From Calabi-Yau dg Categories to Frobenius manifolds via Primitive Forms

It is one of the most important problems in mirror symmetry to obtain functorially Frobenius manifolds from smooth compact Calabi-Yau $A_\infty$-categories. This paper gives an approach to this problem based on the theory of primitive forms. Under an assumption on the formality of a certain homotopy algebra, a formal primitive form for a smooth compact Calabi-Yau dg algebra can be constructed, which enable us to have a formal Frobenius manifold.Comment: 22 pages. This article is based on a talk given in Kavli IPMU at the workshop "Primitive forms and related subjects" on February 11th 2014. Corrected typos and minor error

Weighted Projective Lines Associated to Regular Systems of Weights of Dual Type

We associate to a regular system of weights a weighted projective line over an algebraically closed field of characteristic zero in two different ways. One is defined as a quotient stack via a hypersurface singularity for a regular system of weights and the other is defined via the signature of the same regular system of weights. The main result in this paper is that if a regular system of weights is of dual type then these two weighted projective lines have equivalent abelian categories of coherent sheaves. As a corollary, we can show that the triangulated categories of the graded singularity associated to a regular system of weights has a full exceptional collection, which is expected from homological mirror symmetries. Main theorem of this paper will be generalized to more general one, to the case when a regular system of weights is of genus zero, which will be given in the joint paper with Kajiura and Saito. Since we need more detailed study of regular systems of weights and some knowledge of algebraic geometry of Deligne--Mumford stacks there, the author write a part of the result in this paper to which another simple proof based on the idea by Geigle--Lenzing can be applied.Comment: 16 pages, improved Section

Matrix Factorizations and Representations of Quivers I

This paper introduces a mathematical definition of the category of D-branes in Landau-Ginzburg orbifolds in terms of $A_\infty$-categories. Our categories coincide with the categories of (graded) matrix factorizations for quasi-homogeneous polynomials. After setting up the necessary definitions, we prove that our category for the polynomial $x^{n+1}$ is equivalent to the derived category of representations of the Dynkin quiver of type $A_{n}$. We also construct a special stability condition for the triangulated category in the sense of T. Bridgeland, which should be the "origin" of the space of stability conditions.Comment: 20 pages, added reference

Lattices for Landau-Ginzburg orbifolds

We consider a pair consisting of an invertible polynomial and a finite abelian group of its symmetries. Berglund, H\"ubsch, and Henningson proposed a duality between such pairs giving rise to mirror symmetry. We define an orbifoldized signature for such a pair using the orbifoldized elliptic genus. In the case of three variables and based on the homological mirror symmetry picture, we introduce two integral lattices, a transcendental and an algebraic one. We show that these lattices have the same rank and that the signature of the transcendental one is the orbifoldized signature. Finally, we give some evidence that these lattices are interchanged under the duality of pairs.Comment: 23 page

Maximally-graded matrix factorizations for an invertible polynomial of chain type

In 1977, Orlik--Randell construct a nice integral basis of the middle homology group of the Milnor fiber associated to an invertible polynomial of chain type and they conjectured that it is represented by a distinguished basis of vanishing cycles. The purpose of this paper is to prove the algebraic counterpart of the Orlik--Randell conjecture. Under the homological mirror symmetry, we may expect that the triangulated category of maximally-graded matrix factorizations for the Berglund--H\"{u}bsch transposed polynomial admits a full exceptional collection with a nice numerical property. Indeed, we show that the category admits a Lefschetz decomposition with respect to a polarization in the sense of Kuznetsov--Smirnov, whose Euler matrix are calculated in terms of the "zeta function" of the inverse of the polarization. As a corollary, it turns out that the homological mirror symmetry holds at the level of lattices, namely, the Grothendieck group of the category with the Euler form is isomorphic to the middle homology group with the intersection form (with a suitable sign).Comment: 19 page

A geometric definition of Gabrielov numbers

Gabrielov numbers describe certain Coxeter-Dynkin diagrams of the 14 exceptional unimodal singularities and play a role in Arnold's strange duality. In a previous paper, the authors defined Gabrielov numbers of a cusp singularity with an action of a finite abelian subgroup $G$ of {\rm SL}(3,\CC) using the Gabrielov numbers of the cusp singularity and data of the group $G$. Here we consider a crepant resolution Y \to \CC^3/G and the preimage $Z$ of the image of the Milnor fibre of the cusp singularity under the natural projection \CC^3 \to \CC^3/G. Using the McKay correspondence, we compute the homology of the pair $(Y,Z)$. We construct a basis of the relative homology group H_3(Y,Z;\QQ) with a Coxeter-Dynkin diagram where one can read off the Gabrielov numbers.Comment: 13 pages, 6 figure

Strange duality between hypersurface and complete intersection singularities

C.T.C. Wall and the first author discovered an extension of Arnold's strange duality embracing on one hand series of bimodal hypersurface singularities and on the other, isolated complete intersection singularities. In this paper, we derive this duality from the mirror symmetry and the Berglund-H\"ubsch transposition of invertible polynomials.Comment: 20 page

On rational Frobenius Manifolds of rank three with symmetries

We study Frobenius manifolds of rank three and dimension one that are related to submanifolds of certain Frobenius manifolds arising in mirror symmetry of elliptic orbifolds. We classify such Frobenius manifolds that are defined over an arbitrary field $\mathbb{K} \subset \mathbb{C}$ via the theory of modular forms. By an arithmetic property of an elliptic curve $\mathbb{E}_\tau$ defined over $\mathbb K$ associated to such a Frobenius manifold, it is proved that there are only two such Frobenius manifolds defined over $\mathbb C$ satisfying a certain symmetry assumption and thirteen Frobenius manifolds defined over $\mathbb Q$ satisfying a weak symmetry assumption on the potential

Mirror symmetry between orbifold curves and cusp singularities with group action

We consider an orbifold Landau-Ginzburg model $(f,G)$, where $f$ is an invertible polynomial in three variables and $G$ a finite group of symmetries of $f$ containing the exponential grading operator, and its Berglund-H\"ubsch transpose $(f^T, G^T)$. We show that this defines a mirror symmetry between orbifold curves and cusp singularities with group action. We define Dolgachev numbers for the orbifold curves and Gabrielov numbers for the cusp singularities with group action. We show that these numbers are the same and that the stringy Euler number of the orbifold curve coincides with the $G^T$-equivariant Milnor number of the mirror cusp singularity.Comment: 29 pages, Table 2 corrected, Assumption g=0 added to Theorem 2

Variance of the exponents of orbifold Landau-Ginzburg models

We prove a formula for the variance of the set of exponents of a non-degenerate weighted homogeneous polynomial with an action of a diagonal subgroup of {\rm SL}_n(\CC).Comment: 17 pages; major revision, gap in the proof of the main result fille