The Fock-Schwartz spin representation space

Abstract

In this thesis, we define and study a family of Sobolev-like subspaces (the “FockSobolev spaces”) and the corresponding Schwartz-like space (the “Fock-Schwartz space”) arising from the infinite-dimensional spin representation constructed by Pressley and Segal. In particular, we study the infinitesimal actions of the group of orientation-preserving diffeomorphisms, Diff⁺(𝑆¹), and the loop group 𝒧Spin(2𝑛), as well as the action of an infinite-dimensional Clifford algebra on the Fock-Sobolev spaces and Fock-Schwartz space. All of this work is motivated by the goal of constructing the Dirac-Ramond operator on the loop space of a string manifold.