Poisson structure and symmetry in the Chern–Simons formulation of (2 1)-dimensional gravity
Open accessAbstract
In the formulation of (2 + 1)-dimensional gravity as a Chern–Simons gauge theory, the phase space is the moduli space of flat Poincaré group connections. Using the combinatorial approach developed by Fock and Rosly, we give an explicit description of the phase space and its Poisson structure for the general case of a genus g oriented surface with punctures representing particles and a boundary playing the role of spatial infinity. We give a physical interpretation and explain how the degrees of freedom associated with each handle and each particle can be decoupled. The symmetry group of the theory combines an action of the mapping class group with asymptotic Poincaré transformations in a nontrivial fashion. We derive the conserved quantities associated with the latter and show that the mapping class group of the surface acts on the phase space via Poisson isomorphisms.
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The quantum group structure of these brackets can be expressed in terms of the quantum double of the Lorentz group.
“several authors have further explored the quantum group structure of these brackets, which can be expressed in terms of the quantum double of the Lorentz group [37, 36, 61, 201].”
From Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe (2005) · cites this paper as [201]
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2 + 1 dimensional gravity as an exactly soluble system
Witten, E. (1988) · Nuclear Physics B · doi:10.1016/0550-3213(88)90143-5 · Cited by 2,562 · see citing works
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Research works printed but not linked: 12
- Carlip S 1998 Quantum Gravity in 2 + 1 Dimensions (Cambridge: Cambridge University Press)
- Fock V V and Rosly A A 1992 Poisson structures on moduli of flat connections on Riemann surfaces and r-matrices ITEP Preprint 72-92 See also Fock V V and Rosly A A 1998 Preprint math.QA/9802054 · arXiv
- Sharpe R W 1996 Differential Geometry (New York: Springer)
- Atiyah M and Bott R 1982 Yang-Mills equations over Riemann surfaces Phil. Trans. R. Soc. A 308 523
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- Birman J S 1975 Braids, Links and Mapping Class Groups (Ann. Math. Stud. 82) (Princeton, NJ: Princeton University Press)
- Giulini D 1995 Asymptotic symmetry groups of long-ranged gauge configurations Mod. Phys. Lett. A 10 2059-70 (Preprint gr-qc/9410042) · arXiv
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- Lu J H 1990 Multiplicative and affine poisson structures on Lie groups PhD thesis University of California, Berkeley, CA
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Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe
Carlip, S. (2005) · Living Reviews in Relativity · doi:10.12942/lrr-2005-1 · Cited by 172 · see citing works
- Authors
- Meusburger, C., & Schroers, B. J.
- Affiliations
- Heriot-Watt University (GB)
- Venue
- Classical and Quantum Gravity
- Publisher
- IOP Publishing
- Published
- 2003-05-08
- Volume
- 20
- Issue
- 11
- Pages
- 2193-2233
- ISSN
- 0264-9381 (ISSN-L)
- 1361-6382
- OpenAlex
- W2077714331
- Cited by
- 79
- References
- 1
- Citations by year
- 2026: 3 · 2025: 2 · 2024: 1 · 2023: 1 · 2021: 6 · 2020: 3 · 2019: 5 · 2018: 4 · 2017: 5 · 2016: 2 · 2015: 5 · 2014: 5 · 2013: 4 · 2012: 4
- FWCI
- 1.62
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Classical and Quantum Gravity (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Studienstiftung des Deutschen Volkes
- Engineering and Physical Sciences Research Council
- Topics
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Physical Sciences › Mathematics › Geometry and Topology › Geometric and Algebraic Topology
- Keywords
- three-dimensional gravity, Chern-Simons theory, Poisson structures, moduli spaces, mapping class group, conserved charges, phase space symmetry, punctured surfaces
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