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12
Bihamiltonian Hierarchies in 2D Topological Field Theory At OneLoop Approximation
, 1997
"... We compute the genus one correction to the integrable hierarchy describing coupling to gravity of a 2D topological field theory. The bihamiltonian structure of the hierarchy is given by a classical Walgebra; we compute the central charge of this algebra. We also express the generating function of ..."
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Cited by 92 (8 self)
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We compute the genus one correction to the integrable hierarchy describing coupling to gravity of a 2D topological field theory. The bihamiltonian structure of the hierarchy is given by a classical Walgebra; we compute the central charge of this algebra. We also express the generating function of elliptic Gromov Witten invariants via taufunction of the isomonodromy deformation problem arising in the theory of WDVV equations of associativity.
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
A descendant relation in genus 2
"... Let Mg,n be the moduli space of DeligneMumford stable, npointed, genus g, complex algebraic curves. There is an algebraic stratification of Mg,n by the underlying topological type of the pointed curve. These strata are naturally indexed by stable, npointed, genus g dual graphs. A relation among t ..."
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Cited by 38 (4 self)
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Let Mg,n be the moduli space of DeligneMumford stable, npointed, genus g, complex algebraic curves. There is an algebraic stratification of Mg,n by the underlying topological type of the pointed curve. These strata are naturally indexed by stable, npointed, genus g dual graphs. A relation among the cycle classes (or homological classes)
Invertible cohomological field theories and WeilPetersson volumes
"... Abstract. We show that the generating function for the higher Weil–Petersson volumes of the moduli spaces of stable curves with marked points can be obtained from Witten’s free energy by a change of variables given by Schur polynomials. Since this generating function has a natural extension to the m ..."
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Cited by 26 (1 self)
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Abstract. We show that the generating function for the higher Weil–Petersson volumes of the moduli spaces of stable curves with marked points can be obtained from Witten’s free energy by a change of variables given by Schur polynomials. Since this generating function has a natural extension to the moduli space of invertible Cohomological Field Theories, this suggests the existence of a “very large phase space”, correlation functions on which include Hodge integrals studied by C. Faber and R. Pandharipande. From this formula we derive an asymptotical expression for the Weil–Petersson volume as conjectured by C. Itzykson. We also discuss a topological interpretation of the genus expansion formula of Itzykson–Zuber, as well as a related bialgebra acting upon quantum cohomology as a complex version of the classical path groupoid. 0. Introduction and summary. The aim of this paper is to record some progress in understanding intersection numbers on moduli spaces of stable pointed curves and their generating functions. Continuing the study started in [KoM], [KoMK] and pursued further in [KaMZ], [KabKi], we work with Cohomological
Recursions for characteristic numbers of genus one plane curves
, 1998
"... Characteristic numbers of families of maps of nodal curves to P² are de ned as intersection of natural divisor classes. (This definition agrees with the usual definition for families of plane curves.) Simple recursions for characteristic numbers of genus one plane curves of all degrees are computed. ..."
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Cited by 17 (10 self)
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Characteristic numbers of families of maps of nodal curves to P² are de ned as intersection of natural divisor classes. (This definition agrees with the usual definition for families of plane curves.) Simple recursions for characteristic numbers of genus one plane curves of all degrees are computed.
Enumerative Geometry of Plane Curve of Low Genus
"... Abstract. We collect various known results (about plane curves and the moduli space of stable maps) to derive new recursive formulas enumerating low genus plane curves of any degree with various behaviors. Recursive formulas are given for the characteristic numbers of rational plane curves, elliptic ..."
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Cited by 13 (0 self)
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Abstract. We collect various known results (about plane curves and the moduli space of stable maps) to derive new recursive formulas enumerating low genus plane curves of any degree with various behaviors. Recursive formulas are given for the characteristic numbers of rational plane curves, elliptic plane curves, and elliptic plane curves with fixed complex structure. Recursions are also given for the number of elliptic (and rational) plane curves with various “codimension 1 ” behavior (cuspidal, tacnodal, triple pointed, etc., as well as the geometric and arithmetic sectional genus of the Severi variety). We compute the latter numbers for genus 2 and 3 plane curves as well. We rely on results of Caporaso, Diaz, Getzler, Harris, Ran, and especially Pandharipande. 1.
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.
MIRZAKHARNI’S RECURSION FORMULA IS EQUIVALENT TO THE WITTENKONTSEVICH THEOREM
, 2009
"... In this paper, we give a proof of Mirzakhani’s recursion formula of WeilPetersson volumes of moduli spaces of curves using the WittenKontsevich theorem. We also describe properties of intersections numbers involving higher degree κ classes. ..."
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Cited by 5 (3 self)
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In this paper, we give a proof of Mirzakhani’s recursion formula of WeilPetersson volumes of moduli spaces of curves using the WittenKontsevich theorem. We also describe properties of intersections numbers involving higher degree κ classes.
A change of coordinates on the large phase space of quantum cohomology
 Comm. Math. Phys
"... Abstract. The GromovWitten invariants of a smooth, projective variety V, when twisted by the tautological classes on the moduli space of stable maps, give rise to a family of cohomological field theories and endow the base of the family with coordinates. We prove that the potential functions associ ..."
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Cited by 3 (0 self)
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Abstract. The GromovWitten invariants of a smooth, projective variety V, when twisted by the tautological classes on the moduli space of stable maps, give rise to a family of cohomological field theories and endow the base of the family with coordinates. We prove that the potential functions associated to the tautological ψ classes (the large phase space) and the κ classes are related by a change of coordinates which generalizes a change of basis on the ring of symmetric functions. Our result is a generalization of the work of Manin–Zograf who studied the case where V is a point. We utilize this change of variables to derive the topological recursion relations associated to the κ classes from those associated to the ψ classes. Notation. All (co)homology are with Q coefficients unless explicitly mentioned otherwise. Summation over repeated upper and lower indices
INTERSECTION NUMBERS ON THE MODULI SPACES OF STABLE MAPS IN GENUS 0
, 1998
"... Abstract. Let V be a smooth, projective, convex variety. We define tautological ψ and κ classes on the moduli space of stable maps M0,n(V), give a (graphical) presentation for these classes in terms of boundary strata, derive differential equations for the generating functions of the GromovWitten i ..."
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Cited by 1 (0 self)
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Abstract. Let V be a smooth, projective, convex variety. We define tautological ψ and κ classes on the moduli space of stable maps M0,n(V), give a (graphical) presentation for these classes in terms of boundary strata, derive differential equations for the generating functions of the GromovWitten invariants of V twisted by these tautological classes, and prove that these intersection numbers are completely determined by the GromovWitten invariants of V. This results in families of Frobenius manifold structures on the cohomology ring of V which includes the quantum cohomology as a special case. There has recently been a great deal of interest in Mg,n(V), the moduli space of stable maps of genus g with n marked points into a smooth, projective variety V, an object whose construction was envisioned by Kontsevich as the proper algebrogeometric setting for GromovWitten