Stable quotients and the holomorphic anomaly equation

TL;DR

This paper establishes a geometric proof of the holomorphic anomaly equation for stable quotient invariants related to local CP2, extending to twisted theories and providing insights into the formal and true quintic theories.

Contribution

It provides a direct geometric proof of the holomorphic anomaly equation for stable quotient invariants and introduces new equations for twisted theories on projective spaces.

Findings

Proves the holomorphic anomaly equation in the form predicted by B-model physics.

Derives new holomorphic anomaly equations for twisted theories on projective spaces.

Shows the formal quintic satisfies the same anomaly equations as the true quintic.

Abstract

We study the fundamental relationship between stable quotient invariants and the B-model for local CP2 in all genera. Our main result is a direct geometric proof of the holomorphic anomaly equation in the precise form predicted by B-model physics. The method yields new holomorphic anomaly equations for an infinite class of twisted theories on projective spaces. An example of such a twisted theory is the formal quintic defined by a hyperplane section of CP4 in all genera via the Euler class of a complex. The formal quintic theory is found to satisfy the holomorphic anomaly equations conjectured for the true quintic theory. Therefore, the formal quintic theory and the true quintic theory should be related by transformations which respect the holomorphic anomaly equations.