Boundary states and broken bulk symmetries in W A(r) minimal models

TL;DR

This paper analyzes boundary states in W symmetry rational conformal field theories, correcting the classification of primary fields, constructing boundary states, and exploring W violating sectors with Verlinde-like formulas.

Contribution

It provides a complete classification of primary fields, constructs boundary states explicitly, and extends the Verlinde formula to W violating sectors in these models.

Findings

Corrected the classification of primary fields for these models.

Constructed boundary states as coherent states satisfying boundary conditions.

Identified W violating boundary states and their Verlinde-like structure.

Abstract

We study the boundary states of (p', p) rational conformal field theories having a W symmetry of the type A(r) using the multi-component free-field formalism. The classification of primary fields for these models given in the literature is shown to be incomplete; we give the correct classification by demanding modular covariance and show that the resulting modular S matrix satisfies all the necessary conditions. Basis states satisfying the boundary conditions are found in the form of coherent states and as expected we find that W violating states can be found for all these models. We construct consistent physical boundary for all the rank 2 (p+1, p) models (of which the already known case of the 3-state Potts model is the simplest example) and find that the W violating sector possesses a direct analogue of the Verlinde formula.