aainstitutetext: Indian Institute of Technology Kanpur, Kalyanpur, Kanpur
208016. India.
bbinstitutetext: Centre de Physique Théorique, Ecole Polytechnique de Paris,
91128 Palaiseau Cedex, France.
ccinstitutetext: NORDITA, Hannes Alfvéns väg 12, 106 91 Stockholm, Sweden
ddinstitutetext: Institute for Theoretical Physics, TU Wien,Wiedner
Hauptstrasse 8-10/136, 1040 Vienna, Austria.

# BMS modular covariance and structure constants

Arjun Bagchi a Saikat Mondal a Sanchari Pal d and Max Riegler
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Two-dimensional (2d) field theories invariant under the Bondi-Metzner-Sachs
algebra, or 2d BMSFTs in short, are putative holographic duals of Einstein
gravity in 3d asymptotically flat spacetimes. When defined on a torus, these
field theories come equipped with a modified modular structure. We use the
modular covariance of the BMS torus two-point function to develop formulae for
different three-point structure constants of the field theory. These structure
constants indicate that BMSFTs follow the eigenstate thermalization
hypothesis, albeit with some interesting changes to usual 2d CFTs. The
singularity structures of the structure constants contain information on
perturbations of cosmological horizons in 3d asymptotically flat spacetimes,
which we show can also be obtained as a limit of BTZ quasinormal modes.

##  1 Introduction

The success of the AdS/CFT correspondence Maldacena:1998zhr ; Witten:1998qj ;
Gubser:1998bc  leading to a deeper understanding of theories of quantum
gravity in terms of lower-dimensional quantum field theories calls for a more
generalized understanding of the holographic principle. In this context,
understanding holography for asymptotically flat spacetimes is particularly
important since most of the real-world physical processes, e.g., processes
involving astrophysical black holes, happen in approximately flat spacetime.

One of the precursors of the AdS/CFT correspondence was the analysis of
asymptotic symmetries of AdS3 in Einstein gravity by Brown and Henneaux
Brown:1986nw . They showed that with suitable boundary conditions, the
symmetries at the asymptotic boundary are enhanced to two copies of the
infinite-dimensional Virasoro algebra, thereby providing a connection between
gravity and two-dimensional conformal field theories. These infinite-
dimensional asymptotic symmetry algebras are not unique to AdS3 but were also
found in asymptotically Minkowski spacetimes in the 1960’s by Bondi, van der
Burg, Metzner Bondi:1962px  and independently by Sachs Sachs:1962zza . They
showed that in asymptotically flat four-dimensional (4d) spacetime, the
symmetry algebra at null infinity gets enhanced from the usual Poincare
algebra to an infinite dimensional algebra now called $\mathrm{BMS}_{4}$
algebra and is given by

$\displaystyle[L_{n},L_{m}]$
$\displaystyle=(n-m)L_{m+n},\quad[\bar{L}_{n},\bar{L}_{m}]=(m-n)\bar{L}_{m+n},$
$\displaystyle[L_{m},M_{p,q}]$
$\displaystyle=\left(\frac{m+1}{2}-p\right)M_{m+p,q},\quad[\bar{L}_{m},M_{p,q}]=\left(\frac{m+1}{2}-q\right)M_{p,m+q},$
$\displaystyle[M_{p,q},M_{r,s}]$ $\displaystyle=0,$ (1)

where $n,m=0,\pm 1$ and all other indices run through all integers.
$M_{r,s}$’s are the generators of supertranslations that depend on the angles
of the celestial sphere at null infinity and form an infinite-dimensional
subgroup. The $L_{n}$’s are the Lorentz generators. The sub-algebra spanned by
$L_{n}$ and $\bar{L}_{n}$ gets enhanced to two copies of the Witt algebra if
one relaxes the condition that generators need to be globally well defined
Barnich:2010eb . This extension of the BMS4 algebra has been used
predominantly in the literature when considering aspects of holography in 4d
asymptotically flat spacetimes. 111Other extensions, including the enhancement
of the global conformal group on the sphere to the full diffeomorphism group
Diff(S2) Campiglia:2014yka  have also been investigated.

The investigation of holography in asymptotically flat spacetimes has seen a
resurgence in recent years and efforts have followed two distinct paths now
called Celestial and Carrollian holography. Celestial holography
Strominger:2013jfa ; He:2014laa ; Strominger:2014pwa  makes use of the fact
that the Lorentz group in the bulk of flat space acts as the global conformal
group on the null boundary and has been very successful in a novel
understanding of S-matrices in terms of a dual 2d Celestial CFT that lives on
the celestial sphere Pasterski:2016qvg ; Pasterski:2017kqt ; Banerjee:2018gce
. This approach yielded lots of new physics about scattering amplitudes and
asymptotic symmetries. We refer the reader to the wonderful reviews
Raclariu:2021zjz ; Pasterski:2021rjz ; Strominger:2017zoo  for more details.

Carrollian holography, on the other hand, seems to be more natural from the
point of view of the original ideas of holography and proposes a co-dimension
one holographic dual where one not only uses the Lorentz sub-group of the
Poincare group but fits translations naturally into the formulation
Bagchi:2016bcd . Recent successes include Donnay:2022wvx ; Bagchi:2022emh ;
Donnay:2022aba ; Bagchi:2023fbj ; Nguyen:2023vfz ; Saha:2023hsl . The initial
success of the formulation was in lower dimensions, specifically in
understanding 3d asymptotically flat spacetime, which we describe below.

In three dimensions, the asymptotic symmetry algebra is infinite-dimensional
and is given by the centrally extended BMS3 algebra Barnich:2006av :

$\displaystyle[L_{n},L_{m}]$
$\displaystyle=(n-m)L_{n+m}+c_{L}(n^{3}-n)\delta_{n+m,0},$
$\displaystyle[L_{n},M_{m}]$
$\displaystyle=(n-m)M_{n+m}+c_{M}(n^{3}-n)\delta_{n+m,0},$
$\displaystyle[M_{n},M_{m}]$ $\displaystyle=0.$ (2)

In the above, $M_{n}$’s are the generators of supertranslations that depend on
one index corresponding to the Fourier modes of the celestial circle at null
infinity (as opposed to the 2-sphere in the 4d case). The $L_{n}$’s are the
superrotations that form the algebra of diffeomorphisms of the celestial
circle at null infinity. This algebra admits central extensions $c_{L}$ and
$c_{M}$. For Einstein gravity, $c_{L}=0$ and $c_{M}=3/G$ Barnich:2006av . One
can have non-zero values of $c_{L}$ when considering theories beyond Einstein
gravity, e.g., Topologically Massive Gravity Bagchi:2012yk .

Motivated by the construction of Brown and Henneaux and its subsequent
understanding in terms of AdS3/CFT2, the basic observation of flat space
holography in this formulation is to propose that the algebra (1) also governs
the putative 2d dual field theory living on the null boundary of flat
spacetime Bagchi:2010zz ; Bagchi:2012cy . This algebra can also be obtained by
taking the speed of light to zero on the dual CFT Bagchi:2012cy , which
motivated the recent nomenclature of Carrollian holography, following the
Carrollian contraction of the Poincare algebra first obtained in the 1970s
galileigroup ; Levy . The process of getting from AdS to flat space by sending
the radius of AdS to infinity corresponds to sending the speed of light to
zero in the dual field theory.

This correspondence between 3d bulk asymptotically flat spacetimes and 2d BMS3
invariant field theories or 2d Carrollian CFTs has been quite successful
during the past decade, with several checks of quantities computed between the
bulk and boundary theories Bagchi:2012xr ; Barnich:2012xq ; Bagchi:2014iea ;
Barnich:2015mui ; Bagchi:2015wna ; Jiang:2017ecm ; Hijano:2017eii . One of the
main avenues of success has been the consideration of modular properties of
these Carroll CFTs and consequences for the analogs of the BTZ black holes
Banados:1992wn ; Banados:1992gq  in 3d flat space which turn out to be
cosmological solutions called Flat Space Cosmologies (FSCs). Modular
invariance of the Carroll partition function leads to the BMS-Cardy formula
and a subsequent matching of the entropy of FSCs counting microstates of the
BMSFT on the boundary Bagchi:2012xr ; Bagchi:2013qva ; Bagchi:2019unf .
Modular covariance of the torus one-point function has resulted in the
understanding of the structure constants, which were also reproduced by a bulk
calculation involving a probe in the FSC geometry Bagchi:2020rwb .

In this paper, we extend our explorations of modular aspects of 2d Carrollian
CFTs to the study of torus two-point functions. Our interest in this study is
two-pronged. The first concerns an exploration of the thermal properties of
Carrollian CFTs. In a 2d CFT, the torus two-point function naturally leads to
the off-diagonal structure constants of the theory Brehm:2018ipf ;
Hikida:2018khg ; Romero-Bermudez:2018dim , which in turn provides an
interpretation in terms of the Eigenstate Thermalization Hypothesis (ETH)
Srednicki:1995pt . ETH is a criterion that determines whether a closed quantum
system thermalizes at late times under unitary evolution. The study of
Carrollian field theories has and continues to reveal many aspects of these
theories that are unfamiliar from the perspective of usual relativistic QFTs.
Among them is the question of thermal equilibrium. Our study of the torus two-
point functions in this work provides some hints of a notion of ETH emerging
in 2d Carroll CFTs, albeit with some interesting differences.

Our final interest in this paper is understanding the singularity structure of
the BMS torus two-point functions that we compute. For the 2d CFT case, the
singularities of the torus two-point function carry information on the quasi-
normal modes of the dual BTZ black hole. In the Carrollian case, which we
primarily focus on in the paper, the structure of poles should shed light on
the quasi-normal modes of FSC solutions in 3d flat spacetimes. We find
interesting singularity structures from different types of torus two-point
functions and comment on their relationship to physics in the bulk.

The paper is organized as follows. We begin in Sec. 2 with a compact review of
3d flat holography from the Carrollian perspective and elaborate modular
aspects of the 2d BMSFTs. Sec. 3 contains the main computations of this work.
Here we derive the off-diagonal structure constants by using modular
covariance of the BMS torus two-point functions. We arrive at two distinctive
formulae. One for temporally separated probes leading to a type of averaged
three-point coefficient, while the other for spatially separated probes leads
to a different class off-diagonal structure constant. In Sec. 4 and 5, we
dwell on the physics arising from these two different types of structure
constants. In Sec. 4, we investigate the Eigenstate Thermalization Hypothesis,
and after a brief detour to 2d CFTs, we show that ETH holds for 2d BMSFTs as
well, with some interesting changes. In Sec. 5, we investigate the analogs of
quasinormal modes (QNM) of FSCs arising from the singularity structure of our
obtained formulae for the structure constants. Rather remarkably, we recover
our very different answers for the singularity structure of these two types of
BMS structure constants as the leading and subleading pieces of the BTZ QNM.
We conclude in Sec. 6 with a summary of our results and some comments.

##  2 Flat holography and BMS modular transformations

The asymptotic symmetries of 3d asymptotically flat spacetimes at null
infinity are given by the infinite-dimensional centrally extended BMS3 algebra
(1). The central message of the Carrollian version of holography, now adapted
for 3d flat spacetimes, is that the dual field theory is a 2d theory that
lives on the entire null boundary $(\mathscr{I}^{\pm})$, not just the circle
at infinity and it inherits the underlying symmetry algebra given above (1).
We call these theories 2d BMS invariant field theories or 2d BMSFTs for short.
These 2d BMSFT defined on $\mathscr{I}^{\pm}$ also inherit the following
metric

$\displaystyle ds^{2}=0\times du^{2}+d\theta^{2}.$ (3)

Here $u$ is the retarded (null) time direction, and $\theta$ is the angular
direction of the celestial circle at null infinity. We now focus on a single
BMSFT living on $\mathscr{I}^{+}$. The topology of the null boundary is that
of a cylinder $\mathbb{R}_{u}\times S^{1}$.

Flat spacetime can be understood as an infinite radius222Or in other words a
limit of vanishing comsmological constant. limit of AdS. Hence it is natural
to attempt a formulation of flat holography in terms of a limit of AdS/CFT
where the AdS radius is taken to infinity. This singular limit in the bulk
amounts to sending the speed of light to zero in the boundary CFT
Bagchi:2012cy , and results in what is called a Carrollian CFT. The
isomorphism between the conformal Carroll algebra that is obtained as a
contraction of the relativistic conformal algebra and the BMS algebra in one
dimension higher Bagchi:2010zz ; Duval:2014uva  is at the heart of this. The
2d Conformal Carroll or BMS3 algebra can be obtained by an Inönü-Wigner
contraction of two copies of the Virasoro algebra

$\displaystyle L_{n}=\mathcal{L}_{n}-\bar{\mathcal{L}}_{-n},\quad
M_{n}=\epsilon(\mathcal{L}_{n}+\bar{\mathcal{L}}_{-n}),$ (4)

where $\mathcal{L}_{n}$ and ${\bar{\mathcal{L}}}_{n}$ are the Virasoro
generators. The parameter $\epsilon$ can be thought of as the inverse of the
AdS radius in the bulk and the speed of light on the boundary theory.

In the construction of Carroll holography for 3d asymptotically flat
spacetimes, it is convenient to work with the highest-weight
representation333Though also other representations have been considered
previously in the literature. See, e.g., Campoleoni:2015qrh ;
Campoleoni:2016vsh ; Oblak:2016eij . of the algebra (1). In this
representation states are labeled by the $L_{0}$ and $M_{0}$ eigenvalues,
given by $\Delta$ and $\xi$ respectively Bagchi:2009ca . We denote the states
as $\ket{\Delta,\xi}$:

$L_{0}\ket{\Delta,\xi}=\Delta\ket{\Delta,\xi},\quad
M_{0}\ket{\Delta,\xi}=\xi\ket{\Delta,\xi}.$ (5)

The action of $L_{n}$ and $M_{n}$ lowers the $\Delta$ eigenvalue of the
states. If we take the spectrum of $\Delta$ to be bounded from below, there
are states $\ket{\Delta_{p},\xi_{p}}\equiv\ket{\Delta,\xi}_{p}$ for which this
eigenvalue cannot be lowered further and

$\displaystyle
L_{n}\ket{\Delta,\xi}_{p}=M_{n}\ket{\Delta,\xi}_{p}=0,\quad\forall n>0.$ (6)

These states $\ket{\Delta,\xi}_{p}$ are called BMS primary states. A tower of
descendant states can be obtained by acting on the state
$\ket{\Delta,\xi}_{p}$ with $L_{-n}$ and $M_{-n}$ with $n>0$. The primary
state $\ket{\Delta,\xi}_{p}$, the infinite tower of descendant states and the
central charges $c_{L}$ and $c_{M}$ together make up a BMS module
$\mathcal{B}(\Delta,\xi,c_{L},c_{M})$.

### Modular properties of BMSFTs

The partition function for a 2d BMSFT is given by

$\displaystyle Z_{\mathrm{BMS}}(\sigma,\rho)=\mathrm{Tr}\left[e^{2\pi
i\sigma\left(L_{0}-\frac{c_{L}}{2}\right)}e^{2\pi
i\rho\left(M_{0}-\frac{c_{M}}{2}\right)}\right].$ (7)

For a 2d BMSFT defined on a null torus, a notion of modular transformations
descends from the parent relativistic 2d CFT Bagchi:2012xr ; Bagchi:2013qva .
This is given by

$\sigma\to\frac{a\sigma+b}{c\sigma+d},\quad\rho\to\frac{\rho}{(c\sigma+d)^{2}}.$
(8)

For the specific case of the $S$-transformation, which will be very useful for
us below, this reads:

$S:\quad(\sigma,\rho)\to\left(-\frac{1}{\sigma},\frac{\rho}{\sigma^{2}}\right).$
(9)

The above can be understood as an exchange of circles of the deformed torus,
where one of the circles becomes infinitesimally small (but crucially does not
completely collapse the torus to a circle).

We assume that the partition function $Z_{\mathrm{BMS}}$ (which is the torus
zero-point function) is invariant under the BMS modular transformations (8)
and exploit the invariance under the $S$-transformation to arrive at the BMS
version of the Cardy formula:

$S_{\mathrm{BMS-Cardy}}=\log
d(\Delta,\xi)=2\pi\left(c_{L}\sqrt{\frac{2\xi}{c_{M}}}+\Delta\sqrt{\frac{c_{M}}{2\xi}}\right).$
(10)

In the above, $d(\Delta,\xi)$ is the asymptotic density of states. One can
further exploit the modular covariance of the torus one-point function to
arrive at a Cardy-like formula for the asymptotic structure constants of a 2d
BMSFT

$\displaystyle C_{\Delta\mathcal{O}\Delta}\approx
C_{\Delta_{\chi}\mathcal{O}\Delta_{\chi}}i^{-\Delta_{\mathcal{O}}}e^{\frac{\xi_{\mathcal{O}}}{2}\left(-\frac{\Delta}{\xi}+\frac{c_{L}}{c_{M}}\right)}e^{-2\pi\left(\sqrt{\frac{2\xi}{c_{M}}}\Delta_{\chi}+\sqrt{\frac{\xi}{2c_{M}}}\left(\frac{\Delta}{\xi}-\frac{c_{L}}{c_{M}}\right)\xi_{\chi}\right)}$
(11)

In this formula above, $(\Delta,\xi)$, $(\Delta_{\chi},\xi_{\chi})$ and
$(\Delta_{\mathcal{O}},\xi_{\mathcal{O}})$ denote the weights of the
heavy444Heavy (Light) in this context means that the BMS weights $\xi$ and
$\Delta$ are large (small) in comparison to the central charges $c_{M}$ and
$c_{L}$. background, the lowest non-vacuum primary ${\chi}$ and a light probe
${\mathcal{O}}$ respectively 555We have taken the degeneracy of the vacuum and
that of $\chi$ to be one here. Otherwise there would be a factor
$\frac{d(\Delta_{\chi},\xi_{\chi})}{d(0,0)}$ on the right-hand-side..

### Bulk implications

In the parlance of AdS3/CFT2, BTZ black holes are dual to thermal states in a
2d CFT. BTZ black holes are also orbifolds of AdS3. There are similar
orbifolds of 3d Minkowski spacetimes. Of particular interest are the shifted-
boost orbifolds, which give rise to cosmological solutions called Flat Space
Cosmologies (FSC) Cornalba:2002fi ; Cornalba:2003kd . These FSCs can be
described by the following locally Ricci-flat metric

$ds^{2}=-\frac{dr^{2}}{f(r)}+f(r)^{2}dt^{2}+r^{2}(d\phi-N_{\phi}(r)dt)^{2},$
(12)

where

$f(r)=\frac{\hat{r}_{+}^{2}(r^{2}-r_{0}^{2})}{r^{2}},\quad
N_{\phi}(r)=\frac{\hat{r}_{+}r_{0}}{r^{2}}.$ (13)

In the above, $r_{0}$ denotes the FSC horizon and $\hat{r}_{+}=\sqrt{8GM}$.
The coordinates $-\infty<t<\infty$ and $0\leq\phi\leq 2\pi$ are spacelike in
the region outside of the horizon $r>r_{0}$. These objects can also be
obtained in the flat limit ($\ell\to\infty$, where $\ell$ is the AdS radius)
of a non-extremal BTZ. An intriguing aspect of the limit is that only the
inner horizon of the BTZ $(r_{-})$ survives the limit, while the outer horizon
$r_{+}$ gets pushed to infinity. More conretely,

$\displaystyle r_{+}\to\ell\sqrt{8GM}=\ell\hat{r}_{+},\hskip
14.22636ptr_{-}\to r_{0}=\sqrt{\frac{2G}{M}}J,$ (14)

where $M$ and $J$ are the mass and angular momentum of the FSC. Like a BTZ
black holes, an FSC can be associated with having a Hawking temperature
$T_{FSC}=\frac{\hat{r}_{+}^{2}}{2\pi r_{0}}$. The BMS-Cardy entropy can be
shown to exactly match the entropy of the FSC horizon Bagchi:2012xr ;
Bagchi:2013qva ; Bagchi:2019unf . In the dual bulk theory, the three-point
coefficient in BMSFT is interpreted as the expectation value of a probe
operator in the background FSC states Bagchi:2020rwb .

##  3 Off-Diagonal Structure Constant in BMSFT

By exploiting modular covariance of BMSFTs on a torus, it is possible to
derive the asymptotic density of states and an asymptotic formula for
structure constants from the zero-point (partition function) and one-point
functions on the torus. It is natural to take this line of inquiry forward and
explore the implications of what physics the torus two-point function can
elucidate. In the context of AdS3/CFT2, the analogous analysis provides some
interesting insights into various phenomena, including the eigenstate
thermalization hypothesis and the BTZ black hole quasinormal mode spectrum.
With the hope of connecting to similar phenomena in the context of flat
holography, we now delve into the details of this computation.

Using the modular covariance of BMS two-point functions we derive the off-
diagonal structure constants in 2D BMSFT. We start with the expression of a
BMS two-point function on the cylinder:

$\langle\mathcal{O}(u_{1},\theta_{1})\mathcal{O}(u_{2},\theta_{2})\rangle={\sin\left(\frac{\theta_{1}-\theta_{2}}{2}\right)^{-2\Delta_{\mathcal{O}}}}\exp\left[-\xi_{\mathcal{O}}(u_{1}-u_{2})\cot\left(\frac{\theta_{1}-\theta_{2}}{2}\right)\right].$
(15)

Here $u$ and $\theta$ are the (null) time and space coordinates on the
cylinder that describes e.g. $\mathscr{I}^{+}$ in asymptotically flat 3d
spacetime. To get to the torus, we identify the ends of the cylinder. We have
already introduced the modified modular transformations for 2d BMSFTs. This is
given by (8). The elliptic coordinates $(u,\theta)$ of the BMS torus also
change under modular transformations:

$u\to\frac{u}{c\sigma+d}-\frac{\theta\rho
c}{(c\sigma+d)^{2}},\quad\theta\to\frac{\theta}{c\sigma+d}.$ (16)

In a 2d BMSFT, a BMS transformation from a set of coordinates
$(x,y)\to(x^{\prime},y^{\prime})$ leads to a transformation on BMS primaries:

$\displaystyle\tilde{\mathcal{O}}(x^{\prime},y^{\prime})=(\partial_{y}y^{\prime})^{-\Delta}\exp{\xi\frac{\partial_{y}x^{\prime}}{\partial_{y}y^{\prime}}}\mathcal{O}(x,y).$
(17)

This means that under BMS transformations, the two-point function of primaries
transforms as:

$\displaystyle\left\langle\mathcal{O}_{1}(x^{\prime}_{1},y^{\prime}_{1})\mathcal{O}_{2}(x^{\prime}_{2},y^{\prime}_{2})\right\rangle$
$\displaystyle=$
$\displaystyle\prod_{i=1}^{2}\left[(\partial_{y}y^{\prime})^{-\Delta_{i}}\exp(\xi_{i}\frac{\partial_{y}x^{\prime}}{\partial_{y}y^{\prime}})\right]_{\begin{subarray}{c}x=x_{i}\\\
y=y_{i}\end{subarray}}\left\langle\mathcal{O}_{1}(x_{1},y_{1})\mathcal{O}_{2}(x_{2},y_{2})\right\rangle$
(18)

We express the two-point function on the torus with modular parameters
$\sigma$ and $\rho$ as a trace taken over highest-weight states and their
descendants:

$\displaystyle\langle\mathcal{O}(u_{1},\theta_{1})\mathcal{O}(u_{2},\theta_{2})\rangle_{(\sigma,\rho)}$
$\displaystyle=$
$\displaystyle\mathrm{Tr}\left[\mathcal{O}(u_{1},\theta_{1})\mathcal{O}(u_{2},\theta_{2})e^{2\pi
i\sigma(L_{0}-c_{L}/2)}e^{2\pi i\rho(M_{0}-c_{M}/2)}\right]$ (19)
$\displaystyle=$ $\displaystyle\sum_{i,j}e^{2\pi
i\sigma(\Delta_{i}-c_{L}/2)}e^{2\pi
i\rho(\xi_{i}-c_{M}/2)}|\langle\Delta_{i},\xi_{i}|\mathcal{O}(0,0)|\Delta_{j},\xi_{j}\rangle|^{2}e^{i(u_{12}\xi_{ij}+\theta_{12}\Delta_{ij})}$
$\displaystyle=$
$\displaystyle\widetilde{\langle\mathcal{O}(u_{1},\theta_{1})\mathcal{O}(u_{2},\theta_{2})\rangle}_{(\sigma,\rho)}e^{-2\pi
i(\sigma c_{L}/2+\rho c_{M}/2)}.$

In the second step, the trace is taken over all primaries and their
descendants. We have also inserted a complete set of states,
$\sum_{j}\ket{\Delta_{j},\xi_{j}}\bra{\Delta_{j},\xi_{j}}$. Finally, time
evolution and spatial translation of the operator ${\mathcal{O}}$ is given by

$\mathcal{O}(u,\theta)=e^{iHu+iP\theta}\mathcal{O}(0,0)e^{-iHu-
iP\theta}=e^{iM_{0}u+iL_{0}\theta}\mathcal{O}(0,0)e^{-iM_{0}u-iL_{0}\theta},$
(20)

where we have used $H=i\partial_{u}=M_{0}$ and $P=i\partial_{\theta}=L_{0}$.
Notice the subscript on the two-point function on the l.h.s of the equation
(19). Here $(\sigma,\rho)$ denotes the modular parameters which we will keep
track of. In the last step of (19), the
$\widetilde{\langle{\mathcal{O}}{\mathcal{O}}\rangle}$ is just
$\langle{\mathcal{O}}{\mathcal{O}}\rangle$ with the central terms stripped
off:

$\widetilde{\langle\mathcal{O}(u_{1},\theta_{1})\mathcal{O}(u_{2},\theta_{2})\rangle}_{(\sigma,\rho)}=\sum_{i,j}e^{2\pi
i\sigma\Delta_{i}}e^{2\pi
i\rho\xi_{i}}|\langle\Delta_{i},\xi_{i}|\mathcal{O}(0,0)|\Delta_{j},\xi_{j}\rangle|^{2}e^{i(u_{12}\xi_{ij}+\theta_{12}\Delta_{ij})}.$
(21)

In the above, we have used abbreviations like $u_{12}=u_{1}-u_{2}$. Changing
the summation to integration and then inverting the integral we can extract
the off-diagonal three-point coefficients, which can be obtained upon solving

$d(\Delta,\xi)d(\Delta^{\prime},\xi^{\prime})\overline{|C_{\Delta\xi\mathcal{O}\Delta^{\prime}\xi^{\prime}}|^{2}}=\int\mathrm{d}\sigma\mathrm{d}\rho\mathrm{d}u\mathrm{d}\theta\,e^{-2\pi
i(\sigma c_{L}+\rho
c_{M})}e^{-i(u_{12}\zeta+\theta_{12}\lambda)}\widetilde{\langle\mathcal{O}(u_{1},\theta_{1})\mathcal{O}(u_{2},\theta_{2})\rangle}_{(\sigma,\rho)},$
(22)

where we have introduced

$\zeta=\xi-\xi^{\prime}\quad\lambda=\Delta-\Delta^{\prime},$ (23)

and $d(\Delta,\xi)$ and $d(\Delta^{\prime},\xi^{\prime})$ are the density of
states with weights $(\Delta,\xi)$ and $(\Delta^{\prime},\xi^{\prime})$
respectively and $d(\Delta^{\prime},\xi^{\prime})$ is given by

$d(\Delta,\xi)=\exp{2\pi\left(c_{L}\sqrt{\frac{2\xi}{c_{M}}}+\Delta\sqrt{\frac{c_{M}}{2\xi}}\right)}.$
(24)

Now, using the BMS transformation behavior of the two-point function one can
relate
$\widetilde{\langle\mathcal{O}(u_{1},\theta_{1})\mathcal{O}(u_{2},\theta_{2})\rangle}_{(\sigma,\rho)}$
with its $S$-modular transformed counterpart. The S-modular transformation on
the elliptic coordinates reads

$S:\quad(u,\theta)\to(u^{\prime},\theta^{\prime})=\left(\frac{u}{\sigma}-\frac{\theta\rho}{\sigma^{2}},\frac{\theta}{\sigma}\right).$
(25)

We now use the change in the two-point function under BMS modular
$S$-transformation:

$\widetilde{\langle\mathcal{O}(u_{1},\theta_{1})\mathcal{O}(u_{2},\theta_{2})\rangle}_{(\sigma,\rho)}=\sigma^{-2\Delta_{\mathcal{O}}}e^{\frac{2\xi_{\mathcal{O}}\rho}{\sigma}}e^{\pi
i\left(\sigma c_{L}+\rho c_{M}+\frac{c_{L}}{\sigma}-\frac{\rho
c_{M}}{\sigma^{2}}\right)}\widetilde{\langle\mathcal{O}(u_{1}^{\prime},\theta_{1}^{\prime})\mathcal{O}(u_{2}^{\prime},\theta_{2}^{\prime})\rangle}_{\left(-\frac{1}{\sigma},\frac{\rho}{\sigma^{2}}\right)}.$
(26)

The modular parameters $\sigma$ and $\tau$ are related to the twist of the
torus $\Omega$ and temperature $\beta$ through the following relations:

$2\pi\sigma=i\Omega\quad,\quad 2\pi\rho=\beta.$ (27)

This $S$-transformation takes $\Omega\to-\frac{1}{\Omega}$. Thus the vacuum
state dominates the $S$-transformed correlation function since the
contributions from non-vacuum states are exponentially suppressed at low
values of the twist. Inserting the expression of the cylinder two-point
function into (26) one obtains

$\displaystyle\widetilde{\langle\mathcal{O}(u_{1},\theta_{1})\mathcal{O}(u_{2},\theta_{2})\rangle}_{(\sigma,\rho)}$
$\displaystyle=$
$\displaystyle\sigma^{-2\Delta_{\mathcal{O}}}e^{2\xi_{\mathcal{O}}\rho/\sigma}e^{2\pi
i\left(\frac{\sigma c_{L}}{2}+\frac{\rho
c_{M}}{2}+\frac{c_{L}}{2\sigma}-\frac{\rho
c_{M}}{2\sigma^{2}}\right)}\left[\sin\left(\frac{\theta^{\prime}_{12}}{2}\right)\right]^{-2\Delta_{\mathcal{O}}}$
(28)
$\displaystyle\quad\times\exp\left[-\xi_{\mathcal{O}}u^{\prime}_{12}\cot\left(\frac{\theta^{\prime}_{12}}{2}\right)\right],$

where $u^{\prime},\theta^{\prime}$ are given by the modular transformed
$u,\theta$ in (25). One now has the following integration to solve to get the
expression for
$\overline{|C_{\Delta\xi\mathcal{O}\Delta^{\prime}\xi^{\prime}}|^{2}}$:

$\displaystyle
d(\Delta,\xi)d(\Delta^{\prime},\xi^{\prime})\overline{|C_{\Delta\xi\mathcal{O}\Delta^{\prime}\xi^{\prime}}|^{2}}$
$\displaystyle=$
$\displaystyle\int\mathrm{d}\sigma\mathrm{d}\rho\mathrm{d}u\mathrm{d}\theta\,e^{-2\pi
i(\sigma c_{L}+\rho
c_{M})}e^{-i(u^{\prime}_{12}\zeta+\theta^{\prime}_{12}\lambda)}\sigma^{-2\Delta_{\mathcal{O}}}e^{2\xi_{\mathcal{O}}\rho/\sigma}$
(29) $\displaystyle e^{2\pi i\left(\frac{\sigma c_{L}}{2}+\frac{\rho
c_{M}}{2}+\frac{c_{L}}{2\sigma}-\frac{\rho
c_{M}}{2\sigma^{2}}\right)}\frac{\exp\left[-\xi_{\mathcal{O}}u^{\prime}_{12}\cot\left(\frac{\theta^{\prime}_{12}}{2}\right)\right]}{\left[\sin\left(\frac{\theta^{\prime}_{12}}{2}\right)\right]^{2\Delta_{\mathcal{O}}}}.$

The above integral is very difficult to solve in full generality and its
solution gives the most general off-diagonal three-point function coefficients
in a BMSFT. To provide tractable solutions we focus on two specific cases, one
where the probes are only temporally separated and one where the probes are
only spatially separated. The temporally separated case is analog to the 2d
CFT case and is an indicator for ETH in 2d CFTs. We will elaborate on this in
detail in the next section. The spatially separated case for BMSFTs has very
interesting features which we will also comment on in the next section. But
before we move on to the two cases, we do a few simplifications to the above
equation (29). First, we focus on Einstein gravity as the dual bulk theory, so
we put $c_{L}=0$. We also position the probes at
$(u^{\prime},\theta^{\prime})$ and $(0,0)$ without loss of generality.
Inserting the modular transformed $(u^{\prime},\theta^{\prime})$ in terms of
$(u,\theta)$, (29) now simplifies to:

$\displaystyle
d(\Delta,\xi)d(\Delta^{\prime},\xi^{\prime})\overline{|C_{\Delta\xi\mathcal{O}\Delta^{\prime}\xi^{\prime}}|^{2}}$
$\displaystyle=$
$\displaystyle\int\mathrm{d}\sigma\mathrm{d}\rho\mathrm{d}u\mathrm{d}\theta\,e^{-2\pi
i\rho
c_{M}}e^{-i((\frac{u}{\sigma}+\frac{\theta\rho}{\sigma^{2}})\zeta+\frac{\theta}{\sigma}\lambda)}\sigma^{-2\Delta_{\mathcal{O}}}e^{2\xi_{\mathcal{O}}\rho/\sigma}e^{\pi
i\rho c_{M}\left(1-\frac{1}{\sigma^{2}}\right)}$ (30)
$\displaystyle\quad\sin\left(\frac{\theta}{2\sigma}\right)^{-2\Delta_{\mathcal{O}}}{\exp\left[-\xi_{\mathcal{O}}\left(\frac{u}{\sigma}-\frac{\theta\rho}{\sigma^{2}}\right)\cot\left(\frac{\theta}{2\sigma}\right)\right]}.$

###  3.1 Temporally Separated Probes

We first consider the case where the probes are temporally separated. To make
sure that there are no divergences in our computations, we start with
$\langle\Delta^{\prime},\xi^{\prime}|\mathcal{O}(u,\theta_{c})\mathcal{O}(0,0)|\Delta,\xi\rangle$
and finally take $\theta_{c}\to 0$. Also, for this calculation, we take
$\Delta=\Delta^{\prime}$, i.e. the initial and final angular momenta of the
heavy background are the same.

At equal $\Delta$, there is no $\theta$ integration and (30) reduces to:

$\displaystyle
d(\Delta,\xi)d(\Delta,\xi^{\prime})\overline{|C_{\xi\mathcal{O}\xi^{\prime}}|^{2}}=\int$
$\displaystyle\mathrm{d}u\
e^{-iu\zeta}\int\mathrm{d}\sigma\mathrm{d}\rho\hskip
5.69054pt\sigma^{-2\Delta_{\mathcal{O}}}e^{2\xi_{\mathcal{O}}\rho/\sigma}e^{2\pi
i\left(\frac{\rho c_{M}}{2}-\frac{\rho c_{M}}{2\sigma^{2}}\right)}e^{-2\pi
i\sigma\Delta}e^{-2\pi i\rho\xi}$
$\displaystyle{\left(\sin{\frac{\theta_{c}}{2\sigma}}\right)^{-2\Delta_{\mathcal{O}}}}\exp{-\xi_{\mathcal{O}}\left(\frac{u}{\sigma}-\frac{\rho\theta_{c}}{\sigma^{2}}\right)\cot\left(\frac{\theta_{c}}{2\sigma}\right)}.$
(31)

We first perform the integration over $\sigma$ and $\rho$. Since the
background is much heavier than the probe $\mathcal{O}$, one can perform a
saddle-point approximation. Computing the saddle points yields
$(\sigma,\rho)$:

$\displaystyle\sigma_{c}=i\sqrt{\frac{c_{M}}{2\xi}},\quad\rho_{c}=-i\frac{\Delta\sqrt{c_{M}}}{(2\xi)^{3/2}}.$
(32)

As $\xi\to\infty$, one can approximate

$\sinh\left(\frac{\theta_{c}\sqrt{\xi}}{\sqrt{2c_{M}}}\right)\to
e^{\theta_{c}\sqrt{\xi}/\sqrt{2c_{M}}},\quad\coth\left(\frac{\theta_{c}\sqrt{\xi}}{\sqrt{2c_{M}}}\right)\to
1.$ (33)

Since $\theta_{c}\to 0$, we require that $\sqrt{\frac{\xi}{2c_{M}}}$ goes to
infinity faster than $\theta_{c}\to 0$. Putting this in equation 3.1 one
obtains,

$\displaystyle
d(\Delta,\xi)d(\Delta,\xi^{\prime})\overline{|C_{\xi\mathcal{O}\xi^{\prime}}|^{2}}$
$\displaystyle\approx
f(\Delta,\xi)\int_{0}^{\infty}e^{-\xi_{\mathcal{O}}u\sqrt{\frac{2\xi}{c_{M}}}}e^{-iu\zeta}\mathrm{d}u=\frac{f(\Delta,\xi)}{\left(i\zeta+\xi_{\mathcal{O}}\sqrt{\frac{2\xi}{c_{M}}}\right)},$
(34)

where the prefactor

$f(\Delta,\xi)=\left(\frac{\xi}{c_{M}}\right)^{\Delta_{\mathcal{O}}}e^{-\xi_{\mathcal{O}}\Delta/\xi}e^{2\pi\left(\frac{\Delta}{2}\left(\frac{c_{M}}{2\xi}\right)^{3/2}+\Delta\sqrt{\frac{c_{M}}{2\xi}}\right)}e^{-\theta_{c}{\sqrt{\frac{2\xi}{c_{M}}}}\left(\Delta_{\mathcal{O}}+\frac{\xi_{\mathcal{O}}\Delta}{2\xi}\right)}.$
(35)

We stress that this is a probe approximation, i.e., we work in the limit when
$\Delta_{\mathcal{O}},\xi_{\mathcal{O}}\ll\xi$ 666Even though we previously
assumed $\sqrt{\frac{\xi}{2c_{M}}}$ goes to infinity faster than
$\theta_{c}\to 0$, the additional factor containing the weight of the probes
justifies our result.. This means one can drop the last term in the above
expression. Inserting the expressions for the density of states one gets

$\overline{|C_{\xi\mathcal{O\xi^{\prime}}}|^{2}}\approx\left(\frac{\xi_{avg}}{c_{M}}\right)^{\Delta_{\mathcal{O}}}e^{-\xi_{\mathcal{O}}\frac{\Delta_{avg}}{\xi_{avg}}}e^{-2\pi\Delta_{avg}\sqrt{\frac{c_{M}}{2\xi_{avg}}}\left(1-\frac{c_{M}}{4\xi_{avg}}\right)}\frac{1}{\left(i\zeta+\xi_{\mathcal{O}}\sqrt{\frac{2\xi_{avg}}{c_{M}}}\right)}.$
(36)

In the above, we have replaced the weights of the states with average weights
given by

$\displaystyle\Delta+\Delta^{\prime}=2\Delta=2\Delta_{\mathrm{avg}},\quad\quad\xi+\xi^{\prime}=2\xi_{\mathrm{avg}},$
(37)

where we have neglected terms
$\mathcal{O}\left(\frac{\zeta}{\xi_{\mathrm{avg}}}\right)$ since
$\xi_{\mathrm{avg}}\gg\zeta$.

###  3.2 Spatially Separated Probes

We now investigate the case of spatially separated probes. The operators are
now inserted at $(0,\theta)$ and $(0,0)$. Thus, one is looking at correlations
of the form
$\langle\Delta^{\prime},\xi^{\prime}|\mathcal{O}(0,\theta)\mathcal{O}(0,0)|\Delta,\xi\rangle$.
We shall also take $\xi=\xi^{\prime}$. After carrying out the saddle point
analysis, one is now left with

$\displaystyle
d(\Delta,\xi)d(\Delta^{\prime},\xi)\overline{|C_{\Delta\mathcal{O}\Delta^{\prime}}|^{2}}\approx
g(\Delta,\xi)\int\mathrm{d}\theta\,\frac{e^{-i\theta\lambda}e^{-\xi_{\mathcal{O}}\frac{\theta\Delta}{\sqrt{2c_{M}\xi}}\coth{\theta\sqrt{\frac{\xi}{2c_{M}}}}}}{\left(\sinh\theta\sqrt{\frac{\xi}{2c_{M}}}\right)^{2\Delta_{\mathcal{O}}}},$
(38)

where

$\displaystyle
g(\Delta,\xi)=e^{-\xi_{\mathcal{O}}\Delta/\xi}\left(\frac{\xi}{c_{M}}\right)^{\Delta_{\mathcal{O}}}\exp{2\pi\left(\Delta\sqrt{\frac{c_{M}}{2\xi}}+\frac{\Delta}{2}\left(\frac{c_{M}}{2\xi}\right)^{3/2}\right)}.$
(39)

Since the background field is very heavy and hence $\xi\gg 1$, so we
approximate $\coth\left({\theta\sqrt{\frac{\xi}{2c_{M}}}}\right)$ by 1.
However, in this case we do not approximate
$\sinh\left(\frac{\theta\sqrt{\xi}}{\sqrt{2c_{M}}}\right)\to
e^{\theta\sqrt{\xi}/\sqrt{2c_{M}}}$. 777This is done in order to be able to
make use of certain Mellin-Barnes integral identities, as we show in what
follows. We also work with de-compactified spatial coordinates. One now has,

$\displaystyle
d(\Delta,\xi)d(\Delta^{\prime},\xi)\overline{|C_{\Delta\mathcal{O}\Delta^{\prime}}|^{2}}$
$\displaystyle\approx
g(\Delta,\xi)\int_{-\infty}^{\infty}\mathrm{d}\theta\left(\sinh\theta\sqrt{\frac{\xi}{2c_{M}}}\right)^{-2\Delta_{\mathcal{O}}}\exp{-i\theta\left(\frac{\xi_{\mathcal{O}}\Delta}{i\sqrt{2c_{M}\xi}}+\lambda\right)}$
$\displaystyle=g(\Delta,\xi)$
$\displaystyle\sqrt{\frac{c_{M}}{\xi}}\exp{\pi\lambda\sqrt{\frac{c_{M}}{2\xi}}}\frac{\bigg{|}\Gamma\left(\Delta_{\mathcal{O}}+\frac{\xi_{\mathcal{O}}\Delta}{2\xi}+i\sqrt{\frac{c_{M}}{2\xi}}\lambda\right)\bigg{|}^{2}}{\Gamma(2\Delta_{\mathcal{O}})}.$
(40)

The $\theta$ integration can be solved by using the Mellin-Barns integral
identity (see, e.g., Jantzen:2012cb ; Becker:2014jla ). Writing
$\overline{|C_{\Delta\mathcal{O}\Delta^{\prime}}|^{2}}$ in terms of
$\Delta_{avg}$ and neglecting terms of $\mathcal{O}(\lambda/\Delta_{avg})$ we
find,

$\displaystyle\overline{|C_{\Delta\mathcal{O}\Delta^{\prime}}|^{2}}$
$\displaystyle\approx$ $\displaystyle
e^{-\xi_{\mathcal{O}}\frac{\Delta_{avg}}{\xi_{avg}}}\left(\frac{\xi_{avg}}{c_{M}}\right)^{\Delta_{\mathcal{O}}-1/2}e^{-2\pi\Delta_{avg}\sqrt{\frac{c_{M}}{2\xi_{avg}}}\left(1-\frac{c_{M}}{4\xi_{avg}}-\frac{\lambda}{2\Delta_{avg}}\right)}e^{-i\pi\frac{\xi_{\mathcal{O}}\Delta_{avg}}{2\xi_{avg}}}$
(41)
$\displaystyle\qquad\qquad\qquad\qquad\frac{\bigg{|}\Gamma\left(\Delta_{\mathcal{O}}+\frac{\xi_{\mathcal{O}}\Delta_{avg}}{2\xi_{avg}}+i\sqrt{\frac{c_{M}}{2\xi_{avg}}}\lambda\right)\bigg{|}^{2}}{\Gamma(2\Delta_{\mathcal{O}})}.$

With $\frac{\lambda}{\Delta_{avg}}\ll 1$, one can omit terms
$\mathcal{O}\left(\frac{\lambda}{\Delta_{\mathrm{avg}}}\right)$ in the
exponential. Dropping
$e^{-i\pi\frac{\xi_{\mathcal{O}}\Delta_{avg}}{2\xi_{avg}}}$ in the probe
approximation, we finally arrive at the expression for the off-diagonal three-
point coefficient for spatially separated probes:

$\overline{|C_{\Delta\mathcal{O}\Delta^{\prime}}|^{2}}\approx\left(\frac{\xi_{avg}}{c_{M}}\right)^{\Delta_{\mathcal{O}}-1/2}e^{-\xi_{\mathcal{O}}\frac{\Delta_{avg}}{\xi_{avg}}}e^{-2\pi\Delta_{avg}\sqrt{\frac{c_{M}}{2\xi_{avg}}}\left(1-\frac{c_{M}}{4\xi_{avg}}\right)}\frac{\bigg{|}\Gamma\left(\Delta_{\mathcal{O}}+\frac{\xi_{\mathcal{O}}\Delta_{avg}}{2\xi_{avg}}+i\sqrt{\frac{c_{M}}{2\xi_{avg}}}\lambda\right)\bigg{|}^{2}}{\Gamma(2\Delta_{\mathcal{O}})}.$
(42)

##  4 Eigenstate Thermalization Hypothesis

The eigenstate thermalization hypothesis provides us with a mechanism to study
the late-time behavior of a thermal system, i.e., thermalization. In the
observable dependent notion of thermalization, one studies the coarse-grained
expectation value of a generic observable $\mathcal{O}$ over the states
$\ket{\psi}$. The states $\ket{\psi}$ are a superposition of the energy
eigenstates of the quantum system and are given by
$\ket{\psi}=\sum_{n}C_{n}\ket{n}$. The expectation value of $\mathcal{O}$ at
time $t$ is given by,

$\displaystyle\langle\mathcal{O}(t)\rangle=\sum_{n}|C_{n}|^{2}\bra{n}\mathcal{O}\ket{n}+\sum_{n,m\neq
n}C_{n}^{*}C_{m}e^{i(E_{n}-E_{m})t}\bra{n}\mathcal{O}\ket{m},$ (43)

where $E_{n}$ denotes the energy of the state $\ket{n}$. For a system with a
large number of degrees of freedom, the late time average of $\mathcal{O}(t)$
is equivalent to the thermal average of the operator Srednicki:1995pt . The
late time average is given by,

$\displaystyle\overline{\langle\mathcal{O}(t)\rangle}=\lim_{\mathcal{T}\to\infty}\frac{1}{\mathcal{T}}\int_{0}^{\mathcal{T}}\mathrm{d}t\langle\mathcal{O}(t)\rangle=|C_{n}|^{2}\bra{n}\mathcal{O}\ket{n}.$
(44)

The fluctuations of the system are given by the difference between the
expectation value of the observable and its late time average. They are
encoded in the off-diagonal part of the expectation value. We have

$\displaystyle\langle\mathcal{O}(t)\rangle-\overline{\langle\mathcal{O}(t)\rangle}=\sum_{m\neq
n}C_{n}^{*}C_{m}e^{i(E_{n}-E_{m})t}\bra{n}\mathcal{O}\ket{m}.$ (45)

For the system to thermalize, this fluctuation has to decay very fast or
become reasonably negligible. ETH states that the off-diagonal element
$\bra{n}\mathcal{O}\ket{m}$ fall of as $e^{-S(E_{\mathrm{avg}})/2}$ where
$S(E_{\mathrm{avg}})$ is the entropy of the system at the average energy
$E_{\mathrm{avg}}=\frac{E_{n}+E_{m}}{2}$.

###  4.1 Structure Constants for 2d CFT

Our BMS analysis has been inspired by the calculations in 2d CFTs
Brehm:2018ipf ; Hikida:2018khg ; Romero-Bermudez:2018dim . Here we first
recount briefly the 2d CFT result and then the physics that can be extracted
from this answer. A 2d CFT has well-known modular properties. The partition
function

$Z_{\text{CFT}}={\text{Tr}}\,e^{2\pi
i\tau({\mathcal{L}}_{0}+\frac{c}{24})}e^{-2\pi
i\bar{\tau}(\bar{{\mathcal{L}}}_{0}+\frac{\bar{c}}{24})},$ (46)

is invariant under modular transformations that take the modular parameter
$(\tau,\bar{\tau})$ and the elliptic coordinates on the torus
$(\omega,\bar{\omega})$ to

$\tau\to\frac{a\tau+b}{c\tau+d}\quad,\quad w\to\frac{w}{c\tau+d},$ (47)

and similarly for the barred components. The correlators on the torus
transform covariantly under the modular group. In particular, the two-point
function transforms as

$\langle\mathcal{O}(w_{1},\bar{w}_{1})\mathcal{O}(w_{2},\bar{w}_{2})\rangle_{\tau}=(c\tau+d)^{-2h_{\mathcal{O}}}(c\bar{\tau}+d)^{-2\bar{h}_{\mathcal{O}}}\langle\mathcal{O}(w_{1}^{\prime},\bar{w}_{1}^{\prime})\mathcal{O}(w_{2}^{\prime},\bar{w}_{2}^{\prime})\rangle_{\frac{a\tau+b}{c\tau+d}}.$
(48)

We are interested in the autocorrelations
$\langle\mathcal{O}(t,0)\mathcal{O}(0,0)\rangle_{\beta}$, where $\beta$ is the
inverse temperature which is related to the modular parameter by
$\beta=i(\tau-\bar{\tau})$. One can exploit the $S$-modular covariance of the
two-point function, do a saddle-point analysis, and arrive at a formula for
the off-diagonal structure constants, similar to what we have done in the
previous section for BMSFTs. The off-diagonal structure constants for a 2d CFT
are given by

$\displaystyle\overline{|C_{E\mathcal{O}E^{\prime}}|^{2}}\sim
e^{-S_{avg}}\left(\frac{12E_{avg}}{c}-1\right)^{E_{\mathcal{O}}+1/4}\bigg{|}\Gamma\left(E_{\mathcal{O}}+i\frac{\omega}{\sqrt{12E_{avg}/c-1}}\right)\bigg{|}^{2}.$
(49)

In the above, we have set the central terms to be equal $c=\bar{c}$ as is the
case for Einstein gravity. We have defined $E=h+{\bar{h}}$ and
$\omega=E^{\prime}-E$. Also $E_{avg}=\frac{1}{2}(E^{\prime}+E)$. $S_{avg}$ is
given by the Cardy formula

$S_{avg}=2\pi\sqrt{\frac{cE_{avg}}{3}}.$

Finally, in (49), $E_{\mathcal{O}}$ is the weight of the probe
${\mathcal{O}}$. It is clear from the above calculation that the off-diagonal
matrix elements in a 2d CFT are suppressed as $e^{-S_{avg}}$, and hence 2d
CFTs follow ETH and thermalize.

###  4.2 ETH and BMS

As highlighted earlier, our main results in this work are expressions for
different types of BMS structure constants given by Eq. (36) and Eq. (42).

#### Temporally separated probes.

We first focus on the answer for temporally separated probes:

$\overline{|C_{\xi\mathcal{O\xi^{\prime}}}|^{2}}\approx\left(\frac{\xi_{avg}}{c_{M}}\right)^{\Delta_{\mathcal{O}}}e^{-\xi_{\mathcal{O}}\frac{\Delta_{avg}}{\xi_{avg}}}e^{-2\pi\Delta_{avg}\sqrt{\frac{c_{M}}{2\xi_{avg}}}\left(1-\frac{c_{M}}{4\xi_{avg}}\right)}\frac{1}{\left(i\zeta+\xi_{\mathcal{O}}\sqrt{\frac{2\xi_{avg}}{c_{M}}}\right)}.$
(50)

This is the analog of the 2d CFT analysis that we presented previously. Notice
that

$\exp{-2\pi\Delta_{\mathrm{avg}}\sqrt{\frac{c_{M}}{2\xi_{avg}}}}=\exp{-S^{\text{BMS}}_{avg}},$
(51)

when the central charge $c_{L}=0$ . This corresponds to asymptotically flat
spacetimes in 3d Einstein gravity as a dual bulk theory. However, the
exponentially suppressed prefactor $p(\Delta_{avg},\xi_{avg},c_{M})$ comes
with another piece:

$p(\Delta_{avg},\xi_{avg},c_{M})=\exp{-S^{\text{BMS}}_{avg}\left(1-\frac{c_{M}}{4\xi_{avg}}\right)}.$
(52)

We are in the Cardy regime, meaning that $\xi_{avg}\gg c_{M}$. So,
$\frac{c_{M}}{4\xi_{avg}}\ll 1$ in this region of parameter space. Hence the
off-diagonal structure constants are indeed exponentially suppressed compared
to the diagonal entries. This suppression, however, is not completely
analogous to a 2d CFT because of this additional piece. We take this to be a
signature of the non-Lorentzian nature of the theory. The fact that there is
exponential suppression means that the system seems to follow ETH and
thermalizes. One can also infer from the change in the exponential structure
that the thermalization is now slower compared to the 2d CFT case.

Before we move on to discuss the other case, it is important to mention that
there are additional prefactors on the r.h.s of (50) multiplying the
exponential suppression. We will deal with the singularity structure arising
from the last term extensively in the next section. For now, we comment on the
first couple of terms:

$s_{1}=\left(\frac{\xi_{avg}}{c_{M}}\right)^{\Delta_{\mathcal{O}}}e^{-\xi_{\mathcal{O}}\frac{\Delta_{avg}}{\xi_{avg}}}.$
(53)

These terms are interpreted as probe dependent sub-leading corrections to the
entropy of the FSC in the bulk, as they explicitly depend on the weights of
the probe ${\mathcal{O}}$.

#### Spatially separated probes.

Let us now consider spatially separated probes. The answer for the off-
diagonal structure constants in this case is given by

$\displaystyle\overline{|C_{\Delta\mathcal{O}\Delta^{\prime}}|^{2}}\approx\left(\frac{\xi_{avg}}{c_{M}}\right)^{\Delta_{\mathcal{O}}-1/2}$
$\displaystyle
e^{-\xi_{\mathcal{O}}\frac{\Delta_{avg}}{\xi_{avg}}}e^{-2\pi\Delta_{avg}\sqrt{\frac{c_{M}}{2\xi_{avg}}}\left(1-\frac{c_{M}}{4\xi_{avg}}\right)}$
(55)
$\displaystyle\frac{\bigg{|}\Gamma\left(\Delta_{\mathcal{O}}+\frac{\xi_{\mathcal{O}}\Delta_{avg}}{2\xi_{avg}}+i\sqrt{\frac{c_{M}}{2\xi_{avg}}}\lambda\right)\bigg{|}^{2}}{\Gamma(2\Delta_{\mathcal{O}})}.$

One can see that the same exponential suppression of the off-diagonal
structure constants with the prefactor $p(\Delta_{avg},\xi_{avg},c_{M})$
occurs. Even though the answer here does not a priori have a relation to late-
time thermalization of eigenstates, the structure of the answer is
interesting. An important point to note is that the bulk dual is an FSC
solution that comes from a limit of a non-extremal BTZ black hole. Here the
outer radius is pushed out to infinity and one is left with the inside of the
black hole where the roles of time and space reverse. The dual field theory
should also carry signatures of this flipping of the temporal and spatial
direction. The indication of the answer above is similar. This is also
reminiscent of the computations of chaos in Carrollian CFTs in Bagchi:2021qfe
, where one found trajectories diverge with Lyapunov coefficients while
looking at spatial evolution instead of temporal ones. We thus interpret the
exponential suppression in the above as a spatial analog of ETH and also
thermalization.

In $d=2$ there is an interesting duality between the $c\to 0$ Carrollian
theories and the $c\to\infty$ Galilean theories. For non-Lorentzian CFTs,
there is an isomorphism between the Carrollian and Galilean conformal
algebras. This isomorphism effectively exchanges the identification of the
temporal and spatial directions. The analysis we have performed in the 2d
BMSFT is mostly agnostic to whether the theory came from a Galilean or a
Carrollian limit. We have used BMS modular invariance, which also holds in a
2d Galilean CFT. The only thing that changes is the identification of space
and time. So, the analysis done in the above sections is equally applicable
for 2d Galilean CFTs, albeit with a flip of temporal and spatial coordinates.
The answer (55) is thus what one would get for off-diagonal structure
constants in the case of temporally separated probes of 2d Galilean CFTs. The
presence of the exponentially suppressing prefactor
$p(\Delta_{avg},\xi_{avg},c_{M})$ indicates that even for 2d Galilean CFTs,
there is a notion of thermalization for late times in terms of ETH.

Before closing this section, we point out the additional prefactors on the
r.h.s of (55) multiplying the exponential suppression now reads:

$s_{2}=\left(\frac{\xi_{avg}}{c_{M}}\right)^{\Delta_{\mathcal{O}}-1/2}e^{-\xi_{\mathcal{O}}\frac{\Delta_{avg}}{\xi_{avg}}}.$
(56)

This is almost identical to the previous prefactor (53), except for a shift of
the exponent of the first term:
$\Delta_{\mathcal{O}}\to\Delta_{\mathcal{O}}-\frac{1}{2}.$ This extra factor
of $-\frac{1}{2}$ comes about from the $\theta$ integration performed in Sec
(3.2). The implications for this additional shift are not yet clear at this
moment.

##  5 Torus 2pt Functions and Bulk Quasinormal Modes

###  5.1 BTZ QNM

When a black hole is perturbed by a probe, the perturbation propagates with
certain discrete frequencies due to fixed boundary conditions at the horizon
and asymptotic infinity Birmingham:2001hc . To study the black hole
perturbation, one needs to solve $(\nabla^{2}-m^{2})\phi=0,$ in the black hole
background. Here $m$ is the mass of the field $\phi$. The frequencies, called
quasinormal modes, depend upon the parameters of the probe and the black hole.
Analytic computations for the decay rate of black hole perturbations were
first carried out in the BTZ background in Birmingham:1997rj . For a non-
extremal BTZ, this computation yields:

$\omega_{L,R}=\pm\frac{k}{\ell}-2i\left(\frac{r_{+}\mp
r_{-}}{\ell^{2}}\right)\left(n+\frac{1}{2}+\frac{1}{2}\sqrt{1+m^{2}\ell^{2}}\right).$
(57)

Here, $k$ is the angular momentum and $m$ the mass of the perturbation, $n$
takes integer values $0,1,2,\ldots$ and $r_{\pm}$ are the outer and inner
horizons of the BTZ black hole. An interpretation of QNM in terms of dual
thermal CFTs was first suggested in Horowitz:1999jd , and the analysis for BTZ
black hole perturbation and the decay of QNM in the context of 2D CFT was
first carried out in Birmingham:2001pj . The scalar probe of mass $m$
corresponds to an operator $\mathcal{O}$ with energy $E_{\mathcal{O}}$ in the
2d CFT, where

$E_{\mathcal{O}}=h_{\mathcal{O}}+\bar{h}_{\mathcal{O}}=1+\sqrt{1+m^{2}\ell^{2}}.$
(58)

Therefore, in AdS/CFT, the problem of black hole perturbation gets reduced to
that of the perturbation of a thermal system by a probe and its eventual
relaxation back to an equilibrium state. For a small perturbation, the process
can be studied under the purview of linear response theory Birmingham:2002ph .
According to this analysis, the frequencies of the decay of perturbation,
i.e., of the black hole quasinormal modes, are given by the poles of the
correlation function of the perturbation operator $\mathcal{O}$ in momentum
space.

The relevant correlation function has the form

$\mathcal{D}(\omega,k)\propto\Big{|}\Gamma\left(h_{\mathcal{O}}+i\frac{p_{-}}{2\pi
T_{R}}\right)\Gamma\left(\bar{h}_{\mathcal{O}}+i\frac{p_{+}}{2\pi
T_{L}}\right)\Big{|}^{2}.$ (59)

Frequencies of the quasinormal modes are given by the poles of the gamma
function and they are given by

$\displaystyle\omega_{L}=k-4\pi
iT_{L}(n+h_{\mathcal{O}}),\quad\omega_{R}=-k-4\pi
iT_{R}(n+\bar{h}_{\mathcal{O}}),$ (60)

where $T_{L/R}$ are related to $r_{\pm}$ as $T_{L,R}=2\pi\left(\frac{r_{+}\pm
r_{-}}{\ell^{2}}\right)$. Thus, the CFT answer (60) matches exactly with the
bulk analysis (57).

The off-diagonal structure constants can be obtained from the 2d CFT torus
two-point function (49). The poles of (49) are given by

$\omega_{n}=-i\left(\frac{12E_{avg}}{c}-1\right)^{1/2}(n+E_{\mathcal{O}}).$
(61)

For a non-rotating black hole, the Hawking temperature is given by

$T_{H}=\frac{r_{+}}{2\pi\ell^{2}}.$ (62)

Relating the conformal weights $E$ with the mass of the black hole $M$

$M\ell=E-\frac{c}{12}=\frac{r_{+}^{2}}{8G\ell^{2}}\Rightarrow
T_{H}=\frac{1}{2\pi\ell}\left(\frac{12E}{c}-1\right)^{1/2}.$ (63)

Hence, the poles can be recast as,

$\frac{\omega_{n}}{\ell}=-\frac{2\pi i}{\beta_{avg}}(n+E_{\mathcal{O}}),$ (64)

where $\beta_{avg}$ is the temperature corresponding to $E_{avg}$ and is
identified with $T^{-1}_{H}$. From (60), setting $T_{L}=T_{R}=T_{H}$ for a
non-rotating solution, one can readily see that

$\omega_{n}=\ell(\omega_{L}+\omega_{R}),$ (65)

and there is no $k$ dependence as the probes are inserted at the same angular
location and are only displaced in time. One thus sees that the off-diagonal
three-point coefficients in a 2d CFT carry the imprints of BTZ QNM and provide
a qualitative understanding of these QNM.

###  5.2 Singularity Structure of BMS Three-Point Coefficients

In an effort to understand the analogs of QNM for the FSC solutions in flat
space, we now focus on the singularity structure of the two different BMS off-
diagonal structure constants that we have derived previously. Understanding
QNM for FSCs has been obstructed up until now as it was not clear what
boundary conditions should be used. The singularity structure of the off-
diagonal structure constants now provides a way to understand QNM for FSCs.

In analogy to the BTZ case, we interpret the average three-point coefficients
as denoting the transition of an FSC solution from an initial state with mass
and angular momentum proportional to $\xi$ and $\Delta$ to a final state with
$\xi^{\prime}$ and $\Delta^{\prime}$. We will consider the two cases
(temporally and spatially separated) treated previously below.

#### Temporally Separated Probes:

For temporally separated probes, we assumed $\Delta_{i}=\Delta_{j}$. This
choice also constrains our analysis to a case where the FSC transition takes
place between two states with the same angular momentum.
$\overline{|C_{\xi\mathcal{O}\xi^{\prime}}|^{2}}$ is thus a measure of the
transition rate of an FSC state from $|\xi,\Delta\rangle$ to
$|\xi^{\prime},\Delta\rangle$. During this transition, the field with energy
$\zeta=\xi-\xi^{\prime}$ is emitted or absorbed (depending on the sign of
$\zeta$). This process bears a close resemblance to the perturbation process
of a black hole. The decay of the perturbation can be inferred from the poles
of the transition rate. The black hole horizon has now been replaced with a
cosmological horizon making the process even more interesting. The singularity
structure of the off-diagonal three-point constant
$\overline{|C_{\xi\mathcal{O}\xi^{\prime}}|^{2}}$, as can be seen from (50),
is given by a localized value of $\zeta$ that reads

$\zeta=i\xi_{\mathcal{O}}\sqrt{\frac{2\xi_{avg}}{c_{M}}}.$ (66)

In terms of FSC parameters, one has,

$\zeta=i\xi_{\mathcal{O}}\sqrt{8GM_{avg}}=i\xi_{\mathcal{O}}{\hat{r}_{+}}.$
(67)

It is important to note that one doesn’t get a discrete set of frequencies but
just a single one. This collapse of the QNM in the flat space limit is
expected from earlier work. If one thinks of this in terms of the flat space
limit, the QNM of the BTZ black hole are frequencies of oscillations of a
probe that is located in the region between $r=r+$ and $r\to\infty$. As one
takes the flat space limit, the pole structure gets squeezed together as the
outer radius gets pushed out. In the strict $\ell\to\infty$ limit, all these
poles converge into a single pole as $r_{+}$ coincides with the boundary. The
remnant of this pole structure is what we find in (67).

#### Spatially Separated Probes:

The expression $\overline{|C_{\Delta\mathcal{O}\Delta^{\prime}}|^{2}}$ denotes
the emission of a particle with energy equivalent to
$\lambda=\Delta-\Delta^{\prime}$. $\xi_{i}=\xi_{j}$ corresponds to a situation
where the mass of the FSC before and after the transition remains the same.
The probe thus carries away/in the angular momentum lost/gained in the
process. The singularity structure in this case differs significantly from the
earlier case. One can see that the frequencies of quasinormal modes are spread
over a discrete spectrum, showing a similar behaviour as the analysis of BTZ
quasinormal modes. As can be seen from the poles of the gamma function in the
expression for the three-point structure constant (55), the singularity
structure reads

$\lambda=\pm
i\sqrt{\frac{2\xi_{avg}}{c_{M}}}\left(n+\Delta_{\mathcal{O}}+\frac{\xi_{\mathcal{O}}\Delta_{avg}}{2\xi_{avg}}\right).$
(68)

Expressing this in terms of the FSC mass and angular momentum one obtains,

$\lambda=\pm
i\left[(n+\Delta_{\mathcal{O}})\sqrt{8GM}+\xi_{\mathcal{O}}J\sqrt{\frac{2G}{M}}\right]=\pm
i\left[(n+\Delta_{\mathcal{O}})\hat{r}_{+}+\xi_{\mathcal{O}}r_{0}\right].$
(69)

One can interpret these as the spatial imprints of a perturbation of the
cosmological horizon of the FSC when a probe ${\mathcal{O}}$ with BMS weights
$(\Delta_{\mathcal{O}},\xi_{\mathcal{O}})$ is thrown at it. The flipping of
the spatial and temporal directions from the point of the BTZ solution is
evident from the fact that this answer mimics the usual BTZ QNM and arises
from the poles of a gamma function, even though this is an answer arising from
the calculation of a two-point function in an FSC background with purely
spatially separated probes. The fact that we have the initial and final states
of the FSC at the same mass and different angular momenta also indicates that
it is the Virasoro sub-algebra of the BMS algebra which is important here and
not the supertranslations. The spatially separated probes also indicate that
the relevant underlying structure is the celestial circle at null infinity and
hence the relevant two-point functions are determined via Diff(S1).

#### Recovering Answers in the Flat Space Limit.

Rather remarkably, as we will see below, one can recover the answers that we
obtained using singularities from the off-diagonal BMS structure constants by
invoking modular covariance of BMS torus two-point functions, from a limit on
the QNM of the BTZ black hole. One can rewrite the left and right frequencies
at zero angular separation of probes as

$\omega^{L}_{n}=-2i\left(\frac{r_{+}+r_{-}}{\ell^{2}}\right)(n+h_{L}),\quad\omega^{R}_{n}=-2i\left(\frac{r_{+}-r_{-}}{\ell^{2}}\right)(n+h_{R}).$
(70)

We now make the following substitutions as $\ell\to\infty$:

$h_{L}=\Delta_{\mathcal{O}}+\ell\xi_{\mathcal{O}},\quad
h_{R}=-\Delta_{\mathcal{O}}+\ell\xi_{\mathcal{O}},\quad
r_{+}\to\ell\hat{r}_{+},\,r_{-}\to r_{0}.$ (71)

Inserting this in (70) one obtains

$\displaystyle\omega^{L}_{n}\to-i\xi_{\mathcal{O}}\hat{r}_{+}-\frac{i}{\ell}\left[(n+\Delta_{\mathcal{O}})\hat{r}_{+}+r_{0}\xi_{\mathcal{O}}\right]+\ldots$
(72)
$\displaystyle\omega^{R}_{n}\to-i\xi_{\mathcal{O}}\hat{r}_{+}+\frac{i}{\ell}\left[(-n+\Delta_{\mathcal{O}})\hat{r}_{+}+r_{0}\xi_{\mathcal{O}}\right]+\ldots$
(73)

and similarly for $\omega_{R}$. In the limit $\ell\to\infty$, one obtains

$\displaystyle\lim_{\ell\to\infty}\omega_{n}^{L,R}=-i\xi_{\mathcal{O}}\hat{r}_{+}\pm\frac{i}{\ell}\left[(\pm
n+\Delta_{\mathcal{O}})\hat{r}_{+}+r_{0}\xi_{\mathcal{O}}\right],$ (74)
$\displaystyle\Rightarrow\zeta_{n}=\omega_{n}^{L}+\omega_{-n}^{R},\quad\lambda_{n}=\ell(\omega_{n}^{L}-\omega_{-n}^{R}),$
(75)

where we have used our previous results (67) and (69). Thus, the leading and
next-to-leading pieces of the BTZ QNM in the flat limit reproduce the pole
structure we obtained from the BMS off-diagonal three-point coefficients. It
is interesting to see that these are encoded in two different off-diagonal
structure constants, one for temporally separated probes and one for spatially
separated probes. It is perhaps expected that the flat limit of the BTZ QNM
would lead us to the answer of the collapse of the QNM in the flat limit and
the answer $\overline{|C_{\xi\mathcal{O}\xi^{\prime}}|^{2}}$ recovered from
the temporally separated probes. But that the subleading piece of the BTZ QNM
in the flat limit, with the non-trivial pole structure, is contained in
$\overline{|C_{\Delta\mathcal{O}\Delta^{\prime}}|^{2}}$ is surprising.

##  6 Concluding Remarks

###  6.1 Summary

In this work, we have investigated modular properties of 2d BMSFTs. Our goal
was to exploit the covariance of the BMS torus two-point function to arrive at
expressions for off-diagonal structure constants of the 2d BMSFT. Like in
usual 2d CFTs, the knowledge of all structure constants of a theory, along
with the spectrum of primaries and the central charges, is enough to
completely characterize the theory using bootstrap methods outlined, e.g., in
Bagchi:2016geg ; Bagchi:2017cpu . We have arrived successfully at (averaged
expressions of) two different classes of structure constants
$\overline{|C_{\xi\mathcal{O}\xi^{\prime}}|^{2}}$ and
$\overline{|C_{\Delta\mathcal{O}\Delta^{\prime}}|^{2}}$. We used the two-point
functions of temporally and spatially separated probes, respectively, and also
assumed equal background FSC angular momenta in the first case and equal FSC
mass in the second case after a perturbation. These two expressions constitute
the main results of our work and are given in (50) and (55).

There is rich physics hidden in the expressions of these off-diagonal
structure constants. We saw that in both cases the off-diagonal structure
constants were exponentially suppressed and this indicates that these BMSFTs
follow a notion of the eigenstate thermalization hypothesis. The exponential
suppression, however, had interesting additional pieces as compared to 2d
CFTs, which led us to conclude that although the systems would thermalize,
this process of thermalization would possibly be slower compared to their 2d
CFT counterparts. The fact that the spatially separated probes also led to
off-diagonal structure constants with exponential suppression was reminiscent
of the spatial version of chaos seen earlier in 2d BMSFTs in Bagchi:2021qfe .
Finally, the singularity structure of these expressions (50) and (55) were
examined and connections were made to QNM of FSC solutions in the
asymptotically flat bulk theory. We recovered the answers of the singularities
through a flat limit of the BTZ QNM. Very surprisingly, we were able to match
both the leading and subleading pieces of the large $\ell$ answer to the
singularities of the two different classes of structure constants.

###  6.2 Discussions

In our pursuit of a holographic dual for asymptotically flat spacetimes, we
have investigated theories where the speed of light goes to zero. These
Carrollian field theories and specifically Carrollian CFTs (see e.g
Bagchi:2019xfx ; Bagchi:2019clu ; deBoer:2021jej ) provide a challenge to
standard rules of relativistic QFTs as lightcones collapse and notions of
causality and unitarity are challenged. These apparently bizarre theories have
now also come up in many real-life scenarios, including the theory of flat
bands Bagchi:2022eui  and fractons Bidussi:2021nmp  in condensed matter
physics, cosmology deBoer:2021jej , as well as in fluids moving with very high
velocities e.g. in modeling of the quark-gluon-plasma Bagchi:2023ysc .
Understanding aspects of thermalization of these theories is thus an important
problem and one where we have made some headway in our present paper. It does
seem that structure constants of the 2d Carroll CFTs obey ETH and thus a
Carroll CFT should reach thermal equilibrium, although with some interesting
changes from that of a usual 2d CFT. This indicates that the possible issues
about the non-existence of an equilibrium state for Carrollian field theories
are perhaps not as serious as initially anticipated. Of course, our analysis
is only for $d=2$, and further investigation is needed. However, our analysis
in this work is a promising start for these future investigations.

Our other major contribution in this work is a proposal for the analogs of
quasinormal modes for the FSC solutions of asymptotically flat spacetimes.
This has been a long-standing problem because of the issue of boundary
conditions. Adding perturbations to an FSC with e.g. a bulk scalar field, it
has been very unclear what boundary conditions to impose on the cosmological
horizon and most obvious attempts have not been successful. We have
circumvented this issue by looking at the dual field theory and the poles of
the thermal two-point functions which were embedded in our formulae for the
off-diagonal structure constants. While this indirect method yielded
surprisingly nice answers which matched with the flat limit of the BTZ QNM, it
would be good to have further cross-checks of the spectrum we have uncovered.
With the help of the expected QNM spectrum, one may be able to reconstruct
what the “proper” boundary conditions for the probe bulk field would be. Other
methods, like using the Selberg zeta-function Keeler:2018lza , would also be
useful wip .

The matching of the leading term of the $\ell\to\infty$ limit of the BTZ QNM
with the singularity structure from the temporally separated probe in the 2d
BMSFT is something that was not unexpected as we have stressed before. But the
matching of the subleading term in the $\ell\to\infty$ limit with the
singularities of the spatially separated probe seems surprising. The off-
diagonal structure constants are very different in terms of the field theory.
They denote temporal and spatially separated probes. They also carry
information on transitions between FSCs of equal angular momentum and unequal
mass in the first case and equal mass and unequal angular momentum in the
second case. Why these two apparently very different three-point coefficients
from different parts of the BMSFT phase space should carry information arising
from the same BTZ QNM is a puzzle that needs further clarification. This looks
similar to the Carroll expansion of the relativistic field theory (say scalar
theory) action in a series where the speed of light $c$ controls the expansion
parameter. There the leading term is the so-called electric term and the sub-
leading one (with certain modifications) is the magnetic one. At first sight,
these electric and magnetic theories don’t seem related to each other at all,
but they arise in the limit of the same relativistic theory. Perhaps there is
something similar at play here as well. We hope to report on this and other
aspects of perturbations of the FSC background and its formulation in terms of
the dual 2d BMSFT in the near future.

### Acknowledgements

We thank Diptarka Das, Daniel Grumiller, Cynthia Keeler, Victoria Martin,
Rahul Poddar for interesting discussions and comments.

During the course of this work, AB was partially supported by a Swarnajayanti
fellowship (SB/SJF/2019-20/08) from the Science and Engineering Research Board
(SERB) India, the SERB grant (CRG/2020/002035), a visiting professorship at
École Polytechnique Paris, a distinguished visiting professorship at NORDITA
Stockholm, and a Royal Society of London international exchange grant with the
University of Edinburgh. AB also acknowledges the warm hospitality of the
Niels Bohr Institute, Copenhagen, the University of Edinburgh, UK, ULB
Brussels and NORDITA, Stockholm, during 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). The research of MR is supported by the
Austrian Science Fund (FWF) project P32581. MR also acknowledges the warm
hospitality of Harvard University, the Okinawa Institute of Science and
Technology (OIST), the Perimeter Institute for Theoretical Physics (PI), and
the Yukawa Institute for Theoretical Physics (YITP) during various stages of
this work.

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