't Hooft line in 4D $U(1)$ lattice gauge theory and a microscopic description of dyon's statistics

TL;DR

This paper introduces a lattice definition of the 't Hooft line in 4D U(1) gauge theory, enabling the study of dyons and their statistics without violating the admissibility condition, thus bridging lattice gauge theory and topological phenomena.

Contribution

It extends the excision method from 2D to 4D Maxwell theory to define 't Hooft lines and analyze dyon statistics on the lattice.

Findings

Demonstrates a lattice analogue of the Witten effect creating dyons.

Shows the dyon's statistics can be directly read from boundary interpretations.

Constructs a finite-lattice genuine loop operator satisfying Dirac quantization.

Abstract

In lattice gauge theory with compact gauge field variables, an introduction of the gauge field topology requires the assumption that lattice field configurations are sufficiently smooth. This assumption is referred to as the admissibility condition. However, the admissibility condition always ensures the Bianchi identity, and thus prohibits the existence of magnetic objects such as the 't~Hooft line. Recently, in 2D compact scalar field theory, Ref.~\cite{Abe:2023uan} proposed a method to define magnetic objects without violating the admissibility condition by introducing holes into the lattice. In this paper, we extend this ``excision method'' to 4D Maxwell theory and propose a new definition of the 't~Hooft line on the lattice. Using this definition, we first demonstrate a lattice counterpart of the Witten effect which endows the 't~Hooft line with electric charge and make it a dyon. Furthermore, we show that by interpreting the 't~Hooft line as a boundary of the lattice system, the statistics of the dyon can be directly read off. We also explain how the dyonic operator which satisfies the Dirac quantization condition becomes a genuine loop operator even at finite lattice spacings.