Constrained Hamiltonian dynamics of 3D gravity coupled to topological matter

TL;DR

This paper develops a Hamiltonian formalism for 3D gravity coupled to topological matter, revealing the constraint structure, gauge symmetries, and symplectic form, confirming the topological nature with no local degrees of freedom.

Contribution

It provides a complete Hamiltonian analysis of 3D gravity with topological matter, including constraints, gauge generators, and Dirac brackets, clarifying the model's symmetries and phase space structure.

Findings

Confirmed absence of local degrees of freedom

Derived full constraint algebra and gauge transformations

Computed Dirac brackets and symplectic structure

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.