Abstract
In the Chern-Simons formulation of Einstein gravity in 2+1 dimensions, the phase space is the moduli space of flat G-connections on a two-dimensional surface, where G is a typically non-compact Lie group which depends on the signature of space-time and the cosmological constant. For Euclidean signature and vanishing cosmological constant, G is the three-dimensional Euclidean group. For this case the Poisson structure of the moduli space is given explicitly in terms of a classical r-matrix. It is shown that the quantum R-matrix of the quantum double D(SU(2)) provides a quantization of that Poisson structure.
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References
A. Achucarro and P. Townsend, A Chern-Simons action for three-dimensional anti-de Sitter supergravity theories, Phys. Lett. B180 (1986), 85–100.
A. Y. Alekseev, H. Grosse and V. Schomerus, Combinatorial quantization of the Hamiltonian Chern-Simons Theory, Commun. Math. Phys. 172 (1995), 317–358.
A. Yu. Alekseev, H. Grosse and V. Schomerus, Combinatorial quantization of the Hamiltonian Chern-Simons Theory II, Commun. Math. Phys. 174 (1995), 561–604.
A. Yu. Alekseev and V. Schomerus, Representation theory of Chern-Simons observables, Duke Math. J. 85 (1996), 447–510.
A. Yu. Alekseev, A. Z. Malkin and E. Meinrenken, Lie group valued moment maps, J. Diff. Geom. 48 (1998), 445–495.
M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Phil. Trans. Roy. Soc. London A308 (1983), 523–615.
[] M. F. Atiyah, The Geometry and Physics of Knots, Cambridge University Press, Cambridge.
F. A. Bais, P. van Driel, and M. de Wild Propitius, Quantum symmetries in discrete gauge theories, Phys. Lett. B 280 (1992), 63–70.
F. A. Bais and N. M. Muller, Topological field theory and the quantum double of SU(2), Nucl. Phys. B530 (1998), 349–400.
[] F. A. Bais, N. M. Muller and B. J. Schroers, Quantum double symmetry and topological interactions in (2+1)-dimensional quantum gravity, in preparation.
P. Bonneau, Topological quantum double, Rev. Math. Phys. 6 (1994), 305–318.
E. Buffenoir and Ph. Roche, Harmonic analysis on the quantum Lorentz group, Commun. Math. Phys. 207 (1999), 499–555.
[] S. Carlip, Lectures on 2+1 dimensional gravity,UCD-95–6 (1995), grgc/9503024.
S. Carlip, Quantum Gravity in (2+1) Dimensions, Cambridge University Press, Cambridge, 1998.
E. Celeghini and R. Giachetti, The three dimensional euclidean quantum group E(3) Q and its R-matrix, J. Math. Phys. 32 (1991), 1159–1165.
V. Chari and A. Pressley Quantum Groups, Cambridge University Press, Cambridge, 1994.
R. Dijkgraaf, C. Vafa, E. Verlinde, and H. Verlinde, The operator algebra of orbifold models, Commun. Math. Phys. 123 (1989), 485–526.
R. Dijkgraaf, V. Pasquier, and P. Roche, Quasi Hopf algebras, group cohomology and orbifold models, Nucl. Phys. Proc. Suppl. 18B (1990), 60–72.
[] V. V. Fock and A. A. Rosly, Poisson structures on moduli of flat connections on Riemann surfaces and r-matrices, ITEP preprint 72–92, 1992, math. QA/9802054.
K. Guruprasad, J. Huebschmann, L. Jeffrey, A. Weinstein, Group systems, groupoids, and moduli spaces of parabolic bundles, Duke Math. J. 89 (1997), 377–412.
[] T. H. Koornwinder and N. M. Muller. The quantum double of a (locally) compact group, J. Lie Theory 7 (1997), 33–52; Err. 8 (1998), 187.
T. H. Koornwinder, F. A. Bais and N. M. Muller, Tensor Product Representations of the Quantum Double of a Compact Group, Commun. Math. Phys. 198 (1998), 157–186.
H. J. Matschull, On the relation between (2+1) Einstein gravity and Chern-Simons Theory, Class. Quant. Gray. 16 (1999), 2599–2609.
J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, Springer-Verlag, New York, 1994.
M. Müger, Quantum Double actions on operator algebras and orbifold quantum field theories, Commun. Math. Phys. 191 (1998), 137–181.
[] N. Muller, Topological interactions and quantum double symmetries, Ph.D. dissertation, University of Amsterdam, 1998.
G. K. Pedersen, C* -algebras and their Automorphism Groups, Academic Press, London, 1979.
R. W. Sharpe, Differential Geometry, Springer Verlag, New York, 1996.
P. Stachura, Poisson-Lie structures on Poincaré and Euclidean groups in three dimensions, J. Phys. A 31 (1998), 4555–4564.
E. Witten, 2+1 dimensional gravity as an exactly soluble system, Nucl. Phys., B311 (1988), 46–78.
E. Witten, Quantization of Chern-Simons gauge theory with complex gauge group, Commun. Math. Phys. 137 (1991), 29–66.
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Schroers, B.J. (2001). Combinatorial quantization of Euclidean gravity in three dimensions. In: Landsman, N.P., Pflaum, M., Schlichenmaier, M. (eds) Quantization of Singular Symplectic Quotients. Progress in Mathematics, vol 198. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8364-1_12
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DOI: https://doi.org/10.1007/978-3-0348-8364-1_12
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