# De Donder-Weyl Hamiltonian formalism of MacDowell-Mansouri gravity

## TL;DR

This paper explores the covariant Hamiltonian formulation of MacDowell-Mansouri gravity, showing how symmetry breaking leads to Einstein's equations and highlighting differences from the Lagrangian approach.

## Contribution

It introduces a De Donder-Weyl Hamiltonian formalism for MacDowell-Mansouri gravity and demonstrates symmetry breaking within this covariant framework.

## Key findings

- Symmetry breaking can be performed before or after variation in the Hamiltonian formalism.
- The formalism recovers Einstein's equations via the Poisson-Gerstenhaber bracket.
- The approach provides a covariant Hamiltonian perspective on gravity theories.

## Abstract

We analyse the behaviour of the MacDowell-Mansouri action with internal symmetry group $\mathrm{SO}(4,1)$ under the covariant Hamiltonian formulation. The field equations, known in this formalism as the De Donder-Weyl equations, are obtained by means of the graded Poisson-Gerstenhaber bracket structure present within the covariant formulation. The decomposition of the internal algebra $\mathfrak{so}(4,1)\simeq\mathfrak{so}(3,1)\oplus\mathbb{R}^{3,1}$ allows the symmetry breaking $\mathrm{SO}(4,1)\to\mathrm{SO}(3,1)$, which reduces the original action to the Palatini action without the topological term. We demonstrate that, in contrast to the Lagrangian approach, this symmetry breaking can be performed indistinctly in the covariant Hamiltonian formalism either before or after the variation of the De Donder-Weyl Hamiltonian has been done, recovering Einstein's equations via the Poisson-Gerstenhaber bracket.