Related papers: The Lagrangian Cubic Equation

The purpose of this paper is to give a survey of the various versions of Floer homology for manifolds with contact type boundary that have so far appeared in the literature. Under the name of ``Symplectic homology'' or ``Floer homology for…

Take a closed monotone symplectic manifold containing a smooth anticanonical divisor. The quantum connection on its cohomology has singularities at zero and infinity (in the quantum parameter). At zero it has a regular singular point, by…

Given a closed, oriented Lagrangian submanifold $L$ in a Liouville domain $\overline{M}$, one can define a Maurer-Cartan element with respect to a certain $L_\infty$-structure on the string homology…

We give a construction of the Floer homology of the pair of {\it non-compact} Lagrangian submanifolds, which satisfies natural continuity property under the Hamiltonian isotopy which moves the infinity but leaves the intersection set of the…

We apply Heegaard Floer homology to study deformations of singularities of plane algebraic curves. Our main result provides an obstruction to the existence of a deformation between two singularities. Generalizations include the case of…

This is an introduction to quantum algebra, from a geometric perspective. The classical spaces $X$, such as the Lie groups, homogeneous spaces, or more general manifolds, are described by various algebras $A$, defined over various fields…

We attempt to define a new invariant I of (almost) Calabi-Yau 3-folds M, by counting special Lagrangian rational homology 3-spheres N in M in each 3-homology class, with a certain weight w(N) depending on the topology of N. This is…

The variance of observables of quantum states of the Laplacian on the modular surface is calculated in the semiclassical limit. It is shown that this hermitian form is diagonalized by the irreducible representations of the modular quotient…

We explain a strategy, based on spectral invariants on symmetric product orbifolds, for proving the smooth closing lemma for Hamiltonian diffeomorphisms of a symplectic manifold when the orbifold quantum cohomologies of its symmetric…

We establish a new version of Floer homology for monotone Lagrangian submanifolds and apply it to prove the following (generalized) version of Audin's conjecture : if $L$ is an aspherical manifold which admits a monotone Lagrangian…

Using quilted Floer cohomology and relative quilt invariants, we define a composition functor for categories of Lagrangian correspondences in monotone and exact symplectic Floer theory. We show that this functor agrees with geometric…

The notion of quantum embedding is considered for two classes of examples: quantum coadjoint orbits in Lie coalgebras and quantum symplectic leaves in spaces with non-Lie permutation relations. A method for constructing irreducible…

Quantum $L_\infty$ algebras are higher loop generalizations of cyclic $L_\infty$ algebras. Motivated by the problem of defining morphisms between such algebras, we construct a linear category of $(-1)$-shifted symplectic vector spaces and…

In a recent paper we defined a new filtration of the mapping class group--the "Lagrangian" filtration. We here determine the successive quotients of this filtration, up to finite index. As an application we show that, for any additive…

We find a relation between Lagrangian Floer pairing of a symplectic manifold and Kapustin-Li pairing of the mirror Landau-Ginzburg model under localized mirror functor. They are conformally equivalent with an interesting conformal factor…

Localization of Floer homology is first introduced by Floer \cite{floer:fixed} in the context of Hamiltonian Floer homology. The author employed the notion in the Lagrangian context for the pair $(\phi_H^1(L),L)$ of compact Lagrangian…

A quantum theory is constructed for the system of a relativistic particle with mass m moving freely on the SL(2,R) group manifold. Applied to the cotangent bundle of SL(2,R), the method of Hamiltonian reduction allows us to split the…

We study hamiltonian actions of compact groups in the presence of compatible involutions. We show that the lagrangian fixed point set on the symplectically reduced space is isomorphic to the disjoint union of the involutively reduced spaces…

We introduce a contravariant functor, called Floer functor, from the category of Lagrangian conductors of a symplectic manifold to the homotopy category of bounded chain complexes of open strings in this manifold. The latter two categories…

We prove that a very general cubic fourfold containing a plane can be embedded into a holomorphic symplectic eightfold as a Lagrangian submanifold. We construct the desired holomorphic symplectic eightfold as a moduli space of Bridgeland…