Completeness of boundary conditions for the critical three-state Potts model

TL;DR

This paper demonstrates that all conformally invariant boundary conditions for the three-state Potts model are known and explores their structure, revealing connections to free field theories and string theory concepts.

Contribution

It shows that the known eight boundary conditions are complete for the three-state Potts model and relates them to higher-spin currents and fusion rules, extending to minimal models.

Findings

All boundary conditions are exhausted by eight solutions.

Fixed and mixed conditions correspond to Neumann conditions.

Free and new boundary conditions relate to Dirichlet conditions for higher-spin currents.

Abstract

We show that the conformally invariant boundary conditions for the three-state Potts model are exhausted by the eight known solutions. Their structure is seen to be similar to the one in a free field theory that leads to the existence of D-branes in string theory. Specifically, the fixed and mixed boundary conditions correspond to Neumann conditions, while the free boundary condition and the new one recently found by Affleck et al [1] have a natural interpretation as Dirichlet conditions for a higher-spin current. The latter two conditions are governed by the Lee\hy Yang fusion rules. These results can be generalized to an infinite series of non-diagonal minimal models, and beyond.