Hamiltonian formulation for scalar and two-form gauge fields coupled to 3d gravity

TL;DR

This paper develops a Hamiltonian formulation for a 3D gravity system coupled with scalar and two-form gauge fields, analyzing constraints, symmetries, and degrees of freedom.

Contribution

It provides a systematic Hamiltonian analysis, including constraint classification, gauge symmetry generators, and phase space structure for the coupled model.

Findings

Full constraint algebra and classification into first- and second-class constraints.

Explicit gauge symmetry generators reproducing spacetime diffeomorphisms and Poincaré symmetries.

Reduced phase-space structure with three reducibility conditions ensuring consistency.

Abstract

We develop a systematic Hamiltonian formulation for a gravitating topological matter system in three-dimensional spacetime, coupling a scalar gauge field and a rank-2 antisymmetric gauge field to Einstein--Cartan gravity. We perform the Dirac--Bergmann analysis, systematically finding the full structure of the constraints, classifying them into first- and second-class ones, and computing their Poisson bracket algebra. Furthermore, we write down the explicit expression for the Hamiltonian generator of gauge symmetries on the full set of canonical variables, containing the exact number of gauge parameters, and demonstrate that, through a mapping of the gauge parameters, these gauge transformations reproduce on-shell the spacetime diffeomorphism and local Poincar\'e symmetries, thereby establishing the full symmetry structure of the coupled model. Our canonical analysis further reveals that the reduced phase-space admits exactly three reducibility conditions for the first-class constraints, which guarantee the consistency of the gravitating matter system by ensuring a correct count of physical degrees of freedom. The fundamental symplectic structure on the reduced phase-space is established through the explicit computation of the Dirac brackets.