Wilson-'t Hooft operators and the theta angle

TL;DR

This paper analyzes how 't Hooft operators in SU(N)/Z_N Yang-Mills theory transform under a 2π shift in the theta angle, revealing a monodromy involving Wilson operators and deepening understanding of gauge theory topological effects.

Contribution

It proves the monodromy of 't Hooft operators under theta angle shifts in SU(N)/Z_N Yang-Mills theory, connecting topological operators with gauge transformations.

Findings

't Hooft operators undergo a monodromy when theta increases by 2π.

The transformed 't Hooft operator relates to the original via Wilson operators.

The result clarifies the topological structure of gauge theories with nontrivial center.

Abstract

We consider $(3+1)$-dimensional $SU(N)/\mathbb Z_N$ Yang-Mills theory on a space-time with a compact spatial direction, and prove the following result: Under a continuous increase of the theta angle $\theta\to\theta+2\pi$, a 't Hooft operator $T(\gamma)$ associated with a closed spatial curve $\gamma$ that winds around the compact direction undergoes a monodromy $T(\gamma) \to T^\prime(\gamma)$. The new 't Hooft operator $T^\prime(\gamma)$ transforms under large gauge transformations in the same way as the product $T(\gamma) W(\gamma)$, where $W(\gamma)$ is the Wilson operator associated with the curve $\gamma$ and the fundamental representation of SU(N).