Homotopy invariants and time evolution in (2+1)-dimensional gravity

1994/01/20 by H. Waelbroeck, Henri Waelbroeck, Federico Zertuche +1 · 2 citations
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Combinatorics #Computer science #Cosmology and Gravitation Theories #Hamiltonian (control theory) #Homotopy #Homotopy group #Mathematical physics #Mathematics #Noncommutative and Quantum Gravity Theories #Polygon (computer graphics) #Pure mathematics #Symplectic geometry #gr-qc

paper · pdf · doi:10.1103/physrevd.50.4966

published as Phys.Rev. D50 (1994) 4966-4981 · 34 pages, Mexico preprint ICN-UNAM-93-11

arxiv created 1994/01/20 · openalex publication_date 1994/10/15 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We establish the relation between the ISO(2,1) homotopy invariants and the polygon representation of (2+1)-dimensional gravity. The polygon closure conditions, together with the SO(2,1) cycle conditions, are equivalent to the ISO(2,1) cycle conditions for the representations \ensuremathρ:\mathrm\ensuremathπ₁(\mathrm\ensuremathΣ_g,N)\ensuremath→ISO(2,1). Also, the symplectic structure on the space of invariants is closely related to that of the polygon representation. We choose one of the polygon variables as internal time and compute the Hamiltonian, then perform the Hamilton-Jacobi transformation explicitly. We make contact with other authors' results for g=1 and g=2 (N=0).

Citations

• Quantum canonical transformations. Physical equivalence of quantum theories
• 2 + 1 dimensional gravity as an exactly soluble system
• New Variables for Classical and Quantum Gravity
• New Hamiltonian formulation of general relativity

Cited by

• Canonical quantization of (2+1)-dimensional gravity
• (2 + 1) covariant lattice theory and 't Hooft's formulation