Holomorphic anomaly equations for the formal quintic

TL;DR

This paper constructs a formal Gromov-Witten theory for the quintic 3-fold using localization, providing a geometric proof of holomorphic anomaly equations that align with predictions from B-model physics, indicating a deep relationship with the true quintic theory.

Contribution

It offers the first geometric proof of holomorphic anomaly equations for the formal quintic, matching the form predicted by physics, and suggests a connection to the true quintic theory through transformations.

Findings

Proves holomorphic anomaly equations for the formal quintic

Shows the equations match B-model physics predictions

Indicates a relationship between formal and true quintic theories

Abstract

We define a formal Gromov-Witten theory of the quintic 3-fold via localization on CP4. Our main result is a direct geometric proof of holomorphic anomaly equations for the formal quintic in precisely the same form as predicted by B-model physics for the true Gromov-Witten theory of the quintic 3-fold. The results suggest that the formal quintic and the true quintic theories should be related by transformations which respect the holomorphic anomaly equations. Such a relationship has been recently found by Q. Chen, S. Guo, F. Janda, and Y. Ruan via the geometry of new moduli spaces.