  A bundle gerbe is constructed from an oriented smooth vector bundle of even
rank with a fiberwise inner product, over a compact connected orientable smooth
manifold with Riemannian metric. From a trivialization of the bundle gerbe is
constructed an irreducible Clifford module bundle, a spinor bundle over the
smooth free loop space of the manifold.
  First, a Clifford algebra bundle over the loop space is constructed from the
vector bundle. A polarization class bundle is constructed, choosing
continuously over each point of the loop space a polarization class of
Lagrangian subspaces of the complexification of the real vector space from
which the Clifford algebra is made. Being unable to choose a Lagrangian
subspace continuously from the polarization class over each point, the thesis
constructs a bundle gerbe over the loop space of the base manifold to encode
over each loop all such subspaces, along with the isomorphisms between the Fock
spaces made from them, resulting from their being in the same polarization
class.
  The vanishing of the Dixmier-Douady class of the bundle gerbe implies that
the latter has a trivialization, from which is constructed a spinor bundle.
