Non-relativistic symmetries in three space-time dimensions and the Nappi-Witten algebra

Abstract

We show that the Extended Bargmann and Newton-Hooke algebras in 2+1 dimensions can be obtained as expansions of the Nappi-Witten algebra. The result can be generalized to obtain two infinite families of non-relativistic symmetries, which include the Maxwellian Exotic Bargmann symmetry, its generalized Newton-Hooke counterpart, and its Hietarinta dual. In each case, the invariant bilinear form on the Nappi-Witten algebra leads to the invariant tensor on the expanded algebra, allowing one to construct the corresponding Chern-Simons gravity theory.

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.…”

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 exotic Newtonian gravity with cosmological constant

“…Nevertheless, it would be interesting to recover it by a possible non-relativistic limit or contraction procedure of a relativistic theory. On the other hand, the algebra expansion procedure introduced in [43] and subsequently defined using Maurer-Cartan forms [44,45] and semigroups [46] have not long ago turned out to be useful to get diverse non-relativistic (super)algebras [47][48][49][50][51]. It would be worth it to explore the possibility to obtain the extended Newtonian algebra and the exotic Newtonian one presented here using the expansion procedure [work in progress].…”