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Topological recursion for symplectic volumes of moduli spaces of curves. (English) Zbl 1266.53078

The authors develop a recursion formula for calculating the symplectic volume of the moduli space of the set \(\mathcal M_{g,n}\) of all smooth algebraic curves of genus \(g\) that have \(n\) distinguished labeled points. It is proved that this recursion formula is equivalent to the DVV equation (Virasoro constraint) for \(\psi\)-class intersections on the compactification \(\overline{\mathcal{M}}_{g,n}\) and also to the Eynard-Orantin recursion for the spectral curve \(x=\frac{1}{2}y^2\).

MSC:

53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
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References:

[1] G. Borot, B. Eynard, M. Mulase, and B. Safnuk, A matrix model for simple Hurwitz numbers, and topological recursion, J. Geom. Phys. 61 (2011), 522-540. · Zbl 1225.14043 · doi:10.1016/j.geomphys.2010.10.017
[2] V. Bouchard, A. Klemm, M. Mariño, and S. Pasquetti, Remodeling the B-model, Comm. Math. Phys. 287 (2009), 117-178. · Zbl 1178.81214 · doi:10.1007/s00220-008-0620-4
[3] V. Bouchard and M. Mariño, Hurwitz numbers, matrix models and enumerative geometry, From Hodge theory to integrability and TQFT tt*-geometry, Proc. Sympos. Pure Math., 78, pp. 263-283, Amer. Math. Soc., Providence, RI, 2008. · Zbl 1151.14335
[4] B. H. Bowditch and D. B. A. Epstein, Natural triangulations associated to a surface, Topology 27 (1988), 91-117. · Zbl 0649.32017 · doi:10.1016/0040-9383(88)90008-0
[5] K. M. Chapman, M. Mulase, and B. Safnuk, The Kontsevich constants for the volume of the moduli of curves and topological recursion, preprint, 2010, arXiv: · Zbl 1259.14028
[6] R. Dijkgraaf, E. Verlinde, and H. Verlinde, Loop equations and Virasoro constraints in nonperturbative two-dimensional quantum gravity, Nuclear Phys. B 348 (1991), 435-456. · Zbl 0783.58088 · doi:10.1016/0550-3213(91)90199-8
[7] J. J. Duistermaat and G. J. Heckman, On the variation in the cohomology of the symplectic form of the reduced phase space, Invent. Math. 69 (1982), 259-268. · Zbl 0503.58015 · doi:10.1007/BF01399506
[8] B. Eynard, Recursion between Mumford volumes of moduli spaces, preprint, 2007, arXiv: · Zbl 1245.14013 · doi:10.1007/s00023-011-0113-4
[9] —, All order asymptotic expansion of large partitions, J. Stat. Mech. Theory Exp. 9 (2008), P07023.
[10] —, A matrix model for plane partitions, J. Stat. Mech. Theory Exp. 10 (2009), P10011.
[11] B. Eynard, A. K. Kashani-Poor, and O. Marchal, A matrix model for the topological string I: Deriving the matrix model, preprint, 2010, arXiv: · Zbl 1408.81027
[12] B. Eynard, M. Mulase, and B. Safnuk, The Laplace transform of the cut-and-join equation and the Bouchard-Marino conjecture on Hurwitz numbers, Publ. Res. Inst. Math. Sci. 47 (2011), 629-670. · Zbl 1225.14022 · doi:10.2977/PRIMS/47
[13] B. Eynard and N. Orantin, Invariants of algebraic curves and topological expansion, Commun. Number Theory Phys. 1 (2007), 347-452. · Zbl 1161.14026 · doi:10.4310/CNTP.2007.v1.n2.a4
[14] —, Weil-Petersson volume of moduli spaces, Mirzakhani’s recursion and matrix models, preprint, 2007, arXiv:
[15] —, Topological recursion in enumerative geometry and random matrices, J. Phys. A 42 (2009), 293001. · Zbl 1177.82049 · doi:10.1088/1751-8113/42/29/293001
[16] —, Geometrical interpretation of the topological recursion, and integrable string theories, preprint, 2009, arXiv:
[17] V. Guillemin and S. Sternberg, Convexity properties of the moment mapping, Invent. Math. 67 (1982), 491-513. · Zbl 0503.58017 · doi:10.1007/BF01398933
[18] J. L. Harer, The cohomology of the moduli space of curves, Theory of moduli (Montecatini Terme, 1985), Lecture Notes in Math., 1337, pp. 138-221, Springer-Verlag, Berlin, 1988. · Zbl 0707.14020
[19] M. E. Kazarian and S. K. Lando, An algebro-geometric proof of Witten’s conjecture, J. Amer. Math. Soc. 20 (2007), 1079-1089. · Zbl 1155.14004 · doi:10.1090/S0894-0347-07-00566-8
[20] Y.-S. Kim and K. Liu, A simple proof of Witten conjecture through localization, preprint, 2005..
[21] M. Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys. 147 (1992), 1-23. · Zbl 0756.35081 · doi:10.1007/BF02099526
[22] K. Liu and H. Xu, New properties of the intersection numbers on moduli spaces of curves, Math. Res. Lett. 14 (2007), 1041-1054. · Zbl 1184.14043 · doi:10.4310/MRL.2007.v14.n6.a12
[23] —, New results of intersection numbers on moduli spaces of curves, Proc. Natl. Acad. Sci. USA 104 (2007), 13896-13900. · Zbl 1190.14056 · doi:10.1073/pnas.0705910104
[24] —, Recursion formulae of higher Weil-Petersson volumes, Int. Math. Res. Not. 5 (2009), 835-859. · Zbl 1186.14059 · doi:10.1093/imrn/rnn148
[25] M. Mirzakhani, Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces, Invent. Math. 167 (2007), 179-222. · Zbl 1125.30039 · doi:10.1007/s00222-006-0013-2
[26] —, Weil-Petersson volumes and intersection theory on the moduli space of curves, J. Amer. Math. Soc. 20 (2007), 1-23. · Zbl 1120.32008 · doi:10.1090/S0894-0347-06-00526-1
[27] M. Mulase and M. Penkava, Ribbon graphs, quadratic differentials on Riemann surfaces, and algebraic curves defined over \(\bar{\mathbb Q},\) Asian J. Math. 2 (1998), 875-919. · Zbl 0964.30023
[28] —, Topological recursion for the Poincaré polynomial of the combinatorial moduli space of curves, preprint, 2010, arXiv:
[29] M. Mulase and B. Safnuk, Mirzakhani’s recursion relations, Virasoro constraints and the KdV hierarchy, Indian J. Math. 50 (2008), 189-218. · Zbl 1144.14030
[30] —, Combinatorial structures in topological recursion, in preparation.
[31] M. Mulase and N. Zhang, Polynomial recursion formula for linear Hodge integrals, Commun. Number Theory Phys. 4 (2010), 267-293. · Zbl 1239.14022
[32] P. Norbury, Counting lattice points in the moduli space of curves, preprint, 2008, arXiv: · Zbl 1225.32023 · doi:10.4310/MRL.2010.v17.n3.a7
[33] —, String and dilaton equations for counting lattice points in the moduli space of curves, Math. Res. Lett. 17 (2010), 467-481. · Zbl 1225.32023
[34] A. Okounkov and R. Pandharipande, Gromov-Witten theory, Hurwitz numbers, and matrix models, Algebraic geometry (Seattle, 2005), Proc. Sympos. Pure Math., 80, pp. 325-414, Amer. Math. Soc., Providence, RI, 2009. · Zbl 1205.14072
[35] R. C. Penner, The decorated Teichmüller space of punctured surfaces, Comm. Math. Phys. 113 (1987), 299-339. · Zbl 0642.32012 · doi:10.1007/BF01223515
[36] B. Safnuk, Integration on moduli spaces of stable curves through localization, Differential Geom. Appl. 27 (2009), 179-187. · Zbl 1162.53063 · doi:10.1016/j.difgeo.2009.01.001
[37] —, Generalizations of topological recursion, in preparation.
[38] D. D. Sleator, R. E. Tarjan, and W. P. Thurston, Rotation distance, triangulations, and hyperbolic geometry, J. Amer. Math. Soc. 1 (1988), 647-681. · Zbl 0653.51017 · doi:10.2307/1990951
[39] E. Witten, Two-dimensional gravity and intersection theory on moduli space, Surveys in differential geometry (Cambridge, MA, 1990), pp. 243-310, Lehigh Univ., Bethlehem, PA, 1991. · Zbl 0757.53049
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