  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.
