• abelian monopoles
• lattice QCD

Colour confinement is the constraint on physical states which interact through strong interactions to be singlets under colour symmetry, the SU(3) local symmetry group which is gauged in QCD. While it has been widely proved both phenomenologically and numerically, a theoretical explanation is still missing.
This thesis wants to be a contribution in the investigation, making use of a relatively new tool which is the study of the condensation of abelian toplogical defects in a non abelian theory: thermal magnetic monopoles. Even though this technique has been proposed as a way to understand colour confinement since the 1970s, it is now starting to be systematically investigated also in full QCD.
In pure gauge theories, a true transition between confined and deconfined phases exists and previous studies of abelian monopoles proved that their condensation occurs at the same temperature as this transition. On the contrary, at the physical point of the theory, including fermions of non zero mass, there is no true phase transition, instead, the system presents a smooth crossover.
The transition is commonly studied by looking at chiral symmetry restoration, which occurs at temperature T ≃ 155 MeV, while in recent studies of thermal monopoles, their condensation temperature was found to be more than 100 MeV higher. Such a discrepancy could suggest something new, but at the same time the chiral symmetry is exact only in the case of zero quark masses not in the physical case, where a true transition does not exist, so different observables can show the transition at different temperatures.
In order to better understand this result, in this thesis, two physical setups will be considered, with the criterion that in both cases the system goes through a true transition with a definite temperature.
The first case is the introduction of a non zero and purely imaginary quark chemical potential. QCD in this scenario has an exact symmetry, Roberge Weiss symmetry, whose breaking induces a transition at around 208 MeV, at precise and periodical values of the chemical potential.
The choice of a non real chemical potential is also relevant since it solves the so-called sign problem, which arises both in continuum and lattice QCD and forbids the possibility of a Monte Carlo simulation.
The second case is full QCD with an external magnetic field, this choice has an analogous reason: it was proved in recent studies that this introduction changes the temperature and order of chiral transition, which becomes a real first order transition in the case of large external fields.
In both cases, the presence of a true transition could have an effect on monopole condensation temperature or could be seen as an additional change in monopole degrees of freedom, thus these two setups can help to better understand the link between confinement and magnetic monopoles.
The numerical simulations in the two cases were implemented using two different approaches, in order to double check results and compare the advantages of the two strategies.
The main observables analyzed were the density of monopole currents wrap- ping multiple times, where, close to condensation, higher numbers of wrappings become more relevant, and the cluster length ratio, which is the ratio between the length of the largest cluster of connected monopole currents and the length of the whole set of currents, which also shows a discontinuous behaviour at condensation temperature.
The results in the case of non zero chemical potential showed that the monopole condensation temperature does not coincide with the Roberge Weiss transition temperature, it is higher also in this case and it is compatible with the one found at zero chemical potential.
The introduction of an external magnetic field affected the condensation temperature as it was shown for the chiral critical temperature: the temperature decreases for high magnetic fields and the transition becomes sharper; even though the value of the temperature is still higher than expected this sug- gests a link between monopole condensation and deconfinement transition.