On modular properties of the AdS3 CFT

TL;DR

This paper investigates the modular properties of the AdS3 WZNW model, focusing on the Lorentzian torus characters, and derives a generalized S matrix that relates to fusion rules and boundary states.

Contribution

It introduces a generalized S matrix for the Lorentzian torus characters of the AdS3 WZNW model, linking modular transformations with fusion algebra and boundary states.

Findings

Derived a generalized S matrix with two diagonal and one off-diagonal block.

Connected the S matrix to fusion rules and boundary states.

Constructed Ishibashi states for D-branes and related one-point functions.

Abstract

We study modular properties of the AdS3 WZNW model. Although the Euclidean partition function is modular invariant, the characters on the Euclidean torus are ill-defined and their modular transformations are unknown. We reconsider the characters defined on the Lorentzian torus, focusing on their structure as distributions. We find a generalized S matrix, depending on the sign of the real modular parameters, which has two diagonal blocks and one off-diagonal block, mixing discrete and continuous representations, that we fully determine. We then explore the relations among the modular transformations, the fusion algebra and the boundary states. We explicitly construct Ishibashi states for the maximally symmetric D-branes and show that the generalized S matrix defines the one-point functions associated to point-like and H2 branes as well as the fusion rules of the degenerate representations of SL(2,R) appearing in the open string spectrum of the point-like D-branes, through a generalized Verlinde theorem.