A Lattice U(1) Chern-Simons Theory via Lattice Deligne-Beilinson Cohomology

TL;DR

This paper develops a lattice version of U(1) Chern-Simons theory using Deligne-Beilinson cohomology, providing a rigorous mathematical framework that captures topological features like self-linking numbers and exhibits level quantization.

Contribution

It introduces a lattice Deligne-Beilinson cohomology framework for U(1) Chern-Simons theory, enabling a gauge-invariant formulation with explicit path integral construction and topological invariants.

Findings

Lattice DB cohomology retains key properties of continuum theory.

The Chern-Simons action is expressed as a quadratic form with level quantization.

Wilson line expectation values relate to self-linking numbers.

Abstract

We define Deligne-Beilinson (DB) cohomology on a cubic lattice and use it to formulate and analyze lattice $U(1)$ Chern-Simons theory at even levels. The continuum DB cohomology provides a refined mathematical framework for continuum $U(1)$ connections constructed in a patchwise manner. The lattice DB cohomology we construct retains many essential properties of the continuum DB cohomology and naturally incorporates a notion of self-linking number. The lattice $U(1)$ Chern-Simons action formulated using the lattice DB cohomology is expressed as a simple quadratic form via the star product, which naturally exhibits level quantization. Framed Wilson lines respecting staggered symmetry are defined in a gauge-invariant manner, and their expectation values are shown to be given by the self-linking number, as follows from completing the square. Using the lattice Hodge decomposition, we explicitly characterize the DB cohomology on a three-dimensional cubic toroidal lattice and present a gauge-fixed, rigorous path integral for the lattice Chern-Simons theory. To regulate divergences in the lattice Chern-Simons path integral arising from staggered symmetry, we introduce a small Maxwell term. The resulting error is controlled by the linear order in the small Maxwell coupling.