The sum over topologies in three-dimensional Euclidean quantum gravity
Open accessAbstract
In the Euclidean path-integral approach to quantum gravity, the partition function for Hawking's 'volume canonical ensemble' is computed by summing contributions from all possible topologies. The behaviour such a sum can be estimated in three spacetime dimensions in the limit of small cosmological constant. The sum over topologies diverges for any sign of Lambda , but for dramatically different reasons: for Lambda )0, the divergent behaviour comes from the contributions of very low-volume, topologically complex manifolds, while for Lambda )0 it is a consequence of the existence of infinite sequences of relatively high-volume manifolds with converging geometries. Possible implications for four-dimensional quantum gravity are discussed.
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For Λ > 0, the number of topologically distinct lens spaces with volumes less than vol(S3) grows fast enough that these spaces dominate, leading to a divergent partition function for closed three-manifolds.
“As shown in [71], though, the number of topologically distinct lens spaces with volumes less than volḡ(S3) grows fast enough that these spaces dominate, leading to a divergent partition function for closed three-manifolds.”
From Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe (2005) · cites this paper as [71]
In §3 (pp. 211–213), Carlip identifies distinct lens spaces with volumes proportional to 1/p and finds that the sum is dominated by manifolds with increasingly large fundamental groups and small volumes. But the paper calls lens-space dominance a suggestion, and its discussion of approximations says not all saddle points or topologies have been included. The claim therefore overstates the result as an established divergence of the partition function for all closed three-manifolds. (LLM, conf=0.84)
Experimental — accuracy not yet measuredFor Λ < 0, manifolds with larger volumes are numerous enough to lead to a divergence in the partition function.
“Here, too, however, manifolds with larger volumes — although individually exponentially suppressed — are numerous enough to lead to a divergence in the partition function [71].”
From Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe (2005) · cites this paper as [71]
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Cites
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: 3
- Goldman W M 1988 Geomeay of Group Repre.smatim ed W M Goldman and R Magid (Providence, Witten E 1989 Commwr Mafh Phys. 121 351 (Cambridge: Cambridge University Press) RI: American Mathematical Society)
- Thomas C B 1986 Ellbtic Srmctures on 3-MmifohLv (Cambridge Cambridge University Press)
- Lot1 J 1989 Recent DevelopmmLr in Geomeby ed S Y Cheng, H Choi and R E Greene (Providence, Hardy G H and Wright E M 1954 An introduction lo the Theory of Numbers (Oxford Ciarendon) Eslermann T 1961 M a t h e " 8 83 RI: American Malhemalical Society)
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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
Self-citationCarlip, S. (2023) · Reports on Progress in Physics · doi:10.1088/1361-6633/acceb4 · Cited by 41 · see citing works
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- Authors
- Carlip, S.
- Affiliations
- University of California, Davis (US)
- Corresponding author
- Carlip, S
- ORCID
- Carlip, S
- Venue
- Classical and Quantum Gravity
- Publisher
- IOP Publishing
- Published
- 1993-02-01
- Volume
- 10
- Issue
- 2
- Pages
- 207-218
- ISSN
- 0264-9381 (ISSN-L)
- 1361-6382
- OpenAlex
- W1995452865
- Cited by
- 40
- References
- 1
- Citations by year
- 2026: 3 · 2025: 4 · 2024: 2 · 2023: 3 · 2022: 3 · 2021: 1 · 2020: 1 · 2017: 1 · 2015: 1 · 2013: 1
- FWCI
- 1.61
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Classical and Quantum Gravity (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- U.S. Department of Energy (FG0391ER40674)
- Topics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
- 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
- Keywords
- three-dimensional quantum gravity, Euclidean quantum gravity, path integral, cosmological constant, topological manifolds, four-dimensional quantum gravity
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