Large-scale quantization of trace I: Finite propagation operators

TL;DR

This paper introduces a family of universal trace formulae that depend solely on large-scale geometric features, generalizing previous results in mathematics and physics for arbitrary dimensions.

Contribution

It extends existing trace formulae to higher dimensions, unifying concepts from coarse geometry and macroscopic quantization in a novel framework.

Findings

Derived universal trace formulae based on large-scale geometry

Generalized Roe's partitioned manifold index theorem

Connected mathematical and physical quantization formulae

Abstract

Inspired by parallel developments in coarse geometry in mathematics and exact macroscopic quantization in physics, we present a family of general trace formulae which are universally quantized and depend only on large-scale geometric features of the input data. They generalize, to arbitrary dimensions, formulae found by Roe in his partitioned manifold index theorem, as well as the Kubo and Kitaev formulae for 2D Hall conductance used in physics.