Categorification of invariants in gauge theory and sypmplectic geometry

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

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 $\tau$ on a symplectic manifold. We introduce the notion of $\tau$-relatively spin structure for an anti-symplectic involution $\tau$, and study how the orientations on the moduli space behave under the involution $\tau$. We also apply this to the study of Lagrangian Floer theory of real Lagrangian submanifolds. In particular, we study unobstructedness of the $\tau$-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 $\R P^{2n+1}$ over $\Lambda_{0,nov}^{\Z}$ 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 and moduli spaces in topological field theories

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

Homological algebra related to surfaces with boundary

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

Unobstructed immersed Lagrangian correspondence and filtered A infinity functor

In this paper we construct a 2-functor from the unobstructed immersed Weinstein category to the category of all filtered A infinity categories. We consider arbitrary (compact) symplectic manifolds and its arbitrary (relatively spin) immersed Lagrangian submanifolds. The filtered A infinity category associated to a symplectic manifold is defined by using Lagrangian Floer theory in such generality by Fukaya-Oh-Ohta-Ono and Akaho-Joyce. The morphism of unobstructed immersed Weinstein category is by definition a pair of immersed Lagrangian submanifold of the direct product and its bounding cochain (in the sense of Fukaya-Oh-Ohta-Ono and Akaho-Joyce). Such a morphism transforms an (immersed) Lagrangian submanifold of one factor to one of the other factor. The key new result proved in this paper shows that this geometric transformation preserves unobstructed-ness of the Lagrangian Floer theory. Thus, this paper generalizes earlier results by Wehrheim-Woodward and Mau's-Wehrheim-Woodward so that it works in complete generality in the compact case. The main idea of the proofs are based on Lekili-Lipiyansky's Y diagram and a lemma from homological algebra, together with systematic use of Yoneda functor. In other words the proofs are based on a different idea from those which are studied by Bottmann-Mau's-Wehrheim-Woodward, where strip shrinking and figure 8 bubble plays the central role.Comment: approximately 260 pages and 100 figures, some errors are corrected from previous version. (Mainly on Section 16. Other places are only typo.) Reference are update