Weil-Petersson volumes and intersection theory on the moduli space of curves

By Maryam Mirzakhani. Journal of The American Mathematical Society, 2006

353 citations recorded by the source corpus.

<p>In this paper, we establish a relationship between the Weil- Petersson volume <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V Subscript g comma n Baseline left-parenthesis b right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>V</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>g</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>b</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">V_{g,n}(b)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the moduli space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M Subscript g comma n Baseline left-parenthesis b right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>g</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>b</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {M}_{g,n}(b)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of hyperbolic Riemann surfaces with geodesic boundary components of lengths <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="b 1"> <mml:semantics> <mml:msub> <mml:mi>b</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">b_{1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, …, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="b Subscript n"> <mml:semantics> <mml:msub> <mml:mi>b</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">b_{n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and the intersection numbers of tautological classes on the moduli space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M overbar Subscript g comma n"> <mml:semantics> <mml:msub> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:mo accent="false">¯<!-- ¯ --></mml:mo> </mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>g</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\overline {\mathcal {M}}_{g,n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of stable curves. As a result, by using the recursive formula for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V Subscript g comma n Baseline left-parenthesis b right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>V</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>g</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>b</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">V_{g,n}(b)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> obtained in the author’s <italic>Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces</italic>, preprint, 2003, we derive a new proof of the Virasoro constraints for a point. This result is equivalent to the Witten-Kontsevich formula.</p>

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1a58ce5a8e874bf55022f33351bf1b0fea1b49c9

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In Geometry and Topology

1. Marcos Mariño
2. M. Mulase
3. Dawei Chen
4. Maryam Mirzakhani
5. R. Pandharipande
6. V. Bouchard
7. Martin Möller
8. Naizhen Zhang

In Mathematical Physics

1. Marcos Mariño
2. R. Pandharipande
3. Dawei Chen
4. Martin Möller
5. Don Zagier
6. M. Mulase
7. V. Bouchard
8. Maryam Mirzakhani

Citations in the corpus

• Recursion Formulae of Higher Weil–Petersson Volumes (2007, methodology): “In a series of innovative papers [18, 19], Mirzakhani obtained a beautiful recursion formula of the Weil-Petersson volumes of the moduli spaces of bordered Riemann surfaces.”
• Polynomial recursion formula for linear Hodge integrals (2009, background): “Each part µi determines a local description of the map f , which is given by z 7−→ zµi in terms of a local coordinate z of X around pi.”
• Hurwitz numbers, matrix models and enumerative geometry (2007, background): “It is known that Mirzakhani’s recursion is essentially equivalent to the KdV/Virasoro structure of 2d topological gravity [MS, Mir2 , LZ].”
• Masur–Veech volumes and intersection theory on moduli spaces of Abelian differentials (2019, result): “f strict brackets (see Section 10.3). Further directions. Our work opens an avenue to study a series of related questions. First, we point out an interesting comparison with the proofs by Mirzakhani ([Mir07]), by Kontsevich ([Kon92]), and by Okounkov-Pandharipande ([OP09]) of Witten’s conjecture: th”
• LOCALIZATION IN GROMOV-WITTEN THEORY AND ORBIFOLD GROMOV-WITTEN THEORY (2011, neutral): “[45] M. Mirzakhani, “Weil-Petersson volumes and intersection theory on the moduli space of curves,” J. Amer.”
• Masur-Veech volumes, frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves (2019, background): “rk 3.4 (M. Kazarian). For one particular pair (g,n), namely for (g,n) = (1,1), Mirzakhani’s normalization (3.13) VolMirzakhani WP M 1, (b) = 1 24 (b 2+4π), (see equation (4.5) on page 116 in [Mi3] or [Mi2]) is twice bigger than the normalization in many other papers. Topologically M1,1 is homeomorph”
• Ergodic Theory of the Earthquake Flow (2010, methodology): “Using the results of this paper, we prove an equidistribution result for the level sets of the lengths of simple closed curves in the moduli space [25].”
• An algebro-geometric proof of Witten's conjecture (2006, neutral): “A number of proofs of the Witten conjecture are known, but all of them exploit techniques that do not seem to be intrinsically related to the initial problem: Kontsevich’s proof [12] makes use of Jenkins–Strebel differentials and matrix integrals, the proof due to Okounkov and Pandharipande [18], wh”
• Low-dimensional de Sitter quantum gravity (2019, background): “hey are determined via the Eynard-Orantin topological recursion relations [54{56], which for the double-scaled density of states above are related to the Mirzakhani recursion relation for the V g;n’s [49,50]. We refer the reader to the nice summary of both in Section 2 of [21]. Indeed, the genus exp”
• Tautological and non-tautological cohomology of the moduli space of curves (2011, background): “Other geometric approaches to the ψ integrals can be found in [48, 51].”