Combinatorial quantization of Euclidean gravity in three dimensions

Bernd Johannes Schroers

doi:10.1007/978-3-0348-8364-1_12

Combinatorial quantization of Euclidean gravity in three dimensions

Bernd Johannes Schroers

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

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.

Original language	English
Title of host publication	Quantization of Singular Symplectic Quotients
Editors	N. P. Landsman, M. Pflaum, M. Schlichenmaier
Publisher	Birkhäuser
Pages	307-327
Number of pages	21
ISBN (Electronic)	978-3-0348-8364-1
ISBN (Print)	978-3-0348-9535-4
DOIs	https://doi.org/10.1007/978-3-0348-8364-1_12
Publication status	Published - 2001
Event	Workshop on Quantization of Singular Symplectic Quotients 1999 - Oberwolfach, Germany Duration: 2 Aug 1999 → 6 Aug 1999

Publication series

Name	Progress in Mathematics
Publisher	Birkhäuser Basel
Volume	198
ISSN (Print)	0743-1643

Conference

Conference	Workshop on Quantization of Singular Symplectic Quotients 1999
Country/Territory	Germany
City	Oberwolfach
Period	2/08/99 → 6/08/99

Keywords

CHERN-SIMONS THEORY
QUANTUM

Access to Document

10.1007/978-3-0348-8364-1_12

Cite this

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title = "Combinatorial quantization of Euclidean gravity in three dimensions",

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.",

keywords = "CHERN-SIMONS THEORY, QUANTUM",

author = "Schroers, {Bernd Johannes}",

year = "2001",

doi = "10.1007/978-3-0348-8364-1_12",

language = "English",

isbn = "978-3-0348-9535-4",

series = "Progress in Mathematics",

publisher = "Birkh{\"a}user",

pages = "307--327",

editor = "Landsman, {N. P.} and M. Pflaum and M. Schlichenmaier",

booktitle = "Quantization of Singular Symplectic Quotients",

address = "Switzerland",

note = "Workshop on Quantization of Singular Symplectic Quotients 1999 ; Conference date: 02-08-1999 Through 06-08-1999",

}

Schroers, BJ 2001, Combinatorial quantization of Euclidean gravity in three dimensions. in NP Landsman, M Pflaum & M Schlichenmaier (eds), Quantization of Singular Symplectic Quotients. Progress in Mathematics, vol. 198, Birkhäuser, pp. 307-327, Workshop on Quantization of Singular Symplectic Quotients 1999, Oberwolfach, Germany, 2/08/99. https://doi.org/10.1007/978-3-0348-8364-1_12

TY - GEN

T1 - Combinatorial quantization of Euclidean gravity in three dimensions

AU - Schroers, Bernd Johannes

PY - 2001

Y1 - 2001

N2 - 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.

AB - 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.

KW - CHERN-SIMONS THEORY

KW - QUANTUM

U2 - 10.1007/978-3-0348-8364-1_12

DO - 10.1007/978-3-0348-8364-1_12

M3 - Conference contribution

SN - 978-3-0348-9535-4

T3 - Progress in Mathematics

SP - 307

EP - 327

BT - Quantization of Singular Symplectic Quotients

A2 - Landsman, N. P.

A2 - Pflaum, M.

A2 - Schlichenmaier, M.

PB - Birkhäuser

T2 - Workshop on Quantization of Singular Symplectic Quotients 1999

Y2 - 2 August 1999 through 6 August 1999

ER -

Combinatorial quantization of Euclidean gravity in three dimensions

Abstract

Publication series

Conference

Keywords

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