Characters for Complex Bundles and their Connections

TL;DR

This paper explores the interplay between complex bundles, bordism theories, and index theorems, providing new insights into invariants of stable complex bundles and their connections via dualities and adiabatic limits.

Contribution

It establishes a novel relationship between stable complex bundles and bordism theories using dualities, and connects invariants to eta invariants through adiabatic limit techniques.

Findings

Stable complex bundles have a complete system of numerical invariants.

These invariants can be computed via integrals of Chern-Weil forms over manifolds with boundary.

The adiabatic limit relates Chern-Simons connections to Levi-Civita connections, linking invariants to eta invariants.

Abstract

The paper combines several fortunate mini miracles to achieve its two objectives. These were woven together in a several year's effort to answer a question raised by Iz Singer a decade ago. Our answer is accessible to the topologist, to the differential geometer and to the analyst who appreciates the statement of the Index theorem of Atiyah,Patodi,Singer for manifolds with boundary. The mini miracles are these: a] The Conner Floyd miracle that complex bordism tensored over the Todd genus and the Bott miracle that stable complex vector bundles respectively satisfy the axioms of a generalized homology theory and of a generalized cohomology theory. b] That these theories, with the covariant and contravariant geometric representations indicated, stably almost complex (SAC) manifolds modulo product relations and stable complex bundles, are not only related by Alexander duality but they are also related by Pontryagin duality. c] The abstract corollary of b] that stable complex bundles have a complete system of numerical invariants and that these can be computed by integrals of chern weil characteristic forms over manifolds with boundary reduced modulo integers, thanks to the APS Index Theorem. d] The adiabatic limit argument of the appendix to the last section showing a direct sum connection on the total space of a riemannian family of Riemannian manifolds with connection is Chern Simons equivalent in the limit to the Levi Civita connection of the direct sum metric. This allows the invariants to be described by the eta invariants of odd SAC manifolds reduced mod integers.