2004/07/14 by Alan L. Carey, Carey, Alan L., Bai-Ling Wang +2
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Homotopy and Cohomology in Algebraic Topology #math-ph #math.DG #math.MP #msc:19K56 #msc:55R65 #msc:57J52 #msc:58J28
paper · pdf · doi:10.48550/arxiv.math/0407243
26 pages, no figures; to appear in Journ. of Geom. and Phys
arxiv created 2006/02/20 · arxiv updated 2009/12/01
The goal of this paper is to apply the universal gerbe of \citeCMi1 and \citeCMi2 to give an alternative, simple and more unified view of the relationship between index theory and gerbes. We discuss determinant bundle gerbes \citeCMMi1 and the index gerbe of \citeL 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.