Non-relativistic strings and membranes

The main focus of this thesis is the derivation of non-relativistic particle, string and membrane actions and equations of motion. In particular, the theories we consider are based on (generalizations of) the Galilean algebra and Newton-Cartan gravity. Our starting point will be computing the beta functions of a non-relativistic string theory with Torsional Newton Cartan symmetries in the target space. In analogy with usual relativistic string theory, the equations obtained by setting these beta functions to zero are then interpreted as the target space equations of motion for (Type I) Torsional Newton Cartan gravity. Subsequently, we derive a target space action for this theory, as well as for other non-Riemannian theories that are closely related to it: Carrollian and Stringy Newton Cartan gravity. These actions correspond to different non-Riemannian limits of the bosonic sector of the usual ten-dimensional supergravity actions. Finally, we study a non-relativistic limit of M-Theory, whose low energy limit gives a theory that we dub Membrane Newton Cartan gravity, which should be thought of as the non-relativistic limit of the bosonic sector of eleven-dimensional supergravity. Two conceptually different dimensional reductions can then be performed on MNC gravity: one of them turns out to be precisely the same SNC gravity mentioned above, while the other one is a novel type of non-relativistic theory associated to D2 branes.