Supersymmetry on curved spaces and holography

This thesis deals with superconformal and supersymmetric field theories on curved spaces with a view toward applications to holography and localisation. It contains two fairly different parts. In the first (and main) part we classify Euclidean and Lorentzian four-manifolds with some preserved N=1,2 supersymmetry, and Euclidean three-manifolds with some preserved N=2 supersymmetry. We take a holographic approach, starting with manifolds that preserve superconformal symmetries. Preserved supersymmetry for asymptotically locally AdS solutions implies the existence of a certain (generalised) ``conformal Killing spinor'' on the boundary. In the non-conformal case a closely related spinor exists, which also will be discussed. In this thesis we classify the manifolds in three and four dimensions that admit such spinors. In particular we find for the case with four supercharges that supersymmetry can be preserved in four dimensions on every Euclidean complex manifold and on any Lorentzian space-time with a null conformal Killing vector. In three Euclidean dimensions we find a condition very similar to complexity in four dimensions. When the field theory has eight supercharges, supersymmetry can generically be preserved on manifolds with time-like conformal Killing vectors; there are singular cases depending on the signature, in the Lorentzian there is a degenerate case reducing to the N=1 analysis, in the Euclidean there is a degenerate case corresponding to the topological twist. The supersymmetric curvature couplings are systematically understood in the rigid limit of supergravity. We give explicit formulae for the background fields that one needs to turn on in order to preserve some supersymmetry. Putting supersymmetric field theories on curved manifolds has led to interesting results over the past years. In the second part of the thesis we analyse a matrix model for the partition function of three dimensional field theories on S^3, which was obtained by supersymmetric localisation. In the large N limit one can evaluate the matrix model, allowing us to perform a non-classical and non-perturbative check of the AdS_4/CFT_3 correspondence and Seiberg duality. In particular, we compute the large-N free energy of various three dimensional quiver gauge theories with arbitrary R-charges, which are dual to M-theory on Sasaki-Einstein seven-manifolds. In particular, we check that the free energy functional depending on the R-charges is minimised for the exact R-symmetry, an extremisation that is dual to the volume-minimisation of the Sasaki-Einstein manifold in the gravity sector.