arXiv subjects

Cheol-Hyun Cho

Publications and source records attributed to Cheol-Hyun Cho.

36 records · Page 2Linked to original sources

arXiv2014

Lagrangian Floer superpotentials and crepant resolutions for toric orbifolds

We investigate the relationship between the Lagrangian Floer superpotentials for a toric orbifold and its toric crepant resolutions. More specifically, we study an open string version of the crepant resolution conjecture (CRC) which states that the Lagrangian Floer superpotential of a Gorenstein toric orbifold $\mathcal{X}$ and that of its toric crepant resolution $Y$ coincide after analytic continuation of quantum parameters and a change of variables. Relating this conjecture with the closed CRC, we find that the change of variable formula which appears in closed CRC can be explained by relations between open (orbifold) Gromov-Witten invariants. We also discover a geometric explanation (in terms of virtual counting of stable orbi-discs) for the specialization of quantum parameters to roots of unity which appears in Y. Ruan's original CRC ["The cohomology ring of crepant resolutions of orbifolds", Gromov-Witten theory of spin curves and orbifolds, 117-126, Contemp. Math., 403, Amer. Math. Soc., Providence, RI, 2006]. We prove the open CRC for the weighted projective spaces $\mathcal{X}=\mathbb{P}(1,\ldots,1,n)$ using an equality between open and closed orbifold Gromov-Witten invariants. Along the way, we also prove an open mirror theorem for these toric orbifolds.

math.SG↗

arXiv2014

Lagrangian Floer potential of orbifold spheres

For each sphere with three orbifold points, we construct an algorithm to compute the open Gromov-Witten potential, which serves as the quantum-corrected Landau-Ginzburg mirror and is an infinite series in general. This gives the first class of general-type geometries whose full potentials can be computed. As a consequence we obtain an enumerative meaning of mirror maps for elliptic curve quotients. Furthermore, we prove that the open Gromov-Witten potential is convergent, even in the general-type cases, and has an isolated singularity at the origin, which is an important ingredient of proving homological mirror symmetry.

math.SG↗

arXiv2013

On orbifold embeddings

For an orbifold, there is a notion of an orbifold embedding, which is more general than the one of sub-orbifolds. We develop several properties of orbifold embeddings. In the case of translation groupoids, we show that such a notion is equivalent to a strong equivariant immersion.

math.GT↗

arXiv2013

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.

math.AT↗

arXiv2013

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.

math.SG↗

arXiv2012

Examples of Matrix Factorizations from SYZ

We find matrix factorization corresponding to an anti-diagonal in ${\mathbb C}P^1 \times {\mathbb C}P^1$, and circle fibers in weighted projective lines using the idea of Chan and Leung of Strominger-Yau-Zaslow transformations. For the tear drop orbifolds, we apply this idea to find matrix factorizations for two types of potential, the usual Hori-Vafa potential or the bulk deformed (orbi)-potential. We also show that the direct sum of anti-diagonal with its shift, is equivalent to the direct sum of central torus fibers with holonomy $(1,-1)$ and $(-1,1)$ in the Fukaya category of ${\mathbb C}P^1 \times {\mathbb C}P^1$, which was predicted by Kapustin and Li from B-model calculations.

math.SG↗

arXiv2012

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.

math.SG↗

arXiv2012

Chern-Weil Maslov index and its orbifold analogue

We give Chern-Weil definitions of the Maslov indices of bundle pairs over a Riemann surface Σwith boundary, which consists of symplectic vector bundle on Σand a Lagrangian subbundle on \partialΣ 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.

math.SG↗

arXiv2010

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.

math.QA↗

arXiv2010

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.

math.GT↗

arXiv2010

Notes on Kontsevich-Soibelman's theorem about cyclic A-infinity algebras

Kontsevich and Soibelman has proved a relation between a non-degenerate cyclic homology element of an A-infinity algebra A and its cyclic inner products on the minimal model of A. We find an explicit formula of this correspondence, in terms of the strong homotopy inner products and negative cyclic cohomology of A. We prove that an equivalence class of the induced strong homotopy inner product depends only on the given negative cyclic cohomology class. Also, we extend such a correspondence to the case of gapped filtered A-infinity algebras.

math.SG↗

arXiv2009

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.

math.SG↗

arXiv2008

Non-displaceable Lagrangian submanifolds and Floer cohomology with non-unitary line bundle

We show that in many examples the non-displaceability of Lagrangian submanifolds by Hamiltonian isotopy can be proved via Lagrangian Floer cohomology with non-unitary line bundle. The examples include all monotone Lagrangian torus fibers in toric Fano manifold (which was also proven by Entov and Polterovich via the theory of symplectic quasi-states), some non-monotone Lagrangian torus fibers. We also extend the results by Oh and the author about the computations of Floer cohomology of Lagrangian torus fibers to the case of all toric Fano manifolds, removing the convexity assumption in the previous work.

math.SG↗

arXiv2007

Strong homotopy inner product of an A-infinity algebra

We introduce a strong homotopy notion of a cyclic symmetric inner product of an A-infinity algebra and prove a characterization theorem in the formalism of the infinity inner products by Tradler. We also show that it is equivalent to the notion of a non-constant symplectic structure on the corresponding formal non-commutative supermanifold. We show that (open Gromov-Witten type) potential for a cyclic filtered A-infinity algebra is invariant under the cyclic filtered A-infinity homomorphism up to reparametrization, cyclization and a constant addition, generalizing the work of Kajiura.

math.AT↗

arXiv2006

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.

math.SG↗

arXiv2005

Products of Floer cohomology of torus fibers in toric Fano manifolds

We compute the ring structure of Floer cohomology groups of Lagrangian torus fibers in some toric Fano manifolds continuing the study of \cite{CO}. Related $\AI$-formulas hold for transversal choice of chains. Two different computations are provided: a direct calculation using the classification of holomorphic discs by Oh and the author in \cite{CO}, and another method by using an {\it analogue of divisor equation} in Gromov-Witten invariants to thecase of discs. Floer cohomology rings are shown to be isomorphic to Clifford algebras, whose quadratic forms are given by the Hessians of functions $W$, which turn out to be the superpotentials of Landau-Ginzburg mirrors. In the case of $\CP^n$ and $ \CP^1 \times \CP^1$, this proves the prediction made by Hori, Kapustin and Li by B-model calculations via physical arguments. The latter method also provides correspondence between higher derivatives of the superpotential of LG mirror with the higher products of $\AI$(or $\LI$)-algebra of the Lagrangian submanifold.

math.SG↗

arXiv2003

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.

math.SG↗

arXiv2003

Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toric manifolds

In this paper, we first provide an explicit description of {\it all} holomorphic discs (``disc instantons'') attached to Lagrangian torus fibers of arbitrary compact toric manifolds, and prove their Fredholm regularity. Using this, we compute Fukaya-Oh-Ohta-Ono's (FOOO's) obstruction (co)chains and the Floer cohomology of Lagrangian torus fibers of Fano toric manifolds. In particular specializing to the formal parameter $T^{2π} = e^{-1}$, our computation verifies the folklore that FOOO's obstruction (co)chains correspond to the Landau-Ginzburg superpotentials under the mirror symmetry correspondence, and also proves the prediction made by K. Hori about the Floer cohomology of Lagrangian torus fibers of Fano toric manifolds. The latter states that the Floer cohomology (for the parameter value $T^{2π} = e^{-1}$) of all the fibers vanish except at a finite number, the Euler characteristic of the toric manifold, of base points in the momentum polytope that are critical points of the superpotential of the Landau-Ginzburg mirror to the toric manifold. In the latter cases, we also prove that the Floer cohomology of the corresponding fiber is isomorphic to its singular cohomology. We also introduce a restricted version of the Floer cohomology of Lagrangian submanifolds, which is a priori more flexible to define in general, and which we call the {\it adapted Floer cohomology}. We then prove that the adapted Floer cohomology of any non-singular torus fiber of Fano toric manifolds is well-defined, invariant under the Hamiltonian isotopy and isomorphic to the Bott-Morse Floer cohomology of the fiber.

math.SG↗