Non-relativistic supergravity in three space-time dimensions

This year Einstein's theory of general relativity celebrates its one hundredth birthday. It supersedes the non-relativistic Newtonian theory of gravity in two aspects: i) there is a limiting velocity, nothing can move quicker than the speed of light and ii) the theory is valid in arbitrary coordinate systems. While point i) is by definition the necessary difference between relativistic and non-relativistic theories, one might wonder if there exists a version of Newtonian gravity thatallows point ii), a theory of non-relativistic gravity that is invariant under general coordinate transformations. Indeed, such a theory was constructed a few years after Einstein's theory of general relativity and it is calledNewton-Cartan gravity. This theory finds applications e.g. in models of condensed matter physics that describe systems which exhibitnon-relativistic symmetries. It is also used in generalizations of the holographic principle to non-relativistic settings.With these motivations in mind we study Newton-Cartan structures in this thesis. We present a non-relativistic limiting procedure that enables us to get Newton-Cartan gravity from Einstein's relativistic theory. In addition we focus our study on supersymmetric extensions of Newton-Cartan gravity. We study non-relativistic versions of cosmological and conformalsupergravity in three dimensions. We also look at off-shell formulations and a non-relativistic version of the superconformal tensor calculus which we call Schroedinger tensor calculus