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

TL;DR

The paper addresses the problem of classifying conformal boundary conditions for $\,sl(2)$ minimal theories, including non-diagonal cases. It introduces a unified framework based on pairs of simply laced Dynkin diagrams $(A,G)$ and the tensor-product graph $A\otimes G$, yielding boundary operator fusion ${\cal N}_{i_1 i_2}{}^{i_3}=N_{r_1 r_2}{}^{r_3}\hat{N}_{a_1 a_2}{}^{a_3}$ and cylinder functions $Z_{i_1|i_2}(q)$ from the boundary fusion algebra. The framework is demonstrated for the critical three-state Potts model, i.e., the folded $(A_4,D_4)$ theory, where twelve boundary sectors and their Virasoro characters are worked out, together with explicit boundary weights for the associated lattice model. This work links conformal boundary data to integrable lattice boundary conditions via the boundary Yang-Baxter equation and suggests extensions to higher rank families, with a more complete treatment to appear.

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.

Figures (1)

• Figure 1: Folding and orbifold duality relating the tensor product graph $T_2\otimes D_4$ to $A_4\otimes D_4$ and $A_4\otimes A_5$. The conformal weights of the 8 conformal boundary conditions of the 3-state Potts model appear in the boxes of the $T_2\otimes D_4$ theory.