tailieunhanh - Đề tài " Higher genus GromovWitten invariants as genus zero invariants of symmetric products "

I prove a formula expressing the descendent genus g Gromov-Witten invariants of a projective variety X in terms of genus 0 invariants of its symmetric product stack S g+1 (X). When X is a point, the latter are structure constants of the symmetric group, and we obtain a new way of calculating the GromovWitten invariants of a point. 1. Introduction Let X be a smooth projective variety. The genus 0 Gromov-Witten invariants of X satisfy relations which imply that they can be completely encoded in the structure of a Frobenius manifold on the cohomology H ∗ (X, C). . | Annals of Mathematics Higher genus Gromov-Witten invariants as genus zero invariants of symmetric products By Kevin Costello Annals of Mathematics 164 2006 561 601 Higher genus Gromov-Witten invariants as genus zero invariants of symmetric products By Kevin Costello Abstract I prove a formula expressing the descendent genus g Gromov-Witten invariants of a projective variety X in terms of genus 0 invariants of its symmetric product stack Sg 1 X . When X is a point the latter are structure constants of the symmetric group and we obtain a new way of calculating the Gromov-Witten invariants of a point. 1. Introduction Let X be a smooth projective variety. The genus 0 Gromov-Witten invariants of X satisfy relations which imply that they can be completely encoded in the structure of a Frobenius manifold on the cohomology H X C . In this paper I prove a formula which expresses the descendent genus g Gromov-Witten invariants of a smooth projective variety X in terms of the descendent genus 0 invariants of the symmetric product stack Sg 1X. The latter are encoded in a Frobenius manifold structure on the orbifold cohomology group HOrb Sg 1 X C . This implies that the Gromov-Witten invariants of X at all genera are described by a sequence of Frobenius manifold structures on the homogeneous components of the Fock space F Sym H X C C iC i d 0H rb Sd X C . Standard properties of genus 0 invariants such as associativity when applied to the symmetric product stacks SdX yield implicit relations among higher-genus Gromov-Witten invariants of X . When X is a point the symmetric product is the classifying stack BSd of the symmetric group. The Frobenius manifold associated to the genus 0 invariants of BSd is in fact a Frobenius algebra which is the center of the group algebra of the symmetric group C SdA. Our result therefore gives a new way of expressing the integrals of tautological classes on Mg n in terms of structure constants of C Sd . 562 KEVIN COSTELLO More generally the .