UC Berkeley

This thesis is primarily concerned with the construction and analysis of free fermion Segal CFTs in arbitrary genus, with emphasis placed on analytic aspects of the construction. In particular, our Segal CFTs take values in the "category" of Hilbert spaces and trace class maps. The main results are:

* A detailed construction of the free fermion Segal CFT, in arbitrary genus, using Hilbert spaces and trace class operators (Chapter 4),

* An explicit identification of the operators assigned to three-punctured spheres with the free fermion vertex operator algebra (Chapter 5.1),

* Preliminary construction of (quotients of) tensor products of $SU(k)_\ell$ WZW vertex modules via descent from fermions. (Chapter 5.2).

As a technical tool, we will develop a Riemann surface generalization of the Cauchy transform for planar domains (Appendix A).