Weil-Petersson volumes and intersection theory on the moduli space of curves
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In this paper, we establish a relationship between the Weil- Petersson volume V g , n ( b ) of the moduli space M g , n ( b ) of hyperbolic Riemann surfaces with geodesic boundary components of lengths b 1 , …, b n , and the intersection numbers of tautological classes on the moduli space M ¯ g , n of stable curves. As a result, by using the recursive formula for V g , n ( b ) obtained in the author’s Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces , preprint, 2003, we derive a new proof of the Virasoro constraints for a point. This result is equivalent to the Witten-Kontsevich formula.
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Mirzakhani's formula gives the Weil-Peterson volumes V_{g,m} appearing in (4.4c).
“The starting point is Mirzakhani's formula for the Weil-Peterson volumes V g;m appearing in (4.4c) [85]:”
From Free energy from replica wormholes (2021) · cites this paper as [85]
Mirzakhani circumvented the mapping class group integration problem by a trick involving shooting geodesics perpendicular to a boundary.
“In Mirzakhani (2007a) this was circumvented by using the following trick.43 First we pick a (geodesic) boundary, labeled as 1 below, and start shooting geodesics perpendicular to the boundary, along the entire circumference of the boundary.”
From Solvable models of quantum black holes: a review on Jackiw–Teitelboim gravity (2023) · cites this paper as Mirzakhani 2007a
There are two approaches to computing Weil-Petersson volumes, both developed by Mirzakhani.
“There are two approaches to this problem, both developed by Mirzakhani (2007a, 2007b).”
From Solvable models of quantum black holes: a review on Jackiw–Teitelboim gravity (2023) · cites this paper as Mirzakhani 2007b
There are two approaches to computing Weil-Petersson volumes, both developed by Mirzakhani.
“There are two approaches to this problem, both developed by Mirzakhani (2007a, 2007b).”
From Solvable models of quantum black holes: a review on Jackiw–Teitelboim gravity (2023) · cites this paper as Mirzakhani 2007a
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References 38 · Cited by 294
- Authors
- Mirzakhani, M.
- Affiliations
- Princeton University (US)
- Corresponding author
- Mirzakhani, Maryam
- ORCID
- Mirzakhani, Maryam
- Venue
- Journal of the American Mathematical Society
- Publisher
- American Mathematical Society (AMS)
- Published
- 2006-03-08
- Volume
- 20
- Issue
- 1
- Pages
- 1-23
- ISSN
- 0894-0347 (ISSN-L)
- 1088-6834
- OpenAlex
- W1526007200
- Cited by
- 294
- References
- 38
- Citations by year
- 2026: 6 · 2025: 12 · 2024: 18 · 2023: 23 · 2022: 23 · 2021: 22 · 2020: 24 · 2019: 14 · 2018: 9 · 2017: 18 · 2016: 10 · 2015: 14 · 2014: 4 · 2013: 8 · 2012: 10
- FWCI
- 23.53
- Type
- journal article
- Language
- en
- Open access
- hybrid
- doi.org
- Licence
- Locations
- Journal of the American Mathematical Society (publishedVersion, public-domain)
- Topics
- Physical Sciences › Mathematics › Geometry and Topology › Algebraic Geometry and Number Theory
- Physical Sciences › Mathematics › Geometry and Topology › Geometric and Algebraic Topology
- Physical Sciences › Mathematics › Algebra and Number Theory › Advanced Mathematical Identities
- Keywords
- Weil-Petersson volume, moduli space of curves, intersection theory, tautological classes, stable curves, hyperbolic Riemann surfaces, geodesic boundary, Virasoro constraints, bordered Riemann surfaces, recursive formula
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