Twisted boundary states and representation of generalized fusion algebra

TL;DR

This paper explores twisted boundary states in rational conformal field theories, demonstrating they form non-negative integer matrix representations of a generalized fusion algebra, extending the understanding of boundary conditions and fusion structures.

Contribution

It introduces the concept of a generalized fusion algebra incorporating automorphisms, and shows how twisted boundary states realize NIM-reps of this algebra, with concrete examples and non-commutative cases.

Findings

Twisted boundary states form NIM-reps of the generalized fusion algebra.

The generalized fusion algebra can be non-commutative for non-abelian automorphism groups.

Graph fusion algebras for simple current extensions match the generalized fusion algebra.

Abstract

The mutual consistency of boundary conditions twisted by an automorphism group G of the chiral algebra is studied for general modular invariants of rational conformal field theories. We show that a consistent set of twisted boundary states associated with any modular invariant realizes a non-negative integer matrix representation (NIM-rep) of the generalized fusion algebra, an extension of the fusion algebra by representations of the twisted chiral algebra associated with the automorphism group G. We check this result for several concrete cases. In particular, we find that two NIM-reps of the fusion algebra for $su(3)_k (k=3,5)$ are organized into a NIM-rep of the generalized fusion algebra for the charge-conjugation automorphism of $su(3)_k$. We point out that the generalized fusion algebra is non-commutative if G is non-abelian and provide some examples for $G = S_3$. Finally, we give an argument that the graph fusion algebra associated with simple current extensions coincides with the generalized fusion algebra for the extended chiral algebra, and thereby explain that the graph fusion algebra contains the fusion algebra of the extended theory as a subalgebra.