Holomorphic disc, spin structures and Floer cohomology of the Clifford torus

We compute the Bott-Morse Floer cohomology of the Clifford torus in \CP^n with all possible spin-structures. Each spin structure is known to determine an orientation of the moduli space of holomorphic discs, and we analyze the change of orientation according to the change of spin structure of the Clifford torus. Also, we classify all holomorphic discs with boundary lying on the Clifford torus by establishing a Maslov index formula for such discs. As a result, we show that in odd dimensions there exist two spin structures which give non-vanishing Floer cohomology of the Clifford torus, and in even dimensions, there is only one such spin structure. When the Floer cohomology is non-vanishing, it is isomorphic to the singular cohomology of the torus (with a Novikov ring as its coefficients). As a corollary, we prove that any Hamiltonian deformation of the Clifford torus intersects with it at least at $2^n$ distinct intersection points, when the intersection is transversal. We also compute the Floer cohomology of the Clifford torus with flat line bundles on it and verify the prediction made by Hori using a mirror symmetry calculation.Comment: 31 pages, 2 figure

On the counting of holomorphic discs in toric Fano manifolds

Open Gromov-Witten invariants in general are not well-defined. We discuss in detail the enumerative numbers of the Clifford torus $T^2$ in \CP^2. For cyclic A-infinity algebras, we show that certain generalized way of counting may be defined up to Hochschild or cyclic boundary elements. In particular we obtain a well-defined function on Hochschild or cyclic homology of a cyclic A-infinity algebra, which has invariance property under cyclic A-infinity homomorphism. We discuss an example of Clifford torus $T^2$ and compute the invariant for a specific cyclic cohomology class.Comment: 17 pages, 2 figures,v2: rewritten using the language of cyclic A-infinity algebra, v3: added an example of a cyclic cohomology class, published versio

On the obstructed Lagrangian Floer theory

Lagrangian Floer homology in a general case has been constructed by Fukaya, Oh, Ohta and Ono, where they construct an \AI-algebra or an \AI-bimodule from Lagrangian submanifolds, and studied the obstructions and deformation theories. But for obstructed Lagrangian submanifolds, standard Lagrangian Floer homology can not be defined. We explore several well-known cohomology theories on these \AI-objects and explore their properties, which are well-defined and invariant even in the obstructed cases. These are Hochschild and cyclic homology of an \AI-objects and Chevalley-Eilenberg or cyclic Chevalley-Eilenberg homology of their underlying \LI objects. We explain how the existence of $m_0$ effects the usual homological algebra of these homology theories. We also provide some computations. We show that for an obstructed \AI-algebra with a non-trivial primary obstruction, Chevalley-Eilenberg Floer homology vanishes, whose proof is inspired by the comparison with cluster homology theory of Lagrangian submanifolds by Cornea and Lalonde. In contrast, we also provide an example of an obstructed case whose cyclic Floer homology is non-vanishing.Comment: 43 pages, 1 figur

Counting real pseudo-holomorphic discs and spheres in dimension four and six

First, we provide another proof that the signed count of the real $J$-holomorphic spheres (or $J$-holomorphic discs) passing through a generic real configuration of $k$ points is independent of the choice of the real configuration and the choice of $J$, if the dimension of the Lagrangian submanifold $L$ (fixed points set of the involution) is two or three, and also if we assume $L$ is orientable and relatively spin, and $M$ is strongly semi-positive. This theorem was first proved by Welschinger in a more general setting, and we provide more natural approach using the degree of evaluation maps from the moduli spaces of $J$-holomorphic discs. Then, we define the invariant count of discs intersecting cycles of a symplectic manifold at fixed interior marked points, and intersecting real points at the boundary under certain assumptions. The last result is new and was not proved by Welshinger's method.Comment: 18 pages, 2 figures, typo correcte

Orbifold Morse-Smale-Witten complex

We construct Morse-Smale-Witten complex for an effective orientable orbifold. For a global quotient orbifold, we also construct a Morse-Bott complex. We show that certain type of critical points of a Morse function has to be discarded to construct such a complex, and gradient flows should be counted with suitable weights. The homology of these complexes are shown to be isomorphic to the singular homology of the quotient spaces under the self-indexing assumptions.Comment: 35 pages, 6 figures; The content of the last section is replaced by the explanation on weak group actions. Some other minor change

Holomorphic orbidiscs and Lagrangian Floer cohomology of symplectic toric orbifolds

We develop Floer theory of Lagrangian torus fibers in compact symplectic toric orbifolds. We first classify holomorphic orbi-discs with boundary on Lagrangian torus fibers. We show that there exists a class of basic discs such that we have one-to-one correspondences between a) smooth basic discs and facets of the moment polytope, and b) between basic orbi-discs and twisted sectors of the toric orbifold. We show that there is a smooth Lagrangian Floer theory of these torus fibers, which has a bulk-deformation by fundamental classes of twisted sectors of the toric orbifold. We show by several examples that such bulk-deformation can be used to illustrate the very rigid Hamiltonian geometry of orbifolds. We define its potential and bulk-deformed potential, and develop the notion of leading order potential. We study leading term equations analogous to the case of toric manifolds by Fukaya, Oh, Ohta and Ono.Comment: 75 pages, 4 figures. shortened by reducing repetition of construction from manifold case

Gradient-like vector fields on a complex analytic variety

Given a complex analytic function f on a Whitney stratified complex analytic variety of complex dimension n, whose real part Re(f) is Morse, we prove the existence of a stratified gradient-like vector field for Re(f) such that the unstable set of a critical point p on a stratum S of complex dimension s has real dimension $m(p)+n-s$ as was conjectured by Goresky and MacPherson.Comment: 23 pages, 4 figure, v2:major revision, v3:added restriction on Morse function and removed fast conditio

Finite group actions on Lagrangian Floer theory

We construct finite group actions on Lagrangian Floer theory when symplectic manifolds have finite group actions and Lagrangian submanifolds have induced group actions. We first define finite group actions on Novikov-Morse theory. We introduce the notion of a {\em spin profile} as an obstruction class of extending the group action on Lagrangian submanifold to the one on its spin structure, which is a group cohomology class in $H^2(G;\Z/2)$. For a class of Lagrangian submanifolds which have the same spin profiles, we define a finite group action on their Fukaya category. In consequence, we obtain the $s$-equivariant Fukaya category as well as the $s$-orbifolded Fukaya category for each group cohomology class $s$. We also develop a version with $G$-equivariant bundles on Lagrangian submanifolds, and explain how character group of $G$ acts on the theory. As an application, we define an orbifolded Fukaya-Seidel category of a $G$-invariant Lefschetz fibration, and also discuss homological mirror symmetry conjectures with group actions.Comment: 81 pages, 12 figures; comments welcome

Potentials of homotopy cyclic \AI-algebras

For a cyclic \AI-algebra, a potential recording the structure constants can be defined. We define an analogous potential for a homotopy cyclic \AI-algebra and prove its properties. On the other hand, we find another different potential for a homotopy cyclic \AI-algebra, which is related to the algebraic analogue of generalized holonomy map of Abbaspour, Tradler and Zeinalian.Comment: 18 pages, 1 figur

Chern-Weil Maslov index and its orbifold analogue

We give Chern-Weil definitions of the Maslov indices of bundle pairs over a Riemann surface \Sigma with boundary, which consists of symplectic vector bundle on \Sigma and a Lagrangian subbundle on \partial{\Sigma} as well as its generalization for transversely intersecting Lagrangian boundary conditions. We discuss their properties and relations to the known topological definitions. As a main application, we extend Maslov index to the case with orbifold interior singularites, via curvature integral, and find also an analogous topological definition in these cases.Comment: 19 page