Index and small bundle gerbes

TL;DR

The paper introduces small bundle gerbes $(M,F,J)$ as finite-dimensional, smooth models for integral 3-classes, and constructs associated smooth Azumaya bundles $\mathcal{A}_J$ to realize twisted $K$-theory $K^*(M;\mathcal{A}_J)$. It develops a semiclassical index theory for fibrewise smoothing operators, including twisted variants $\operatorname{ind}_{\operatorname{sl},J}$, and proves an Atiyah–Singer type push-forward result identifying the index with the twisted K-theory Gysin map. The work extends to extensions of semiclassical quantization, range-twisted quantization, and twisted Chern characters, culminating in a twisted index formula in cohomology and an obstruction to metrics with large positive scalar curvature for projective families of Dirac operators. The framework applies to aspherical 3-manifolds (via Thurston geometries and connected sums) and yields topological obstructions to positive scalar curvature through the twisted index and twisted Chern character calculations.

Abstract

By a small bundle gerbe we mean a bundle gerbe in the sense of Murray defined on a smooth, finite-dimensional, fibre bundle over a manifold. We construct such gerbes over compact oriented aspherical 3-manifolds, as well as in higher dimensions, generalizing the construction of decomposable bundle gerbes in earlier work with Singer. For these small bundle gerbes there is a direct index map given in terms of either fibrewise pseudodifferential operators, or more conveniently fibrewise semiclassical smoothing operators, twisted by the simplicial line bundle. We prove the Atiyah-Singer type theorem that this realizes the push-forward into twisted K-theory. We also give an application via the index of projective families of Spin_c Dirac operators, to show the existence of obstructions to metrics with large positive scalar curvature.

Theorems & Definitions (46)

• Theorem 1
• Proposition 3: Bigerbes
• Theorem 1.1
• proof
• Theorem 2.1
• Proposition 4
• proof
• Lemma 5
• proof
• Lemma 6