Factorization constraints and boundary conditions in rational CFT

TL;DR

This paper reviews how rational conformal field theories construct correlators using modular tensor categories and Frobenius algebras, emphasizing boundary conditions and factorization constraints.

Contribution

It elucidates the role of Frobenius algebras and gluing homomorphisms in RCFT, linking boundary conditions to the algebra's representation theory.

Findings

Correlators satisfy all factorization constraints.

Boundary conditions are classified by the semisimple algebra A.

Annulus partition functions derive from A's representation theory.

Abstract

Among (conformal) quantum field theories, the rational conformal field theories are singled out by the fact that their correlators can be constructed from a modular tensor category C with a distinguished object, a symmetric special Frobenius algebra A in C, via the so-called TFT-construction. These correlators satisfy in particular all factorization constraints, which involve gluing homomorphisms relating correlators of world sheets of different topology. We review the action of the gluing homomorphisms and discuss the implications of the factorization constraints for boundary conditions. The so-called classifying algebra A for a RCFT is a semisimple commutative associative complex algebra, which classifies the boundary conditions of the theory. We show that the annulus partition functions can be obtained from the representation theory of A.