• AdS-CFT
• gauge-gravity duality.
• Quantum Chromodynamics
• string theory
• SU(N) gauge theory
• theta dependence

The dynamics of gluons, the particles mediating strong interactions, is described by a Yang-Mills gauge theory with group SU(3). The high energy behaviour of this model is well understood, as in this regime the coupling is weak and the investigations can be performed by means of perturbation theory. This property is known as asymptotic freedom and implies that in the opposite regime the theory becomes strongly coupled. As a result, perturbation theory fails in the study of large distances physics, where relevant phenomena occur such as confinement, the appearance of bound states (called glueballs) and effects related to topologically non-trivial configurations of the gauge fields.

The parameter coupled to the so-called topological term is known as theta. Studying the theta dependence of physical observables in Yang-Mills is interesting because it allows to enlarge our knowledge of the vacuum structure of the theory and, phenomenologically, because when we couple Yang-Mills to quark matter fields (the theory called Quantum Chromodynamics) it provides us with relevant informations on the eta' sector: topological effects give a mass term to this meson, solving the so-called U(1)A problem of Quantum Chromodynamics, and govern its interactions.

At present, the only direct approach to non-perturbative Yang-Mills and QCD is the use of numerical simulations performed on lattice. In this context, studying theta dependence is challenging: these simulations are performed using Monte Carlo techniques on euclidean spacetime, but the theta term in the euclidean lagrangian contains an imaginary factor. Because of this complication, lattice results are limited to small values of theta, mainly to the first non-trivial orders in the Taylor expansion around zero.

A famous proposal, due to 't Hooft, could help overcoming this and other limits of the lattice approach, at the price of replacing the SU(3) gauge group with SU(N). The theory simplifies when N goes to infinity and one can try to study it in a 1/N expansion. In this limit the perturbative series of the gauge theory can be interpreted as a (yet to be determined) theory of relativistic one-dimensional objects, called strings. The hope is that the non-perturbative features of the gauge theory can be captured by means of a dual description involving strings.

The first concrete realization of 't Hooft idea goes under the name of AdS-CFT correspondence. In its most understood form, it is a conjectured (but widely sustained by many theoretical evidences) equivalence between a Conformal Field Theory, the 4-dimensional maximally supersymmetric SU(N) Yang-Mills, and a 10-dimensional superstring theory on a curved spacetime, whose non-compact part is a 5-dimensional Anti de Sitter space. Remarkably, in the large N limit and strong coupling, the string
theory effectively reduces to a classical theory of gravity: by means of computations in classical gravity in 5 non-compact dimensions we can learn about a strongly interacting 4-dimensional gauge theory. Furthermore, the correspondence is also an explicit realization of the holographic principle, a conjectured property of quantum gravity coming from black hole physics stating that a d-dimensional quantum field theory contains the same amount of information as a (d+1)-dimensional theory with gravity.

Soon after this first example of gauge-gravity duality, there have been several attempts to enlarge it to a more realistic theory,
non conformal and non supersymmetric, that could be the prototype for the gravity dual of an SU(N) gauge theory and fulfil 't Hooft idea. One of the proposals, which we refer to as Holographic Yang-Mills, is due to Witten. It describes a peculiar Yang-Mills coupled to massive matter, but shares with realistic SU(3) gauge theory the relevant low energy features. In this model these properties
can be analitically studied in the regime where the classical gravity is an appropriate approximation.

With the aim to provide new insights into the $\theta$ dependence, in this thesis work we have considered the inclusion of the theta term in Witten's Holographic Yang-Mills model. One of the advantages of this approach is that it works, in principle, at any value of theta. After having rivisited the gravity solution, using the holographic prescriptions coming from string theory we converted these informations into gauge theory results. These are valid at strong coupling and large N. The physical observables we have computed are: the ground state energy density, the string tension, the light scalar glueball mass, the critical temperature for deconfinement (and thus the whole T-theta phase diagram), the entanglement entropy (which has not been studied on the lattice yet). Our results show qualitative agreement with the available lattice data and signal an overall mass scale reduction due to the topological parameter. Moreover, since they are exact in theta, they provide further informations on higher order corrections in the expansion around theta=0 of large N Yang-Mills.