††institutetext: 1Science Institute University of Iceland
Dunhaga 3, 107 Reykjavík, Iceland††institutetext: 2Department of Applied
Mathematics and Theoretical Physics
University of Cambridge, Cambridge CB3 0WA, United Kingdom††institutetext:
3Isaac Newton Institute for Mathematical Sciences
University of Cambridge, Cambridge CB3 0EH, United Kingdom

# Carroll Strings with an Extended Symmetry Algebra

Matthias Harksen1 Diego Hidalgo1 Watse Sybesma1,2,3 and Lárus Thorlacius1
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Starting from the Polyakov action we consider two distinct Carroll limits in
target space, keeping the string worldsheet relativistic. The resulting
magnetic and chiral Carroll string models exhibit different symmetries and
dynamics. Both models have an infinite dimensional symmetry algebra with
Carroll symmetry included in a finite dimensional subalgebra. For the magnetic
model, this is the so-called string Carroll algebra. The chiral model realises
an extended version of the string Carroll algebra. The magnetic model does not
have any transverse string excitations. The chiral model is less restrictive
and includes arbitrary left-moving modes that carry transverse momentum but do
not contribute to the energy in target space.

##  1 Introduction

String theory provides a robust framework for investigating quantum gravity in
various decoupling limits. In this regard, non-relativistic limits of string
theories Gomis:2000bd ; Danielsson:2000gi ; Danielsson:2000mu  form a self-
contained sector within string theory governed by a relativistic worldsheet
metric with Galilean global symmetries on the target spacetime. We refer the
interested readers to Oling:2022fft  for a recent review of non-relativistic
string theory. In this note, we consider bosonic string theory in another non-
Lorentzian target space limit called the Carroll limit Levy1965 ; Bacry:1968zf
; Leblonde:1965 ; Gupta1966OnAA . A pictorial way to highlight the contrasting
extremes of these limits is that rather than opening up the light cone, as
happens in the Galilean limit, the lightcone collapses to a line in the
Carroll limit. As a result, timelike particles are confined to the temporal
axis. Tachyons still move in the Carroll limit but have no rest frame.
Furthermore, Carroll boost symmetry requires energy flux to vanish. Particles
that move are thus forced to have vanishing energy while particles at rest can
have non-vanishing energy deBoer:2017ing ; deBoer:2021jej . Particle
properties in the Carroll limit were further studied in Bergshoeff:2014jla ;
Casalbuoni:2023bbh ; deboer:2023fnj , where it was demonstrated that non-
trivial dynamics can arise for coupled Carroll particles. Our primary
motivation for developing a Carroll limit of string theory is to see if
Carroll string dynamics is less constraining than Carroll particle dynamics.
Carroll strings with worldsheet Carroll symmetry were studied in Isberg:1993av
; Bagchi:2015nca ; Bidussi:2023rfs ; Gomis:2023eav , whereas target space
Carroll symmetry was studied in Cardona:2016ytk ; Bagchi:2023cfp ;
Bidussi:2023rfs . As we proceed, we will comment on similarities and
differences between our work and these earlier papers.

We study two inequivalent Carroll limits in the target space of bosonic string
theory by introducing different sets of auxiliary worldsheet fields into the
Polyakov action. One of these limits corresponds to what is known throughout
the literature as a magnetic limit Duval:2014uoa ; Henneaux:2021yzg ;
deBoer:2021jej ; Bergshoeff:2022qkx . Our other Carroll limit breaks
worldsheet chirality via a novel auxiliary field configuration. In parallel
with the non-relativistic string Gomis:2000bd , the Carroll symmetry appears
as a global symmetry of the target spacetime, while the worldsheet remains
relativistic. The magnetic limit reproduces an action that is invariant under
the string Carroll algebra previously derived by Cardona, Gomis, and Pons
(CGP) by considering a Carroll limit of the canonical form of the Nambu-Goto
action Cardona:2016ytk .

The worldsheet dynamics of our magnetic Carroll string model is similar to the
CGP model in that no motion is allowed in the transverse directions. The
transverse target space momentum is unconstrained, but does not enter into the
magnetic Carroll string dispersion relation. This is reminiscent of the
behaviour of Carroll particles Bergshoeff:2014jla ; Casalbuoni:2023bbh ;
deBoer:2021jej . The chiral Carroll model also has unconstrained transverse
momentum and a simple dispersion relation, where energy does not depend on
transverse momentum. Unlike the magnetic model, transverse motion is allowed
in the chiral model via arbitrary left-moving string excitations, thus
departing from the Carroll particle picture. The symmetry algebra of the
chiral model is an extension of the string Carroll algebra of the magnetic
model. Furthermore, an infinite dimensional symmetry extension is realised in
both our models in a similar manner as for strings with Galilean symmetry
Bergshoeff:2019pij ; Batlle:2016iel .

The remainder of the paper is organised as follows. In Section 2, we establish
notation and construct the magnetic Carroll limit by introducing suitable
auxiliary fields in the Polyakov action. The symmetry algebra of the magnetic
Carroll model and its string dispersion relation are derived. In Section 3 we
present the chiral Carroll model and its extended Carroll string algebra. We
then show how to extend the symmetry to an infinite dimensional algebra for
both the chiral and magnetic models. In Section 4 we summarise our results and
discuss some future directions.

##  2 Magnetic Carroll string model

This section introduces the first of two Carroll limits for strings we
consider. The Carroll structure of the model is observed on the target space,
while the worldsheet metric remains relativistic. As we will see in Section
2.1, the magnetic Carroll string arises from relativistic string theory via an
asymmetric scaling of the longitudinal and transverse target space coordinate
fields in the Polyakov action that renders the transverse sector ‘dominant’
over the longitudinal sector. The naive divergence that appears when taking
this Carroll limit can be controlled by introducing auxiliary fields along the
lines of the corresponding construction for non-relativistic strings in
Gomis:2000bd . The presence of the auxiliary fields severely constrains the
dynamics of the magnetic Carroll string model.

###  2.1 Relativistic origin of the magnetic Carroll model

Our starting point is the Polyakov action, given by

$S_{\text{P}}=-\frac{1}{4\pi\alpha^{\prime}}\int
d^{2}\sigma\sqrt{-\gamma}\gamma^{ab}\partial_{a}\hat{X}^{\mu}\partial_{b}\hat{X}^{\nu}\eta_{\mu\nu}\,.$
(1)

Here lowercase Latin indices run over the worldsheet coordinates
$a,b\in\left\\{\sigma,\tau\right\\}$ and $\gamma^{ab}$ denotes the inverse of
the worldsheet metric. The flat $D$-dimensional target-space metric is denoted
by $\eta_{\mu\nu}$ with Lorentzian signature. The coupling constant
$\alpha^{\prime}$ is related to the string tension $T$ via
$T=1/(2\pi\alpha^{\prime})$. The worldsheet fields in (1) are labelled by hats
in anticipation of a rescaling to be carried out immediately below. We reserve
a hat-free notation for the rescaled fields that we work with in the Carroll
limit.

We want to take a Carroll limit on the target space while allowing the
worldsheet theory to retain its relativistic and Weyl symmetries. In order to
achieve this, we introduce an effective coupling constant
$\alpha^{\prime}_{\text{eff}}$ and perform an asymmetric scaling of the
worldsheet fields by the dimensionless factor of
$\alpha^{\prime}/\alpha^{\prime}_{\text{eff}}$, where longitudinal coordinates
in target space, denoted by uppercase Latin indices $A,B=0,1$, are scaled
differently from the transverse coordinates, denoted by uppercase primed
indices $A^{\prime},B^{\prime}=2,\ldots,D-1$,

$\displaystyle
X^{A}=\sqrt{\frac{\alpha^{\prime}_{\text{eff}}}{\alpha^{\prime}}}\,\hat{X}^{A}\,,\hskip
28.45274ptX^{A^{\prime}}=\frac{\alpha^{\prime}_{\text{eff}}}{\alpha^{\prime}}\,\hat{X}^{A^{\prime}}.$
(2)

With this scaling, the Polyakov action (1) can be rewritten as

$\displaystyle S_{\text{P}}=$ $\displaystyle\leavevmode\nobreak\
-\frac{1}{4\pi\alpha^{\prime}_{\text{eff}}}\int
d^{2}\sigma\sqrt{-\gamma}\gamma^{ab}\left(\partial_{a}X^{A}\partial_{b}X_{A}+\frac{\alpha^{\prime}}{\alpha^{\prime}_{\text{eff}}}\partial_{a}X^{A^{\prime}}\partial_{b}X_{A^{\prime}}\right)$
(3) $\displaystyle=$ $\displaystyle\leavevmode\nobreak\
-\frac{1}{4\pi\alpha^{\prime}_{\text{eff}}}\int
d^{2}\sigma\sqrt{-\gamma}\gamma^{ab}\left(\partial_{a}X^{A}\partial_{b}X_{A}+2\lambda_{a}^{A^{\prime}}\partial_{b}X_{A^{\prime}}-\frac{\alpha^{\prime}_{\text{eff}}}{\alpha^{\prime}}\lambda_{a}^{A^{\prime}}\lambda_{bA^{\prime}}\right)\,.$

In the second line we have introduced auxiliary fields
$\lambda_{a}^{A^{\prime}}$, which can be integrated out to recover the first
line. Taking the $\alpha^{\prime}\to\infty$ limit, we obtain

$\displaystyle S_{\text{M}}=-\frac{1}{4\pi\alpha^{\prime}_{\text{eff}}}\int
d^{2}\sigma\sqrt{-\gamma}\gamma^{ab}\left(\partial_{a}X^{A}\partial_{b}X_{A}+2\lambda_{a}^{A^{\prime}}\partial_{b}X_{A^{\prime}}\right)\,,$
(4)

which we refer to as the _magnetic_ Carroll model.111A corresponding
_electric_ Carroll model is obtained by a different scaling,
$\hat{X}^{A}=X^{A}$,
$\hat{X}^{A^{\prime}}=\sqrt{\frac{\alpha^{\prime}}{\alpha^{\prime}_{\text{eff}}}}X^{A^{\prime}}$.
In this case, taking the $\alpha^{\prime}\to\infty$ limit leads to
$S_{\text{E}}=-\frac{1}{4\pi\alpha^{\prime}_{\text{eff}}}\int
d^{2}\sigma\sqrt{-\gamma}\gamma^{ab}\partial_{a}X^{A^{\prime}}\partial_{b}X_{A^{\prime}}\,,$
which corresponds to the Polyakov action for the transverse coordinate fields
alone. Upon adopting a conformal gauge for the worldsheet metric,

$ds^{2}=-d\sigma^{+}d\sigma^{-}\,,\qquad\sigma^{\pm}=\tau\pm\sigma\,,\qquad\partial_{\pm}=\frac{1}{2}\left(\partial_{\tau}\pm\partial_{\sigma}\right)\,,$
(5)

we can write the magnetic action (4) as

$S_{\text{M}}=2\,T_{\text{eff}}\,\int
d^{2}\sigma\left(\partial_{+}X^{A}\partial_{-}X^{B}\eta_{AB}+\lambda_{+}^{A^{\prime}}\partial_{-}X_{A^{\prime}}+\lambda_{-}^{A^{\prime}}\partial_{+}X_{A^{\prime}}\right)\,,$
(6)

where we have introduced the effective string tension
$T_{\text{eff}}=1/(2\pi\alpha^{\prime}_{\text{eff}})$, and defined

$\displaystyle\lambda_{\pm}^{A^{\prime}}=\frac{1}{2}\left(\lambda_{\tau}^{A^{\prime}}\,\pm\,\lambda_{\sigma}^{A^{\prime}}\right)\,.$
(7)

In conformal gauge, the field equations of the magnetic Carroll model are

$\begin{array}[]{rr}\partial_{+}\partial_{-}X^{A}=0\,,&\partial_{+}X^{A^{\prime}}=0\,,\\\
\partial_{-}X^{A^{\prime}}=0\,,&\hskip
28.45274pt\partial_{+}\lambda_{-}^{A^{\prime}}+\partial_{-}\lambda_{+}^{A^{\prime}}=0\,,\end{array}$
(8)

supplemented by constraints of the form

$\displaystyle\partial_{\pm}X^{A}\partial_{\pm}X_{A}+2\lambda_{\pm}^{A^{\prime}}\partial_{\pm}X_{A^{\prime}}=0\,.$
(9)

It is immediately apparent that $X^{A^{\prime}}$ is constant, i.e. there is no
transverse motion in the magnetic Carroll model, and the constraints reduce to

$\displaystyle\partial_{\pm}X^{A}\partial_{\pm}X_{A}=0\,.$ (10)

The dispersion relation of magnetic Carroll strings,

$\displaystyle(P^{0})^{2}=(P^{1})^{2}\,,$ (11)

follows from the worldsheet field equations and constraints in a
straightforward way. To see this we consider null combinations of the
longitudinal coordinate fields, $X^{\pm}=\frac{1}{\sqrt{2}}(X^{0}\pm X^{1})$,
and fix one of them by the usual lightcone gauge condition,

$X^{+}=x^{+}+\alpha^{\prime}_{\text{eff}}\,P^{+}\tau\,.$ (12)

This is manifestly a solution of the two-dimensional wave equation as required
by (8). The other longitudinal field is a solution of the wave equation and a
periodic function of $\sigma$, which results in a general mode expansion of
the form

$\displaystyle
X^{-}=x^{-}+\alpha^{\prime}_{\text{eff}}\,P^{-}\,\tau+i\,\sqrt{\frac{\alpha^{\prime}_{\text{eff}}}{2}}\,\sum_{n\neq
0}\frac{1}{n}\left(\alpha_{n}^{-}e^{-in\sigma^{-}}+\tilde{\alpha}_{n}^{-}e^{-in\sigma^{+}}\right)\,.$
(13)

The longitudinal fields must also satisfy the constraints (10), which reduce
to

$\displaystyle P^{+}\partial_{\pm}X^{-}=0\,,$ (14)

when the lightcone gauge condition (12) is satisfied. There are two cases to
consider. If $P^{+}\neq 0$ then the constraints require $X^{-}$ to be a
constant, which implies $P^{-}=0$. On the other hand, if $P^{+}=0$ then
$X^{-}$ is unrestricted. In both cases the product of lightcone momenta
vanishes, $P^{+}P^{-}=0$, which is equivalent to (11).

We note that the energy of the magnetic Carroll string is independent of the
transverse momentum. The transverse momentum itself, however, remains
unconstrained as it only appears in

$P^{A^{\prime}}=T_{\text{eff}}\int d\sigma\lambda^{A^{\prime}}_{\tau}\,,$ (15)

where the zero mode of $\lambda^{A^{\prime}}_{\tau}$ remains undetermined by
the constraints.

Finally, if the spatial longitudinal coordinate is compactified on a circle of
radius $R$,

$X^{1}\sim X^{1}+2\pi R\,,$ (16)

then the energy takes discrete values $P^{0}=n/R$.

###  2.2 Canonical formulation

In this subsection, we present the first-order formulation of the magnetic
model to facilitate comparison with the Carroll string model found by Cardona,
Gomis, and Pons (CGP) in Cardona:2016ytk . The canonical momentum densities
conjugate to the dynamical fields $X^{A}$ and $X^{A^{\prime}}$ of the magnetic
Carroll action (4) are given by

$\displaystyle\Pi_{A}=\frac{\delta\mathcal{L}_{\text{M}}}{\delta\dot{X}^{A}}=-T_{\text{eff}}\,\sqrt{-\gamma}\gamma^{a\tau}\partial_{a}X_{A}\,,\hskip
28.45274pt\Pi_{A^{\prime}}=\frac{\delta\mathcal{L}_{\text{M}}}{\delta\dot{X}^{A^{\prime}}}=-T_{\text{eff}}\,\sqrt{-\gamma}\gamma^{a\tau}\lambda_{aA^{\prime}}\,,$
(17)

where the dot denotes a partial derivative with respect to $\tau$. The
canonical conjugate momenta of the worldsheet metric $\gamma_{ab}$ and the
auxiliary field $\lambda_{aA^{\prime}}$ are identically zero and thus
correspond to the primary constraints of the model. The canonical Hamilton
density $\mathcal{H}_{\text{M}}=\Pi_{\mu}\dot{X}^{\mu}-\mathcal{L}_{\text{M}}$
can be written as a sum of a longitudinal and transverse part,

$\displaystyle\mathcal{H}_{\text{M}}^{\|}$ $\displaystyle=$
$\displaystyle\Pi_{A}\partial_{\tau}X^{A}+\frac{1}{2}T_{\text{eff}}\sqrt{-\gamma}\gamma^{ab}\partial_{a}X^{A}\partial_{b}X_{A}$
(18) $\displaystyle=$
$\displaystyle\frac{\sqrt{-\gamma}}{2\,T_{\text{eff}}\,\gamma_{\sigma\sigma}}\left(\Pi_{A}\Pi^{A}+T_{\text{eff}}^{2}\,\partial_{\sigma}X_{A}\partial_{\sigma}X^{A}\right)+\frac{\gamma_{\sigma\tau}}{\gamma_{\sigma\sigma}}\,\Pi_{A}\partial_{\sigma}X^{A}\,\,,$
$\displaystyle\mathcal{H}_{\text{M}}^{\perp}$ $\displaystyle=$
$\displaystyle\Pi_{A^{\prime}}\partial_{\tau}X^{A^{\prime}}+T_{\text{eff}}\sqrt{-\gamma}\gamma^{ab}\lambda_{a}^{A^{\prime}}\partial_{b}X_{A^{\prime}}$
(19) $\displaystyle=$
$\displaystyle\frac{T_{\text{eff}}\,\sqrt{-\gamma}}{\gamma_{\sigma\sigma}}\lambda_{\sigma
A^{\prime}}\partial_{\sigma}X^{A^{\prime}}+\frac{\gamma_{\sigma\tau}}{\gamma_{\sigma\sigma}}\,\Pi_{A^{\prime}}\partial_{\sigma}X^{A^{\prime}}\,.$

We note that the longitudinal part has the standard form obtained from the
Polyakov action. It follows that the Hamilton equations for the magnetic
Carroll model are given by

$\displaystyle\dot{\Pi}_{A}-\partial_{\sigma}\left(\frac{\gamma_{\sigma\tau}}{\gamma_{\sigma\sigma}}\Pi_{A}+\frac{T_{\text{eff}}\sqrt{-\gamma}}{\gamma_{\sigma\sigma}}\,\partial_{\sigma}X_{A}\right)$
$\displaystyle=$ $\displaystyle 0\,,$ (20)
$\displaystyle\dot{X}^{A}-\frac{\gamma_{\sigma\tau}}{\gamma_{\sigma\sigma}}\partial_{\sigma}X^{A}-\frac{\sqrt{-\gamma}}{T_{\text{eff}}\,\gamma_{\sigma\sigma}}\,\Pi_{B}\,\eta^{AB}$
$\displaystyle=$ $\displaystyle 0\,,$ (21)
$\displaystyle\dot{\Pi}_{A^{\prime}}-\partial_{\sigma}\left(\frac{\gamma_{\sigma\tau}}{\gamma_{\sigma\sigma}}\,\dot{\Pi}_{A^{\prime}}+\frac{T_{\text{eff}}\sqrt{-\gamma}}{\gamma_{\sigma\sigma}}\,\lambda_{\sigma
A^{\prime}}\right)$ $\displaystyle=$ $\displaystyle 0\,,$ (22)
$\displaystyle\dot{X}^{A^{\prime}}-\frac{\gamma_{\sigma\tau}}{\gamma_{\sigma\sigma}}\,\partial_{\sigma}X^{A^{\prime}}$
$\displaystyle=$ $\displaystyle 0\,.$ (23)

As expected, these equations reduce to (8) in the conformal gauge (5).

The action of the CGP Carroll string action in Cardona:2016ytk  is given by

$\displaystyle S_{\text{CGP}}=\int
d^{2}\sigma\left(\Pi\cdot\dot{X}-\mu\,\Pi\cdot
X^{\prime}-\frac{e}{2}\,\left(\Pi_{A}\Pi^{A}+T_{\text{eff}}^{2}\,\partial_{\sigma}X^{A^{\prime}}\partial_{\sigma}X_{A^{\prime}}\right)\right)\,,$
(24)

where
$\Pi\cdot\dot{X}\equiv\Pi_{A}\partial_{\tau}X^{A}+\Pi_{A^{\prime}}\partial_{\tau}X^{A^{\prime}}$
and $\Pi\cdot
X^{\prime}\equiv\Pi^{A}\partial_{\sigma}X_{A}+\Pi^{A^{\prime}}\partial_{\sigma}X_{A^{\prime}}$.
Here $\mu$ and $e$ are Lagrange multipliers enforcing the primary constraints.
The action of the magnetic Carroll model (4) takes a similar form in first-
order variables,

$\displaystyle S_{\text{M}}=\int d^{2}\sigma$
$\displaystyle\left[\Pi\cdot\dot{X}-\frac{\gamma_{\sigma\tau}}{\gamma_{\sigma\sigma}}\Pi\cdot
X^{\prime}-\frac{\sqrt{-\gamma}}{2\,T_{\text{eff}}\,\gamma_{\sigma\sigma}}\left(\Pi_{A}\Pi^{A}+T_{\text{eff}}^{2}\left({X^{\prime}}^{A}X^{\prime}_{A}+2\lambda_{\sigma}^{A^{\prime}}X^{\prime}_{A^{\prime}}\right)\right)\right]\,,$
(25)

but there are significant differences between the two models. The longitudinal
part of the magnetic Carroll action retains the standard Polyakov form while
for the CGP action the longitudinal part
$T_{\text{eff}}{X^{\prime}}^{A}X^{\prime}_{A}$ is absent and in the transverse
fields of the magnetic Carroll model are constrained by the auxiliary fields
$\lambda_{\sigma}^{A^{\prime}}$. Notably, neither model has a term quadratic
in transverse momenta $\Pi^{A^{\prime}}\Pi_{A^{\prime}}$.

The Hamilton equations governing the dynamics of the two models in a conformal
gauge are listed in Table 1. For completeness, we also include the dynamical
equations (also in a conformal gauge) of the chiral Carroll model studied in
Section 3 below.

Carroll string |  $\dot{X}^{A}=\partialderivative{{H}}{\Pi_{A}}$ | $\dot{\Pi}^{A}=-\partialderivative{{H}}{X_{A}}$ | $\dot{X}^{A^{\prime}}=\frac{\partial{H}}{\partial\Pi_{A^{\prime}}}$ |  $\dot{\Pi}^{A^{\prime}}=-\partialderivative{{H}}{X_{A^{\prime}}}$
---|---|---|---|---
CGP Cardona:2016ytk  | $T_{\text{eff}}\dot{X}^{A}=\Pi^{A}$ | $\dot{\Pi}^{A}=0$ | $\dot{X}^{A^{\prime}}=0$ |  $\dot{\Pi}^{A^{\prime}}=T_{\text{eff}}\,X^{\prime\prime A^{\prime}}$
Magnetic model | $T_{\text{eff}}\dot{X}^{A}=\Pi^{A}$ | $\dot{\Pi}^{A}=T_{\text{eff}}\,X^{\prime\prime A}$ | $\dot{X}^{A^{\prime}}=0$ |  $\dot{\lambda}_{\tau}^{A^{\prime}}=\lambda_{\sigma}^{\prime A^{\prime}}$
Chiral model | $T_{\text{eff}}\dot{X}^{A}=\Pi^{A}$ | $\dot{\Pi}^{A}=T_{\text{eff}}\,X^{\prime\prime A}$ | $\dot{X}^{A^{\prime}}=X^{\prime A^{\prime}}$ |  $\dot{\lambda}_{+}^{A^{\prime}}=\lambda_{+}^{\prime A^{\prime}}$
Table 1: Dynamical equations for three different Carroll string models in
conformal gauge.

The field equations place strong restrictions on the dynamics in all three
models. In particular, there is no transverse motion at all allowed in the CGP
and magnetic Carroll models. The chiral model is less constrained in this
respect and supports left-moving transverse modes. Longitudinal momentum is
constant in the CGP Carroll model but the magnetic and the chiral Carroll
models do not have that restriction.

###  2.3 Symmetries of the magnetic Carroll model

The magnetic Carroll action (4) is invariant under infinitesimal symmetry
transformations of the form

$\displaystyle\delta X^{A}$ $\displaystyle=$ $\displaystyle
K^{A}+\Lambda^{A}{}_{A^{\prime}}X^{A^{\prime}}+\Lambda^{A}{}_{B}X^{B}\,,$
(26a) $\displaystyle\delta X^{A^{\prime}}$ $\displaystyle=$ $\displaystyle
K^{A^{\prime}}+\Lambda^{A^{\prime}}{}_{B^{\prime}}X^{B^{\prime}}\,,$ (26b)
$\displaystyle\delta\lambda_{a}^{A^{\prime}}$ $\displaystyle=$
$\displaystyle-{\Lambda_{B^{\prime}}}^{A^{\prime}}\lambda_{a}^{B^{\prime}}-{\Lambda_{A}}^{A^{\prime}}\partial_{a}X^{A}\,,$
(26c)

where $K^{A}$ and $K^{A^{\prime}}$ denote longitudinal and transverse target
space translations respectively, $\Lambda^{A}{}_{B}$ and
$\Lambda^{A^{\prime}}{}_{B^{\prime}}$ denote a longitudinal boost and a
transverse rotation222Here both parameters are skew-symmetric, i.e.
$\Lambda_{AB}=-\Lambda_{BA}$ and
$\Lambda_{A^{\prime}B^{\prime}}=-\Lambda_{B^{\prime}A^{\prime}}$.
respectively, and $\Lambda^{A}{}_{A^{\prime}}$ denotes a Carroll boost from
the longitudinal sector to the transverse sector. The corresponding Noether
charges are

$\displaystyle P_{A}$ $\displaystyle=\int d\sigma\,\Pi_{A}\,,\hskip
69.70915ptP_{A^{\prime}}=\int d\sigma\,\Pi_{A^{\prime}}\,,$ (27)
$\displaystyle{M_{AB}}$ $\displaystyle=2\int d\sigma\,\Pi_{[A}X_{B]}\,,\hskip
28.45274ptM_{A^{\prime}B^{\prime}}=2\int
d\sigma\,\Pi_{[A^{\prime}}X_{B^{\prime}]}\,,$ $\displaystyle{C_{AB^{\prime}}}$
$\displaystyle=\int d\sigma\,\Pi_{A}X_{B^{\prime}}\,.$

The charges form a closed algebra with the non-vanishing commutators given by

$\displaystyle\left[P_{A^{\prime}}\,,{C_{AB^{\prime}}}\right]$
$\displaystyle=$ $\displaystyle-\delta_{A^{\prime}B^{\prime}}P_{A}\,,$ (28a)
$\displaystyle\left[M_{AB}\,,C_{CA^{\prime}}\right]$ $\displaystyle=$
$\displaystyle 2\eta_{C[B}\,C_{A]A^{\prime}}\,,$ (28b)
$\displaystyle\left[{M_{A^{\prime}B^{\prime}}}\,,C_{AC^{\prime}}\right]$
$\displaystyle=$
$\displaystyle-2\delta_{C^{\prime}[A^{\prime}}\,C_{|A|B^{\prime}]}\,,$ (28c)
$\displaystyle\left[M_{A^{\prime}B^{\prime}}\,,P_{C^{\prime}}\right]$
$\displaystyle=$ $\displaystyle
2\delta_{C^{\prime}[B^{\prime}}P_{A^{\prime}]}\,,$ (28d)
$\displaystyle\left[M_{AB}\,,P_{C}\right]$ $\displaystyle=$ $\displaystyle
2\eta_{C[B}P_{A]}\,,$ (28e) $\displaystyle\left[M_{AB}\,,M_{CD}\right]$
$\displaystyle=$ $\displaystyle 4\,\eta_{[C[B}M_{A]D]}\,,$ (28f)
$\displaystyle\left[M_{A^{\prime}B^{\prime}}\,,M_{C^{\prime}D^{\prime}}\right]$
$\displaystyle=$ $\displaystyle
4\,\delta_{[C^{\prime}[B^{\prime}}M_{A^{\prime}]D^{\prime}]}\,.$ (28g)

This algebra was proposed in Cardona:2016ytk , where it was obtained as a
contraction of the Poincaré algebra and referred to as the _string Carroll
algebra_. We adopt the same terminology here.

The magnetic Carroll action (4) is also invariant under an infinitesimal
dilatation with scale parameter $\ell$ that acts only on the transverse
sector,

$\displaystyle\delta X^{A^{\prime}}=\ell\,X^{A^{\prime}}\,,\hskip
28.45274pt\delta\lambda_{a}^{A^{\prime}}=-\ell\,\lambda_{a}^{A^{\prime}}\,.$
(29)

The corresponding Noether charge is

$\displaystyle D=\frac{1}{2\pi\alpha^{\prime}_{\text{eff}}}\int
d\sigma\,\lambda_{\tau}^{A^{\prime}}X_{A^{\prime}}=\int
d\sigma\,\Pi^{A^{\prime}}X_{A^{\prime}}\,,$ (30)

and the non-vanishing commutators with the rest of the symmetry generators are

$\left[D,\,P_{A^{\prime}}\right]=P_{A^{\prime}}\,,\hskip
28.45274pt\left[D,\,C_{AA^{\prime}}\right]=-C_{AA^{\prime}}\,.$ (31)

This is reminiscent of the additional longitudinal dilatation symmetry found
in non-relativistic string theory Bergshoeff:2019pij , which gives rise to an
additional gauge field in the gauging procedure for the string Newton-Cartan
algebra. Here, the transverse dilatation symmetry (29) is expected to be
associated with an extra gauge field when gauging the string Carroll algebra
(28).

The non-relativistic string in flat spacetime admits an infinite dimensional
symmetry algebra, which contains the string Newton-Cartan algebra as a finite
dimensional subalgebra Bergshoeff:2019pij ; Batlle:2016iel . A similar story
plays out in the CGP Carroll string model Cardona:2016ytk  and also in the
Carroll string models considered here. The magnetic and chiral Carroll models
admit infinite dimensional symmetries, which contain as finite dimensional
subalgebras the string Carroll algebra and the extended string Carroll algebra
(defined in Section 3.3 below), respectively. As it turns out, the symmetries
of the magnetic Carroll model are a subset of those of the chiral Carroll
model and therefore we postpone the discussion of the infinite dimensional
symmetries until we have introduced the chiral Carroll model in Section 3.

##  3 Chiral Carroll string model

In this section, we consider a novel Carroll limit on target space. The
construction involves the same asymmetric scaling of longitudinal and
transverse coordinates as in the magnetic Carroll model but this time around
an extra set of auxiliary fields is introduced that allows us to impose a
worldsheet chirality on the transverse coordinate fields. The resulting model
realises a non-trivial extension of string Carroll symmetry and the transverse
dynamics turn out to be less constrained than in the magnetic Carroll model.

###  3.1 Relativistic origin of the chiral Carroll model

Our starting point is once again the Polyakov action (1) with a rescaling of
the coordinate fields as in (2) but now we use two sets of auxiliary fields
$\lambda_{aA^{\prime}}$ and $\chi_{aA^{\prime}}$ to rewrite the action,333The
equations of motion for $\lambda_{a}^{A^{\prime}}$ and $\chi_{a}^{A^{\prime}}$
implied by (33) are covariantly given by:
$\displaystyle\chi_{a}^{A^{\prime}}=\left(\frac{\alpha^{\prime}}{\alpha^{\prime}_{\text{eff}}}\right)^{2}\gamma_{ab}\Gamma^{bc}_{+}\partial_{c}X^{A^{\prime}},\hskip
28.45274pt\lambda_{a}^{A^{\prime}}=\left(\frac{\alpha^{\prime}}{\alpha^{\prime}_{\text{eff}}}\right)\gamma_{ab}\Gamma_{-}^{bc}\partial_{c}X^{A^{\prime}}$
(32)

$\displaystyle S_{\text{P}}=$ $\displaystyle\leavevmode\nobreak\
-\frac{1}{4\pi\alpha_{\text{eff}}^{\prime}}\int
d^{2}\sigma\sqrt{-\gamma}\gamma^{ab}\left[\partial_{a}X^{A}\partial_{b}X_{A}+\frac{\alpha^{\prime}}{\alpha^{\prime}_{\text{eff}}}\partial_{a}X^{A^{\prime}}\partial_{b}X_{A^{\prime}}\right]$
(33) $\displaystyle=$ $\displaystyle\leavevmode\nobreak\
-\frac{1}{4\pi\alpha_{\text{eff}}^{\prime}}\int
d^{2}\sigma\sqrt{-\gamma}\left[\gamma^{ab}\partial_{a}X^{A}\partial_{b}X_{A}+2\,\Gamma_{+}^{ab}\lambda_{aA^{\prime}}\partial_{b}X^{A^{\prime}}\right.$
$\displaystyle\left.\qquad\qquad\qquad\qquad\qquad+\,\frac{2\alpha^{\prime}_{\text{eff}}}{\alpha^{\prime}}\,\Gamma_{-}^{ab}\chi_{aA^{\prime}}\partial_{b}X^{A^{\prime}}-2\left(\frac{\alpha_{\text{eff}}^{\prime}}{\alpha^{\prime}}\right)^{2}\gamma^{ab}\lambda_{a}^{A^{\prime}}\chi_{bA^{\prime}}\right]\,,$

where we have introduced the chiral projection operators

$\Gamma^{ab}_{\pm}=\frac{1}{2}\left(\gamma^{ab}\pm\epsilon^{ab}\right)\,,$
(34)

with $\epsilon^{ab}$ the Levi-Civita tensor. We employ the conventions
$\epsilon^{ab}={\overline{\epsilon}}^{ab}/\sqrt{-\gamma}\,,$
$\epsilon_{ab}=\overline{\epsilon}_{ab}\sqrt{-\gamma}$, where
$\overline{\epsilon}^{ab}$ denotes the Levi-Civita symbol,
$\overline{\epsilon}^{01}=+1=-\overline{\epsilon}^{10}$. We can now readily
take the limit $\alpha^{\prime}\to\infty$ to obtain a _chiral Carroll string
model_ ,

$\displaystyle S_{\text{C}}=-\frac{1}{4\pi\alpha^{\prime}_{\text{eff}}}\int
d^{2}\sigma\sqrt{-\gamma}\left(\gamma^{ab}\partial_{a}X^{A}\partial_{b}X_{A}\;+\;2\,\Gamma_{+}^{ab}\lambda_{a}^{A^{\prime}}\partial_{b}X_{A^{\prime}}\right)\,.$
(35)

Upon adopting the conformal gauge (5) the chiral Carroll action reduces to

$S_{\text{C}}=\frac{1}{\pi\alpha^{\prime}_{\text{eff}}}\int
d^{2}\sigma\left(\partial_{+}X^{A}\partial_{-}X_{A}+\lambda_{+}^{A^{\prime}}\partial_{-}X_{A^{\prime}}\right)\,.$
(36)

Compared to the magnetic model (6) we have effectively projected out one
worldsheet chirality of the transverse fields $X^{A^{\prime}}$ and
$\lambda_{aA^{\prime}}$.444We have chosen to retain the left-moving fields
$\lambda_{+A^{\prime}}$. We could of course just as well have projected onto
the other chirality. The new auxiliary fields $\chi_{aA^{\prime}}$ do not
appear in the final action in the $\alpha^{\prime}\to\infty$ limit but they
nevertheless play an important role in implementing the projection onto a
single worldsheet chirality. In conformal gauge the equations of motion
obtained from the chiral Carroll action reduce to

$\partial_{+}\partial_{-}X^{A}=\partial_{-}\lambda_{+}^{A^{\prime}}=\partial_{-}X^{A^{\prime}}=0\,.$
(37)

This is a subset of the field equations of the magnetic model (8) and
therefore the dynamics is now less restrictive than before. In particular, the
transverse fields $X^{A^{\prime}}$ can have non-trivial dependence on the
$\sigma^{+}$ direction in the chiral model, in contrast to the magnetic model
where the transverse fields are only allowed to take constant values. The
energy momentum tensor constraints are now given by

$\displaystyle\begin{split}0&=\partial_{+}X^{A}\partial_{+}X_{A}+\lambda_{+}^{A^{\prime}}\partial_{+}X_{A^{\prime}}\,,\\\
0&=\partial_{-}X^{A}\partial_{-}X_{A}\,.\end{split}$ (38)

By implementing the lightcone gauge condition

$\displaystyle X^{+}=x^{+}+\alpha^{\prime}_{\text{eff}}P^{+}\tau\,,$ (39)

and inserting it into the constraints we find that

$\displaystyle\begin{split}\partial_{+}X^{-}&=\frac{1}{\alpha^{\prime}_{\text{eff}}P^{+}}\lambda_{+}^{A^{\prime}}\partial_{+}X_{A^{\prime}}\,,\\\
\partial_{-}X^{-}&=0\,.\end{split}$ (40)

From the latter it follows that $X^{-}$ is also chiral and hence the mode
expansion for $X^{-}$ can be written in the following form

$\displaystyle
X^{-}(\sigma^{-},\sigma^{+})=X^{-}_{L}(\sigma^{+})=x^{-}+\frac{1}{2}\alpha^{\prime}_{\text{eff}}P^{-}\sigma^{+}+i\sqrt{\frac{\alpha^{\prime}_{\text{eff}}}{2}}\sum_{n\neq
0}\frac{1}{n}\alpha_{n}^{-}e^{-in\sigma^{+}}\,.$ (41)

However, since $X^{-}$ is periodic in $\sigma$ this then implies that
$P^{-}=0$ i.e. that $E=P_{1}$. Therefore, although we have transverse
excitations, they do not contribute to the energy. Now consider the transverse
fields which are left-moving and periodic in $\sigma$,

$\displaystyle\begin{split}X^{A^{\prime}}(\sigma^{+})&=x^{A^{\prime}}+i\sqrt{\frac{\alpha^{\prime}_{\text{eff}}}{2}}\sum_{n\neq
0}\frac{1}{n}\alpha_{n}^{A^{\prime}}e^{-in\sigma^{+}},\\\
\lambda_{+}^{A^{\prime}}(\sigma^{+})&=\sqrt{\frac{\alpha^{\prime}_{\text{eff}}}{2}}\sum_{n}\lambda_{n}^{A^{\prime}}e^{-in\sigma^{+}},\end{split}$
(42)

where we note in particular that the transverse momenta $P^{A^{\prime}}$ are
related to the zero modes of the auxiliary fields $\lambda^{A^{\prime}}$ via

$\displaystyle
P^{A^{\prime}}=\int_{0}^{2\pi}d\sigma\Pi^{A^{\prime}}=\frac{1}{\sqrt{2\alpha^{\prime}_{\text{eff}}}}\lambda_{0}^{A^{\prime}}.$
(43)

As usual, the longitudinal $\alpha_{n}^{-}$ are obtained by inserting the mode
expansions into the constraint in (40). We find that for $n\neq 0$ we have

$\displaystyle\alpha_{n}^{-}=\frac{1}{P^{+}\sqrt{2\alpha^{\prime}_{\text{eff}}}}\sum_{m}\lambda_{m}^{A^{\prime}}\alpha_{n-m}^{B^{\prime}}\delta_{A^{\prime}B^{\prime}},$
(44)

where we have additionally defined $\alpha_{0}^{A^{\prime}}=0$ and
$\alpha_{0}^{-}=0$ for notational convenience. The zero mode of the constraint
then implies that

$\displaystyle\sum_{n}\lambda^{A^{\prime}}_{n}\alpha^{B^{\prime}}_{-n}\delta_{A^{\prime}B^{\prime}}=0\,,$
(45)

which says the total left-moving excitation level must be zero after all.

In order to have non-zero string excitations one needs to compactify a
transverse direction, say $A^{\prime}=D-1$, such that

$X^{D-1}(\sigma+2\pi)=X^{D-1}(\sigma)+2\pi w\mathcal{R}\,.$ (46)

where $w$ is the winding number. We now have a zero mode and a quantised
momentum in the compact direction,

$\alpha_{0}^{D-1}=\sqrt{\frac{2}{\alpha^{\prime}_{\text{eff}}}}w\mathcal{R}\,,\qquad
P^{D-1}=\frac{n}{\mathcal{R}}\,,$ (47)

and the two constraint equations in (40) become

$E=P_{1}\,,\qquad\sum_{n>0}\lambda^{A^{\prime}}_{n}\alpha^{B^{\prime}}_{-n}\delta_{A^{\prime}B^{\prime}}=-wn\,,$
(48)

where we can interpret the second equation as a level matching condition that
determines the left-moving excitation level in terms of the winding number.

###  3.2 Canonical formulation

The canonical conjugate momenta for the longitudinal and transverse worldsheet
fields of the chiral Carroll action (35) are given by

$\displaystyle\Pi_{A}=\frac{\delta\mathcal{L}_{\text{C}}}{\delta\dot{X}^{A}}=-T_{\text{eff}}\,\sqrt{-\gamma}\gamma^{a\tau}\partial_{a}X_{A}\,,\hskip
28.45274pt\Pi_{A^{\prime}}=\frac{\delta\mathcal{L}_{\text{C}}}{\delta\dot{X}^{A^{\prime}}}=-T_{\text{eff}}\sqrt{-\gamma}\,\Gamma_{+}^{a\tau}\lambda_{aA^{\prime}}\,.$
(49)

The canonical Hamilton density
$\mathcal{H}_{\text{M}}=\Pi_{\mu}\dot{X}^{\mu}-\mathcal{L}_{\text{M}}$ can be
written as a sum of a longitudinal and transverse part,

$\displaystyle\mathcal{H}_{\text{C}}^{\|}$ $\displaystyle=$
$\displaystyle\Pi_{A}\partial_{\tau}X^{A}+\frac{1}{2}T_{\text{eff}}\sqrt{-\gamma}\gamma^{ab}\partial_{a}X^{A}\partial_{b}X_{A}$
(50) $\displaystyle=$
$\displaystyle\frac{\sqrt{-\gamma}}{2\,T_{\text{eff}}\,\gamma_{\sigma\sigma}}\left(\Pi_{A}\Pi^{A}+T_{\text{eff}}^{2}\,\partial_{\sigma}X_{A}\partial_{\sigma}X^{A}\right)+\frac{\gamma_{\sigma\tau}}{\gamma_{\sigma\sigma}}\,\Pi_{A}\partial_{\sigma}X^{A}\,,$
$\displaystyle\mathcal{H}_{\text{C}}^{\perp}$ $\displaystyle=$
$\displaystyle\Pi_{A^{\prime}}\partial_{\tau}X^{A^{\prime}}+T_{\text{eff}}\sqrt{-\gamma}\gamma^{ab}\lambda_{a}^{A^{\prime}}\partial_{b}X_{A^{\prime}}$
(51) $\displaystyle=$
$\displaystyle\frac{(\gamma_{\sigma\tau}+\epsilon_{\sigma\tau})}{\gamma_{\sigma\sigma}}\,\Pi_{A^{\prime}}\partial_{\sigma}X^{A^{\prime}}\,.$

The longitudinal part is given by the same Polyakov expression as we found for
the magnetic model in (18). The transverse part differs from the magnetic
model in that it does not involve the auxiliary fields at all and incorporates
chirality through the anti-symmetric Levi-Civita tensor $\epsilon_{ab}$. The
Hamilton equations for the chiral model are

$\displaystyle\dot{\Pi}_{A}-\partial_{\sigma}\left(\frac{\gamma_{\sigma\tau}}{\gamma_{\sigma\sigma}}\Pi_{A}+\frac{T_{\text{eff}}\sqrt{-\gamma}}{\gamma_{\sigma\sigma}}\,\partial_{\sigma}X_{A}\right)$
$\displaystyle=$ $\displaystyle 0\,,$ (52)
$\displaystyle\dot{X}^{A}-\frac{\gamma_{\sigma\tau}}{\gamma_{\sigma\sigma}}\partial_{\sigma}X^{A}-\frac{\sqrt{-\gamma}}{T_{\text{eff}}\,\gamma_{\sigma\sigma}}\,\Pi_{B}\,\eta^{AB}$
$\displaystyle=$ $\displaystyle 0\,,$ (53)
$\displaystyle\dot{\Pi}_{A^{\prime}}-\partial_{\sigma}\left(\frac{(\gamma_{\sigma\tau}+\epsilon_{\sigma\tau})}{\gamma_{\sigma\sigma}}\,\dot{\Pi}_{A^{\prime}}\right)$
$\displaystyle=$ $\displaystyle 0\,,$ (54)
$\displaystyle\dot{X}^{A^{\prime}}-\frac{(\gamma_{\sigma\tau}+\epsilon_{\sigma\tau})}{\gamma_{\sigma\sigma}}\,X^{\prime
A^{\prime}}$ $\displaystyle=$ $\displaystyle 0\,.$ (55)

In the conformal gauge (5), these Hamilton equations reproduce (37) and they
are also listed in Table 1 above for comparison with the CGP and magnetic
Carroll models. The field equations of the chiral Carroll model only require
the transverse fields $X^{A^{\prime}}$ and their conjugate momenta to be
constant along the $\sigma^{-}$ worldsheet direction. Therefore, unlike the
CGP and magnetic Carroll models, the chiral model allows for transverse
excitations moving in one direction along the string.

###  3.3 Symmetries of the chiral Carroll model

The chiral Carroll action (35) is invariant under infinitesimal symmetry
transformations on the target space of the form

$\displaystyle\delta X^{A}=\leavevmode\nobreak$
$\displaystyle\Lambda^{A}_{\;\;\;A^{\prime}}X^{A^{\prime}}+\Lambda^{A}{{}_{B}}X^{B}+K^{A}\,,$
(56) $\displaystyle\delta X^{A^{\prime}}=\leavevmode\nobreak$
$\displaystyle\Lambda^{A^{\prime}}_{\;\;\;B^{\prime}}X^{A^{\prime}}+K^{A^{\prime}}\,,$
$\displaystyle\delta\lambda_{aA^{\prime}}=\leavevmode\nobreak$
$\displaystyle-\Lambda^{B^{\prime}}{}_{A^{\prime}}\lambda_{aB^{\prime}}-\Lambda^{A}_{\;\;\;A^{\prime}}\partial_{a}X_{A}\,,$

where
$\\{K^{A},K^{A^{\prime}},\Lambda^{A}{}_{B},\Lambda^{A^{\prime}}{}_{B^{\prime}},\Lambda^{A}{}_{A^{\prime}}\\}$
denote respectively longitudinal and transverse translations, longitudinal and
transverse rotations, and Carroll string boosts. The corresponding Noether
charges are given by

$\displaystyle P_{A}$ $\displaystyle=\int d\sigma\,\Pi_{A}\,,\hskip
69.70915ptP_{A^{\prime}}=\int d\sigma\,\Pi_{A^{\prime}}\,,$ (57)
$\displaystyle{M_{AB}}$ $\displaystyle=2\int d\sigma\,\Pi_{[A}X_{B]}\,,\hskip
28.45274ptM_{A^{\prime}B^{\prime}}=2\int
d\sigma\,\Pi_{[A^{\prime}}X_{B^{\prime}]}\,,$ $\displaystyle{C_{AB^{\prime}}}$
$\displaystyle=\int
d\sigma\left(\Pi_{A}X_{B^{\prime}}-T_{\text{eff}}\,X_{A}\partial_{\sigma}X_{B^{\prime}}\right)\,,$

where we note that the extra term in the Carroll string boosts
$C_{AB^{\prime}}$ compared with (27) arises from a total derivative in the
symmetry invariance of the action. The non-vanishing commutators are

$\displaystyle\commutator{Z_{A^{\prime}B^{\prime}}}{P_{C^{\prime}}}$
$\displaystyle=$ $\displaystyle
2\delta_{C^{\prime}[A^{\prime}}Z_{B^{\prime}]}$ (58a)
$\displaystyle\commutator{M_{A^{\prime}B^{\prime}}}{Z_{C^{\prime}}}$
$\displaystyle=$ $\displaystyle
2\delta_{C^{\prime}[A^{\prime}}Z_{B^{\prime}]},$ (58b)
$\displaystyle\commutator{M_{A^{\prime}B^{\prime}}}{Z_{C^{\prime}D^{\prime}}}$
$\displaystyle=$ $\displaystyle
4\delta_{[C^{\prime}[A^{\prime}}Z_{B^{\prime}]D^{\prime}]},$ (58c)
$\displaystyle\commutator{M_{AB}}{P_{C}}$ $\displaystyle=$ $\displaystyle
2\eta_{C[B}P_{A]},$ (58d) $\displaystyle\commutator{C_{AB^{\prime}}}{P_{C}}$
$\displaystyle=$ $\displaystyle-\eta_{CA}Z_{B^{\prime}},$ (58e)
$\displaystyle\commutator{C_{AB^{\prime}}}{P_{C^{\prime}}}$ $\displaystyle=$
$\displaystyle\delta_{C^{\prime}B^{\prime}}P_{A},$ (58f)
$\displaystyle\commutator{M_{A^{\prime}B^{\prime}}}{P_{C^{\prime}}}$
$\displaystyle=$
$\displaystyle-2\delta_{C^{\prime}[A^{\prime}}P_{B^{\prime}]},$ (58g)
$\displaystyle\commutator{M_{AB}}{M_{CD}}$ $\displaystyle=$ $\displaystyle
4\eta_{[C[A}M_{B]D]}$ (58h)
$\displaystyle\commutator{M_{A^{\prime}B^{\prime}}}{M_{C^{\prime}D^{\prime}}}$
$\displaystyle=$ $\displaystyle
4\delta_{[C^{\prime}[A^{\prime}}M_{B^{\prime}]D^{\prime}]}$ (58i)
$\displaystyle\commutator{M_{AB}}{C_{CD^{\prime}}}$ $\displaystyle=$
$\displaystyle-2\eta_{C[A}C_{B]D^{\prime}},$ (58j)
$\displaystyle\commutator{M_{A^{\prime}B^{\prime}}}{C_{CD^{\prime}}}$
$\displaystyle=$
$\displaystyle-2C_{C[A^{\prime}}\delta_{B^{\prime}]D^{\prime}},$ (58k)
$\displaystyle\commutator{C_{AB^{\prime}}}{C_{CD^{\prime}}}$ $\displaystyle=$
$\displaystyle\eta_{CA}Z_{B^{\prime}D^{\prime}},$ (58l)

where we have defined additional charges,

$Z_{A^{\prime}}=T_{\text{eff}}\,\int
d\sigma\,\partial_{\sigma}X_{A^{\prime}}\,,\hskip
42.67912ptZ_{A^{\prime}B^{\prime}}=Z_{[A^{\prime}B^{\prime}]}=2\,T_{\text{eff}}\,\int
d\sigma\,X_{[A^{\prime}}\,\partial_{\sigma}X_{B^{\prime}]}\,.$ (59)

With these extensions included, we have a closed algebra that we refer to as
the _extended string Carroll algebra_. This algebra can also be derived from
an $\dot{\text{I}}$nönü-Wigner contraction of the coadjoint Poincaré algebra
as we will explicitly show in subsection 3.5 below.

Finally, we mention that the algebra (58) can be further enlarged by an
emergent scaling symmetry $D$ as in (30). This adds the following non-
vanishing commutators

$\commutator{D}{P_{A^{\prime}}}\,=\,P_{A^{\prime}}\,,\hskip
12.23447pt\commutator{D}{Z_{A^{\prime}}}\,=\,-Z_{A^{\prime}}\,,\hskip
12.23447pt\commutator{D}{C_{AA^{\prime}}}\,=\,-C_{AA^{\prime}}\,,\hskip
12.23447pt\commutator{D}{Z_{A^{\prime}B^{\prime}}}=-2Z_{A^{\prime}B^{\prime}}\,.$
(60)

In the next subsection, we will see how the scaling symmetry arises from an
infinite dimensional extension of (58).

###  3.4 Infinite dimensional symmetries of the chiral Carroll action

The chiral Carroll action (35) is in fact invariant under an infinite
dimensional symmetry,555These symmetry transformations can be derived through
the conservation of a generator of canonical symmetry transformations $G$,
given by  $G=\int
d\sigma\left(\xi^{A}\Pi_{A}+\xi^{A^{\prime}}\Pi_{A^{\prime}}+\psi_{a}^{A^{\prime}}\Pi_{aA^{\prime}}+\Lambda\pi_{e}+\gamma\pi_{\mu}\right)$
(61) such that $\delta X^{A}=\\{G,X^{A}\\}=\xi^{A},\delta
X^{A^{\prime}}=\\{G,X^{A^{\prime}}\\}=\xi^{A^{\prime}}$, and
$\delta\lambda_{a}^{A^{\prime}}=\\{G,\lambda_{a}^{A^{\prime}}\\}=\psi^{A^{\prime}}_{a}$,
where the functions $\xi^{A},\xi^{A^{\prime}},\psi^{A^{\prime}}_{a},\Lambda$,
and $\gamma$ satisfy differential equations obtained from $dG/d\tau=0$.

$\displaystyle\delta X^{A}$ $\displaystyle=$ $\displaystyle
f^{A}(X^{A^{\prime}})+\Omega^{A}{}_{B}(X^{A^{\prime}})X^{B}\,,$ (62a)
$\displaystyle\delta X^{A^{\prime}}$ $\displaystyle=$
$\displaystyle\xi^{A^{\prime}}(X^{B^{\prime}})\,,$ (62b)
$\displaystyle\delta\lambda_{aA^{\prime}}$ $\displaystyle=$
$\displaystyle-\partial_{a}X_{A}\partial_{A^{\prime}}f^{A}-\partial_{a}X_{A}\partial_{A^{\prime}}\Omega^{A}{}_{B}X^{B}-\lambda_{aB^{\prime}}\partial_{A^{\prime}}\xi^{B^{\prime}}\,,$
(62c)

where $f^{A},\Omega^{A}{}_{B}$ and $\xi^{A^{\prime}}$ are arbitrary functions
that depend only on the transverse coordinates $X^{A^{\prime}}$ and
$\Omega^{A}{}_{B}$ is anti-symmetric, $\Omega_{AB}=-\Omega_{BA}$. In the
specific case of

$\displaystyle
f^{A}(X^{A^{\prime}})=K^{A}+\Lambda^{A}{}_{A^{\prime}}X^{A^{\prime}}\,,\hskip
14.22636pt\Omega^{A}{}_{B}(X^{B^{\prime}})=\Lambda^{A}{}_{B}\,,\hskip
14.22636pt\xi^{A^{\prime}}(X^{B^{\prime}})=K^{A^{\prime}}+\Lambda^{A^{\prime}}{}_{B^{\prime}}X^{B^{\prime}}\,,$
(63)

we recover the target space symmetries (56) discussed above. Note that under
the infinitesimal transformation $\delta X^{A}=f^{A}(X^{A^{\prime}})$ the
action is quasi-symmetric since under such a transformation we obtain a total
derivative term,

$\displaystyle\delta S_{\text{C}}=\int
d^{2}\sigma\,\partial_{a}\left(T_{\text{eff}}\sqrt{-\gamma}\,\epsilon^{ab}\,\partial_{A^{\prime}}f^{A}X_{A}\partial_{b}X^{A^{\prime}}\right)\,,$
(64)

which gives an additional contribution to the conserved charges associated
with $f^{A}$. Now consider expanding the functions $f^{A},\Omega^{A}{}_{B}$
and $\xi^{A^{\prime}}$ in a Taylor series of the transverse coordinates,

$\displaystyle f^{A}(X^{A^{\prime}})$
$\displaystyle=a^{A}+a^{A}{}_{A^{\prime}}X^{A^{\prime}}+a^{A}{}_{A^{\prime}B^{\prime}}X^{A^{\prime}}X^{B^{\prime}}+\ldots=\
a^{A}{}_{A^{\prime}_{1}A^{\prime}_{2}\cdots
A^{\prime}_{n}}X^{A^{\prime}_{1}A^{\prime}_{2}\cdots A^{\prime}_{n}},$ (65a)
$\displaystyle\Omega^{A}{}_{B}(X^{A^{\prime}})$
$\displaystyle=\omega^{A}{}_{B}+\omega^{A}{}_{BA^{\prime}}X^{A^{\prime}}+\omega^{A}{}_{BA^{\prime}B^{\prime}}X^{A^{\prime}}X^{B^{\prime}}+\ldots=\omega^{A}{}_{BA^{\prime}_{1}A^{\prime}_{2}\cdots
A^{\prime}_{n}}X^{A^{\prime}_{1}A^{\prime}_{2}\ldots A^{\prime}_{n}},$ (65b)
$\displaystyle\xi^{A^{\prime}}(X^{B^{\prime}})$
$\displaystyle=\varphi^{A^{\prime}}+\varphi^{A^{\prime}}{}_{B^{\prime}}X^{B^{\prime}}+\varphi^{A^{\prime}}{}_{B^{\prime}C^{\prime}}X^{B^{\prime}}X^{C^{\prime}}+\ldots=\varphi^{A^{\prime}}{}_{A^{\prime}_{1}A^{\prime}_{2}\cdots
A^{\prime}_{n}}X^{A^{\prime}_{1}A^{\prime}_{2}\ldots A^{\prime}_{n}},$ (65c)

where we have introduced the short hand notation
$X^{A^{\prime}_{1}A^{\prime}_{2}\cdots
A^{\prime}_{n}}=X^{A^{\prime}_{1}}X^{A^{\prime}_{2}}\cdots
X^{A^{\prime}_{n}}$. For each of the functions $f^{A},\Omega^{A}{}_{B}$ and
$\xi^{A^{\prime}}$, there is an infinite tower of conserved charges given by

$\displaystyle\xi^{A^{\prime}}:\hskip
23.33147ptK^{(n)}_{A^{\prime}A^{\prime}_{1}A^{\prime}_{2}\cdots
A^{\prime}_{n}}$ $\displaystyle=$ $\displaystyle\int
d\sigma\,\Pi_{A^{\prime}}X_{A^{\prime}_{1}A^{\prime}_{2}\cdots
A^{\prime}_{n}}\,,$ (66a) $\displaystyle f^{A}:\hskip
28.45274ptL^{(n)}_{AA^{\prime}_{1}A^{\prime}_{2}\cdots A^{\prime}_{n}}$
$\displaystyle=$ $\displaystyle\int
d\sigma\,\left(\Pi_{A}X_{A^{\prime}_{1}A^{\prime}_{2}\cdots
A^{\prime}_{n}}-T_{\text{eff}}\,X_{A}\partial_{\sigma}\left(X_{A^{\prime}_{1}\cdots
A^{\prime}_{n}}\right)\right)\,,$ (66b) $\displaystyle\Omega^{A}{}_{B}:\hskip
20.48601ptM^{(n)}_{ABA^{\prime}_{1}A^{\prime}_{2}\cdots A^{\prime}_{n}}$
$\displaystyle=$ $\displaystyle 2\int
d\sigma\,\Pi_{[A}X_{B]}X_{A^{\prime}_{1}A^{\prime}_{2}\cdots
A^{\prime}_{n}}\,.$ (66c)

where the index $n\geq 0$ denotes the number of additional transverse indices.
For instance, we recover the charges in (57) as follows,

$\displaystyle P_{A^{\prime}}=K_{A^{\prime}}^{(0)}\,,\hskip
14.22636ptP_{A}=L_{A}^{(0)}\,,\hskip 14.22636ptM_{AB}=M_{AB}^{(0)}\,,\hskip
14.22636ptC_{AB^{\prime}}=L_{AB^{\prime}}^{(1)}\,,\hskip
14.22636ptM_{A^{\prime}B^{\prime}}=2K_{[A^{\prime}B^{\prime}]}^{(1)}\,.$ (67)

The infinite tower of charges in (66) satisfies an infinite dimensional Lie
algebra,

$\displaystyle\commutator{K_{A^{\prime}A^{\prime}_{1}\cdots
A^{\prime}_{n}}^{(n)}}{K_{B^{\prime}B^{\prime}_{1}\cdots
B^{\prime}_{m}}^{(m)}}$
$\displaystyle=\sum_{k=1}^{n}\delta_{B^{\prime}A^{\prime}_{k}}K_{A^{\prime}A^{\prime}_{1}\cdots
A^{\prime}_{k-1}A^{\prime}_{k+1}\cdots A^{\prime}_{n}B^{\prime}_{1}\cdots
B^{\prime}_{m}}^{(n+m-1)}$ (68a) $\displaystyle\hskip
28.45274pt-\sum_{k=1}^{m}\delta_{A^{\prime}B^{\prime}_{k}}K_{B^{\prime}A^{\prime}_{1}\cdots
A^{\prime}_{n}B^{\prime}_{1}\cdots B^{\prime}_{k-1}B^{\prime}_{k+1}\cdots
B^{\prime}_{m}}^{(n+m-1)}\,,$ (68b)
$\displaystyle\commutator{K^{(n)}_{A^{\prime}A^{\prime}_{1}\cdots
A^{\prime}_{n}}}{L_{BB^{\prime}_{1}\cdots B^{\prime}_{m}}^{(m)}}$
$\displaystyle=-\sum_{k=1}^{m}\delta_{A^{\prime}B^{\prime}_{k}}L^{(m+n-1)}_{BA^{\prime}_{1}\cdots
A^{\prime}_{n}B^{\prime}_{1}\cdots B^{\prime}_{k-1}B^{\prime}_{k+1}\cdots
B^{\prime}_{m}}\,,$ (68c)
$\displaystyle\commutator{K_{A^{\prime}A^{\prime}_{1}\cdots
A^{\prime}_{n}}^{(n)}}{M_{BCB^{\prime}_{1}\cdots B^{\prime}_{m}}^{(m)}}$
$\displaystyle=-\sum_{k=1}^{m}\delta_{A^{\prime}B^{\prime}_{k}}M_{BCA^{\prime}_{1}\cdots
A^{\prime}_{n}B^{\prime}_{1}\cdots B^{\prime}_{k-1}B^{\prime}_{k+1}\cdots
B^{\prime}_{m}}^{(n+m-1)}\,,$ (68d)
$\displaystyle\commutator{L_{AA^{\prime}_{1}\cdots
A^{\prime}_{n}}^{(n)}}{L_{BB^{\prime}_{1}\cdots B^{\prime}_{m}}^{(m)}}$
$\displaystyle=\eta_{AB}\,Z^{(n,m)}_{A^{\prime}_{1}\cdots
A^{\prime}_{n}B^{\prime}_{1}\cdots B^{\prime}_{m}}\,,$ (68e)
$\displaystyle\commutator{L_{AA^{\prime}_{1}\cdots
A^{\prime}_{n}}^{(n)}}{M_{BCB^{\prime}_{1}\cdots B^{\prime}_{m}}^{(m)}}$
$\displaystyle=2\eta_{A[B}L^{(n+m)}_{C]A^{\prime}_{1}\cdots
A^{\prime}_{n}B^{\prime}_{1}\cdots B^{\prime}_{m}}\,,$ (68f)
$\displaystyle\commutator{M_{ABA^{\prime}_{1}\cdots
A^{\prime}_{n}}^{(n)}}{M_{CDB^{\prime}_{1}\cdots B^{\prime}m}^{(m)}}$
$\displaystyle=4\eta_{[C[B}M^{(n+m)}_{A]D]A^{\prime}_{1}\cdots
A^{\prime}_{n}B^{\prime}_{1}\cdots B^{\prime}_{m}}\,,$ (68g)

where we have again introduced a set of extension generators,

$\displaystyle Z^{(n,m)}_{A^{\prime}_{1}\cdots
A^{\prime}_{n}B^{\prime}_{1}\cdots B^{\prime}_{m}}$
$\displaystyle=T_{\text{eff}}\int d\sigma\left(X_{A^{\prime}_{1}\cdots
A^{\prime}_{n}}\partial_{\sigma}\left(X_{B^{\prime}_{1}\cdots
B^{\prime}_{m}}\right)-X_{B^{\prime}_{1}\cdots
B^{\prime}_{n}}\partial_{\sigma}\left(X_{A^{\prime}_{1}\cdots
A^{\prime}_{m}}\right)\right)\,.$ (69)

In the specific case of

$\displaystyle Z^{(0,1)}_{A^{\prime}}$ $\displaystyle=T_{\text{eff}}\int
d\sigma\partial_{\sigma}X_{A^{\prime}}\,,\hskip
28.45274ptZ^{(1,1)}_{A^{\prime}B^{\prime}}=T_{\text{eff}}\int d\sigma
X_{[A^{\prime}}\partial_{\sigma}X_{B^{\prime}]}\,,$ (70)

we recover the extensions defined in (59). The only additional non-vanishing
commutators that arise from adding these extensions are given by

$\displaystyle\commutator{Z^{(n,m)}_{A^{\prime}_{1}\cdots
A^{\prime}_{n}B^{\prime}_{1}\cdots
B^{\prime}_{m}}}{K^{(\ell)}_{C^{\prime}C^{\prime}_{1}\cdots
C^{\prime}_{\ell}}}$
$\displaystyle=\sum_{k=1}^{m}\delta_{C^{\prime}B^{\prime}_{k}}\,Z_{B^{\prime}_{1}\cdots
B^{\prime}_{k-1}B^{\prime}_{k+1}\cdots B^{\prime}_{m}C_{1}^{\prime}\cdots
C_{l}^{\prime}A_{1}^{\prime}\cdots A_{n}^{\prime}}^{(m-1,l+n)}$ (71)
$\displaystyle\hskip
28.45274pt+\sum_{k=1}^{n}\delta_{C^{\prime}A^{\prime}_{k}}\,Z_{A_{1}^{\prime}\cdots
A^{\prime}_{k-1}A^{\prime}_{k+1}\cdots A_{n}^{\prime}C_{1}^{\prime}\cdots
C_{l}^{\prime}B_{1}^{\prime}\cdots B_{m}^{\prime}}^{(n-1,l+m)}\,.$ (72)

The magnetic Carroll model (4) also has an infinite dimensional symmetry
algebra which is simply the subalgebra of (68) obtained by setting the
extensions (69) to zero.

The symmetry algebras of both the chiral and magnetic Carroll models include
the following generators,

$\displaystyle D$
$\displaystyle\equiv\delta^{A^{\prime}B^{\prime}}K^{(1)}_{A^{\prime}B^{\prime}}=\int
d\sigma\,\Pi_{A^{\prime}}X^{A^{\prime}}\,,$ (73a) $\displaystyle
K_{A^{\prime}}$ $\displaystyle\equiv
2\delta^{B^{\prime}C^{\prime}}K_{B^{\prime}C^{\prime}A^{\prime}}^{(2)}-\delta^{B^{\prime}C^{\prime}}K^{(2)}_{A^{\prime}B^{\prime}C^{\prime}}=\int
d\sigma\,\left(2\Pi_{B^{\prime}}X^{B^{\prime}}X_{A^{\prime}}-\Pi_{A^{\prime}}X^{B^{\prime}}X_{B^{\prime}}\right)\,.$
(73b)

The former corresponds to spatial dilatations and the latter corresponds to
spatial special conformal transformations also found for the CGP Carroll
string Cardona:2016ytk . Furthermore, we can construct a longitudinal
conformal transformation generator

$\displaystyle
K_{A}=\delta^{A^{\prime}B^{\prime}}L^{(2)}_{A\,A^{\prime}B^{\prime}}=\int
d\sigma\,\left(\Pi_{A}X_{A^{\prime}}X^{A^{\prime}}-T_{\text{eff}}X_{A}\partial_{\sigma}\left(X^{A^{\prime}}X_{A^{\prime}}\right)\right)\,.$
(74)

The three generators in (73) and (74) are somewhat similar to the generators
of the conformal Carroll algebra given by Basu:2018dub ; Bagchi:2019xfx ,

$\displaystyle\tilde{D}$ $\displaystyle=\int
d\sigma\left(\Pi^{A}X_{A}+\Pi^{A^{\prime}}X_{A^{\prime}}\right)\,,$ (75a)
$\displaystyle\tilde{K}_{A}$ $\displaystyle=\int d\sigma
X^{A^{\prime}}X_{A^{\prime}}\Pi_{A}\,,$ (75b)
$\displaystyle\tilde{K}_{A^{\prime}}$ $\displaystyle=\int
d\sigma\left(2X_{A^{\prime}}(X^{A}\Pi_{A}+X^{B^{\prime}}\Pi_{B^{\prime}})-(X^{B^{\prime}}X_{B^{\prime}})\Pi_{A^{\prime}}\right)\,.$
(75c)

but there are important differences. In particular, the longitudinal parts of
$\tilde{D}$ and $\tilde{K}_{A^{\prime}}$ are absent from our spatial special
conformal transformations in equation (73) and we have an additional term
associated with the quasi-symmetry in comparing $\tilde{K}_{A}$ with (74).

###  3.5 Extended Carroll algebra from the coadjoint Poincaré algebra

In this section, we derive the extended string Carroll algebra (58) through an
$\dot{\text{I}}$nönü-Wigner contraction of the coadjoint Poincaré algebra. The
procedure is as follows: Let us consider an extension of the Poincaré algebra
enlarged by the antisymmetric two-tensor $\hat{Z}_{\mu\nu}$ and the vector
$\hat{Z}_{\mu}$ (see, for example, Barducci:2019jhj ), which consists of the
following non-vanishing relativistic brackets666The hatted generators will be
the ones associated to the relativistic algebra.

$\displaystyle\commutator{\hat{M}_{\mu\nu}}{\hat{M}_{\rho\sigma}}$
$\displaystyle=4\eta_{[\rho[\mu}\hat{M}_{\nu]\sigma]}\,,$ (76)
$\displaystyle\commutator{\hat{M}_{\mu\nu}}{\hat{P}_{\rho}}$
$\displaystyle=2\eta_{\rho[\mu}\hat{P}_{\nu]},$ (77)
$\displaystyle\commutator{\hat{M}_{\mu\nu}}{\hat{Z}_{\rho}}$
$\displaystyle=2\eta_{\rho[\mu}\hat{Z}_{\nu]},$ (78)
$\displaystyle\commutator{\hat{M}_{\mu\nu}}{\hat{Z}_{\rho\sigma}}$
$\displaystyle=4\eta_{[\rho[\mu}\hat{Z}_{\nu]\sigma]}\,,$ (79)
$\displaystyle\commutator{\hat{Z}_{\mu\nu}}{\hat{P}_{\rho}}$
$\displaystyle=2\eta_{\rho[\mu}\hat{Z}_{\nu]}\,,$ (80)

along with the following vanishing brackets

$\displaystyle\commutator{\hat{P}_{\mu}}{\hat{P}_{\nu}}=\commutator{\hat{Z}_{\mu}}{\hat{P}_{\nu}}=\commutator{\hat{Z}_{\mu}}{\hat{Z}_{\nu}}=\commutator{\hat{Z}_{\mu\nu}}{\hat{Z}_{\rho}}=\commutator{\hat{Z}_{\mu\nu}}{\hat{Z}_{\rho\sigma}}=0\,.$
(81)

In order to obtain the extended string Carroll algebra from the algebra above,
we scale the generators in the following manner

$\displaystyle\hat{P}_{A}$ $\displaystyle=\omega\,P_{A}\,,\hskip
128.0374pt\hat{Z}_{A}=\omega\,Z_{A},$ (82) $\displaystyle\hat{P}_{A^{\prime}}$
$\displaystyle=P_{A^{\prime}}+\frac{\omega^{2}}{2}\,Z_{A^{\prime}}\,,\hskip
85.35826pt\hat{Z}_{A^{\prime}}=\omega\,Z_{A^{\prime}}\,,$ (83)
$\displaystyle\hat{M}_{AB}$ $\displaystyle=M_{AB}\,,\hskip
122.34692pt\hat{Z}_{AB}=Z_{AB}\,,$ (84)
$\displaystyle\hat{M}_{A^{\prime}B^{\prime}}$
$\displaystyle=M_{A^{\prime}B^{\prime}}\,,\hskip
110.96556pt\hat{Z}_{A^{\prime}B^{\prime}}=\omega\,Z_{A^{\prime}B^{\prime}}\,,$
(85) $\displaystyle\hat{M}_{AB^{\prime}}$
$\displaystyle=\omega\,C_{AB^{\prime}}-\frac{\omega^{2}}{2}Z_{AB^{\prime}}\,,\hskip
56.9055pt\hat{Z}_{AB^{\prime}}=\omega\,Z_{AB^{\prime}}\,,$ (86)

where we introduced longitudinal indices $A=0,1$ and transverse indices
$A^{\prime}=2,\ldots,D-1$, and $\omega$ a dimensionless parameter. These
scaling choices are inspired by Galilean results of ‘case 2b’ appearing in
Barducci:2019jhj . The contraction is carried out by taking the
$\omega\to\infty$ limit, and we obtain the extended string Carroll algebra
(58) supplemented with the following additional brackets

$\displaystyle\left[M_{AB},Z_{C}\right]$
$\displaystyle=2\eta_{C[A}Z_{B]}\,,\hskip
49.79231pt\left[Z_{AB^{\prime}},P_{C^{\prime}}\right]=-\delta_{C^{\prime}B^{\prime}}Z_{A}\,,$
(87) $\displaystyle\left[M_{AB},Z_{CD}\right]$
$\displaystyle=4\eta_{[C[A}Z_{B]D]}\,,\hskip
44.10185pt\commutator{Z_{AB}}{P_{C}}=2\eta_{C[A}Z_{B]}\,,$ (88)
$\displaystyle\left[M_{AB},Z_{CD^{\prime}}\right]$
$\displaystyle=2\eta_{C[A}Z_{B]D^{\prime}}\,,\hskip
36.98866pt\commutator{C_{AB^{\prime}}}{Z_{CD}}=-2\eta_{A[C}Z_{D]B^{\prime}}\,,$
(89) $\displaystyle\left[M_{A^{\prime}B^{\prime}},Z_{CD^{\prime}}\right]$
$\displaystyle=2Z_{C[A^{\prime}}\delta_{B^{\prime}]D^{\prime}}\,,\hskip
28.45274pt\commutator{M_{AB}}{Z_{CD^{\prime}}}=2\eta_{A[C}Z_{B]D^{\prime}}\,.$
(90)

Therefore, the resulting algebra can be written as
$\mathcal{A}\oplus\mathcal{I}$ such that

$\displaystyle\left[\mathcal{A},\,\mathcal{I}\right]\,\subseteq\,\mathcal{I}\,,\hskip
28.45274pt\left[\mathcal{A},\,\mathcal{A}\right]\,\subseteq\,\mathcal{A}\,,\hskip
28.45274pt\left[\mathcal{I},\,\mathcal{I}\right]\,\subseteq\,\mathcal{I}\,.$
(91)

where the subalgebra $\mathcal{A}$ and ideal $\mathcal{I}$ are

$\displaystyle\mathcal{A}=\left\\{M_{AB},M_{A^{\prime}B^{\prime}},C_{AB^{\prime}},P_{A},P_{A^{\prime}},Z_{A^{\prime}},Z_{A^{\prime}B^{\prime}}\right\\}\,,\hskip
14.22636pt\text{and}\hskip
14.22636pt\mathcal{I}=\left\\{Z_{AB^{\prime}},Z_{AB},Z_{A}\right\\}\,.$ (92)

Taking the quotient of $\mathcal{A}/\mathcal{I}$ we recover the extended
string Carroll algebra (58).

##  4 Discussion

We have presented two inequivalent ways of taking a Carroll limit of the flat
target space of the Polyakov action, yielding two Carroll string models that
we refer to as magnetic and chiral. Both models make use of auxiliary fields
that render the action finite in the Carroll limit. In the chiral model a
second set of auxiliary fields projects out one worldsheet chirality, leaving
only left-moving string excitations.

The magnetic model, described in Section 2, has a global symmetry algebra on
the target space that coincides with the string Carroll algebra found for the
CGP Carroll string model of Cardona et al. in Cardona:2016ytk . The magnetic
model and the CGP model also share the feature that the Carroll strings can
carry arbitrary transverse momentum but do not move in the transverse
directions, reminiscent of Carroll particles.

The dynamics of the chiral model, described in Section 3, is less constrained
and allows arbitrary left-moving transverse string excitations, which does not
have a counterpart when studying Carroll particles. The global symmetry of the
chiral Carroll model is an extension of the string Carroll algebra, enlarged
by a vector generator $Z_{A}$ and an anti-symmetric generator $Z_{AB}$. We
show how this extended string Carroll algebra can be obtained from a
contraction of the coadjoint Poincaré algebra.

We further show how the global symmetries of chiral Carroll string model are
embedded in an infinite dimensional symmetry algebra, which contains the
infinite dimensional symmetry of the magnetic Carroll string as a subalgebra.
An analogous infinite dimensional symmetry structure was found previously for
strings with Galilean symmetry in Bergshoeff:2019pij ; Batlle:2016iel .

Here we have only considered closed strings and we required at least one of
the transverse directions to be compactified in order to have non-trivial
(left-moving) transverse string excitations in the chiral Carroll model. As a
result, open strings will not have transverse modes in the models and it
remains an open question whether open Carroll strings can have interesting
dynamics via some other construction.

We considered a flat target space throughout. The authors of Hartong:2021ekg ;
Hartong:2022dsx  considered a series expansion for a curved target space in
the Galilean limit and were able to recover, amongst others, the Gomis-Ooguri
model. It would be curious to see if our Carroll string models can be obtained
in an analogous Carroll expansion and explore how the models couple to curved
spacetime. Finally, relaxing the relativistic symmetry on the worldsheet might
lead to interesting interplay between the target space and the worldsheet,
yielding novel string spectra.

## Acknowledgements

We thank Oscar Fuentealba, Joaquim Gomis, Troels Harmark, Jelle Hartong, Emil
Have, Niels Obers, Gerben Oling, and Ziqi Yan for discussions and insightful
comments. Research supported by the Icelandic Research Fund Grant 228952-053
and by the University of Iceland Research Fund. LT would like to thank the
Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and
hospitality during the program “Black Holes: Bridges Between Number Theory and
Holographic Quantum Information” where work on this paper was undertaken. This
work was supported by EPSRC grant no EP/K032208/1. WS acknowledges support of
the Simons Foundation through the INI-Simons Postdoctoral Fellowship.

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# Spacetime Foam and Solution of the Cosmological Constant Problem

Alexander I Nesterov<EMAIL_ADDRESS>Departamento de Física,
CUCEI, Universidad de Guadalajara, Av. Revolución 1500, Guadalajara, CP 44430,
Jalisco, México

###### Abstract

The cosmological constant problem is a fundamental issue that has puzzled
researchers in the fields of theoretical physics and cosmology for a long
time. It arises from the discrepancy between the observed value of the
cosmological constant and the value predicted by quantum field theory. A new
spacetime model based on nonassociative geometry and statistical physics of
complex networks offers a fresh perspective on the problem. Our research
indicates that spacetime topology plays a crucial role in solving the
cosmological constant problem and addressing the dark energy issue. We
discovered that spacetime foam significantly impacts the effective
cosmological constant, which is determined by the density of topological
geons. Furthermore, we demonstrate that the source of dark energy is
topological geons

Emergent spacetime, Discrete spacetime, Cosmological constant, Dark energy,
Topological geons, Nonassociative geometry, Euler characteristic, Complex
networks

The equivalence principle of general relativity requires that every form of
energy gravitate similarly. According to this principle, the enormous energy
of the quantum vacuum fluctuations must produce a sizeable gravitational
effect. In particular, it implies that the virtual particles’ contribution to
the vacuum energy density, commonly referred to as the zero-point energy
density, should affect the value of the cosmological constant Zel’dovich
(1968, 1967). Quantum field theory (QFT) allows us to calculate the
contribution of the zero-point energy to the cosmological constant.
Unfortunately, the estimates disagree with observational data by $10^{122}$
factor. This discrepancy between theory and observation is a cosmological
constant problem. It is also known as the worst prediction of theoretical
physics Weinberg (2000); Carroll  _et al._ (1992); Dolgov (1997); Carroll
(2001); Peebles and Ratra (2003a); Padmanabhan (2003); Copeland  _et al._
(2006); Rugh and Zinkernagel (2002); O’Raifeartaigh  _et al._ (2018); Sahni
and Starobinsky (2000); Peebles and Ratra (2003b); Weinberg (1989).

Despite extensive research, the cosmological constant problem remains an open
and unsolved puzzle in physics. Some proposed solutions to this problem invoke
the anthropic principle, suggesting that we live in a universe where the
cosmological constant is small enough to allow for the existence of
intelligent life Weinberg (1989, 1987); Carr and Rees (1979); Wilczek (1982);
John D. Barrow (1986); Ed. (2007). Others involve modifications to quantum
field theory or general relativity, including emergent gravity, that implies
omitting the use of continuum concepts a priori. It is assumed that at the
Planck scale, spacetime is fundamentally discrete Butterfield and Isham
(2000); Majid  _et al._ (2012); Ambjørn (2009); Loll (1998); ’t Hooft (2022);
Van Raamsdonk (2010); Swingle (2018); Padmanabhan (2016); Dowker (2014);
Padmanabhan (2015).

In the middle of the last century, Pauli suggested that the vacuum energies of
bosons and fermions might compensate for each other Pauli (2000). This
assumption is based on the fact that the vacuum energy of fermions and bosons
have opposite signs: the energy of bosons has a positive sign, while that of
fermions has a negative one. Later, Zel’dovich developed this approach to link
the vacuum energy to the cosmological constant Zel’dovich (1968, 1967).

Using the Pauli-Zel’dovich idea of cancellation of vacuum bosonic and
fermionic degrees of freedom, we have recently shown that the “foamy”
structure, analogous to Wheeler’s “spacetime foam” Wheeler (1957); DeWitt C.M.
(1968), significantly contributes to the effective cosmological constant. The
latter is defined by the universe’s Euler characteristic Nesterov (2023). In
this Letter, we describe this approach in detail and explain the contribution
of the topology and curvature to the cosmological constant. We show that the
density of topological geons determines the cosmological constant.

We use the natural units, setting $c=\hbar=k_{B}=1$.

Building discrete spacetime. – Our approach is based on nonassociative
geometry and statistical physics of complex networks and suggests a unified
algebraic description of both continuum and discrete spacetime as well
Nesterov and Sabinin (2000a); Sabinin (1999); Nesterov and Sabinin (2000b);
Sabinin (2001); Nesterov and Sabinin (2006); Nesterov (2006); Nesterov and
Mata (2019); Nesterov (2023). We treat space as a complex network presented by
simplicial 3-complex. Nodes of the network (0-simplices) are the atoms of
spacetime (pre-quarks or preons) that represent the universe’s building
blocks, just as atoms are the building blocks of matter. They are assumed to
be fermions with spin one-half and size $\sim\ell_{p}$. An edge, or 1-simplex,
has two vertices; therefore, it is a boson composed of two preons. A 2-simplex
being built from three preons is a fermion. A 3-simplex is a boson formed by
four preons.

The curvature of the simplicial complex is described by an elementary holonomy
– non-local analog of curvature Sabinin (1999); Nesterov and Sabinin (2000a,
b); Sabinin (2001); Nesterov and Sabinin (2006); Nesterov (2006); Nesterov and
Mata (2019). The elementary holonomy is associated with the faces of 2-simplex
and can be written as $h=1\pm\Delta S/R^{2}$, where $\Delta S$ is its area,
and $1/R^{2}$ is the curvature. Upper/lower sign corresponds to
positive/negative curvature, respectively Nesterov and Mata Villafuerte
(2020); Nesterov (2023).

A finite simplicial complex is pure if each facet has the same dimension. In
other words, a pure simplicial complex is formed only by simplices of the same
dimension. A pure simplicial complex with slight variations of the edge length
can be regarded as a discrete approximation to a smooth manifold. However,
some simplices can collapse to lower dimension simplices. In this case, the
simplicial complex can not be considered as a decomposition of a smooth
manifold Hawking (1978). As an example, consider the simplicial network with
positive curvature presented in Fig. 1. One can observe a non-triangulated,
completely disconnected space (top left), a partially triangulated space (top
right), and a pure simplicial 2-complex (bottom) that can be thought of as a
triangulation of the two-dimensional sphere.

Figure 1: Discrete space with positive curvature. Left top: a simplicial
0-complex (a non-triangulated, completely disconnected space). Right top: a
simplicial 2-complex (a partially triangulated space). Together with isolated
0-simplices and 1-simplices, one can observe the formation of two-dimensional
clusters connected by simplicial 1-complexes. Bottom: a pure simplicial
2-complex (a triangulated space). Adapted with permission from Ref. Nesterov
and Mata (2019).

Statistical description of complex networks. – The statistical description of
complex networks is based on the Shannon-Gibbs entropy,
$S=-\sum_{G\in\mathcal{G}}P(G)\ln P(G)$, where $P(G)$ is the probability of
obtaining a graph $G$, belonging to ensemble of graphs $\mathcal{G}$ Park and
Newman (2004); Squartini and Garlaschelli (2017); van der Hoorn  _et al._
(2018). For an undirected network with a fixed number of vertices and a
varying number of links, $P(G)$ is given by Park and Newman (2004);
Garlaschelli and Loffredo (2008); Garlaschelli  _et al._ (2013); Cimini  _et
al._ (2019)

$\displaystyle P(G)=\frac{1}{\mathcal{Z}}\exp\big{(}\beta(\mu
L(G)-H(G))\big{)},$ (1)

where $\mathcal{Z}=\sum_{G\in\mathcal{G}}\exp\big{(}\beta(\mu
L(G)-H(G))\big{)},$ is the partition function, $\beta=1/T$ is an inverse
network temperature, $\mu$ is the chemical potential, and $L(G)$ is number of
links in the graph $G$.

Let us assign the “energy”, $\varepsilon_{ij}$, to each edge $\langle
i,j\rangle$. Then the graph Hamiltonian can be written as
$H(G)=\sum_{i<j}\varepsilon_{ij}a_{ij}$, where $a_{ij}$ is an adjacency
matrix. It takes value 0 or 1 in the $i,j$ entry for each existing or non-
existing link between pairs of nodes ($ij$). The computation of the partition
function yields Garlaschelli  _et al._ (2013)

$\displaystyle\mathcal{Z}=$
$\displaystyle\sum_{G\in\mathcal{G}}\prod_{i<j}e^{\beta\left(\mu-\varepsilon_{ij}\right)a_{ij}}.$
(2)

The network connectivity is characterized by the connection probability
$p_{ij}$, i.e., the probability that a pair of nodes $(ij)$ is connected.
Using the partition function (2), one can obtain the connection probability of
the link between nodes $i$ and $j$ as the derivative of the partition
function: $p_{ij}=-{\partial\ln\mathcal{Z}}/{\partial(\beta\varepsilon_{ij})}$
Watts and Strogatz (1998); Newman  _et al._ (2001); Boccaletti  _et al._
(2006); Albert and Barabási (2002).

For a fermionic graph, when only one edge is allowed between any pair of
vertices, we obtain the Fermi-Dirac distribution Park and Newman (2004):

$\displaystyle p_{ij}=\frac{1}{e^{\beta\left(\varepsilon_{ij}-\mu\right)}+1},$
(3)

Model. – In our concept, spacetime comprises entangled quanta (preons),
similar to how quantum spin liquids arise from the collective behavior of
entangled spins. Gravity and geometry are emergent phenomena occurring in the
low-energy corner of the physical fermionic vacuum. At high temperatures,
$T\gg T_{p}$, where $T_{p}$ is the Planck temperature, spacetime behaves like
a quantum Fermi liquid. At the Planck temperature, the topological phase
transition occurs, resulting in the formation of a simplicial three-complex.
Emergent properties of geometry and gravity arise due to this transition.

The model can be described by the Hamiltonian, which is a sum of two parts:
$H_{\tau}=H_{0}+H_{s}$. The first term, $H_{0}$, represents the Hamiltonian of
the Fermi liquid of preons, and the second one, $H_{s}$, describes the
spacetime network. The Fermi liquid obeys the Fermi-Dirac distribution:

$\displaystyle\bar{n}_{i}=\frac{1}{e^{\beta\left(\varepsilon_{i}-\mu_{q}\right)}+1},$
(4)

where $\mu_{q}$ is the chemical potential, $\bar{n}_{i}$ refers to the average
occupation number with the normalization $N=\sum_{i}\bar{n}_{i}$ and $N$ is
the total number of particles, representing 0-simplices of the network. The
Hamiltonian, $H_{0}$, can be written as
$H_{0}=-m_{p}\sum_{i}\bar{n}_{i}+\sum_{i}\varepsilon_{i}n_{i}$, where $n_{i}$
denotes the occupation state at site $i$, The first term in $H_{0}$ accounts
for the zero-point energy contribution.

A simplicial complex allows for only one link between any two given nodes.
Therefore, space can be represented only by fermionic graphs, where only one
edge is permitted between any pair of vertices. To describe the spacetime
network, we consider the modified version of the graph Hamiltonian proposed in
Nesterov (2023)

$\displaystyle H_{s}=m_{p}\sum_{\langle ij\rangle}a_{ij}-m_{p}\sum_{\langle
ijk\rangle}{\lambda}_{ijk}a_{ij}a_{jk}a_{ki}+m_{p}\sum_{\langle
ijkp\rangle}a_{ij}a_{jk}a_{kp}a_{pi}.$ (5)

Here the summation is over 1- simplices, 2-simplices, and 3-simplices denoted
as $\langle ij\rangle$, $\langle ijk\rangle$ and $\langle ijkp\rangle$,
respectively. In (5), ${\lambda}_{ijk}=1+\epsilon_{\langle ijk\rangle}\Delta
S_{\langle ijk\rangle}/R_{\langle ijk\rangle}^{2}$ denotes the elementary
holonomy of 2-simplex associated with the triplet of sites $\langle
i,j,k\rangle$ forming a left triangle concerning the edge $\langle ij\rangle$,
and $\epsilon_{\langle ijk\rangle}=0,\pm 1$.

The first term in Eq.(5) describes the contribution of the bosonic degree of
freedom related to the two-atom entangled states, the second term yields the
contribution of the quantum fermionic vacuum fluctuations and curvature of the
entangled three-atom states, and the third one represents the quantum bosonic
degree of freedom related to the entangled four-atom states.

Within the mean-field approximation, the graph Hamiltonian $H_{s}$ is replaced
by the effective Hamiltonian

$\displaystyle{\mathcal{H}}_{s}=\sum_{\langle
ij\rangle}\varepsilon_{ij}a_{ij}+{2m_{p}}\sum_{\langle
ijk\rangle}{\lambda}_{ijk}p_{ij}p_{jk}p_{ki}-3{m_{p}}\sum_{\langle
ijkp\rangle}p_{ij}p_{jk}p_{kp}p_{pi},$ (6)

where

$\displaystyle\varepsilon_{ij}=m_{p}-{3m_{p}}\sum_{k}\lambda_{ijk}p_{ik}p_{kj}+{4m_{p}}\sum_{k,p}p_{ik}p_{kp}p_{pj}.$
(7)

The network is described by the generalized Fermi-Dirac distribution (3) that
can be recast as follows:

$\displaystyle
p_{ij}=\frac{1}{2}\bigg{(}1+\tanh\bigg{(}\frac{\beta}{2}\Big{(}\mu-
m_{p}+{3m_{p}}\sum_{k}\lambda_{ijk}p_{ik}p_{kj}-{4m_{p}}\sum_{k,p}p_{ik}p_{kp}p_{pj}\Big{)}\bigg{)}\bigg{)}$
(8)

At the Planck temperature, $T_{p}$, the system experiences the phase
transition, leading to fundamental structural changes in the network topology
Nesterov (2023). The entanglement of the system is significantly reduced at
temperatures above $T_{p}$. It is completely lost for the high temperatures
($T\gg T_{p}$), so the graph becomes disconnected, and a simpilicial 0-complex
presents the space. Below the Planck temperature, the entanglement increases
significantly, peaking at the system’s ground state. Thus, the high
dimensional simplicial complices emerge with decreasing the network’s
temperature, and the maximally triangulated space appears at zero temperature.
For $T\gg T_{p}$, the ground state is defined by the Fermi liquid’s ground
state; for $T\ll T_{p}$, the system is in the bosonic ground state. Finally,
it’s worth noting that on a lattice, the motion of the preons is restricted,
and their energy spectrum becomes discretized.

Figure 2: Formation of a cluster from 200 initial randomly distributed
tetrahedrons. Adapted with permission from Ref. Nesterov and Mata (2019).

The energy of the total system is given by

$\displaystyle E=-Nm_{p}+\sum_{i}\varepsilon_{i}\bar{n}_{i}+m_{p}\sum_{\langle
ij\rangle}p_{ij}-m_{p}\sum_{\langle
ijk\rangle}{\lambda}_{ijk}p_{ij}p_{jk}p_{ki}+m_{p}\sum_{\langle
ijkp\rangle}p_{ij}p_{jk}p_{kp}p_{pi}.$ (9)

In the limit of $T\rightarrow 0$, we obtain the ground state with the energy

$\displaystyle
E_{vac}=-m_{p}\chi(\mathfrak{M})-m_{p}N_{2}\frac{\Delta\theta}{a^{2}},$ (10)

where $\chi(\mathfrak{M})=\sum^{3}_{i=0}(-1)^{i}N_{i}-$ is the Euler
characteristic of the simplicial complex $\mathfrak{M}$ with $N_{i}$ being a
number of $i$-simplices Klaus (2016), and $\Delta\theta$ is the average angle
deficit of 2-simplices:

$\displaystyle\Delta\theta=\frac{a^{2}}{N_{2}}\sum_{\langle
ijk\rangle}\epsilon_{\langle ijk\rangle}\frac{\Delta S_{\langle
ijk\rangle}}{R_{\langle ijk\rangle}^{2}}.$ (11)

Here $a$ is the cosmic scale factor.

To find the effective pressure in our model, we use the relation $P=-\partial
F/\partial V|_{T}$, where $F=E-TS$ is the Helmholtz free energy. In the limit
of $T\rightarrow 0$, we obtain $P=-\rho_{vac}$, where $\rho_{vac}=E_{vac}/V$
is the density of vacuum energy. Thus, the pressure of the vacuum in the
network is negative and should lead to the universe’s accelerating expansion.

Evolution of the universe. – In our approach, space and time are quantized in
the Planck units, and a random/stochastic process governs spacetime evolution.
A random walk with a reflecting barrier $R_{0}$ at $n=0$ and a step $\ell_{0}$
defines the cosmological scale factor. After $n$ steps, the average
cosmological scale factor, $\left\langle a_{n}\right\rangle$, grows as follows
Nesterov and Mata (2019)

$\displaystyle\left\langle a_{n}\right\rangle=a_{0}+\sqrt{2n/\pi},$ (12)

where $a_{0}=R_{0}/\ell_{0}$ is the scale factor related to the initial
3-simplex defining the minimal size of the universe. After $\sim 10^{8}$ steps
in Planck units of time, which corresponds to pre-inflationary time (time of
the beginning of inflation), $t_{pr}\sim 10^{-35}\mathrm{sec}$ the
cosmological scale factor would be $a_{pr}\sim a_{0}+\sqrt{2/\pi}10^{4}$.

One can expect the formation of clusters of various dimensions in the process
of evolution. It is reasonable to assume that spacetime consists of many small
fluctuating simplicial regions. These fluctuations may result in forming mini-
clusters, as illustrated in Fig. 2. They can be created in different areas and
grow as mini-universes, forming spatial foam.

Thus, for $T<T_{p}$, the graph is represented by a simplicial 3-complex formed
by groups or clusters of various dimensions. Within the same group, node-node
connections are dense, but connections are less dense between groups. A group
of nodes forms a component when all of them are connected, directly or
indirectly Newman  _et al._ (2001); Li  _et al._ (2021). A “giant component”
contains a significant part of the total number of nodes. For instance, if the
pure simplicial complex presents the spacetime network, the whole network is a
giant component. The current universe’s formation occurs in the inflation
period as a merging of the mini-universes and the formation of a giant
component as the result of the structural phase transition occurring at the
temperature $T_{c}\sim 10^{28}\rm K$ (Fig. 3).

Figure 3: Reconstruction of the Universe after inflation. Cosmic inflation
predicts many big bangs, creating multiple expanding universes being part of a
multiverse. Adapted with permission from Ref. Nesterov and Mata (2019).

The cosmological constant problem. – In classical general relativity, the
cosmological constant, $\Lambda$, added in the Einstein equations, governs the
Universe’s accelerated expansion. Writing the Einstein equations with the
cosmological constant as

$\displaystyle R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R={8\pi
G}T_{\mu\nu}-g_{\mu\nu}\Lambda.$ (13)

one can observe that space with non-zero cosmological constant produces the
same gravitational field as the matter with mass density
$\rho_{\Lambda}=\Lambda/8\pi G$ and pressure $P_{\Lambda}=-\rho_{\Lambda}$.
Thus, one can speak about the energy density of the vacuum and its pressure.

According to the equivalence principle of general relativity, the energy of
the quantum vacuum fluctuations must produce a gravitational field, and the
semiclassical Einstein equations describe their contribution:

$\displaystyle R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R={8\pi G}\langle
T_{\mu\nu}\rangle-g_{\mu\nu}\Lambda,$ (14)

where $\langle T_{\mu\nu}\rangle=\langle 0|T_{\mu\nu}|0\rangle$ is the
expectation value of the quantum vacuum energy-momentum tensor. Lorentz
invariance requires that $\langle T_{\mu\nu}\rangle$ takes the form $\langle
T_{\mu\nu}\rangle=-\rho_{vac}g_{\mu\nu}$ where $\rho_{vac}=\langle
0|T_{00}|0\rangle$ is the expectation value of the energy density of the
matter fields in the vacuum state. Employing Eq.(14), one can recast Eq.(13)
as

$\displaystyle R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R={8\pi
G}T_{\mu\nu}-g_{\mu\nu}\Lambda_{\mathrm{eff}},$ (15)

where $\Lambda_{\mathrm{eff}}=\Lambda+8\pi\rho_{vac}\ell_{p}^{2}$ is the
effective (observable) cosmological constant.

The current Hubble expansion rate $H_{0}$ yields an upper bound for
$\Lambda_{\mathrm{eff}}$ Wang (2020),

$\displaystyle\Lambda_{\mathrm{eff}}=3H^{2}\leq
3H_{0}^{2}\sim\Lambda_{0}\ell_{p}^{-2},$ (16)

where $\Lambda_{0}=10^{-122}$ is the dimensionless cosmological constant, and
$H=\dot{a}/a=\sqrt{\Lambda_{\mathrm{eff}}/3}$ is the Hubble constant
describing acceleration of the universe. The vacuum density energy in the QFT
is estimated as $\rho_{vac}\sim m_{p}^{4}$, where $m_{p}$ is the Planck mass.
Using this result in (16), we obtain $\Lambda_{\mathrm{eff}}\sim
10^{-122}\rho_{vac}\ell_{p}^{2}$. Thus, one has a huge disagreement between
the observed value of the cosmological constant and the theoretical prediction
of its value resulting from the quantum field theory.

It is worth noting that the standard computation of the cosmological constant
assumes that the spacetime is homogeneous and isotropic, and the theory is
Lorentz invariant. These assumptions are reasonable at a cosmological scale
but questionable at a small (Planck) scale Wang  _et al._ (2017); Wang (2020);
Carlip (2019).

To resolve the cosmological constant problem, we use the statistical
description of the spacetime, treating the latter as a complex network. We
define the observable (effective) cosmological constant as

$\displaystyle\Lambda=8\pi\ell_{p}^{2}\rho_{vac}\equiv\Lambda_{0}\ell_{p}^{-2},$
(17)

where $\rho_{vac}$ is the energy density of the vacuum state defined by the
contribution of all bosonic and fermionic fields of the simplicial manifold
$\mathfrak{M}$.

Employing Eq. (10), we obtain

$\displaystyle\rho_{vac}=m_{p}n_{g}-\frac{m_{p}N_{2}}{V}\frac{\Delta\theta}{a^{2}},$
(18)

where $n_{g}$ denotes the number density of topological geons:

$\displaystyle n_{g}=$ $\displaystyle-\frac{1}{V}\chi(\mathfrak{M})$ (19)

Substituting $V=N_{3}V_{0}$ in Eq. (18), where
$V_{0}={\ell_{p}^{3}}/{6\sqrt{2}}$ is an average volume of the 3-simplex, and
employing Eq.(17), we obtain

$\displaystyle\Lambda_{0}=8\pi\ell^{3}_{p}n_{g}-\frac{8\pi\ell^{3}_{p}N_{2}}{V_{0}N_{3}}\cdot\frac{\Delta\theta}{a^{2}}.$
(20)

Assume that $\mathfrak{M}$ is a pure 3-dimensional simplicial complex, then
$\chi(\mathfrak{M})=0$ and $N_{2}=2N_{3}$ Klaus (2016). Using these results,
we obtain

$\displaystyle\Lambda_{0}=-\frac{16\pi\ell^{3}_{p}}{V_{0}}\cdot\frac{\Delta\theta}{a^{2}}.$
(21)

Thus, if the simplicial manifold is represented by a pure 3-dimensional
simplicial complex before inflation, the cosmological constant is determined
by its average curvature. In another limit case, when the contribution of the
second term in Eq. (20) is negligible in comparison with the first term, we
obtain $\Lambda_{0}={8\pi\ell^{3}_{p}}n_{g}$. The possible scenario of the
cosmological constant emerging is as follows. Before inflation, the
cosmological constant is determined by the average curvature of the universe.
During the inflation period, the term with $\chi(\mathfrak{M})$ begins to
dominate so that at the end of the inflation, the cosmological constant is
determined by the density of topological geons.

The density of the topological geons determines the effective cosmological
constant. The current observables yield $n_{g}\approx 10^{-25}\rm cm^{-3}$.
Taking the size of the observable universe as 93 billion light-years, we find
that the number of the geons in our universe, defined by the Euler
characteristic, is $\sim 10^{61}$. This enormous number implies that the
topology of spacetime is highly non-trivial due to vacuum fluctuations.

In summary, we point out that the standard formulation of the cosmological
constant problem is problematic since spacetime and vacuum energy density are
highly inhomogeneous and fluctuate wildly at the Planck scale. The vacuum
fluctuations can result in the foamy structure of spacetime, formed by micro-
universes connected by the Einstein-Rosen bridges with wormhole topology. We
show that the spacetime foam significantly contributes to the effective
cosmological constant defined by the density of topological geons. Our
research reveals the role of spacetime topology in solving the cosmological
constant problem and demonstrates that the source of the dark energy is
topological geons.

The proposed model challenges the current Quantum Field Theory approach by
offering a statistical description of spacetime and suggests a new approach to
address the cosmological constant problem. The implications of this
statistical description of spacetime are significant and could reshape our
understanding of the universe’s fundamental nature.

The author acknowledges the support of the CONAHCYT.

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