$\theta$ dependence of $T_c$ in SU(2) Yang-Mills theory

We present an exploratory study to determine the confinement-deconfinement transition temperature at finite $\theta$, $T_c(\theta)$, for the 4d \SU(2) pure Yang-Mills theory. Lattice numerical simulations are performed on three spatial sizes $N_S=24$, $32$, $48$ with a fixed temporal size $N_T=8$. We introduce a non-zero $\theta$-angle by the sub-volume method to mitigate the sign problem. By taking advantage of the universality in the second order phase transition and the Binder cumulant of the order parameter, the $\theta$-dependence of $T_c$ is determined to be $T_c(\theta)/T_c(0)=1-0.16(2)\,(\theta/\pi)^2-0.03(4)\,(\theta/\pi)^4$ up to $\theta\sim 0.9\,\pi$. The reliability of the extrapolations in the sub-volume method is extensively checked. We also point out that the temperature dependence of the topological susceptibility should exhibit a singularity with the exponent for the specific heat.