Counting genus zero real curves in symplectic manifolds

Abstract

There are two types of $J$-holomorphic spheres in a symplectic manifold invariant under an anti-symplectic involution: those that have a fixed point locus and those that do not. The former are described by moduli spaces of $J$-holomorphic disks, which are well studied in the literature. In this paper, we first study moduli spaces describing the latter and then combine the two types of moduli spaces to get a well-defined theory of counting real curves of genus 0. We use equivariant localization to show that these invariants (unlike the disk invariants) are essentially the same for the two (standard) involutions on ${\mathbb{P}}^{4n-1}$.

Publishing version

References in corpus (9)

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Cited by in corpus (14)

• Point-like bounding chains in open Gromov-Witten theory
• WDVV-Type Relations for Disk Gromov-Witten Invariants in Dimension 6
• Enumeration of real curves in CP^{2n-1} and a WDVV relation for real Gromov-Witten invariants
• Orientability in real Gromov-Witten theory
• A Klein TQFT: the local Real Gromov-Witten theory of curves
• Fixed-point Localization for ${\mathbb{RP}}^{2m}\subset {\mathbb{CP}}^{2m}$
• Equivariant Open Gromov-Witten Theory of $\mathbb{R}{\mathbb{P}}^{2m}\hookrightarrow \mathbb{C}{\mathbb{P}}^{2m}$
• WDVV-Type Relations for Welschinger's Invariants: Applications
• Notes on genus one real Gromov-Witten invariants
• A recursion for counts of real curves in CP^{2n-1}: another proof
• Real Ruan-Tian Perturbations
• Relative quantum cohomology of the Chiang Lagrangian
• Spin/Pin-Structures and Real Enumerative Geometry
• Real Topological Recursions and WDVV Relations