Free boson formulation of boundary states inW3minimal models and the critical Potts model

We develop a Coulomb gas formalism for boundary conformal field theory havinga $W$ symmetry and illustrate its operation using the three state Potts model.We find that there are free-field representations for six $W$ conservingboundary states, which yield the fixed and mixed physical boundary conditions,and two $W$ violating boundary states which yield the free and new boundaryconditions. Other $W$ violating boundary states can be constructed but theydecouple from the rest of the theory. Thus we have a complete free-fieldrealization of the known boundary states of the three state Potts model. Wethen use the formalism to calculate boundary correlation functions in variouscases. 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 conditionsare indeed the last two solutions of the sixth order differential equationgenerated by the singular vector