A generalized action for (2 + 1)-dimensional Chern–Simons gravity

Abstract

We show that the so-called semi-simple extended Poincaré (SSEP) algebra in D dimensions can be obtained from the anti-de Sitter algebra so (D − 1, 2) by means of the S-expansion procedure with an appropriate semigroup S. A general prescription is given for computing Casimir operators for S-expanded algebras, and the method is exemplified for the SSEP algebra. The S-expansion method also allows us to extract the corresponding invariant tensor for the SSEP algebra, which is a key ingredient in the construction of a generalized action for Chern-Simons gravity in 2 + 1

On stabilization of Maxwell-BMS algebra

“…The AdS-Lorentz symmetry has been studied in [67,77,95,102] and can be seen as a deformation of the Maxwell algebra. Further extensions of the AdS-Lorentz algebra in higher dimensions have been studied in [103][104][105] in order to recover the pure Lovelock theory.…”

On the supersymmetric extension of asymptotic symmetries in three spacetime dimensions

“…Indeed, the set of generators J m , P m , Z m , T a 0 , B a 0 , Z a 0 , G i,± r , H i,± r with m, n = 0, ±1 and r = ± 1 2 reproduces the N = 4 Maxwell superalgebra. It is worth it to mention that the N = 4 Maxwell superalgebra can also be obtained by applying an S-expansion to the N = 4 super Lorentz algebra considering the same semigroup S gravity theory A semi-simple enlargement of the Poincaré symmetry has been introduced in [35][36][37][38] which can be seen as the direct sum of the Lorentz and AdS algebra. In three spacetime dimensions, the so-called AdS-Lorentz algebra can be written in the basis {J a , P a , Z a } whose generators satisfy…”

“…Such limit can be reproduced not only at the bosonic level but also at the supersymmetric [53,76,93], non-relativistic [42,94], higher-spin [95] and asymptotic level [34]. The three-dimensional CS gravity action invariant under the AdS-Lorentz algebra (4.28) reads [28,34,37,42,96] where, R a = dω a + 1 2 ǫ abc ω b ω c is the Lorentz curvature two-form, T a = D ω e a is the torsion two-form and F a = D ω σ a + 1 2ℓ 2 ǫ abc σ b σ c is the curvature two-form related to σ a . An explicit realisation of the asymptotic symmetry at null infinity was presented in [34] and turned out to be a semi-simple enlargement of the BM S 3 algebra: Such enlarged BM S 3 algebra results to be isomorphic to three copies of the Virasoro algebra.…”

“…Subsequently, a semi-simple enlargement of the BM S 3 algebra has been introduced in [34] corresponding to the asymptotic symmetry of a gravity theory invariant under the so-called AdS-Lorentz algebra which can be seen as a semi-simple enlargement of the Poincaré algebra. The AdS-Lorentz symmetry has been introduced in [35][36][37][38] and has lead to diverse applications in the context of Lovelock gravity [39][40][41] and non-relativistic gravity theory [42]. An interesting feature of the enlarged BM S 3 algebra obtained in [34] is the connection to the deformed BM S 3 algebra through the flat limit ℓ → ∞.…”

Resonant algebras in Chern-Simons model of topological insulators

“…This algebra was introduced in the 1970's [12,13] and is defined by generators {J ab , P a , Z ab }. It was thoroughly analyzed in a wide range of contexts (see [14,15,16,17,18,19,20,21,22], along with many further generalizations [23,24,25,26,27,29]). Its novelty concerns the commutator of two translations, which takes the form…”