Radial SLE martingale-observables

Co-author: Nikolai Makarov (Caltech)
After implementing a version of radial conformal field theory in the OPE family of statistical fields generated by background charge modification of the Gaussian free field, I present an analytical and probabilistic proof of a well-known statement in physics that the correlation functions of such fields under the insertion of one-leg operator form a collection of radial SLE martingale-observables. In the construction of one-leg operator, the so-called neutrality conditions on the charges play an important role. I explain two neutrality conditions: first, the linear combination of bosonic fields is to be a well-defined Fock space field; second, the Coulomb gas correlation function is to be conformally invariant. To reconcile these two neutrality conditions, one needs to place the background charge at the marked interior point, the target point of SLE. This is a joint work with Nikolai Makarov.