Topological observables and $\theta$ dependence in high temperature QCD from lattice simulations

We study topology in Quantum Chromodynamics at high temperatures by means of lattice calculations. Simulations are performed with $N_f=2+1+1$ Wilson twisted mass fermions at maximal twist with physical quark masses, and temperatures $T~\gtrsim~180$ MeV. The results obtained with three lattice spacings ranging between $0.057$ and $0.080$ fm are extrapolated to the continuum limit. We compare the results for the topological susceptibility obtained with the field-theoretic definition with those obtained from an observable constructed with the disconnected part of the chiral susceptibility, and we confirm their agreement -- within the largish errors -- on our range of temperatures. We also study the topological charge distribution, the next order cumulant $b_2$ and, for the first time, the Free Energy as a function of the $\theta$ angle. We find a rapid crossover to the Dilute Instanton Gas behaviour above $ T \simeq 300$ MeV for all the observables we have considered.