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

We prove the Bershadsky-Cecotti-Ooguri-Vafa's conjecture for all genus Gromov-Witten potentials of the quintic 3-folds, by identifying the Feynman graph sum with the NMSP stable graph sum via an R-matrix action. The Yamaguchi-Yau functional equations (HAE) are direct consequences of the BCOV Feynman sum rule. Contents 0. Introduction 1 1. The Main Theorems 5 2. Cohomological field theory and R-matrix action 9 3. Expressing NMSP-[0, 1] theory via CohFTs 13 4.

We prove an explicit formula for the genus-one Fan-Jarvis-Ruan-Witten invariants associated to the quintic threefold, verifying the genus-one mirror conjecture of Huang, Klemm, and Quackenbush. The proof involves two steps. The first step uses localization on auxiliary moduli spaces to compare the usual Fan-Jarvis-Ruan-Witten invariants with a semisimple theory of twisted invariants. The second step uses the genus-one formula for semisimple cohomological field theories to compute the twisted invariants explicitly.

We prove the genus-one restriction of the all-genus Landau-Ginzburg/Calabi-Yau conjecture of Chiodo and Ruan, stated in terms of the geometric quantization of an explicit symplectomorphism determined by genus-zero invariants. This gives the first evidence supporting the higher-genus Landau-Ginzburg/Calabi-Yau correspondence for the quintic threefold, and exhibits the first instance of the "genus zero controls higher genus" principle, in the sense of Givental's quantization formalism, for non-semisimple cohomological field theories.1 genus-zero correspondence through an explicit quantization procedure. While such a principle has been studied extensively and proved in many cases for semisimple cohomological field theories, for example [Tel12, Giv01b, BCR13, Zon16, CI14, HLSW15, IMRS16], this is the first significant evidence for such a principle in the non-semisimple case.1.1. Plan of the Paper. We begin in Section 2 by recalling the basic definitions in Gromov-Witten and Fan-Jarvis-Ruan-Witten theory. We recall some previously known results, including the genuszero mirror theorems, the genus-zero Landau-Ginzburg/Calabi-Yau correspondence, and the genusone mirror theorems. In Section 3, we discuss the Birkhoff factorization of the symplectomorphism U and recall Givental's quantization formulas in order to make Theorem 3.3 precise. We also apply the string and dilaton equations to reduce the main theorem to the one parameter 'small statespace'. In Section 4, we provide a proof of the genus-zero restriction of the quantization conjecture, mostly in order to set up notation for the genus-one correspondence. The proof of the genus-one correspondence occupies Sections 5, 6, and 7, where we carefully analyze the vertex-and loop-type graphs that appear in the quantization formula. 7

We prove a higher genus version of the genus $0$ local-relative correspondence of van Garrel-Graber-Ruddat: for $(X,D)$ a pair with X a smooth projective variety and D a nef smooth divisor, maximal contact Gromov-Witten theory of $(X,D)$ with $\lambda _g$ -insertion is related to Gromov-Witten theory of the total space of ${\mathcal O}_X(-D)$ and local Gromov-Witten theory of D. Specializing to $(X,D)=(S,E)$ for S a del Pezzo surface or a rational elliptic surface and E a smooth anticanonical divisor, we show that maximal contact Gromov-Witten theory of $(S,E)$ is determined by the Gromov-Witten theory of the Calabi-Yau 3-fold ${\mathcal O}_S(-E)$ and the stationary Gromov-Witten theory of the elliptic curve E. Specializing further to $S={\mathbb P}^2$ , we prove that higher genus generating series of maximal contact Gromov-Witten invariants of $({\mathbb P}^2,E)$ are quasimodular and satisfy a holomorphic anomaly equation. The proof combines the quasimodularity results and the holomorphic anomaly equations previously known for local ${\mathbb P}^2$ and the elliptic curve. Furthermore, using the connection between maximal contact Gromov-Witten invariants of $({\mathbb P}^2,E)$ and Betti numbers of moduli spaces of semistable one-dimensional sheaves on ${\mathbb P}^2$ , we obtain a proof of the quasimodularity and holomorphic anomaly equation predicted in the physics literature for the refined topological string free energy of local ${\mathbb P}^2$ in the Nekrasov-Shatashvili limit.