Abstract
We investigate the deformations and rigidity of boundary Heisenberg-like algebras. In particular, we focus on the Heisenberg and Heisenberg ⊕ 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 Heisenberg ⊕ mathfrak{witt} and the asymptotic symmetry algebra Weyl-bms3 are not connected via single deformation but in a more subtle way.
Highlights
We focus on the Heisenberg and Heisenberg ⊕ witt algebras which arise as symmetry algebras in three-dimensional gravity theories
As a byproduct of our analysis, we obtain that Heisenberg ⊕ witt and the asymptotic symmetry algebra Weyl-bms3 are not connected via single deformation but in a more subtle way
It has been shown that certain near horizon and asymptotic symmetry algebras of three-dimensional and four-dimensional asymptotically flat and Friedman spacetimes form part of the same multi-parametric families of deformation algebras, denoted as W -algebras1 [9, 10, 18, 19]. Another motivation for studying deformations of infinite dimensional algebras is that they provide us with a path to construct new algebras, which can possibly be realized as symmetry algebras under new boundary conditions and in different geometric settings
Summary
In the recent years, pioneering analysis of asymptotically flat spacetimes performed by Bondi, van der Burg, Metzner and Sachs (BMS) [1–3] has been refined and extended to many other dimensions, spacetimes and boundaries. The structure of spacetime near generic null surfaces (including event and cosmological horizons), but mostly in three spacetime dimensions, has been intensively investigated in the last few years [4–7] In this context, boundary Heisenberg algebras have played a major role all the way through and constitute a fundamental piece behind the different symmetry algebras popping up in a variety of boundary symmetry analysis. It has been shown that certain near horizon and asymptotic symmetry algebras of three-dimensional and four-dimensional asymptotically flat and Friedman spacetimes form part of the same multi-parametric families of deformation algebras, denoted as W -algebras1 [9, 10, 18, 19] Another motivation for studying deformations of infinite dimensional algebras is that they provide us with a path to construct new algebras, which can possibly be realized as symmetry algebras under new boundary conditions and in different geometric settings. We will be using “W (a, b) family” of algebras to denote a set of algebras for different values of the a, b parameters
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