K-theoretic invariants of Hamiltonian fibrations

TL;DR

This paper introduces new K-theoretic invariants for Hamiltonian fibrations derived from geometric quantization techniques, providing novel tools to detect non-trivial fibrations through analytic index theory.

Contribution

It develops a new family index invariant in twisted K-theory for Hamiltonian fibrations and constructs a homotopy equivalence map related to Fredholm operators.

Findings

New invariants detect previously unseen non-trivial fibrations.

Construction of a homotopy equivalence map from $BU$ to Fredholm operators.

Application of geometric quantization techniques to topological invariants.

Abstract

We introduce new invariants of Hamiltonian fibrations with values in the suitably twisted K-theory of the base. Inspired by techniques of geometric quantization, our invariants arise from the family analytic index of a family of natural $Spin^c$-Dirac operators. As an application we give new examples of non-trivial Hamiltonian fibrations, that have not been previously detected by other methods. As one crucial ingredient we construct a potentially new homotopy equivalence map, with a certain naturality property, from $BU$ to the space of index $0$ Fredholm operators on a Hilbert space, using elements of modern theory of homotopy colimits.