格点维兰哈密顿形式下麦克斯韦理论的\(\mathrm{SL}(2,\mathbb{Z})\)西塔子群结构
$\mathrm{SL}(2,\mathbb{Z})$ Theta Subgroup Structure of Maxwell theory in the Lattice Villain Hamiltonian Formulation
AI总结：
研究哈密顿维兰形式下带西塔项的格点麦克斯韦理论对偶结构，聚焦玻色子理论中\(\mathcal{S}\)和\(\mathcal{T}^{2}\)变换生成的西塔子群，构建变换实现其结构，扩展到电和磁荷扇区，还讨论了相关不可逆缺陷及融合规则。
AI中文摘要：
我们研究哈密顿维兰形式下带西塔项的格点麦克斯韦理论的对偶结构。由于奇数级陈 - 西蒙斯理论依赖于自旋结构的选择，完整实现整个\(\mathrm{SL}(2,\mathbb{Z})\)结构需要费米自由度。因此我们将分析限制在玻色子理论，聚焦于由\(\mathcal{S}\)和\(\mathcal{T}^{2}\)变换生成的西塔子群。我们在算符层面构建这些变换并表明它们实现了\(\mathrm{SL}(2,\mathbb{Z})\)的西塔子群结构。我们还将分析扩展到分别通过违反高斯定律约束和比安基恒等式引入的电和磁荷扇区。我们表明\(\mathcal{S}\)变换交换电和磁荷，而\(\mathcal{T}^{2}\)变换实现维滕效应。最后，我们讨论通过对全局\(\mathrm{U}(1)\)\(1\)-形式对称性的\(\mathbb{Z}_N\)子群进行规范得到的相关不可逆缺陷，并表明其融合规则再现了预期的坦巴拉 - 亚马加米结构。
英文摘要：
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.
