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Modified Villain formulation of abelian Chern-Simons theory

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arxiv 2303.06160 v2 pith:JEDCN6IK submitted 2023-03-10 hep-th cond-mat.str-elhep-lat

Theodore Jacobson , Tin Sulejmanpasic This is my paper
classification hep-thcond-mat.str-elhep-lat
keywords theorychern-simonslatticemodifiedsymmetryvillainactionanomaly
verification ladder T0 review T1 audit T2 compute T3 formal
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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.

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Cited by 5 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. $\mathrm{SL}(2,\mathbb{Z})$ Theta Subgroup Structure of Maxwell theory in the Lattice Villain Hamiltonian Formulation

    hep-lat 2026-07 conditional novelty 7.0 of 10

    Lattice Villain Maxwell theory realizes the theta subgroup of SL(2,Z) via Hamiltonian operators S and T2, with charge exchange, the Witten effect, and a non-invertible defect with Tambara-Yamagami fusion.

  2. Exactly Solvable 1+1d Chiral Lattice Gauge Theories

    hep-th 2026-01 conditional novelty 7.0 of 10

    Anomaly-free chiral U(1) gauge theories in 1+1 dimensions can be written as quadratic, exactly solvable lattice Hamiltonians, with the 34-50 model as an explicit example.

  3. Symmetry theta angles and topological Witten effects

    hep-th 2025-06 conditional novelty 7.0 of 10

    Symmetry theta angles are intrinsic, Lagrangian-independent parameters of any non-anomalous Abelian invertible symmetry, and their universal effect is a topological Witten effect that shuffles twisted sectors while pr...

  4. Bosonization versus the Nielsen-Ninomiya theorem

    hep-th 2026-07 conditional novelty 6.5 of 10

    The 2D modified Villain model's Weyl operators reproduce free Dirac correlations in the continuum, and their no-doubler Dirac kernel is non-local, showing exactly how bosonization evades the Nielsen-Ninomiya theorem.

  5. 2d Cardy-Rabinovici model with the modified Villain lattice: Exact dualities and symmetries

    hep-th 2025-05 conditional novelty 6.0 of 10

    The 2d Cardy-Rabinovici model is constructed on a modified Villain lattice, making rescaled theta periodicity and strong-weak duality exact at finite spacing and giving a symmetry-based phase classification.

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