Weil-Petersson volume of moduli spaces, Mirzakhani's recursion and matrix models

By B. Eynard, N. Orantin. 2007

154 citations recorded by the source corpus.

We show that Mirzakhani's recursions for the volumes of moduli space of Riemann surfaces are a special case of random matrix loop equations, and therefore we confirm again that Kontsevitch's integral is a generating function for those volumes. As an application, we propose a formula for the Weil-Petersson volume Vol(M_{g,0}).

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3bbbeac79a1720bb9ea1bbecc8215924158d190b

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In Geometry and Topology

1. B. Eynard
2. N. Orantin
3. D. Stanford
4. Marcos Mariño
5. G. Turiaci
6. L. Spitz
7. E. Witten
8. Sergey Shadrin

In Mathematical Physics

1. B. Eynard
2. N. Orantin
3. D. Stanford
4. E. Witten
5. L. Spitz
6. Sergey Shadrin
7. Petr Dunin-Barkowski
8. G. Turiaci

Citations in the corpus

• The path integral of 3D gravity near extremality; or, JT gravity with defects as a matrix integral (2020, background): “The Weil-Petersson volumes are simply related to the expansion of the resolvent in JT gravity by a Laplace transform [54],”
• JT gravity as a matrix integral (2019, background): “This was noted and proven in general in work by Eynard and Orantin [33], who showed that after Laplace transform, Mirzakhani’s recursion relation takes the form of (56) with the spectral curve y = sin(2πz) 4π .”
• JT gravity and the ensembles of random matrix theory (2019, methodology): “1) is the correspondence [36] between Mirzakhani’s recursion relation for the volumes of moduli space [37] and Eynard-Orantin “topological recursion” [38] which gives the genus expansion of a Hermitian matrix integral [39].”
• Invariants of spectral curves and intersection theory of moduli spaces of complex curves (2011, background): “- for the spectral curve y = 1 2π sin (2π √ x), the W (g) n ’s are the Laplace transforms of Weil-Petersson volumes of the moduli spaces Mg,n (see [24, 13]).”
• The Laplace transform of the cut-and-join equation and the Bouchard-Marino conjecture on Hurwitz numbers (2009, unknown): “The topological recursion”
• Computation of Open Gromov–Witten Invariants for Toric Calabi–Yau 3-Folds by Topological Recursion, a Proof of the BKMP Conjecture (2012, background): “It was also noticed that the topological recursion implies the holomorphic anomaly equations [20] as well as some properties very similar to Givental’s formalism [53], but the converse has not been proved.”
• Identification of the Givental Formula with the Spectral Curve Topological Recursion Procedure (2012, background): “β 2αh1 2g+n−2 βg+n−1 X∞ m=0 (−1)m m! X α~∈N∗m Ym k=1 (2αk+1)!! h2α k+1 h1 Yn i =1 (2di +1)!!dzi z2d i+2 i * Yn j τd j Ym k τα k+1 + g,n+m . Proof. Once again the proof can be found in the literature [10, 14, 11]. However, let us study a graphical interpretation of this result when considering an arb”
• Recursion Formulae of Higher Weil–Petersson Volumes (2007, background): “per [16], it is shown that the Witten-Kontsevich theorem implies the MulaseSafnuk form of Mirzakhani’s recursion formula. Its relationship with matrix integrals has been studied by Eynard and Orantin [5, 6]. More discussions about computations of Weil-Petersson or higher Weil-Petersson volumes can b”