Wilson-'t Hooft operators and the theta angle

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 $θ\toθ+2π$, a 't Hooft operator $T(γ)$ associated with a closed spatial curve $γ$ that winds around the compact direction undergoes a monodromy $T(γ) \to T^\prime(γ)$. The new 't Hooft operator $T^\prime(γ)$ transforms under large gauge transformations in the same way as the product $T(γ) W(γ)$, where $W(γ)$ is the Wilson operator associated with the curve $γ$ and the fundamental representation of SU(N).