# Lattice Gauge Fields and Noncommutative Geometry

## TL;DR

This paper introduces a novel algebraic formulation of lattice gauge theories that incorporates spacetime topology using noncommutative geometry, providing a more geometrically meaningful framework for gauge and Dirac fields.

## Contribution

It develops a topologically aware lattice gauge theory framework based on cochains and noncommutative geometry, improving upon conventional methods.

## Key findings

- Derived Wilson's actions from algebraic principles.
- Proposed natural candidates for QCD theta and Chern-Simons terms.
- Presented simpler formulations for gauge theories on discrete spaces.

## Abstract

Conventional approaches to lattice gauge theories do not properly consider the topology of spacetime or of its fields. In this paper, we develop a formulation which tries to remedy this defect. It starts from a cubical decomposition of the supporting manifold (compactified spacetime or spatial slice) interpreting it as a finite topological approximation in the sense of Sorkin. This finite space is entirely described by the algebra of cochains with the cup product. The methods of Connes and Lott are then used to develop gauge theories on this algebra and to derive Wilson's actions for the gauge and Dirac fields therefrom which can now be given geometrical meaning. We also describe very natural candidates for the QCD theta term and Chern-Simons action suggested by this algebraic formulation. Some of these formulations are simpler than currently available alternatives. The paper treats both the functional integral and Hamiltonian approaches.