High-Energy Scattering in strongly coupled $\mathcal{N}=4$ super Yang-Mills Theory

This thesis concerns itself with the analytic structure of scattering amplitudes in stronglycoupled N = 4 super Yang-Mills theory (abbreviated N = 4 SYM) in the multi-Reggelimit. Through the AdS/CFT-correspondence observables in strongly coupled N = 4 SYMare accessible via dual calculations in a weakly coupled string theory on an AdS_5×S^5 -geometry, in which observables can be calculated using standard perturbation theory. Inparticular, the calculation of the leading order of the n-gluon amplitude in N = 4 SYMat strong coupling corresponds to the calculation of a minimal surface embedded intoAdS_5 . This surface ends on the concatenation of the gluon momenta, which is a light-like curve. The calculation of the minimal surface area can be reduced to finding thesolution of a set of non-linear, coupled integral equations, which have no analytic solutionin arbitrary kinematics. In this thesis, we therefore specialise to the multi-Regge limit,the n-particle generalisation of the Regge limit. This limit is especially interesting as evenin the description of scattering amplitudes in weakly coupled N = 4 SYM in this limit acertain set of Feynman diagrams has to be resummed. This description organises itself intoorders of logarithms of the energy involved in the scattering process. In this expansioneach order in logarithms includes terms from every order in the coupling constant andtherefore contains information about the strong coupling sector of the theory, albeit in avery specific way. This raises the central question of this thesis, which is how much of theanalytic structure of the scattering amplitudes in the multi-Regge limit is preserved as wego to the strong coupling regime. We show that the equations governing the area of theminimal surface simplify drastically in the multi-Regge limit, which allows us to obtainanalytic results for the scattering amplitudes. We develop an algorithm for the calculationof scattering amplitudes in the multi-Regge limit and apply it to the special cases of the 6-and 7-gluon amplitude. Our results show that for the cases under study the factorisationof the amplitude as predicted by Regge theory is also preserved in the strong couplinglimit