Exact SL(2, ℤ)-structure of lattice Maxwell theory with θ-term in modified Villain formulation

A bstract We study the duality of lattice Maxwell theory in the modified Villain formulation, employing an ultra-local action with a theta term. Although this action is known to become non ultra-local through the Poisson resummation formula, we show that this non ultra-locality can be removed by incorporating a non-local transformation procedure into the definition of the $$ \mathcal{S} $$ S -transformation. As a result, the ultra-local action with a theta term exhibits an exact SL(2, ℤ )-duality. We further analyze the SL(2, ℤ )-structure of Wilson and ’t Hooft loops, demonstrating that they transform properly up to a nontrivial phase factor arising from the nontrivial self-linking of the loops. This effect originates from the non-local transformation procedure in the $$ \mathcal{S} $$ S -transformation. Remarkably, the resulting SL(2, ℤ )-structure closely resembles that of non-spin Maxwell theory.