Localized Standard Versus Reduced Formula and Genus 1 Local Gromov–Witten Invariants

Abstract

For local Calabi–Yau (CY) manifolds which are total spaces of vector bundle over algebraic Goresky-Kottwitz-MacPherson (GKM) manifolds, we propose a formal definition of reduced genus 1 Gromov–Witten (GW) invariants, by assigning contributions from the refined decorated rooted trees. We show that this definition satisfies a localized version of the standard versus reduced (LSvR) formula, whose global version in the compact cases is due to Zinger. As an application, we prove the conjecture in a previous article on the genus 1 GW invariants of local CY manifolds which are total spaces of concave splitting vector bundles over projective spaces. In particular, we prove the mirror formulae for genus 1 GW invariants of |$K_{\mathbb {P}^{2}}$| and |$K_{\mathbb {P}^{3}}$|⁠, conjectured by Klemm, Zaslow, and Pandharipande. In the appendix, we derive the modularity of genus 1 GW invariants for the local |$\mathbb {P}^{2}$| as a consequence of the results on Ramanujan's cubic transformation. Inspired by the LSvR formula, we show that the ordinary genus 1 GW invariants of CY hypersurfaces in projective spaces can be computed by virtual localization and quantum hyperplane property after the contribution of a genus 1 vertex is replaced by a modified one.