# Polysymplectic formulation for BF gravity with Immirzi parameter

## TL;DR

This paper develops a covariant Hamiltonian (polysymplectic) formulation of the CMPR BF gravity model with an Immirzi parameter, deriving Einstein equations and recovering the De Donder-Weyl Hamiltonian for the Holst action.

## Contribution

It introduces a polysymplectic framework for the CMPR model, providing a covariant Hamiltonian analysis and connecting it to the Holst action's Hamiltonian formulation.

## Key findings

- Derived Einstein equations from the CMPR model.
- Recovered the De Donder-Weyl Hamiltonian formulation of the Holst action.
- Established a covariant Hamiltonian approach using polysymplectic geometry.

## Abstract

The polysymplectic formulation of the CMPR action, which is a BF-type formulation of General Relativity that involves an arbitrary Immirzi parameter, is performed. We implement a particular scheme within this covariant Hamiltonian approach to analyze the constraints that characterize the CMPR model. By means of the privileged $(n-1)$-forms and the Poisson-Gerstenhaber bracket, inherent to the polysymplectic framework, the BF field equations associated to the CMPR action are obtained and, in consequence, the Einstein equations naturally emerge by solving the simplicity constraints of the theory. Further, from the polysymplectic analysis of the CMPR action the De Donder-Weyl Hamiltonian formulation of the Holst action is recovered, which is consistent with the Lagrangian analysis of this model as reported in the literature.