Integrable Boundaries, Conformal Boundary Conditions and A-D-E Fusion Rules

TL;DR

This paper explores the classification of conformal boundary conditions in $sl(2)$ minimal theories, linking them to A-D-E fusion rules and integrable boundary conditions via the boundary Yang-Baxter equation, exemplified by the 3-state Potts model.

Contribution

It proposes a complete set of conformal boundary conditions labeled by tensor product graph nodes and connects them to integrable boundary conditions through the boundary Yang-Baxter equation.

Findings

Boundary conditions labeled by nodes of $A\otimes G$ graph.

Partition functions derived from graph fusion algebra.

Existence of integrable boundary conditions confirmed for models like the 3-state Potts.

Abstract

The $sl(2)$ minimal theories are labelled by a Lie algebra pair $(A,G)$ where $G$ is of $A$-$D$-$E$ type. For these theories on a cylinder we conjecture a complete set of conformal boundary conditions labelled by the nodes of the tensor product graph $A\otimes G$. The cylinder partition functions are given by fusion rules arising from the graph fusion algebra of $A\otimes G$. We further conjecture that, for each conformal boundary condition, an integrable boundary condition exists as a solution of the boundary Yang-Baxter equation for the associated lattice model. The theory is illustrated using the $(A_4,D_4)$ or 3-state Potts model.