Modified Villain formulation of abelian Chern-Simons theory

TL;DR

This paper develops a lattice formulation of abelian Chern-Simons theory using a modified Villain approach, capturing key continuum features like level quantization, framing, and anomalies through lattice-specific constructs.

Contribution

It introduces a novel lattice action for U(1)_k Chern-Simons theory that incorporates the Pontryagin square and higher cup products, enabling a faithful representation of continuum topological properties.

Findings

Successfully reproduces continuum topological features on the lattice

Projects out unframed Wilson loops, leaving topologically correct ribbon loops

Provides a new lattice theta term and anomaly inflow mechanism

Abstract

We formulate $U(1)_k$ Chern-Simons theory on a Euclidean spacetime lattice using the modified Villain approach. Various familiar aspects of continuum Chern-Simons theory such as level quantization, framing, the discrete 1-form symmetry and its 't Hooft anomaly, as well as the electric charge of monopole operators are manifest in our construction. The key technical ingredient is the cup product and its higher generalizations on the (hyper-)cubic lattice, which recently appeared in the literature. All unframed Wilson loops are projected out by a peculiar subsystem symmetry, leaving topological, ribbon-like Wilson loops which have the correct correlation functions and topological spins expected from the continuum theory. Our action can be obtained from a new definition of the theta term in four dimensions which improves upon previous constructions within the modified Villain approach. This bulk action coupled to background fields for the 1-form symmetry is given by the Pontryagin square, which provides anomaly inflow directly on the lattice.