Curvatures and Non-metricities in the Non-Relativistic Limit of Bosonic Supergravity

TL;DR

The paper address the challenge of formulating a diffeomorphism-covariant non-relativistic limit of bosonic supergravity by introducing a torsionless, yet non-metric, Newton–Cartan–type geometry with fixed non-metricities that remain consistent with the relativistic metric. It then provides explicit covariant decompositions of the relativistic curvatures into NR data and derives a fully covariant NR two-derivative bosonic Lagrangian, while outlining systematic procedures to extract finite contributions to higher-derivative $\alpha'$ corrections and to general $f(R,Q)$ NR theories. A key result is the demonstration that the NR formulation is equivalent to the intrinsic torsion framework of string NC geometry and that the approach yields a practical and covariant path to organize NR higher-derivative terms. The framework promises a robust geometric toolkit for NR gravity and string theory in the NR limit, with potential extensions to heterotic theories and more general non-relativistic geometries, though full control of divergences at four-derivative order remains an open challenge.

Abstract

We construct a diffeomorphism-covariant formulation of the non-relativistic (NR) limit of bosonic supergravity. This formulation is particularly useful for decomposing relativistic tensors, such as powers of the Riemann tensor, in a manifest covariant form with respect to the NR degrees of freedom. The construction is purely geometrical and is based on a torsionless connection. The non-metricities are associated with the gravitational fields of the theory, $τ_{μν}, h_{μν}, τ^{μν}$ and $h^{μν}$, and are fixed by requiring compatibility with the relativistic metric. We provide a fully covariant decomposition of the relativistic Riemann tensor, Ricci tensor, and scalar curvature. Our results establish an equivalence between the proposed construction and the intrinsic torsion framework of string Newton-Cartan geometry. We also discuss potential applications, including a manifestly diffeomorphism-covariant rewriting of the two-derivative finite bosonic supergravity Lagrangian under the NR limit, a powerful simplification in deriving bosonic $α'$-corrections under the same limit, and extensions to more general $f(R,Q)$ Newton-Cartan geometries.