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From conformal field theory on the Riemann sphere, we implement its boundaryversion in a simply-connected domain using the Schottky double construction. Weconsider the statistical fields generated by background charge modification ofthe Gaussian free field with Dirichlet boundary condition under the OPEmultiplications. We prove that the correlation functions of such fields withsymmetric background charges form a collection of martingale-observables for(forward) chordal/radial SLE with force points and spins. We also present theconnection between conformal field theory with Neumann boundary condition andthe theory of backward SLE.

Conformal field theory on the Riemann sphere and its boundary version for SLE