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26
Moduli spaces of higher spin curves and integrable hierarchies
 Compositio Math
"... Abstract. We prove the genus zero part of the generalized Witten conjecture relating moduli spaces of spin curves to GelfandDickey hierarchies. That is, we show that intersection numbers on the moduli space of stable rspin curves assemble into a generating function which yields a solution of the s ..."
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Cited by 52 (8 self)
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Abstract. We prove the genus zero part of the generalized Witten conjecture relating moduli spaces of spin curves to GelfandDickey hierarchies. That is, we show that intersection numbers on the moduli space of stable rspin curves assemble into a generating function which yields a solution of the semiclassical limit of the KdVr equations. We formulate axioms for a cohomology class on this moduli space which allow one to construct a cohomological field theory of rank r −1 in all genera. In genus zero it produces a Frobenius manifold which is isomorphic to the Frobenius manifold structure on the base of the versal deformation of the singularity Ar−1. We prove analogs of the puncture, dilaton, and topological recursion relations by drawing an analogy with the construction of GromovWitten invariants and quantum cohomology. The moduli space of stable curves of genus g with n marked points Mg,n is a fascinating object. Mumford [37] introduced tautological cohomology classes associated to the universal curve Cg,n
The GromovWitten potential of a point, Hurwitz numbers, and Hodge integrals
 Proc. London Math. Soc
, 1999
"... 1.1. Recursions and GromovWitten theory 2 ..."
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MIRZAKHANI’S RECURSION RELATIONS, VIRASORO CONSTRAINTS AND THE KDV HIERARCHY
"... Abstract. We present in this paper a differential version of Mirzakhani’s recursion relation for the WeilPetersson volumes of the moduli spaces of bordered Riemann surfaces. We discover that the differential relation, which is equivalent to the original integral formula of Mirzakhani, is a Virasoro ..."
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Cited by 28 (8 self)
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Abstract. We present in this paper a differential version of Mirzakhani’s recursion relation for the WeilPetersson volumes of the moduli spaces of bordered Riemann surfaces. We discover that the differential relation, which is equivalent to the original integral formula of Mirzakhani, is a Virasoro constraint condition on a generating function for these volumes. We also show that the generating function for ψ and κ1 intersections on Mg,n is a 1parameter solution to the KdV hierarchy. It recovers the WittenKontsevich generating function when the parameter is set to be 0. 1.
WeilPetersson geometry on moduli space of polarized CalabiYau manifolds
 J. Inst. Math. Jussieu
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An explicit upper bound for WeilPetersson volumes of the moduli spaces of punctured Riemann surfaces
"... Abstract. An explicit upper bound for the WeilPetersson volumes of punctured Riemann surfaces is obtained using Penner’s combinatorial integration scheme from [Pe1]. It is shown that for a fixed number of punctures n and for genus g increasing, volWP(Mg,n) lim ≤ 2, g→∞, n fixed g ln g while this li ..."
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Cited by 13 (0 self)
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Abstract. An explicit upper bound for the WeilPetersson volumes of punctured Riemann surfaces is obtained using Penner’s combinatorial integration scheme from [Pe1]. It is shown that for a fixed number of punctures n and for genus g increasing, volWP(Mg,n) lim ≤ 2, g→∞, n fixed g ln g while this limit is exactly equal to two for n = 1. After Wolpert in [Wo] computed the cohomology of the moduli space of Riemann surfaces as a graded vector space, the question of computing the cohomology ring structure (aka the intersection theory) on the moduli arose. The problem has been intensively studied since then.
Recursion formulae of higher WeilPetersson volumes
 Inter. Math. Res. Notices
"... Abstract. In this paper we study effective recursion formulae for computing intersection numbers of mixed ψ and κ classes on moduli spaces of curves. By using the celebrated WittenKontsevich theorem, we generalize MulaseSafnuk form of Mirzakhani’s recursion and prove a recursion formula of higher ..."
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Cited by 13 (4 self)
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Abstract. In this paper we study effective recursion formulae for computing intersection numbers of mixed ψ and κ classes on moduli spaces of curves. By using the celebrated WittenKontsevich theorem, we generalize MulaseSafnuk form of Mirzakhani’s recursion and prove a recursion formula of higher WeilPetersson volumes. We also present recursion formulae to compute intersection pairings in the tautological rings of moduli spaces of curves. 1.
Estimates of WeilPetersson volumes via effective divisors
 Comm. Math. Phys
"... The total free energy of two dimensional gravity, which is a generating function for certain intersection numbers on the compactified moduli spaces Mg,n of stable npointed curves, was conjectured by Witten (and proved by Kontsevich) to satisfy certain KdV equations. This ..."
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Cited by 12 (0 self)
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The total free energy of two dimensional gravity, which is a generating function for certain intersection numbers on the compactified moduli spaces Mg,n of stable npointed curves, was conjectured by Witten (and proved by Kontsevich) to satisfy certain KdV equations. This
KontsevichWitten model from 2+1 gravity: new exact combinatorial solution
, 2008
"... In previous publications ( J. Geom.Phys.38 (2001) 81139 and references therein) the partition function for 2+1 gravity was constructed for the fixed genus Riemann surface.With help of this function the dynamical transition from pseudoAnosov to periodic (Seifertfibered) regime was studied. In this ..."
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Cited by 10 (4 self)
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In previous publications ( J. Geom.Phys.38 (2001) 81139 and references therein) the partition function for 2+1 gravity was constructed for the fixed genus Riemann surface.With help of this function the dynamical transition from pseudoAnosov to periodic (Seifertfibered) regime was studied. In this paper the periodic regime is studied in some detail in order to recover major results of Kontsevich (Comm.Math.Phys. 147 (1992) 123) inspired by earlier work of Witten on topological two dimensional quantum gravity.To achieve this goal some results from enumerative combinatorics have been used. The logical developments are extensively illustrated using geometrically convincing figures. This feature is helpful for development of some non traditional applications (mentioned through the entire text) of obtained results to fields other than theoretical particle physics.
Descendant GromovWitten Invariants, Simple Hurwitz Numbers, and the Virasoro Conjecture for IP¹
, 1999
"... In this “experimental” research, we use known topological recursion relations in generazero,one, andtwo to compute the npoint descendant GromovWitten invariants of IP¹ for arbitrary degrees and low values of n. The results are consistent with the Virasoro conjecture and also lead to explicit com ..."
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Cited by 9 (5 self)
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In this “experimental” research, we use known topological recursion relations in generazero,one, andtwo to compute the npoint descendant GromovWitten invariants of IP¹ for arbitrary degrees and low values of n. The results are consistent with the Virasoro conjecture and also lead to explicit computations of all Hodge integrals in these genera. We also derive new recursion relations for simple Hurwitz numbers similar to those of Graber and Pandharipande.