BPS invariants of symplectic log Calabi-Yau fourfolds

TL;DR

This paper investigates genus zero and higher relative Gromov-Witten invariants of symplectic log Calabi-Yau fourfolds, providing new proofs, detailed calculations, and insights into higher genus cases.

Contribution

It offers a short proof of a conjecture relating invariants to integral invariants, revisits localization calculations, and explores higher genus invariants with decomposition analysis.

Findings

Proof of conjecture relating invariants to integral invariants

Recalculation of localization contributions with detailed deformation space analysis

Decomposition of genus one invariants into distinct contributions

Abstract

Using the Fredholm setup of [12], we study genus zero (and higher) relative Gromov-Witten invariants with maximum tangency of symplectic log Calabi-Yau fourfolds. In particular, we give a short proof of [23, Conjecture 6.2] that expresses these invariants in terms of certain integral invariants by considering generic almost complex structures to obtain a geometric count. We also revisit the localization calculation of the multiple-cover contributions in [23, Proposition 6.1] and recalculate a few terms differently to provide more details and illustrate the computation of deformation/obstruction spaces for maps that have components in a destabilizing (or rubber) component of the target. Finally, we study a higher genus version of these invariants and explain a decomposition of genus one invariants into different contributions.