Deformation of Asymptotic Symmetry Algebras and Their Physical Realizations

TL;DR

This thesis investigates the deformation and classification of infinite-dimensional symmetry algebras relevant to gravity theories, revealing their non-rigidity and proposing a new extension of a classical theorem for such algebras.

Contribution

It introduces a comprehensive analysis of deformations of asymptotic symmetry algebras, classifies the resulting algebras, and extends the Hochschild-Serre theorem to infinite-dimensional cases.

Findings

Algebras considered are not rigid and can be deformed into new structures.

Classified all deformed algebras and their central extensions.

Proposed a conjecture extending Hochschild-Serre theorem for infinite-dimensional algebras.

Abstract

This thesis is devoted to the study of the deformation and rigidity of infinite dimensional Lie algebras which are not subject to the Hochschild-Serre factorization theorem. In particular, we consider $bms_{3}$, Virasoro-Kac-Moody and $bms_{4}$ algebras and their central extensions which are respectively obtained as asymptotic and/or near horizon symmetry algebras for Einstein gravity on $3d$ flat, AdS$_{3}$ and $4d$ flat spacetimes. We also explore possible deformations of the Maxwell-BMS algebra, which is obtained as asymptotic symmetry algebra of the Chern-Simons gravity theory invariant under the $2+1$ dimensional Maxwell algebra. We find that these algebras are not rigid and can be deformed into new non isomorphic infinite dimensional (family of) algebras. We study these deformations by direct computations and also by cohomological analysis. We then classify all the algebras obtained through deformation of these algebras as well as all possible central extensions thereof. We propose/conjecture an extension of the Hochschild-Serre factorization theorem for infinite dimensional algebras as well as introducing a new notion of rigidity for family algebras obtained through deformation. We also explore physical realizations and significance of the family of algebras we obtain through the deformation procedure.