Structure of Higher Genus Gromov-Witten Invariants of Quintic 3-folds

TL;DR

This paper proves key conjectures about the structure of higher genus Gromov-Witten invariants of quintic 3-folds, revealing universal properties and deriving holomorphic anomaly equations through A-model techniques.

Contribution

It establishes the finite generation, orbifold regularity, and holomorphic anomaly equations for the Gromov-Witten theory of quintic 3-folds using a novel combinatorial and geometric approach.

Findings

Proof of Yamaguchi--Yau's finite generation conjecture

Derivation of holomorphic anomaly equations

Verification of orbifold regularity

Abstract

There is a set of remarkable physical predictions for the structure of BCOV's higher genus B-model of mirror quintic 3-folds which can be viewed as conjectures for the Gromov-Witten theory of quintic 3-folds. They are (i) Yamaguchi--Yau's finite generation, (ii) the holomorphic anomaly equation, (iii) the orbifold regularity and (iv) the conifold gap condition. Moreover, these properties are expected to be universal properties for all the Calabi-Yau 3-folds. This article is devoted to proving first three conjectures. The main geometric input to our proof is a log GLSM moduli space and the comparison formula between its reduced virtual cycle (reproducing Gromov--Witten invariants of quintic 3-folds) and its nonreduced virtual cycle. Our starting point is a Combinatorial Structural Theorem expressing the Gromov-Witten cohomological field theory as an action of a generalized $R$-matrix in the sense of Givental. An $R$-matrix computation implies a graded finite generation property. Our graded finite generation implies Yamaguchi-Yau's (nongraded) finite generation, as well as the orbifold regularity. By differentiating the Combinatorial Structural Theorem carefully, we derive the holomorphic anomaly equations. Our technique is purely A-model theoretic and does not assume any knowledge of B-model. Finally, above structural theorems hold for a family of theories (the extended quintic family) including the theory of quintic as a special case.