The Fermionic integral on loop space and the Pfaffian line bundle

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.