Self-dual U(1) lattice field theory with a $\theta$-term

TL;DR

This paper develops a lattice U(1) gauge theory with a $ heta$-term that maintains electric-magnetic duality and $ heta$-periodicity, resulting in a local action suitable for studying dualities and finite $ heta$ effects.

Contribution

It introduces a duality-preserving lattice formulation of U(1) gauge theories with a $ heta$-term, extending previous ultra-local models to include local actions with exact duality.

Findings

The theory exhibits exact electric-magnetic duality on the lattice.

SL(2,Z) duality group emerges for dyonic matter and $ heta$-shifts.

The formulation enables finite $ heta$ studies without a sign problem.

Abstract

We study U(1) gauge theories with a modified Villain action. Such theories can naturally be coupled to electric and magnetic matter, and display exact electric-magnetic duality. In their simplest formulation without a $\theta$-term, such theories are ultra-local. We extend the discussion to U(1) gauge theories with $\theta$-terms, such that $\theta$ periodicity is exact for a free theory, and show that imposing electric-magnetic duality results in a local, but not ultra-local lattice action, which is reminiscent of the L\"uscher construction of axial-symmetry preserving fermions in 4d. We discuss the coupling to electric and magnetic matter as well as to dyons. For dyonic matter the electric-magnetic duality and shifts of the $\theta$-angle by $2\pi$ together generate an SL$(2,\mathbb Z)$ duality group of transformations, just like in the continuum. We finally illustrate how the SL$(2,\mathbb Z)$ duality may be used to explore theories at finite $\theta$ without a sign problem.