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Developments in the Mathematics of the A-Model: Constructing Calabi-Yau Structures and Stability Conditions on Target Categories

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Developments in the Mathematics of the A-Model: Constructing Calabi-Yau Structures and Stability Conditions on Target Categories Developments in the mathematics of the A-model: constructing Calabi-Yau structures and stability conditions on target categories by Alex Atsushi Takeda A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Vivek Shende, Co-chair Professor Mina Aganagic, Co-chair Professor Ori Ganor Professor David Nadler Summer 2019 Developments in the mathematics of the A-model: constructing Calabi-Yau structures and stability conditions on target categories Copyright 2019 by Alex Atsushi Takeda 1 Abstract Developments in the mathematics of the A-model: constructing Calabi-Yau structures and stability conditions on target categories by Alex Atsushi Takeda Doctor of Philosophy in Physics University of California, Berkeley Professor Vivek Shende, Co-chair Professor Mina Aganagic, Co-chair This dissertation is an exposition of the work conducted by the author in the later years of graduate school, when two main projects were completed. Both projects concern the application of sheaf-theoretic techniques to construct geometric structures on categories ap- pearing in the mathematical description of the A-model, which are of interest to symplectic geometers and mathematicians working in mirror symmetry. This dissertation starts with an introduction to the aspects of the physics of mirror symmetry that will be needed for the exposition of the techniques and results of these two projects. The first project concerns the construction of Calabi-Yau structures on topological Fukaya categories, using the microlocal model of Nadler and others for these categories. The second project introduces and studies a similar local-to-global technique, this time used to construct Bridgeland stability conditions on Fukaya categories of marked surfaces, extending some results of Haiden, Katzarkov and Kontsevich on the relation between stability of Fukaya categories and geometry of holomor- phic differentials. i To all the friends, family members and lovers that supported me along this journey. ii Contents Contents ii List of Figures iv 1 Overview of mirror symmetry 1 1.1 Introduction . 1 1.2 Mirror symmetry for Calabi-Yau manifolds . 2 1.3 The homological mirror symmetry conjecture: D-branes, Fukaya categories and coherent sheaves . 7 2 TCFTs, Calabi-Yau structures, microlocal models for Fukaya categories 16 2.1 The TQFT/TCFT perspective . 16 2.2 A-infinity categories and Calabi-Yau structures . 19 2.3 Liouville manifolds and the wrapped Fukaya category . 20 2.4 Topological Fukaya categories . 24 2.5 Arboreal singularities . 28 3 Calabi-Yau structures on topological Fukaya categories 33 3.1 Summary of results . 34 3.2 DG categories . 37 3.3 Calabi-Yau structures and local orientations . 40 3.4 A local orientation on an arboreal singularity . 53 3.5 Global orientations . 64 3.6 Examples . 68 4 Stability and Fukaya categories 78 4.1 Bridgeland stability conditions and the Thomas-Yau conjecture . 79 4.2 Fukaya categories of marked surfaces . 81 4.3 Lemmas about stability conditions . 85 4.4 The HKK construction . 96 5 Relative stability conditions 99 5.1 Relative stability conditions on Fukaya categories of surfaces . 101 iii 5.2 Cutting and gluing relative stability conditions . 109 5.3 Calculations . 122 5.4 Conclusions . 131 Bibliography 133 iv List of Figures 2.1 Wrapped Fukaya category of the cylinder T ∗S1 . 23 2.2 Cylinder with skeleta S1 and Λ . 26 2.3 Gluing A3 from discs . 30 2.4 Arboreal singularity A2 ................................ 31 2.5 Arboreal singularity A3 ................................ 32 3.1 Coarser stratification of the arboreal singularity A2 . 59 3.2 The subsets A3(•; •) ................................. 61 3.3 A pair of discs in !A3 ................................. 62 3.4 Front projection for d =2 .............................. 67 3.5 The comb X ...................................... 69 3.6 The cylinder with concentric circles . 71 3.7 Restriction to boundaries . 71 3.8 Legendrian links for irregular singularities . 74 3.9 Irregular type 1=z3=2 ................................. 75 4.1 Schematic depiction of the data defining a graded curve . 82 4.2 A marked surface with a system of arcs in red . 83 4.3 A truly immersed Lagrangian L ........................... 86 4.4 The annulus mirror to Db(Coh(P1)) . 87 4.5 A chain of stable intervals with N=4. 88 4.6 Prohibited polygons . 89 ∗ 1 4.7 The annulus ∆(1;1) mirror to P ........................... 90 4.8 Four possible cases for extensions within a self-overlapping chain. 92 4.9 One example where AN extends XN−1 ....................... 94 5.1 A system of arcs and its closure . 102 5.2 The two `modified’ surfaces . 103 ∼ ~ ∼ ~0 ∼ 5.3 The surfaces S = ∆2 and S = S = ∆3 . 106 5.4 A lozenge of stable objects . 115 5.5 Unobstructed vs obstructed lozenge . 115 5.6 A decomposition of the surface Σ . 122 v 5.7 Prohibited polygon . 128 5.8 The sets Xp;q ..................................... 130 5.9 Stable circle L and stable interval I . 131 vi Acknowledgments I would like to especially thank my advisor Vivek Shende for being so willing in sharing his time, guidance and original ideas. I would also like to thank other members of UC Berkeley faculty such as my co-advisor Mina Aganagic, David Nadler, Ori Ganor and others for being so formative in my education here. I would like to thank as well Tom Bridgeland, Fabian Haiden, Chris Brav and Tobias Dyckerhoff for being so patient and clear in explaining to me some of their work, and Ky¯ojiSait¯ofor very enjoyable lessons. Last but not least, I would like to thank my friends Dori Bejleri, Ilyas Bayramov, Benjamin Gammage, Tatsuki Kuwagaki, Michael Levy, Ryan Thorngren and Gjergji Zaimi, without whom mathematics would be no fun. 1 Chapter 1 Overview of mirror symmetry 1.1 Introduction For the past 30 years, the study of mirror symmetry has been one a very active and fruitful front of the interaction between physics and mathematics. One can trace the lineage of mirror symmetry to the very fruitful study of QFT- and string-theoretic dualities of the 80s and 90s; the physical argument for mirror symmetry can be phrased as an application of T-duality in string theory to type II string theories compactified on Calabi-Yau threefolds. Mirror symmetry provides a striking example of the power of physical arguments in mathematics, since it suggests a deep relation between different types of geometry that would not apparent for mathematicians working on either one of these fields. Perhaps the most classical and prominent example of this phenomenon is the pioneering work of Candelas, de la Ossa, Green and Parkes [26], where the authors conjecture a precise formula for the number of rational curves of a fixed degree in the quintic threefold X ⊂ P4, given in terms of periods of a different Calabi-Yau manifold (the mirror quintic). On the mathematical side, this formula was proven by the work of Lian, Liu and Yau [76]. For mathematicians, one of the important developments in the study of mirror symmetry is Kontsevich's homological mirror symmetry conjecture [71]. This conjecture relates the fields of symplectic geometry and complex geometry; roughly it can be interpreted to say that the symplectic geometry of a Calabi-Yau manifold is very closely related to the complex geometry of its mirror manifold. One can assign certain categorical invariants to these manifolds, which should be non- trivially related for mirror pairs of manifolds. On the complex side, the category to be considered is the derived category of coherent sheaves, which is a very familiar object in algebraic geometry. On the symplectic side, the category to be considered is the Fukaya category, which first appeared in the work of Fukaya on Morse theory for Lagrangians. To a given symplectic manifold X, this construction assigns a category whose objects are given by Lagrangians in X, and whose morphisms come from intersection points; for the structure of this category to be made precise, one needs to qualify these statements properly. CHAPTER 1. OVERVIEW OF MIRROR SYMMETRY 2 This dissertation will focus on some recent mathematical developments on geometric structures on certain Fukaya categories. The results presented here are from my own work [100, 107] completed during my graduate studies in collaboration with my academic advisor. These papers address the construction of two kinds of geometric structures on Fukaya cat- egories, namely Calabi-Yau structures and Bridgeland stability conditions. In this chapter we will review some of the background material on these topics, with an emphasis on the physics behind them; most of the material in here is based on the now-classical references [57, 8] There are many different versions of Fukaya categories developed over the past decades, and to provide a comprehensive review of the field would be beyond the scope of this thesis; the interested reader can consult some more comprehensive monographs and reviews such as [46, 10]. Here we will limit ourselves to a schematic overview of some specific topics that will be important for the exposition of original research. 1.2 Mirror symmetry for Calabi-Yau manifolds Generalities about SCFTs Here we will follow the exposition in Kapustin and Orlov's 2003 lectures on mirror symmetry [63] and their work on this subject [65]. Mirror symmetry can be seen algebraically as a symmetry of the N = (2; 2) supercon- formal algebra, without necessarily a relation to geometry. The full set of generators and relations for this superconformal algebra is quite complicated, and can be found eg. in [65] (where it is referred to as the N = 2 super-Virasoro algebra). This algebra contains two ¯ ¯ copies Lm; Lm of the Virasoro generators, together with another set of even generators Jn; Jn, ± ¯± and as odd generators, two copies of the set N = 2 supersymmetry generators Qn ; Qn .
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