The Extremal Nature of Membrane Newton-Cartan Formulations with Exotic Supergravity Theories

We construct a non-relativistic limit of eleven and ten-dimensional supergravity theories from the point of view of the fundamental symmetries, the higher-dimensional effective action, and the equations of motion. This fundamental limit can only be realized in a supersymmetric way provided we impose by hand a set of geometric constraints, invariant under all the symmetries of the non-relativistic the- ory, that define a so-called Dilatation-invariant Superstring Newton-Cartan geometry and Membrane Newton-Cartan expansion. In order to obtain a finite fundamental limit, the field strength of the eleven-dimensional four-form is required to obey a transverse self-duality constraint, ultimately due to the presence of the Chern-Simons term in eleven dimensions. The present research consider a non- relativistic fundamental limit of the bosonic sector of eleven-dimensional supergravity, leading to a theory based on a Covariant Membrane Newton-Cartan Supergeometry. We further show that the Membrane Newton-Cartan theory can be embedded in the U-duality symmetric formulation of excep- tional field theory, demonstrating that it shares the same exceptional Lie algebraic symmetries as the relativistic supergravity, and providing an alternative derivation of the extra Poisson equation