Modular invariance of string theory on AdS_3

TL;DR

This paper investigates the modular invariance properties of the SL(2,R) WZW model in string theory, analyzing characters based on discrete series representations and their implications for unitarity and ghost issues.

Contribution

It provides a detailed analysis of modular invariants using sl(2,R) characters, showing conditions for their existence and limitations related to unitarity bounds.

Findings

Modular invariants exist only for level k<2 with large spins included.

Explicit form of modular invariant with three variables Z(z,, u) is provided.

Discrete series characters alone are insufficient for unitarity-compatible invariants.

Abstract

We discuss the modular invariance of the SL(2,R) WZW model. In particular, we discuss in detail the modular invariants using the \hat{sl}(2,R) characters based on the discrete unitary series of the SL(2,R) representations. The explicit forms of the corresponding characters are known when no singular vectors appear. We show, for example, that from such characters modular invariants can be obtained only when the level k < 2 and infinitely large spins are included. In fact, we give a modular invariant with three variables Z(z,\tau, u) in this case. We also argue that the discrete series characters are not sufficient to construct a modular invariant compatible with the unitarity bound, which was proposed to resolve the ghost problem of the SL(2,R) strings.