Higher genus Gromov-Witten invariants of the Grassmannian, and the Pfaffian Calabi-Yau threefolds

TL;DR

This work computes higher-genus Gromov–Witten invariants for a derived-equivalent pair of Calabi–Yau threefolds $X$ and $X'$ arising from $Gr(2,7)$ and $Pf(7)$ by solving the BCOV holomorphic anomaly equations for $g\le 5$ using polynomiality (YY) and gap conditions (HKQ). Leveraging mirror symmetry, Picard–Fuchs equations, and mirror maps, the genus-0 and genus-1 data are established and shown to align across the pair, while the genus-$g$ potentials ${\tt F}_g(t)$ are reconstructed recursively up to $g=5$ with holomorphic ambiguities fixed through conifold-gap analyses and GV-invariant constraints. The results yield explicit Gopakumar–Vafa invariants $n_g(d)$ up to genus 5 (Tables 1–2) and support the derived-equivalence picture, including the conjectured two-FM-partner structure and the interpretation of $X'$ as a moduli space of stable sheaves on $X$. The study also highlights a special conifold structure at $x=3$ and suggests additional gap-based refinements to extend to higher genus.

Abstract

We solve Bershadsky-Cecotti-Ooguri-Vafa (BCOV) holomorphic anomaly equation to determine the higher genus Gromov-Witten invariants ($g \leq 5$) of the derived equivalent Calabi-Yau threefolds, which are of the appropriate codimensions in the Grassmannian Gr(2,7) and the Pfaffian Pf(7).