Free boson formulation of boundary states in W_3 minimal models and the critical Potts model

TL;DR

This paper develops a Coulomb gas formalism for boundary conformal field theories with W symmetry, providing a complete free-field realization of boundary states in the three-state Potts model and calculating associated correlation functions.

Contribution

It introduces a Coulomb gas approach to boundary W-symmetric conformal field theories, explicitly constructing all boundary states for the three-state Potts model.

Findings

Constructed six W-conserving boundary states corresponding to physical conditions.

Identified two W-violating boundary states with free and new boundary conditions.

Calculated boundary correlation functions matching differential equation solutions.

Abstract

We develop a Coulomb gas formalism for boundary conformal field theory having a $W$ symmetry and illustrate its operation using the three state Potts model. We find that there are free-field representations for six $W$ conserving boundary states, which yield the fixed and mixed physical boundary conditions, and two $W$ violating boundary states which yield the free and new boundary conditions. Other $W$ violating boundary states can be constructed but they decouple from the rest of the theory. Thus we have a complete free-field realization of the known boundary states of the three state Potts model. We then use the formalism to calculate boundary correlation functions in various cases. We find that the conformal blocks arising when the two point function of $\phi_{2,3}$ is calculated in the presence of free and new boundary conditions are indeed the last two solutions of the sixth order differential equation generated by the singular vector.