Exact SL(2,Z)-Structure of Lattice Maxwell Theory with $\theta$-term in Modified Villain Formulation

TL;DR

This paper demonstrates that lattice Maxwell theory with a theta term can exhibit an exact SL(2,Z) duality when a non-local transformation is used, revealing a structure similar to non-spin Maxwell theory.

Contribution

It shows how to maintain exact SL(2,Z) duality in lattice Maxwell theory with a theta term by incorporating a non-local transformation, clarifying the duality structure.

Findings

Exact SL(2,Z) duality achieved with a non-local transformation.

Wilson and 't Hooft loops transform with a phase factor from self-linking.

SL(2,Z) structure resembles that of non-spin Maxwell theory.

Abstract

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 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 S-transformation. Remarkably, the resulting SL(2,Z)-structure closely resembles that of non-spin Maxwell theory.