Localized standard versus reduced formula and genus one local Gromov-Witten invariants

Abstract: For local Calabi-Yau manifolds which are total spaces of vector bundle over balloon manifolds, we propose a formal definition of reduced Genus one Gromov-Witten invariants, by assigning contributions from the refined decorated rooted trees. We show that this definition satisfies a localized version of the standard versus reduced formula, whose global version in the compact cases is due to A. Zinger. As an application we prove the conjecture in a previous article on the genus one Gromov-Witten invariants of local Calabi-Yau manifolds which are total spaces of concave splitting vector bundles over projective spaces. In particular, we prove the mirror formulae for genus one Gromov-Witten 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 one Gromov-Witten invariants for the local $\mathbb{P}^{2}$ as a consequence of the results on Ramanujan's cubic transformation. Inspired by the localized standard versus reduced formula, we show that the ordinary genus one Gromov-Witten invariants of Calabi-Yau hypersurfaces in projective spaces can be computed by virtual localization after the contribution of a genus one vertex is replaced by a modified one.