Gromov-Witten Invariants of Symplectic Sums

Abstract

The natural sum operation for symplectic manifolds is defined by gluing along codimension two submanifolds. Specifically, let X be a symplectic 2n-manifold with a symplectic (2n − 2)-submanifold V . Given a similar pair (Y, V ) with a symplectic identification V = V and a complex anti-linear isomorphism between the normal bundles of V and V , we can form the symplectic sum Z = X# V =V Y . This note announces a general formula for computing the Gromov-Witten invariants of the sum Z in terms of relative Gromov-Witten invariants of (X, V ) and (Y, V ).Section 1 is a review of the GW invariants for symplectic manifolds and the associated invariants, which we call T W invariants, that count reducible curves. The corresponding relative invariants of a symplectic pair (X, V ) are defined in section 2. The sum formula is stated in a special case in section 3, and in general as Theorem 4.1. The last section presents two applications: a short derivation of the Caporaso-Harris formula [CH], and new proof that the rational enumerative invariants of the rational elliptic surface are given by the "modular form" (5.3).Related results, involving symplectic sums along contact manifolds, are being developed by Li and Ruan [LR] and by Eliashberg and Hofer. Symplectic invariantsThe moduli space of (J, ν)-holomorphic maps from genus g curves with n marked points representing a class A in the free part of H 2 (X; Z) has a compactification M g,n (X, A). This comes with a mapwhere the first factor is the "stabilization" map st to the Deligne-Mumford moduli space (defined by collapsing all unstable components of the domain curve), the second factor records the images of the marked points, and the last one keeps track of the homology class A of the image. After perturbation (cf.[LT]), the image defines a rational homology class,for each choice g, n, A. The last term in (1.2) can be identified with the rational group ring RH 2 (X) of H free 2 (X; Z), that is, with finite sums c A t A over A ∈ H free 2 (X; Z) where c A ∈ Q and the t A are variables satisfying t A t B = t A+B .

Open string instantons and relative stable morphisms

“…The construction of relative stable maps was first developed in the symplectic category [LR,IP1,IP2]. Recently in [Lil,Li2] the first author of the present paper has given an algebro-geometric definition of the moduli space of relative stable morphisms and has constructed relative Gromov-Witten invariants in the algebraic category.…”

Open string instantons and relative stable morphisms

“…Hence we have (5)(6)(7)(8)(9)(10) where in the last equality we have used the fact that the moduli space M g,1 of DeligneMumford stable curves has dimension dim C M g,1 = 3g − 2. The Hodge integral above can be easily evaluated by using Faber and Pandharipande's generating function for Hodge integrals over the moduli space M g,1 [2].…”

“…It will become clear later in the paper that the theory of relative stable maps is tailor-made for studying topological open string theory. The construction of relative stable maps was first developed in the symplectic category (Li-Ruan [11], Ionel-Parker [6,7]). Recently in [13,12] the first author of the present paper has given an algebro-geometric definition of the moduli space of relative stable morphisms and has constructed relative Gromov-Witten invariants in the algebraic category.…”

“…According to the general principle of virtual moduli cycles, the expected (virtual) number of maps in M rel g,µ (W r ) should be (3)(4)(5)(6)(7)(8)(9)(10) [ Extending V to M rel g,µ (P 1 ) needs more work. There is an obvious extension as follows: Let f ∈ M rel g,µ (P 1 ) be any relative stable morphism with domain C and distinguished divisor D d as before.…”

Symplectic gluing and family Gromov-Witten invariants

“…With one more application of the sum formula we can obtain the corresponding generating function for curves of each genus g > 0 in E(1) (see [IP1]). …”