On Rigidity of 3d Asymptotic Symmetry Algebras

TL;DR

The paper investigates rigidity and deformations of infinite-dimensional asymptotic symmetry algebras in 3d spacetimes, focusing on BMS3, its central extension, and nearby KM/U(1) structures. Using explicit cocycle calculations and cohomological tools (including Hochschild-Serre sequences) it classifies infinitesimal and formal deformations, finding that deformations of the centerless BMS3 algebra yield the W(a,b) family and the Witt sum, while central extensions modify integrability, leading to the rigid Virasoro direct sums. The analysis shows that the BS3 and KM_u(1) algebras are not rigid in general; their stabilized forms reveal a landscape of rigid deformations tied to W(a,b) and Witt⊕Witt (or Vir⊕Vir) structures, with central terms playing a crucial role. The results have physical implications for holography and the interpretation of deformation versus contraction in 3d gravity, potentially connecting asymptotic symmetries to RG-flow-like behavior in dual theories.

Abstract

We study rigidity and stability of infinite dimensional algebras which are not subject to the Hochschild-Serre factorization theorem. In particular, we consider algebras appearing as asymptotic symmetries of three dimensional spacetimes, the BMS3, u(1) Kac-Moody and Virasoro algebras. We construct and classify the family of algebras which appear as deformations of BMS3, u(1) Kac-Moody and their central extensions by direct computations and also by cohomological analysis. The Virasoro algebra appears as a specific member in this family of rigid algebras; for this case stabilization procedure is inverse of the Inönü-Wigner contraction relating Virasoro to BMS3 algebra. We comment on the physical meaning of deformation and stabilization of these algebras and relevance of the family of rigid algebras we obtain

Figures (1)

• Figure 1: Parametric space of formal deformations of $\mathfrak{bms}_{3}$ and $\mathfrak{KM}_{\mathfrak{u}(1)}$. $W(a,b)$ algebra appears as the formal deformation of both $\mathfrak{bms}_{3}$ and $\mathfrak{KM}_{\mathfrak{u}(1)}$. As discussed below \ref{['Fourier-expand-a']} parameter $a$ ranges in $[0,1/2]$ while $b,\nu$ can be any real number. Besides the $W(a,b)$, $\mathfrak{KM}_{\mathfrak{u}(1)}$ can be deformed to $\mathfrak{KM}(a,\nu)$ which is spanned by $b=0$ plane. The $\mathfrak{bms}_3$ algebra can be deformed to $\mathfrak{witt}\oplus \mathfrak{witt}$ which again is out of the $(a,b,\nu)$ space displayed above.

Theorems & Definitions (3)

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