Harmonic Analysis and Superconformal Gauge Theories in Three Dimensions   from AdS/CFT Correspondence
  In this thesis I review various aspects of the AdS_4/CFT_3 correspondence, where AdS_4 supergravity arises from compactification of M-theory on a coset space G/H and preserves N<8 supersymmetries. One focal point of my review is that the complete spectrum of such N-extended supergravity can be determined by means of harmonic analysis on the homogeneous space G/H. This spectrum can be matched with the candidate conformal theory on the boundary, in this way providing very non-trivial checks of the AdS/CFT correspondence. Furthermore, this spectrum can be useful to study the representation theory of N-extended supersymmetry on AdS_4, namely representation theory for the superalgebra of Osp(N|4). I review Osp(N|4) representation theory, and derive the translation vocabulary between states of AdS_4 supergravity and conformal superfields on the boundary, by means of the double interpretation of Osp(N|4) unitary irreducible representations. In the cases of N=2,3, using results from harmonic analysis I give the complete structure of all supermultiplets. Harmonic analysis as a method to determine spectra of supergravity compactifications is explained. Calculations are explicitly performed in the case G/H=M^111=(SU(3)xSU(2)xU(1))/(SU(2)xU(1)xU(1)), preserving N=2 supersymmetries. For this manifold, and also for the case G/H=Q^111=(SU(2)x SU(2)xSU(2))/(U(1)xU(1)), I describe the construction of a candidate dual superconformal theory on the boundary. This construction is based on geometrical insight provided by the properties of the metric cone C(G/H) transverse to the M2-brane worldvolume. 
Leonardo Gualtieri