Browsing by Subject "Index theorems"
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Item An index theorem in differential K-theory(2008-05) Klonoff, Kevin Robert, 1972-; Freed, Daniel S.We construct a geometric model for differential K-theory, and prove it is isomorphic to the model proposed in [25]. We construct differential K-orientations for families and elucidate the pushforward map given in [25] in detail. We prove a geometric index theorem for odd dimensional manifolds. Finally, using this index theorem and the holonomy theorem of Bismut and Freed from [10], we prove what may be considered a special case of a geometric refinement of the Aityah-Singer index theorem.Item Superconnections and index theory(2008-08) Kahle, Alexander Rudolf; Freed, Daniel S.This document presents a systematic investigation of the geometric index theory of Dirac operators coupled superconnections. A local version of the index theorem for Dirac operators coupled to superconnection is proved, and extended to families. An [eta]-invariant is defined, and it is shown to satisfy an APS-like theorem. A geometric determinant line bundle with section, metric, and connection is associated to a family of Dirac operators coupled to superconnections, and its holonomy is calculated in terms of the [eta]-invariant.