Higher genus Gromov-Witten theory of one-parameter Calabi-Yau threefolds II: Feynman rule and anomaly equations

TL;DR

This paper proves conjectured Feynman rules and anomaly equations for higher genus Gromov-Witten invariants of specific Calabi-Yau threefolds, enabling recursive computation of these invariants from lower-genus data.

Contribution

It establishes the Feynman rule and anomaly equations for certain Calabi-Yau threefolds, providing a recursive method to compute Gromov-Witten invariants.

Findings

Proved Feynman rule conjecture for specified Calabi-Yau threefolds.

Validated anomaly equations for these threefolds.

Enabled recursive calculation of genus g Gromov-Witten invariants.

Abstract

We prove the Feynman rule conjectured by Bershadsky-Cecotti-Ooguri-Vafa arXiv:hep-th/9309140 and the anomaly equations conjectured by Yamaguchi-Yau arXiv:hep-th/0406078 for the Gromov-Witten theory of the Calabi-Yau threefolds $Z_6 \subset \mathbb{P}(1,1,1,1,2)$, $Z_8 \subset \mathbb{P}(1,1,1,1,4)$, and $Z_{10} \subset \mathbb{P}(1,1,1,2,5)$. These determine the generating series $F_g$ of genus $g$ Gromov-Witten invariants recursively from the lower-genus $F_{h<g}$ up to $3g-3$ unknown parameters.