††institutetext: Indian Institute of Technology Gandhinagar, Palaj 382355
Gujarat. India.

# Circuit Complexity for Carrollian Conformal (BMS) Field Theories

Arpan Bhattacharyya  Poulami Nandi111corresponding author.
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We systematically explore the construction of Nielsen’s circuit complexity to
a non-Lorentzian field theory keeping in mind its connection with flat
holography. We consider a $2d$ boundary field theory dual to $3d$
asymptotically flat spacetimes with infinite-dimensional BMS3 as the
asymptotic symmetry algebra. We compute the circuit complexity functional in
two distinct ways. For the Virasoro group, the complexity functional resembles
the geometric action on its co-adjoint orbit. Using the limiting approach on
the relativistic results, we show that it is possible to write BMS complexity
in terms of the geometric action on BMS co-adjoint orbit. However, the
limiting approach fails to capture essential information about the conserved
currents generating BMS supertranslations. Hence, we refine our analysis using
the intrinsic approach. Here, we use only the symmetry transformations and
group product laws of BMS to write the complexity functional. The refined
analysis shows a richer structure than only the geometric action. Lastly, we
extremize and solve the equations of motion (for a simple solution) in terms
of the group paths and connect our results with available literature.

##  1 Introduction

The entropy of the black hole, containing the information of its microstates,
is proportional to the area of its event horizon and not its volume. With this
simple yet, counter-intuitive statement, we witnessed the first notion of the
Holographic Principle tHooft:1993dmi ; Susskind:1994vu . The holographic
principle says that we can study a theory of quantum gravity by looking into a
field theory without gravity, living on its boundary in one lower dimension.
The most powerful and successful example of the Holographic Principle was
given by the AdS/CFT correspondence Maldacena:1997re ; Gubser:1998bc ;
Witten:1998qj . To date, it has found many interesting applications in various
branches of physics Kovtun:2003wp ; McGreevy:2009xe ; Hartnoll:2009sz ;
Harlow:2014yka ; Ammon:2015wua ; 2017LNP…931…..R ; 2022EPJC…82..458B . Even
after the enormous success, our physical world can not be described by a
constant negative curvature spacetime. For many astrophysical purposes, it is
best approximated as asymptotically flat instead of AdS.

There has been quite some advancement in the program of flat space holography
over the past few years Bagchi:2010zz ; Barnich:2006av ; Barnich:2010eb . It
is done mainly by looking into the symmetries of the bulk asymptotically flat
spacetimes and proposing the boundary field theory to inherit them. The
symmetries at the boundary of $3d$ and $4d$ asymptotically flat spacetimes are
enhanced from the Poincaré symmetries and are given by the infinite-
dimensional Bondi-Metzner-Sachs (BMS) group Bondi:1962px ; Sachs:1962zza .
Hence, the dual boundary field theories are also expected to be BMS invariant.
The initial attempts to build the holographic principle for flat spacetimes
consider taking the AdS radius to infinity, which in turn introduces an ultra-
relativistic (Carrollian Leblond65 ; Sengupta65 ) limit on the dual boundary
Conformal Field Theory Bagchi:2012cy . This line of approach is widely known
as Carrollian holography.222There exists another parallel approach to flat
space holography, called the Celestial holography. For $4d$ asymptotically
flat spacetimes, the structure at null infinity $\mathscr{I}^{\pm}$ is
$\mathbb{R}\times\mathbb{S}^{2}$. On the sphere at infinity, called the
celestial sphere, the structure is very much like $2d$ CFT. Hence, the physics
of $4d$ flat spacetimes can be expressed in terms of $2d$ CFT, giving rise to
a correspondence between the asymptotic symmetries and scattering amplitudes.
For more details, the readers are referred to the excellent reviews
Strominger:2017zoo ; Pasterski:2021rjz ; Raclariu:2021zjz , and also the
recent works on bridging these two considerably different flat holography
pictures, Carrollian and Celestial Donnay:2022aba ; Bagchi:2022emh . It was
also shown that the BMS algebra is isomorphic to the Carrollian version of
Conformal Algebra, or Carrollian Conformal Algebra (CCA). The field theories
having Carrollian Conformal symmetries live on the null boundaries
($\mathscr{I}^{\pm}$) of the asymptotically flat spacetimes. In the Carrollian
limit, the lightcone collapses, and the speed of light goes to zero. On the
null boundary, the metric becomes degenerate, and the underlying geometry is
no longer (pseudo) Riemannian. The ultra-relativistic limit replaces the
relativistic Poincare symmetry to Carrollian symmetries. However, it is well
known now that Carrollian symmetries are associated with any null manifold,
e.g. the event horizon of the black hole Donnay:2019jiz . Thus, Carrollian and
Conformal Carrollian symmetries are vividly related to flat space holography
and black hole physics. There has been significant success to this
formulation, specifically in $3d$ bulk and $2d$ boundary Bagchi:2012yk , BMS
Cardy type formula Bagchi:2012xr ; Barnich:2012xq , entanglement entropy
Bagchi:2014iea ; Jiang:2017ecm ; Hijano:2017eii , holographic construction of
correlation functions Bagchi:2015wna  etc., to name a few. Apart from its
importance in the flat holography program, Carrollian physics has found
important applications in condensed matter physics through the theory of
fractons Bidussi:2021nmp , flat bands Bagchi:2022eui , cosmology, dark energy
and inflation deBoer:2021jej , Carroll fluids deBoer:2017ing ;
Ciambelli:2018wre , the tensionless limit of string theory Bagchi:2013bga ;
Bagchi:2019cay  and Carrollian gravity Bergshoeff:2017btm ; Duval:2017els .

On the other hand, using quantum information theory, one aims to understand
the entanglement between two or more subsystems. The entanglement entropy (EE)
is one of the useful measurements of the entanglement between two subsystems,
which can be computed by tracing out one of the subsystems and computing the
Von Neumann entropy for the resulting reduced density matrix. The relation
between quantum information theory and the holographic principle came into
prominence with the formulation of the Ryu-Takayanagi (RT) proposal Ryu:2006bv
; Ryu:2006ef  (later, a covariant version of this proposal had been put
forward in Hubeny:2007xt ) of entanglement entropy using AdS/CFT. It says that
the quantum correlations between two regions in a CFT are given by the areas
of the minimal surfaces in asymptotically AdS spacetimes. The geometrization
of EE has been insightful in studying the encoding of boundary information in
the bulk and vice versa. Even though it partially addresses the long-standing
questions in theoretical physics, e.g. the famous information loss paradox
Penington:2019npb ; Almheiri:2019psf ; Penington:2019kki ; Almheiri:2019qdq
333For an interesting review, please see Raju:2020smc  and the references
therein., it is now understood that EE alone is not sufficient to capture the
entire information on quantum correlations in a state Susskind:2014moa .

In the last few years, another measurement of quantum information rose into
eminence, called the quantum computational complexity 2006Sci…311.1133N ;
nielsen2005geometric . In simplistic notion, quantum computational complexity
determines how difficult it is to reach from a certain reference state to a
target state using unitary operations (gates) nielsen2005geometric ;
watrous2008quantum ; aaronson2016complexity ; 2006Sci…311.1133N . It
essentially describes an extremal circuit consisting of quantum gates starting
from the reference to the target state for discrete systems or a geodesic
distance in the manifold of unitary (group) operators for field theories . In
the context of AdS/CFT, the growth of holographic complexity can capture the
growth of volume behind blackhole interiors. The two most notable holographic
complexity proposals are known as CV (complexity=volume) Stanford:2014jda  and
CA (complexity=action) Brown:2015bva . These two geometric quantities have
been conjectured to be dual to the circuit complexity of the field theory
state. After that, there has been an upsurge in understanding complexity in
the context of field theory Jefferson:2017sdb ; Chapman:2017rqy ;
Hackl:2018ptj ; Khan:2018rzm ; Bhattacharyya:2018bbv ; Bhattacharyya:2019kvj ;
me1 ; Magan:2018nmu ; Caputa:2018kdj ; Erdmenger:2020sup ; Flory:2020eot ;
Flory:2020dja ; Erdmenger:2021wzc ; Chagnet:2021uvi ; Koch:2021tvp ;
Bhattacharyya:2022ren ; 2022PhRvD.106h6010B ; Erdmenger:2022lov ;
Rabambi:2022jwu ; Jiang:2018nzg ; Guo:2020dsi  444This list is by no means
exhaustive. Interested readers are referred to these reviews Chapman:2021jbh ;
Bhattacharyya:2021cwf , and citations therein for more details..

Despite the rise of quantum complexity in theoretical physics, it is mostly
focused on applications in AdS/CFT and quantum many-body systems. Hence, it is
an opportune moment to systematically explore the notion of the circuit
complexity to other non-Lorentzian field theories and flat space holography.
For earlier work in this direction, see Fareghbal:2018ngr ; Banerjee:2022ime .
These works mainly focus either on the holographic side or explore circuit
complexity for some simple quantum mechanical models. So far, a study of
circuit complexity, purely from the field theory for BMS field theories, has
yet to be done. With this aim, we take the first step towards this direction
in our current work. Also, we are considering a semi-direct product group
instead of the usual direct product (e.g., centrally extended Virasoro and
Kac-Moody) groups to compute the circuit complexity.

#### Objectives of the paper:

In this paper, we systematically explore the construction of Nielsen’s circuit
complexity for a non-Lorentzian field theory keeping in mind its connection
with flat holography. We consider a $2d$ boundary field theory dual to $3d$
asymptotically flat spacetimes with BMS3 as the asymptotic symmetry algebra.
We aim to keep this paper as self-contained as possible. In section 2, we
briefly review the Bondi-Metzner-Sachs (BMS3) group, its algebra,
representation theory, and also the contraction from $2d$ relativistic
Conformal algebra. We discuss the isomorphism of BMS3 algebra with the
Carrollian Conformal Algebra (CCA). It is possible to reach CCA as an ultra-
relativistic ($c\to 0$) limit555The ultra-relativistic limit of the
relativistic CFT algebra gives only CCA of a particular level $N=2$. The
isomorphism between BMS and CCA is also for the level $N=2$. In this paper, we
only focus on CCA(N=2). For more details, the readers are referred to
Duval:2014uva ; Bagchi:2019clu . to the relativistic CFT algebra. The field
theories having Carrollian Conformal symmetries (Carrollian CFTs) are putative
dual to asymptotically flat spacetimes. We also mention that for $2d$
boundary, there exists another isomorphism between BMS3 and the Galilean
Conformal Algebra (GCA), which is the non-relativistic limit ($c\to\infty$) of
CFT algebra. We use this isomorphism in the later section. Next, in section 3,
we revisit the construction of circuit complexity for $2d$ CFTs, the infinite-
dimensional Virasoro algebra. We discuss that Nielsen’s complexity functional
(with essential modifications for an infinite dimensional system) for the
Virasoro group can be given by the geometric action on the co-adjoint orbit of
the Virasoro group. We review the construction of the gates and instantaneous
velocity $\epsilon(\tau,\sigma)$ from group elements product law and draw a
parallel with the Maurer-Cartan form before adding the extra contribution from
the central extension of the Virasoro group. Then, we move towards the main
objective of this paper, i.e. the circuit complexity calculation for $2d$ BMS3
invariant field theories (or, equivalently, $2d$ Carrollian CFTs). First, in
section 4, we follow the limiting analysis on the relativistic CFT results to
reach the complexity functional of $2d$ Carrollian CFT. We find that following
the non-relativistic contraction of the CFT complexity functional, it is
possible to obtain the BMS complexity functional, resembling the BMS geometric
action. However, this analysis does not include all of the information about
the BMS conserved currents (namely, the supertranslations), and hence we
refine our analysis in section 5. We complete this section purely from the
symmetries of BMS3 and construct the complexity functional intrinsically. We
see that the BMS complexity functional has a richer structure and can be
reduced to the geometric action on the co-adjoint orbit only for specific
values of the instantaneous velocity. Later, we extremize the complexity
functional and find the optimal path (for the simplest solution) in the
manifold of unitary transformations to reach the target state from the chosen
reference state (BMS primaries). We comment on our results and their
similarities with available literature in parallel with the Virasoro circuit
complexity. We end this paper with a discussion and future extensions in
section 6.

##  2 Carrollian Conformal (BMS3) physics: prerequisites

Let us start with the necessary backgrounds on the $2d$ BMS3 invariant field
theories (Carrollian CFTs). This will set the stage for the main objectives of
the paper. The asymptotic structure at the null boundary $\mathscr{I}^{\pm}$
of $3d$ asymptotically flat spacetimes are given by
$\mathbb{R}_{u}\times\mathbb{S}^{1}$. Here, the null direction is
$\mathbb{R}_{u}$ with $u,v=t\pm x$, and $\mathbb{S}^{1}$ gives the circle at
infinity. The symmetries at the boundary of the spacetimes are given by the
asymptotic symmetry groups (ASG), and the associated algebra is called the
asymptotic symmetry algebra (ASA). It was shown by Bondi, van der Burgh,
Metzner, and Sachs that the ASG for $3d$ asymptotically flat spacetimes are
given by the infinite-dimensional Bondi-Metzner-Sachs (BMS) group Bondi:1962px
; Sachs:1962zza , which is different from the usual Poincaré group. BMS group
can be realised as the semi-direct product of all local conformal
transformations (diffeomorphisms), and the angle-dependent translations of the
(null) time direction on the circle $\mathbb{S}^{1}$ called the
supertranslations. In this section, we review some useful features of the $3d$
BMS group.

###  2.1 Group algebra and contraction

The BMS3 algebra is given by,

$\displaystyle\left[L_{m},L_{n}\right]$ $\displaystyle=$
$\displaystyle(m-n)L_{m+n}+\frac{c_{L}}{12}(n^{3}-n)\delta_{m+n,0},$ (1a)
$\displaystyle\left[L_{m},M_{n}\right]$ $\displaystyle=$
$\displaystyle(m-n)M_{m+n}+\frac{c_{M}}{12}(n^{3}-n)\delta_{m+n,0},$ (1b)
$\displaystyle\left[M_{n},M_{m}\right]$ $\displaystyle=$ $\displaystyle 0.$
(1c)

Here, $L_{n}$’s, which define the diffeomorphisms on the circle
$\mathbb{S}^{1}$, are the generators of the superrotations. On the other hand,
$M_{n}$’s are the generators of the supertranslations or the angle-dependent
translations on the circle at infinity. $c_{L}$ and $c_{M}$ are the two
central extensions. For the Einstein gravity, they turn out to be the
following

$c_{L}=0\quad\textrm{and}\quad c_{M}=\frac{3}{G},$

where $G$ is Newton’s Constant Barnich:2006av . This is obtained by performing
an asymptotic symmetry analysis and computing the Poisson bracket for the
charges.

In a seminal analysis by Brown and Henneaux Brown:1986nw , the ASA for $3d$
Anti-de Sitter spacetime was also found to be infinite-dimensional, comprising
of two copies of the Virasoro algebra.

$\displaystyle\left[\mathcal{L}_{m},\mathcal{L}_{n}\right]$ $\displaystyle=$
$\displaystyle(m-n)\mathcal{L}_{m+n}+\frac{c}{12}(n^{3}-n)\delta_{m+n,0},$
(2a) $\displaystyle\left[\bar{\mathcal{L}}_{m},\bar{\mathcal{L}}_{n}\right]$
$\displaystyle=$
$\displaystyle(m-n)\bar{\mathcal{L}}_{m+n}+\frac{\bar{c}}{12}(n^{3}-n)\delta_{m+n,0}.$
(2b)

Here, the central terms for Einstein gravity in AdS3 are given by,

$c=\bar{c}=\frac{3l}{2G}.$ (3)

with $l$ being the AdS radius. The BMS3 algebra (1) of asymptotically flat
spacetime can also be reached from the ASA of the AdS3 (2) by an Inönü-Wigner
contraction,

$L_{n}=\mathcal{L}_{n}-\bar{\mathcal{L}}_{-n},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
M_{n}=\epsilon(\mathcal{L}_{n}+\bar{\mathcal{L}}_{-n}),\leavevmode\nobreak\
\leavevmode\nobreak\ \epsilon\to 0.$ (4)

In literature, this is referred to as the ultra-relativistic or Carrollian
limit Leblond65 ; Sengupta65 . In the Carrollian limit, the light cone closes
up, and the speed of light goes to zero ($c\to 0$). This can be understood by
the following spacetime contraction

$t\to\epsilon t,\leavevmode\nobreak\ \leavevmode\nobreak\ x^{i}\to
x^{i},\leavevmode\nobreak\ \leavevmode\nobreak\ \epsilon\to
0.\implies\frac{v}{c}=\frac{1}{c}\frac{x}{t}\to\infty.$ (5)

The Carrollian limit can be interpreted as taking the AdS radius $(l)$ to
infinity in bulk, or equivalently taking the inverse radius
$\epsilon=\frac{1}{l}\to 0$ Bagchi:2012cy . Also, we can recover the BMS
central terms from the Brown-Henneaux central charges of AdS3 using the
following contraction,

$c_{L}=(c-\bar{c})=0,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
c_{M}=\lim_{l\to\infty}\frac{1}{l}(c+\bar{c})=\frac{3}{G}.$ (6)

On the other hand, the Virasoro generators on the cylinder (corresponding to
the conformal boundary of $3d$ global AdS)

$\mathcal{L}_{m}=e^{im\omega}\partial_{\omega},\leavevmode\nobreak\
\leavevmode\nobreak\
\bar{\mathcal{L}}_{m}=e^{-im\bar{\omega}}\partial_{\bar{\omega}},\leavevmode\nobreak\
\leavevmode\nobreak\ \text{where,\leavevmode\nobreak\
}\omega,\bar{\omega}=t\pm\sigma$ (7)

under the Inönü-Wigner contraction

$\sigma\to\sigma,\leavevmode\nobreak\ \leavevmode\nobreak\ t\to\epsilon
t,\leavevmode\nobreak\ \leavevmode\nobreak\ \epsilon\to
0,\implies\frac{v}{c}\sim\frac{1}{c}\frac{\sigma}{t}\to\infty$ (8)

gives the BMS3 generators on the null cylinder $(\sigma,t)$

$L_{n}=e^{in\sigma}(\partial_{\sigma}+int\partial_{t}),\leavevmode\nobreak\
\leavevmode\nobreak\ M_{n}=ie^{in\sigma}\partial_{t}.$ (9)

This is known as the cylinder representation of BMS, where only the coordinate
$\sigma$ (representing the circle at infinity) is compact. The null direction
$\tau$, also corresponding to the time direction in the dual field theory, is
non-compact. There exists another representation known as the plane
representation, where both the directions on the plane $(x,\tau)$ are non-
compact.

$L_{n}=-x^{n+1}\partial_{x}-(n+1)x^{n}\tau\partial_{\tau},\leavevmode\nobreak\
\leavevmode\nobreak\ M_{n}=x^{n+1}\partial_{\tau}.$ (10)

The plane-to-cylinder mapping takes the following coordinate transformations,

$\tau=ite^{i\sigma},\leavevmode\nobreak\ \leavevmode\nobreak\ x=e^{i\sigma}.$
(11)

In general, a finite BMS transformation takes the form Barnich:2012xq

$\tilde{\sigma}=f(\sigma),\leavevmode\nobreak\ \leavevmode\nobreak\
\tilde{t}=f^{\prime}(\sigma)t+\alpha(\sigma).$ (12)

###  2.2 Short review of the representation theory

Now, we will take a brief look into the representation theory of $2d$ BMS3
invariant field theories (BMSFT). First, let us summarise the highest weight
representation in $2d$ CFTs. In $2d$ CFTs, the eigenvalues $h,\bar{h}$ of the
operators $\mathcal{L}_{0},\bar{\mathcal{L}}_{0}$ label the states,

$\mathcal{L}_{0}|h,\bar{h}\rangle=h|h,\bar{h}\rangle,\leavevmode\nobreak\
\leavevmode\nobreak\
\bar{\mathcal{L}}_{0}|h,\bar{h}\rangle=\bar{h}|h,\bar{h}\rangle.$ (13)

The CFT primaries are defined as

$\mathcal{L}_{n}|h,\bar{h}\rangle_{p}=\bar{\mathcal{L}}_{n}|h,\bar{h}\rangle_{p}=0\leavevmode\nobreak\
\leavevmode\nobreak\ \forall n>0.$ (14)

The spectrum is bounded from below, and the CFT highest weight module is
constructed by acting raising operators
$\mathcal{L}_{-n},\bar{\mathcal{L}}_{-n}\leavevmode\nobreak\ (\forall n>0)$ on
the primaries. The primary states are the basis of the highest weight
representation, and the descendant states are obtained by acting with
arbitrary numbers of raising operators on them.

Now, we construct the highest weight representation of BMS in a similar
fashion. We label the BMS states by the eigenvalues of dilatation and boost,
$L_{0}$ and $M_{0}$, respectively.

$L_{0}|\Delta,\xi\rangle=\Delta|\Delta,\xi\rangle,\leavevmode\nobreak\
\leavevmode\nobreak\ M_{0}|\Delta,\xi\rangle=\xi|\Delta,\xi\rangle.$ (15)

The BMS primary is defined by assuming the spectrum is bounded from below.

$L_{n}|\Delta,\xi\rangle_{p}=M_{n}|\Delta,\xi\rangle_{p}=0,\leavevmode\nobreak\
\leavevmode\nobreak\ \forall n>0.$ (16)

Similar to $2d$ CFT, the BMS highest weight modules are constructed by acting
the raising operators $L_{-n},M_{-n}\leavevmode\nobreak\ (\forall n>0)$ on the
primaries.

The Virasoro states $|h,\bar{h}\rangle$ goes over to the BMS states
$|\Delta,\xi\rangle$ under the UR (Carrollian) contraction (4) Bagchi:2020rwb

$\displaystyle(\mathcal{L}_{0}-\bar{\mathcal{L}}_{0})|h,\bar{h}\rangle$
$\displaystyle\implies$ $\displaystyle
L_{0}|\Delta,\xi\rangle=\Delta|\Delta,\xi\rangle,\text{with\leavevmode\nobreak\
}\Delta=h-\bar{h},$
$\displaystyle(\mathcal{L}_{0}+\bar{\mathcal{L}}_{0})|h,\bar{h}\rangle$
$\displaystyle\implies$ $\displaystyle
M_{0}|\Delta,\xi\rangle=\xi|\Delta,\xi\rangle,\text{with\leavevmode\nobreak\
}\xi=\epsilon(h+\bar{h})$ (17)

with the following identifications
$\Delta=h-\bar{h},\>\>\xi=\epsilon(h+\bar{h})$. However, it is interesting to
point out here that the UR contraction (4) of the Virasoro highest weight
representation does not end up giving the BMS highest weight module. It is
easier to see this from the conditions on the primaries.

$\displaystyle\mathcal{L}_{n}|h,\bar{h}\rangle$ $\displaystyle=$
$\displaystyle(L_{n}+\frac{1}{\epsilon}M_{n})|h,\bar{h}\rangle=0,\>\>\forall
n>0,\leavevmode\nobreak\ \leavevmode\nobreak\ \implies
M_{n}|\Delta,\xi\rangle=0,\forall n>0,$
$\displaystyle\bar{\mathcal{L}}_{n}|h,\bar{h}\rangle$ $\displaystyle=$
$\displaystyle(-L_{-n}+\frac{1}{\epsilon}M_{-n})|h,\bar{h}\rangle=0,\>\>\forall
n>0,\leavevmode\nobreak\ \leavevmode\nobreak\ \implies
M_{-n}|\Delta,\xi\rangle=0,\forall n>0.$ (18)

Thus, the UR contractions on the Virasoro highest weight representation give
the following conditions

$L_{0}|\Delta,\xi\rangle=\Delta|\Delta,\xi\rangle,\leavevmode\nobreak\
\leavevmode\nobreak\
M_{0}|\Delta,\xi\rangle=\xi|\Delta,\xi\rangle,\leavevmode\nobreak\
\leavevmode\nobreak\ M_{n}|\Delta,\xi\rangle=0\leavevmode\nobreak\ \forall
n\neq 0\,.$ (19)

This is known as the induced representation of BMS in literature, explored in
detail in Barnich:2014kra ; Barnich:2015uva ; Campoleoni:2016vsh . Even though
the BMS’s highest weight and induced representations have very different
characteristics, the operations of $L_{0},M_{0}$ on the primaries are the same
for both.666In our calculation, we have only used the action of $L_{0},M_{0}$
on the BMS primaries $|\Delta,\xi\rangle$. Hence, our results are valid in
both choices of representations.

However, it is possible to reach the BMS highest weight representation from
Virasoro’s highest weight through a different limit, reproducing the same
algebra in (1). This is known as the non-relativistic (NR) contraction. This
is opposite to the UR contraction described above, in the sense that the light
cone opens up and the speed of light $c\to\infty$. The NR contraction on the
Virasoro generators is realised as,

$L_{n}=\mathcal{L}_{n}+\bar{\mathcal{L}}_{n},\>\>M_{n}=\epsilon(\mathcal{L}_{n}-\bar{\mathcal{L}}_{n})\,.$
(20)

The readers can find more details on the NR contraction and the differences
between the two singular limits in Bagchi:2009my ; Bagchi:2019xfx . In the
non-relativistic limit, the CFT primary goes over to the BMS primaries
$|\Delta,\xi\rangle$, with the identifications
$\Delta=h+\bar{h},\>\xi=\epsilon(h-\bar{h})$.

$\displaystyle(\mathcal{L}_{0}+\bar{\mathcal{L}_{0}})|h,\bar{h}\rangle$
$\displaystyle=(h+\bar{h})|h,\bar{h}\rangle,\Rightarrow
L_{0}|\Delta,\xi\rangle=\Delta|\Delta,\xi\rangle,\text{\leavevmode\nobreak\
with\leavevmode\nobreak\ }\Delta=h+\bar{h},$ (21a)
$\displaystyle(\mathcal{L}_{0}-\bar{\mathcal{L}_{0}})|h,\bar{h}\rangle$
$\displaystyle=(h-\bar{h})|h,\bar{h}\rangle,\Rightarrow
M_{0}|\Delta,\xi\rangle=\xi|\Delta,\xi\rangle,\text{\leavevmode\nobreak\
with\leavevmode\nobreak\ }\xi=\epsilon(h-\bar{h})\,.$ (21b)

The NR limit on the CFT algebra produces the Galilean Conformal Algebra (GCA).
In terms of spacetime coordinate, the NR limit reflects as

$t\to t,\leavevmode\nobreak\ \leavevmode\nobreak\ x^{i}\to\epsilon
x^{i},\implies,\frac{v}{c}\sim\frac{1}{c}\frac{x}{t}\to 0,\leavevmode\nobreak\
\leavevmode\nobreak\ \text{equivalently,\leavevmode\nobreak\ }c\to\infty\,.$
(22)

It is also fascinating to note that the isomorphism between the CCA and GCA
exists only when the boundary spacetime dimensions ($d$) is two. In $2d$, we
contract only one direction; the temporal one for the UR (Carrollian)
contraction and the spatial direction for the NR (Galilean) one (as opposed to
more than one contracted spatial direction for $d>2$), reproducing the same
algebra (1)Bagchi:2019xfx . We will exploit this isomorphism in a later part
of the paper.

##  3 Revisiting circuit complexity construction for Virasoro groups

After briefly reviewing the necessary details of the BMS3 algebra and its
contractions, we now focus on the other component of this paper, the circuit
complexity. In this section, we revisit circuit complexity for the
relativistic cousin of the BMS group, the infinite-dimensional Virasoro group.
This review will come in handy before we dive into the details of BMS
complexity. Note that, here, we are only considering one copy of the Virasoro
algebra (the holomorphic part) similar to Caputa:2018kdj ; Erdmenger:2020sup .
The contribution from the anti-holomorphic part comes in a similar fashion,
and we will address the same in the next section. First, let us discuss the
modifications necessary to use the notion of Nielsen’s geometric approach
nielsen2005geometric ; nielsen2006quantum ; dowling2006geometry  in computing
the circuit complexity for an continuum field theory Caputa:2018kdj ;
Erdmenger:2020sup .

The geometric approach of Nielsen’s complexity counts the discrete gates to
reach from a reference to a target state, resulting in the construction of a
path in the space of the unitary operators and then extremizes it
nielsen2005geometric ; nielsen2006quantum . In general, one also needs to
associate a penalty factor with various gates, and then perform the
extremization of complexity functional. This extremization essentially gives
geodesics on the symmetry group manifold. The penalty factor determines how
hard it is to perform the given set of transformations. In our case, the gates
are built out of the symmetry transformation generators for a continuous
symmetry group. Hence, all gates are equally difficult to perform. Thus, the
notion of the penalty factor does not bear any contribution (other than an
overall pre-factor) in the complexity functional. Due to this choice, we can
reach the target states that are only related to the reference state by group
transformations. Also, the complexity depends on the choice of the reference
state.

Now, let us briefly review the circuit complexity for the Virasoro group
following Caputa:2018kdj ; Erdmenger:2020sup . The Virasoro group is denoted
by the group elements $(f(\sigma),\mathfrak{a})$. Here, $f(\sigma)$ with
$\sigma\in\mathbb{S}^{1}$ denotes diffeomorphisms on the circle
$\mathbb{S}^{1}$. In other words, considering $\sigma$ as the light cone
coordinate, $\sigma\to f(\sigma)$ gives the conformal transformations in $2d$.
The extra contribution $\mathfrak{a}\in\mathbb{R}$ comes from the central
extension $c$ of the Virasoro group. We will first focus on the contribution
from the conformal transformations $f(t,\sigma)$.

Let us start with a reference state $|\psi_{R}\rangle$ of our choice, and the
target state $|\psi_{T}\rangle$ is related to the reference state by unitary
transformations of the group.

$|\psi_{T}\rangle=U_{f(T)}|\psi_{R}\rangle\,.$ (23)

Here, $U_{f(t)}$ denotes the unitary group transformations after time $t$. At
$t=0,$ the initial transformation $U_{f(t=0)}=\mathrm{1}$ denotes the
reference state itself, and $U_{f(t=T)}$ gives the final target state. Thus,
we can write the finite transformation $U_{f(T)}$ in terms of infinitesimal
group transformations

$U_{f(T)}=U_{\epsilon(T)}U_{\epsilon(T-dT)}\cdots U_{\epsilon(dt)}\mathrm{1}.$
(24)

Therefore, the infinitesimal symmetry generators play the role of unitary
gates of Nielsen’s complexity. In $2d$, the conserved energy-momentum tensor
$T(\sigma)$ of the Virasoro group can be used to write the gates,

$Q(t)=\frac{1}{2\pi}\int d\sigma\>\epsilon(t,\sigma)\>T(\sigma).$ (25)

Here, the gates $Q(t)$ are path ordered in the sense that earlier gates are
applied first, and $\epsilon(t,\sigma)$ is the instantaneous group velocities.
The geometric notion of Nielsen’s complexity defines a path (geodesic, after
extremization) in the manifold of the group transformations. Two
infinitesimally close points on the path are related to each other by
Caputa:2018kdj ,

$U(t+dt)=e^{-Q(t)dt}\>U(t),\leavevmode\nobreak\ \leavevmode\nobreak\
\text{where,\>\>}U_{f(t)}=U(t).$ (26)

Now, we calculate the instantaneous group velocities $\epsilon(t,\sigma)$ from
the group elements (diffeomorphisms) $f(t,\sigma)$.

$f(t+dt,\sigma)=e^{\epsilon(t,\sigma)dt}\cdot f(t,\sigma)$ (27)

Here, ‘$\cdot$’ refers to the group operation of Virasoro group $(f_{1}\cdot
f_{2}=f_{1}\circ f_{2})$. Expanding upto the first order on both sides,

$\displaystyle f(t,\sigma)+dt\>\frac{\partial f(t,\sigma)}{\partial t}$
$\displaystyle=$ $\displaystyle(\mathrm{1}+dt\>\epsilon(t,s))\circ
f(t,\sigma).$ $\displaystyle\implies\epsilon(t,f(t,\sigma))$ $\displaystyle=$
$\displaystyle\frac{\partial f(t,\sigma)}{\partial t}.$

Our aim is to extract the group velocity $\epsilon(t,\sigma)$. To do so, we
define the inverse diffeomorphism $F(t,\sigma)$ at each point to the
diffeomorphisms $f(t,\sigma)$, such that

$F(t,f(t,\sigma))=\sigma.$ (28)

Thus,

$\epsilon(t,\sigma)=\frac{\partial f(t,F(t,\sigma))}{\partial t}.$ (29)

To find the expression of $\epsilon(t,\sigma)$ we use the following

$\frac{\partial\>F(t,f)}{\partial t}+\frac{\partial F(t,f)}{\partial
f(t,\sigma)}\frac{\partial\>f(t,\sigma)}{\partial t}=0.$ (30)

Next, we change the variable from $f(t,\sigma)$ to $\sigma$. Thus,

$\displaystyle\frac{\partial\>F(t,\sigma)}{\partial t}+\frac{\partial
F(t,\sigma)}{\partial\sigma}\frac{\partial\>f(t,F(t,\sigma))}{\partial t}=0.$
$\displaystyle\implies\frac{\partial f(t,F(t,\sigma))}{\partial
t}=-\frac{\frac{\partial\>F(t,\sigma)}{\partial
t}}{\frac{\partial\>F(t,\sigma)}{\partial\sigma}}.$ (31)

This evaluates the group velocity in terms of the inverse diffeomorphism
$F(t,\sigma)$ Caputa:2018kdj ; Erdmenger:2020sup

$\epsilon(t,\sigma)=-\frac{\dot{F}(t,\sigma)}{F^{\prime}(t,\sigma)}.$ (32)

Here, ‘$\leavevmode\nobreak\ \dot{}\leavevmode\nobreak\ $’ and
‘$\leavevmode\nobreak\ \prime\leavevmode\nobreak\ $’ denotes the derivative
wrt $t$ and $\sigma$.

After evaluating the group velocity, we are left with writing the cost
functional $\mathcal{F}$. We adapt Nielsen’s approach to write the cost
functional of a finite-dimensional system and modify it by imposing state-
dependence in order to account for an infinite dimensional system. For this
purpose, we use the density matrix $\rho(t)$, which is evolved from the
initial density matrix $\rho_{0}$ by the symmetry transformations $U(t)$.

$\rho(t)=U(t)\rho_{0}U^{\dagger}(t),\leavevmode\nobreak\ \leavevmode\nobreak\
\rho_{0}=|\psi_{R}\rangle\langle\psi_{R}|$ (33)

We chose the one-norm cost function with the explicit state-dependence
according to Caputa:2018kdj ; Erdmenger:2020sup ,

$\mathcal{F}=|tr[\rho(t)Q(t)]|=|\langle\psi_{R}|U^{\dagger}(t)Q(t)U(t)|\psi_{R}\rangle|=\int
d\sigma\>\epsilon(t,\sigma)\langle\psi_{R}|U^{\dagger}(t)T(\sigma)U(t)|\psi_{R}\rangle.$
(34)

There are a plethora of choices for cost-functional. For more details of
possible choices for cost-functional interested readers are referred to
nielsen2005geometric ; Guo:2018kzl . It is worth reminding here that the issue
of penalty factors associated with the possible paths is already taken care of
by choosing symmetry transformations as the gates $Q(t)$. Hence, the
complexity functional $\mathcal{C}[f]$ gives the total cost of a constructed
path from the reference to the target state in the group manifold in terms of
group element $f(t)$.

$\mathcal{C}[f]=\int dt\mathcal{F}=\frac{1}{2\pi}\int dt\int
d\sigma\>\epsilon(t,\sigma)\langle\psi_{R}|U^{\dagger}(t)T(\sigma)U(t)|\psi_{R}\rangle\,.$
(35)

Minimizing the complexity functional $C[f]$ gives the equation of motion in
terms of the group path $f(t)$, the solution of which gives the optimized
circuit. We then put the solution back into the complexity functional to
obtain the on-shell value of the complexity arising from the optimal circuit
starting from the chosen reference to the desired target state. For
completeness, we show the construction of the complexity functional explicitly
for the Virasoro group Caputa:2018kdj ; Erdmenger:2020sup .

First, we write the transformation law of the energy-momentum tensor $T$ under
the conformal transformation in terms of the inverse diffeomorphism $F$,
consisting of the usual Schwarzian $\\{F,\sigma\\}$.

$U_{f}^{\dagger}TU_{f}=F^{\prime
2}T(F)+\frac{c}{12}\\{F,\sigma\\},\leavevmode\nobreak\ \leavevmode\nobreak\
\text{where,\>}\\{F,\sigma\\}=\frac{F^{\prime\prime\prime}(\sigma)}{F^{\prime}(\sigma)}-\frac{3}{2}\left(\frac{F^{\prime\prime}(\sigma)}{F^{\prime}(\sigma)}\right)^{2}\,.$
(36)

We also choose the reference state to be the CFT primary,
$|\psi_{R}\rangle=|h\rangle$. Putting the transformation of $T$ and the
expression for the group velocity $\epsilon(t,\sigma)$ back into (35), we
write the complexity functional $\mathcal{C}[F]$ in terms of the inverse
diffeomorphism,

$\mathcal{C}[F]=\int dt\>\mathcal{F}=\frac{1}{2\pi}\int dt\int
d\sigma\left[-\dot{F}F^{\prime}\langle
h|T(F)|h\rangle-\frac{c}{12}\frac{\dot{F}}{F^{\prime}}\\{F,\sigma\\}\right]\,.$
(37)

In terms of the diffeomorphism $f(t,\sigma)$, the complexity functional takes
the form

$\mathcal{C}[f]=\frac{1}{2\pi}\int^{T}_{0}dt\int_{0}^{2\pi}d\sigma\frac{\dot{f}}{f^{\prime}}\left(-\langle
h|T(\sigma)|h\rangle+\frac{c}{12}\\{f,\sigma\\}\right)\,.$ (38)

Next, we will be accounting for the contribution due to the central extension
$\mathfrak{a}$ in the complexity.

#### Contribution due to the central extension:

In Caputa:2018kdj ; Erdmenger:2020sup , the instantaneous velocity
$\epsilon(t,\sigma)$ is identified with the non-centrally extended Maurer-
Cartan form $\theta_{f^{-1}}$ of the Virasoro group. This can be easily
understood by writing the symmetry transformation $f(t)$ of the Virasoro group
in terms of group velocity $\epsilon(t,\sigma)$ and its infinitesimal close
point on the path of unitary manifold, followed by multiplying both sides with
the inverse element $F$, and then taking derivative,

$f(t)=e^{\int_{t}^{s}\>\epsilon{(s^{\prime})ds^{\prime}}}f(s),\quad\implies\frac{d}{ds}\Big{|}_{s=t}(f(t)\cdot
F(s))=-\epsilon(t).$ (39)

The non-centrally extended Maurer-Cartan form of the Virasoro group in terms
of $f^{-1}$ (rather than $f$) is the group velocity,

$\theta_{f^{-1}}=\frac{d}{ds}\Big{|}_{s=t}(f(t)\circ F(s))=-\epsilon(t).$ (40)

However, in Erdmenger:2020sup , the analysis was carried forward by arguing
that it is also necessary to use the generalized (centrally extended) Maurer-
Cartan form in order to find the complete contribution of the central
extension $c$ in the circuit complexity. The authors argued not only to use
the path through the group transformations $f(t)$, but also the path through
the real number $\mathfrak{a}(t)$. This statement translates to the
generalized version of (39),

$(f(t),\mathfrak{a}(t))=e^{\int_{t}^{s}\>(\epsilon{(s^{\prime}),\>\beta(s^{\prime}))ds^{\prime}}}\big{(}f(s),\mathfrak{a}(s)\big{)}\,.$
(41)

Multiplying from the right side by
$\big{(}f,\mathfrak{a}\big{)}^{-1}=(F,-\mathfrak{a})$ and taking the
derivative wrt $s$, we obtain the Maurer-Cartan form $\theta$ and its central
extension $m_{\theta}$ with respect to the inverse element $f^{-1}$ of the
Virasoro group,

$\displaystyle\big{(}\theta,m_{\theta}\big{)}_{f^{-1}}=-\big{(}\epsilon(t),\beta(t)\big{)}$
$\displaystyle=$
$\displaystyle\frac{d}{ds}\Big{(}\big{(}f(t),\mathfrak{a}(t))\cdot(F(s),-\mathfrak{a}(s))\Big{)},$
$\displaystyle=$ $\displaystyle\Bigg{(}\frac{d}{ds}\Big{|}_{s=t}(f(t)\circ
F(s)),\>\>\frac{d}{ds}\Big{|}_{s=t}\mathfrak{C}\big{(}f(t),F(s)\big{)}+a(t)-a(s)\Bigg{)}\,.$

Here, in the last line, we have used the group multiplication rule of the
Virasoro group $(g_{1},\alpha_{1})\cdot(g_{2},\alpha_{2})=(g_{1}\circ
g_{2},\alpha_{1}+\alpha_{2}+\mathfrak{C}(g_{1},g_{2}))$. Also, the real number
valued part is assumed to be constant $\mathfrak{a}(t)=\mathfrak{a}(s)=$const.
The term $\mathfrak{C}\big{(}f(t),F(s)\big{)}$ is the 2-cocycle defining the
central extensions. The centrally extended Maurer-Cartan form takes the form
for the Virasoro group Erdmenger:2020sup ; Oblak:2017ect ,

$(\theta,m_{\theta})=\Bigg{(}\frac{\dot{F}}{F^{\prime}},\frac{1}{48\pi}\int_{0}^{2\pi}d\sigma\frac{\dot{F}}{F^{\prime}}\>\Big{(}\frac{F^{\prime\prime}}{F^{\prime}}\Big{)}^{{}^{\prime}}\>\Bigg{)}\,.$
(44)

This begs the following modification in the cost function (34) due to the
velocities $\epsilon(t)$ and $\beta(t)$ of centrally extended Virasoro group,

$\mathcal{F}=c\beta(t)\>\>+\>\>\int
d\sigma\>\epsilon(t,\sigma)\langle\psi_{R}|U^{\dagger}(t)T(\sigma)U(t)|\psi_{R}\rangle.$
(45)

Putting both the velocities $\epsilon(t)=-\theta$ and $\beta(t)=-m_{\theta}$
from (44) back in the complexity functional (37), we find

$\displaystyle\mathcal{C}[F]$ $\displaystyle=$ $\displaystyle\int
dt\>\mathcal{F}$ (46) $\displaystyle=$ $\displaystyle\int dt\int
d\sigma\left[-\dot{F}F^{\prime}\frac{\langle
h|T(F)|h\rangle}{2\pi}+\frac{c}{24\pi}\frac{\dot{F}}{F^{\prime}}\\{F,\sigma\\}-\frac{c}{48\pi}\frac{\dot{F}}{F^{\prime}}\>\Big{(}\frac{F^{\prime\prime}}{F^{\prime}}\Big{)}^{{}^{\prime}}\right]$
$\displaystyle=$ $\displaystyle\int
d\sigma\>dt\left[-j_{0}(F)\dot{F}{F^{\prime}}+\frac{c}{48\pi}\frac{\dot{F}}{F^{\prime}}\>\Bigg{(}\frac{F^{\prime\prime\prime}}{F^{\prime}}-2\Big{(}\frac{F^{\prime\prime}}{F^{\prime}}\Big{)}^{{}^{2}}\Bigg{)}\right].$

Here, $j_{0}$ is a constant. The complexity functional (46), including the
contribution from the central term, is equal to the geometric action for the
Virasoro group Alekseev:1988ce ; Alekseev:2018pbv ; Alekseev:1990mp ;
Barnich:2017jgw . This is one of the central claims of Erdmenger:2020sup ,
where the authors show the equivalence of the complexity functional with the
on-shell value of the geometric action for centrally extended direct product
groups, e.g. Virasoro and Kac-Moody. To construct the circuit complexity for
symmetry groups in the case of non-Lorentzian field theories, we start our
analysis from here in the next section.777For extremization of the complexity
functional to find the optimal path $f$ and the corresponding value of the
complexity, readers are referred to Erdmenger:2020sup .

##  4 From CFT to BMS complexity

In the last section, we briefly familiarised ourselves with the formulation of
circuit complexity using symmetry transformations as the unitary gates for the
infinite-dimensional centrally extended Virasoro group. Strikingly, the
modified cost function (due to the contribution from the central extension
(45)) gives the complexity functional (46), which is equivalent to the
geometric action of the centrally extended Virasoro group. Now, we will
translate the above formulation for the BMS3 group ($2d$ Carrollian Conformal
group) and find the circuit complexity. We will approach the problem in two
ways: first, as a limit to the formulation used in the context of the Virasoro
group in section 3; and then using the intrinsic symmetry transformations of
BMS.

BMS3 is the asymptotic symmetry group at the boundary of $3d$ asymptotically
flat spacetime. The null boundary has a structure
$\mathbb{R}\times\mathbb{S}^{1}$. BMS3 group is the semi-direct product of the
local co-ordinate transformations (diffeomorphisms) on the circle at
$\mathscr{I}^{+}$ (also, known as the superrotations) and the angle-dependent
supertranslations. Also, $c_{L}$ and $c_{M}$ are the central charges in the
$[L,L]$ and $[L,M]$ brackets in (1).

$\text{BMS}_{3}=\underbrace{\text{Diff}(S^{1})}_{\text{Super-
rotation}}\ltimes\underbrace{\text{Vec}(S^{1})}_{\text{Supertranslation}}\,.$
(47)

The elements of the BMS3 groups are the diffeomorphisms $f(t,\sigma)$ with
central extension $c_{1}$ and the supertranslations $\alpha(t,\sigma)$ with
central extension $c_{2}$. They are given by the periodicity conditions:

$\displaystyle f(\sigma+2\pi)$ $\displaystyle=$ $\displaystyle
f(\sigma)+2\pi,\leavevmode\nobreak\ \leavevmode\nobreak\
f^{\prime}(\sigma)>0,$ (48) $\displaystyle\alpha(\sigma+2\pi)$
$\displaystyle=$ $\displaystyle\alpha(\sigma).$ (49)

In section 2, we showed that it is possible to reach the BMS3 algebra starting
from the Virasoro algebra. Hence, we will first use the limiting analysis to
find the BMS3 circuit complexity from the analogous results of the centrally
extended Virasoro group. We start with the Virasoro cost functional (34)
without the contribution from the central extension,

$\mathcal{C}_{1}=\int dt\>\mathcal{F}=\frac{1}{2\pi}\int dt\int
d\sigma\>\epsilon(t,\sigma)\big{\langle}\psi_{R}\big{|}U^{\dagger}(t)\left(\>T+\bar{T}\>\right)U(t)\big{|}\psi_{R}\big{\rangle}\,.$
(50) $\mathcal{C}=\mathcal{C}_{1}+\mathcal{C}_{2}\,,$ (51)

where $\mathcal{C}_{2}$ is the contribution from the central extension to the
circuit complexity $\mathcal{C}$, which we discuss later. Here, we have also
explicitly written the contribution from the holomorphic and anti-holomorphic
parts $T,\bar{T}$ of the stress tensors for $2d$ CFT. Now, we use the non-
relativistic limit (20) on the CFT stress tensors.888In $2d$, the non-
relativistic and the ultra-relativistic contractions of the Virasoro algebra
gives the GCA and the CCA, respectively, which are isomorphic to the BMS3
algebra. The isomorphism between GCA and CCA is not seen in $d>2$.  For the
non-relativistic limit the contraction is given by Bagchi:2015nca ,

$T+\bar{T}=T_{1},\leavevmode\nobreak\ \leavevmode\nobreak\ \lim_{\epsilon\to
0}\epsilon(T-\bar{T})=T_{2}.$ (52)

Here, $T_{1},T_{2}$ are the stress tensors for BMS3. The reference state for
$2d$ CFT is chosen to be the primary $|h,\bar{h}\rangle$. In the non-
relativistic limit, the CFT primary goes to the BMS primaries
$|\Delta,\xi\rangle$ Bagchi:2020rwb . Thus,

$|\psi_{R}\rangle=|h,\bar{h}\rangle\xRightarrow[\text{limit}]{\text{NR}}|\Delta,\xi\rangle\,.$
(53)

Hence the contribution from (50) in the non-relativistic limit gives,

$\mathcal{C}_{1}=\int dt\>\mathcal{F}=\frac{1}{2\pi}\int dt\int
d\sigma\>\epsilon(t,\sigma)\big{\langle}\Delta,\xi\big{|}U^{\dagger}(t)T_{1}(t,\sigma)U(t)\big{|}\Delta,\xi\big{\rangle}\,.$
(54)

Next, we expand the BMS3 stress tensors with modes on the cylinder
Bagchi:2015nca ,

$T_{1}(t,\sigma)=\sum_{n}\left(L_{n}-in\sigma
M_{n}\right)e^{-int}+\frac{c_{L}}{12}\delta_{n,0},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
T_{2}(\tau,\sigma)=\sum_{n}M_{n}e^{-int}+\frac{c_{M}}{12}\delta_{n,0}.$ (55)

Here, $L_{n}$ and $M_{n}$’s are generators of the BMS3 algebra in (1). Thus,

$\mathcal{C}_{1}=\frac{1}{2\pi}\int dt\int
d\sigma\>\epsilon(t,\sigma)\sum_{n}\big{\langle}\Delta,\xi\big{|}U^{\dagger}(t)\Big{(}\left(L_{n}-in\sigma
M_{n}\right)e^{-int}+\frac{c_{L}}{12}\delta_{n,0}\Big{)}U(t)\big{|}\Delta,\xi\big{\rangle}$
(56)

On the cylinder, the generators are written in terms of the conserved currents
$j^{cyl}$ and $p^{cyl}$ Jiang:2017ecm ,

$L_{n}^{cyl}=-\frac{1}{2\pi}\int_{0}^{2\pi}d\sigma
e^{in\sigma}j^{cyl}(\sigma),\leavevmode\nobreak\ \leavevmode\nobreak\
M_{n}=-\frac{1}{2\pi}\int_{0}^{2\pi}d\sigma e^{in\sigma}p^{cyl}(\sigma)$ (57)

Under the finite BMS transformations,

$\tilde{\sigma}=F(\sigma),\leavevmode\nobreak\ \leavevmode\nobreak\
\tilde{t}=tF^{\prime}(\sigma)+\alpha(\sigma)$ (58)

the transformation relations of the conserved currents are Basu:2015evh ;
Jiang:2017ecm

$\displaystyle\tilde{p}(\sigma)$
$\displaystyle=F^{\prime}(\sigma)^{2}p(\tilde{\sigma})+\frac{c_{M}}{12}\\{F,\sigma\\},$
(59a) $\displaystyle\tilde{j}(\sigma)$
$\displaystyle=F^{\prime}(\sigma)^{2}j({\tilde{\sigma}})+2F^{\prime}(\sigma)\alpha^{\prime}(\sigma)p(\tilde{\sigma})+F^{\prime}(\sigma)^{2}\alpha\>\partial_{\tilde{\sigma}}p(\tilde{\sigma})+\frac{c_{L}}{12}\\{F,\sigma\\}+\frac{c_{M}}{12}\llbracket(F,\alpha),\sigma\rrbracket\,.$
(59b)

Here, prime ‘$\>\prime$ ’ denotes the derivative wrt $\sigma$. Also, we have
the Schwarzian $\\{F,\sigma\\}$ and the BMS Schwarzian
$\llbracket(F,\alpha),\sigma\rrbracket$,

$\displaystyle\\{F,\sigma\\}$
$\displaystyle=\frac{F^{\prime\prime\prime}}{F^{\prime}}-\frac{3}{2}\left(\frac{F^{\prime\prime}}{F^{\prime}}\right)^{2},$
(60a) $\displaystyle\llbracket(F,\alpha),\sigma\rrbracket$
$\displaystyle=\frac{[3(F^{\prime\prime})^{2}-F^{\prime}F^{\prime\prime\prime}]\alpha^{\prime}-3F^{\prime}F^{\prime\prime}\alpha^{\prime\prime}+(F^{\prime})^{2}\alpha^{\prime\prime\prime}}{(F^{\prime})^{3}}\,.$
(60b)

We choose the reference state to be a BMS primary $|\Delta,\xi\rangle$ (15).
Hence, the only mode that will contribute is $n=0$, or $L_{0}$ in the sandwich
between two BMS primaries (35). We are rewriting the transformation relation
of the currents (59) in a slightly different notation following Merbis:2019wgk
; Barnich:2017jgw .

$\displaystyle\tilde{p}(\sigma)$
$\displaystyle=F^{\prime}(\sigma)^{2}p(F)-\frac{c_{2}}{24\pi}\\{F,\sigma\\},$
(61a) $\displaystyle\tilde{j}(\sigma)$
$\displaystyle=F^{\prime}(\sigma)^{2}\Big{(}\partial_{F}p(F)\alpha(F)+2\partial_{F}\alpha(F)p(F)-\frac{c_{2}}{24\pi}\partial_{F}^{3}\alpha(F)\Big{)}+F^{\prime}(\sigma)^{2}j(F)-\frac{c_{1}}{24\pi}\\{F,\sigma\\}\,.$
(61b)

Here, $\prime$ denotes the derivatives wrt $\sigma$. Also, the BMS central
charges are $c_{1}=-2\pi c_{L},c_{2}=-2\pi c_{M}$. The BMS Schwarzian is
reflected in the term $\partial_{F}^{3}\alpha(F)$,

$\partial_{F}^{3}\alpha(F)=\frac{\llbracket(F,\alpha),\sigma\rrbracket}{(F^{\prime})^{5}}.$
(62)

In the limiting analysis, we find that only the transformation of the current
$j(\sigma)$ (corresponding to the superrotations) contributes to the unitary
transformation $U^{\dagger}T_{1}(\sigma,t)U$. In terms of the inverse
diffeomorphisms,

$F(t,f(t,\sigma))=\sigma$ (63)

the group velocity $\epsilon(t,\sigma)$ is

$\epsilon(t,\sigma)=-\frac{\dot{F}(t,\sigma)}{F^{\prime}(t,\sigma)}$ (64)

Finally, using (61) we can rewrite the complexity functional (35) as,

$\displaystyle\mathcal{C}_{1}$ $\displaystyle=\frac{1}{2\pi}\int dt\int
d\sigma\>\Big{[}-\dot{F}F^{\prime}\left(j_{0}(F)+\partial_{F}p_{0}(F)\alpha(F)+2\partial_{F}\alpha(F)p_{0}(F)-\frac{c_{2}}{24\pi}\partial_{F}^{3}\alpha(F)\right)$
(65) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
+\frac{c_{1}}{24\pi}\frac{\dot{F}}{F^{\prime}}\left(\frac{F^{\prime\prime\prime}}{F^{\prime}}-\frac{3}{2}\left(\frac{F^{\prime\prime}}{F^{\prime}}\right)^{2}\right)\Big{]}\,.$

Here, $j_{0}$ and $p_{0}$ are two constants.

#### Contribution from the central extensions:

Similar to the arguments discussed in section 3, we also add here the
contribution due to the central extension $c_{1}$,999We do not add here the
other central term $c_{2}$ due to absence of the supertranslation current
$p(\sigma)$ in (54).

$\mathcal{C}_{2}=\frac{1}{2\pi}\int dt\int
d\sigma\>\left(-\frac{\dot{F}}{F^{\prime}}\right)\>\left[\left(\frac{c_{1}}{48\pi}\right)\left(\frac{F^{\prime\prime}}{F^{\prime}}\right)^{\prime}\right]\,.$
(66)

Thus, the complexity functional of BMS3 as a limit of the Virasoro is given
as,

$\displaystyle\mathcal{C}$ $\displaystyle=$
$\displaystyle\mathcal{C}_{1}+\mathcal{C}_{2}$ (67) $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int dt\int
d\sigma\>\Big{[}-\dot{F}F^{\prime}\left(j_{0}(F)+\partial_{F}p_{0}(F)\alpha(F)+2\partial_{F}\alpha(F)p_{0}(F)-\frac{c_{2}}{24\pi}\partial_{F}^{3}\alpha(F)\right)$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
+\frac{c_{1}}{24\pi}\frac{\dot{F}}{F^{\prime}}\left(\frac{F^{\prime\prime\prime}}{F^{\prime}}-\frac{3}{2}\left(\frac{F^{\prime\prime}}{F^{\prime}}\right)^{2}-\frac{1}{2}\left(\frac{F^{\prime\prime}}{F^{\prime}}\right)^{\prime}\right)\Big{]}$

The terms in the last line of (67) with the central charge $c_{1}$ can be
expressed as,

$\displaystyle\int
dtd\sigma\>\frac{c_{1}}{48\pi}\frac{\dot{F}}{F^{\prime}}\left(\frac{F^{\prime\prime\prime}}{F^{\prime}}-2\left(\frac{F^{\prime\prime}}{F^{\prime}}\right)^{2}\right)$
$\displaystyle=$ $\displaystyle\int
dtd\sigma\>\frac{c_{1}}{48\pi}\frac{\dot{F}}{F^{\prime}}\Big{(}\left(\frac{F^{\prime\prime}}{F^{\prime}}\right)^{\prime}+F^{\prime\prime}\left(\frac{1}{F^{\prime}}\right)^{\prime}\Big{)}$
$\displaystyle=$ $\displaystyle-\int
dtd\sigma\>\frac{c_{1}}{48\pi}\Big{(}\frac{\dot{F}^{\prime\prime}}{F^{\prime}}\Big{)}+\frac{c_{1}}{48\pi}\int
dtd\sigma\>\Bigg{(}\frac{\dot{F}}{F^{\prime}}\frac{F^{\prime\prime}}{F^{\prime}}+\frac{\dot{F}^{\prime}}{F^{\prime}}\Bigg{)}^{\prime}$

The second term in (4) is a total derivative term. So, the final form of the
complexity functional is

$\displaystyle\mathcal{C}$ $\displaystyle=$ $\displaystyle-\frac{1}{2\pi}\int
dt\>\>d\sigma\>\Big{[}\dot{F}F^{\prime}j_{0}(F)+\frac{c_{1}}{48\pi}\frac{\dot{F}^{\prime\prime}}{F^{\prime}}$
(69) $\displaystyle\leavevmode\nobreak\
+\dot{F}F^{\prime}\Big{(}\partial_{F}p_{0}(F)\alpha(F)+2\partial_{F}\alpha(F)p_{0}(F)-\frac{c_{2}}{24\pi}\partial_{F}^{3}\alpha(F)\Big{)}$

The final expression for BMS3 circuit complexity (69) is equal to the
geometric (co-adjoint orbit) action of BMS3 in literature Barnich:2017jgw ;
Merbis:2019wgk . This is the expression we obtain from the limiting analysis.

Before moving into the next section, let us spare a moment here to understand
(69). The geometric action corresponds to generic (massive) co-adjoint orbits
of BMS3 when the orbit representatives $j_{0}$ and $p_{0}$ are constant and
non-zero ($j_{0}\neq 0,p_{0}\geq-\frac{c_{2}}{48\pi}$). For, the Flat Space
Cosmology (FSC) solutions 101010The Flat Space Cosmologies (FSC) are shifted
boost orbifolds of $3d$ flat spacetimes Cornalba:2003kd . Alternatively, they
can be understood as the flat space limit (AdS radius $l\to\infty$) of the
non-extremal BTZ black holes. As a consequence of this limit, the outer
horizon of the BTZ black hole is pushed to infinity, whereas the inner horizon
stays at a finite distance, known as the cosmological horizon. The nature of
the spacelike and timelike coordinates are interchanged due to the limit
Riegler:2014bia . More details regarding the thermodynamics of these
cosmologies can be found in Barnich:2012xq ; Bagchi:2013lma ;
Detournay:2014fva , and the BMS-Cardy formula in the context of flat space
holography reproducing the Bekenstein-Hawking entropy for these can be found
in Bagchi:2012xr .) we have $p_{0}>0$ and $j_{0}\neq 0$. Whereas, for conical
deficit solutions we have $-\frac{c_{2}}{48\pi}<p_{0}<0$ and $j_{0}\neq 0$.
Both of these two types (FSC and conical deficit solutions) of generic
(massive) orbits of BMS3 have their energy bounded from below Merbis:2019wgk .

There exists another type of BMS3 orbits, called the vacuum BMS3 orbit (e.g
Minkowski vaccuum with $j_{0}=0$ and $p_{0}=-\frac{c_{2}}{48\pi}$). Here also,
the energy is bounded from below. But the geometric action on these vacuum
BMS3 co-adjoint orbits are different than (69) and we will not discuss them in
our work. The readers are requested to go through Merbis:2019wgk  for more
details on the vacuum BMS3 orbits.

##  5 BMS3 complexity: Intrinsic way

In the previous section, we used the limiting analysis to find complexity
functional for the BMS3 group from the circuit complexity of the Virasoro
group. In the limiting analysis, the coadjoint orbit action of BMS gives the
complexity functional. However, the information of the transformation of
$p(\sigma)$, the current corresponding to the supertranslation generators
$M_{n}$, did not enter the complexity functional. It is always possible to add
a Hamiltonian $\int\>dt\>H$, preserving the global symmetries of the theory to
the geometric action on the coadjoint orbit. However, it does not seem
convincing from the point of view of calculating the complexity functional
only (if our sole intention is not to match the complexity functional with the
geometric action). In this section, we try to modify our computation by taking
the intrinsic approach. First, we calculate the complexity functional strictly
using the symmetry generators of BMS3. Later, we extremize the complexity
functional and find the optimal path in the space of group transformations.

###  5.1 Constructing the BMS Complexity functional

We start with the general expression for cost functional using the conserved
currents $J(t,\sigma)$ for symmetry gates $Q(t)=\frac{1}{2\pi}\int
d\sigma\>\epsilon(t,\sigma)\>J(\sigma)$. Here, $\epsilon(t,\sigma)$ is the
instantaneous velocity and the conserved currents for BMS supertranslations
and superrotations are $j(\sigma),p(\sigma)$. Their transformation relations
are given in (61). Thus, the cost functional for BMS3 takes the form,

$\mathcal{F}=\int
d\sigma\>\>\Big{[}\epsilon_{L}(t,\sigma)\langle\psi_{R}|U^{\dagger}(t)j(\sigma)U(t)|\psi_{R}\rangle+\epsilon_{M}(t,\sigma)\langle\psi_{R}|U^{\dagger}(t)p(\sigma)U(t)|\psi_{R}\rangle\Big{]}\,.$
(70)

We have used two different instantaneous velocities $\epsilon_{L}$ and
$\epsilon_{M}$ corresponding to the transformations $f(t,\sigma)$
(diffeomorphisms) and $\alpha(t,\sigma)$ (supertranslations). Thus, the
complexity functional constructed from the cost functional (70) is,

$\displaystyle\mathcal{C}=\int dt\mathcal{F}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int dt\int
d\sigma\>\Big{[}\epsilon_{L}(t,\sigma)\langle\psi_{R}|U^{\dagger}(t)j(\sigma)U(t)|\psi_{R}\rangle+\epsilon_{M}(t,\sigma)\langle\psi_{R}|U^{\dagger}(t)p(\sigma)U(t)|\psi_{R}\rangle\Big{]},$
(71) $\displaystyle=$ $\displaystyle\frac{1}{2\pi}\int dt\int
d\sigma\>\Big{[}\epsilon_{L}(t,\sigma)\langle\psi_{R}|\tilde{j}(\sigma)|\psi_{R}\rangle+\epsilon_{M}(t,\sigma)\langle\psi_{R}|\tilde{p}(\sigma)|\psi_{R}\rangle\Big{]}\,.$

We are also using the inverse diffeomorphism $F(t,\sigma)$ at each point
$f(t,\sigma)$ such that,

$F(t,f(t,\sigma))=\sigma\,.$ (72)

Using the transformations of the currents (61),

$\displaystyle\mathcal{C}[F]$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int
dt\>d\sigma\Bigg{[}\epsilon_{M}(t,\sigma)\Big{(}F^{\prime}(\sigma)^{2}p_{0}(F)-\frac{c_{2}}{24\pi}\\{F,\sigma\\}\Big{)}$
$\displaystyle+\epsilon_{L}(t,\sigma)\Big{(}F^{\prime}(\sigma)^{2}\Big{[}\partial_{F}p_{0}(F)\alpha(F)+2\partial_{F}\alpha(F)p_{0}(F)-\frac{c_{2}}{24\pi}\partial_{F}^{3}\alpha(F)\Big{]}+F^{\prime}(\sigma)^{2}j_{0}(F)-\frac{c_{1}}{24\pi}\\{F,\sigma\\}\Big{)}\Bigg{]}.$

Here, $p_{0}$ and $j_{0}$ are constants. Now, we will calculate the
instantaneous velocities $\epsilon_{L},\epsilon_{M}$. For this, we use the BMS
group transformation rules. For a finite BMS transformation, we have

$\sigma\to f(\sigma),\leavevmode\nobreak\ \leavevmode\nobreak\ t\to
tf^{\prime}(\sigma)+\alpha(\sigma)\,.$ (74)

For two such consecutive transformations, we go from $(\sigma,t)$ to
$(\sigma_{2},t_{2})$,

$(\sigma,t)\xrightarrow{(f_{1},\alpha_{1})}(\sigma_{1},t_{1})\xrightarrow{(f_{2},\alpha_{2})}(\sigma_{2},t_{2})$
(75)

where,

$\sigma_{1}=f_{1}(\sigma),\leavevmode\nobreak\ \leavevmode\nobreak\
\text{and,\leavevmode\nobreak\
}\sigma_{2}=f_{2}(\sigma_{1})=f_{2}(f_{1}(\sigma))=(f_{2}\circ
f_{1})(\sigma).$ (76)

Also,

$\displaystyle t_{1}$ $\displaystyle=$ $\displaystyle
tf^{\prime}_{1}(\sigma)+\alpha_{1}(\sigma),$ $\displaystyle t_{2}$
$\displaystyle=$ $\displaystyle
t_{1}f^{\prime}_{2}(\sigma_{1})+\alpha_{2}(\sigma_{1})=t\frac{\partial(f_{2}\circ
f_{1})(\sigma)}{\partial\sigma}+\alpha_{1}\frac{\partial(f_{2}\circ
f_{1})(\sigma)}{\partial f_{1}(\sigma)}+\alpha_{2}(f_{1}(\sigma))\,.$ (77)

Now, if we directly go from $(\sigma,t)$ to $(\sigma_{2},t_{2})$ the following
also holds true.

$\displaystyle(\sigma,t)$ $\displaystyle\xrightarrow{(f_{3},\alpha_{3})}$
$\displaystyle(\sigma_{2},t_{2}).$ $\displaystyle\text{such
that,\leavevmode\nobreak\ }\sigma_{2}$ $\displaystyle=f_{3}(\sigma)$
$\displaystyle,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ t_{2}=t\frac{\partial
f_{3}(\sigma)}{\partial\sigma}+\alpha_{3}(\sigma).$ (78)

Comparing, (76) and (5.1) with (5.1), we get the following transformation law
of the group elements

$\displaystyle f_{3}(\sigma)$ $\displaystyle=$ $\displaystyle(f_{2}\circ
f_{1})(\sigma)=f_{2}(f_{1}(\sigma)).$ $\displaystyle\alpha_{3}(\sigma)$
$\displaystyle=$ $\displaystyle(\alpha_{2}\circ
f_{1})(\sigma)+\alpha_{1}(\sigma)\frac{\partial(f_{2}\circ
f_{1})(\sigma)}{\partial
f_{1}(\sigma)}=\alpha_{2}(f_{1}(\sigma))+\alpha_{1}(\sigma)\frac{\partial
f_{2}(f_{1}(\sigma))}{\partial f_{1}(\sigma)}.$ (79)

We will use (5.1) to find the instantaneous velocities. The symmetry
transformation $(f(t,\sigma),\alpha(t,\sigma))$ for two infinitesimally close
points in the path of the circuit is related as

$\big{(}f(t+dt,\sigma),\alpha(t+dt,\sigma)\big{)}=e^{(\epsilon_{L}(t,\sigma),\epsilon_{M}(t,\sigma))dt}\>\>(f(t,\alpha),\alpha(t,\sigma)).$
(80)

Expanding in the first order,

$\displaystyle\implies$
$\displaystyle\big{(}f(t,\alpha),\alpha(t,\sigma)\big{)}+dt\>\>(\partial_{t}f(t,\sigma),\partial_{t}\alpha(t,\sigma))=\big{(}\mathbb{I}+dt\>\>(\epsilon_{L}(t,\sigma),\epsilon_{M}(t,\sigma))\big{)}\cdot\big{(}f(t,\sigma),\alpha(t,\sigma)\big{)}$
(81) $\displaystyle\implies$ $\displaystyle
dt\>\>(\partial_{t}f(t,\sigma),\partial_{t}\alpha(t,\sigma))=dt\>\>\big{(}\epsilon_{L}(t,\sigma)),\epsilon_{M}(t,\sigma)\big{)}\cdot(f(t,\sigma),\alpha(t,\sigma))\,.$

Here, ‘$\cdot$’ represents the group operation in (5.1). Thus, we have

$\displaystyle\partial_{t}f(t,\sigma)$ $\displaystyle=$
$\displaystyle\epsilon_{L}(t,f(t,\sigma)),$ (82)
$\displaystyle\partial_{t}\alpha(t,\sigma)$ $\displaystyle=$
$\displaystyle\epsilon_{M}(t,f(t,\sigma))+\frac{\partial\epsilon_{L}(t,f(t,\sigma))}{\partial
f(t,\sigma)}\alpha(t,\sigma).$ (83)

Now, we will use the inverse diffeomorphism $F(t,f(t,\sigma))=\sigma$ to find
the instantaneous velocities $\epsilon_{L},\epsilon_{M}$ at $(t,\sigma)$.
Thus,

$\epsilon_{L}(t,\sigma)=\frac{\partial f(t,F(t,\sigma))}{\partial
t}=-\frac{\dot{F}(t,\sigma)}{F^{\prime}(t,\sigma)}.$ (84)

The calculation of $\epsilon_{L}$ is similar to (30). Similarly, we have from
(83),

$\displaystyle\epsilon_{M}(t,\sigma)$ $\displaystyle=$
$\displaystyle\partial_{t}\alpha(t,F(t,\sigma))-\alpha(t,F(t,\sigma))\frac{\partial\epsilon_{L}(t,\sigma)}{\partial\sigma},$
(85) $\displaystyle=$
$\displaystyle\dot{\alpha}(t,F)+\alpha(t,F)\left(\frac{\dot{F}^{\prime}}{F^{\prime}}-\frac{\dot{F}F^{\prime\prime}}{F^{\prime^{2}}}\right).$

We use group velocities $\epsilon_{L}(t,\sigma)$ (84) and
$\epsilon_{M}(t,\sigma)$ (85) in the complexity functional (69) to write

$\displaystyle\mathcal{C}[F]$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int
dt\>d\sigma\Bigg{[}-\dot{F}F^{\prime}\Big{[}\partial_{F}p_{0}(F)\alpha(F)+2\partial_{F}\alpha(F)p_{0}(F)-\frac{c_{2}}{24\pi}\partial_{F}^{3}\alpha(F)+j_{0}(F)\Big{]}+\frac{c_{1}}{24\pi}\frac{\dot{F}}{F^{\prime}}\\{F,\sigma\\}$
(86) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
+\Bigg{(}\dot{\alpha}(t,F)+\alpha(t,F)\left(\frac{\dot{F}^{\prime}}{F^{\prime}}-\frac{\dot{F}F^{\prime\prime}}{F^{\prime^{2}}}\right)\Bigg{)}\Big{(}F^{\prime}(\sigma)^{2}p_{0}(F)-\frac{c_{2}}{24\pi}\\{F,\sigma\\}\Big{)}\Bigg{]}.$

Due to the presence of the central extension $c_{1}$ we also add the
modification in the cost function similar to (45),

$\displaystyle\begin{split}&\mathcal{F}=\int
d\sigma\>\Big{[}\epsilon_{L}(t,\sigma)\langle\psi_{R}|\tilde{j}(\sigma)|\psi_{R}\rangle+\epsilon_{M}(t,\sigma)\langle\psi_{R}|\tilde{p}(\sigma)|\psi_{R}\rangle\Big{]}\leavevmode\nobreak\
\leavevmode\nobreak\ +c_{1}\beta(t)\,,\,\\\ &\mathcal{C}=\int
dt\mathcal{F},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\text{where,}\leavevmode\nobreak\ \leavevmode\nobreak\
\beta=-m_{\theta}=-\frac{1}{48\pi}\int_{0}^{2\pi}d\sigma\frac{\dot{F}}{F^{\prime}}\Big{(}\frac{F^{\prime\prime}}{F^{\prime}}\Big{)}^{{}^{\prime}}.\end{split}$
(87)

Thus, the final form for the complexity functional for BMS3 is given by

$\displaystyle\mathcal{C}[F]$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int
dt\>d\sigma\Bigg{[}-\dot{F}F^{\prime}\Big{[}\partial_{F}p_{0}(F)\alpha(F)+2\partial_{F}\alpha(F)p_{0}(F)-\frac{c_{2}}{24\pi}\partial_{F}^{3}\alpha(F)+j_{0}(F)\Big{]}-\frac{c_{1}}{48\pi}\frac{\dot{F}^{\prime\prime}}{F^{\prime}}$
(88) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
+\Big{(}\dot{\alpha}(F)+\alpha(F)\left(\frac{\dot{F}^{\prime}}{F^{\prime}}-\frac{\dot{F}F^{\prime\prime}}{F^{\prime^{2}}}\right)\Big{)}\Big{(}F^{\prime}(\sigma)^{2}p_{0}(F)-\frac{c_{2}}{24\pi}\\{F,\sigma\\}\Big{)}\Bigg{]}.$

The first line of (88) is the one obtained as a limit to Virasoro complexity
functional in (69). Putting, $\epsilon_{M}=0$, we recover the geometric action
on the generic (massive) coadjoint orbit for BMS3 Barnich:2017jgw ;
Merbis:2019wgk . However, the terms in the second line in (88) is not
recovered by deforming the geometric action with the addition of Hamiltonian.
We will comment on this in a moment. The second line is essential because it
is not convincing to ignore the contribution from the supertranslation
generators in the construction of the gates $Q(t)$. Next, we extremize the
complexity functional (88) to obtain the equations of motions and solve for
$f(t,\sigma),\alpha(t,\sigma)$. It is to note here, however, if we extremize
the functional with $\epsilon_{M}=0$, $\alpha$ remains undetermined.

Before proceeding further, we connect the BMS complexity functional with some
of the previous studies. Identifying $j_{0},p_{0},c_{1}$ in the following way
and setting the instantaneous velocity $\epsilon_{M}$ to be a
constant,131313There is no a priori reason to choose $\epsilon_{M}(t,\sigma)$
as a constant for the complexity calculation.

$p_{0}=k\frac{\mathfrak{M}_{0}}{2},\>\>j_{0}=k\mathfrak{L}_{0},\>\>c_{1}=0,\>\>c_{2}=-2\pi
c_{M}=24\pi k,\>\>\epsilon_{M}=1.$ (89)

Here we have used the fact, that for Einstein gravity $c_{1}$ can be set zero.
Then we see that (88) becomes the deformed geometric action (by addition of
Hamiltonian) of BMS3 in Merbis:2019wgk . Here, $\mathfrak{M}_{0}$ and
$\mathfrak{L}_{0}$ (the orbit representatives) are proportional to the angular
momentum and mass of the gravitational saddle point. We use the notations from
the authors Merbis:2019wgk  to write,

$\displaystyle\mathcal{C}[F]$ $\displaystyle=$
$\displaystyle\mathcal{I}_{CS}\left[F,\alpha,\mathfrak{L}_{0},\mathfrak{M}_{0}\right]$
(90) $\displaystyle=$ $\displaystyle-\frac{k}{2\pi}\int
dt\>d\sigma\>\>\Bigg{[}\Big{(}\mathfrak{L}_{0}+\mathfrak{M}_{0}\partial_{F}\alpha(F)-\partial_{F}^{3}\alpha(F)\Big{)}\dot{F}F^{\prime}-\frac{1}{2}\Big{(}\mathfrak{M}_{0}F^{\prime
2}-2\\{F,\sigma\\}\Big{)}\Bigg{]}.\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ $

This is the same as the effective two-dimensional action obtained from bulk
Chern-Simons classical gravity action for three-dimensional flat spacetime,
after using the BMS3 boundary conditions mentioned in Barnich:2006av . Here,
the boundary theory is equivalent to the coadjoint action on the geometric
orbit of BMS3 with non-trivial bulk holonomies.

###  5.2 Extremisation of the complexity functional

Now, we extremize the BMS complexity functional (88) wrt
$f(t,\sigma),\alpha(t,\sigma)$. Then we solve the equations of motion to find
the optimal path in terms of $f$ and $\alpha$. We put them back in the
complexity functional to find the circuit complexity of the optimized path
from reference to the target state in the manifold of unitary transformations.
First, we write the complexity functional in terms of $f$ instead of the
inverse diffeomorphism $F$.

$\displaystyle\mathcal{C}[f,\alpha]$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int d\sigma
dt\Biggl{[}\left(\frac{\dot{f}}{f^{\prime}}\right)\bigg{\\{}\biggl{(}j_{0}(\sigma)+\frac{c_{1}}{48\pi}\left(\frac{f^{\prime\prime}}{f^{\prime}}\right)^{\prime}\biggr{)}+\biggl{(}2\alpha^{\prime}(\sigma)p_{0}(\sigma)+2p_{0}^{\prime}\alpha(\sigma)-\frac{c_{2}}{24\pi}\alpha^{\prime\prime\prime}(\sigma)\biggr{)}\bigg{\\}}$
(91) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
+\frac{1}{f^{\prime^{2}}}\left(\dot{\alpha}(\sigma)+\alpha(\sigma)\dot{f}^{\prime}\right)\left(p_{0}(\sigma)+\frac{c_{2}}{24\pi}\\{f,\sigma\\}\right)\Biggr{]}.$

Next, we extremize wrt $\alpha$ and obtain the equation of motion for
$\alpha(t,\sigma)$.

$\frac{\partial\mathcal{C}}{\partial\alpha}-\frac{\partial}{\partial
t}\left(\frac{\partial\mathcal{C}}{\partial\dot{\alpha}}\right)-\frac{\partial}{\partial\sigma}\left(\frac{\partial\mathcal{C}}{\partial{\alpha}^{\prime}}\right)+\frac{\partial^{2}}{\partial\sigma^{2}}\left(\frac{\partial\mathcal{C}}{\partial{\alpha}^{\prime\prime}}\right)-\frac{\partial^{3}}{\partial\sigma^{3}}\left(\frac{\partial\mathcal{C}}{\partial{\alpha}^{\prime\prime\prime}}\right)=0\,.$
(92)

Such that,

$2p_{0}^{\prime}\frac{\dot{f}}{f^{\prime}}+\frac{\dot{f}^{\prime}}{f^{\prime^{2}}}\left(p_{0}+\frac{c_{2}}{24\pi}\\{f,\sigma\\}\right)-\frac{\partial}{\partial
t}\bigg{\\{}\frac{1}{f^{\prime^{2}}}\left(p_{0}+\frac{c_{2}}{24\pi}\\{f,\sigma\\}\right)\bigg{\\}}-2\biggl{(}p_{0}\;\frac{\dot{f}}{{f}^{\prime}}\biggr{)}^{\prime}+\frac{c_{2}}{24\pi}\biggl{(}\frac{\dot{f}}{f^{\prime}}\biggr{)}^{\prime\prime\prime}=0\,.$
(93)

Without the loss of generality, we assume

$j_{0}=\textrm{const},\quad p_{0}=\textrm{const}.$ (94)

Then the simplest solution comes as,

$\frac{\dot{f}}{f^{\prime}}=\textrm{const}^{},\leavevmode\nobreak\
\leavevmode\nobreak\ \dot{f}^{\prime}=0,\leavevmode\nobreak\
\leavevmode\nobreak\ f^{\prime}\neq 0,\leavevmode\nobreak\
\leavevmode\nobreak\ f^{\prime\prime}=0\,.$ (95)

Thus,

$f(\sigma,t)=a_{0}\sigma+a_{1}t\,.$ (96)

Also, periodicity demands that

$\displaystyle f(\sigma+2\pi)$ $\displaystyle=$ $\displaystyle
f(\sigma)+2\pi,\leavevmode\nobreak\ \leavevmode\nobreak\ \implies a_{0}=1$
$\displaystyle\text{Thus,\leavevmode\nobreak\ }f(\sigma,t)$ $\displaystyle=$
$\displaystyle\sigma+a_{1}t\,.$ (97)

Also, extremizing wrt $f(t,\sigma)$, we get the other equation of motion,

$\frac{\partial\mathcal{C}}{\partial f}-\frac{\partial}{\partial
t}\left(\frac{\partial\mathcal{C}}{\partial\dot{f}}\right)-\frac{\partial}{\partial\sigma}\left(\frac{\partial\mathcal{C}}{\partial{f}^{\prime}}\right)+\frac{\partial^{2}}{\partial
t\partial\sigma}\left(\frac{\partial\mathcal{C}}{\partial\dot{f}^{{}^{\prime}}}\right)+\frac{\partial^{2}}{\partial\sigma^{2}}\left(\frac{\partial\mathcal{C}}{\partial{f}^{\prime\prime}}\right)-\frac{\partial^{3}}{\partial\sigma^{3}}\left(\frac{\partial\mathcal{C}}{\partial{f}^{\prime\prime\prime}}\right)=0\,.$
(98)
$\displaystyle\frac{2}{f^{\prime}}\Biggl{[}j_{0}\left(\frac{\dot{f}}{f^{\prime}}\right)^{\prime}-\frac{c_{1}}{48\pi}\left(\frac{\dot{f}}{f^{\prime}}\right)^{\prime\prime\prime}\Biggr{]}+\frac{1}{f^{\prime}}\Bigg{(}2\dot{f}^{{}^{\prime}}-\frac{\dot{f}}{f^{\prime}}\biggl{\\{}\frac{2f^{\prime\prime}}{f^{\prime}}-\partial_{\sigma}\biggr{\\}}+\partial_{t}\Biggr{)}\Bigg{(}2\alpha^{\prime}\>p_{0}+2p_{0}^{\prime}\>\alpha-\frac{c_{2}}{24\pi}\alpha^{\prime\prime\prime}\Biggr{)}$
$\displaystyle+\frac{\partial}{\partial\sigma}\Biggl{[}\frac{1}{f^{\prime^{2}}}(\dot{\alpha}+\alpha\dot{f}^{{}^{\prime}})\biggl{\\{}\frac{2}{f^{\prime}}(p_{0}+\frac{c_{2}}{24\pi}\\{f,\sigma\\})-\frac{c_{2}}{24\pi}\left(-\frac{f^{\prime\prime\prime}f^{\prime\prime}}{f^{\prime^{2}}}+3\frac{f^{\prime\prime^{3}}}{f^{\prime^{3}}}\right)\biggr{\\}}\Biggr{]}$
$\displaystyle-\frac{\partial^{2}}{\partial
t\partial\sigma}\Biggl{\\{}-\frac{\alpha}{f^{\prime^{2}}}(p_{0}+\frac{c_{2}}{24\pi}\\{f,\sigma\\})\Biggr{\\}}-\frac{c_{2}}{24\pi}\Biggl{[}\frac{\partial^{2}}{\partial\sigma^{2}}\Biggl{\\{}3\frac{f^{\prime\prime}}{f^{\prime^{3}}}(\dot{\alpha}+\alpha\dot{f}^{{}^{\prime}})\Biggr{\\}}-\frac{\partial^{3}}{\partial\sigma^{3}}\Biggl{\\{}\frac{1}{f^{\prime^{3}}}(\dot{\alpha}+\alpha\dot{f}^{{}^{\prime}})\Biggr{\\}}\Biggr{]}=0\,.$

The first term in parentheses is the equation of motion for the diffeomorphism
in the case of Virasoro circuit complexity Erdmenger:2020sup . This (5.2) is a
highly non-linear and coupled PDE and we write only the simplest solution,

$\dot{\alpha}^{\prime}=0,\leavevmode\nobreak\ \leavevmode\nobreak\
\alpha^{\prime\prime}=0\leavevmode\nobreak\ \leavevmode\nobreak\
\implies\alpha(\sigma,t)=a_{2}\sigma+a_{3}(t).$ (100)

Imposing the periodicity condition,

$\alpha(\sigma+2\pi)=\alpha(\sigma),\leavevmode\nobreak\ \leavevmode\nobreak\
\implies a_{2}=0,\text{thus,\leavevmode\nobreak\ }\alpha=a_{3}(t).$ (101)

The constant $a_{1}$ and the function $a_{3}(t)$ remain undetermined.
Interestingly, the simplest solutions found in (5.2) and (101) agree with the
solutions to gravitational saddle points with constant orbit representatives
$j_{0},p_{0}$ in Merbis:2019wgk . For flat space cosmologies (FSC), the
solution is given by

$f=\sigma,\leavevmode\nobreak\ \leavevmode\nobreak\ \alpha=t.$ (102)

The FSC solutions, together with its BMS3 descendents, correspond to a generic
(massive) coadjoint orbits of the BMS3 group, in the same way that the BTZ
solutions correspond to the Virasoro coadjoint orbits. For the FSC solutions,
we have the orbit representatives $p_{0}>0$ and $j_{0}\neq 0$. We only
consider evolution inside such orbits, consisting of descendants of the
initial orbit representatives, obtained by acting with BMS generators.
151515Field theory states representing FSC solutions cannot be obtained from a
generic primary, but only from a primary that already has the correct
expectation values for the zero modes of the BMS algebra $j_{0},p_{0}$
Merbis:2019wgk .

Putting the solution back to the complexity functional we obtain, the circuit
complexity for the optimal path

$\displaystyle\mathcal{C}[T]$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi}\>d\sigma\>\int_{0}^{T}dt\big{[}j_{0}a_{1}+\dot{a_{3}}p_{0}\big{]}$
(103) $\displaystyle=$ $\displaystyle a_{1}\;j_{0}T+p_{0}(a_{3}(T)-a_{3}(0)).$
(104)

Here,
$j_{0}=|\Delta-\frac{c_{1}}{24\pi}|,\>\>p_{0}=|\xi-\frac{c_{2}}{24\pi}|$, the
expectation values of $L_{0}$ and $M_{0}$ on the cylinder. $\Delta$ and $\xi$
are the eigenvalues of $L_{0}$ and $M_{0}$ respectively. Similar to the
Virasoro (as well as Warped CFT) case, the BMS complexity (104) depends on the
central charges and is proportional to $T$. For FSC, $a_{1}=0$ and
$a_{3}(t)=t$, hence,

$\mathcal{C}[T]\propto\;p_{0}\;T.$ (105)

##  6 Conclusions

We have computed Nielsen’s complexity for a non-Lorentzian field theory in
this paper. We have considered $2d$ Carrollian CFT (BMS3 invariant field
theory), a putative dual to $3d$ asymptotically flat spacetimes. We have taken
two distinct approaches here, namely limiting and intrinsic. For the
relativistic cousin of the said field theories, the circuit complexity
functional takes the form of the geometric action on the co-adjoint orbit of
the Virasoro group. Using the limiting approach on the relativistic CFT
complexity functional, we show that it is possible to write the geometric
action on the co-adjoint orbit of the BMS3 group as its complexity functional
for BMS circuit complexity. However, this does not give us the full picture.
161616The notion of “complexity=geometric action” holds for BMS3 only through
the limiting analysis, and not in the intrinsic approach. This result, for a
semi-direct product group like BMS3, is unlike the “complexity=geometric
action” proposal for direct product groups (e.g Virasoro and Kac-Moody
Erdmenger:2020sup ).Hence, we refine our analysis using the intrinsic
approach. We consider the BMS symmetry transformations as the required gates
and find the instantaneous velocities from the BMS group product law. As a
result, the complexity functional acquires a more involved structure than only
the geometric action for BMS. We show that even after the addition of the
Hamiltonian to the BMS geometric action, it is not possible to reproduce the
exact form of the complexity functional. However, for a particular value of
the instantaneous velocity $\epsilon_{M}(\tau,\sigma)$, we can relate the
complexity functional with available literature (modified bulk Chern-Simons
classical gravity action for three-dimensional flat spacetime reduced with BMS
boundary conditions to a two-dimensional boundary theory). Next, we extremise
the complexity functional with the group paths and find the equations of
motion to be highly non-linear and coupled PDEs. We write the simplest
solutions and find the similarity with the solutions to gravitational saddle
points for flat space cosmology. This provides a consistency check for our
computations and connects with (flat-space) holography. It will be interesting
to investigate whether there are any more non-trivial solutions of (5.2).
Also, the result in equation (105) shows that the BMS circuit complexity
(calculated using the symmetry generators as the gates) grows linearly in
time. This reflects the similar behaviour for the Virasoro complexity in
Caputa:2018kdj . This also supports the general claim arising from holography
about the linear growth of the complexity. The holographic complexity proposal
typically suggests a volume divergence. But this way of computing the
complexity does not give rise to the divergence, which is similar to the
earlier results from the findings in Caputa:2018kdj ; Erdmenger:2020sup ;
Bhattacharyya:2022ren  for Virasoro and warped CFT. This absence of the
divergence structure needs to be understood better. It will be interesting to
find the exact reason behind this in future.

#### Discussion and future directions

#### Complexity and Berry Phases:

In Caputa:2018kdj ; Erdmenger:2020sup , it was shown that the complexity
functional for the Virasoro group and the complexity for all possible
diffeomorphisms $f(t,\sigma)$ of the Virasoro group is equivalent to a Berry
phase upto a boundary term.171717The authors also mentioned that not all Berry
Phases are equivalent to complexity, particularly the non-optimal closed paths
$f(t,\sigma)$, which do not solve the equations of motion for the Virasoro
case. We briefly comment on the relation between BMS Berry Phases and
complexity in parallel to the relativistic case. The BMS Berry phase is given
by Oblak:2017ptc

$\displaystyle\mathcal{B}_{\text{scalar}}[f,\alpha]$ $\displaystyle=$
$\displaystyle-\frac{M}{2\pi}\int\>dt\>d\sigma\>\frac{\dot{\alpha}(t,f(t,\sigma))}{f^{\prime}}-\frac{c_{2}}{24\pi}\int\>dt\>d\sigma\>\frac{\dot{\alpha}(t,f(t,\sigma))}{f^{\prime}}\left(\\{f,\sigma\\}-\frac{1}{2}\right).$
(106) $\displaystyle\mathcal{B}_{\text{spin}}[f]$ $\displaystyle=$
$\displaystyle-\frac{1}{2\pi}\int\>dt\>d\sigma\>\frac{\dot{f}}{f^{\prime}}\left[s-\frac{c_{1}}{24}+\frac{c_{1}}{24}\left(\frac{f^{\prime\prime}}{f^{\prime}}\right)^{\prime}\>\right].$
(107)

Even though the spin part of the BMS Berry phase is seen in the complexity
functional (91), the scalar part is not apparent due to the change of argument
of $\dot{\alpha}$. Also, we obtain an extra part (the second term in
parentheses of the complexity expression (91)) that is unclear in the context
of the BMS Berry Phase. We leave this for future investigation.

#### Higher dimensional exploration:

It would be interesting to look into aspects of circuit complexity for $d>2$
BMS invariant field theories. Recently, in 2022JHEP…12..154B  the geometric
action for BMS4 was given. A natural extension would be to investigate whether
any similarity exists for the higher dimensional geometric action and
complexity functional.

#### Other applications:

There are other notions of complexity based of various other methods, e.g.
Fubini Study distance, Path Integral Erdmenger:2021wzc ; Flory:2020eot ;
Caputa:2017yrh ; Bhattacharyya:2018wym ; Camargo:2019isp . It will be very
interesting to generalize these notions (particularly the Fubini-Study one) of
complexity for BMS invariant field theory. Furthermore, in recent times, the
notion of ‘Krylov Complexity’ has gained much attraction. It is related to the
notion of the complexity associated with the operator growth 2019PhRvX…9d1017P
. It has been recently studied for CFT Dymarsky:2021bjq ; 2022arXiv221214429A
; 2022arXiv221214702C  and for some quantum mechanical models where Carroll
symmetry emerges Banerjee:2022ime . It would be exciting to explore different
notions of complexities in other areas, e.g. Carrollian physics in the context
of flat bands Bagchi:2022eui , or with the inclusion of Super Carrollian
structures Bagchi:2022owq . It will also be of importance to find other
optimal solutions of the equations of motion (93), (5.2) for the group paths
$f,\alpha$ for Carrollian circuit complexity. We hope to make a systematic of
these for BMS field theories in the near future.

## Acknowledgements

A.B thanks Shubho Roy and Aritra Banerjee for useful discussions and
collaboration on related projects. PN is thankful to Daniel Grumiller and Max
Reigler for their helpful comments. Research of A.B and PN is supported by the
Relevant Research Project grant (202011BRE03RP06633-BRNS) by the Board Of
Research In Nuclear Sciences (BRNS), Department of Atomic Energy (DAE), India.
Research of A.B is also supported by the Mathematical Research Impact Centric
Support Grant (MTR/2021/000490) by the Department of Science and Technology
Science and Engineering Research Board (India). PN is also indebted to the
Science and Engineering Research Board (SERB-India) for the International
Travel Support Grant (ITS/2022/000812) during the progress of this work, and
also to the people of India for their generous support to research in basic
sciences.

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