Constrained Hamiltonian dynamics of 3D gravity-coupled topological matter

TL;DR

This work delivers a complete constrained Hamiltonian treatment of the BCEA theory, a 3D gravity system minimally coupled to two one-form topological matter fields. Using Dirac's formalism and a 2+1 decomposition, it identifies 36 primary and 12 secondary constraints, classifies 24 as first-class and 24 as second-class, and shows the physical phase space has zero degrees of freedom per space point, confirming the topological character. A generating functional for gauge transformations is constructed, from which the full diffeomorphism and Poincaré symmetries are recovered on-shell, and the Dirac brackets of the reduced phase space are explicitly computed, revealing the symplectic structure. The results illuminate the gauge structure and constraint algebra of the model, providing a solid foundation for any future canonical quantization and for understanding how torsion and contorsion mediate gravity-matter interactions in a topological setting.

Abstract

We analyze the dynamics of two one-form topological matter fields minimally coupled to first-order gravity in three-dimensional spacetime using the Dirac Hamiltonian formalism. Working in the full phase space, we systematically identify the complete set of constraints of the system, classify them into first- and second-class, and compute their Poisson bracket algebra. The constraint analysis confirms the absence of physical degrees of freedom, consistent with the system's topological character. Furthermore, we construct the generating functional for gauge transformations and demonstrate that, with appropriate gauge parameter mappings, these transformations recover the full diffeomorphism and Poincaré symmetries of the theory. Finally, we explicitly compute the Dirac brackets, establishing the symplectic structure of the reduced phase space.