The Pre-geometric Origin of Geometric Trinity of Gravity

Abstract: The so-called Geometric Trinity of Gravity is based on three distinct geometric features of spacetime, i.e.\ curvature, torsion and non-metricity, which give rise to equivalent dynamics for General Relativity (GR), Teleparallel Equivalent of General Relativity (TEGR) and Symmetric Teleparallel Equivalent of General Relativity (STEGR). Pre-geometric gravity, on the other hand, offers a unifying framework from which all metric-affine theories can emerge. Starting from a gauge formulation \textit{à la} Yang--Mills with a Higgs-like field, a mechanism of spontaneous symmetry breaking can give rise to an effective metric as well as to the classical dynamics of the gravitational field. In particular, the emergence of gravity in the spontaneously broken phase is shown to be consistent with all the different formulations of the Geometric Trinity of Gravity, in terms both of actions and of gauge choices for the affine connection. This general result is achieved by deriving and analysing suitable expressions in the unbroken phase for pre-geometric actions and for pre-geometric gauge-fixing conditions respectively.

• The paper demonstrates that spontaneous symmetry breaking in a pre-geometric gauge theory yields effective metric-affine actions representing curvature (GR), torsion (TEGR), and non-metricity (STEGR).
• It constructs explicit mappings between pre-geometric fields and standard gravitational variables, clarifying the gauge-fixing required to manifest the geometric trinity.
• The framework unifies distinct gravitational formulations and suggests novel high-energy insights that could impact quantum gravity research.

The Pre-geometric Origin of the Geometric Trinity of Gravity

Introduction

The paper addresses a foundational aspect of gravity: the emergence of the so-called "geometric trinity"—curvature, torsion, and non-metricity—as effective descriptions within the metric-affine paradigm, proposing their deeper origin in pre-geometric gauge theory. The authors demonstrate that all three avatars of gravity—General Relativity (GR), its teleparallel equivalent (TEGR), and the symmetric teleparallel equivalent (STEGR)—can be understood as effective low-energy limits of a pre-geometric phase, itself described via spontaneous symmetry breaking (SSB) in a gauge-theoretic framework with a Higgs-like field. They establish explicit mappings between the pre-geometric action and the standard metric-affine actions, elucidating the gauge-fixing that yields the familiar geometric trinity.

Pre-geometric and Metric-affine Gravity

Metric-affine gravity, formulated in the Palatini approach, treats the metric $g_{\mu\nu}$ and the affine connection $\Gamma^\lambda_{\mu\nu}$ as independent dynamical variables, with the possibility of nonzero curvature $R$, torsion $T^\lambda_{\mu\nu}$, and non-metricity $Q_{\lambda\mu\nu}$. Conventionally, these geometric structures are realized via the Levi-Civita, Weitzenböck, or coincident connections, underlying GR, TEGR, and STEGR, respectively.

By contrast, pre-geometric gravity postulates no prior metric structure; it is formulated as a gauge theory for the (anti-)de Sitter or general linear group, built upon a differentiable manifold with a gauge potential and a Higgs-like field. The effective metric and gravitational dynamics emerge only after SSB, when the Higgs-like field acquires a vacuum expectation value, breaking the gauge group to the Lorentz or $GL(4)$ subgroup.

Emergence of the Geometric Trinity from Pre-geometry

A principal result demonstrated is that the process of SSB generically yields the field content and dynamics of metric-affine gravity, with curvature, torsion, and non-metricity emerging depending on the gauge fixing and group structure imposed in the unbroken phase.

The authors detail a precise "pre-geometric dictionary" relating pre-geometric structures (built from the gauge potential and the Higgs field) to the metric, tetrads, and affine/spin connections arising after SSB. For example, the components of the gauge field $A_{\mu}^{AB}$ with internal index along the broken direction correspond to tetrads, while those within the residual symmetry become the spin connection. The identification is formalized for both the Einstein–Hilbert and more general invariant structures, including the torsion and non-metricity scalars, by appropriate gauge fixing and construction of pre-geometric Lagrangians.

Figure 1: The octet of gravity—a diagrammatic classification of metric-affine theories based on the number $s$ of nonzero geometric invariants ($R$, $T$, $\Gamma^\lambda_{\mu\nu}$0). The geometric trinity corresponds to $\Gamma^\lambda_{\mu\nu}$1.

In the case of non-metricity, which requires a non-Lorentzian, generally $\Gamma^\lambda_{\mu\nu}$2 spin connection, the unbroken phase must be endowed with the higher $\Gamma^\lambda_{\mu\nu}$3 symmetry, subsequently reduced to $\Gamma^\lambda_{\mu\nu}$4 by SSB. The pre-geometric action includes the Higgs mechanism as an essential ingredient, and additional gauge-fixing ensures the absence of spurious degrees of freedom from the enlarged group.

Construction of Pre-geometric Actions

The paper systematically builds pre-geometric analogs for the gravitational actions corresponding to the trinity of gravity:

• Curvature-based (GR): The pre-geometric Wilczek Lagrangian, possibly augmented to eliminate the cosmological constant, produces the Einstein–Hilbert action under SSB. The Planck mass and cosmological constant emerge dynamically from the VEV and coupling constants.
• Torsion-based (TEGR): Suitable scalar invariants constructed in the pre-geometric phase, via antisymmetrized double covariant derivatives of the Higgs field, yield the torsion scalar $\Gamma^\lambda_{\mu\nu}$5 after symmetry breaking.
• Non-metricity-based (STEGR): By generalizing the pre-geometric gauge potential to $\Gamma^\lambda_{\mu\nu}$6 and imposing the required gauge conditions, the construction gives the non-metricity scalar $\Gamma^\lambda_{\mu\nu}$7.

The approach is fully generalizable. For any $\Gamma^\lambda_{\mu\nu}$8 theory, one can define an associated pre-geometric action, which flows to the corresponding metric-affine theory in the broken phase, as dictated by the form of the scalar invariants in the pre-geometric sector.

Gauge Choices and Physical Implications

The correspondence between gauge choices for the affine connection in the metric-affine setting and gauge-fixing conditions in the pre-geometric theory is explicitly developed. For instance, the Levi-Civita, Weitzenböck, and coincident gauges correspond to specific configurations of the gauge potential in the pre-geometric phase, realized by targeted gauge-fixing aligned with the spontaneous breaking.

The framework demonstrates that metric-affine gravity with any combination of curvature, torsion, or non-metricity (as illustrated by the "octet" in Figure 1) can be unified as emergent phenomena within pre-geometric gauge theory. This synthesis points toward a broader perspective in quantum gravity, wherein the low-energy geometric notions are secondary to pre-geometric, gauge-theoretic origins.

Theoretical and Phenomenological Implications

Theoretically, the pre-geometric scenario recasts gravity as a gauge-Higgs phenomenon, demoting the metric to a derived concept. The unification and mutual equivalence of the trinity indicate that differences between curvature, torsion, and non-metricity are not fundamental, but mere facets of the emergent metric-affine geometry.

Phenomenologically, the framework introduces additional degrees of freedom, specifically those associated with the Higgs-like field, which are inert at low energy but potentially significant near the Planck scale. The emergent view suggests that observable deviations from GR, particularly in the UV/early universe regime, may serve as tests of the underlying pre-geometric scenario. Moreover, the approach circumvents Ostrogradsky instabilities, as the pre-geometric actions—by construction—avoid higher-derivative pathologies.

Future directions include a detailed Hamiltonian analysis to ascertain constraint structure and the true dynamical content; the extension to quantum corrections and the viability of resolutions to the cosmological constant problem; and explorations of coupling to Standard Model matter within the pre-geometric scheme.

Conclusion

The work rigorously establishes that any metric-affine theory of gravity—classified by its nonzero curvature, torsion, and non-metricity—can be interpreted as an emergent low-energy phase of a pre-geometric gauge theory. Spacetime geometry itself is revealed as a dynamical consequence of SSB in a gauge-Higgs system, with the geometric trinity arising from different gauge choices and group reductions. This comprehensive correspondence not only unifies the trinity but generalizes to all $\Gamma^\lambda_{\mu\nu}$9-type theories, advocating for a genuinely pre-geometric, gauge-theoretic foundation for gravitational physics.