Connections Between Supergravity And Gauge Theory From Supersymmetry And String Theory

In this thesis, I will try to explore some connections between supergravity and gauge theory. I will focus on three aspects: the BPS bound in supersymmetric theories, nonlinear Kaluza-Klein (KK) reductions on spheres and the AdS-CFT correspondence. In the first topic, the study of rigidly supersymmetric field theories gives insight into supergravity theories. In a supersymmetric theory, the BPS bound is a consequence of the susy algebra. We study whether the BPS bound remains satisfied at the quantum level and for this purpose we evaluate quantum corrections to the mass and charge of a soliton in two dimensional toy models. To avoid regularization and renormalization dependence, I introduce a physical principle to fix any ambiguities. The methods described here might be of use for higher dimensional theories and more interesting solitons, like D-branes, if one understands by D-branes solutions of supergravity theories. The second topic is nonlinear KK reductions on spheres of supergravity theories. Nonlinear KK reductions on spheres give gauged supergravities. The main result of the analysis is to find the nonlinear KK ansatz of 11d sugra on S4 and to show that it gives a consistent truncation to 7d gauged sugra. This calculation is checked by reducing to previously known particular cases, and several generalizations for other supergravity theories on spheres are also presented. KK reductions give insight into gauge theories in two ways. One can either take the viewpoint that the Standard Model is embedded in a higher dimensional sugra or string theory, and then a consistent truncation to a gauged sugra gives a way to avoid the complication of working with the whole tower of massive KK states. Or one can use the nonlinear embedding of gauged sugra into string theory for the AdS-CFT correspondence. The AdS-CFT correspondence relates classical gauged sugra correlation functions (in AdS space) to large N and large 't Hooft coupling g2YM N correlation functions in a CFT on the boundary of AdS space. We prove this assertion for the case of 2- and 3- point functions of R-currents in d = 4 N = 4 Super Yang- Mills, dual to AdS space correlation functions of gauge fields in d = 5 N = 8 sugra. I show that the existence of a consistent truncation to gauged sugra implies that one has to use just the gauged sugra action for the calculation of certain CFT correlators at large N