Physics Letters B 804 (2020) 135392
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Three-dimensional exotic Newtonian gravity with cosmological
constant
Patrick Concha
a,∗
, Lucrezia Ravera
b
, Evelyn Rodríguez
c
a
Departamento de Matemática y Física Aplicadas, Universidad Católica de la Santísima Concepción, Alonso de Ribera 2850, Concepción, Chile
b
INFN, Sezione di Milano, Via Celoria 16, I-20133 Milano, Italy
c
Departamento de Ciencias, Facultad de Artes Liberales, Universidad Adolfo Ibáñez, Viña del Mar, Chile
a r t i c l e i n f o a b s t r a c t
Article history:
Received 18 February 2020
Received in revised form 19 March 2020
Accepted 20 March 2020
Available online 23 March 2020
Editor: N. Lambert
In this work we introduce a cosmological constant in the extended Newtonian gravity theory. To this
end, we extend the exotic Newton-Hooke algebra by introducing new generators and central charges.
The new algebra obtained here has been denoted as exotic Newtonian algebra and reproduces the
extended Newtonian one in the flat limit →∞. A three-dimensional Chern-Simons action for the
exotic Newtonian algebra is presented. We show that the non-relativistic gravity theory proposed here
reproduces the most general extended Newtonian gravity theory in the flat limit.
© 2020 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
Newtonian (super)gravity theories have received a growing in-
terest
in the recent years due to their utility in condensed matter
systems [1–8] and non-relativistic effective field theories [9–12].
The underlying symmetry of Newtonian gravity is given by the
so-called Bargmann algebra [13–15], also known as centrally ex-
tended
Galilean algebra. The construction of such gravity theory
requires a geometric framework called Newton-Cartan geometry
[13,14,16–25]. This subsequently allowed to formulate a super-
symmetric
extension of Newton-Cartan gravity [26,27]. Despite
the great progress achieved at the bosonic level, an action prin-
ciple
for the Newtonian gravity model has been presented only
recently, in [28]. Subsequently, in [29]the authors constructed,
using the Chern-Simons (CS) formalism, a three-dimensional (su-
per)gravity
theory based on the extended Newtonian algebra. Such
algebra requires to extend the so-called extended Bargmann al-
gebra
[30–36]by including new generators and central charges.
The advantage of adopting a three-dimensional CS formalism re-
lies
in the fact that it not only provides with a simpler frame-
work
to formulate non-relativistic (super)gravities but also repro-
duces
interesting toy models to approach higher-dimensional the-
ories.
*
Corresponding author.
E-mail addresses: patrick.concha@ucsc.cl (P. Concha), lucrezia.ravera@mi.infn.it
(L. Ravera),
evelyn.rodriguez@edu.uai.cl (E. Rodríguez).
On the other hand, non-relativistic models that include a cos-
mological
constant are described through the Newton-Hooke sym-
metry,
which in the flat limit, leads to the Galilei symmetry
[37–45]. However, the incorporation of a cosmological constant in
the extended Newtonian gravity remains an open issue.
In this work, we present a novel non-relativistic algebra that
allows us to construct a three-dimensional CS exotic Newtonian
gravity in the presence of a cosmological constant. To this end, we
extend the so-called extended Newton-Hooke algebra [46,47]by
introducing new generators. Furthermore, in order to have a well-
defined
non-degenerate invariant tensor, we require the presence
of two central charges. The new algebra is denoted as exotic New-
tonian
algebra and reproduces the extended Newtonian algebra
[29]in the flat limit →∞. The CS action obtained contains the
extended Newton-Hooke gravity Lagrangian as a sub-case. Further-
more,
we show that the flat limit reproduces not only the extended
Newtonian gravity but also an exotic term. To our knowledge, this
is the first report showing an action principle for Newtonian grav-
ity
including a cosmological constant. Our results could be useful
in the formulation of a non-relativistic supergravity model in the
presence of a cosmological constant.
The paper is organized as follows: In Section 2, following [29],
we briefly review the extended Newtonian gravity theory and in-
troduce
an exotic term in the model. In Section 3, we present a
new non-relativistic algebra that we have called exotic Newtonian
algebra. Then, the explicit construction of a CS action invariant un-
der
this algebra is presented. Section 4 concludes our work with a
discussion about possible future developments.
https://doi.org/10.1016/j.physletb.2020.135392
0370-2693/© 2020 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.
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