# The Hamilton-Jacobi analysis and Canonical Covariant description for three dimensional Palatini theory plus a Chern-Simons term

## TL;DR

This paper applies Hamilton-Jacobi and covariant phase space methods to analyze three-dimensional Palatini gravity with a Chern-Simons term, revealing symmetries, constraints, and a gauge-invariant symplectic structure.

## Contribution

It provides a complete Hamilton-Jacobi analysis and a canonical covariant formulation for the theory, highlighting the role of a Barbero-Immirzi-like parameter.

## Key findings

- Identified all Hamilton-Jacobi constraints and symmetries.
- Showed dependence of brackets on a Barbero-Immirzi-like parameter.
- Constructed a gauge-invariant symplectic two-form.

## Abstract

By using the Hamilton-Jacobi [HJ] framework the three dimensional Palatini theory plus a Chern-Simons term [PCS] is analyzed. We report the complete set of $HJ$ Hamiltonians and a generalized $HJ$ differential from which all symmetries of the theory are identified. Moreover, we show that in spite of PCS Lagrangian produces Einstein's equations, the generalized $HJ$ brackets depend on a Barbero-Immirzi like parameter. In addition we complete our study by performing a canonical covariant analysis, and we construct a closed and gauge invariant two form that encodes the symplectic geometry of the covariant phase space.