10-plectic formulation of gravity and Cartan connections

TL;DR

This work develops a covariant Hamiltonian formulation of Weyl--Einstein--Cartan gravity within a multisymplectic framework, treating the Cartan connection as a $rak{p}$-valued $1$-form on the total space of the frame bundle. By moving to the De Donder--Weyl setting with a $10$-form $\theta^{(10)}$ and its exterior derivative $\pmb{\omega}=d\theta^{(10)}$, the paper derives the Hamilton equations (HVDW) and shows that the local equivariance of the Cartan connection emerges as a dynamical constraint rather than an imposed one. The resulting equations reproduce Einstein--Cartan dynamics with torsion, expressed through multimomenta $p_a^{bj}$ and $p_a^{cj}$ and encoded by the Einstein and Spin 3-forms $\Upsilon$ and $\Sigma$, with a clear separation between spacetime dynamics and bundle dynamics. Under appropriate decay or compactness assumptions, the right-hand side sources can vanish, yielding vacuum Einstein--Cartan relations. The framework provides a coordinate-free, covariant unification of spacetime and gauge dynamics with potential implications for cosmology and dark-energy interpretations within a geometrically rigorous multimomenta formalism.

Abstract

We give a Hamiltonian formulation of %the first order Weyl--Einstein--Cartan gravity which is covariant from the viewpoint of the geometry of the principal fiber bundle. The connection is represented by a $1$-form with values in the Poincaré Lie algebra, which is defined on the total space of the orthonormal frame bundle fibered over the space-time. Within the $10$-plectic framework we discover that the local equivariance property of the Cartan connection is a consequence of the Hamilton equations.

Theorems & Definitions (15)

• Proposition 4.1
• Proposition 4.2
• Lemma 5.1
• Definition 5.1
• Lemma 5.2
• Lemma 5.3
• Lemma 5.4
• Definition 5.2
• Lemma 5.5
• Lemma 5.6