A classifying algebra for boundary conditions

TL;DR

This paper introduces a finite-dimensional algebra that classifies boundary conditions in conformal field theories, especially those with Z_2 symmetry, linking twisted sector structures to modular invariance.

Contribution

It presents a new algebraic framework for understanding boundary conditions in conformal field theories with specific symmetries, including twisted sectors.

Findings

The algebra controls boundary conditions in conformal field theories.

It incorporates the fusion algebra of the untwisted sector.

It reveals structures related to modular invariance in twisted sectors.

Abstract

We introduce a finite-dimensional algebra that controls the possible boundary conditions of a conformal field theory. For theories that are obtained by modding out a Z_2 symmetry (corresponding to a so-called D_odd-type, or half-integer spin simple current, modular invariant), this classifying algebra contains the fusion algebra of the untwisted sector as a subalgebra. Proper treatment of fields in the twisted sector, so-called fixed points, leads to structures that are intriguingly close to the ones implied by modular invariance for conformal field theories on closed orientable surfaces.