SLE(κ,ρκ,ρ) and boundary Coulomb gas

We consider the Coulomb gas model on the upper half-plane with different boundary conditions, namely, Dirichlet, Neumann and mixed. We relate this model to SLE(κ,ρκ,ρ) theories. We derive a set of conditions connecting the total charge of the Coulomb gas, the boundary charges, the parameters κ and ρ. Also we study a free fermion theory in presence of a boundary and show with the same methods that it would lead to logarithmic boundary changing operators.