SLE($\kappa ,\rho$)and Boundary Coulomb Gas

We consider the coulomb gas model on the upper half plane with different boundary conditions, namely Drichlet, Neuman and mixed. We related this model to SLE($\kappa ,\rho$) theories. We derive a set of conditions connecting the total charge of the coulomb gas, the boundary charges, the parameters $\kappa$ and $\rho$. 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.