Boundary Heisenberg Algebras and Their Deformations

TL;DR

This paper explores the deformations of boundary Heisenberg-like algebras in three-dimensional gravity, revealing new algebraic structures and their interconnections, which enhances understanding of symmetry algebras in gravitational theories.

Contribution

It identifies a large class of deformed algebras, including new ones, and investigates their relationships, showing connections beyond traditional constructions like Sugawara.

Findings

Discovered new algebraic structures through deformation.

Established relationships between boundary symmetry algebras.

Showed certain algebras are not connected via single deformation.

Abstract

We investigate the deformations and rigidity of boundary Heisenberg-like algebras. In particular, we focus on the Heisenberg and $\text{Heisenberg}\oplus\mathfrak{witt}$ algebras which arise as symmetry algebras in three-dimensional gravity theories. As a result of the deformation procedure we find a large class of algebras. While some of these algebras are new, some of them have already been obtained as asymptotic and boundary symmetry algebras, supporting the idea that symmetry algebras associated to diverse boundary conditions and spacetime loci are algebraically interconnected through deformation of algebras. The deformation/contraction relationships between the new algebras are investigated. In addition, it is also shown that the deformation procedure reaches new algebras inaccessible to the Sugawara construction. As a byproduct of our analysis, we obtain that $\text{Heisenberg}\oplus\mathfrak{witt}$ and the asymptotic symmetry algebra Weyl-$\mathfrak{bms}_3$ are not connected via single deformation but in a more subtle way.