The $\theta$-dependence of the $\mathrm{S}\mathrm{U}(N)$ critical temperature at large $N$

We investigate, by means of numerical lattice simulations, the $\theta$-dependence of the critical deconfinement temperature of $\mathrm{S}\mathrm{U}(N)$ gauge theories at large $N$: $T_c(\theta )=T_c(0)[1-R{\theta }^2+O({\theta }^4)]$, with $R\sim O(1\mathrm{/}{N}^2)$. We follow two different strategies to determine $R$, one based on the calculation of the latent heat of the transition and on the jump of the topological susceptibility at the $\theta =0$ critical point, the other relying on a direct probe of $T_c(\theta )$ by means of imaginary-$\theta$ Monte Carlo simulations. Our results show that $R$ follows the expected large-$N$ scaling.