Lagrangian Floer Theory and Its Deformations: An Introduction to Filtered Fukaya Category (KIAS Springer Series in Mathematics, 2)

A-infinity structure was introduced by Stasheff in the 1960s in his homotopy characterization of based loop space, which was the culmination of earlier works of Sugawara's homotopy characterization of H-spaces and loop spaces. At the beginning of the 1990s, a similar structure was introduced by Fukaya in his categorification of Floer homology in symplectic topology. This structure plays a fundamental role in the celebrated homological mirror symmetry proposal by Kontsevich and in more recent developments of symplectic topology. A detailed construction of A-infinity algebra structure attached to a closed Lagrangian submanifold is given in Fukaya, Oh, Ohta, and Ono's two-volume monograph Lagrangian Intersection Floer Theory (AMS-IP series 46 I & II), using the theory of Kuranishi structures―a theory that has been regarded as being not easily accessible to researchers in general. The present lecture note is provided by one of the main contributors to the Lagrangian Floer theory and is intended to provide a quick, reader-friendly explanation of the geometric part of the construction. Discussion of the Kuranishi structures is minimized, with more focus on the calculations and applications emphasizing the relevant homological algebra in the filtered context. The book starts with a quick explanation of Stasheff polytopes and their two realizations―one by the rooted metric ribbon trees and the other by the genus-zero moduli space of open Riemann surfaces―and an explanation of the A-infinity structure on the motivating example of the based loop space. It then provides a description of the moduli space of genus-zero bordered stable maps and continues with the construction of the (curved) A-infinity structure and its canonical models. Included in the explanation are the (Landau–Ginzburg) potential functions associated with compact Lagrangian submanifolds constructed by Fukaya, Oh, Ohta, and Ono. The book explains calculations of potential functions for toric fibers in detail and reviews several explicit calculations in the literature of potential functions with bulk as well as their applications to problems in symplectic topology via the critical point theory thereof. In the Appendix, the book also provides rapid summaries of various background materials such as the stable map topology, Kuranishi structures, and orbifold Lagrangian Floer theory. Preface Contents Introduction Acknowledgements Conventions Notations List of Figures 1 Based Loop Space and upper A Subscript normal infinityAinfty Space 1.1 Based Loop Space and Stasheff Polytope 1.2 Rooted Ribbon Trees 1.3 Stasheff Polytopes and upper A Subscript nAn Space 1.4 Two Realizations of Stasheff Polytopes upper K Subscript nKn 1.4.1 Metric Ribbon Trees 1.4.2 Moduli Space of Bordered Stable Curves 1.4.3 Configuration Space of upper S Superscript 1S1 and ModifyingAbove script upper M With quotation dash Subscript k plus 1overlinemathcalMk+1 1.4.4 Duality Between the Cell Structures of ModifyingAbove script upper M With quotation dash Subscript k plus 1overlinemathcalMk+1 and ModifyingAbove upper G r With quotation dash Subscript k plus 1overlineGrk+1 1.5 The Based Loop Space Is an upper A Subscript normal infinityAinfty Space 2 upper A Subscript normal infinityAinfty Algebras and Modules: Unfiltered Case 2.1 Definition of upper A Subscript normal infinityAinfty Algebra 2.1.1 Definition of upper A Subscript normal infinityAinfty Structure Maps German m Subscript kmathfrakmk 2.1.2 Coalgebra and Bar Complex 2.2 Massey Product and upper A Subscript normal infinityAinfty Algebra 2.2.1 Higher-Order Linking of Borromean Ring 2.2.2 upper A Subscript normal infinityAinfty-algebra Interpretation of Massey Product 2.3 Hochschild Complex of upper A Subscript normal infinityAinfty Algebras 2.4 upper A Subscript normal infinityAinfty Homomorphisms 2.5 Right upper A Subscript normal infinityAinfty Modules 2.6 upper A Subscript upper KAK Modules and upper A Subscript upper KAK Homomorphisms 2.7 Hochschild Cohomology and Whitehead Theorem 2.7.1 Hochschild Cohomology of upper A Subscript normal infinityAinfty Homomorphisms 2.7.2 Hochschild Cohomology and upper A Subscript normal infinityAinfty Whitehead Theorem 2.8 Obstruction Theory and upper A Subscript normal infinityAinfty Whitehead Theorem 2.8.1 upper A Subscript upper K plus 1AK+1-obstruction Class upper O Subscript upper K plus 1 Baseline left parenthesis psi right parenthesisOK+1(ψ) 2.8.2 Wrap-Up of the Proof of Whitehead Theorem 2.9 upper A Subscript normal infinityAinfty Bimodules 3 Obstruction-Deformation Theory of Filtered upper A Subscript normal infinityAinfty Bimodules 3.1 Gapped Filtered upper A Subscript normal infinityAinfty Algebras and Homomorphisms 3.1.1 Universal Novikov Ring 3.1.2 Energy Filtration and Floer-Novikov Monoids 3.1.3 Filtered upper A Subscript normal infinityAinfty Homomorphism 3.1.4 Filtered upper A Subscript normal infinityAinfty Bimodules and Homomorphisms 3.2 Homological Perturbation Theory and Canonical Model 3.2.1 Unfiltered Cases: Statement 3.2.2 Unfiltered Cases: Proof 3.2.3 Canonical Model: Filtered Cases 3.3 Bounding Cochains and Deformations of upper A Subscript normal infinityAinfty Algebra 3.3.1 The (Strict) Bounding Cochains 3.3.2 The Weak Bounding Cochains, Gauge Equivalence and Potential Function 3.4 Boundary Deformations of upper A Subscript normal infinityAinfty Bimodules 3.4.1 The left parenthesis upper G 0 comma upper G 1 right parenthesis(G0,G1)-sets of Monoid Pair left parenthesis upper G 0 comma upper G 1 right parenthesis(G0,G1) 3.4.2 Deformations of Filtered upper A Subscript normal infinityAinfty Bimodules 3.5 Deformations of Filtered upper A Subscript normal infinityAinfty Bimodule Homomorphisms 3.5.1 The Case of upper A Subscript normal infinityAinfty Algebra Homomorphisms 3.5.2 The Case of Filtered Bimodule Homomorphisms 4 Symplectic Geometry and Hamiltonian Dynamics 4.1 Definition of Symplectic Manifolds 4.2 Symplectic Linear Algebra 4.2.1 Lagrangian Grassmanian 4.2.2 Arnold Stratification 4.3 Lagrangian Submanifolds 4.3.1 Basic Definitions 4.3.2 Calculus of Lagrangian Submanifolds 4.4 Hamiltonian Diffeomorphisms and Hamiltonian Calculus 4.4.1 Hofer's Geometry of Ham left parenthesis upper M comma omega right parenthesisHam(M,ω) 4.4.2 Family of Hamiltonian Diffeomorphisms 4.5 Hamiltonian Displacement of Lagrangian Submanifolds Chapter 5 Analysis of Pseudoholomorphic Curves and Bordered Stable Maps 5.1 Almost Complex Manifolds and Hermitian Metric upper A Subscript normal infinity . Definition 5.1.1 upper S squared ]) Let is called an almost complex structure on a manifold . since the bracket is skew symmetric. Note that . . upper S squared . upper A Subscript normal infinity . . upper A Subscript normal infinity . as generalization of classical Cauchy–Riemann equation to a system thereof on an almost complex manifold. upper S squared Observation When upper A Subscript normal infinity Theorem 5.1.7 (Nijenhuis–Woolf [ A Riemannian metric 5.2 Pseudoholomorphic Curves and Their Boundary Value Problem . be given, and consider a 2-dimensional surface is called the fundamental two-form of upper S squared left parenthesis upper L 0 comma upper L 1 right parenthesis . . . . . , the exponential map . . is a diffeomorphism. is a contractible infinite- dimensional (Frechet) manifold (in the weak . . 5.1.7 not necessarily satisfying the integrability condition. . (1) orientable? . For the study of transversality, one should study the deformation problem of the moduli space under the change of For the study of compactness, it is essential to obtain a uniform bound for the derivative upper S squared left parenthesis upper L 0 comma upper L 1 right parenthesis . Once such a bound is achieved, the Ascoli-Arzela theorem can be applied for the uniform convergence. Once this convergence is achieved, the deriva-tive convergence can be obtained by the a priori estimates based on the ellipticity of the . . (5.2.3) . . left parenthesis upper L 0 comma upper L 1 right parenthesis . . , Exercise 5.2.12 Consider the dilatation upper S squared -norm, is a borderline case because a domain upper A Subscript normal infinity . or even the symplectic area below, proves the following boundary analog to Corollary for upper S squared is a nonlinear elliptic boundary value problem for any such . . . . , when there is no non-constant . . . carries an a priori energy bound . . upper A Subscript normal infinity . . on . is a Euclidean vector bundle with inner product . . The following is proved in almost complex geometry. (See [ Here we denote by upper S squared . the trace Laplacian and . (5.2.11) . upper S squared . . . Remark 5.2.28 We have used the canonical connection associated to . . ] for complete details applied to more complicated equation of contact instantons.) upper S squared . ). If the connection 5.2.4 Local a Priori Elliptic Estimates . .) We use the (linear) ellipticity of the operator To proceed with the higher regularity estimate, we choose an isothermal coordinate upper S squared , we obtain the equation associated to the complex coordinate . (with respect to . . . left parenthesis upper L 0 comma upper L 1 right parenthesis ], [ 2. An important application of the above boundary regularity result is the following removable boundary singularity theorem [ contained in a compact subset . , upper S squared left parenthesis upper L 0 comma upper L 1 right parenthesis Theorem 5.2.37 (Removable singularity theorem) Let . . has no sub- sequence . is is uniformly bounded. of pseudo-holomorphic maps, when the domain complex structure is fixed, the genus 0 case is the most nontrivial to study. Because of this, we will focus on the case and assume :For . Let upper A Subscript normal infinity ’s arising in Theorem considered as a density converges to a density upper A Subscript normal infinity . as . upper S squared left parenthesis upper L 0 comma upper L 1 right parenthesis upper A Subscript normal infinity -holomorphic curves should involve singular curves involving sphere and disc components. . upper A Subscript normal infinity , i.e., be a convergent sequence in ], Parker-Wolfson [ ].) However, bearing intuitive Gromov’s original statements in mind is also useful and often enough for the application to various problems in symplectic topology, especially in the beginning stage of learning Gromov’s com- pactification and its symplectic topology applications. . 5.3 Bordered Stable Maps upper A Subscript normal infinity . Then for any sequence of upper S squared so that the following hold: -holomorphic maps . . . We start with the definition of real algebraic curves (aka symmetric Riemann surfaces). . Although the case upper S squared . will not be needed explicitly in this book except in our mentioning of the Cardy relation with regard to the bulk-boundary map (in the paragraph after Definition The genus . upper A Subscript normal infinity . , which is defined to be . is said to be of type upper A Subscript normal infinity , Sect. 2] for example: the . . By taking one of the two possible such orientations on . . . and left parenthesis upper L 0 comma upper L 1 right parenthesis upper A Subscript normal infinity , Sect. 2.1.2], [ . . . From now on throughout the book, for the simplicity of notations, we will just write Proof There exists an obvious fibration . . upper A Subscript normal infinity . ], which we will just denote by . . upper S squared -holomorphic prestable map. , . (b) (Stability Condition) . upper S squared left parenthesis upper L 0 comma upper L 1 right parenthesis . 5.3.3 for a map (Fig. . upper S squared . . . topology of the evaluation map . . upper S squared B.1.2 upper A Subscript normal infinity . (5.3.6) . . intersects transversally with . left parenthesis upper L 0 comma upper L 1 right parenthesis at . . . . upper A Subscript normal infinity is a symplectic vector bundle and upper S squared ], [ is a Lagrangian subbundle of For the orientability issue, it is shown in [ -skeleton. There exists a real vector bundle Definition 5.3.20 We define a . . to be a choice of left parenthesis upper L 0 comma upper L 1 right parenthesis . Applying Proposition . is defined over any smooth map of maps ) is trivial and hence finishes the proof of the theorem. . which is connected. We consider the determinant real line bundle ). This is proved in [ for from a bordered Riemann surface . 5.4 Lagrangian Submanifolds and Filtered (5.4.1) , Chap. 8] and [ upper A Subscript normal infinity 5.7.3 . 1. is the completed tensor product. We equip this module with the cohomo- logical degree. Then the grading is given as follows: 5.4.2 Definition of Fukaya Algebra as a Filtered . upper S squared left parenthesis upper L 0 comma upper L 1 right parenthesis upper A Subscript normal infinity , Chap. 21] for the details of construction. We refer readers to Appendix C for the summary of relevant definitions. We define the operations . upper S squared . upper A Subscript normal infinity . . upper A Subscript normal infinity over upper S squared and Gromov’s compactness theorem. . Here, we remark that for the right-hand side of ( .▢ as in [ 5.5 Definition and Properties of the Fukaya Algebra upper S squared . upper A Subscript normal infinity Fig. 5.4 , Proposition 8.5.1].) Let upper S squared , Proposition 8.3.3] for the consideration of the precise sign. According to their sign convention, the sign factor be the diagonal. Here, we assume that can be written as , upper S squared left parenthesis upper L 0 comma upper L 1 right parenthesis upper A Subscript normal infinity preserve the filtration. The oper- ators . 5.5.1 upper S squared (1) , Proposition 8.5.1] for the sign consideration.)▢ . . Then there exists . . Now we are ready to prove the following gap theorem for the areas of nonconstant . left parenthesis upper L 0 comma upper L 1 right parenthesis upper A Subscript normal infinity . . . . upper A Subscript normal infinity and so 5.6 Example: On . for all sufficiently large . is in this example, we collect a few facts on the relevant moduli space of disks: in [ upper A Subscript normal infinity A Maslov index upper S squared left parenthesis upper L 0 comma upper L 1 right parenthesis . . . . . upper S squared is primitive. . . of the cohomology, respectively. upper A Subscript normal infinity as a upper S squared and -chain. Then is diffeomorphic to . . As we’ve seen in the calculation of upper A Subscript normal infinity which coincides with the computation above via singular chains. upper S squared , where . Fig. 5.6 . where and upper S squared If are given in Fig. By the Lagrangian property of . upper S squared 5.7 Geometry of Lagrangian Pairs ] and many others in the literature. . We call such an . We now define a covering space of . . (5.7.1) upper A Subscript normal infinity from upper A Subscript normal infinity . upper S squared Definition 5.7.4 (Novikov covering space) We say that . push down to homomorphisms is a commutative ring with unit. , i.e., satisfies the finiteness condition (2.2.7) in its definition. Thus . upper S squared . . be the action one-form on . is the constant path with upper S squared . upper A Subscript normal infinity . . . . defined by satisfying: upper S squared left parenthesis upper L 0 comma upper L 1 right parenthesis . is somewhat resembles that of . . . upper A Subscript normal infinity is transverse to the Maslov cycle 5.7.4 left parenthesis upper L 0 comma upper L 1 right parenthesis , and with . . We denote by upper A Subscript normal infinity for any . . If we identify the tangent space upper A Subscript normal infinity for a map . upper S squared 5.7.7 upper A Subscript normal infinity 5.2.38 . . , then First, since . . satisfies ( . . . . upper A Subscript normal infinity as . . Finally the hypothesis . By the triangle inequality, we have . , the following lemma is standard. . . upper S squared for all We also denote by We denote by . ] (see Appendix . the compactified map of upper S squared left parenthesis upper L 0 comma upper L 1 right parenthesis .Byanabuse of notation, we also denote by of Lagrangian subspaces in . . Here left parenthesis upper L 0 comma upper L 1 right parenthesis upper A Subscript normal infinity 5.7.13 -parameterized family of vector spaces (5.7.15) in . Following [ . We need to find a systematic way to orient ( an orientation of the determinant bundle is , Sect. 8.1]. We glue the end . 5.7.13 with left parenthesis upper L 0 comma upper L 1 right parenthesis . . upper A Subscript normal infinity 5.7.4 left parenthesis upper L 0 comma upper L 1 right parenthesis 5.8 Filtered . -equivalence defined in Definition . This filtration obviously descends to one on and so upper S squared left parenthesis upper L 0 comma upper L 1 right parenthesis defines an equivalence relation on . . ,which is be a module. From now on, we will suppress the choice upper S squared of Lagrangian submanifolds equipped with base path (Choice from the notation of to be . upper S squared are . . . . upper A Subscript normal infinity (5.8.9) . upper A Subscript normal infinity 5.9 Floer Continuation Map Associated to Moving Lagrangian Boundary , and upper A Subscript normal infinity with 5.8.6 upper S squared . By writing its generating Hamiltonian by , Sects. 5.3.2, 5.3.3 & 5.3.5]. We try to streamline the exposition thereof and to make it easier to follow by systematically utilizing the Hamiltonian calculus laid out in Sect. , we introduce this additional notation. (2) is an elongation function defined by . upper S squared . . . bimodule homomorphism) Floer’s continuation map can be promoted to be a upper A Subscript normal infinity 5.9.5 satisfying the boundary condition upper S squared by definition (cf. §3), and to the one satisfying the moving boundary condition (2) above. Therefore both terms in the equivalence (4) satisfy the boundary conditions on 5.9.6 . . upper S squared . . be any Hamiltonian isotopies with the fixed Recall the definition of upper S squared (5.9.11) upper A Subscript normal infinity upper A Subscript normal infinity (5.9.14) upper S squared left parenthesis upper L 0 comma upper L 1 right parenthesis Now we would like to estimate the difference 5.9.8 upper A Subscript normal infinity . upper S squared left parenthesis upper L 0 comma upper L 1 right parenthesis . upper A Subscript normal infinity upper S squared left parenthesis upper L 0 comma upper L 1 right parenthesis upper A Subscript normal infinity 6 Critical Points of Potential Functions and Floer Cohomology 6.1 BRST Anomaly and Holomorphic Discs 6.2 Weak Unobstructedness of Monotone Lagrangian Submanifolds 6.3 Twisting of Floer Cohomology by Local Systems 6.3.1 Construction of Floer Cohomology of upper LL with Local Systems 6.3.2 Holonomy-Twisted Canonical Model and Potential Functions 6.4 Lagrangian Floer Theory on Toric Manifolds 6.4.1 Geometry of Compact Toric Manifolds 6.4.2 Structure Theorem of Moduli Spaces 6.4.3 Sphere Bubbles and upper T Superscript nTn-Equivariant Kuranishi Perturbations 6.4.4 One-Point Open Gromov–Witten Invariants; Toric Case 6.4.5 Forgetful Map Compatibility and upper T Superscript nTn-Equivariance 6.4.6 upper T Superscript nTn-Invariant Canonical Model on upper H left parenthesis upper L left parenthesis u right parenthesis comma normal upper Lamda 0 right parenthesisH(L(u), Λ0) 6.4.7 Evaluation of Potential Function for the Toric Case 6.4.8 Gauge Equivalence on Maurer–Cartan Moduli Space and Its Linear Sector 6.4.9 Relationship with the Landau–Ginzburg upper BB-Model 6.4.10 Extending Potential Functions Over normal upper Lamda 0 minus normal upper Lamda Subscript plusΛ0 Λ+: Algebraic versus Geometric 6.4.11 Critical Points of upper W Superscript uWu and Nondisplaceable Fiber upper L left parenthesis u right parenthesisL(u) 6.5 Calculation of Potential Functions: Examples 6.5.1 The Case of upper S squaredS2 6.5.2 The Case of double struck upper C upper P Superscript nmathbbCPn 6.5.3 The Case of upper S squared times upper S squaredS2 timesS2 6.5.4 The Case of Hirzerbruch Surface upper F 2 left parenthesis alpha right parenthesisF2(α) 7 Filtered Fukaya Category and Its Bulk Deformations 7.1 Definitions of upper A Subscript normal infinityAinfty Categories, Functors and Modules 7.1.1 Unfiltered upper A Subscript normal infinityAinfty Categories 7.1.2 Curved Filtered upper A Subscript normal infinityAinfty Categories 7.1.3 Strictification of Curved Filtered upper A Subscript normal infinityAinfty Category 7.1.4 upper A Subscript normal infinityAinfty Functors 7.1.5 upper A Subscript normal infinityAinfty Bimodules 7.2 Construction of Filtered Fukaya Category 7.2.1 Description of the Cocycle Problems to Solve 7.2.2 Off-Shell Geometry I: Abstract Index 7.2.3 Off-Shell Geometry II: Polygonal Maslov Index 7.2.4 On-Shell Geometry I: Coherent Orientations of the Moduli Spaces 7.2.5 On-Shell Geometry II: Products—Structure Maps German m Subscript k Baseline comma k greater than or equals 2mathfrakmk, k 2 7.2.6 Relationship with Graded Lagrangian Submanifolds 7.2.7 Strictification 7.3 Bulk Deformations 7.3.1 Bulk-Boundary Map (Aka Closed-Open Map) 7.3.2 Bulk Deformations of Filtered Fukaya Category 7.3.3 Bulk Deformations on Toric Manifolds 7.4 Potential Functions with Bulk and Their Applications 7.4.1 Bulk-Deformed Potential Functions 7.4.2 Continuum Family of Nondisplaceable Tori in upper S squared times upper S squaredS2 timesS2 7.4.3 Gelfand–Cetlin Systems and Their Bulk Deformations 7.4.4 Mak and Smith's Work on Lagrangian Links in upper S squaredS2 Appendix A Minimal-Area Metric Representation of Genus Zero Bordered Stable Curves A.1 Dual Cell Decomposition A.2 The Isomorphism Property of (A.0.1) A.3 Proof of the Smooth Invertibility of the Map upper Theta Appendix B Stable Map Topology B.1 For the Closed Case B.1.1 Stable Cases B.1.2 General Cases B.2 For the Bordered Case B.2.1 Topology of the Moduli Space of Bordered Riemann Surfaces B.2.2 Compactification of Bordered Stable Maps and an Exceptional Case Appendix C Kuranishi Structures, Multisections and CF-Perturbations C.1 Kuranishi Structure and Good Coordinate System C.1.1 Basics on Effective Orbifolds and Orbibundles C.1.2 Kuranishi Structure C.1.3 Good Coordinate System C.2 Multivalued Perturbation C.2.1 Multisections of an Orbifold C.2.2 Multisection on a Good Coordinate System C.3 CF-Perturbations C.3.1 CF-Perturbation on One Orbifold Chart C.3.2 CF-Perturbation on a Single Kuranishi Chart C.4 Integration Along the Fiber (Push-Out) C.4.1 Integration Along the Fiber (Push-Out) on a Single Kuranishi Chart C.4.2 Differential Forms on a Good Coordinate System and a Kuranishi Structure Appendix D Relative Spin Structures D.1 Definition of Relative Spin Structures D.2 Action of upper H squared left parenthesis upper M comma upper L semicolon double struck upper Z 2 right parenthesisH2(M,L;mathbbZ2) on upper S p i n left parenthesis upper M comma upper L right parenthesisSpin(M,L) D.3 Orientation Data for Floer's Trajectories D.3.1 Universal Bundle upper U Superscript ori Baseline left parenthesis double struck upper R Superscript 2 n Baseline right parenthesisU ori(mathbbR2n) of upper Lamda Superscript ori Baseline left parenthesis double struck upper R Superscript 2 n Baseline right parenthesis ori(mathbbR2n) D.3.2 Extending the Spin Structure to the Connector Appendix E Lagrangian Floer Theory on Symplectic Orbifolds E.1 Definition of Orbifolds and Twisted Sectors E.2 Compatible Systems and Good Maps E.3 Moduli Space of Ghost Maps and Chen-Ruan Cohomology E.4 Orbifold Disc Maps and Their Moduli Spaces E.5 Orbifold Lagrangian Floer Theory E.5.1 Maslov Indices of Orbibundle Pairs and of Orbicurves E.5.2 Fredholm Theory and the Formulae for Virtual Dimensions E.5.3 Bulk Deformations of Floer Cohomology and Orbi-Potential Appendix References Index