Construction of the unitary free fermion Segal CFT

TL;DR

This paper constructs and analyzes the mathematical conformal field theory of free fermions following Segal's framework, focusing on vertex operators and their analytic properties using Cauchy transforms on Riemann surfaces.

Contribution

It provides a detailed construction of the free fermion Segal CFT, verifying vertex operator properties and establishing new analytic tools for Riemann surfaces.

Findings

Verification of vertex operators for disks with two disks removed

Establishment of properties of the Cauchy transform on Riemann surfaces

Analytic characterization of free fermion conformal field theory

Abstract

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.