Star products on extended massive non-rotating BTZ black holes

TL;DR

This paper constructs non-commutative deformations of D-string worldsheet algebras in BTZ black hole backgrounds using star products, and explores their geometric and spectral properties.

Contribution

It introduces new star products invariant under a subgroup of SL(2,R) on BTZ black hole spacetimes and initiates the development of spectral triples in non-commutative AdS contexts.

Findings

Constructed non-formal star products on D-string worldsheets.

Identified two maximal extensions of the BTZ spacetime.

Initiated the framework for spectral triples in non-commutative AdS.

Abstract

$AdS_3$ space-time admits a foliation by two-dimensional twisted conjugacy classes, stable under the identification subgroup yielding the non-rotating massive BTZ black hole. Each leaf constitutes a classical solution of the space-time Dirac-Born-Infeld action, describing an open D-string in $AdS_3$ or a D-string winding around the black hole. We first describe two nonequivalent maximal extensions of the non-rotating massive BTZ space-time and observe that in one of them, each D-string worldsheet admits an action of a two-parameter subgroup ($\ca \cn$) of $\SL$. We then construct non-formal, $\ca \cn$-invariant, star products that deform the classical algebra of functions on the D-string worldsheets and on their embedding space-times. We end by giving the first elements towards the definition of a Connes spectral triple on non-commutative $AdS$ space-times.