On the relationship of gerbes to the odd families index theorem

Abstract

The goal of this paper is to apply the universal gerbe of [A. Carey, J. Mickelsson, A gerbe obstruction to quantization of fermions on odd dimensional manifolds, Lett. Math. Phys. 51 (2000) 145–160] and [A.L. Carey, J. Mickelsson, The universal gerbe, Dixmier–Douady classes and gauge theory, Lett. Math. Phys. 59 (2002) 47–60] to give an alternative, simple and more unified view of the relationship between index theory and gerbes. We discuss determinant bundle gerbes [A. Carey, J. Mickelsson, M. Murray, Index theory, gerbes, and Hamiltonian quantization, Comm. Math. Phys. 183 (1997) 707–722] and the index gerbe of [J. Lott, Higher-degree analogs of the determinant line bundle, Comm. Math. Phys. 230 (1) (2002) 41–69] 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 and Piazza’s C l ( 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.