The Fermionic integral on loop space and the Pfaffian line bundle

TL;DR

This paper extends finite-dimensional formulas to infinite-dimensional loop spaces, constructing a top degree component of forms as sections of the Pfaffian line bundle, crucial for fermionic path integrals in supersymmetry.

Contribution

It generalizes a finite-dimensional formula to infinite-dimensional loop spaces, linking the top degree form to the Pfaffian line bundle and transgression of the spin gerbe.

Findings

Constructs a top degree component as a section of the Pfaffian line bundle.

Identifies this section with the transgression of the spin lifting gerbe.

Provides a foundation for fermionic path integrals in supersymmetric theories.

Abstract

As the loop space of a Riemannian manifold is infinite-dimensional, it is a non-trivial problem to make sense of the "top degree component" of a differential form on it. In this paper, we show that a formula from finite dimensions generalizes to assign a sensible "top degree component" to certain composite forms, obtained by wedging with the exponential (in the exterior algebra) of the canonical 2-form on the loop space. The result is a section on the Pfaffian line bundle on the loop space. We then identify this with a section of the line bundle obtained by transgression of the spin lifting gerbe. These results are a crucial ingredient for defining the fermionic part of the supersymmetric path integral on the loop space.