Boundary Conditions in Rational Conformal Field Theories

TL;DR

<3-5 sentence high-level summary>: The paper develops boundary conformal field theory for rational theories by foregrounding a triplet of algebras—the Verlinde, graph fusion, and Pasquier algebras—and shows that the Cardy equation on a cylinder is equivalent to finding integer-valued representations of the Verlinde fusion algebra. It naturalizes a graph-based labeling of boundary conditions, yielding complete ADE-type boundary data for $\widehat{sl}(2)$ theories and clarifying the status for $\widehat{sl}(3)$; it also generalizes Cardy-Lewellen sewing to multiplicities, linking bulk-boundary coefficients to modular data. The work unifies boundary and bulk structures, deriving duality relations (Moore-Seiberg) in both diagonal and non-diagonal cases and revealing how boundary data encode torus partition functions and extended algebras. The resulting framework paves the way for systematic graph-based classifications of RCFTs, with potential extensions to higher rank algebras, twisted sectors, and non-unitary theories, and connects to subfactor and operator-algebra approaches for a broader mathematical understanding.

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.

Figures (14)

• Figure 1: The two computations of the partition function $Z_{b|a}$: (a) on the cylinder, between the boundary states $a$ and $b$, (b) as a periodic time evolution on the strip, with boundary conditions $a$ and $b$.
• Figure 2: The $A$-$D$-$E$-$T$ graphs.
• Figure 3: The torus partition function reconstructed from two cylinder partition functions.
• Figure 4: Graphical representation of (\ref{['moorese']}). To stress the presence of two types of vertices, we distinguish them explicitly on this particular figure only
• Figure 5: (a) and (b): boundary field 2- and 3-point functions