A non-relativistic limit of M-theory and 11-dimensional membrane Newton-Cartan geometry

TL;DR

This paper develops a non-relativistic limit of eleven-dimensional supergravity, leading to a membrane Newton-Cartan geometry with Galilean symmetries, and explores its reduction and symmetry properties within exceptional field theory.

Contribution

It introduces a novel non-relativistic M-theory framework based on membrane Newton-Cartan geometry, extending stringy Newton-Cartan concepts to eleven dimensions.

Findings

Derived a finite non-relativistic limit with a covariant membrane Newton-Cartan geometry.

Connected the theory to known stringy Newton-Cartan and D2-brane theories via dimensional reduction.

Embedded the non-relativistic theory into exceptional field theory, revealing shared symmetries.

Abstract

We consider a non-relativistic limit of the bosonic sector of eleven-dimensional supergravity, leading to a theory based on a covariant `membrane Newton-Cartan' (MNC) geometry. The local tangent space is split into three `longitudinal' and eight `transverse' directions, related only by Galilean rather than Lorentzian symmetries. This generalises the ten-dimensional stringy Newton-Cartan (SNC) theory. In order to obtain a finite 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 finite action then gives a set of equations that is invariant under longitudinal and transverse rotations, Galilean boosts and local dilatations. We supplement these equations with an extra Poisson equation, coming from the subleading action. Reduction along a longitudinal direction gives the known SNC theory with the addition of RR gauge fields, while reducing along a transverse direction yields a new non-relativistic theory associated to D2 branes. We further show that the MNC 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.