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 theory , 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 exceptional 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.