The QCD phase diagram at imaginary chemical potential

The properties of strongly interacting matter in extreme conditions of temperature and baryon density have been the subject of intense experimental and theoretical research efforts for decades. The features of the phase diagram of Quantum Chromodynamics are intimately connected with these properties. Most notably, the change of state between hadronic matter and quark gluon plasma defines the so-called "pseudocritical" line of QCD. While heavy ion collision experiments are probing the phase diagram of QCD, from the theoretical point of view a lot has been achieved with numerical simulations of lattice QCD (LQCD) with Montecarlo methods. However, although these techniques have been used successfully to study the thermodynamic and time-independent properties of the theory, at finite density severe limitations appear due to the so-called "sign problem". Since the sign problem is absent when the quark chemical potential is purely imaginary, one of the possible ways out is to simulate LQCD in this case and exploit analytic continuation to recover results of physical interest. The pseudocritical line of QCD is studied in the first part of this Thesis making use of this technique. An estimate of the curvature of the pseudocritical line relevant to continuum physics is obtained and is compared with the most recent results in the literature, carefully assessing the systematics involved. The structure of the QCD phase diagram at imaginary chemical potential is also interesting per se. Most notably, Roberge and Weiss predicted that a phase transition occurs. In the second part of this Thesis, the location and properties of this transition are studied for physical quark masses, and preliminary results are presented for reduced quark masses. These preliminary results have been obtained with a LQCD code running on GPU clusters, the implementation of which is described in the Appendices.