Quantum Field Theory, Lattice and Strings

The development of non-perturbative methods to extract informations from the strong coupling regime of Quantum Field Theories (QFT) still represents a challenging problem. In particular it lacks an analytical description of important aspects of gauge theories, as the confinement of quarks or the phase structure of quantum chromodynamics. At the same time a "natural" formulation of the physics beyond the Standard Model would require a new symmetry, exchanging fermions and bosons, known as Supersymmetry. Supersymmetry provides also a favorable arena in which to study, in a simplified context, the mentioned longstanding problems, due to the severe constraints imposed by its deep mathematical structure. We study non perturbative analytical approaches to QFT, in supersymmetric and non-supersymmetric models, using methods and advanced mathematical techniques as localization of path-integrals, integrability, matrix-models, conformal field theory and resurgence. We also examine the related perturbative aspects in scattering amplitudes, Wilson loops and correlation functions of composite operators. We also explore the possible insertions of defects, boundaries and non-trivial background fields.

The lattice regularization provides one of the possible formulation of Quantum Field Theories (QFT). This fully captures the spirit of the functional integral: with a (euclidean) discretization of space-time in place, the conceptual connection to statistical mechanics and the renormalization group is evident. Being a fully non-perturbative formulation of QFT, the lattice opens the way to tackling problems for which perturbative methods (so successful in certain domains) have little to say. The typical example is the strongly interacting regime of QCD, with topics including confinement, mass spectrum, matrix elements setting the strength of certain interactions, thermodynamics properties of QCD and its phase diagram. The latter is a very good example of the rules of the game. The lattice formulation opens the way to computer simulations via Monte Carlo methods and much of the success of lattice gauge theories depends on the effectiveness of brilliant  algorithmic solutions implemented on powerful (super)computers (the most powerful we can think of, actually). Still, this is not at all enough. Problems like the study of the phase diagram requires more powerful theoretical solutions: we need different formulation of QFT to overcome what is known as the sign problem. This is one of the topic on which the Parma group has been working in recent years: the so-called thimble regularization complexifies the manifold on which the functional integral is defined, with the result that the original domain of integration is traded for another (non-trivial) one, on which the sign problem disappears. Another theoretical tool that was born in Parma is a numerical-stochastic formulation of perturbation theory (NSPT), which enables to reach unprecedented high orders in computing perturbative series. This gives access to the region in which non-perturbative physics is deeply connected to the asymptotic nature of perturbation theory, a regime that in recent years has been investigated in the framework of the so-called Resurgence theory.

The quantization of gravity is probably the most elusive problem in modern high energy physics: the usual quantum field theoretical methods seem to fail when energies of the order of Planck scale are approached. A possible solution is represented by String Theory, where point particles appear as the low-energy limit of fundamental one-dimensional objects call "strings", whose excitations describe all the known elementary particles and include graviton. The quantum theory of strings is mathematically difficult and largely unknown but recently it has provided a number of tools to explore aspects of quantum gravity (black hole entropy) and of gauge theories (gauge/gravity duality). The famous AdS/CFT correspondence (and example of such duality) suggests to describe gauge theories in the strong coupling regime as weakly coupled gravitational systems and produced a large number of theoretical results. We study in this context the behavior of computable observables and their strong coupling limit: for example the potentials between particle-antiparticle systems, scattering amplitudes, Wilson loops, thermodynamical entropies and their gravitational dual description.