On generalizations of asymptotically AdS_3 spaces and geometry of SL(N)

TL;DR

This paper explores the geometric and symmetry properties of asymptotically AdS spaces, focusing on the Lie group structures of SL(2), SL(3), and SL(N), and aims to understand their boundary theories.

Contribution

It revisits AdS_3 geometry from a Lie group perspective and sets the groundwork for analyzing higher SL(N) spaces and their asymptotic symmetries.

Findings

Identifies the boundary energy-momentum tensor for AdS_3

Shows the infinite-dimensional symmetry groups relate to SL(N) structures

Prepares methods for analyzing SL(3) and SL(N) cases

Abstract

In three and two dimensions the asymptotic symmetry groups of $AdS$ spaces are infinite dimensional. This can be explained easily by noting the relations $AdS_3 \simeq SL(2)$ and $AdS_2 \simeq SL(2)/SO(2)$, i.e. that the asymptotic symmetries are in fact that of the Lie group SL(2). As show in the author's previous work, similar infinite dimensional asymptotic symmetry groups can be found in the case of SL(3) and probably also for other noncompact Lie groups and their homogeneous spaces. The purpose of the present work is to revisit the $AdS_3$ space in detail from the Lie group point of view by finding the boundary theory energy-momentum tensor and to prepare to tackle the SL(3) and SL(N) cases.