On the relationship of gerbes to the odd families index theorem

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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.