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

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 ($τ_{μν}$, $h_{μν}$, $τ^{μν}$, $h^{μν}$) 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 intrinsic torsion framework of string Newton–Cartan geometry and the proposed construction and. 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.

💡 Research Summary

The paper presents a fully diffeomorphism‑covariant formulation of the non‑relativistic (NR) limit of bosonic supergravity, addressing a long‑standing obstacle: the relativistic Levi‑Civita connection diverges when expressed solely in terms of Newton‑Cartan variables. Starting from the 26‑dimensional bosonic NS‑NS sector (metric ĝ_{μν}, Kalb‑Ramond two‑form \hat B_{μν}, and dilaton \hat φ), the authors expand these fields in powers of the speed of light c: \