Results 1  10
of
37
Semisimple Frobenius structures at higher genus
, 2000
"... We describe genus g ≥ 2 potentials of semisimple Frobenius structures. Our formula can be considered as a definition in the axiomatic context of Frobenius manifolds. In GromovWitten theory, it becomes a conjecture expressing higher genus GWinvariants in terms of genus 0 GWinvariants of symplect ..."
Abstract

Cited by 74 (4 self)
 Add to MetaCart
(Show Context)
We describe genus g ≥ 2 potentials of semisimple Frobenius structures. Our formula can be considered as a definition in the axiomatic context of Frobenius manifolds. In GromovWitten theory, it becomes a conjecture expressing higher genus GWinvariants in terms of genus 0 GWinvariants of symplectic manifolds with generically semisimple quantum cupproduct. The conjecture is supported by the corresponding theorem about equivariant GWinvariants of tori actions with isolated fixed points. The parallel theory of gravitational descendents is also presented.
Relative maps and tautological classes
"... 0.1. Tautological classes. Let Mg,n be the moduli space of stable curves of genus g with n marked points defined over C. Let A ∗ (Mg,n) denote the Chow ring (always taken here with Qcoefficients). The system of tautological rings ..."
Abstract

Cited by 62 (9 self)
 Add to MetaCart
(Show Context)
0.1. Tautological classes. Let Mg,n be the moduli space of stable curves of genus g with n marked points defined over C. Let A ∗ (Mg,n) denote the Chow ring (always taken here with Qcoefficients). The system of tautological rings
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 ..."
Abstract

Cited by 48 (8 self)
 Add to MetaCart
(Show Context)
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
COUNTING CURVES ON RATIONAL SURFACES
"... Abstract. In [CH3], Caporaso and Harris derive recursive formulas counting nodal plane curves of degree d and geometric genus g in the plane (through the appropriate number of fixed general points). We rephrase their arguments in the language of maps, and extend them to other rational surfaces, and ..."
Abstract

Cited by 28 (0 self)
 Add to MetaCart
(Show Context)
Abstract. In [CH3], Caporaso and Harris derive recursive formulas counting nodal plane curves of degree d and geometric genus g in the plane (through the appropriate number of fixed general points). We rephrase their arguments in the language of maps, and extend them to other rational surfaces, and other specified intersections with a divisor. As applications, (i) we count irreducible curves on Hirzebruch surfaces in a fixed divisor class and of fixed geometric genus, (ii) we compute the higher genus GromovWitten invariants of (or equivalently, counting curves of any genus and divisor class on) del Pezzo surfaces of degree at least 3. In the case of the cubic surface in (ii), we first use a result of Graber to enumeratively interpret highergenus GromovWitten invariants of certain Knef surfaces, and then apply this to a degeneration of a cubic surface.
The moduli space of curves and GromovWitten theory
, 2006
"... The goal of this article is to motivate and describe how GromovWitten theory can and has provided tools to understand the moduli space of curves. For example, ideas and methods from GromovWitten theory have led to both conjectures and theorems showing that the tautological part of the cohomology r ..."
Abstract

Cited by 26 (4 self)
 Add to MetaCart
(Show Context)
The goal of this article is to motivate and describe how GromovWitten theory can and has provided tools to understand the moduli space of curves. For example, ideas and methods from GromovWitten theory have led to both conjectures and theorems showing that the tautological part of the cohomology ring has a remarkable and profound structure. As an illustration, we describe a new approach to Faber’s intersection number conjecture via branched covers of the projective line (work with I.P. Goulden and D.M. Jackson, based on work with T. Graber). En route we review the work of a large number of mathematicians.
Invariance of tautological equations I: conjectures and applications
"... Abstract. The main goal of this paper is to introduce a set of conjectures on the relations in the tautological rings. In particular, the conjectures gives an efficient algorithm to calculate, conjecturally, all tautological equations using only finite dimensional linear algebra. Other applications ..."
Abstract

Cited by 23 (8 self)
 Add to MetaCart
(Show Context)
Abstract. The main goal of this paper is to introduce a set of conjectures on the relations in the tautological rings. In particular, the conjectures gives an efficient algorithm to calculate, conjecturally, all tautological equations using only finite dimensional linear algebra. Other applications include the proofs of Witten’s conjecture on the relations between higher spin curves and Gelfand– Dickey hierarchy and Virasoro conjecture for target manifolds with conformal semisimple quantum cohomology, both for genus up to two. 1.
Witten’s conjecture, Virasoro conjecture, and semisimple Frobenius manifolds
, 2002
"... Abstract. The main goal of this paper is to prove the following two conjectures for genus up to two: (1) Witten’s conjecture on the relations between higher spin curves and Gelfand–Dickey hierarchy. (2) Virasoro conjecture for target manifolds with conformal semisimple Frobenius manifolds. The main ..."
Abstract

Cited by 21 (7 self)
 Add to MetaCart
(Show Context)
Abstract. The main goal of this paper is to prove the following two conjectures for genus up to two: (1) Witten’s conjecture on the relations between higher spin curves and Gelfand–Dickey hierarchy. (2) Virasoro conjecture for target manifolds with conformal semisimple Frobenius manifolds. The main technique used in the proof is the invariance of tautological equations under loop group action. 1.
Virasoro constraints for quantum cohomology
 J. Diff. Geom
, 1998
"... In [EHX2], Eguchi, Hori and Xiong, proposed a conjecture that the partition function of topological sigma model coupled to gravity is annihilated by infinitely many differential operators which form half branch of the Virasoro algebra. A similar conjecture was also proposed by S. Katz [Ka] (See also ..."
Abstract

Cited by 17 (7 self)
 Add to MetaCart
(Show Context)
In [EHX2], Eguchi, Hori and Xiong, proposed a conjecture that the partition function of topological sigma model coupled to gravity is annihilated by infinitely many differential operators which form half branch of the Virasoro algebra. A similar conjecture was also proposed by S. Katz [Ka] (See also [EJX]). Assuming this conjecture is true, they were able to reproduce certain instanton numbers of some projective spaces known before (cf. the above cited references and [EX] for details). This conjecture is also referred to as the Virasoro conjecture by some authors. The main purpose of this paper is to give a proof of this conjecture for the genus zero part. The theory of topological sigma model coupled to gravity has been extensively studied recently by both mathematicians and physicists. This theory is built on the intersection theory of moduli spaces of stable maps from Riemann surfaces to a fixed manifold V 2d,which is a smooth projective variety (or more generally, a symplectic manifold). To each cohomology class of V (denoted by O) and a nonnegative integer n, there is associated a quantum field theory operator, denoted by τn(O). When n = 0, the corresponding operator is simply denoted by O and is called a primary field. For n> 0, τn(O) is called the nth (gravitational) descendent of O. The so called kpoint genusg correlators in topological field theory can be defined via the GromovWitten invariants as follows:
Enumerative geometry of hyperelliptic plane curves
 J. Algebraic Geom
"... In recent years there has been a tremendous amount of progress on classical problems in enumerative geometry. This has largely been a result of new ideas and motivation for these problems coming from theoretical physics. In particular, the theory of GromovWitten invariants ..."
Abstract

Cited by 16 (2 self)
 Add to MetaCart
(Show Context)
In recent years there has been a tremendous amount of progress on classical problems in enumerative geometry. This has largely been a result of new ideas and motivation for these problems coming from theoretical physics. In particular, the theory of GromovWitten invariants
A genus3 topological recursion relation
"... In this paper, we give a new genus3 topological recursion relation for GromovWitten invariants of compact symplectic manifolds. This formula also applies to intersection numbers on moduli spaces of spin curves. A byproduct of the proof of this formula is a new relation in the tautological ring of ..."
Abstract

Cited by 15 (2 self)
 Add to MetaCart
(Show Context)
In this paper, we give a new genus3 topological recursion relation for GromovWitten invariants of compact symplectic manifolds. This formula also applies to intersection numbers on moduli spaces of spin curves. A byproduct of the proof of this formula is a new relation in the tautological ring of the moduli space of 1pointed genus3 stable curves. Let Mg,n be the moduli space of genusg stable curves with n marked points. It is well known that relations in the tautological rings on Mg,n produce universal equations for the GromovWitten invariants of compact symplectic manifolds. Examples of genus1 and genus2 universal equations were given in [Ge1], [Ge2], and [BP]. Relations among known universal equations were discussed in [L2]. It is expected that for manifolds with semisimple quantum cohomology, such universal equations completely determine all higher genus GromovWitten invariants in terms of its genus0 invariants. This has been proven for the genus1 case in [DZ] and for the genus2 case in [L1]. However for genus bigger than 2, no explicit universal equations had been found except for those which follow from