2026/07/28 by Shoto Aoki, Yoshio Kikukawa, Toshinari Takemoto
#hep-lat
We study the duality structure of lattice Maxwell theory with a theta term in the Hamiltonian Villain formulation. Reflecting the fact that odd-level Chern--Simons theory depends on a choice of spin structure, a complete realization of the full SL(2,ℤ) structure would require fermionic degrees of freedom. We therefore restrict our analysis to the bosonic theory and focus on the theta subgroup generated by the S and T2 transformations. We construct these transformations at the operator level and show that they realize the theta subgroup structure of SL(2,\mathbb Z). We also extend the analysis to sectors with electric and magnetic charges, introduced as violations of the Gauss-law constraint and the Bianchi identity, respectively. We show that the \mathcal S transformation exchanges electric and magnetic charges, while the \mathcal T2 transformation realizes the Witten effect. Finally, we discuss a related non-invertible defect obtained by gauging a \mathbb ZN subgroup of the global U(1) 1-form symmetry, and show that its fusion rule reproduces the expected Tambara--Yamagami structure.