The Universal Gerbe and Local Family Index Theory

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Abstract
The goal of this paper is to apply the universal gerbe developed in \cite{CMi1} and \cite{CMi2} and the local family index theorems to give a unified viewpoint on the known examples of geometrically interesting gerbes, including the determinant bundle gerbes in \cite{CMMi1}, the index gerbe in \cite{L} for a family of Dirac operators on odd dimensional closed manifolds. We also discuss the associated gerbes for a family of Dirac operators on odd dimensional manifolds with boundary, and 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. 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 an exact form) arising from the local family index theorem.

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