• confinement
• lattice gauge theory
• monopoles
• QCD

The study of the confinement properties of the Yang-Mills theories is a notoriously difficult subject, confinement being a completely nonperturbative phenomenon, and this prevents a complete understanding of the physics of the Standard Model of particles in the strongly coupled regime.

The lattice formulation of QCD introduced by Wilson \cite{Wilson} is an invaluable tool for studying strong coupling gauge theories both analytically and by means of numerical simulations. During the years enormous numerical evidence has been collected to support the idea that in non-abelian gauge theories the color degrees of freedom are confined, \ie that only color singlet states are present in the spectrum. Nevertheless a satisfactory understanding of the
physical mechanism responsible for color confinement is still lacking.
Topologically stable configurations are though to be involved in the color confinement mechanism, however there is no consensus on the choice of the relevant topological defects, the two main candidates being vortices and monopoles.

The ideas behind the two proposal of vortex- or monopole-related confinement are very different in spirit although they are both aimed at explaining the presence of a linearly rising potential between a quark-antiquark static pair
or, equivalently, the area law behaviour of the Wilson loops.

In the vortex-related theory the area scaling of the Wilson loops is explained by what is usually called ``center disorder'': if in the confined phase a large number of sufficiently randomly distributed vortices are present, a given Wilson loop will be pierced by a large number of independent vortices and, depending on the even or odd number of piercings, the sign of the Wilson loop will strongly fluctuate, with large cancellations occurring and a net exponential behaviour will result (for details see \eg \cite{ELRT}).
In the monopole-related confinement scenario the assumption is that the monopole degrees of freedom are condensed in the confined phase and the linearly rising potential is generated by the dual analogue of the Abrikosov flux tubes,
\ie the vacuum behaves as a dual superconductor.

While in the vortex scenario confinement is related to the existence of a percolating vortex cluster, in the monopole scenario confinement is connected to the realization of the magnetic $U(1)$ symmetry and the absence of colored
asymptotic states in the spectrum is enforced by the vacuum symmetry, thus avoiding naturalness problems. This is not the only theoretically appealing feature of the dual superconductivity model, since it leaves open the door to the possibility of a duality symmetry between the electric and the magnetic degrees of freedom, \ie
to the possibility of describing the QCD strong coupling regime by means of an effective weakly interacting theory of monopoles.

Effective weakly interacting theories which describe the strong coupling regime of a physical systems by means of effective degrees of freedom are ubiquitous in condensed matter physics (\eg the Landau theory of Fermi liquids),
however there are very few examples of systems for which the duality transformation is explicitly known. These are
typically simple spin systems, like the 2d Ising model \cite{KC} or the 2d XY model \cite{XY}.
A notable exception is the solution by Seiberg and Witten of the $\mathcal{N}=2$ supersymmetric Yang-Mills theory \cite{SeibergWitten} in which the duality transformation can be explicitly performed and confinement is described by
monopole condensation.

The effective degrees of freedom are typically introduced by the topologically nontrivial behaviour of the fields at spatial infinity. For example, for a Yang-Mills theory living in $D+1$ dimension, the effective degrees of freedom would be associated to the $\pi^{D-1}$ homotopy group and in the ordinary $3+1$ dimensional
space-time monopoles thus appear as the natural choice.

The prototype monopole configuration for gauge theories is the soliton solution of the $SU(2)$ Higgs model with the Higgs field in the adjoint representation \cite{tHooft74, Polyakov}. The general behaviour of this solution can easily be computed when the gauge symmetry is broken to $U(1)$ by the Higgs vacuum expectation value,
the magnetic degree of freedom being the massless unbroken component of the gauge field. Since we do not know the explicit form of the duality transformation, when the gauge symmetry is unbroken it is not clear how to select
the $U(1)$ magnetic subgroup of the gauge group.

In \cite{tHooft81} the possibility was advocated that all the choices of the residual $U(1)$ magnetic gauge symmetry (abelian projections) are equivalent, motivated by the apparent absence of a preferred direction in color space. In particular a convenient way to define monopoles is to use a composite field in the adjoint
representation of the gauge group: monopoles can then be identified with the points in which two eigenvalues of the composite field becomes degenerate.

After the seminal work by DeGrand and Toussaint \cite{DT}, in which a method to detect monopoles in numerically generated lattice configurations was proposed, it was noted that the number and the position of the observed monopoles in a given
configuration strongly depend on the abelian projection adopted. Monopoles thus seem to be gauge invariant objects.
This is unacceptable from a physical point of view: for condensation of monopoles to be at the origin of color confinement, monopoles have to be gauge invariant object, independent of the projection used to define them.

While most of the numerical work related to monopoles in lattice gauge theories was aimed to detect monopoles, in order to confirm or disprove the dual superconductivity picture a better strategy is to compute the vacuum expectation
value of a magnetically charged operator. To define such an operator we have to choose an abelian projection, so also this second strategy can give indications on the equivalence (or not) of the various abelian projections. The
numerical results indicate that, in contrast to monopole detection, monopole condensation is a gauge invariant phenomenon.
We thus have two apparently conflicting results

1) monopole detection depends on the abelian projection
2) monopole condensation is abelian projection independent

In order to reconcile the two points of view and, more important, to gain a better understanding of the role played by the abelian projection in the definition of monopoles, it is convenient to investigate if a gauge covariant quantity
exists that is related to the magnetic monopole. In the first chapter of this thesis we will show that such a quantity is the violation of the non-abelian Bianchi identity and by using its relation to the magnetic current we will show
from a theoretical point of view that monopole condensation is indeed abelian projection independent. In the following we will also analyze the DeGrand-Toussaint recipe to detect monopoles on the lattice and we will show that the
gauge dependence of the number of observed monopole is not in contradiction with the gauge independence of monopoles.

We mentioned above the possibility of constructing a magnetically charged operator to be used to detect monopole condensation. The construction of such an operator in abelian lattice gauge theory is well understood, however the generalization to the non-abelian ones turn out to be far from trivial: the operator
proposed in \cite{dualsup1} seemed to satisfy all the needed requirements, however a more accurate analysis revealed that it is not well defined. This was interpreted as a signal of the failure of the dual superconductivity picture in \cite{GL} but we will show it is just a consequence of the nonlocal nature of the operator, that requires some care in dealing with the $O(a^2)$ lattice artefacts. An improved version of the operator proposed in \cite{dualsup1}
will be presented, together with numerical simulations that show that the problem of the original formulation does not affect the improved version of the monopole operator.

When fermions are coupled to the gauge field the dual superconductivity picture of the vacuum does not require any \emph{ad hoc} modifications, however also the chiral degrees of freedom can play a predominant role in the determination
in the phase diagram, thus making the theoretical analysis more difficult.

An accurate understanding of the QCD phase diagram at non zero temperature is clearly of the utmost importance for its considerable phenomenological implications. Nevertheless there are still some points that are not settled and deserve further investigations.
Among these is the determination of the order of the chiral transition for the case of two massless quark flavours.
Theoretical arguments based on effective chiral Lagrangians restrict the possibilities for the transition to be first order or second order in the 3d $O(4)$ universality class; which of these two possibilities is realized in QCD is a non universal features that need to be investigated by means of numerical simulations. We will present in the following the state of the art of this problem and the investigations we are performing in order to give it a
definite answer.

Simulations of QCD are however extremely computationally demanding and require dedicated machines to be performed.
Since both a precise verification of the theoretical expectation and the numerical evaluation of phenomenologically relevant quantities depend on the precision of these simulations (which in turn depends on the acquired statistics), in order to speed up the computations much attention is being devoted to improve the algorithms and to find new
architectures on which to perform simulations.

In the last few years it was shown that the modern graphics cards, also known as graphic processing units (GPUs), can be used in the high performance computing field with surprisingly good results. For QCD simulations they are typically used together with more traditional architectures in order to speed up just some steps of the computations, mainly the analysis of the generated configuration. We showed that it is
possible to use GPUs to perform complete simulations without the need to rely on the traditional expensive dedicated machines; in the last section we will analyze some of the problems encountered in using GPUs for QCD simulations and the solution strategies adopted to circumvent them.