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

TL;DR

This work analyzes the modular properties of a simple non-rational CFT, the $SL(2,\mathbb{R})/U(1)$ coset, by introducing extended characters that sum over winding sectors to stabilize $U(1)$ charges under modular transformations. The authors embed the coset into an enlarged $N=2$ superconformal framework, derive the modular $S$ and $T$ matrices for both extended continuous and discrete characters, and show that discrete representations mix with continuous ones under modular transformations. They obtain explicit decompositions of the modular transformations and demonstrate that the resulting S-matrix satisfies charge-conjugation properties (including $S^2=C$), highlighting new features of non-rational CFT modularity. The results provide a non-rational analogue of string-function modular properties, with potential implications for modular invariants, boundary states, and dualities in non-compact CFTs.

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.

Figures (2)

• Figure 1: Choice of contour of integration (for $\tau_1>0$).
• Figure 2: Change of contour of integration (case $r>\frac{k \nu_2}{2}$).