On the Relationship of Gerbes to the Odd Families Index Theorem

The goal of this paper is to apply the universal gerbe of [CMi1] and [CMi2] to give an alternative, simple and more unified view of the relationship between index theory and gerbes. We discuss determinant bundle gerbes [CMMi1] and the index gerbe of [L] for the case of families of Dirac operators on odd dimensional closed manifolds. The method also works for a family of Dirac operators on odd dimensional manifolds with boundary, for a pair of Melrose-Piazza’s Cl(1)-spectral sections for a family of Dirac operators on even dimensional closed manifolds with vanishing index in K-theory and, in a simple case, for manifolds with corners. The common feature of these bundle gerbes is that there exists a canonical bundle gerbe connection whose curving is given by the degree 2 part of the even eta-form (up to a locally defined exact form) arising from the local family index theorem.