Construction of the Unitary Free Fermion Segal CFT

In this article, we provide a detailed construction and analysis of the mathematical conformal field theory of the free fermion, defined in the sense of Graeme Segal. We verify directly that the operators assigned to disks with two disks removed correspond to vertex operators, and use this to deduce analytic properties of the vertex operators. One of the main tools used in the construction is the Cauchy transform for Riemann surfaces, for which we establish several properties analogous to those of the classical Cauchy transform in the complex plane.