Dirac operator index and topology of lattice gauge fields

TL;DR

This paper demonstrates that the fermionic topological charge on lattice gauge fields converges to the continuum topological charge in the classical limit, establishing a key link between lattice and continuum gauge theories.

Contribution

It shows that the fermionic topological charge reduces to the continuum topological charge, bridging lattice gauge theory and continuum topology, with implications for 4-manifold invariants.

Findings

Fermionic topological charge has properties analogous to geometrical charge.

It reduces to the continuum topological charge in the classical limit.

Potential application to combinatorial construction of 4-manifold invariants.

Abstract

The fermionic topological charge of lattice gauge fields, given in terms of a spectral flow of the Hermitian Wilson--Dirac operator, or equivalently, as the index of Neuberger's lattice Dirac operator, is shown to have analogous properties to L\"uscher's geometrical lattice topological charge. The main new result is that it reduces to the continuum topological charge in the classical continuum limit. (This is sketched here; the full proof will be given in a sequel to this paper.) A potential application of the ideas behind fermionic lattice topological charge to deriving a combinatorial construction of the signature invariant of a 4-manifold is also discussed.