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

Abstract

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 $\mathrm{SL}(2,\mathbb{Z})$ 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 $\mathcal{S}$ and $\mathcal{T}^{2}$ transformations. We construct these transformations at the operator level and show that they realize the theta subgroup structure of $\mathrm{SL}(2,\mathbb Z)$. 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 $\mathcal T^2$ transformation realizes the Witten effect. Finally, we discuss a related non-invertible defect obtained by gauging a $\mathbb Z_N$ subgroup of the global $\mathrm{U}(1)$ $1$-form symmetry, and show that its fusion rule reproduces the expected Tambara--Yamagami structure.