Pfaffian line bundles over loop spaces, spin structures and the index theorem

We study the relation between different types of real line bundles with metric defined over the free loop space of a smooth manifold M. These line bundles can be naturally constructed from an oriented real vector bundle with metric and connection E → M.;On one hand, the loop space LM parametrizes families of (twisted) Dirac operators over the circle. These are families of real skew-symmetric elliptic operators and their Pfaffian lines and elements fit into smooth real line bundles with metric over LM together with canonical smooth sections. These sections can be interpreted as the integrand of the partition functions of certain 1|1-dimensional SUSY non-linear sigma-models with target M, and hence the non-triviality of the Pfaffian line bundles represents an anomaly of the sigma-model.;On the other hand, we consider variations of a real line bundle with metric over loop space suggested by Stolz and Teichner that can be constructed using finite dimensional Clifford algebras, Fock space representations and their intertwiners. These line bundles are canonically embedded into bundles of finite dimensional exterior algebras and hence carry canonical smooth sections via the orthogonal projection of the vacuum and volume elements.;In this thesis, we construct explicit isometries between Pfaffian line bundles and the line bundles of Stolz and Teichner and explain the exact relation between their canonical sections.;Stolz and Teichner also noticed that a choice of spin structure on E induces a fusion preserving trivialization of their line bundle and viceversa. We compute the functions induced by the previous sections in terms of traces and supertraces of the parallel transport on the spinor bundle. When E = TM, they are related to the integrand of the path-integral representation of the heat kernel of the Dirac operator, and hence to its index and to the A-genus of M.;We also explain the relation between these line bundles and the transgression to LM of the lifting bundle gerbe over M associated to the oriented frame bundle PSO( E) and the group extension 0 → Z /2 → Spin(n) → SO( n) → 0. More generally, we show how to construct sections of line bundles associated to the transgression of the lifting gerbe corresponding to a principal G-bundle with connection P → M and a central extension 0 → Z /2 → G → G → 0, starting from representations of G.;Hence, this thesis provides a better understanding of the role that a spin structure plays in the trivialization of the Pfaffian line bundle and we hope that the techniques applied here can be adapted to obtain a well defined integrand for the partition function of a 2|1-dimensional SUSY non-linear sigma-model with target M in the case that M is endowed with a (geometric) string structure, and to understand the relation between this partition function and the Witten genus of M.