Direct numerical simulation of the 't Hooft partition function and (de)confining phases

The ‘t Hooft partition function $Z_{\mathrm{t}\mathrm{H}}[E_i;B_{ij}]$ is a discrete Fourier transform of Yang–Mills partition functions in background ${\mathbb{Z}}_N$ 2-form gauge fields and encodes information on confinement, Higgs, Coulomb and oblique-confining phases. We report a direct Monte Carlo strategy to measure $Z_{\mathrm{t}\mathrm{H}}$ without reweighting, by extending hybrid Monte Carlo to include dynamical updates of the background flux variables. As a first application we measure all flux sectors of four-dimensional $SU(2)$ lattice Yang–Mills on $T^4$ and observe the characteristic ``light/heavy’’ behavior expected in the confining phase, together with the shift implied by the Witten effect at $\theta =2\pi$. We also present a preliminary finite-temperature study and discuss outstanding issues on thermalization and separability between different flux sectors.