Extended SL(2,R)/U(1) characters, or modular properties of a simple non-rational conformal field theory

TL;DR

This paper introduces extended characters for the SL(2,R)/U(1) conformal field theory, demonstrating their modular properties and explicit modular matrices, revealing mixing of discrete and continuous representations under modular transformations.

Contribution

It defines extended SL(2,R)/U(1) characters with winding sectors and embeds them into N=2 algebra characters, showing their modular transformation properties and calculating the modular matrices explicitly.

Findings

Discrete and continuous SL(2,R) representations mix under modular transformations.

Extended characters have well-behaved modular properties.

Explicit modular matrices for the non-rational CFT are obtained.

Abstract

We define extended SL(2,R)/U(1) characters which include a sum over winding sectors. By embedding these characters into similarly extended characters of N=2 algebras, we show that they have nice modular transformation properties. We calculate the modular matrices of this simple but non-trivial non-rational conformal field theory explicitly . As a result, we show that discrete SL(2,R) representations mix with continuous SL(2,R) representations under modular transformations in the coset conformal field theory. We comment upon the significance of our results for a general theory of non-rational conformal field theories.