Constrained Hamiltonian dynamics of 3D gravity coupled to topological matter

Abstract

We present the Dirac Hamiltonian formalism for a pair of $1$-form fields with a topological-like potential coupled to first-order gravity in three-dimensional spacetime. By considering the complete phase space, we derive the full structure of the physical constraints, including both primary and secondary ones; analyze their consistency conditions; classify them into first- and second-class; and compute their Poisson-bracket algebra. Our analysis confirms the absence of local degrees of freedom, consistent with the topological nature of the model's action. Furthermore, we construct the canonical generator for gauge transformations and demonstrate that, through appropriate gauge parameter mappings, these transformations recover the full diffeomorphism and Poincar\'e symmetries of the Lagrangian formulation. Finally, we explicitly compute the Dirac brackets, establishing the symplectic structure of the reduced phase space.