Energy Bounds and Vanishing Results for the Gromov-Witten Invariants of the Projective Space

TL;DR

This paper provides explicit generating functions for all genus Gromov-Witten invariants of projective space, establishing energy bounds and vanishing results that connect these invariants to symplectic geometry and moduli space dimensions.

Contribution

It introduces explicit generating functions for arbitrary-genus Gromov-Witten invariants of projective space, revealing structural properties and conjectural links to symplectic geometry.

Findings

Uniform energy bounds for Gromov-Witten invariants

Vanishing results for certain invariants

Conjectural relations between invariants, energy, and moduli space dimension

Abstract

We describe generating functions for arbitrary-genus Gromov-Witten invariants of the projective space with any number of marked points explicitly. The structural portion of this description gives rise to uniform energy bounds and vanishing results for these invariants. They suggest deep conjectures relating Gromov-Witten invariants of symplectic manifolds to the energy of pseudo-holomorphic maps and the expected dimension of their moduli space.