Lagrangian Intersection Floer Theory: Anomaly and Obstruction, Part II by Kenji Fukaya (PDF)

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This is a two-volume series research monograph on the general Lagrangian Floer theory and on the accompanying homological algebra of filtered $A_infty$-algebras. This book provides the most important step towards a rigorous foundation of the Fukaya category in general context. In Volume I, general deformation theory of the Floer cohomology is developed in both algebraic and geometric contexts. An essentially self-contained homotopy theory of filtered $A_infty$ algebras and $A_infty$ bimodules and applications of their obstruction-deformation theory to the Lagrangian Floer theory are presented. Volume II contains detailed studies of two of the main points of the foundation of the theory: transversality and orientation. The study of transversality is based on the virtual fundamental chain techniques (the theory of Kuranishi structures and their multisections) and chain level intersection theories. A detailed analysis comparing the orientations of the moduli spaces and their fiber products is carried out. A self-contained account of the general theory of Kuranishi structures is also included in the appendix of this volume. Titles in this series are co-published with International Press, Cambridge, MA.

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⭐Anyone who approaches this book needs to have a solid understanding of symplectic geometry, differential and algebraic topology, algebraic geometry, abstract algebra, global analysis, and complex manifolds. It will also help if prospective readers had some knowledge of topics from physics and mathematics such as mirror symmetry, string theory, the theory of operads, and non-Archimedean geometry. The book is not written with the goal of explaining the key concepts, but rather as a report detailing the contributions of the authors to the subject. There are an enormous number of references included at the end of the Volume 2, and a study of some of these will be needed if the contents of these volumes are to be appreciated beyond that of a mere formal understanding. The books are another example of the typical way in which modern mathematical monographs are written, namely that of expounding and not of explaining, and this makes it difficult for those who sincerely want to understand the subject matter, even though they have no intentions of conducting research in the relevant area. Therefore readers who want to be challenged intellectually will definitely find ample opportunity for such in these two volumes.Readers will need an understanding of Floer homology, which is essentially an infinite-dimensional version of Morse-Novikov for the ‘symplectic action functional’. The goal is to interpret the contractible 1-periodic solutions of a Hamiltonian as the critical points of the symplectic action functional on the universal cover of the space L(M) of contractible loops in a symplectic manifold. The universal cover is the unique of covering space of L(M) whose group of deck transformations is the image of the Hurewicz homomorphism from the second homotopy group of M to the second homology group of M.The symplectic action function acts on the covering space and takes values in the real numbers. The critical points of this action correspond to periodic solutions and the focus is then on the upwards gradient flow lines of the symplectic action functional with respect to the L^2 metric on LM induced by an almost complex structure on M. These are the solutions u(s,t) of the Floer equation that have domain 2-dimensional Euclidean space and have period 1 in t. These solutions approach the critical points as s approaches infinity in the distant past and future, and these solutions form the moduli space of solutions of finite energy, the latter of which is defined to be difference of the action functional on the respective critical points in the distant past and future. The dimension of this moduli space is the difference in the Maslov index (some readers may understand this to be the Conley-Zehnder index) of the respective critical points in the distant past and future.Along these lines readers will need to have an appreciation of what it means to formulate and study moduli spaces. Generally speaking, a moduli problem is one of classifying certain geometric objects up to a notion of equivalence. Constructing a moduli space involves putting an an algebraic structure of M that should capture the way objects very in families. The ‘moduli problem’ is very familiar in the area of algebraic geometry where one has a family of objects over a base scheme, and with a notion of a pullback of families along morphisms. The pullback along the identity morphism gives the same family, and pullbacks of compositions of morphisms are merely the pullback of the first followed by the second. Two families over the base are equivalent if there is a map between the bases and the pullbacks of the map on these two families are equivalent. A ‘universal family’ is a family U where for every family X over the base there exists a unique morphism K from the base to M such that the pullback of K on U is equivalent to X as families over the base. The base is then called a ‘fine moduli space’. For every base B there exists a bijection between the set of families over B and the set of morphisms from B to M.Along these same lines, and also motivated by what is done in algebraic geometry, the theme of compactification of the moduli space plays an important role in these two volumes. The compactification relies on an important concept called ‘stability’ and in algebraic geometry typically arises when studying what are called ‘n-pointed smooth rational curves’. These are projective smooth rational curves with n marked points and maps between these curves preserves marked points. One can also have families of n-pointed smooth rational curves where each fiber will have marked points. There exists a fine moduli space M(0, n) for classifying n-pointed smooth rational curves up to isomorphism, and there exists a universal family of n-point curves U(0, n) with respect to M(0, n). This means that for every family of projective smooth rational curves over a base B with n disjoint sections is induced by a pullback along a unique morphism from B to M(0, n).But M(0, n) is not necessarily compact (for example M(0, 4)) and to compactify correctly one must include curves that are ‘reducible’. These are the ‘stable’ curves, and intuitively involve configurations where there is a “break” in the curve. Prospective readers of these volumes can find ample pictorial examples of stable curves in the literature on algebraic geometry. More formally an n-pointed curve is ‘stable’ if does not have a map that fixes each marked point: some of these marked points, or all, must move under this map. ‘Stabilization’ of a curve involves adding points so as to obtain a stable curve. Of particular relevance to the content of these volumes is that a stable curve of genus zero is a finite collection of 2-spheres that are glued together at certain points in these spheres and have marked points coming from these spheres.The notion of stable curves in algebraic geometry carries over to symplectic geometry and the theory of J-holomorphic curves, and therefore in these volumes, but with an added complication that is referred to as “bubbling”. The presence of bubbling is important because it contributes to an obstruction in defining Floer cohomology. Generally speaking, J-holomorphic curves are maps from Riemann surfaces to smooth manifolds M of even dimension whose exterior derivatives satisfy a Cauchy-Riemann equation involving an almost complex structure J on M. Similar to what is done in algebraic geometry, one constructs moduli spaces of J-holomorphic curves for the case where M is a compact almost complex manifold, and then attempts to show these moduli spaces are compact.Showing compactness entails studying sequences of J-holomorphic curves, and if the gradient of these curves is uniformly bounded in the L^p norm on compact subsets then a successful compactification of the moduli space means there exists a subsequence converging uniformly on compact subsets to a J-holomorphic curve. It is the case where the gradient is unbounded and the energy is uniformly bounded that results in the bubbling of a non-constant J-holomorphic curve. The use of the word “bubbling” here comes from the fact that one can find a subsequence which converges to a J-holomorphic curve on the 2-sphere. There are many interesting examples of bubbling in the research literature as prospective readers will find, but it must be said that some of these examples do not look sphere-like in appearance. A J-holomorphic curve can have several different bubbles, depending on the the scaling that is chosen, and if the genus of the curve is to be preserved during the limiting process, then this is referred to as stability, as in the case of algebraic geometry.The phenomenon of bubbling puts a constraint on the construction of Floer cohomology which the authors handle via the use of A-infinity algebras. The discussion of these algebras (more precisely that of filtered A-infinity algebras) is the most interesting part of these volumes, and is a reflection of the fact that to study Lagrangian intersections one must develop a theory at the cochain level. Filtered A-infinity algebras have an infinite number of operations that are must satisfy certain identities, and because of the complexity of these operations readers who are familiar with operads will be more comfortable with them. The authors show how to associate to a Lagrangian submanifold a filtered A-infinity algebra over a particular type of Novikov ring, and this algebra is interpreted as a quantum deformation of the rational homotopy type of L. The physicist reader may object to the use of the term ‘quantum’ to describe what are essentially all nonlinear contributions coming from pseudo-holomorphic disks, and such an objection would be valid. There is nothing “quantum” about these deformations in the sense that they exhibit interference or entanglement, two properties that are manifest in quantum physics. One could perhaps view them as a “semi-classical” phenomena manifested as “disk instantons.”The authors spend a considerable amount of time detailing when it is possible to define the cohomology of the filtered A-infinity algebra, which essentially involves defining it relative to a particular type of bounding cochain that satisfies an A-infinity version of the Maurer-Cartan equations. The obstructions to defining Floer cohomology for Lagrangian submanifolds involve deformations of this bounding cochain. The authors gives the details on how to define Floer obstruction classes and show a connection with the famous Gromov-Witten inveriants.A special type of algebraic object that arises in Morse theory for multi-valued functions on loop spaces is the Novikov ring, and is used throughout these two volumes. A perusal of Novikov’s original paper on the Morse theory of closed 1-forms will assist in gaining an intuition on this ring, and the authors of this work spend a fair amount of time in discussing it, even though for the most part their discussion is purely formal. Essentially what Novikov did is to take a closed 1-form w on a manifold M and integrate it over paths in M. This results in a multivalued function S which becomes single-valued on some covering p of M with a free abelian monodromy group given by taking the pullback p* of w. The number of generators of this monodromy group are the number of rationally independent integrals of w over integral cycles in M. The surfaces of steepest descent for critical points of S on the cover define cell complexes C which are invariant with respect to the action of a generator t: covering space-> M of the integers Z. This implies that C is a free complex of Z[t, 1/t] modules generated by S on the covering space. The object Z[t, 1/t] is the ring of Laurent polynomials and serves as an elementary example of the Novikov ring. A Novikov complex is then the collection of finitely generated modules over the Novikov ring whose generators are in 1-1 correspondence with the critical points of the 1-form.Another concept in these volumes that readers may find challenging to understand is that of a Kuranishi structure. To get some insight into what a Kuranishi structure is, the reviewer had to perform a fairly time-intensive consultation of the research literature. Loosely speaking, a Kuranishi structure is generalization of an ordinary manifold in that it allows the presence of singularities. In the context of symplectic topology, one is interested in the construction of algebraic structures by extracting homological information from the moduli spaces of holomorphic curves in general compact symplectic manifolds. This is referred to as ‘regularization’ and is to be contrasted with other regularization approaches such as where almost complex structures are perturbed but are unable to yield a smooth structure on the moduli space. Such a regularization procedure makes used of some kind of perturbative technique that will permit transversality. The construction of the algebraic structure also should be independent of the choices involved in obtaining transversality. The elaboration of these requirements leads to the performing of regularization via Kuranishi structures.

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