{R}_{i}\sigma+\frac{i}{\sqrt{2}}\sum_{n\neq
0}\frac{1}{n}\left\\{j(\alpha)^{i}_{n}e^{-2\pi{in(\kappa\tau+\sigma)}}+\bar{j}(\alpha)^{i}_{n}e^{-2\pi{in(\kappa\tau-\sigma)}}\right\\}.$
(60)

Note that when $\kappa\to 0$, the oscillatory nature of the solution gets
disturbed and it leads to an expansion linear in $\tau$ when the limit is
taken carefully 111111See Bagchi:2015nca  for a detailed discussion on this
limit on the mode expansion in the context of null strings. One could find the
near-BMS version of our Bogoliubov transformations by taking $\kappa\to 0$
limit on these mode expansions and redefining oscillators..

We now focus on the flow of spectrum. Away from the end point of $\kappa\to
0$, the underlying symmetry algebra is unaltered. Similar to the undeformed
case, the winding and KK momenta can still be related to the zero modes of
(60):

$\displaystyle\bar{m}^{i}=j(\alpha)^{i}_{0}-\bar{j}(\alpha)^{i}_{0}\,,\quad\bar{n}^{i}=j(\alpha)^{i}_{0}+\bar{j}(\alpha)^{i}_{0}.$
(61)

Using the explicit Bogoliubov transformations between the two sets of
oscillatory modes as defined in (48), the relations between two sets of
quantities can be expressed as:

$\displaystyle\bar{m}^{i}=(\cosh{\theta}-\sinh{\theta})m^{i}=\kappa
m^{i},\quad\bar{n}^{i}=(\cosh{\theta}+\sinh{\theta})n^{i}=\frac{1}{\kappa}n^{i},$
(62)

with $\kappa$ being the effective speed of light. We also assume a highest
weight representation for the CFT vacuum to start with, so that for a generic
state in the undeformed theory is labeled by weights $h,\bar{h}$:

$\mathcal{L}_{0}|h,\bar{h}\rangle=h|h,\bar{h}\rangle,~{}~{}\mathcal{\bar{L}}_{0}|h,\bar{h}\rangle=\bar{h}|h,\bar{h}\rangle.$
(63)

From the deformed mode expansion of the fields (60) one can easily find the
zero modes of the Virasoro generators:

$\displaystyle\mathcal{L}_{0}(\alpha)$ $\displaystyle=$
$\displaystyle\frac{1}{4}\Big{(}\frac{\bar{n}_{i}}{\bar{R}_{i}}+\bar{m}_{i}\bar{R}_{i}\Big{)}^{2}+\frac{1}{2}\sum_{n\neq
0}j(\alpha)^{i}_{-n}~{}j(\alpha)^{i}_{n}$ (64a)
$\displaystyle\bar{\mathcal{L}}_{0}(\alpha)$ $\displaystyle=$
$\displaystyle\frac{1}{4}\Big{(}\frac{\bar{n}_{i}}{\bar{R}_{i}}-\bar{m}_{i}\bar{R}_{i}\Big{)}^{2}+\frac{1}{2}\sum_{n\neq
0}\bar{j}{(\alpha)}^{i}_{-n}~{}\bar{j}{(\alpha)}^{i}_{n}.$ (64b)

Acting on a deformed state, the conformal dimension is given by the deformed
winding and KK momenta:

$\displaystyle
h(\alpha)=\frac{1}{4}\Big{(}\frac{\bar{n}_{i}}{\bar{R}_{i}}+\bar{m}_{i}\bar{R}_{i}\Big{)}^{2},~{}~{}~{}\bar{h}(\alpha)=\frac{1}{4}\Big{(}\frac{\bar{n}_{i}}{\bar{R}_{i}}-\bar{m}_{i}\bar{R}_{i}\Big{)}^{2}$
(65a)

Given the above, one can check the combinations

$\displaystyle
h(\alpha)-\bar{h}(\alpha)=\bar{n_{i}}\bar{m_{i}}=n_{i}m_{i}=h-\bar{h}$ (66)
$\displaystyle
h(\alpha)+\bar{h}(\alpha)=\frac{1}{2}\bigg{(}\left(\frac{\bar{n}_{i}}{\bar{R}_{i}}\right)^{2}+(\bar{m_{i}}\bar{R}_{i})^{2}\bigg{)}=\frac{1}{2}\bigg{(}\frac{1}{\kappa^{2}}\left(\frac{n_{i}}{\bar{R}_{i}}\right)^{2}+\kappa^{2}(m_{i}\bar{R}_{i})^{2}\bigg{)}.$
(67)

Clearly, the spectrum of the theory will be dominated by the KK momenta in the
Carroll limit $\kappa\to 0$. 121212This is again in keeping with results from
compactified tensionless string literature Bagchi:2022iqb ; Banerjee:2023ekd ,
where it was already found that there is no winding number contribution to the
mass formula. Moreover, one could note that the factors of speed of light in
the mass spectrum can be re-absorbed by rescaling the radii of
compactification $\bar{R}_{i}\to\kappa\bar{R}_{i}$ when $\kappa$ is finite.
For $\kappa\to 0$, this implies again, a de facto contraction of the
$(h+\bar{h})$ combination.

The take home message from this section is that the diagonal $J\bar{J}$-like
deformations dial the speed of light of a field theory, thereby smoothly
interpolating between the relativistic theory to the Carrollian theory at the
end of the flow. The presence of such deformations implies a mixing of modes
in the form of Bogoliubov transformations, and hence the physical vacuum and
the spectrum continuously evolve in the space of couplings as the theory
approaches the BMS point. This can also be interpreted as a flow in the space
of $SL(2,\mathbb{R})$ unitaries, with the BMS point attributed to a singular
point in this parameter space Banerjee:2022ime . In the next section, we focus
on a non-diagonal current-current deformation, which will also induce a
spectral flow in the space of couplings, however there would be no continuous
change in the physical vacuum and the appearance of BMS will be abrupt.
Intriguingly, as we will see, that scenario can also be justified as a
singular limit in $SL(2,\mathbb{R})$ transformation space.

##  4 The antisymmetric deformation

In this section, we turn our attention to the free relativistic scalar theory
together with antisymmetric deformations of form
$\epsilon_{ab}J^{a}\bar{J}^{b}$ that are exactly marginal. A salient feature
of such deformations is these mix spacetime indices on the fields, as opposed
to the symmetric $J\bar{J}$ case we discussed previously where spacetime
indices just decouple from each other. We will see that this feature leads to
interesting modifications at the level of boundary conditions.

###  4.1 The classical theory

Instead of any number of scalar fields like we considered previously, we now
have to consider an even number of them. We start with the simplest case, viz.
the two dimensional Lagrangian density of two bosonic fields $\phi_{1,2}$
deformed by a $J^{1}\wedge\bar{J}^{2}$ operator. Here the currents
$J^{i}=\partial_{z}\phi^{i}$ and $\overline{J}^{i}=\partial_{\bar{z}}\phi^{i}$
are the holomorphic and antiholomorphic $U(1)$ currents of the theory.
131313Addition of this operator can be equivalently seen as turning on a
constant B-field (see Appendix C) on the worldsheet of a string moving on a
flat Euclidean 2-dimensional target space. Our notations are in sync with
those in Guica:2021fkv , where the deformation was defined recursively and the
full summed action is:

$\mathcal{L}^{(g)}=\frac{1}{1+g^{2}}\left[\frac{1}{2}\sum_{i=1,2}\left(\left(\partial_{\tau}\phi_{i}\right)^{2}-\left(\partial_{\sigma}\phi_{i}\right)^{2}\right)-g\left(\partial_{\tau}\phi_{1}\partial_{\sigma}\phi_{2}-\partial_{\sigma}\phi_{1}\partial_{\tau}\phi_{2}\right)\right].$
(68)

Starting from the deformed Lagrangian above, we now find momentum densities
corresponding to the two fields:

$\Pi_{1}=\frac{1}{1+g^{2}}\left({\dot{\phi}_{1}}-g\phi_{2}^{\prime}\right),\qquad\Pi_{2}=\frac{1}{1+g^{2}}\left({\dot{\phi}_{2}}+g\phi_{1}^{\prime}\right).$
(69)

Note that $g$ here is a coupling constant which is not necessarily small, and
in fact it will be some upper limit on the value of this coupling that we will
be interested in, mirroring our computations in the preceding section. The
Hamiltonian density is then obtained in the standard way

$\displaystyle\mathcal{H}^{(g)}=\sum_{i=1,2}\Pi_{i}\dot{\phi}_{i}-\mathcal{L}^{(g)}=\frac{1}{2}\left(\left(1+g^{2}\right)\left(\Pi_{1}^{2}+\Pi_{2}^{2}\right)+{\phi_{1}^{\prime}}^{2}+{\phi_{2}^{\prime}}^{2}\right)+g\left(\Pi_{1}\phi^{\prime}_{2}-\Pi_{2}\phi^{\prime}_{1}\right).$
(70)

This density can be represented in a matrix form141414To be precise, the off
diagonal component is not unique, but here is chosen by a requirement for the
matrix to be symmetric.

$\mathcal{H}^{(g)}=\,\begin{pmatrix}\Pi_{1}\\\ \Pi_{2}\\\ \phi_{1}^{\prime}\\\
\phi_{2}^{\prime}\end{pmatrix}^{\rm T}M^{(g)}\begin{pmatrix}\Pi_{1}\\\
\Pi_{2}\\\ \phi_{1}^{\prime}\\\ \phi_{2}^{\prime}\end{pmatrix}\
,\qquad\text{where}\qquad
M^{(g)}=\begin{pmatrix}\frac{1+g^{2}}{2}&0&0&\frac{g}{2}\\\
0&\frac{1+g^{2}}{2}&-\frac{g}{2}&0\\\ 0&-\frac{g}{2}&\frac{1}{2}&0\\\
\frac{g}{2}&0&0&\frac{1}{2}\end{pmatrix}\ .$ (71)

The matrix $M^{(g)}$ has two sets of eigenvalues:

$\lambda^{(g)}_{\pm}=\frac{2+g^{2}\pm
g\sqrt{g^{2}+4}}{4}=\frac{\gamma_{\pm}^{2}}{2}\
,\qquad\text{where}\qquad\gamma_{\pm}=\frac{\sqrt{g^{2}+4}\pm
g}{2}=\frac{1}{\gamma_{\mp}}\ .$ (72)

Now one similarly calculates the corresponding orthonormal eigenvectors:

$u^{+}_{1}=c^{+}\begin{pmatrix}1\\\ 0\\\ 0\\\ \gamma_{-}\end{pmatrix}\quad
u^{+}_{2}=c^{+}\begin{pmatrix}0\\\ 1\\\ -\gamma_{-}\\\ 0\end{pmatrix}\quad
u^{-}_{1}=c^{-}\begin{pmatrix}1\\\ 0\\\ 0\\\ -\gamma_{+}\end{pmatrix}\quad
u^{-}_{2}=c^{-}\begin{pmatrix}0\\\ 1\\\ \gamma_{+}\\\ 0\end{pmatrix}$ (73)

where we have the parameters

${}c^{\pm}=\frac{1}{\sqrt{1+2\lambda_{\mp}}},$ (74)

and thus $M^{(g)}$ is diagonalised as

$\Lambda={\rm
diag}(\lambda^{(g)}_{+},\lambda^{(g)}_{+},\lambda^{(g)}_{-},\lambda^{(g)}_{-})=S^{\rm
T}M^{(g)}S$ (75)

by an element in SO(4)

$S=\begin{pmatrix}u^{+}_{1}&u^{+}_{2}&u^{-}_{1}&u^{-}_{2}\end{pmatrix}.$ (76)

Then the Hamiltonian density can be represented in a simple bilinear form in
terms of redefined field variables $\Xi$:

$\mathcal{H}^{(g)}=\,\Xi^{\rm
T}\Lambda\Xi=\,\lambda^{(g)}_{+}\left\\{\Pi_{+}^{2}+(\chi^{\prime}_{-})^{2}\right\\}+\,\lambda^{(g)}_{-}\left\\{\Pi_{-}^{2}+(\chi^{\prime}_{+})^{2}\right\\}$
(77)

where the set of fields $\Xi$ is defined by

$\Xi=\begin{pmatrix}\Pi_{+}\\\ \chi^{\prime}_{-}\\\ -\Pi_{-}\\\
\chi^{\prime}_{+}\end{pmatrix}:=S^{\rm T}\begin{pmatrix}\Pi_{1}\\\ \Pi_{2}\\\
\phi_{1}^{\prime}\\\
\phi_{2}^{\prime}\end{pmatrix}=\begin{pmatrix}c^{+}(\Pi_{1}+\gamma_{-}\phi_{2}^{\prime})\\\
c^{+}(\Pi_{2}-\gamma_{-}\phi_{1}^{\prime})\\\
c^{-}(\Pi_{1}-\gamma_{+}\phi_{2}^{\prime})\\\
c^{-}(\Pi_{2}+\gamma_{+}\phi_{1}^{\prime})\end{pmatrix}$ (78)

Note that at $g=0$, the above redefined fields do not reduce to the original
variables, but rather to:

$\left.\Xi\right|_{g=0}=\begin{pmatrix}\frac{1}{\sqrt{2}}\left(\dot{\phi}_{1}+\phi_{2}^{\prime}\right)\\\
\frac{1}{\sqrt{2}}\left(\dot{\phi}_{2}-\phi_{1}^{\prime}\right)\\\
\frac{1}{\sqrt{2}}\left(\dot{\phi}_{1}-\phi_{2}^{\prime}\right)\\\
\frac{1}{\sqrt{2}}\left(\dot{\phi}_{2}+\phi_{1}^{\prime}\right)\end{pmatrix}.$
(79)

This field re-definition turns out to be very useful. As we will see below,
these are precisely the combinations that will allow us to track the flow of
the theory as the deformation is dialled and will be crucial to see how even
at the end point of the flow, the formalism does not break down but there is
the emergence of a different spacetime algebra.

Some comments are needed for consistency of this field re-definition. Note
that the transformation mixes canonical variables for different fields. This
is expected in a theory with an antisymmetric field contribution. This
redefinition re-diagonalises the deformed Hamiltonian density. However it is
not clear at the outset whether this is a symplectomorphism for the dynamical
variables as well. We discuss this point at length below.

####  4.1.1 Dynamics of new conjugate variables

We now elaborate on the structure of redefined field variables (79) introduced
above. The dynamics of the deformed Hamiltonian system is still governed by
the canonical relation

$\\{\phi_{i}({\sigma}),\Pi_{j}(\sigma^{\prime})\\}_{\rm
PB}=-\\{\Pi_{i}(\sigma),\phi_{j}(\sigma^{\prime})\\}_{\rm
PB}=\delta_{ij}\delta(\sigma-\sigma^{\prime})$ (80)

as well as the time evolution equation

$\dot{\mathcal{O}}=\\{\mathcal{O},H\\}_{\rm PB}\qquad\quad H=\int
d\sigma\,\mathcal{H}^{(g)}$ (81)

for any physical observable $\mathcal{O}$. We may first examine the equations
of motion for fields $\phi_{i}$ as a consistency check for (80), and then,
figure out conjugate pairs among components of the field variables in $\Xi$.
The time evolutions of variables ${\phi}_{i}$ are given by

$\displaystyle\dot{\phi}_{1}$ $\displaystyle=\\{\phi_{1},H\\}_{\rm
PB}=(1+g^{2})\Pi_{1}+g\phi^{\prime}_{2},$ (82) $\displaystyle\dot{\phi}_{2}$
$\displaystyle=\\{\phi_{2},H\\}_{\rm PB}=(1+g^{2})\Pi_{2}-g\phi^{\prime}_{1}.$

These reproduce the relations of momentum densities with the fields (69), as
expected. Further, the time evolutions of momentum densities are

$\dot{\Pi}_{1}=\\{\Pi_{1},H\\}_{\rm
PB}=\phi^{\prime\prime}_{1}-g\Pi^{\prime}_{2},\quad\dot{\Pi}_{2}=\\{\Pi_{1},H\\}_{\rm
PB}=\phi^{\prime\prime}_{2}+g\Pi^{\prime}_{1}.$ (83)

Combining the above sets of equations together gives rise to the equations of
motion for fields $\phi_{i}$, which simply turn out to be wave equations:

$\ddot{\phi}_{i}-\phi^{\prime\prime}_{i}=0.$ (84)

Note here that unlike the symmetric deformation, the antisymmetric one does
not gradually change the speed of light in the deformed theory. This is clear
in hindsight since the deformation term can be regarded as a boundary term,
and hence should not change the equations of motion.

Next, we specify conjugate pairs among the new fields, i.e. components of
$\Xi$. In order to do that, we work out the Poisson brackets for them using
definitions as in (78). This leads to,

$\\{\Pi_{+}(\sigma),\Pi_{-}(\sigma^{\prime})\\}_{\rm
PB}=\\{\chi^{\prime}_{+}(\sigma),\chi^{\prime}_{-}(\sigma^{\prime})\\}_{\rm
PB}=0,$ (85)

while

$\displaystyle\\{\Pi_{+}(\sigma),\chi^{\prime}_{-}(\sigma^{\prime})\\}_{\rm
PB}$
$\displaystyle=\left(c^{+}\right)^{2}\gamma_{-}\left(\partial_{\sigma}+\partial_{\sigma^{\prime}}\right)\delta(\sigma-\sigma^{\prime})=0,$
(86)
$\displaystyle\\{\Pi_{+}(\sigma),\chi^{\prime}_{+}(\sigma^{\prime})\\}_{\rm
PB}$
$\displaystyle=c^{+}c^{-}\left(\gamma_{+}+\gamma_{-}\right)\partial_{\sigma}\delta(\sigma-\sigma^{\prime})=\partial_{\sigma}\delta(\sigma-\sigma^{\prime}).$
(87)

Thus, integrating over $\sigma$, we conclude that $\chi_{+}$ and $\Pi_{+}$ are
indeed a pair of conjugate variables that satisfies

$\\{\chi_{+}(\sigma),\Pi_{+}(\sigma^{\prime})\\}_{\rm
PB}=\delta(\sigma-\sigma^{\prime})$ (88)

It is worth emphasising that although we considered $\chi_{+}$ to be the
dynamical variable and $\Pi_{+}$ as its conjugate, this is a choice. Another
viable choice is $\tilde{\chi}_{+}=\Pi_{+}$ as the dynamical variable and
$\tilde{\Pi}_{+}=-\chi_{+}$ as its conjugate. This arbitrariness comes from a
freedom of choice of Lagrangian submanifold on a symplectic manifold. For
completeness, we also see

$\displaystyle\\{\chi^{\prime}_{-}(\sigma),\Pi_{-}(\sigma^{\prime})\\}_{\rm
PB}$ $\displaystyle=\partial_{\sigma}\delta(\sigma-\sigma^{\prime})$ (89)
$\displaystyle\\{\Pi_{-}(\sigma),\chi^{\prime}_{+}(\sigma^{\prime})\\}_{\rm
PB}$ $\displaystyle=0$ (90)

which shows that $\chi_{-}$ and $\Pi_{-}$ will indeed be another pair of
conjugate variables:

$\\{\chi_{-}(\sigma),\Pi_{-}(\sigma^{\prime})\\}_{\rm
PB}=\delta(\sigma-\sigma^{\prime}).$ (91)

Furthermore, since the resulting system is still Hamiltonian, we write the
equations of motion:

$\dot{\chi}_{+}=\frac{\partial\mathcal{H}^{(g)}}{\partial\Pi_{+}}=2\,\lambda^{(g)}_{+}\Pi_{+}\qquad\quad\dot{\chi}_{-}=\frac{\partial\mathcal{H}^{(g)}}{\partial\Pi_{-}}=2\,\lambda^{(g)}_{-}\Pi_{-}$
(92)

which are consistent with the Poisson brackets in (88) and (91), as well as
the time flow equation (81), i.e., we also consistently find
$\dot{\chi}_{\pm}=\\{{\chi}_{\pm},H\\}_{\rm
PB}=2\,\lambda^{(g)}_{\pm}\Pi_{\pm}$. All of this leads to:

$\displaystyle\dot{\Pi}_{\pm}$
$\displaystyle=\left\\{{\Pi}_{\pm},H\right\\}_{\rm
PB}=2\,\lambda^{(g)}_{\mp}\chi^{\prime\prime}_{\pm}$ (93)

and thus we find that the equation of motion of $\chi_{\pm}$ is the same wave
equation as that of $\phi_{i}\,(i=1,2)$, whatever be the value of the coupling
$g$:

$\displaystyle\ddot{\chi}_{+}$
$\displaystyle\overset{\eqref{time.evol}}{=}\left\\{\dot{\chi}_{+},H\right\\}_{\rm
PB}\overset{\eqref{time.evol.chipm}}{=}2\,\lambda^{(g)}_{+}\left\\{\Pi_{+},H\right\\}_{\rm
PB}$
$\displaystyle\overset{!}{=}2\,\lambda^{(g)}_{+}\dot{\Pi}_{+}\overset{\eqref{time.evol.Pipm}}{=}4\,\lambda^{(g)}_{+}\lambda^{(g)}_{-}\chi^{\prime\prime}_{+}=\chi^{\prime\prime}_{+}$
(94)

where at $!$ we see that it is the relation obtained by simply taking a
differential of (92) with respect to $\tau$, providing a check for self-
consistency of our system. The resultant equations of motion are expected,
because in any case $\Pi_{\pm}$ and $\chi^{\prime}_{\pm}$ are linear
combinations of
$\\{\dot{\phi}_{1},\dot{\phi}_{2},{\phi}^{\prime}_{1},{\phi}^{\prime}_{2}\\}$.

Before going further, we may convince the reader of the consistency of the
relations in (92) with the definition (78). For example, take the fourth
component in the column matrix $\Xi$

$\chi^{\prime}_{+}=c^{-}(\Pi_{2}+\gamma_{+}\phi_{1}^{\prime}),$ (95)

which implies

$\displaystyle\chi_{+}(\sigma)$
$\displaystyle=c^{-}\int^{\sigma}_{\sigma_{0}}d\sigma^{\prime}(\Pi_{2}+\gamma_{+}\phi_{1}^{\prime})(\sigma^{\prime})$
$\displaystyle=c^{-}\frac{g+(1+g^{2})\gamma_{+}}{1+g^{2}}\phi_{1}(\sigma)+\frac{c^{-}}{(1+g^{2})}\int^{\sigma}_{\sigma_{0}}d\sigma^{\prime}\,\dot{\phi}_{2}(\sigma^{\prime})$
(96)

which shows $\chi_{+}$ is almost a linear combination of $\phi_{1}$ and
$\phi_{2}$ but involves a integro-differential system. Applying to (96) a
differential with respect to $\tau$, we find

$\displaystyle\dot{\chi}_{+}=c^{-}\frac{g+(1+g^{2})\gamma_{+}}{1+g^{2}}\dot{\phi}_{1}+\frac{c^{-}}{1+g^{2}}{\phi}^{\prime}_{2}$
(97)

while, by definition of $\Pi_{+}$ in (78)

$\displaystyle 2\,\lambda^{(g)}_{+}\Pi_{+}$
$\displaystyle=2\,\lambda^{(g)}_{+}c^{+}(\Pi_{1}+\gamma_{-}\phi_{2}^{\prime})$
$\displaystyle=\frac{2\lambda^{(g)}_{+}c^{+}}{1+g^{2}}\dot{\phi}_{1}+2\,\lambda^{(g)}_{+}c^{+}\frac{-g+(1+g^{2})\gamma_{-}}{1+g^{2}}{\phi}^{\prime}_{2}$
(98)

After some manipulation, we see that each of the coefficients in front of
$\dot{\phi}_{1}$ and ${\phi}^{\prime}_{2}$, respectively, in equations (97)
and (98) are identical, thus ensuring (92). With this we can conclusively
claim that our diagonalized system as described by the $\pm$ field variables
are completely self consistent as a Hamiltonian system.

Now why did we need to painstakingly make sure of using the right dynamical
variables? We then ask the reader to pay attention to our Hamiltonian
$\mathcal{H}^{(g)}$ in the (77). When we crank up the coupling $g\to\infty$,
it is clear that $\lambda_{+}^{(g)}\to\infty$, while $\lambda_{-}^{(g)}\to 0$.
Since we have established $\pm$ variables as canonical ones, one can easily
see

$\mathcal{H}^{(g\to\infty)}\sim\left\\{\Pi_{+}^{2}+(\chi^{\prime}_{-})^{2}\right\\}$
(99)

which clearly commutes with itself at two different spatial points. This is,
in fact, Carroll dynamics emerging again in this system 151515One may be
tempted to compare this with the case of Non-relativistic string theory
Gomis:2000bd  where the B-field in the sigma model becomes critical.
161616Note that using our earlier wisdom, we seem to be unable to quantify
this as either electric or magnetic limit, since two different kind of
dynamical variables survive. However this is not an issue as we have mentioned
earlier choosing $\tilde{\chi}_{+}^{\prime}=\Pi_{+}$ and
$\tilde{\Pi}_{+}=-\chi_{+}^{\prime}$ as conjugate variables in (77) is allowed
by the algebra. So one could interpolate between a purely electric limit with
only momenta surviving to purely magnetic limit with only spatial gradients
surviving, and in all cases the symmetry algebra stays the same. One could
also demand $g\to-\infty$, which in turn makes $\lambda_{-}^{(g)}\to\infty$,
while $\lambda_{+}^{(g)}\to 0$, as opposite to the previous case. This could
be thought of as a “dual” limit on (77). In the next section we will make this
more concrete from the algebraic point of view..

We now comment on the generalisation of our construction above. The question
is whether, if at all, there is a guarantee that one can always diagonalize a
current-current deformed scalar field theory in terms of symplectic field
redefinitions. If so, it is likely that we would always be able to dial our
parameters in a way to reach a truncated regime of the phase space where
Carroll symmetry prevails. The answer to this question is unknown from a
classical field theory perspective, as far as we understand. From a case-by-
case basis, it turns out this can happen for all marginal deformations borne
out of $U(1)$ current bilinears, but we do not have a proof for the same. See
Appendix (D) for some more details and further comments along this direction.

####  4.1.2 Virasoro algebra to BMS algebra

As in the case of the symmetric deformation, there are clear indications that
the symmetries of the system change drastically as $g\to\infty$. We now make
this concrete. Both redefined fields $\chi_{\pm}$ are subject to the wave
equation and conserved currents are

$\mathcal{J}^{\pm}=\dot{\chi}_{\pm}+{\chi}^{\prime}_{\pm}=2\lambda^{(g)}_{\pm}\Pi_{\pm}+{\chi}^{\prime}_{\pm},\quad\overline{\mathcal{J}}^{\pm}=\dot{\chi}_{\pm}-{\chi}^{\prime}_{\pm}=2\lambda^{(g)}_{\pm}\Pi_{\pm}-{\chi}^{\prime}_{\pm}$
(100)

which obey the conservation law
$\bar{\partial}\mathcal{J}^{\pm}=\partial\overline{\mathcal{J}}^{\pm}=0$ with
$\partial=\partial_{\tau}+\partial_{\sigma}$ and
$\bar{\partial}=\partial_{\tau}-\partial_{\sigma}$. The algebra of these
currents is as follows:

$\displaystyle\left\\{\mathcal{J}^{+}(\sigma),\mathcal{J}^{+}(\sigma^{\prime})\right\\}_{\rm
PB}$
$\displaystyle=4\lambda^{(g)}_{+}\partial_{\sigma}\delta(\sigma-\sigma^{\prime}),$
(101a)
$\displaystyle\left\\{\overline{\mathcal{J}}^{+}(\sigma),\overline{\mathcal{J}}^{+}(\sigma^{\prime})\right\\}_{\rm
PB}$
$\displaystyle=-4\lambda^{(g)}_{+}\partial_{\sigma}\delta(\sigma-\sigma^{\prime}),$
(101b)
$\displaystyle\left\\{\mathcal{J}^{-}(\sigma),\mathcal{J}^{-}(\sigma^{\prime})\right\\}_{\rm
PB}$
$\displaystyle=4\lambda^{(g)}_{-}\partial_{\sigma}\delta(\sigma-\sigma^{\prime}),$
(101c)
$\displaystyle\left\\{\overline{\mathcal{J}}^{-}(\sigma),\overline{\mathcal{J}}^{-}(\sigma^{\prime})\right\\}_{\rm
PB}$
$\displaystyle=-4\lambda^{(g)}_{-}\partial_{\sigma}\delta(\sigma-\sigma^{\prime}),$
(101d)

and all other brackets vanish. To make sure these currents have the standard
Kac-Moody commutation relation, we normalize them

$J^{\pm}=\frac{\mathcal{J}^{\pm}}{\sqrt{4\lambda^{(g)}_{\pm}}},\qquad\quad\overline{J}^{\pm}=\frac{\overline{\mathcal{J}}^{\pm}}{\sqrt{4\lambda^{(g)}_{\pm}}}.$
(102)

Now, applying the Sugawara construction we can again obtain an independent
pair of stress tensors

$\displaystyle T(\sigma)$
$\displaystyle=\frac{1}{2}\left\\{J^{+}(\sigma)J^{+}(\sigma)+J^{-}(\sigma)J^{-}(\sigma)\right\\},$
(103) $\displaystyle\overline{T}(\sigma)$
$\displaystyle=\frac{1}{2}\left\\{\overline{J}^{+}(\sigma)\overline{J}^{+}(\sigma)+\overline{J}^{-}(\sigma)\overline{J}^{-}(\sigma)\right\\}.$

As a consistency check, we examine $T+\overline{T}$:

$\displaystyle T+\overline{T}$
$\displaystyle=\frac{1}{2}\left\\{J^{+}J^{+}+J^{-}J^{-}+\overline{J}^{+}\overline{J}^{+}+\overline{J}^{-}\overline{J}^{-}\right\\}$
$\displaystyle=\lambda^{(g)}_{+}\left\\{\Pi_{+}^{2}+\left({\chi}^{\prime}_{-}\right)^{2}\right\\}+\lambda^{(g)}_{-}\left\\{\Pi_{-}^{2}+\left({\chi}^{\prime}_{+}\right)^{2}\right\\}$
(104)

which reproduces the Hamiltonian density (77) as expected, while the other
orthogonal operator is expressed in a form

$\displaystyle T-\overline{T}$
$\displaystyle=\frac{1}{2}\left\\{J^{+}J^{+}+J^{-}J^{-}+\overline{J}^{+}\overline{J}^{+}+\overline{J}^{-}\overline{J}^{-}\right\\}=\Pi_{+}{\chi}^{\prime}_{+}+\Pi_{-}{\chi}^{\prime}_{-}.$
(105)

Let us introduce the generators in a way with suitable scalings, such that the
$g\to\infty$ limit can be taken consistently:

$L^{(g)}=T-\overline{T}\qquad\qquad
M^{(g)}=\frac{1}{\lambda^{(g)}_{+}}(T+\overline{T}).$ (106)

It is easy to see the algebra of $L^{(g)}$ is just the Virasoro algebra:

$\displaystyle\left\\{L^{(g)}(\sigma),L^{(g)}(\sigma^{\prime})\right\\}_{\rm
PB}=2L^{(g)}(\sigma)\partial_{\sigma}\delta(\sigma-\sigma^{\prime})+\left(\partial_{\sigma}L^{(g)}(\sigma)\right)\delta(\sigma-\sigma^{\prime}).$
(107)

In a similar manner we find

$\displaystyle\left\\{L^{(g)}(\sigma),M^{(g)}(\sigma^{\prime})\right\\}_{\rm
PB}$
$\displaystyle=2M^{(g)}(\sigma)\partial_{\sigma}\delta(\sigma-\sigma^{\prime})+\left(\partial_{\sigma}M^{(g)}(\sigma)\right)\delta(\sigma-\sigma^{\prime})$
(108)
$\displaystyle\left\\{M^{(g)}(\sigma),M^{(g)}(\sigma^{\prime})\right\\}_{\rm
PB}$
$\displaystyle=\frac{1}{(\lambda^{(g)}_{+})^{2}}\left[2L^{(g)}(\sigma)\partial_{\sigma}\delta(\sigma-\sigma^{\prime})+\left(\partial_{\sigma}L^{(g)}(\sigma)\right)\delta(\sigma-\sigma^{\prime})\right]$
$\displaystyle\overset{g\to\infty}{\longrightarrow}\ 0$ (109)

which manifestly shows the transition from a pair of Virasoro algebras to BMS3
algebra as $g$ is dialed up to infinity.

We emphasise, that although these equations are similar to (33), we have
arrived at the Carrollian sector in a path very different from the previous
symmetric deformation. Unlike the $J.J$ deformation, in this case there was no
effective dialling of the speed of light, equations of motion were unchanged
and most importantly the generators in (44) and (106), especially the second
equalities are completely different. In fact, when written in terms of modes,
(106) implies a Inönü-Wigner contraction of the two copies of Virasoro algebra
(see (37)), while the second equation of (44) is clearly something different.

Note that since $\lambda^{(g\to\infty)}\sim g^{2}$, the contractions in (106)
as read using the modes (26) for very large values of $g$ will be written as:
$L^{(\infty)}_{n}=\mathcal{L}_{n}-{\mathcal{\bar{L}}_{-n}}$ and
$M^{(\infty)}_{n}=\frac{1}{g^{2}}\left(\mathcal{L}_{n}+{\mathcal{\bar{L}}_{-n}}\right)$,
which one should again compare with the nature of (37). This standard form of
BMS3 generators will be important later on when we go ahead and look at the
fate of the spectrum as one goes to $g\to\infty$.

###  4.2 Quantization and details of the flow

We will now move towards a quantization of our deformed system. For the
present case, we will consider compactification on circles for both the scalar
fields 171717In terms of string theory, the above condition is akin to
considering compactified boundary conditions in the target space in order to
make sure the effect of B-field is visible on the spectrum.. Let us define the
periodic boundary conditions on the original fields as:

$\phi_{i}\sim\phi_{i}+m_{i}R_{i}.\qquad\quad(i=1,2)$ (110)

Here $R_{i}$ are the compactification radii, associated to which we have
Kaluza-Klein (KK) momenta $n_{i}$ and winding modes $m_{i}$:

$p_{i}=\frac{n_{i}}{R_{i}}\qquad\quad w_{i}=R_{i}\,m_{i}\qquad\quad
n_{i},m_{i}\in\mathbb{Z}.$ (111)

We have also rescaled the periodicity in this case $\sigma\in[0,1]$. The
deformation term acts as a boundary term and we will need to be very careful
about boundary conditions. This is what we address next.

####  4.2.1 Boundary conditions

We now carefully implement boundary conditions while quantizing the system.
Consider a variation over fields $\phi_{i}\to\phi_{i}+\delta\phi_{i}$. Under
this, the Lagrangian density in (68) varies as

$\displaystyle\delta\mathcal{L}^{(g)}$
$\displaystyle=\frac{1}{1+g^{2}}\left[\sum_{i=1,2}\left(\partial_{\tau}\left[\delta\phi_{i}\partial_{\tau}\phi_{i}\right]-\partial_{\sigma}\left[\delta\phi_{i}\partial_{\sigma}\phi_{i}\right]\right)+\overbrace{\left(\partial^{2}_{\sigma}\phi_{i}-\partial^{2}_{\tau}\phi_{i}\right)}^{\text{e.o.m.}}\delta\phi_{i}\right]$
$\displaystyle\qquad-\frac{g}{1+g^{2}}\left(\partial_{\tau}\left[\delta\phi_{1}\partial_{\sigma}\phi_{2}-\delta\phi_{2}\partial_{\sigma}\phi_{1}\right]+\partial_{\sigma}\left[\delta\phi_{2}\partial_{\tau}\phi_{1}-\delta\phi_{1}\partial_{\tau}\phi_{2}\right]\right).$
(112)

The spatial derivatives leave boundary terms after integration:

$\displaystyle\int_{0}^{1}d\sigma\,\delta\mathcal{L}^{(g)}$
$\displaystyle=\frac{1}{1+g^{2}}\left[-\delta\phi_{1}\left(\partial_{\sigma}\phi_{1}-g\,\partial_{\tau}\phi_{2}\right)-\delta\phi_{2}\left(\partial_{\sigma}\phi_{2}+g\,\partial_{\tau}\phi_{1}\right)\right]_{\sigma=0}^{1}$
(113)

vanishing of which requires the following boundary conditions:

$\displaystyle\left.\delta\phi_{1}\left[\partial_{\sigma}\phi_{1}-g\,\partial_{\tau}\phi_{2}\right]\right|_{\sigma=1}-\left.\delta\phi_{1}\left[\partial_{\sigma}\phi_{1}-g\,\partial_{\tau}\phi_{2}\right]\right|_{\sigma=0}=0.$
(114a)
$\displaystyle\left.\delta\phi_{2}\left[\partial_{\sigma}\phi_{2}+g\,\partial_{\tau}\phi_{1}\right]\right|_{\sigma=1}-\left.\delta\phi_{2}\left[\partial_{\sigma}\phi_{2}+g\,\partial_{\tau}\phi_{1}\right]\right|_{\sigma=0}=0.$
(114b)

Now for clarity, let’s see what these conditions imply:

  * (i)

Undeformed theory $g=0$: For the undeformed theory, the condition are simply:

$\left.\delta\phi_{i}\partial_{\sigma}\phi_{i}\right|_{\sigma=1}-\left.\delta\phi_{i}\partial_{\sigma}\phi_{i}\right|_{\sigma=0}=0.$
(115)

One can use Neumann ($\left.\partial_{\sigma}\phi_{i}\right|_{\sigma=0,1}=0$)
or Dirichlet ($\left.\delta\phi_{i}\right|_{\sigma=0,1}=0$) boundary
conditions to solve this. This is similar to the analysis for the open string.

One can also employ periodic boundary conditions
($\phi_{i}(\tau,\sigma+1)=\phi_{i}(\tau,\sigma)$), however up to windings,
($\phi_{i}(\tau,\sigma+1)-\phi_{i}(\tau,\sigma)=R_{i}m_{i},\
m_{i}\in\mathbb{Z})$. This is reminiscent of the analysis in closed string
theory.

  * (ii)

Deformed theory $g\neq 0$: Since the periodic boundary condition with windings
can be written as

$\int_{\sigma}^{\sigma+1}d\sigma^{\prime}\,\partial_{\sigma^{\prime}}\phi_{i}(\tau,\sigma^{\prime})=R_{i}m_{i},$
(116)

a natural extension of this condition in the presence of the coupling $g$ is
clearly of the following form:

$\int_{\sigma}^{\sigma+1}d\sigma^{\prime}\,\partial_{\sigma^{\prime}}\widetilde{\phi}_{i}:=\int_{\sigma}^{\sigma+1}d\sigma^{\prime}\,\left(\partial_{\sigma^{\prime}}\phi_{i}-g\epsilon_{ij}\partial_{\tau}\phi_{j}\right)(\tau,\sigma^{\prime})=:\gamma_{i}$
(117)

where we use the tilde to distinguish a deformed field and $\gamma_{i}$ is an
effective winding number for the theory (see also Apolo:2019zai ). This in
turn implies the boundary condition when written in terms of the undeformed
quantum numbers:

$\widetilde{\phi}_{i}(\tau,\sigma+1)-\widetilde{\phi}_{i}(\tau,\sigma)=m_{i}R_{i}-g\epsilon_{ij}\frac{n_{j}}{R_{j}},$
(118)

which means the effective deformed winding is a combination of undeformed
winding and momenta, albeit coming from different fields.

####  4.2.2 Equations of motion

It is straightforward to see that the deformed fields will still satisfy wave
equations of motion. Considering the relation (117) should be valid for any
$\sigma$ as the origin of the periodic integral, implies that the integrands
on both sides should be equated

$\partial_{\sigma}\widetilde{\phi}_{i}=\partial_{\sigma}\phi_{i}-g\epsilon_{ij}\partial_{\tau}\phi_{j}\
.$ (119)

Note that this is a very non-local boundary condition that links the different
fields and their temporal and spatial derivatives. This gives rise to
relations of the dynamical variables $\widetilde{\phi}_{i}$ constructed in
terms of $\phi_{i}$:

$\widetilde{\phi}_{i}(\tau,\sigma)=\phi_{i}(\tau,\sigma)-g\epsilon_{ij}\int_{\sigma_{0}}^{\sigma}d\sigma^{\prime}\,\partial_{\tau}\phi_{j}(\tau,\sigma^{\prime})+F_{i}(\tau,\sigma_{0})$
(120)

where $F_{i}$ are $\sigma$-independent functions

$F_{i}(\tau,\sigma_{0})=\widetilde{\phi}_{i}(\tau,\sigma_{0})-\phi_{i}(\tau,\sigma_{0}).$
(121)

Now since we are dealing with just a boundary term, the equations of motion
coming from this action should not change, i.e. the redefined field
$\widetilde{\phi}_{i}$ obeys the same equation of motion as $\phi_{i}$ as
mentioned before,

$\displaystyle(\partial^{2}_{\sigma}-\partial^{2}_{\tau})\widetilde{\phi}_{i}$
$\displaystyle=(\partial^{2}_{\sigma}-\partial^{2}_{\tau}){\phi}_{i}-g\epsilon_{ij}(\partial^{2}_{\sigma}-\partial^{2}_{\tau})\int_{\sigma_{0}}^{\sigma}d\sigma^{\prime}\,\partial_{\tau}\phi_{j}(\tau,\sigma^{\prime})-\partial_{\tau}^{2}F_{i}$
$\displaystyle=-g\epsilon_{ij}\int^{\sigma}d\sigma^{\prime}\,\partial_{\tau}(\partial^{2}_{\sigma^{\prime}}-\partial^{2}_{\tau})\phi_{j}(\tau,\sigma^{\prime})=0$
(122)

by simply using the equations of motion for $\phi_{i}$181818Note that this
mandates the unknown function to be at most linear in $\tau$:
$F_{i}=F_{i}^{(0)}+\tau F_{i}^{(1)}$.. Hence, $\widetilde{\phi}_{i}$ will have
a Fourier mode expansion, however with the boundary condition (118) taken care
of, which is solved by

$\displaystyle\widetilde{\phi}_{i}(\tau,\sigma)$
$\displaystyle=\widetilde{\phi}_{i}^{(0)}+\widetilde{\phi}_{i}^{(1)}\tau+\left(m_{i}R_{i}-g\epsilon_{ij}\frac{n_{j}}{R_{j}}\right)\sigma+\sum_{n\neq
0}\frac{1}{n}\left\\{\widetilde{\alpha}^{i}_{n}e^{-2\pi{in(\tau+\sigma)}}+\widetilde{\widetilde{\alpha}}^{i}_{n}e^{-2\pi{in(\tau-\sigma)}}\right\\}$
(123)

One can see the linear piece in $\tau$, in this case
$\widetilde{\phi}_{i}^{(1)}$ can be checked for self-consistency using
boundary conditions, noting that (119) implies:

$0=\partial_{\sigma}\left(\partial_{\sigma}\widetilde{\phi}_{i}-\partial_{\sigma}\phi_{i}+g\epsilon_{ij}\partial_{\tau}\phi_{j}\right)\overset{\rm
e.o.m.}{=}\partial_{\tau}\left(\partial_{\tau}\widetilde{\phi}_{i}-\partial_{\tau}\phi_{i}+g\epsilon_{ij}\partial_{\sigma}\phi_{j}\right)$
(124)

as well as

$0=\partial_{\tau}\left(\partial_{\sigma}\widetilde{\phi}_{i}-\partial_{\sigma}\phi_{i}+g\epsilon_{ij}\partial_{\tau}\phi_{j}\right)\overset{\rm
e.o.m.}{=}\partial_{\sigma}\left(\partial_{\tau}\widetilde{\phi}_{i}-\partial_{\tau}\phi_{i}+g\epsilon_{ij}\partial_{\sigma}\phi_{j}\right)$
(125)

thus one could easily find a solution where,

$\partial_{\tau}\widetilde{\phi}_{i}-\partial_{\tau}\phi_{i}+g\epsilon_{ij}\partial_{\sigma}\phi_{j},=C_{i}\,(\rm
const.)$ (126)

which can be heuristically chosen as $C_{i}=g\epsilon_{ij}R_{j}n_{j}$,
resulting in

$\displaystyle\widetilde{\phi}_{i}^{(1)}=\int_{0}^{1}d\sigma\,\partial_{\tau}\widetilde{\phi}_{i}(\tau,\sigma)=\frac{n_{i}}{R_{i}},$
(127)

which one readily expects for zero modes since only the winding modes are
affected by the new boundary conditions. All of this leads to the following
final mode expansion of our deformed fields:

$\phi_{i}(\tau,\sigma)=\phi_{i}^{(0)}+\frac{n_{i}}{R_{i}}\tau+(m_{i}R_{i}-g\epsilon_{ij}\frac{n_{j}}{R_{j}})\sigma+\sum_{n\neq
0}\frac{1}{n}\left\\{\alpha^{i}_{n}e^{-{2\pi
in(\tau+\sigma)}}+\tilde{\alpha}^{i}_{n}e^{-{2\pi in(\tau-\sigma)}}\right\\}$
(128)

where we have now removed the extra tildes to simplify the notation. This mode
expansion for the fields will be used below.

####  4.2.3 Deformed generators

We make the important observation that although the zero modes of the
expansion for the field have now changed, the oscillator modes are unchanged.
Comparing with generic conformal field theory mode expansion for a
compactified target space 191919See DiFrancesco:1997nk  for example., we can
define the shifted (or “flowed”) modes:

$\displaystyle\alpha^{i}_{n}\rightarrow\alpha^{i}_{n}-\frac{g}{\sqrt{2}}\epsilon_{ij}\frac{n_{j}}{R_{j}}\delta_{n,0},\quad\tilde{\alpha}^{i}_{n}\rightarrow\tilde{\alpha}^{i}_{n}+\frac{g}{\sqrt{2}}\epsilon_{ij}\frac{n_{j}}{R_{j}}\delta_{n,0}$
(129)

which keep KK momenta same, while deforming the periodicity. Such shifted
modes do not change the mode algebra or the definition of the vacuum, which
remains the highest weight vacuum:

${}\alpha_{n}|0\rangle=0=\tilde{\alpha}_{n}|0\rangle,~{}~{}~{}~{}~{}\text{for}\,\,n>0.$
(130)

We emphasise that this is markedly different from the $J\overline{J}$ case,
where the vacuum changed via a one parameter Bogoliubov transformation
throughout the flow (51). From the mode expansion, we can construct usual
generators:

$\displaystyle\mathcal{L}_{n}=\frac{1}{2}\sum_{p\in\mathbb{Z}}\alpha^{i}_{n-p}\alpha^{i}_{p}\quad\text{for}\,\,n\neq
0,\,\,\text{and}\quad\mathcal{L}_{0}=\frac{1}{2}\sum_{p\neq
0}\alpha^{i}_{-p}\alpha^{i}_{p}+\frac{1}{2}\alpha^{i}_{0}\alpha^{i}_{0},$
(131)

which extends similarly to the antiholomorphic sector. Now using shifted zero
modes, we can easily get the expression for deformed $\mathcal{L}_{0}$ and
$\bar{\mathcal{L}}_{0}$

$\displaystyle\mathcal{L}_{0}^{(g)}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\Big{(}\frac{n_{i}}{R_{i}}+m_{i}R_{i}-g\epsilon_{ij}\frac{n_{j}}{R_{j}}\Big{)}^{2}+\frac{1}{2}\sum_{n\neq
0}\alpha^{i}_{-n}\alpha^{i}_{n},$ (132a)
$\displaystyle\bar{\mathcal{L}}_{0}^{(g)}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\Big{(}\frac{n_{i}}{R_{i}}-m_{i}R_{i}+g\epsilon_{ij}\frac{n_{j}}{R_{j}}\Big{)}^{2}+\frac{1}{2}\sum_{n\neq
0}\tilde{\alpha}^{i}_{-n}\tilde{\alpha}^{i}_{n}.$ (132b)

In the above, we use the $(g)$ superscript to identify the “flow”-ed
quantities. Similarly, the shifted conformal dimension is given by action of
Virasoro zero modes on a primary state $|h^{(g)},\bar{h}^{(g)}\rangle$:

$\mathcal{L}_{0}^{(g)}|h^{(g)},\bar{h}^{(g)}\rangle=h^{(g)}|h^{(g)},\bar{h}^{(g)}\rangle,\quad\bar{\mathcal{L}}_{0}^{(g)}|h^{(g)},\bar{h}^{(g)}\rangle=\bar{h}^{(g)}|h^{(g)},\bar{h}^{(g)}\rangle,$
(133)

where the explicit shifted weights are

$\displaystyle h^{(g)}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\Big{(}\frac{n_{1}}{R_{1}}+m_{1}R_{1}-g\frac{n_{2}}{R_{2}}\Big{)}^{2}+\frac{1}{4}\Big{(}\frac{n_{2}}{R_{2}}+m_{2}R_{2}+g\frac{n_{1}}{R_{1}}\Big{)}^{2}$
(134a) $\displaystyle\bar{h}^{(g)}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\Big{(}\frac{n_{1}}{R_{1}}-m_{1}R_{1}+g\frac{n_{2}}{R_{2}}\Big{)}^{2}+\frac{1}{4}\Big{(}\frac{n_{2}}{R_{2}}-m_{2}R_{2}-g\frac{n_{1}}{R_{1}}\Big{)}^{2}.$
(134b)

We would be interested in the combinations

$\displaystyle h^{(g)}-\bar{h}^{(g)}=h-\bar{h}=n_{i}m_{i},$ (135a)
$\displaystyle
h^{(g)}+\bar{h}^{(g)}=\Bigg{[}\frac{1}{2}\Big{(}(1+g^{2})\frac{n_{i}^{2}}{R_{i}^{2}}+m_{i}^{2}R_{i}^{2}\Big{)}+g\Big{(}\frac{R_{2}}{R_{1}}n_{1}m_{2}-\frac{R_{1}}{R_{2}}n_{2}m_{1}\Big{)}\Bigg{]}.$
(135b)

We see that $h^{(g)}-\bar{h}^{(g)}$ does not flow, while the other linear
combination $h^{(g)}+\bar{h}^{(g)}$ does. A similar computation with shifted
zero modes leads to expressions for flowed
$\mathcal{L}_{n},\bar{\mathcal{L}}_{n}$ with $n\neq 0$:

$\displaystyle\mathcal{L}_{n}^{(g)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{\begin{subarray}{c}p\neq 0\\\ p\neq
n\end{subarray}}\alpha^{i}_{n-p}\alpha^{i}_{p}+\frac{1}{\sqrt{2}}\Big{(}\frac{n_{i}}{R_{i}}+m_{i}R_{i}-g\epsilon_{ij}\frac{n_{j}}{R_{j}}\Big{)}\alpha^{i}_{n},$
(136a) $\displaystyle\bar{\mathcal{L}}_{n}^{(g)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{\begin{subarray}{c}p\neq 0\\\ p\neq
n\end{subarray}}\tilde{\alpha}^{i}_{n-p}\tilde{\alpha}^{i}_{p}+\frac{1}{\sqrt{2}}\Big{(}\frac{n_{i}}{R_{i}}-m_{i}R_{i}+g\epsilon_{ij}\frac{n_{j}}{R_{j}}\Big{)}\tilde{\alpha}^{i}_{n}.$
(136b)

Combining everything, we can see the Virasoro generators $\mathcal{L}_{n}$ and
$\bar{\mathcal{L}}_{n}$ flows as

$\displaystyle\mathcal{L}_{n}$ $\displaystyle\rightarrow$
$\displaystyle\mathcal{L}_{n}-\frac{g}{\sqrt{2}}\epsilon_{ij}\frac{n_{j}}{R_{j}}\alpha^{i}_{n}+\frac{g^{2}}{4}\frac{n_{i}^{2}}{R_{i}^{2}}\delta_{n,0}$
(137a) $\displaystyle\bar{\mathcal{L}}_{n}$ $\displaystyle\rightarrow$
$\displaystyle\bar{\mathcal{L}}_{n}+\frac{g}{\sqrt{2}}\epsilon_{ij}\frac{n_{j}}{R_{j}}\tilde{\alpha}^{i}_{n}+\frac{g^{2}}{4}\frac{n_{i}^{2}}{R_{i}^{2}}\delta_{n,0}.$
(137b)

Note that despite the shifted nature, all the generators here still correspond
to the conformal algebra for all values of finite $g$. This is of course
expected since the deformation that we started out with is an exactly marginal
deformation and although the spectrum of the theory changes, the theory still
remains conformal.

We can compare this with the usual spectral flow for WZW models 202020See
Maldacena:2000hw  and Appendix (E) for a brief introduction to such spectral
flows.. In this case the spectral flow parameter is the KK momentum associated
to the compactified theory. A similar result has also been derived in
Apolo:2019zai  by considering TsT transformations in a AdS3 target space that
induce current-current deformations on a string worldsheet. In what follows,
we will go a bit further and show that in the large $g$ limit the flowed
algebra in this case conveniently rearranges itself in the form of a BMS
invariant theory.

####  4.2.4 $g\to\infty$ limit: Emergent BMS

We now wish to go to the edge of the parameter space by dialling the coupling
$g$. We know from the classical analysis, that we expect a change of the
spacetime algebra from relativistic to Carrollian conformal symmetries. We
will now see this explicitly in the quantum mechanical theory. Unlike the
classical analysis, now that we have interpreted the quantum mechanical
deformation first as a change of boundary condition and then as a spectral
flow, we now will see that there exists an extreme limit of these boundary
conditions or the spectral flow which deforms the relativistic symmetries to
Carrollian. We should emphasise that these are completely novel observations.
The fact that changing boundary conditions drastically could significantly
change a theory is perhaps expected, but changing relativistic symmetries to
non-relativistic (Carrollian in this case) by a continuous deformation of
boundary condition is remarkable. We now explicitly show this.

To take the $g\to\infty$ limit on the results of the last section, especially
of (137), let us define a new convenient basis of generators of the form
$\mathcal{J}_{n}={\mathcal{L}}_{n}-\bar{{\mathcal{L}}}_{-n}$ and
$\mathcal{H}_{n}=\frac{1}{g^{2}}({\mathcal{L}}_{n}+\bar{{\mathcal{L}}}_{-n})$,
and observe that they also flow as in the following way:

$\displaystyle\mathcal{J}^{(g)}_{n}$
$\displaystyle={\mathcal{L}}^{(g)}_{n}-\bar{{\mathcal{L}}}^{(g)}_{-n}={\mathcal{L}}^{(g=0)}_{n}-\bar{{\mathcal{L}}}^{(g=0)}_{-n}-\frac{g}{\sqrt{2}}\epsilon_{ij}\frac{n_{j}}{R_{j}}(\alpha^{i}_{n}+\tilde{\alpha}^{i}_{-n})$
$\displaystyle\mathcal{H}^{(g)}_{n}$
$\displaystyle=\frac{1}{g^{2}}({\mathcal{L}}^{(g)}_{n}+\bar{{\mathcal{L}}}^{(g)}_{-n})=\frac{1}{g^{2}}({\mathcal{L}}^{(g=0)}_{n}+\bar{{\mathcal{L}}}^{(g=0)}_{-n})-\frac{1}{\sqrt{2}g}\epsilon_{ij}\frac{n_{j}}{R_{j}}(\alpha^{i}_{n}-\tilde{\alpha}^{i}_{-n})+\frac{1}{2}\frac{n_{i}^{2}}{R_{i}^{2}}\delta_{n,0}$

So, we have

$\displaystyle\mathcal{J}^{(g)}_{n}=\mathcal{\overline{J}}_{n}-\frac{1}{\sqrt{2}g}\epsilon_{ij}\frac{n_{j}}{R_{j}}B^{i}_{n},\quad\mathcal{H}^{(g)}_{n}=\mathcal{\overline{H}}_{n}-\frac{1}{\sqrt{2}g}\epsilon_{ij}\frac{n_{j}}{R_{j}}A^{i}_{n}+{\frac{1}{2}\frac{n_{i}^{2}}{R_{i}^{2}}\delta_{n,0}}$
(138)

where we have redefined:

$\alpha^{i}_{n}-\tilde{\alpha}^{i}_{-n}=A^{i}_{n},\quad\alpha^{i}_{n}+\tilde{\alpha}^{i}_{-n}=\frac{1}{g^{2}}B^{i}_{n}$
(139)

as the new combination of oscillator modes and the superscript $(g=0)$
emphasizes the undeformed generators, so that
$\mathcal{\overline{J}}_{n}=\mathcal{J}^{(g=0)}_{n}$ and
$\mathcal{\overline{H}}_{n}=\frac{1}{g^{2}}\mathcal{H}^{(g=0)}_{n}$ are the
non-flowing parts of the deformed generators. Note that the definition of the
stress tensor modes $\mathcal{J}^{(g)}_{n}$ and $\mathcal{H}^{(g)}_{n}$ are
commensurate with (106) in large $g$ limit. Further notice that under the flow
of zero modes as in (129), $A^{i}_{0},B^{i}_{0}$ also shift in the following
way:

$A_{0}^{i}\rightarrow
A_{0}^{i}-\sqrt{2}g\epsilon_{ij}\frac{n_{j}}{R_{j}}\delta_{n,0},~{}~{}~{}~{}B_{0}^{i}\rightarrow
B_{0}^{i}.$ (140)

This immediately reminds us of the spectral flow in 3D flat spacetime as
described in Basu:2017aqn  for a holographic setting. In fact one could easily
identify (138) as an automorphism of a BMS$-U(1)$ theory, where the parent
Virasoro-Kac-Moody algebras had the same level. In that case $A,B$ are the
generators of the affine current algebra. Now focussing on the deformed Kac-
Moody algebra between $\mathcal{H},\mathcal{J},A,B$, we observe:

$\displaystyle\left[\mathcal{H}^{(g)}_{m},A^{k}_{n}\right]$
$\displaystyle=\left[\mathcal{\overline{H}}_{m},A^{k}_{n}\right]$
$\displaystyle\left[\mathcal{H}^{(g)}_{m},B^{k}_{n}\right]$
$\displaystyle=\left[\mathcal{\overline{H}}_{m},B^{k}_{n}\right]-\sqrt{2}{g}\epsilon_{ij}\frac{n_{j}}{R_{j}}m\delta^{ik}\delta_{m+n,0}$
$\displaystyle\left[\mathcal{J}^{(g)}_{m},A^{k}_{n}\right]$
$\displaystyle=\left[\mathcal{\overline{J}}_{m},A^{k}_{n}\right]-\sqrt{2}{g}\epsilon_{ij}\frac{n_{j}}{R_{j}}m\delta^{ik}\delta_{m+n,0}$
$\displaystyle\left[\mathcal{J}^{(g)}_{m},B^{k}_{n}\right]$
$\displaystyle=\left[\mathcal{\overline{J}}_{m},B^{k}_{n}\right]$ (141)

Also, we can easily work out the Kac-Moody algebra involving
$\mathcal{\overline{J}}_{n}$ and $\mathcal{\overline{H}}_{n}$ separately,

$\displaystyle\left[\mathcal{\overline{H}}_{m},B^{k}_{n}\right]=\left[{\mathcal{L}}^{(g=0)}_{m}+\bar{{\mathcal{L}}}^{(g=0)}_{-m},\alpha^{k}_{n}+\tilde{\alpha}^{k}_{-n}\right]=-nA^{k}_{m+n},$
$\displaystyle\left[\mathcal{\overline{H}}_{m},A^{k}_{n}\right]=\frac{1}{g^{2}}\left[{\mathcal{L}}^{(g=0)}_{m}+\bar{{\mathcal{L}}}^{(g=0)}_{-m},\alpha^{k}_{n}-\tilde{\alpha}^{k}_{-n}\right]=-\frac{1}{g^{4}}nB^{k}_{m+n},$
$\displaystyle\left[\mathcal{\overline{J}}_{m},B^{k}_{n}\right]=g^{2}\left[{\mathcal{L}}^{(g=0)}_{m}-\bar{{\mathcal{L}}}^{(g=0)}_{-m},\alpha^{k}_{n}+\tilde{\alpha}^{k}_{-n}\right]=-nB^{k}_{m+n},$
$\displaystyle\left[\mathcal{\overline{J}}_{m},A^{k}_{n}\right]=\left[{\mathcal{L}}^{(g=0)}_{m}-\bar{{\mathcal{L}}}^{(g=0)}_{-m},\alpha^{k}_{n}-\tilde{\alpha}^{k}_{-n}\right]=-nA^{k}_{m+n}.$
(142)

Note that the flow of zero modes do not affect these commutation relations,
and moreover the flowed terms on the RHS of (4.2.4) are simply cancelled by
the flow terms in $A_{0},B_{0}$ coming from the undeformed algebra as defined
in (140). Collecting the results from the above computation and taking the
$g\to\infty$ limit, everything falls into place and we finally reach at the
following Kac-Moody algebra:

$\displaystyle\left[\mathcal{H}^{(\infty)}_{m},A^{k}_{n}\right]$
$\displaystyle=$ $\displaystyle
0,\quad\left[\mathcal{H}^{(\infty)}_{m},B^{k}_{n}\right]=-nA^{k}_{m+n},$
$\displaystyle\left[\mathcal{J}^{(\infty)}_{m},A^{k}_{n}\right]$
$\displaystyle=$ $\displaystyle-
nA^{k}_{m+n},\quad\left[\mathcal{J}^{(\infty)}_{m},B^{k}_{n}\right]=-nB^{k}_{m+n}.$
(143)

This is the BMS$-U(1$) Kac-Moody algebra as found in Bagchi:2023dzx . Note
that unlike in that work, we did not arrive here by contractions of the seed
algebra, but by just deforming the generators with $g$ to the extreme value.

Coming back to the stress tensor modes, we can easily see the barred
generators follow the algebra

$\displaystyle\left[\mathcal{\overline{J}}_{n},\mathcal{\overline{J}}_{m}\right]=(n-m)\mathcal{\overline{J}}_{m+n}$
(144)
$\displaystyle\left[\mathcal{\overline{J}}_{n},\mathcal{\overline{H}}_{m}\right]=(n-m)\mathcal{\overline{H}}_{m+n}$
(145)
$\displaystyle\left[\mathcal{\overline{H}}_{n},\mathcal{\overline{H}}_{m}\right]=(n-m)\mathcal{\overline{J}}_{m+n}\frac{1}{g^{4}}\xrightarrow{g\rightarrow\infty}0$
(146)

This would have easily served as a cousin of the interpolating algebras we
have seen before, going from two copies of Virasoro algebra at finite $g$ to
BMS3 in infinite $g$. However there is a caveat, since the $g=0$ point is not
simply connected in this flow. This makes sense as our basis itself is
designed to work in the large $g$ limit so that we can smoothly go up to
(106). Using the above results, the full flowed generators can be found to
obey:

$\displaystyle\left[\mathcal{J}^{(g)}_{n},\mathcal{J}^{(g)}_{m}\right]$
$\displaystyle=(n-m)\mathcal{\overline{J}}_{m+n}+\frac{1}{\sqrt{2}g}\epsilon_{ij}\frac{n_{j}}{R_{j}}mB^{i}_{m+n}-\frac{1}{\sqrt{2}g}\epsilon_{ij}\frac{n_{j}}{R_{j}}nB^{i}_{m+n}$
$\displaystyle=(n-m)\mathcal{J}^{(g)}_{m+n}$
$\displaystyle\left[\mathcal{J}^{(g)}_{n},\mathcal{H}^{(g)}_{m}\right]$
$\displaystyle=(n-m)\mathcal{\overline{H}}_{m+n}-\frac{1}{\sqrt{2}g}\epsilon_{ij}\frac{n_{j}}{R_{j}}mA^{i}_{m+n}+\frac{1}{\sqrt{2}g}\epsilon_{ij}\frac{n_{j}}{R_{j}}nA^{i}_{m+n}+n\delta_{n+m,0}\frac{n_{i}^{2}}{R_{i}^{2}}$
$\displaystyle=(n-m)\mathcal{H}^{(g)}_{m+n}$
$\displaystyle\left[\mathcal{H}^{(g)}_{n},\mathcal{H}^{(g)}_{m}\right]$
$\displaystyle=\bigg{(}(n-m)\mathcal{\overline{J}}_{m+n}+\frac{1}{\sqrt{2}g}\epsilon_{ij}\frac{n_{j}}{R_{j}}mB^{i}_{m+n}-\frac{1}{\sqrt{2}g}\epsilon_{ij}\frac{n_{j}}{R_{j}}nB^{i}_{m+n}\bigg{)}\frac{1}{g^{4}}$
$\displaystyle=(n-m)\mathcal{J}^{(g)}_{m+n}\frac{1}{g^{4}}\xrightarrow{g\rightarrow\infty}0$
(147)

So in the strict $g\to\infty$ limit, we have obtained the BMS3 or the 2d
Conformal Carroll algebra as promised. Since this is the centerless version,
we should also mention same logic applies to the central charges of the
theory, although we can confirm from (137) that flowed Virasoro generators
satisfy the algebra with same central charges, the consistent $g\to\infty$
limit readily leads to central charges of the flowed BMS algebra:

$c_{\mathcal{J}}=c-\bar{c},~{}~{}~{}c_{\mathcal{H}}=\frac{1}{g^{2}}(c+\bar{c}).$
(148)

We believe this is a remarkable result in the sense that we arrived here just
using spectral flow associated to the system, and going to the large coupling
regime.

##  5 Discussions and Conclusions

###  5.1 Quick summary

We believe we have made a rather important and general discovery in this
paper. We have shown that at the very edges of parameter space of a
relativistic quantum field theory, there are emergence of non-Lorentzian
symmetry. After having laid out the reasons why we believe this to be true in
general in the introduction, we have focussed on a theory of free massless
scalars in two dimensions and shown that for current-current deformations on
this theory. For generic values of the deformation parameter, the theory of
course remains conformal but at the very end of the flow, the spacetime
algebra changes from the relativistic to the Carrollian conformal algebra. We
dealt with two different types of current-current deformations, the symmetric
and the antisymmetric, and carefully considered both the classical and quantum
versions of the flow. There were some very interesting differences between
them, indicating that the flow proceeds in very different routes, but both end
up with Carrollian conformal symmetries at the end of the flow. The
differences were particularly visible when we considered the quantum
mechanical deformed theories. The symmetric deformation proceeded through a
continuous change of the vacuum state, dictated by a Bogoliubov
transformation, which the anti-symmetric deformation manifested itself as a
change of boundary condition leading to an interpretation in terms of spectral
flow. We summarized our findings at the beginning in Sec.(2) and in particular
in Table (1). That a spectral flow is able to generate a change in the
spacetime algebra was a particularly intriguing outcome of our analysis.

Our analysis in the paper raises the intriguing possibility that
renormalisation group flows as pioneered by Wilson RevModPhys.47.773  may have
its limitations and may need re-examination as there are intriguing features
that appear in relativistic QFTs when they are deformed to either very low or
very high energies.

###  5.2 Discussions

There are numerous questions, confusions and a very large number of directions
of open research emerging from our present work. We discuss some of them
below.

  * $\square$

The Galilean limit. In our introduction, we led with the observation that at
very low energies, the effective description of a relativistic QFT should be
Galilean. However, we have offered precious little in terms of concretising
this point. It turns out that the Galilean limit is more subtle than the
Carrollian one (see Appendix B) and the places where the symmetry emerges may
be given by complex values of the deformation parameter. The $d=2$ problem is
further complicated by the fact that the Carrollian and Galilean algebras are
isomorphic leading to issues with identification of ultra-high v/s ultra-low
energies. We are currently investigating this and hope to report on it in the
near future when some of the associated subtlety is suitably addressed.

  * $\square$

Other QFTs. We believe the story we are trying to tell is a generic one, i.e.
there is the emergence of Galilean symmetries at very low energies and
Carrollian symmetries at very high energies for any relativistic quantum field
theory. So, we need to move beyond our present example and repeat our analysis
for other dimensions and other QFTs or understand a general scheme which
allows us make this argument.

  * $\square$

Other kind of deformations. Even within the realms of the 2d scalar theory,
one needs to understand the emergence of Carroll in the extreme UV by
considering irrelevant deformations. This is of course a very difficult
problem, but the recent progress in integrable deformations like the
$T\bar{T}$ deformations can be studied carefully and there are indications
that Carroll structures would emerge even there. In this case, reinterpreting
the current-current deformations as deformations of the worldsheet theory
would be useful.

  * $\square$

$\sqrt{T\overline{T}}$ deformation. Recently in Rodriguez:2021tcz , it was
argued that a $\sqrt{T\overline{T}}$ operator added to a 2d CFT changed the
symmetries to BMS3 and this was not to be thought of in terms of any flow, but
a non-linear map. Later, following Bagchi:2022nvj , it was realised
Tempo:2022ndz ; Parekh:2023xms  that these are indeed interpretable as flows.
This $\sqrt{T\overline{T}}$ type deformation is different Ferko:2022cix  from
$T\bar{T}$ like irrelevant deformations, since here the stress tensors are
just those of the undeformed seed theory, and do not generate iteratively
defined flows. Such a theory with a square-root operator is hard to quantize.
However, from the understanding that $\sqrt{T\overline{T}}$ can be written as
a current-current bilinear operator, our way of looking into flows triggered
by marginal deformations provides insight into the quantum nature of these
square-root deformations. The relation between $\sqrt{T\overline{T}}$ and the
current bilinears at the quantum level of course depends on operator ordering
and this needs to be carefully treated.

  * $\square$

Holography. Carrollian symmetries are generically associated with null
manifolds. Two of the most important ones are the horizons of black holes and
the null boundary of asymptotically flat spacetimes. The deformation of the
relativistic 2d scalar theory can be taken as an example of something generic
happening to any 2d CFT. In the context of AdS/CFT, if we think of the
deformation taking the theory on the boundary and moving it into the bulk, the
Carroll CFTs that emerge should be the end points of these flows. The very
deep bulk region of AdS can be thought of as flat space where the effects of
the AdS radius cannot be felt. From the point of view of symmetries, this can
be thought of as a change of symmetries from $SO(d,2)$ to $ISO(d,1)$. This is
precisely the global subgroup of the conformal Carroll algebra. For
asymptotically AdS spacetimes with black holes, the flow can be thought to end
on the horizon of the black hole in the bulk. The symmetries there again would
be Carrollian Donnay:2019jiz .

  * $\square$

c-theorem. We are advocating that the degrees of freedom of a relativistic QFT
should decrease at very high energies, where the slow moving modes can be
disregarded. This seems at complete odds with the c-theorem
Zamolodchikov:1986gt ; Cardy:1996xt . The answer possibly is that the
transition from relativistic to Carroll symmetries, which is the restriction
of a relativistic QFT to its high energy subsector, breaks inherent
assumptions of the c-theorem, like Poincare invariance and unitarity. We would
like to examine this in more detail going forward.

  * $\square$

Renormalisation group flows in Carrollian theories: The “top down” approach.
In the intriguing new work Banerjee:2023jpi , a RG procedure customized to
suit intrinsic Carrollian scalar field theories has been examined. For Carroll
theories, energy scales and spatial scales are separated in contradiction to
Wilsonian intuition, leading to drastic changes in the notion of canonical
mass dimensions of operators. As a consequence, a spatial derivative added to
a purely Carrollian electric scalar ($\mathcal{L}\sim\dot{\phi}^{2}$) becomes
relevant as we move towards the IR, ultimately reinstating Lorentz invariance
in the process. One can think of this as a complimentary version of the
picture presented here, where we have marginal deformations of scale invariant
scalars triggering flows which eventually kill Lorentz invariance, and leads
to a Carrollian electric scalar theory. It would be of fundamental importance
to understand this dual interplay between local and ultra-local theories while
going forward.

  * $\square$

Re-interpreting Wilson. Wilson’s original formulation of RG flows
RevModPhys.47.773  was in terms of “slow” and “fast” modes, where one
integrated over the fast modes in order to get an effective theory of slow
modes valid in the lower energy regime. We have just mentioned that in the
intrinsically Carrollian theory, the notions of operator dimension and hence
RG flows change drastically Banerjee:2023jpi . The reduction of degrees of
freedom of the relativistic theory, as it enters a very high energy domain
where only modes traveling at or near lightspeed are of importance, might be
an observer dependent redistribution of what are “slow” modes and “fast” modes
to this high energy observer. Perhaps the Wilsonian paradigm can still be made
to work, but the observer dependence has to be carefully reconsidered212121We
thank Shahin Sheikh-Jabbari for illuminating conversations on this point and
stressing this to us..

It is clear that our current work has raised numerous interesting issues and
indicated new directions of research. We hope to be back with more details on
the points discussed above as well as other related problems very soon.

## Acknowledgements

The authors would like to thank Rudranil Basu, Joydeep Chakrabortty, Ritankar
Chatterjee, Stephane Detourney, Priyadarshini Pandit and especially Shahin
Sheikh-Jabbari for many fruitful discussions.

AB is partially supported by a Swarnajayanti Fellowship of the Department of
Science and Technology and Science and Engineering Research Board (SERB) and
also by the following SERB grants SB/SJF/2019-20/08, CRG/2022/006165. ArB is
supported in part by an OPERA grant and a seed grant NFSG/PIL/2023/P3816 from
BITS-Pilani. ArB would like to thank Kyoto University, TU Wien, CNRS Paris and
IIT Kanpur for kind hospitality during the various stages of this work. SM is
supported by grant number 09/092(1039)/2019-EMR-I from Council of Scientific
and Industrial Research (CSIR). DM is supported by a Young Scientist Training
Program (YST) Fellowship from Asia Pacific Center for Theoretical Physics
(APCTP). The work of HM is supported by the Institute for Basic Science
(IBS-R003-D1).

## APPENDICES

##  Appendix A Computations of Poisson brackets

### a) Sugawara construction for stress tensors

To touch base, let us explicitly show how we arrive at the algebra of
undeformed Sugawara stress tensors, in this case, for the anti-holomorphic
one:

$\displaystyle\left\\{\overline{T}(\sigma),\overline{T}\left(\sigma^{\prime}\right)\right\\}_{\mathrm{PB}}$
$\displaystyle=\frac{1}{4}\sum_{i,j}\left\\{\overline{{J}}_{i}^{2}(\sigma),\overline{{J}}_{j}^{2}(\sigma^{\prime})\right\\}_{\mathrm{PB}}$
$\displaystyle=\sum_{i,j}\left\\{\overline{{J}}_{i}(\sigma),\overline{{J}}_{j}(\sigma^{\prime})\right\\}_{\mathrm{PB}}\overline{{J}}_{i}(\sigma)\overline{{J}}_{j}(\sigma^{\prime})$
$\displaystyle=-\sum_{i}\left[\partial_{\sigma}\delta(\sigma-\sigma^{\prime})\right]\overline{{J}}_{i}(\sigma)\overline{{J}}_{i}(\sigma^{\prime})$
$\displaystyle=-\sum_{i}\left[\partial_{\sigma}\delta(\sigma-\sigma^{\prime})\right]\overline{{J}}_{i}(\sigma)\left\\{\overline{{J}}_{i}(\sigma)-(\sigma-\sigma^{\prime})\partial_{\sigma}\overline{{J}}_{i}(\sigma)+\mathcal{O}((\sigma-\sigma^{\prime})^{2})\right\\}$
$\displaystyle=-2\overline{T}(\sigma)\partial_{\sigma}\delta\left(\sigma-\sigma^{\prime}\right)-\left(\partial_{\sigma}\overline{T}(\sigma)\right)\delta\left(\sigma-\sigma^{\prime}\right)$
(149)

### b) Emergent BMS

We show here the Poisson bracket computation for the deformed generators as
described in (106). We only focus on one bracket and others can follow:

$\displaystyle\left\\{L^{(g)}(\sigma),L^{(g)}(\sigma^{\prime})\right\\}_{\rm
PB}$
$\displaystyle=\left\\{(\Pi_{+}{\chi}^{\prime}_{+}+\Pi_{-}{\chi}^{\prime}_{-})(\sigma),(\Pi_{+}{\chi}^{\prime}_{+}+\Pi_{-}{\chi}^{\prime}_{-})(\sigma^{\prime})\right\\}_{\rm
PB}$
$\displaystyle=\Pi_{+}(\sigma){\chi}^{\prime}_{+}(\sigma^{\prime})\left\\{{\chi}^{\prime}_{+}(\sigma),\Pi_{+}(\sigma^{\prime})\right\\}_{\rm
PB}+{\chi}^{\prime}_{+}(\sigma)\Pi_{+}(\sigma^{\prime})\left\\{\Pi_{+}(\sigma),{\chi}^{\prime}_{+}(\sigma^{\prime})\right\\}_{\rm
PB}+({\,}_{+}\,\to\,{\,}_{-})$
$\displaystyle=\left\\{\Pi_{+}(\sigma){\chi}^{\prime}_{+}(\sigma^{\prime})+{\chi}^{\prime}_{+}(\sigma)\Pi_{+}(\sigma^{\prime})\right\\}\partial_{\sigma}\delta(\sigma-\sigma^{\prime})+({\,}_{+}\,\to\,{\,}_{-})$
$\displaystyle=\left\\{2\Pi_{+}(\sigma){\chi}^{\prime}_{+}(\sigma)-(\sigma-\sigma^{\prime})\Pi_{+}(\sigma){\chi}^{\prime\prime}_{+}(\sigma)-(\sigma-\sigma^{\prime}){\chi}^{\prime}_{+}(\sigma)\Pi^{\prime}_{+}(\sigma)\right\\}\partial_{\sigma}\delta(\sigma-\sigma^{\prime})+({\,}_{+}\,\to\,{\,}_{-})$
$\displaystyle=2\Pi_{+}{\chi}^{\prime}_{+}\partial_{\sigma}\delta(\sigma-\sigma^{\prime})+(\Pi_{+}{\chi}^{\prime}_{+})^{\prime}\delta(\sigma-\sigma^{\prime})+({\,}_{+}\,\to\,{\,}_{-})$
$\displaystyle=2L^{(g)}(\sigma)\partial_{\sigma}\delta(\sigma-\sigma^{\prime})+\left(\partial_{\sigma}L^{(g)}(\sigma)\right)\delta(\sigma-\sigma^{\prime})$
(150)

In both of the above cases we have used the formula for simplification

$(x-a)^{n+1}\partial_{x}\delta(x-a)=-\delta_{n,0}\,\delta(x-a).$ (151)

Rest of our poisson brackets can be calculated likewise.

##  Appendix B Going Galilean with $J\overline{J}$

As we discussed in the main text, going to the Galilean theory using a
$J\bar{J}$ deformation is much more subtle. Remember that classically (41)
makes sure the action goes to a Galilean boost invariant one when $\alpha=-1$
and the characteristic speed of light goes to infinity. In the quantum theory,
things are not so quick since in this case, there is an inherent flip between
space and time directions as compared to Carroll. Hence one has to consider
generators and their contractions accordingly. The most important ingredient
in this story is an automorphism at the level of Virasoro algebra, viz,
$\mathcal{L}_{n}\to\mathcal{L}_{n}$ and
$\overline{\mathcal{L}}_{n}\to-\overline{\mathcal{L}}_{-n}$ which keeps the
commutation relations (27) unchanged. Starting with this definition one can
build what is known as the “flipped representation” of the Virasoro algebra
Bagchi:2020fpr . At the level of stress tensors (26), this results in
$\overline{T}(\sigma)=-\overline{T}(\sigma)$ and can be equivalently taken
care of by replacing:

$\bar{j}_{n}\to i\bar{j}_{-n}$ (152)

in the Sugawara construction. Note that under this automorphism, the nature of
the erstwhile relativistic highest-weight vacuum changes to a hybrid of
highest-weight-lowest-weight case.

Following through with the addition of the $J\bar{J}$ term and the computation
of deformed stress tensors, one can find the generators of the symmetry
algebra in this case reads:

$\displaystyle\mathcal{J}_{n}(\alpha)^{NR}=\mathcal{L}_{n}(\alpha)+\bar{\mathcal{L}}_{n}(\alpha),\quad\mathcal{H}_{n}(\alpha)^{NR}=\frac{1}{(\cosh^{2}\theta+\sinh^{2}\theta)}\bigg{(}\mathcal{L}_{n}(\alpha)-\bar{\mathcal{L}}_{n}(\alpha)\bigg{)}.$
(153)

Note that these definitions flip between the canonical Hamiltonian and angular
momentum generators, a fact which we alluded to in the main text. Also note
these generators do not contain any mixing of modes unlike the ones in (53).
This stems from the fact that when we crank up the coupling of the $J\bar{J}$
term, the vacuum structure does not change as (48) does not imply a Bogoliubov
transformation owing to (152). However, the above generators still lead us to
the interpolating algebra in (54), which still degenerate to BMS3 at the same
point as in the Carrollian case. This is expected, since as we mentioned
before, in two dimensions, the algebras of Carrollian and Galilean conformal
group are isomorphic, and these above generators (153) then just become well
known Galilean Conformal Field Theory (GCFT) generators Bagchi:2009my .

##  Appendix C String action in the presence of B-field

The string sigma model in a general background with a NS-NS flux $(G,B)$ is
described by

$S=\frac{1}{2}\int
d^{2}\sigma\left(\sqrt{-h}\,h^{ab}\partial_{a}X^{\mu}\partial_{b}X^{\nu}G_{\mu\nu}+\epsilon^{ab}\partial_{a}X^{\mu}\partial_{b}X^{\nu}B_{\mu\nu}\right)$
(154)

where the string tension $\alpha^{\prime}$ is appropriately normalized. Let’s
choose a conformal gauge for the worldsheet

$h_{ab}=\begin{pmatrix}e^{2\omega}&0\\\ 0&-e^{2\omega}\end{pmatrix}$ (155)

and noting $\epsilon^{01}=-\epsilon^{10}=1$, then

$S_{\rm c.g.}=\int
d^{2}\sigma\left(\frac{1}{2}\dot{X}^{\mu}\dot{X}^{\nu}G_{\mu\nu}-\frac{1}{2}{X^{\prime}}^{\mu}{X^{\prime}}^{\nu}G_{\mu\nu}+\dot{X}^{\mu}{X^{\prime}}^{\nu}B_{\mu\nu}\right).$
(156)

Assuming the background field $B$ to be a constant, the conjugate momenta are

$\Pi_{\mu}=\frac{\delta
S_{c.g.}}{\delta\dot{X}^{\mu}}=\dot{X}^{\nu}G_{\mu\nu}+{X^{\prime}}^{\nu}B_{\mu\nu}$
(157)

which leads to:

$\dot{X}^{\mu}=G^{\mu\rho}\Pi_{\rho}-G^{\mu\rho}B_{\rho\sigma}{X^{\prime}}^{\sigma}$
(158)

The associated Hamiltonian density is obtained by performing the Legendre
transform

$\mathcal{H}=\Pi_{\mu}\dot{X}^{\mu}-\mathcal{L}=\frac{1}{2}\dot{X}^{\mu}\dot{X}^{\nu}G_{\mu\nu}+\frac{1}{2}{X^{\prime}}^{\mu}{X^{\prime}}^{\nu}G_{\mu\nu}$
(159)

which turns out to be only implicitly dependent on $B$. Re-expressing the
Hamiltonian density in terms of a set of canonical variables,

$\displaystyle\mathcal{H}^{(G,B)}$
$\displaystyle=\frac{1}{2}G^{\mu\nu}\Pi_{\mu}\Pi_{\nu}-\Pi_{\mu}G^{\mu\rho}B_{\rho\nu}{X^{\prime}}^{\nu}+\frac{1}{2}\left(-B_{\sigma\nu}G^{\nu\alpha}B_{\alpha\rho}\right){X^{\prime}}^{\rho}{X^{\prime}}^{\sigma}+\frac{1}{2}{X^{\prime}}^{\mu}{X^{\prime}}^{\nu}G_{\mu\nu}$
(160)

For a particular setting of our particular interest: two-dimensional target
space with the B-field given by

$B_{\mu\nu}=\Lambda\,\epsilon_{\mu\nu}\qquad{\rm
with}\qquad\epsilon_{12}=-\epsilon_{21}=1$ (161)

one finds

$-B_{\sigma\nu}G^{\nu\alpha}B_{\alpha\rho}=\Lambda^{2}\,G^{-1}G_{\rho\sigma}\qquad\quad
G={\rm det}\,G_{\mu\nu}$ (162)

thus the Hamiltonian density can be written as:

$\mathcal{H}^{(G,\Lambda)}=\frac{1}{2}G^{\mu\nu}\Pi_{\mu}\Pi_{\nu}-\Lambda\,\Pi_{\mu}G^{\mu\rho}\epsilon_{\rho\nu}{X^{\prime}}^{\nu}+\frac{1}{2}{X^{\prime}}^{\mu}{X^{\prime}}^{\nu}G_{\mu\nu}(1+\Lambda^{2}\,G^{-1})$
(163)

Furthermore, setting $G_{\mu\nu}=\delta_{\mu\nu}$ with $\delta_{\mu\nu}={\rm
diag}\,(1,1)$ yields the Hamiltonian density

$\mathcal{H}^{(\delta,\Lambda)}=\frac{1}{2}\Pi_{\mu}\Pi^{\mu}+\Lambda\,\epsilon_{\mu\nu}{X^{\prime}}^{\mu}\Pi^{\nu}+\frac{1}{2}{X^{\prime}}^{\mu}{X^{\prime}}_{\mu}(1+\Lambda^{2}).$
(164)

By rescaling the above the dynamical fields $X\to X/\sqrt{1+\Lambda^{2}}$
while $\Pi\to\sqrt{1+\Lambda^{2}}\,\Pi$, and replacing $\Lambda\to-g$ as well
as redefining $X^{\mu}=X_{\mu}\to\phi_{i}$, we see

$\mathcal{H}^{(g)}=\frac{1}{2}\sum_{i=1,2}\left\\{(1+g^{2})(\Pi_{i})^{2}+({\phi}^{\prime}_{i})^{2}\right\\}+g\,\left(\Pi_{1}{\phi}^{\prime}_{2}-\Pi_{2}{\phi}^{\prime}_{1}\right)$
(165)

which coincides with our current-current deformed case in (71). A detailed
discussion of the $g\to\infty$ case for the above, i.e. that of a null string
with a Kalb-Ramond field will appear soon in upcoming .

##  Appendix D Hamiltonian structure of general deformations

A general current-current deformation to the multiple scalar field action,
when put into the relevant Hamiltonian form, looks like:

${}\mathcal{H}=A_{ij}\pi^{i}\pi^{j}+B_{ij}\phi^{\prime i}\phi^{\prime
j}+C_{ij}\pi^{i}\phi^{\prime j}.$ (166)

Where $\phi^{i}$ and $\pi^{i}$ are canonically conjugate variables in the
undeformed system and $A,B,C$ are functions of a deformation parameter
$\lambda$, say. We also have

${}\lim_{\lambda\to 0}A_{ij}=\delta_{ij},~{}~{}\lim_{\lambda\to
0}B_{ij}=\delta_{ij},~{}~{}\lim_{\lambda\to 0}C_{ij}=0.$ (167)

Note that even with a general current-current deformation, the Hamiltonian
remains quadratic in conjugate variables $\pi^{i}$ and $\phi^{\prime i}$ since
only these quadratics have the dimension of energy density. The question we
ask at this point: What is the criterion that the above Hamiltonian will
reduce to a Carrollian dynamical system?. Or put into a different form, when
would we have,

${}\\{\mathcal{H}(x),\mathcal{H}(x^{\prime})\\}=0,$ (168)

which is characteristic of Carroll-invariant dynamics. A way to address this
problem would be to put the above Hamiltonian in a normal form (assuming we
work with two scalar fields)

$\mathcal{H}_{d}=a\Pi^{2}_{+}+b\Pi^{2}_{-}+c\phi^{\prime 2}_{+}+d\phi^{\prime
2}_{-},$ (169)

such that $\Pi_{+},\phi^{\prime}_{+}$ and $\Pi_{-},\phi^{\prime}_{-}$ are
canonically conjugate set of variables. Now we demand that in some asymptotic
limit, the eigenvalues $a,b,c,d$, having quadratic conjugate structure amongst
them, will either pairwise vanish or pairwise diverge, so that we are only
left with (at most) two out of the four dynamical variables. Now consider the
cases:

Case I: a and b (or c and d) survive $\rightarrow$ Electric (Magnetic) Carroll
dynamics.
Case II: a and d (or b and c) survive$\rightarrow$ Mix of Electric/Magnetic
Carroll dynamics.
Case III: a and c (or b and d) survive $\rightarrow$ A single scalar field.

For example, consider the asymptotic structure of a generic Hamiltonian
density concerning two scalar fields:

$\mathcal{H}=\frac{\mathcal{N}_{1}}{2}\left\\{\epsilon_{1}\Pi_{1}^{2}+\frac{1}{\epsilon_{1}}(\phi^{\prime}_{1})^{2}\right\\}+\frac{\mathcal{N}_{2}}{2}\left\\{\epsilon_{2}\Pi_{2}^{2}+\frac{1}{\epsilon_{2}}(\phi^{\prime}_{2})^{2}\right\\}.$
(170)

Clearly, depending on the eigenvalues of the general Hessian matrix and the
asymptotic behaviour of $\mathcal{N}_{i}$ and $\epsilon_{i}$, we can reach
either of the three cases mentioned above. This logic is readily extended to
higher number of scalar fields as well. For this to work, one needs to be able
to symplectically diagonalize Hamiltonians of the form (169) in terms of
canonically conjugate variables. In classical mechanics, such a symplectic
diagonalization is possible in case the criterion of Williamson’s Theorem
Arnold:1989who  are met, but an extension to field theories seem absent from
the literature.

##  Appendix E Spectral flow for WZW currents

Let us review the central results pertaining to the spectral flow of
$SL(2,\mathbb{R})$ WZW theory following Maldacena:2000hw ; McElgin:2015eho ;
Hemming:2004gd  to better appreciate the results in Sec.(4). The currents in
the WZW model can be expressed in terms of the components $t^{a}$ of the Lie
algebra as

$J=J^{a}t^{a}\quad\text{and}\quad J^{a}=\text{Tr}(t^{a}J)\ ,$ (171)

where the currents are explicitly given by

$J(z)=-\frac{k}{2}\partial_{z}gg^{-1}\quad\text{and}\quad\bar{J}(\bar{z})=\frac{k}{2}g^{-1}\partial_{\bar{z}}g\
.$ (172)

$g$ is a group element of the Lie group which in our case is
$SL(2,\mathbb{R})$. Choosing the specific parametrization of the group element
$g$222222This parametrization helps in recovering the BTZ geometry from the
WZW action outside the black hole.

$g=e^{u\sigma_{3}}e^{\rho\sigma_{1}}e^{-v\sigma_{3}}\ ,$ (173)

and choosing a convenient basis for $SL(2,\mathbb{R})$ as,

$t^{+}=t^{0}+t^{1}=\frac{1}{2}(-\sigma^{2}+i\sigma^{1})\ ,\
t^{-}=t^{0}-t^{1}=\frac{1}{2}(-\sigma^{2}-i\sigma^{1})\ ,\
t^{3}=\frac{i}{2}\sigma^{3}\ ,$ (174)

we see that the current follows the algebra:

$\displaystyle[J^{3}_{n},J^{\pm}_{m}]$ $\displaystyle=$ $\displaystyle\pm
iJ^{\pm}_{n+m}\ ,$ (175) $\displaystyle{[J^{+}_{n},J^{-}_{m}]}$
$\displaystyle=$ $\displaystyle-2iJ^{3}_{n+m}-kn\delta_{n+m,0}\ ,$ (176)
$\displaystyle{[J^{3}_{n},J^{3}_{m}]}$ $\displaystyle=$
$\displaystyle\frac{k}{2}n\delta_{n+m,0}\ .$ (177)

In the above set of definitions, the $\sigma^{i}$’s are the Pauli matrices and
$u,\rho$ and $v$ can be interpreted as target space coordinates. Note that the
_abelian sector_ of the above algebra generated by $J^{3}_{n}$-s follows an
_abelian Kac-Moody algebra_. The chiral symmetry of the WZW model now enables
us to generate new solutions as

$g(x^{+},x^{-})\rightarrow
e^{-iw_{+}x^{+}t^{3}}g(x^{+},x^{-})e^{iw_{-}x^{-}t^{3}}\ ,$ (178)

where $x^{\pm}=\tau\pm\sigma$ and $w_{\pm}$ parametrize the _spectral flow_.
Under this spectral flow, the _abelian_ component of the currents232323There
are of course two non-abelian components which transform under spectral flow
as $J^{\pm}\rightarrow\tilde{J}^{\pm}\equiv J^{\pm}e^{\mp w_{+}x^{+}}$ and
$\bar{J}^{\pm}\rightarrow\tilde{\bar{J}}^{\pm}\equiv\bar{J}^{\pm}e^{\mp
w_{-}x^{-}}$ while their modes transform as $J_{n}^{\pm}\rightarrow
J^{\pm}_{n\pm iw_{+}}$ and $\bar{J}_{n}^{\pm}\rightarrow\bar{J}^{\pm}_{n\pm
iw_{-}}$ respectively. However, we for the purpose of our current work, we
focus on the sector following an _abelian_ Kac-Moody algebra. transform as

$\displaystyle J^{3}\rightarrow\tilde{J}^{3}\equiv J^{3}+\frac{k}{2}w_{+}\ $ ,
$\displaystyle\
\bar{J}^{3}\rightarrow\tilde{\bar{J}}^{3}\equiv\bar{J}^{3}-\frac{k}{2}w_{-}\
.$ (179)

The transformation of the corresponding modes are given by

$\displaystyle J^{3}_{n}\rightarrow J^{3}_{n}+\frac{k}{2}w_{+}\delta_{n,0}\ $
, $\displaystyle\
\bar{J}^{3}_{n}\rightarrow\bar{J}^{3}_{n}-\frac{k}{2}w_{-}\delta_{n,0}\ .$
(180)

Using the usual Sugawara construction, we can further define the stress tensor
as

$T=\frac{1}{k}\eta_{ab}J^{a}J^{b}=\frac{1}{2k}(J^{+}J^{-}+J^{-}J^{-}-2J^{3})\
.$ (181)

Under spectral flow the above transforms as

$T\rightarrow
T+w_{+}J^{3}+\frac{k}{4}w_{+}^{2}\quad,\quad\bar{T}\rightarrow\bar{T}-w_{-}\bar{J}^{3}+\frac{k}{4}w_{-}^{2}\
.$ (182)

Finally, the transformation rule for the Virasoro generators are given by

$L_{n}\rightarrow