A matrix S for all simple current extensions

TL;DR

This paper introduces a formula for the modular S matrix in simple current extensions of conformal field theories, providing an algorithm to resolve fixed points and ensuring unitarity and symmetry of the matrix.

Contribution

It presents a general formula for the S matrix in simple current extensions, including an algorithm for fixed point resolution applicable to any conformal field theory.

Findings

Provides a universal formula for the S matrix in simple current extensions.

Ensures the S matrix is unitary and symmetric, forming a modular group representation.

Specializes to WZW models and coset models, matching known results.

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.