Calabi-Yau Geometry and Higher Genus Mirror Symmetry A dissertation presented by Si Li to The Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Mathematics Harvard University Cambridge, Massachusetts May 2011 © 2011 { Si Li All rights reserved. iii Dissertation Advisor: Professor Shing-Tung Yau Si Li Calabi-Yau Geometry and Higher Genus Mirror Symmetry Abstract We study closed string mirror symmetry on compact Calabi-Yau manifolds at higher genus. String theory predicts the existence of two sets of geometric invariants, from the A-model and the B-model on Calabi-Yau manifolds, each indexed by a non-negative inte- ger called genus. The A-model has been mathematically established at all genera by the Gromov-Witten theory, but little is known in mathematics for B-model beyond genus zero. We develop a mathematical theory of higher genus B-model from perturbative quantiza- tion techniques of gauge theory. The relevant gauge theory is the Kodaira-Spencer gauge theory, which is originally discovered by Bershadsky-Cecotti-Ooguri-Vafa as the closed string field theory of B-twisted topological string on Calabi-Yau three-folds. We generalize this to Calabi-Yau manifolds of arbitrary dimensions including also gravitational descen- dants, which we call BCOV theory. We give the geometric description of the perturbative quantization of BCOV theory in terms of deformation-obstruction theory. The vanishing of the relevant obstruction classes will enable us to construct the higher genus B-model. We carry out this construction on the elliptic curve and establish the corresponding higher genus B-model. Furthermore, we show that the B-model invariants constructed from BCOV theory on the elliptic curve can be identified with descendant Gromov-Witten invariants on the mirror elliptic curve. This gives the first compact Calabi-Yau example where mirror symmetry can be established at all genera. iv Contents Acknowledgements vi 1. Introduction 1 1.1. The A-model and Gromov-Witten theory 3 1.2. The B-model and BCOV theory 5 1.3. Main results 8 2. Classical Geometry of Calabi-Yau Moduli Space 10 2.1. Polyvector fields 10 2.2. Deformation theory of Calabi-Yau manifolds 16 2.3. Special geometry and tt∗ Equations 20 2.4. Yukawa coupling and prepotential 26 2.5. Kodaira-Spencer gauge theory 30 3. Perturbative Quantization of Gauge Theory 33 3.1. Feynman Diagrams 33 3.2. Batalin-Vilkovisky geometry 36 3.3. Effective field theory and renormalization 47 3.4. Gauge theory and quantum master equation 54 3.5. Feynman graph integral for holomorphic theory on C 65 4. Quantum Geometry of Calabi-Yau Manifolds 72 4.1. Classical BCOV theory 72 4.2. Quantization and higher genus B-model 77 5. Quantization of BCOV theory on elliptic curves 84 5.1. Deformation-obstruction complex 84 5.2. Uniqueness of the quantization 89 5.3. Existence of the quantization 94 5.4. Holomorphicity 99 5.5. Local renormalization group flow 113 5.6. Virasoro equations 116 6. Higher genus mirror symmetry on elliptic curves 121 6.1. BCOV propagator on elliptic curves 123 v 6.2. Boson-Fermion correspondence 128 6.3. Gromov-Witten invariants on elliptic curves 135 6.4.τ ¯ ! 1 limit 143 6.5. Proof of mirror symmetry 157 Appendix A. L1 algebra 159 A.1. L1 structure 159 A.2. Maurer-Cartan equation 163 A.3. Homotopy 165 A.4. Deformation functor 166 Appendix B. D-modules and Jets 167 B.1. D-module 167 B.2. Jet bundles 169 B.3. Local functionals 172 References 173 vi Acknowledgements Foremost, I would like to express my deep and sincere gratitude to my thesis advisor Professor Shing-Tung Yau. I have benefited incredibly in the past five years from his broad and thoughtful view in mathematics and physics, his generosity in sharing his ideas, and his invaluable support and encouragement in my research projects. I am deeply grateful to Professor Kevin Costello, who had kindly explained to me his work on renormalization theory. Part of the thesis grew out of a joint project with him and was carried out while visiting the math department of Northwestern University. I thank for their hospitality. I wish to pay special thanks to Professor Cumrun Vafa for many stimulating discussions and his wonderful lectures in string theory, which has always been an important origin of my curiosity throughout my research. I also want to pay sincere thanks to Professor Kefeng Liu and Professor Sen Hu for their help and support in the past years both in my career and in my life. There are many people who have helped me financially for my graduate study. I want to express my deep thanks to Mr. Sze-Lim Li, who had supported my first year life in the United States. Without him I wouldn't even have this wonderful chance to come to Harvard. I also want to thank Mr. Raphael Salem. The fellowship at Harvard under his name has helped me survive in the academic year 2008-2009. I sincerely appreciate the useful conversations at Harvard with Chien-Hao Liu, Valentino Tosatti, Chen-Yu Chi, Yu-Shen Lin, at Brandeis with Bong H. Lian, at Northwestern with Yuan Shen, and at various workshops and online MSN with Huai-Liang Chang. I would also like to thank the participants of Yau's student seminar, who have helped to realize my ignorance in diverse branches of mathematics. This includes in addition: Wenxuan Lu, Ming-Tao Chuan, PoNing Chen, Ruifang Song, Aleksandar Suboti´c,Yi Li, Jie Zhou, Yih Sung, Li-Sheng Tseng, Baosen Wu and Peng Zhang. Special thank goes to Chen-Yu Chi, who has always been available to discuss math anytime during the 24 hours a day, 7 days a week. Moreover, without his generous and constant provision of chocolates and desserts for the alcove around our offices, I wouldn't gain so much weight in the past few years. vii Among my graduate friends at Harvard math department, I would like to thank specially to Le Anh Vinh and Henry Shin. Without them, I would not be so poor after we invented our strange bill system to pay so many enjoyable dinners. In addition, I appreciate the work of many secretaries at Harvard math department, in- cluding Maureen Armstrong, Susan Gilbert, Robbie Miller, Nancy Miller and Irene Minder. I'm also grateful to Robin Gottlieb, Janet Chen and Jameel Al-Aidroos for their helps on various aspects of my teaching at Harvard. There are many other friends at Harvard outside math department contributing to my meaningful life here. My special gratitude is due to my roommates, Nan Sun and Qu Zhang, together with whom we have maintained a three-year \gelivable" life style, the \Mellen System" named by our friends. In addition, I thank Nan Sun for his constant bothering of interesting math problems in engineering that I've never been able to solve, which has helped me to understand the power and the limitation of math tools in real world problems. I thank Qu Zhang for our long discussion of a project on modelling the high proportion of fixed nucleotide substitutions in orangutan, which has never really started, and from which I have learned so many interesting aspects of population genetics and coalescent theory. Lastly and most importantly, I wish to thank my family. I'm deeply indebted to my parents for their selfless love and enormous care since I was born. Without their support, I could not have gone this far. I want to tell them that I miss you so much. I owe my loving thanks to my wife Xuejing. She has sacrificed so much due to my personal choice on career, my working style on research, and our long-distance relationship in the past seven years. I thank wholeheartedly for her tremendous love and exclusive understanding, without which this work would not be possible. I would like to take this opportunity to say that I love you. 1 1. Introduction Mirror symmetry originated from string theory as a duality between superconformal field theories (SCFT). The natural geometric background involved is the Calabi-Yau manifold, and SCFTs can be realized from twisting the so-called σ-models on Calabi-Yau manifolds in two different ways [Wit88, Wit92]: the A-model and the B-model. The physics statement of mirror symmetry says that the A-model on a Calabi-Yau manifold X is equivalent to the B-model on a different Calabi-Yau manifold X˘, which is called the mirror. The mathematical interests on mirror symmetry started from the work [CdlOGP91], where a remarkable mathematical prediction was extracted from the physics statement of mirror symmetry: the counting of rational curves on the Quintic 3-fold is equivalent to the period integrals on the mirror Quintic 3-fold. Motivated by this example, people have conjectured that such phenomenon holds for general mirror Calabi-Yau manifolds. The counting of rational curves is refered to as the genus 0 A-model, which has now been mathematically established [RT94, LT98] as Gromov-Witten theory. The period integral is related to the variation of Hodge structure, and this is refered to as the genus 0 B- model. Mirror conjecture at genus 0 has been proved by Givental [Giv98] and Lian-Liu-Yau [LLY97] for a large class of Calabi-Yau manifolds inside toric varieties. In the last twenty years, mirror symmetry has lead to numerous deep connections between various branches of mathematics and has been making a huge influence on both mathematics and physics. The fundamental mathematical question is to understand mirror symmetry at higher genus.