Transgression of the index gerbe

Abstract

 The holonomy of an unitary line bundle with connection over some base space B is a U(1)-valued function on the loop space LB. In a parallel manner, the holonomy of a gerbe with connection on B is a line bundle with connection over LB.

Given a family of graded Dirac operators on B and some additional geometric data one can define the determinant line bundle with Quillen metric and Bismut-Freed connection. According to Witten, Bismut-Freed the holonomy of this determinant bundle can be expressed in terms of an adiabatic limit of eta invariants of an associated family of Dirac operators over LB.

Recently, for a family of ungraded Dirac operators on B. Lott constructed an index gerbe with connection. In the present paper we show, in analogy to the holonomy formula for the determinant bundle, that the holonomy of the index gerbe coincides with an adiabatic limit of determinant bundles of the associated family of Dirac operators over LB.