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

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.

• The paper presents a covariant geometric framework for the non-relativistic limit of bosonic supergravity, focusing on decomposing relativistic curvature into Newton–Cartan structures.
• It introduces a torsionless affine connection with specific non-metricities to maintain finite action and local boost symmetry in the c → ∞ limit.
• The study extends the framework to higher-derivative corrections and generalized non-relativistic gravitational theories, enabling systematic analyses of NR limits.

Covariant Geometric Structures in Non-Relativistic Limits of Bosonic Supergravity

Overview and Motivation

The paper "Curvatures and Non-metricities in the Non-Relativistic Limit of Bosonic Supergravity" (2601.03342) offers a systematic, diffeomorphism-covariant reformulation of the non-relativistic (NR) limit of bosonic supergravity, with an emphasis on the geometric decomposition of relativistic curvature invariants into non-relativistic structures. Starting from the bosonic NS-NS sector (metric, Kalb-Ramond two-form, dilaton), the study addresses the technical hurdles associated with performing a $c \to \infty$ expansion—namely, the divergences encountered in the Levi–Civita connection and Riemann tensor—and provides general formalism for organizing the emergent curvatures and non-metricities in Newton–Cartan geometry.

Torsionless Connections and Non-Metric Geometry

A major technical result of the study is the explicit construction of a torsionless affine connection adapted to the Newton–Cartan decomposition of the metric. Unlike the relativistic Levi–Civita connection, this connection is not metric compatible with respect to the non-relativistic fields. Crucially, the covariant derivatives of the fundamental longitudinal and transverse tensors ($\tau_{\mu\nu}$, $h_{\mu\nu}$, and their inverses) are governed by unique non-metricity tensors, fixed by imposing compatibility with the relativistic metric in the NR limit. This prescription is essential to maintain the finite, well-defined structure of the action and its curvature invariants throughout the contraction process.

The paper demonstrates that, under this construction, the non-metricities are generally nonvanishing and transform nontrivially under local boosts, implying they cannot be freely set to zero without explicitly breaking boost invariance. This is a key departure from prior approaches that imposed strict metric compatibility in NR geometry.

Decomposition of Relativistic Curvatures

The relativistic Riemann, Ricci, and scalar curvature tensors are expanded in powers of $c$. The paper provides explicit covariant expressions for each order in the expansion:

• The leading and subleading contributions—designated as $c^n$ terms—are systematically written using the new connection, so that each piece is manifestly a tensor with respect to diffeomorphisms.
• An explicit mapping is established between the various orders in the Riemann tensor and the non-relativistic covariant data ($\tau_\mu^a, h_{\mu\nu},$ etc.), including the proper treatment of nontrivial non-metricities at each order.
• The study identifies a structural interchange symmetry in the curvature decomposition under exchange $\tau_{\mu\nu} \leftrightarrow h_{\mu\nu}$ and $h^{\mu\nu} \leftrightarrow \tau^{\mu\nu}$ between the highest and lowest order terms.

This computational machinery facilitates the explicit and systematic identification of both divergent and finite curvature expressions in the NR limit.

Applications to Supergravity Lagrangians and Higher-Derivative Corrections

Covariant Non-Relativistic Lagrangian

The analysis yields a fully covariant reformulation of the bosonic two-derivative supergravity Lagrangian in the NR limit. Every term—curvature, dilaton, and flux sector—is expressed in terms of the Newton–Cartan fields and their non-metric covariant derivatives. The main contributions consist of the Ricci tensor, terms quadratic in non-metricity, and modified field strengths, all arranged to persist as finite, well-defined objects as $c \to \infty$.

An explicit equivalence is drawn between this construction and the intrinsic torsion framework of string Newton–Cartan geometry, establishing firm geometric consistency with the existing literature.

Four-Derivative and $\alpha'$-Corrections

The formalism is directly applied to derive the covariant structure of higher-derivative (notably $\alpha'$) corrections. The Metsaev–Tseytlin Lagrangian, which incorporates four-derivative terms (e.g., $R_{\mu\nu\rho\sigma} R^{\mu\nu\rho\sigma}$, mixed $H^2 R$, and quartic $H$-flux couplings), is decomposed in the NR limit via the presented framework, permitting systematic identification of all finite contributions at each order in $c$. The explicit treatment of non-metricities proves instrumental in organizing the finite and divergent components.

While control over all sources of divergences in these higher-derivative actions is not yet fully achieved, the work enables concrete progress by demonstrating a tractable, covariant method for further investigation.

Generalizations: $f(R,Q)$ Theories and Arbitrary Non-Metricities

The paper discusses extensions toward more general NR gravitational theories (beyond bosonic supergravity), in which non-metricities may be allowed to vary arbitrarily or set to zero—resulting in Lagrangians with more general dependence on curvature invariants, typically of the schematic form $$L_{NR} \sim a_1 R_{\mu\nu}h^{\mu\nu} + a_2 R_{\mu\nu} \tau^{\mu\nu}$$. However, these choices generally result in explicit breaking of local boost symmetry, in contrast with the physically motivated supergravity limit.

Theoretical and Practical Implications

This methodology closes a persistent technical gap in the non-relativistic contraction of supergravity, providing a unique, geometric foundation for formulating NR gravity theories with manifest diffeomorphism invariance. Practically, it offers a streamlined route to compute NR limits of curvature invariants and higher-derivative corrections, which are otherwise cumbersome using standard geometric tools. The manifestly covariant formalism is particularly relevant for string theory and double field theory, where $c \to \infty$ limits—and resulting non-Riemannian and Newton–Cartan structures—are integral to non-relativistic string backgrounds and their dualities.

The formalism is also expected to facilitate further studies of the cancellation of divergences in higher-derivative corrections, as well as potential generalizations to the heterotic and type II supergravity sectors, and non-relativistic generalizations of $f(R,Q)$-type actions.

Conclusion

The paper presents a rigorous, covariant geometric formalism for the NR limit of bosonic supergravity, anchored in Newton–Cartan geometry with non-metricities fixed by relativistic compatibility. It establishes covariant decompositions for metric expansions, connections, and curvatures; clarifies the role of non-metricity in maintaining finite, boost-invariant actions; and applies the framework to the NR dynamics of both leading and higher-derivative sectors. This work not only organizes and systematizes the construction of NR supergravity actions but also provides essential tools for future exploration of NR limits in string theory and related areas.