Families index theory for Overlap lattice Dirac operator. I

TL;DR

This paper develops a topological framework for the index bundle of the Overlap lattice Dirac operator, linking lattice gauge theory anomalies with continuum topological charges, and providing a finite-dimensional lattice perspective.

Contribution

It introduces an index bundle over lattice gauge fields, derives a formula for its topological charge, and connects lattice and continuum topological properties of Dirac operators.

Findings

Derived a formula for the topological charge of the index bundle.

Connected lattice gauge theory anomalies with continuum topological invariants.

Showed the topology of the index bundle can be captured on a finite lattice.

Abstract

The index bundle of the Overlap lattice Dirac operator over the orbit space of lattice gauge fields is introduced and studied. Obstructions to the vanishing of gauge anomalies in the Overlap formulation of lattice chiral gauge theory have a natural description in this context. Our main result is a formula for the topological charge (integrated Chern character) of the index bundle over even-dimensional spheres in the orbit space. It reduces under suitable conditions to the topological charge of the usual (continuum) index bundle in the classical continuum limit (this is announced and sketched here; the details will be given in a forthcoming paper). Thus we see that topology of the index bundle of the Dirac operator over the gauge field orbit space can be captured in a finite-dimensional lattice setting.