On the relationship of gerbes to the odd families index theorem

Details On the relationship of gerbes to the odd families index theorem On the relationship of gerbes to the odd families index theorem

2004-07-14

arxiv.org/abs/math/0407243v2

Mathematics

Differential Geometry

26 pages, no figures; to appear in Journ. of Geom. and Phys

Scientific paper

The goal of this paper is to apply the universal gerbe of \cite{CMi1} and \cite{CMi2} to give an alternative, simple and more unified view of the relationship between index theory and gerbes. We discuss determinant bundle gerbes \cite{CMMi1} and the index gerbe of \cite{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.

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