Quantum Hilbert Space of GCChern-Simons-Witten Theory and Gravity
Abstract
Euclidean Chern-Simons-Witten theory with a complex Lie group GC is discussed. When GC is a complexification of SU(2), this theory gives an important physical application. It describes Euclidean gravity with negative cosmological constant. In the present paper, we consider a quantum Hilbert space of the Chern-Simons-Witten theory with GC on a torus, and show that it is finite-dimensional at special values of coupling constants. In such cases, we find that any physical state is given by a product of the holomorphic Weyl-Kac character of the Wess-Zumino-Novikov-Witten model and the anti-holomorphic one. Moreover, it is also shown that each of topological invariants of 3-dimensional manifolds which admit the genus one Heegaard splitting is factorized into a product of two parts. One of them coincides with a topoligical invariant of the Chern-Simons-Witten theory with the maximal compact subgroup of GC and the other is closely related to its complex conjugate.
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For noncompact groups such as those appearing in gravity, much less is known, though there has been promising work.
“For noncompact groups such as those that appear in gravity, much less is known, though there has been some promising work [280, 155, 43, 150].”
From Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe (2005) · cites this paper as [155]
For Λ > 0, the relevant gauge group is SO(3,1) or its cover SL(2,C), a complex gauge group whose Chern-Simons theory is somewhat better understood.
“For Λ > 0, the relevant gauge group is SO(3, 1) or its cover SL(2, ℂ), a complex gauge group whose Chern-Simons theory is somewhat better understood [280, 155, 43].”
From Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe (2005) · cites this paper as [155]
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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
- Hayashi, N.
- Affiliations
- The University of Tokyo (JP)
- Corresponding author
- Hayashi, Nobuharu
- Venue
- Progress of Theoretical Physics Supplement
- Publisher
- Oxford University Press (OUP)
- Published
- 1993
- Volume
- 114
- Pages
- 125-147
- ISSN
- 0375-9687 (ISSN-L)
- OpenAlex
- W2052589295
- Cited by
- 17
- References
- 0
- Citations by year
- 2012: 1
- FWCI
- 0.58
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Progress of Theoretical Physics Supplement (publishedVersion)
- HighWire Press Open Archive (submittedVersion)
- Topics
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Physical Sciences › Mathematics › Geometry and Topology › Algebraic structures and combinatorial models
- Physical Sciences › Mathematics › Geometry and Topology › Geometric and Algebraic Topology
- Keywords
- quantum Hilbert space, Euclidean gravity, negative cosmological constant, torus, Wess-Zumino-Witten model, topological invariants, 3-manifolds, finite-dimensional Hilbert space
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