Non-relativistic gravity theory based on an enlargement of the extended Bargmann algebra

Abstract

In this work we study a non-relativistic three dimensional Chern-Simons gravity theory based on an enlargement of the Extended Bargmann algebra. A finite non-relativistic Chern-Simons gravity action is obtained through the non-relativistic contraction of a particular U (1) enlargement of the so-called AdS-Lorentz algebra. We show that the non-relativistic gravity theory introduced here reproduces the Maxwellian Exotic Bargmann gravity theory when a flat limit ℓ → ∞ is applied. We also present an alternative procedure to obtain the non-relativistic versions of the AdS-Lorentz and Maxwell algebras through the semigroup expansion method.

Three-dimensional Maxwellian extended Bargmann supergravity

“…One could follow the procedure used in [89][90][91] and consider the expansion of a relativistic Maxwell superalgebra. Alternatively, one might also extend the results obtained in [84,92] in which NR algebras appear as semigroup expansions of the so-called Nappi-Witten algebra.…”

“…An alternative limit which could be used to recover the MEB algebra is the vanishing cosmological constant limit. In particular, one could conjecture that a supersymmetric extension of the recent enlarged extended Bargmann gravity, introduced in [84], reproduces the present MEB supergravity in a flat limit [work in progress].…”

Newton-Hooke/Carrollian expansions of (A)dS and Chern-Simons gravity

“…This method provides an infinite sequence of non-relativistic algebras extending the Galilei algebra with an increasing number of generators, which have been used in [9,10,8,11] to construct various gravitational actions. The Lie algebra expansion method can also be related to a sequence of post-Newtonian limits as shown in [12], and has also been applied to derive diverse non-relativistic symmetries in the context of (super-)gravity [13][14][15][16][17]. Another method is based on a Galilean free Lie algebra [18] that can be thought as the most general extension of the Galilei algebra and, upon taking quotients, has a connection to Lie algebra expansions and Kac-Moody algebras.…”

Three-dimensional higher-order Schrödinger algebras and Lie algebra expansions

“…where we identify the generators as together with our previous definitions (3.4). Similarly, the gauge fields identified as (4) θ µ =x 1µ ,…”

“…In particular, there are many different non-relativistic gravity theories that are based on the extensions of the Bargmann algebra that cannot be accessed from the Poincaré algebra by means of an Inonu-Wigner contraction. A wellknown example of this kind, called the three-dimensional extended Bargmann gravity [1], has been studied from various angles including its supersymmetric completion and matter couplings [2], its cosmological and non-relativistic conformal [3] and Maxwellian [4] extensions as well as the construction of its parity-odd cousin, the exotic extended Bargmann supergravity [5]. Another important example that goes beyond the Bargmann symmetries, which was put forward in [6,7], is based on the argument that although the Bargmann symmetries are sufficient to establish the Poisson's equation for Newtonian gravity in a covariant manner, an action principle for Newtonian gravity requires extended symmetries beyond the standard Bargmann symmetries.…”