Stringy mirror symmetry

Stringy zeta functions for Q-Gorenstein varieties

The stringy Euler number and stringy E-function are interesting invariants of log terminal singularities, introduced by Batyrev. He used them to formulate a topological mirror symmetry test for pairs of certain Calabi-Yau varieties, and to show a version of the McKay correspondence. It is a natural question whether one can extend these invariants beyond the log terminal case. Assuming the Minimal Model Program, we introduce very general stringy invariants, associated to 'almost all' singularities, more precisely to all singularities which are not strictly log canonical. They specialize to the …

The stringy Euler number of Calabi-Yau hypersurfaces in toric varieties and the Mavlyutov duality

We show that minimal models of nondegenerated hypersufaces defined by Laurent polynomials with a $d$-dimensional Newton polytope $Δ$ are Calabi-Yau varieties $X$ if and only if the Fine interior of $Δ$ consists of a single lattice point. We give a combinatorial formula for computing the stringy Euler number of $X$. This formula allows to test mirror symmetry in cases when $Δ$ is not a reflexive polytope. In particular we apply this formula to pairs of lattice polytopes $(Δ, Δ^{\vee})$ that appear in the Mavlyutov's generalization of the polar duality for reflexive polytopes. Some examples of M…

关于弦霍奇数非负性的巴蒂列夫猜想的一个反例

A counter-example to Batyrev's conjecture on the non-negativity of stringy Hodge numbers

该研究针对巴蒂列夫关于弦霍奇数非负性的猜想，借助ChatGPT给出反例，此猜想在多个数学领域意义重大，此次发现为相关研究带来新情况。

Topological mirror symmetry for rank two character varieties of surface groups

The moduli spaces of flat $\mathrm{SL}_2$- and $\mathrm{PGL}_2$-connections are known to be singular SYZ-mirror partners. We establish the equality of Hodge numbers of their intersection (stringy) cohomology. In rank two, this answers a question raised by Tamás Hausel in Remark 3.30 of "Global topology of the Hitchin system".

Stringy Hodge numbers and p-adic Hodge theory

The aim of this paper is to give an application of p-adic Hodge theory to stringy Hodge numbers introduced by V. Batyrev for a mathematical formulation of mirror symmetry. Since the stringy Hodge numbers of an algebraic variety are defined by choosing a resolution of singularities, the well-definedness is not clear from the definition. We give a proof of the well-definedness based on arithmetic results such as p-adic integration and p-adic Hodge theory. Note that another proof of the well-definedness was already obtained by V. Batyrev himself by motivic integration. This is a generalization of…

Chiral rings of vertex algebras of mirror symmetry

We calculate chiral rings of the N=2 vertex algebras constructed from the combinatorial data of toric mirror symmetry and show that they coincide with the description of stringy cohomology conjectured previously in a joint work with A. Mavlyutov.

Some integrality properties in local mirror symmetry

We prove some integrality properties of the open-closed mirror maps, inverse open-closed mirror maps and mirror curves of some local Calabi-Yau geometries.

From primitive form to mirror symmetry

This is a report on the recent work "Mirror symmetry for exceptional unimodular singularities" joint with Changzheng Li, Si Li and Yefeng Shen.

Quantum Mirror Symmetry for Borcea-Voisin Threefolds

Borcea-Voisin threefolds provided some of the first examples of mirror pairs in the Hodge-theoretic sense, but their mirror symmetry at the quantum level have not previously been shown. We prove a Givental-style quantum mirror theorem for certain Borcea-Voisin threefolds: by means of certain birational models, we show that their Gromov-Witten J-functions are related by a mirror map to solutions of the multi-parameter Picard-Fuchs equations coming from the variation of Hodge structures of their mirror partners.

Stability Conditions and Mirror Symmetry of K3 Surfaces in Attractor Backgrounds

We study the space of stability conditions on $K3$ surfaces from the perspective of mirror symmetry. It is done in the so called attractor backgrounds (moduli) which can be far from the conventional large complex limits and are selected by the attractor mechanism for certain black holes. We find certain highly non-generic behaviors of stability walls (a key notion in the study of wall crossings) in the space of stability conditions. They correspond via mirror symmetry to some non-generic behaviors of special Lagrangians in an attractor background. The main results can be understood as a mirror…

Local Mirror Symmetry for the Topological Vertex

For three-partition triple Hodge integrals related to the topological vertex, we derive Eynard-Orantin type recursion relations from the cut-and-join equation. This establishes a version of local mirror symmetry for the local $C^3$ geometry with three D-branes, as proposed by Marino and Bouchard-Klemm-Marino-Pasquetti.

足球的全亏格开镜像对称

All genus open mirror symmetry for footballs

本研究证明了足球的全亏格完全后代开镜像对称，其B模型采用镜像曲线上的Chekhov-Eynard-Orantin拓扑递归，为开镜像对称领域提供了新的理论结果

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-Hübsch 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 $G^T$-…

Quantum Deformation Theory of the Airy Curve and Mirror Symmetry of a Point

We present a quantum deformation theory of the Airy curve and use it to establish a version of mirror symmetry of a point.

三维镜像对称就是镜像对称

3d Mirror Symmetry is Mirror Symmetry

本文将三维镜像对称解释为二维镜像对称的三维类比，类比 SYZ 用一维对偶描述二维镜像对称的思路，揭示其与二维镜像对称的关系。

广义K3曲面的镜像对称性与刚性结构

Mirror symmetry and rigid structures of generalized K3 surfaces

本文针对广义K3曲面的镜像对称性问题，受相关学者工作启发，通过Mukai格极化提出新表述，解决传统表述缺陷及奇异K3曲面镜像对称性问题，并研究其复与凯勒刚性结构。

Mirror Symmetry Constructions

These expository notes are based on lectures by Yongbin Ruan during a special semester on the B-model at the University of Michigan in Winter 2014. They outline and compare the mirror symmetry constructions of Batyrev-Borisov, Hori-Vafa, and Berglund-Hubsch-Krawitz.

三维镜像对称

3-dimensional mirror symmetry

本文综述了三维镜像对称的最新研究进展，并介绍了希格斯分支与库仑分支的数学定义。

Remarks on mirror symmetry of Donaldson-Thomas theory for Calabi-Yau 4-folds

Motivated by Strominger-Yau-Zaslow's mirror symmetry proposal and Kontsevich's homological mirror symmetry conjecture, we study mirror phenomena (in A-model) of certain results from Donaldson-Thomas theory for Calabi-Yau 4-folds.

Mirror symmetry for moduli spaces of Higgs bundles via p-adic integration

We prove the Topological Mirror Symmetry Conjecture by Hausel-Thaddeus for smooth moduli spaces of Higgs bundles of type $\operatorname{SL}_n$ and $\operatorname{PGL}_n$. More precisely, we establish an equality of stringy Hodge numbers for certain pairs of algebraic orbifolds generically fibred into dual abelian varieties. Our proof utilises p-adic integration relative to the fibres, and interprets canonical gerbes present on these moduli spaces as characters on the Hitchin fibres using Tate duality. Furthermore we prove for $d$ coprime to $n$, that the number of rank $n$ Higgs bundles of deg…

SIMIS同调镜像对称课程：导论与应用

Homological Mirror Symmetry Course at SIMIS: Introduction and Applications

该作品整理了作者于SIMIS开设的同调镜像对称课程的相关主题内容。

Mirror symmetry and K3 surfaces

We review some of the interplay between mirror symmetry and K3 surfaces.

Log mirror symmetry and local mirror symmetry

We study Mirror Symmetry of log Calabi-Yau surfaces. On one hand, we consider the number of ``affine lines'' of each degree in the complement of a smooth cubic in the projective plane. On the other hand, we consider coefficients of a certain expansion of a function obtained from the integrals of dxdy/xy over 2-chains whose boundaries lie on B_ϕwhere {B_ϕ} is a family of smooth cubics. Then, for small degrees, they coincide. We discuss the relation between this phenomenon and local mirror symmetry for projective plane in a Calabi-Yau 3-fold by Chiang-Klemm-Yau-Zaslow.

镜像停止的单值作用：关于紧致Calabi-Yau面的单值作用

Monodromy action of mirror stops for toric Calabi-Yau surfaces

本文研究了镜像对称中通过猜测的弦性Kähler模空间的基本群作用于代数簇导出范畴的行动，提出通过Legendrian模空间研究，并在A_n-1奇点上构造了部分包裹的Fukaya范畴的圆柱辫群作用。

The Berglund-Hübsch-Chiodo-Ruan mirror symmetry for K3 surfaces

We prove that the mirror symmetry of Berglund-Hübsch-Chiodo-Ruan, applied to K3 surfaces with a non-symplectic involution, coincides with the mirror symmetry described by Dolgachev and Voisin.

Mirror symmetry for K3 surfaces

For certain K3 surfaces, there are two constructions of mirror symmetry that are very different. The first, known as BHK mirror symmetry, comes from the Landau-Ginzburg model for the K3 surface; the other, known as LPK3 mirror symmetry, is based on a lattice polarization of the K3 surface in the sense of Dolgachev's definition. There is a large class of K3 surfaces for which both versions of mirror symmetry apply. In this class we consider the K3 surfaces admitting a certain purely nonsymplectic automorphism of order 4, 8, or 12, and we complete the proof that these two formulations of mirror …

On a conjecture by N. Takahashi on log mirror symmetry for the projective plane

The relationship between local and relative BPS state counts is explored in light of a conjecture on log mirror symmetry by N. Takahashi.

A note on bimodal singularities and mirror symmetry

We discuss the relation between transposition mirror symmetry of Berlund and Hübsch for bimodal singularities and polar duality of Batyrev for associated toric K3 hypersurfaces. We also show that homological mirror symmetry for singularities implies the geometric construction of Coxeter-Dynkin diagrams of bimodal singularities by Ebeling and Ploog.

Degenerations and mirror contractions of Calabi-Yau complete intersections via Batyrev-Borisov Mirror symmetry

We show that the dual of the Cayley cone, associated to a Minkowski sum decomposition of a reflexive polytope, contains a reflexive polytope admitting a nef-partition. This nef-partition corresponds to a Calabi-Yau complete intersection in a Gorenstein Fano toric variety degenerating to an ample Calabi-Yau hypersurface in another Fano toric variety. Using the Batyrev-Borisov mirror symmetry construction, we found the mirror contraction of a Calabi-Yau complete intersection to the mirror of the ample Calabi-Yau hypersurface.

Mirror Symmetry for Calabi-Yau complete intersections in Fano toric varieties

Generalizing the notions of reflexive polytopes and nef-partitions of Batyrev and Borisov, we propose a mirror symmetry construction for Calabi-Yau complete intersections in Fano toric varieties.

BHK mirror symmetry for K3 surfaces with non-symplectic automorphism

In this paper we consider the class of K3 surfaces defined as hypersurfaces in weighted projective space, and admitting a non-symplectic automorphism of non-prime order, excluding the orders 4, 8, and 12. We show that on these surfaces the Berglund-Hübsch-Krawitz mirror construction and mirror symmetry for lattice polarized K3 surfaces constructed by Dolgachev agree; that is, both versions of mirror symmetry define the same mirror K3 surface.

Mirror Symmetry and Projective Geometry of Fourier-Mukai Partners

This is a survey article on mirror symmetry and Fourier-Mukai partners of Calabi-Yau threefolds with Picard number one based on recent works by the authors [HoTa1,2,3,4]. For completeness, mirror symmetry and Fourier-Mukai partners of K3 surfaces are also discussed.

The SYZ mirror symmetry and the BKMP remodeling conjecture

The Remodeling Conjecture proposed by Bouchard-Klemm-Mariño-Pasquetti (BKMP) relates the A-model open and closed topological string amplitudes (open and closed Gromov-Witten invariants) of a symplectic toric Calabi-Yau 3-fold to Eynard-Orantin invariants of its mirror curve. The Remodeling Conjecture can be viewed as a version of all genus open-closed mirror symmetry. The SYZ conjecture explains mirror symmetry as $T$-duality. After a brief review on SYZ mirror symmetry and mirrors of symplectic toric Calabi-Yau 3-orbifolds, we give a non-technical exposition of our results on the Remodeling C…

Homological mirror symmetry for singularities

We give a survey on results related to the Berglund-Hübsch duality of invertible polynomials and the homological mirror symmetry conjecture for singularities.

Mirror symmetry and Exact Solution of 4D N=2 Gauge Theories I

Using geometric engineering in the context of type II strings, we obtain exact solutions for the moduli space of the Coulomb branch of all N=2 gauge theories in four dimensions involving products of SU gauge groups with arbitrary number of bi-fundamental matter for chosen pairs, as well as an arbitrary number of fundamental matter for each factor. Asymptotic freedom restricts the possibilities to SU groups with bi-fundamental matter chosen according to ADE or affine ADE Dynkin diagrams. Many of the results can be derived in an elementary way using the self-mirror property of K3. We find that i…

An example of Berglund-Hübsch mirror symmetry for a Calabi-Yau complete intersection

We study an example of complete intersection Calabi-Yau threefold due to Libgober and Teitelbaum arXiv:alg-geom/9301001, and verify mirror symmetry at a cohomological level. Direct computations allow us to propose an analogue to the Berglund-Hübsch mirror symmetry setup for this example arXiv:hep-th/9201014. We then follow the approach of Krawitz to propose an explicit mirror map arXiv:0906.0796.

Homological Mirror Symmetry and Simple Elliptic Singularities

We give a full exceptional collection in the triangulated category of singularities in the sense of Orlov for a hypersurface singularity of Fermat type, and discuss its relation with homological mirror symmetry for simple elliptic hypersurface singularities.

String cohomology of Calabi-Yau hypersurfaces via Mirror Symmetry

We propose a construction of string cohomology spaces for Calabi-Yau hypersurfaces that arise in Batyrev's mirror symmetry construction. The spaces are defined explicitly in terms of the corresponding reflexive polyhedra in a mirror-symmetric manner. We draw connections with other approaches to the string cohomology, in particular with the work of Chen and Ruan.

Batyrev 镜像对的同调镜像对称

Homological mirror symmetry for Batyrev mirror pairs

本文证明了环面簇中由 Batyrev 构造的一大类 Calabi--Yau 超曲面镜像对的 Kontsevich 同调镜像对称猜想，且该定理在特征零及几乎所有正特征下成立。

Quantum $K$-theory of toric varieties, level structures, and 3d mirror symmetry

We introduce a new version of 3d mirror symmetry for toric stacks, inspired by a 3d $\mathcal{N} = 2$ abelian mirror symmetry construction in physics. Given some toric data, we introduce the $K$-theoretic $I$-function with effective level structure for the associated toric stack. When a particular stability condition is chosen, it restricts to the $I$-function for the particular toric GIT quotient. The mirror of a toric stack is defined by the Gale dual of the original toric data. We then proved the mirror conjecture that the $I$-functions of a mirror pair coincide, under the mirror map, which…

关于同调镜像对称

About Homological Mirror Symmetry

本文探讨了Kontsevich同调镜像对称猜想的B侧，自成体系地研究了导出范畴、同调代数、概形与Calabi--Yau流形的层理论，并讨论了椭圆曲线和$K3$曲面的最低维情形。

A framework for tropical mirror symmetry

Applying tropical geometry a framework for mirror symmetry, including a mirror construction for Calabi-Yau varieties, was proposed by the author. We discuss the conceptual foundations of this construction based on a natural mirror map identifying deformations and divisors. We show how the construction specializes to that by Batyrev for hypersurfaces and its generalization by Batyrev and Borisov to complete intersections. Based on an explicit example we comment on the implementation in the Macaulay2 package SRdeformations.

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 enumerativ…

Mirror Symmetry, Mirror Map and Applications to Calabi-Yau Hypersurfaces

Mirror Symmetry, Picard-Fuchs equations and instanton corrected Yukawa couplings are discussed within the framework of toric geometry. It allows to establish mirror symmetry of Calabi-Yau spaces for which the mirror manifold had been unavailable in previous constructions. Mirror maps and Yukawa couplings are explicitly given for several examples with two and three moduli.

Rational Curves on Calabi-Yau Threefolds: Verifying Mirror Symmetry Predictions

In this paper, the numbers of rational curves on general complete intersection Calabi-Yau threefolds in complex projective spaces are computed up to degree six. The results are all in agreement with the predictions made from mirror symmetry.

Recent Developments in (0,2) Mirror Symmetry

Mirror symmetry of the type II string has a beautiful generalization to the heterotic string. This generalization, known as (0,2) mirror symmetry, is a field still largely in its infancy. We describe recent developments including the ideas behind quantum sheaf cohomology, the mirror map for deformations of (2,2) mirrors, the construction of mirror pairs from worldsheet duality, as well as an overview of some of the many open questions. The (0,2) mirrors of Hirzebruch surfaces are presented as a new example.

The Gamma Conjecture for Tropical Curves in Local Mirror Symmetry

We relate a coherent sheaf supported on a holomorphic curve with its mirror Langrangian submanifold in local mirror symmetry through a tropical curve by interpreting their central charges using the combinatorial information of the tropical curve, which proves the Gamma conjecture for local mirror symmetry in this specific case. Furthermore, we put this description in the Gross-Siebert model of local mirror symmetry and confirm that the parameters in the Gross-Siebert model are the canonical coordinates in mirror symmetry.

Meet homological mirror symmetry

In this paper, we introduce the interested reader to homological mirror symmetry. After recalling a little background knowledge, we tackle the simplest cases of homological mirror symmetry: curves of genus zero and one. We close by outlining the current state of the field and mentioning what homo- logical mirror symmetry has to say about other aspects of mirror symmetry.

Dimer models and homological mirror symmetry for triangles

We prove a conjecture on the relation between dimer models, coamoebas and vanishing cycles for the mirrors of two-dimensional toric Fano stacks of Picard number one. As a corollary, we obtain a torus-equivariant version of homological mirror symmetry for such stacks.

Mirror Symmetry, D-Branes and Counting Holomorphic Discs

We consider a class of special Lagrangian subspaces of Calabi-Yau manifolds and identify their mirrors, using the recent derivation of mirror symmetry, as certain holomorphic varieties of the mirror geometry. This transforms the counting of holomorphic disc instantons ending on the Lagrangian submanifold to the classical Abel-Jacobi map on the mirror. We recover some results already anticipated as well as obtain some highly non-trivial new predictions.

J functions, non-nef toric varieties and equivariant local mirror symmetry of curves

We develop techniques for computing the equivariant local mirror symmetry of curves, i.e. mirror symmetry for O(k)+O(-2-k) over P^1 for k greater than 0. We also describe related methods for dealing with mirror symmetry of non-nef toric varieties. The basic tools are equivariant I functions and their Birkhoff factorization.

Greene--Plesser 镜像的镜像映射整性与算术同调镜像对称性

Integrality of mirror maps and arithmetic homological mirror symmetry for Greene--Plesser mirrors

本文通过同调镜像对称性的算术精化证明了 Greene--Plesser 镜像对的镜像映射整性猜想，并利用新的通用性论证在所有具有良好约化的特征下推广了同调镜像对称性。

Homological Mirror Symmetry for Toric del Pezzo Surfaces

We prove the homological mirror conjecture for toric del Pezzo surfaces. In this case, the mirror object is a regular function on an algebraic torus. We show that the derived Fukaya category of this mirror coincides with the derived category of coherent sheaves on the original manifold.

Mirror Symmetry For Zeta Functions

In this paper, we study the relation between the zeta function of a Calabi-Yau hypersurface and the zeta function of its mirror. Two types of arithmetic relations are discovered. This motivates us to formulate two general arithmetic mirror conjectures for the zeta functions of a mirror pair of Calabi-Yau manifolds.