Direct Monte Carlo Computation of the ’t Hooft Partition Function

Abstract The ’t Hooft partition function $\mathcal {Z}_{\text{tH}}[E;B]$ of an $SU(N)$ gauge theory with the $\mathbb {Z}_N$ 1-form symmetry is defined as the Fourier transform of the partition function $\mathcal {Z}[B]$ with respect to the spatial-temporal components of the ’t Hooft flux $B$. Its large volume behavior detects the quantum phase of the system. When the integrand of the functional integral is real-positive, the latter partition function $\mathcal {Z}[B]$ can be numerically computed by a Monte Carlo simulation of the $SU(N)/\mathbb {Z}_N$ gauge theory, just by counting the number of configurations of a specific ’t Hooft flux $B$. We carry out this program for the $SU(2)$ pure Yang–Mills theory with the vanishing $\theta$-angle by employing a newly developed hybrid Monte Carlo (HMC) algorithm (the halfway HMC) for the $SU(N)/\mathbb {Z}_N$ gauge theory. The numerical result clearly shows that all nonelectric fluxes are “light” as expected in the ordinary confining phase with the monopole condensate. Invoking the Witten effect on $\mathcal {Z}_{\text{tH}}[E;B]$, this also indicates the oblique confinement at $\theta =2\pi$ with the dyon condensate.