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

Yo Ikeda

TL;DR

This work constructs Deligne--Beilinson (DB) cohomology on a cubic lattice and uses it to formulate a lattice U(1) Chern--Simons theory at even levels via a star product. The lattice framework preserves essential continuum DB properties, introduces a lattice framing through Wilson lines, and defines mod 2k linking and mod 4k self-linking numbers, all within a gauge-invariant, differential-cohomology setting. The authors develop a gauge-fixed path integral for lattice DB CS, regularize divergences with a Maxwell term, and show Wilson line expectation values reproduce self-linking numbers up to controllable Maxwell-parameter error. They further compare the lattice DB construction with the modified Villain formalism, analyze the Pontrjagin-duality structure, and illustrate non-invertible chiral defects in lattice massless QED arising from this topological framework. The approach offers a rigorous lattice realization of topological gauge theories, with potential extensions to BF theory, odd-level CS, and higher-dimensional generalizations.

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.

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

TL;DR

This work constructs Deligne--Beilinson (DB) cohomology on a cubic lattice and uses it to formulate a lattice U(1) Chern--Simons theory at even levels via a star product. The lattice framework preserves essential continuum DB properties, introduces a lattice framing through Wilson lines, and defines mod 2k linking and mod 4k self-linking numbers, all within a gauge-invariant, differential-cohomology setting. The authors develop a gauge-fixed path integral for lattice DB CS, regularize divergences with a Maxwell term, and show Wilson line expectation values reproduce self-linking numbers up to controllable Maxwell-parameter error. They further compare the lattice DB construction with the modified Villain formalism, analyze the Pontrjagin-duality structure, and illustrate non-invertible chiral defects in lattice massless QED arising from this topological framework. The approach offers a rigorous lattice realization of topological gauge theories, with potential extensions to BF theory, odd-level CS, and higher-dimensional generalizations.

Abstract

We define Deligne-Beilinson (DB) cohomology on a cubic lattice and use it to formulate and analyze lattice Chern-Simons theory at even levels. The continuum DB cohomology provides a refined mathematical framework for continuum 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 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.
Paper Structure (95 sections, 364 equations, 13 figures)

This paper contains 95 sections, 364 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Review
  3. Continuous Chern--Simons theory formulated using DB cohomology
  4. DB $2$-cohomology on $3$-manifolds
  5. Star product in continuum DB cohomology
  6. Continuum Chern--Simons theory via DB cohomology
  7. Expectation values of Wilson lines
  8. Lattice Chern--Simons theory formulated using the modified Villain formalism
  9. Modified Villain formalism
  10. Staggered symmetry and Wilson line
  11. The lattice DB cohomology on $d$-dim cubic torus
  12. Lattice DB cochain
  13. Lattice Čech--de Rham chain
  14. Lattice Čech--de Rham cochain
  15. Coboundary operations
  16. ...and 80 more sections

Figures (13)

  • Figure 1: Examples of elements of $\mathrm{cube}_{(0,0)}(\mathbb{L}^2)$
  • Figure 2: Examples of elements of $\mathrm{cube}_{(1,0)}(\mathbb{L}^2)$
  • Figure 3: An example of an element of $\mathrm{cube}_{(2,0)}(\mathbb{L}^2)$
  • Figure 4: Examples of elements of $\mathrm{cube}_{(0,1)}(\mathbb{L}^2)$
  • Figure 5: Another example of an element of $\mathrm{cube}_{(0,1)}(\mathbb{L}^2)$
  • ...and 8 more figures