Noncommutative $AdS^3$ with Quantized Cosmological Constant

TL;DR

This paper explores a noncommutative deformation of 3D anti-deSitter gravity, revealing a quantized cosmological constant and constructing noncommutative $AdS^3$ vacua using Seiberg-Witten maps.

Contribution

It introduces a noncommutative version of 3D gravity with gauge group $U(1,1) imes U(1,1)$ and demonstrates the quantization of the cosmological constant.

Findings

Cosmological constant is quantized as minus one over an integer squared.

Constructed noncommutative $AdS^3$ vacuum from the commutative case.

Extended the analysis to conical spaces from massive spinning particles.

Abstract

We examine a recent deformation of three-dimensional anti-deSitter gravity based on noncommutative Chern-Simons theory with gauge group $U(1,1)\times U(1,1)$. In addition to a noncommutative analogue of 3D gravity, the theory contains two addition gauge fields which decouple in the commutative limit. It is well known that the level is quantized in noncommutative Chern-Simons theory. Here it implies that the cosmological constant goes like minus one over an integer-squared. We construct the noncommutative $AdS^3$ vacuum by applying a Seiberg-Witten map from the commutative case. The procedure is repeated for the case of a conical space resulting from a massive spinning particle.