Einstein–Cartan–Palatini Gravity

• Einstein–Cartan–Palatini gravity is a metric-affine formulation that unifies Einstein–Hilbert, Cartan, and Palatini approaches by treating the vielbein and connection as independent fields.
• The framework harnesses gauge theoretic and multisymplectic methods to analyze gravitational dynamics in the presence of spinorial matter and higher-curvature corrections.
• Extensions of the theory address effective four-fermion interactions, non-geometric fluxes, and quantum corrections relevant to Regge theory and inflationary cosmology.

Einstein–Cartan–Palatini gravity denotes the class of metric-affine gravitational theories where the independent dynamical variables are a vielbein (coframe) field and a linear connection, typically in the presence of a Lorentz-structure. This formalism unifies and generalizes the Einstein–Hilbert (metric), Einstein–Cartan (with torsion), and Palatini (first-order, independent connection) approaches. It provides a natural framework for gauge-theoretic, variational, and geometric analysis of gravitational dynamics, especially in the presence of spinorial matter, higher-curvature corrections, or string-theoretic flux backgrounds.

1. Fundamental Formulation and Variational Principles

The Einstein–Cartan–Palatini (ECP) action employs independent vielbein $e^a$ and spin connection $\omega^{ab}$ fields, where $a,b=0,1,2,3$ index the local Lorentz frame. The Palatini action in four spacetime dimensions is

$$S_{\mathrm{P}}[e,\omega] = m_p^2 \int \epsilon_{abcd}\; e^a \wedge e^b \wedge R^{cd}(\omega)$$

where $R^{cd}$ is the $\mathfrak{so}(3,1)$-curvature two-form. Variation with respect to $\omega$ yields the Cartan equation for torsion: $T^a := De^a = d e^a + \omega^a{}_b \wedge e^b = 0$ implying that $\omega$ is the (unique) Levi–Civita connection for non-degenerate $e^a$. Variation with respect to $e^a$ produces the Einstein field equation (in vacuum),

$$\epsilon_{abcd}\,e^b\wedge R^{cd}=0 \implies G_{\mu\nu}=0$$

Thus, for purely gravitational systems, ECP theory is classically equivalent to metric GR; the formalism only departs from standard GR in the presence of matter fields—particularly those with spin (Dadhich et al., 2010, Catren, 2014, Capriotti, 2019, Robinson, 1 May 2025).

2. Geometric, Gauge-Theoretic, and Multisymplectic Structures

ECP theory is fundamentally a gauge theory for the Poincaré group (or de Sitter, in the appropriate generalization). The variables $e^a$ and $\omega^{ab}$ unify as a Cartan connection $A = \omega + \theta$ valued in $\mathfrak{so}(1,3)\oplus \mathbb{R}^4$ on an $SO(1,3)$-principal bundle; the curvature $F(A) = R + T$ splits into rotational (Lorentz) and translational (torsion) components. Partial gauge fixing (reduction of the principal bundle from Poincaré to Lorentz) leaves local Lorentz invariance unbroken, while the broken translational symmetry reappears as diffeomorphism invariance on the spacetime manifold (Catren, 2014).

The multisymplectic (covariant Hamiltonian) formulation treats the frame $e$, the connection $\omega$, and their multimomenta within the jet bundle formalism, subject to primary constraints enforcing vanishing torsion and metric-compatibility. The entire system is gauge-invariant under local Lorentz and projective transformations of the connection, the latter being gauge and not corresponding to physical degrees of freedom (Gaset et al., 2018, Dadhich et al., 2010, G. et al., 2021).

3. Torsion, Spin, and Coupling to Matter

When minimally coupled to Dirac spinor fields, the independent variation of $\omega$ yields a nontrivial algebraic equation: $$T^\lambda{}_{\mu\nu} + 2\delta^\lambda{}_{[\mu} T^\sigma{}_{\nu]\sigma} = \kappa S_{\mu\nu}{}^\lambda$$ where $S^{\mu\nu\lambda}$ is the spin density. For Dirac fields,

$$S^{\mu\nu\lambda} = -\frac{i\hbar c}{4}\bar\psi \gamma^{[\mu}\gamma^\nu\gamma^{\lambda]}\psi$$

So torsion is completely determined, non-propagating, and algebraic in the spin density ("Cartan's equation"). Substituting the solution for torsion back into the gravitational and matter actions produces effective four-fermion interactions: the Hehl–Datta equation for the Dirac sector, and a spin–spin contact interaction quadratic in $S_{\mu\nu}{}^\lambda$ in the Einstein equation. In vacuum (no spin), torsion vanishes (Dadhich et al., 2010, Khanapurkar, 2018, Dereli et al., 2010).

4. Projective Symmetry, Gauge Redundancy, and Equivalence with Metric GR

The ECP formalism reveals that the connection equation only determines $\Gamma^\lambda{}_{\mu\nu}$ up to a projective (pure-trace) transformation: $$\Gamma^\lambda{}_{\mu\nu} \to \Gamma^\lambda{}_{\mu\nu} + \delta^\lambda_\mu U_\nu$$ where $U_\mu$ is an arbitrary one-form. This freedom is a genuine gauge symmetry and leaves the action invariant, ensuring that the non-metric parts of the affine connection do not contribute physical degrees of freedom. The equivalence with Einstein–Hilbert gravity follows: classical solutions of ECP theory modulo projective symmetry correspond bijectively to Einstein-metric solutions (Dadhich et al., 2010, G. et al., 2021, Gaset et al., 2018, Capriotti, 2012).

Boundary conditions and the covariant phase space (using the relative variational bicomplex) are fully consistent with this equivalence: the presymplectic form and thus the physical phase space are unaltered by reformulations from pure metric to metric-affine variables, provided gauge and projective degrees of freedom are properly accounted for (G. et al., 2021).

5. Extensions: Higher-Curvature, Stringy Fluxes, and Lovelock–Cartan Gravity

The ECP formalism generalizes to accommodate higher-curvature terms (e.g., Gauss–Bonnet, Lovelock invariants, or $\alpha'$ corrections in string theory), as well as torsional deformations involving three-form fluxes $H_{abc}$. In the Palatini–Lovelock–Cartan class, the action, when independently varied with respect to metric and connection, enforces strong "flux-Bianchi" constraints: $$\nabla_a H_{bcd} = 0,\quad H_{[ab}{}^{e} H_{c]de}=0$$ These ensure Bianchi identities and Jacobi relations essential for the consistency of higher-order string-effective actions and for the description of non-geometric $Q$- and $R$-flux backgrounds. The structure unifies geometric, torsional, and non-geometric fluxes, providing a natural geometric explanation for T-duality-covariant string effective actions (Blumenhagen et al., 2012).

6. Modern Developments: Regge Theory, $L_\infty$-Algebras, and Inflationary Cosmology

High-Energy Scattering and Reggeon Field Theory

The ECP framework admits an effective Regge-field-theory extension, describing multi–Regge gravitational scattering using auxiliary reggeon degrees of freedom (generalized spin–connection and vielbein fields coupled via nonlocal light–cone Wilson lines). This generalization captures collective phenomena in the high-energy, $s\gg |t|$ regime, where multiple soft-graviton exchanges exponentiate to single reggeon exchanges. Such extensions necessarily invoke dynamical torsion, even though the minimal Palatini theory is classically torsionless (Bondarenko et al., 2020).

$L_\infty$ Algebras and BRST/BV Formalism

The gauge structure, field equations, and Noether identities of ECP theory are encompassed within a cyclic $L_\infty$ algebra, encoding both local symmetries (diffeomorphisms, local Lorentz) and the full BV–BRST quantization structure. For $d=3$ (i.e., 3D gravity), the ECP $L_\infty$ reduces to a dg-Lie algebra isomorphic to that of Chern–Simons theory, whereas in $d\geq4$, higher brackets are essential. No off-shell morphism exists to 4D $BF$ theory (Ćirić et al., 2020).

Cosmological Models: Higgs Inflation and Quantum Corrections

Higgs inflation within ECP gravity leverages nonminimal couplings of the scalar field to curvature, as well as Holst and Nieh–Yan terms, leading to a family of models interpolating between the standard metric and Palatini inflaton dynamics. The effective kinetic term for the scalar generally receives corrections from integrating out torsion, yielding inflationary observables such as the spectral index $n_s$ and tensor-to-scalar ratio $r$ that depend on model parameters. Quantum corrections induce generalized $R^2$ terms and a new scalaron degree of freedom, extending the unitarity cutoff and resolving strong-coupling issues at scales near reheating (Shaposhnikov et al., 2020, He et al., 2023).

7. Outlook: Generalizations, Mathematical Tools, and Open Questions

ECP gravity provides a versatile and geometrically transparent framework for studying classical and quantum aspects of gravity and its couplings to matter and higher-curvature or flux-corrected sectors. It admits deep connections with the geometry of Cartan and Ehresmann connections, analysis on jet bundles, Lagrange–Poincaré and Routh reduction schemes, and modern $L_\infty$ and BV–BRST quantization techniques (Catren, 2014, Gaset et al., 2018, Capriotti, 2019, Ćirić et al., 2020).

Current research investigates richer torsional models, topologically nontrivial solutions, nonpropagating vs. propagating torsion, interplay with stringy non-geometric fluxes, and the full quantum theory beyond effective loop expansions. The geometric principles underlying projective symmetry and their implications for fundamental observables, as well as the complete classification of allowable matter couplings consistent with these extended symmetries, remain ongoing subjects of investigation.