Boundary Conditions in Rational Conformal Field Theories

TL;DR

This paper advances the understanding of boundary conditions in Rational Conformal Field Theories by linking algebraic structures to graph classifications, providing complete solutions for $sl(2)$ theories and extending the formalism to more general cases.

Contribution

It introduces a new algebraic framework connecting boundary conditions to graph theory and generalizes Cardy-Lewellen formalism for complex representations in RCFTs.

Findings

Complete classification of boundary conditions for $sl(2)$ theories

Association of graphs to RCFTs via Verlinde algebra matrices

Extension of formalism to include multiplicities and general representations

Abstract

We develop further the theory of Rational Conformal Field Theories (RCFTs) on a cylinder with specified boundary conditions emphasizing the role of a triplet of algebras: the Verlinde, graph fusion and Pasquier algebras. We show that solving Cardy's equation, expressing consistency of a RCFT on a cylinder, is equivalent to finding integer valued matrix representations of the Verlinde algebra. These matrices allow us to naturally associate a graph $G$ to each RCFT such that the conformal boundary conditions are labelled by the nodes of $G$. This approach is carried to completion for $sl(2)$ theories leading to complete sets of conformal boundary conditions, their associated cylinder partition functions and the $A$-$D$-$E$ classification. We also review the current status for WZW $sl(3)$ theories. Finally, a systematic generalization of the formalism of Cardy-Lewellen is developed to allow for multiplicities arising from more general representations of the Verlinde algebra. We obtain information on the bulk-boundary coefficients and reproduce the relevant algebraic structures from the sewing constraints.