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

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 [7]. 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. S. GUO, F. JANDA, AND Y. RUAN 4.2. The extended quintic family as a generalized R-matrix action 20 4.3. Proof of Theorem 4.4 21 5. Graded finite generation and orbifold regularity 25 5.1. CohFT of the λ-twisted invariants 26 5.2. Differential equations for S-matrix and R-matrix 27 5.3. Finite generation for S-matrix and R-matrix 29 5.4. Proof of Theorem 5.1 30 5.5. Yamaguchi-Yau's prediction 33 5.6. Grading and orbifold regularity 33 6. Holomorphic anomaly equations (HAE) 34 6.1. Derivations acting on the R-matrix 34 6.2. Explicit formulae of PDEs for the R-matrix 36 6.3. Proof of the holomorphic anomaly equations 37 6.4. Examples of HAEs 38 7. A technical result of the formal quintic theory 39 7.1. Computing the asymptotic expansion by using stationary phase method 40 7.2. Strengthening using the Picard-Fuchs equations 43 References 45

Quasi-modularity and holomorphic anomaly equation for the twisted Gromov-Witten theory: $\mathcal{O}(3)$ over $\mathbb{P}^2$

“…Recently, it is proved that these two properties are satisfied for Quintic 3-fold (Ref. [9]). Instead, the twisted Gromov-Witten theory is much easier and also expected to satisfy the two properties in the above predictions.…”

“…[10], [3] and [4]). In this paper, inspired by the calculations in [9], we discuss one simple example O(3) over P 2 and prove the quasimodularity property and holomorphic anomaly equation.…”

Finite generation and holomorphic anomaly equation for equivariant Gromov-Witten invariants of $K_{\mathbb{P}^1\times\mathbb{P}^1}$

“…In the past years, many works has been done about the finite generation and holomorphic anomaly equation for (non-)compact Calabi-Yau 3-fold and also twisted theory of Calabi-Yau type (c.f. [3], [5], [8], [9], [10], [11], [15]). Most of the examples studied are about the models of one Kähler parameter.…”

The logarithmic gauged linear sigma model

“…This leads to a very powerful technique for computing higher genus GW/FJRW-invariants of complete intersections in GIT quotients. Applications include computing higher genus invariants of quintic 3-folds [34,35] 1 , and the cycle of holomorphic differentials [43,Conjecture A.1] by establishing a localization formula of r-spin cycles conjectured by the second author [23]. This conjectural localization formula was the original motivation of this project.…”