BCOV's Feynman rule of quintic $3$-folds

Chang, Huai-Liang; Guo, Shuai; Li, Jun

doi:10.48550/arxiv.1810.00394

Cited by 11 publications

(22 citation statements)

References 44 publications

(73 reference statements)

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“…There is a different approach to the higher genus theory of quintic 3-folds by Chang-Guo-Li-Li-Liu. Recently, they posted a series of articles [5,4,3]. Among other things, they proved the original Yamaguchi-Yau (non-graded) prediction independently.…”

Section: Introductionmentioning

confidence: 94%

“…The appearance of extra generators is similar to the simpler case of the Gromov-Witten theory of an elliptic curve, where F g is a quasi-modular form of SL 2 (Z). On the other hand, the corresponding twisted theory of O(3) on P 2 is a quasi-modular form for Γ 0 (3). The ring of quasimodular forms of Γ 0 (3) has more generators than that of SL 2 (Z)!…”

Section: Introductionmentioning

confidence: 99%

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

Guo¹,

Janda²,

Ruan³

2018

Preprint

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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

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Section: Introductionmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

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

Guo¹,

Janda²,

Ruan³

2018

Preprint

Self Cite

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“…For other approaches one may consult the papers [3], [4], [5], [6], [7], [18], [20] and the reference there.…”

Section: Introductionmentioning

confidence: 99%

Splitting of the virtual class for genus one stable quasimaps

Lee¹,

Li²

2019

Preprint

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“…In [NMSP2], we will prove Yamaguchi-Yau's finite generation conjecture for F g , which implies the convergence of F g . In [NMSP3], we will prove the Bershadsky-Cecotti-Ooguri-Vafa (BCOV) Feynman graph sum formula. One of its consequences is the Yamaguchi-Yau Functional equation [YY,Sect.…”

Section: Introductionmentioning

confidence: 99%

The theory of N-Mixed-Spin-P fields

Chang,

Guo,

et al. 2018

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BCOV's Feynman rule of quintic $3$-folds

Cited by 11 publications

References 44 publications

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

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

Splitting of the virtual class for genus one stable quasimaps

The theory of N-Mixed-Spin-P fields

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