String Theory on AdS3: Some Open Questions

TL;DR

The paper surveys string theory on $AdS_3$ through the $SU(1,1)$ WZW framework and its coupling to 2D gravity, highlighting fundamental open questions about a consistent spectrum on a non-compact current algebra. It analyzes the affine $SU(1,1)$ structure, its representations (discrete ${\cal D}^{\pm}(j)$ and continuous ${\cal C}_{\rm p}(b,a)$, ${\cal C}_{\rm s}(j,a)$) and the issues of unitarity and modular invariance, showing that naive constructions yield ghost states or tachyons. The discussion reveals a delicate tension between unitarity constraints (e.g., $k/2 \le j<0$ leading to $N_{ m max}$ bounds) and modular invariance, suggesting the need for boundary-condition refinements, path-integral treatments, or current-algebra deformations. These insights bear on ADS/CFT, 3D black-hole physics, and the broader quest to understand string dynamics in curved backgrounds beyond flat space.

Abstract

String theory on curved backgrounds has received much attention on account of both its own interest, and of its relation with gauge theories. Despite the progress made in various directions, several quite elementary questions remain unanswered, in particular in the very simple case of three-dimensional anti-de Sitter space. I will very briefly review these problems, discuss in some detail the important issue of constructing a consistent spectrum for a string propagating on ADS3 plus torsion background, and comment on potential solutions.