Gauge-Theoretic Gravity: Concepts & Formulations

• Gauge-theoretic gravity is a reformulation of gravitational interactions using local symmetry groups, aligning gravity with gauge theory principles.
• It employs gauge connections for translations, Lorentz, and conformal transformations to generate curvature, torsion, and non-metricity in geometric frameworks.
• This approach integrates topological and quantum corrections through actions like Einstein–Cartan and MacDowell–Mansouri, offering a unified pathway to quantum gravity.

Gauge-theoretic gravity encompasses a hierarchy of frameworks in which gravity is reformulated as a gauge theory associated with local symmetry groups, paralleling the structure of internal gauge theories such as those underlying the Standard Model. In these constructions, gravitational interactions are mediated by gauge connections corresponding to spacetime symmetries—translations, Lorentz rotations, dilatations, or more general linear and conformal transformations. The resulting geometric descriptions, such as Riemann–Cartan and metric-affine geometry, naturally accommodate both curvature and torsion and motivate approaches to quantum gravity, unification, and symmetry breaking. The following systematically develops the principles, formulations, and key results of gauge-theoretic gravity, including its quantum and topological aspects.

1. Symmetry Groups and Gauge Principles in Gravity

The foundational insight is the promotion of rigid spacetime symmetry groups to local gauge groups. The principal cases are:

• Poincaré group $\operatorname{ISO}(1,3)$: Local translations and Lorentz transformations. Gauging this leads to the Einstein–Cartan framework or more general Poincaré gauge theories, with independent coframe (tetrad) and spin connection fields (Hehl, 2012, Santos, 2019, Nester et al., 2016).
• De Sitter/Anti-de Sitter groups $\operatorname{SO}(4,1)$, $\operatorname{SO}(3,2)$: Embedding gravity in a larger gauge structure captures the cosmological constant and allows for spontaneous symmetry breaking to the Lorentz subgroup (Randono, 2010, Sobreiro et al., 2012).
• Affine and Metric-Affine groups $\operatorname{Aff}(n, \mathbb{R})$, $\operatorname{GL}(n, \mathbb{R})$: Yielding Riemann–Cartan or metric-affine spaces with independent torsion and non-metricity (Tjandra et al., 30 Sep 2025, Sardanashvily, 2016).
• Conformal (Weyl and SO(2,4)) group: Including local scale and special conformal transformations leads to frameworks for conformal gravity and Weyl–Cartan geometry (Manolakos et al., 2019, Sobreiro et al., 2012, Fukuyama, 30 May 2026).

The gauge principle operates analogously to Yang–Mills: for each generator, a corresponding gauge field one-form is introduced; the field strengths (curvature, torsion, non-metricity) are defined by the structure equations and Bianchi identities (Hehl, 2012, Nester et al., 2016). Physical motivations include the natural coupling of fermion spin to torsion, emergence of higher-derivative and parity-odd invariants (e.g. quadratic curvature or torsion terms), and the prospect of unification with other gauge interactions in a principal bundle framework (Manolakos et al., 2019, Tjandra et al., 30 Sep 2025).

2. Gauge Connections, Field Strengths, and Geometric Structures

For the spacetime manifold $M$, the canonical gauge-theoretic variables are:

• Coframe (tetrad) $e^{a} = e^{a}{}_{\mu}\,dx^{\mu}$: Gauging translations.
• Spin connection $\omega^{ab} = -\omega^{ba}$: Gauging local Lorentz.
• General linear connection $\omega^{\alpha}{}_{\beta}$ or affine connection $\widetilde{\omega}$: For $\operatorname{SO}(4,1)$0 or $\operatorname{SO}(4,1)$1.

Their field strengths are:

$\operatorname{SO}(4,1)$2

$\operatorname{SO}(4,1)$3

More generally, in metric-affine gauge, the non-metricity 2-form

$\operatorname{SO}(4,1)$4

arises as the field strength for metric compatibility (Tjandra et al., 30 Sep 2025, Sardanashvily, 2016, Hehl, 2012).

In higher or lower spacetime dimensions, and for alternative gauge algebras, this basic structure adapts with suitable index ranges and gauge groups (Randono, 2010, Kerr, 2014). In the teleparallel ("Weitzenböck") case, the connection is globally flat ($\operatorname{SO}(4,1)$5), and gravity is attributed solely to torsion (Hehl, 2012, Tjandra et al., 30 Sep 2025).

3. Gauge-Invariant Actions and Dynamical Principles

Dynamical actions are constructed as diffeomorphism-invariant integrals of Lagrangian $\operatorname{SO}(4,1)$6-forms built from the gauge field strengths and wedge products of coframes. Prototypical forms include:

• Einstein–Cartan (Palatini) action: $\operatorname{SO}(4,1)$7
• General quadratic action: $\operatorname{SO}(4,1)$8 In special limits, one recovers Standard GR, Einstein–Cartan theory, or the teleparallel equivalent of General Relativity (TEGR) (Hehl, 2012, Nester et al., 2016, Tjandra et al., 30 Sep 2025).
• MacDowell–Mansouri action (for (A)dS gauge group): $\operatorname{SO}(4,1)$9 with $\operatorname{SO}(3,2)$0. After symmetry breaking, this yields EC + cosmological term + topological invariants (Randono, 2010).
• Pure-connection (SU(2)) gauge-theoretic gravity: $\operatorname{SO}(3,2)$1 where $\operatorname{SO}(3,2)$2 is a homogeneous, adjoint-invariant function of the $\operatorname{SO}(3,2)$3 curvature, organizing infinite-parameter families of gravity theories with two propagating polarizations (Krasnov, 2012, Krasnov, 2011).
• Affine Gauge Theory (AGT) action: $\operatorname{SO}(3,2)$4 where $\operatorname{SO}(3,2)$5 is the curvature and torsion 2-form of the affine connection (Tjandra et al., 30 Sep 2025).

Topological terms (Nieh–Yan, Pontryagin, Euler, Holst) can be included; classically they do not affect the field equations but impact the symplectic structure and quantum sector labels (Immirzi parameter, $\operatorname{SO}(3,2)$6-angles) (Sengupta, 2011, Randono, 2010).

4. Field Equations,