arXiv subjects

Cheol-Hyun Cho

Publications and source records attributed to Cheol-Hyun Cho.

At least 19 recordsLinked to original sources

arXiv2025

Geometric models of simple Lie algebras via singularity theory

It is well-known that ADE Dynkin diagrams classify both the simply-laced simple Lie algebras and simple singularities. We introduce a polygonal wheel in a plane for each case of ADE, called the Coxeter wheel. We show that equivalence classes of edges and spokes of a Coxeter wheel form a geometric root system isomorphic to the classical root system of the corresponding type. This wheel is in fact derived from the Milnor fiber of corresponding simple singularities of two variables, and the bilinear form on the geometric root system is the negative of its symmetrized Seifert form. Furthermore, we give a completely geometric definition of simple Lie algebras using arcs, Seifert form and variation operator of the singularity theory.

math.RT↗

arXiv2025

Vanishing arcs for isolated plane curve singularities

The variation operator associated with an isolated hypersurface singularity is a classical topological invariant that relates relative and absolute homologies of the Milnor fiber via a non trivial isomorphism. Here we work with a topological version of this operator that deals with proper arcs and closed curves instead of homology cycles. Building on the classical framework of geometric vanishing cycles, we introduce the concept of vanishing arcsets as their counterpart using this geometric variation operator. We characterize which properly embedded arcs are sent to geometric vanishing cycles by the geometric variation operator in terms of intersections numbers of the arcs and their images by the geometric monodromy. Furthermore, we prove that for any distinguished collection of vanishing cycles arising from an A'Campo's divide, there exists a topological exceptional collection of arcsets whose variation images match this collection.

math.GT↗

arXiv2024

Berglund-H\"ubsch mirrors of invertible curve singularities via Floer theory

We find a Floer theoretic approach to obtain the transpose polynomial $W^T$ of an invertible curve singularity $W$. This gives an intrinsic construction of the mirror transpose polynomial and enables us to define a canonical $A_\infty$-functor that takes Lagrangians in the Milnor fiber of W and converts them into matrix factorizations of $W^T$. We find Lagrangians in the Milnor fiber of $W$ that are mirror to the indecomposable matrix factorizations of $W^T$ when $W^T$ is ADE singularity and discover that Auslander-Reiten exact sequences can be realized as surgery exact triangles of Lagrangians in the mirror. There are two primary steps in the Floer theoretic method for obtaining a transposition polynomial: To get a Lagrangian $L$ and corresponding disc potential function $W_L$, we first determine the quotient $X$ by the maximal symmetry group for the Milnor fiber. Second, we define a class $\Gamma$ of symplectic cohomology of $X$ based on the monodromy of the singularity $W$. Another disc counting function, $g$, is defined by the closed-open image of $\Gamma$ on $L$. We demonstrate that restricting to the hypersurface $g = 0$ transforms the disc potential function $W_L$ into the transpose polynomial W T. This second step is the mirror of taking the cone of quantum cap action by the monodromy class $\Gamma$.

math.SG↗

arXiv2024

Canonical form of matrix factorizations from Fukaya category of surface

This paper concerns homological mirror symmetry for the pair-of-pants surface (A-side) and the non-isolated surface singularity $xyz=0$ (B-side). Burban-Drozd classified indecomposable maximal Cohen-Macaulay modules on the B-side. We prove that higher-multiplicity band-type modules correspond to higher-rank local systems over closed geodesics on the A-side, generalizing our previous work for the multiplicity one case. This provides a geometric interpretation of the representation tameness of the band-type maximal Cohen-Macaulay modules, as every indecomposable object is realized as a geometric object. We also present an explicit canonical form of matrix factorizations of $xyz$ corresponding to Burban-Drozd's canonical form of band-type maximal Cohen-Macaulay modules. As applications, we give a geometric interpretation of algebraic operations such as AR translation and duality of maximal Cohen-Macaulay modules as well as certain mapping cone operations.

math.RT↗

arXiv2024

Topological Fukaya category of tagged arcs

A tagged arc on a surface is introduced by Fomin, Shapiro, and Thurston to study cluster theory on marked surfaces. Given a tagged arc system on a graded marked surface, we define its $\mathbb{Z}$-graded $\mathcal{A}_\infty$-category, generalizing the construction of Haiden, Katzarkov, and Kontsevich for arc systems. When a tagged arc system arises from a non-trivial involution on a marked surface, we show that this $\mathcal{A}_\infty$-category is quasi-isomorphic to the invariant part of the topological Fukaya category under the involution. In particular, this identifies tagged arcs with non-geometric idempotents of Fukaya category.

math.SG↗

arXiv2023

Floer theory for the variation operator of an isolated singularity

The variation operator in singularity theory maps relative homology cycles to compact cycles in the Milnor fiber using the monodromy. We construct its symplectic analogue for an isolated singularity. We define the monodromy Lagrangian Floer cohomology, which provides categorifications of the standard theorems on the variation operator and the Seifert form. The key ingredients are a special class $\Gamma$ in the symplectic cohomology of the inverse of the monodromy and its closed-open images. For isolated plane curve singularities whose A'Campo divide has depth zero, we find an exceptional collection consisting of non-compact Lagrangians in the Milnor fiber corresponding to a distinguished collection of vanishing cycles under the variation operator.

math.SG↗

arXiv2022

Fukaya category for Landau-Ginzburg orbifolds

For a weighted homogeneous polynomial and a choice of a diagonal symmetry group, we define a new Fukaya category for a Landau-Ginzburg orbifold (of Fano or Calabi-Yau type). The construction is based on the wrapped Fukaya category of its Milnor fiber together with the monodromy of the singularity, and it is analogous to the variation operator in singularity theory. The new $\AI$-structure is constructed using popsicle maps with interior insertions of the monodromy orbit. This requires new compactifications of popsicle moduli spaces where conformal structures of some of the spheres and discs are aligned due to the popsicle structures. In particular, codimension one popsicle sphere bubbles might exist and become obstructions to define the $\AI$-structure. For log Fano and Calabi-Yau cases, we show that the sphere bubbles do not arise from action and degree estimates, together with the computation of indices of twisted Reeb orbits for Milnor fiber quotients.

math.SG↗

arXiv2022

Homological mirror symmetry of indecomposable Cohen-Macaulay modules for some degenerate cusp singularities

Burban-Drozd showed that the degenerate cusp singularities have tame Cohen-Macaulay representation type, and classified all indecomposable Cohen-Macaulay modules over them. One of their main example is the non-isolated singularity $W=xyz$. On the other hand, Abouzaid-Auroux-Efimov-Katzarkov-Orlov showed that $W=xyz$ is mirror to a pair of pants. In this paper, we investigate homological mirror symmetry of these indecomposable Cohen-Macaulay modules for $xyz$. Namely, we show that closed geodesics (with a flat $\mathbb{C}$-bundle) of a hyperbolic pair of pants have a one-to-one correspondence with indecomposable Cohen-Macaulay modules for $xyz$ with multiplicity one that are locally free on the punctured spectrum. In particular, this correspondence is established first by a geometric $A_{\infty}$-functor from the Fukaya category of the pair of pants to the matrix factorization category of $xyz$, and next by the correspondence between Cohen-Macaulay modules and matrix factorizations due to Eisenbud. For the latter, we compute explicit Macaulayfications of modules from Burban-Drozd's classification and find a canonical form of the corresponding matrix factorizations. In the sequel, we will show that indecomposable modules with higher multiplicity correspond to twisted complexes of closed geodesics. We also find mirror images of rank $1$ indecomposable Cohen-Macaulay modules (of band type) over the singularity $W = x^{3} + y^{2} - xyz$ as closed loops in the orbifold sphere $\mathbb{P}^1_{3,2,\infty}$.

math.SG↗

arXiv2022

Ungraded matrix factorizations as mirrors of non-orientable Lagrangians

We introduce the notion of ungraded matrix factorization as a mirror of non-orientable Lagrangian submanifolds. An ungraded matrix factorization of a polynomial $W$, with coefficients in a field of characteristic 2, is a square matrix $Q$ of polynomial entries satisfying $Q^2 = W \cdot \mathrm{Id}$. We then show that non-orientable Lagrangians correspond to ungraded matrix factorizations under the localized mirror functor and illustrate this construction in a few instances. Our main example is the Lagrangian submanifold $\mathbb{R}P^2 \subset \mathbb{C}P^2$ and its mirror ungraded matrix factorization, which we construct and study. In particular, we prove a version of Homological Mirror Symmetry in this setting.

math.SG↗

arXiv2021

Noncommutative homological mirror functor

We formulate a constructive theory of noncommutative Landau-Ginzburg models mirror to symplectic manifolds based on Lagrangian Floer theory. The construction comes with a natural functor from the Fukaya category to the category of matrix factorizations of the constructed Landau-Ginzburg model. As applications, it is applied to elliptic orbifolds, punctured Riemann surfaces and certain non-compact Calabi-Yau threefolds to construct their mirrors and functors. In particular it recovers and strengthens several interesting results of Etingof-Ginzburg, Bocklandt and Smith, and gives a unified understanding of their results in terms of mirror symmetry and symplectic geometry. As an interesting application, we construct an explicit global deformation quantization of an affine del Pezzo surface as a noncommutative mirror to an elliptic orbifold.

math.AG↗

arXiv2020

Kodaira-Spencer map, Lagrangian Floer theory and orbifold Jacobian algebras

A version of mirror symmetry predicts a ring isomorphism between quantum cohomology of a symplectic manifold and Jacobian algebra of the Landau-Ginzburg mirror, and for toric manifolds Fukaya-Oh-Ohta-Ono constructed such a map called Kodaira-Spencer map using Lagrangian Floer theory. We discuss a general construction of Kodaira-Spencer ring homomorphism when LG mirror potential $W$ is given by $J$-holomorphic discs with boundary on a Lagrangian $L$: we find an $A_{\infty}$-algebra $\mathcal{B}$ whose $m_1$-complex is a Koszul complex for $W$ under mild assumptions on $L$. Closed-open map gives a ring homomorphism from quantum cohomology to cohomology algebra of $\mathcal{B}$ which is Jacobian algebra of $W$. We also construct an equivariant version for orbifold LG mirror $(W,H)$. We construct a Kodaira-Spencer map from quantum cohomology to another $A_{\infty}$-algebra $(\mathcal{B}\rtimes H)^H$ whose cohomology algebra is isomorphic to the orbifold Jacobian algebra of $(W,H)$ under an assumption. For the $2$-torus whose mirror is an orbifold LG model given by Fermat cubic with a $\mathbb{Z}/3$-action, we compute an explicit Kodaira-Spencer isomorphism.

math.SG↗

arXiv2020

A Note on Disk Counting in Toric Orbifolds

We compute orbi-disk invariants of compact Gorenstein semi-Fano toric orbifolds by extending the method used for toric Calabi-Yau orbifolds. As a consequence the orbi-disc potential is analytic over complex numbers.

math.SG↗

arXiv2020

Big Quantum cohomology of orbifold spheres

We construct a Kodaira-Spencer map from the big quantum cohomology of a sphere with three orbifold points to the Jacobian ring of the mirror Landau-Ginzburg potential function. This is constructed via the Lagrangian Floer theory of the Seidel Lagrangian and we show that Kodaira-Spencer map is a ring isomorphism.

math.SG↗

arXiv2018

Gluing Localized Mirror Functors

We develop a method of gluing the local mirrors and functors constructed from immersed Lagrangians in the same deformation class. As a result, we obtain a global mirror geometry and a canonical mirror functor. We apply the method to construct the mirrors of punctured Riemann surfaces and show that our functor derives homological mirror symmetry.

math.SG↗

arXiv2016

Localized mirror functor constructed from a Lagrangian torus

Fixing a weakly unobstructed Lagrangian torus in a symplectic manifold X, we define a holomorphic function W known as the Floer potential. We construct a canonical A-infinity functor from the Fukaya category of X to the category of matrix factorizations of W. It provides a unified way to construct matrix factorizations from Lagrangian Floer theory. The technique is applied to toric Fano manifolds to transform Lagrangian branes to matrix factorizations. Using the method, we also obtain an explicit expression of the matrix factorization mirror to the real locus of the complex projective space.

math.SG↗

arXiv2015

Gross fibrations, SYZ mirror symmetry, and open Gromov-Witten invariants for toric Calabi-Yau orbifolds

For a toric Calabi-Yau (CY) orbifold $\mathcal{X}$ whose underlying toric variety is semi-projective, we construct and study a non-toric Lagrangian torus fibration on $\mathcal{X}$, which we call the Gross fibration. We apply the Strominger-Yau-Zaslow (SYZ) recipe to the Gross fibration of $\mathcal{X}$ to construct its mirror with the instanton corrections coming from genus 0 open orbifold Gromov-Witten (GW) invariants, which are virtual counts of holomorphic orbi-disks in $\mathcal{X}$ bounded by fibers of the Gross fibration. We explicitly evaluate all these invariants by first proving an open/closed equality and then employing the toric mirror theorem for suitable toric (partial) compactifications of $\mathcal{X}$. Our calculations are then applied to (1) prove a conjecture of Gross-Siebert on a relation between genus 0 open orbifold GW invariants and mirror maps of $\mathcal{X}$ -- this is called the open mirror theorem, which leads to an enumerative meaning of mirror maps, and (2) demonstrate how open (orbifold) GW invariants for toric CY orbifolds change under toric crepant resolutions -- an open analogue of Ruan's crepant resolution conjecture.

math.SG↗

arXiv2015

Localized mirror functor for Lagrangian immersions, and homological mirror symmetry for P^1_{a,b,c}

This paper gives a new way of constructing Landau-Ginzburg mirrors using deformation theory of Lagrangian immersions motivated by the works of Seidel, Strominger-Yau-Zaslow and Fukaya-Oh-Ohta-Ono. Moreover we construct a canonical functor from the Fukaya category to the mirror category of matrix factorizations. This functor derives homological mirror symmetry under some explicit assumptions. As an application, the construction is applied to spheres with three orbifold points to produce their quantum-corrected mirrors and derive homological mirror symmetry. Furthermore we discover an enumerative meaning of the (inverse) mirror map for elliptic curve quotients.

math.SG↗

arXiv2014

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.

math.SG↗