A matrix S for all simple current extensions

TL;DR

This work constructs a general, unitary, and symmetric modular S-matrix for simple current extensions of rational conformal field theories by introducing fixed-point resolution through matrices S^J_{ab}. A Fourier-decomposition framework ties tilde S to S^J and group characters of untwisted stabilizers, enabling a universal formula for the extended theory’s S-matrix and ensuring modular-consistent Verlinde fusion when the S^J data satisfy specified conditions. The approach is concretely realized for WZW-models via twining characters and orbit Lie algebras, yielding explicit S^J matrices and clarifying their relation to diagram automorphisms and fixed points. While demonstrations are comprehensive, the paper notes remaining open questions about positivity, uniqueness, and a full conformal-field-theoretic construction of the extended algebras. Overall, the method provides a practical, unified route to compute modular data for a broad class of simple current invariants and coset models.

Abstract

A formula is presented for the modular transformation matrix S for any simple current extension of the chiral algebra of a conformal field theory. This provides in particular an algorithm for resolving arbitrary simple current fixed points, in such a way that the matrix S we obtain is unitary and symmetric and furnishes a modular group representation. The formalism works in principle for any conformal field theory. A crucial ingredient is a set of matrices S^J_{ab}, where J is a simple current and a and b are fixed points of J. We expect that these input matrices realize the modular group for the torus one-point functions of the simple currents. In the case of WZW-models these matrices can be identified with the S-matrices of the orbit Lie algebras that we introduced in a previous paper. As a special case of our conjecture we obtain the modular matrix S for WZW-theories based on group manifolds that are not simply connected, as well as for most coset models.