The Gopakumar-Vafa formula for symplectic manifolds

Abstract

Abstract. The Gopakumar-Vafa conjecture predicts that the Gromov-Witten invariants of a Calabi-Yau 3-fold can be canonically expressed in terms of integer invariants called BPS numbers. Using the methods of symplectic Gromov-Witten theory, we prove that the Gopakumar-Vafa conjecture holds for any symplectic Calabi-Yau 6-manifold, and hence for Calabi-Yau 3-folds. The results extend to all symplectic 6-manifolds and to the genus zero GW invariants of semipositive manifolds.The Gopakumar-Vafa conjecture [GV] predicts that the Gromov-Witten invariants GW A,g of a Calabi-Yau 3-fold can be expressed in terms of some other invariants n A,h , called BPS numbers, by a transform between their generating functions:The content of the conjecture is that, while the GW A,g are rational numbers, the BPS numbers n A,h are integers. (Gopakumar and Vafa also conjectured that for each A ∈ H 2 (X, Z), the coefficients of (0.1) satisfy n A,h = 0 for large h; we do not address this finiteness statement here.) It is natural to enlarge the context by regarding this as a conjecture about the Gromov-Witten invariants of any closed symplectic 6-manifold X that satisfies the topological Calabi-Yau condition c 1 (X) = 0. Formula (0.1) can be viewed as a statement about the structure of the space of solutions to the Jholomorphic map equation. For a generic almost complex structure J, each J-holomorphic map is the composition f = ϕ • ρ of a multiple-cover ρ and an embedding ϕ. The embeddings are well-behaved: they have no nontrivial automorphisms, and the moduli space of J-holomorphic embeddings is a manifold. But multiply-covered maps cause severe analytical problems with transversality. In the symplectic construction of the GW invariants, these problems are avoided by lifting to a cover of the moduli space and turning on a lift-dependent perturbation ν of the equation; this destroys the multiple-cover structure and only shows that the numbers GW A,g are rational. But it also suggests an interpretation of the GV formula: the righthand side of (0.1) might be a sum over embeddings, with the sum over k counting the contributions of the multiple covers of each embedding.This viewpoint is very similar to C. Taubes' work [T] relating Gromov invariants to the SeibergWitten invariants of 4-manifolds, and our approach has been fundamentally influenced by Taubes. It is also similar to the 4-dimensional situation described by Lee and Parker in [LP1] and [LP2]. In both cases, the set of J-holomorphic embeddings in each homology class is discrete and compact for generic J -a simplifying circumstance that does not appear to be true in the context of formula (0.1). Rather, for generic J and with a fixed bound E on area and genus, the moduli space M emb (X) of embeddings is a countable set, possibly with accumulation points. With this picture in mind, our proof is based on three main ideas.The first is the observation that, again for fixed J and E, the full moduli space M(X) can be decomposed (in many ways) into finitely many "clusters" O j . Each cluster...

Generic Transversality for Unbranched Covers of Closed Pseudoholomorphic Curves

“…Some special cases of this conjecture have been proved previously by Lee and Parker [16,17] and Eftekhary [5]. The techniques used in the present paper are related to those of [16,17], which also play a role in the announced solution by Ionel and Parker to the Gopakumar-Vafa conjecture [13].…”

“…On any closed symplectic manifold .M; !/ of real dimension at least 4, there exists a Baire subset J reg in the space of smooth !-tame almost complex structures such that for all J 2 J reg , every closed, connected, and simple J -holomorphic curve with deformation index 0 is super-rigid.Some special cases of this conjecture have been proved previously by Lee and Parker [16,17] and Eftekhary [5]. The techniques used in the present paper are related to those of [16,17], which also play a role in the announced solution by Ionel and Parker to the Gopakumar-Vafa conjecture [13].For an unbranched cover of a simple curve, the super-rigidity condition is equivalent to the usual notion of Fredholm regularity, and our main result (stated as Theorem 1.3 below) is that this can always be achieved by choosing J generically. This may be seen as an initial step toward a proof of Conjecture 1.1 in full generality.…”

“…Results of this nature are desirable for several reasons: one is that the resulting definition of the Gromov-Witten invariants is simpler to understand and to apply. Another is that the relationship between simple curves and their multiple covers can reveal nontrivial relations among Gromov-Witten invariants that cannot be seen by more abstract techniques; one example of this phenomenon is the Gopakumar-Vafa conjecture on symplectic Calabi-Yau 3-folds; see [2,3,9,13]. While moduli spaces of multiply covered curves cannot With Remark 1.2 in mind, from now on we fix a symplectic form !…”

Some Questions in the Theory of Pseudoholomorphic Curves

“…The approach of [46] to the integrality of the numbers n X g,A determined by (2.13) should be adaptable to other situations when the GW-invariants in question are expected to arise entirely from isolated J-holomorphic curves. These situations include the real genus 0 GW-invariants of many real symplectic manifolds and the real arbitrary genus GW-invariants of real symplectic CY sixfolds constructed in [26] and [30], respectively.…”

“…These situations include the real genus 0 GW-invariants of many real symplectic manifolds and the real arbitrary genus GW-invariants of real symplectic CY sixfolds constructed in [26] and [30], respectively. In fact, the integrality of the numbers E X 0,A (µ) determined by (2.15) is already a (secondary) subject of [46]. On the other hand, the approach of [46] does not appear readily adaptable to situations when positive-dimensional families of J-holomorphic curves in X are expected to contribute to the GW-invariants in question.…”