Higgsing N = 4 Super Yang-Mills theory: massive scattering and defect theory

The subject of this thesis is the study of N=4 Super Yang-Mills (SYM), which is the maximally super-conformal field theory in four dimensions, in presence of spontaneous symmetry breaking. When the scalar fields of the theory have a non-trivial vacuum expectation value (vev), the infrared regime is regularized, while preserving a very important property of scattering amplitudes, the dual conformal invariance. It is also possible to partially preserve, in particular cases, the space-time conformal invariance. Interestingly, the prescriptions of the AdS/CFT correspondence can be used and the behavior of the theory at strong coupling is examined by considering a weakly coupled supergravity theory. In particular, two different kinds of symmetry breaking have been studied here. First, it has been analyzed the case of a constant vev, extending some known results for scattering amplitudes by giving different vevs to all the possible scalar directions. In this setting, we considered the four scalars scattering amplitude up to three loops to extract the generalized cusp anomalous dimension. This quantity, describing the ultraviolet divergencies of Wilson loop operators, has been discovered to characterize also the infrared divergencies of scattering amplitudes. We then utilized the cusp anomalous dimension to study the behavior of the spectrum of the bound states of Higgsed N=4 SYM, by considering the amplitudes in their Regge limit. Moreover, we studied the string dual of N=4 SYM through the AdS/CFT correspondence: in this case, the dual of the Wilson Loop operator turns out to be the minimal area swept by the string whose boundary lives on the Wilson loop contour located on the border of AdS. Thus, the cusp anomalous dimension at strong coupling can be determined by a classical string calculation. In particular, the results obtained in literature have been exploited to obtain the spectrum of N=4 SYM also at strong coupling. The second case considered in this thesis concerns conformal defect theories (dCFT). In the usual four-dimensional space-time, a defect is introduced as an hyperplane placed, let's say, at x=0. From one side of the defect, we have the standard N=4 SYM theory with gauge group SU(N), while, on the other side, we have an Higgsed version of N=4 SYM with gauge group SU(N-k). In particular, only three scalars of the theory have a non-trivial vev, that depends on the coordinate x. The theory still preserves a three-dimensional conformal symmetry and we can refer to it as a conformal defect theory. From the string theory point of view, an AdS/dCFT conjecture can be applied. In particular, the string dual geometry corresponds to a stack of D3-branes intersected by an orthogonal D5-brane, that accounts for the defect in the dCFT. In this framework, a circular Wilson loop parallel to the defect has been considered from both sides of the conjecture. The quantum expectation value of the Wilson loop has been firstly derived in the dCFT, considering two different kind of parametrization for the scalar couplings. One case corresponds to the Maldacena-Wilson loop, while the other one is a realization of the Zarembo-Wilson loop in a defect theory. The results obtained from the dCFT calculation have been compared with the solutions obtained in the dual string theory. In the first case, we achieved a general solution of the string action minimization problem, extending previous investigations considered in literature.