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

The Fermionic integral on loop space and the Pfaffian line bundle