Wilson-'t Hooft operators and the theta angle

We consider $(3+1)$-dimensional $SU(N)/\mathbb Z_N$ Yang-Mills theory on aspace-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 'tHooft operator $T(\gamma)$ associated with a closed spatial curve $\gamma$ thatwinds around the compact direction undergoes a monodromy $T(\gamma) \toT^\prime(\gamma)$. The new 't Hooft operator $T^\prime(\gamma)$ transformsunder 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).Comment: 7 page