Nam-Gyu Kang: Gaussian Free Field, Conformal Field Theory, and Schramm–Loewner Evolution

I will present an elementary introduction to conformal field theory in the context of complex analysis and probability theory. Introducing Ward functional as an insertion operator under which the correlation functions are transformed into their Lie derivatives, I will explain several formulas in conformal field theory including Ward's equations. This presentation will also include relations between conformal field theory and Schramm–Loewner evolutions. For these relations, the so-called neutrality conditions of vertex fields 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 joint work with Nikolai Makarov.