Categories of Lagrangian Correspondences in Super Hilbert Spaces and Fermionic Functorial Field Theory

TL;DR

This paper develops a category of Lagrangian correspondences in super Hilbert spaces, establishing a composition law and constructing a functorial field theory related to free fermionic particles in curved spacetime.

Contribution

It introduces a composition framework for Lagrangian correspondences respecting polarizations and constructs a functorial field theory in this setting.

Findings

A well-defined composition law for Lagrangian correspondences.

Construction of a functorial field theory on geometric spin manifolds.

Formal Wick rotation of fermionic particle theory in curved spacetime.

Abstract

In this paper, we study Lagrangian correspondences between Hilbert spaces. A main focus is the question when the composition of two Lagrangian correspondences is again Lagrangian. Our answer leads in particular to a well-defined composition law in a category of Lagrangian correspondences respecting given polarizations of the Hilbert spaces involved. As an application, we construct a functorial field theory on geometric spin manifolds with values in this category of Lagrangian correspondences, which can be viewed as a formal Wick rotation of the theory associated to a free fermionic particle in a curved spacetime.