De Donder–Weyl Hamiltonian formalism of MacDowell–Mansouri gravity

We analyse the behaviour of the MacDowell–Mansouri action with internal symmetry group $SO(4,1)$ under the De Donder–Weyl 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 De Donder–Weyl formulation. The decomposition of the internal algebra $so(4,1)\simeq so(3,1)\oplus R^{3,1}$ allows the symmetry breaking $SO(4,1)\to SO(3,1)$, 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 polysymplectic 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. (paper)