167 research outputs found

    Categorification of invariants in gauge theory and sypmplectic geometry

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    This is a mixture of survey article and research anouncement. We discuss Instanton Floer homology for 3 manifolds with boundary. We also discuss a categorification of the Lagrangian Floer theory using the unobstructed immersed Lagrangian correspondence as a morphism in the category of symplectic manifolds. During the year 1998-2012, those problems have been studied emphasising the ideas from analysis such as degeneration and adiabatic limit (Instanton Floer homology) and strip shrinking (Lagrangian correspondence). Recently we found that replacing those analytic approach by a combination of cobordism type argument and homological algebra, we can resolve various difficulties in the analytic approach. It thus solves various problems and also simplify many of the proofs.Comment: 50 pages, mostly the same as one distributed as a abstract of author's Takagi Lectur

    Anti-symplectic involution and Floer cohomology

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    The main purpose of the present paper is a study of orientations of the moduli spaces of pseudo-holomorphic discs with boundary lying on a \emph{real} Lagrangian submanifold, i.e., the fixed point set of an anti-symplectic involutions τ\tau on a symplectic manifold. We introduce the notion of τ\tau-relatively spin structure for an anti-symplectic involution τ\tau, and study how the orientations on the moduli space behave under the involution τ\tau. We also apply this to the study of Lagrangian Floer theory of real Lagrangian submanifolds. In particular, we study unobstructedness of the τ\tau-fixed point set of symplectic manifolds and in particular prove its unobstructedness in the case of Calabi-Yau manifolds. And we also do explicit calculation of Floer cohomology of RP2n+1\R P^{2n+1} over Λ0,novZ\Lambda_{0,nov}^{\Z} which provides an example whose Floer cohomology is not isomorphic to its classical cohomology. We study Floer cohomology of the diagonal of the square of a symplectic manifold, which leads to a rigorous construction of the quantum Massey product of symplectic manifold in complete generality.Comment: 85 pages, final version, to appear in Geometry and Topolog

    Homological algebra related to surfaces with boundary

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    In this article we describe an algebraic framework which can be used in three related but different contexts: string topology, symplectic field theory, and Lagrangian Floer theory of higher genus. It turns out that the relevant algebraic structure for all three contexts is a homotopy version of involutive bi-Lie algebras, which we call IBL_\infty-algebras.Comment: 127 pages, 22 figures. Some references added in version 2. Fixed a tex problem in version

    Homological algebra and moduli spaces in topological field theories

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    This is a survey of various types of Floer theories (both in symplectic geometry and gauge theory) and relations among them.Comment: 56 pages 12 Figure