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.
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.
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 …
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.
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.
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.
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.
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 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…
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.
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.
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.
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…
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.
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.
We describe recent progress in a program to understand the classification of three-dimensional Fano varieties with $\mathbb{Q}$-factorial terminal singularities using mirror symmetry. As part of this we give an improved and more conceptual understanding of Laurent inversion, a technique that sometimes allows one to construct a Fano variety $X$ directly from a Laurent polynomial $f$ that corresponds to it under mirror symmetry.
We describe a notion of lattice polarization for rational elliptic surfaces and weak del Pezzo surfaces, and describe the complex moduli of the former and the Kähler cone of the latter. We then propose a version of mirror symmetry relating these two objects, which should be thought of as a form of Fano-LG correspondence. Finally, we relate this notion to other forms of mirror symmetry, including Dolgachev-Nikulin-Pinkham mirror symmetry for lattice polarized K3 surfaces and the Gross-Siebert program.
In this paper, we will describe the mathematical foundation of topological Landau-Ginzburg (LG) models coupled to gravity at genus 0 in terms of primitive forms. We also discuss the mirror symmetry for Calabi-Yau manifolds and CP^1 in our context. We will show that the mirror partner of CP^1 is the theory of primitive form associated to f=z+qz^{-1}.
The goal of this paper is to make the vertex operator algebra approach to mirror symmetry accessible to algebraic geometers. Compared to better-known approaches using moduli spaces of stable maps and special Lagrangian fibrations, this approach follows more closely the original line of thinking that lead to the discovery of mirror symmetry by physicists. The ultimate goal of the vertex algebra approach is to give precise mathematical definitions of N=(2,2) superconformal field theories called A and B models associated to any Calabi-Yau variety and then show that thus constructed theories are r…
This is the extended abstract of the talk given at the workshop "Geometry and physics of Higgs bundles", held at the Mathematisches Forschungsinstitut Oberwolfach in May 2019.
Given X a K3 surface, a mirror dual to X can be identified with a component of the moduli space of semistable sheaves on X. We consider fibrations by K3 surfaces over a one dimensional base that are Calabi-Yau and we charaterize the dual fibration that turns to be derived equivalent to the original one relating the problem to mirror symmetry.
Studying the mirror symmetry of a Calabi-Yau threefold $X$ of the Reye congruence in $\mP^4$, we conjecture that $X$ has a non-trivial Fourier-Mukai partner $Y$. We construct $Y$ as the double cover of a determinantal quintic in $\mP^4$ branched over a curve. We also calculate BPS numbers of both $X$ and $Y$ (and also a related Calabi-Yau complete intersection $\tilde X_0$) using mirror symmetry.
We study aspects related to Kontsevich's homological mirror symmetry conjecture in the case of Calabi-Yau complete intersections in toric varieties. In a 1996 lecture at Rutgers University, Kontsevich indicated how his proposal implies that the groups of automorphisms of the two types of categories involved in the homological mirror symmetry conjecture should also be identified. Our main results provide an explicit geometric construction of the correspondence between the automorphisms of the two types of categories.
In this note we illustrate the Fanosearch programme of Coates, Corti, Galkin, Golyshev, and Kasprzyk in the example of the anticanonical cone over the smooth del Pezzo surface of degree 6.
We show that there is an extra dimension to the mirror duality discovered in the early nineties by Greene-Plesser and Berglund-Hübsch. Their duality matches cohomology classes of two Calabi--Yau orbifolds. When both orbifolds are equipped with an automorphism $s$ of the same order, our mirror duality involves the weight of the action of $s^*$ on cohomology. In particular, it matches the respective $s$-fixed loci, which are not Calabi-Yau in general. When applied to K3 surfaces with non-symplectic automorphism $s$ of odd prime order, this provides a proof that Berglund-Hübsch mirror symmetry im…
We modify Gross's construction of mirror symmetry for $\mathbb{P}^2$ by introducing a descendent tropical Landau-Ginzburg potential. The period integrals of this potential compute a modification of Givental's J-function, explicitly encoding a larger sector of the big phase space. As a byproduct of this construction, new tropical methods for computing certain descendent Gromov-Witten invariants are defined.
In this paper, we first prove the parabolic Beauvile-Narasimhan-Ramanan correspondence over an arbitrary field which generalizes the corresponding results over algebraically closed fields in [SWW22]. We use the correspondence and the p-adic integration methods developed by Groechenig- Wyss-Ziegler [GWZ20b] to prove the topological mirror symmetry for parabolic Hitchin systems on curves with arbitrary parabolic structures.
In this paper we complete the proof began by A. Polishchuk and E. Zaslow (math.AG/9801119) of a weak version of Kontsevich's homological mirror symmetry conjecture for elliptic curves. The main difference to the work of Polishchuk and Zaslow is that we consider morphisms between any pair of objects, not only in the transversal case. This enables us to show the conjectured equivalence of categories.
We give a survey on results related to the Berglund-Hübsch duality of invertible polynomials and the homological mirror symmetry conjecture for singularities.
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 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.
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.
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.
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.
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.
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.
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.
Motivated by recent works on localizations in A-twisted gauged linear sigma models, we discuss a generalization of toric residue mirror symmetry to complete intersections in Grassmannians.
We give an expository discussion of recent work using Berglund-Huebsch-Krawitz mirror symmetry to describe the structure of point counts on algebraic varieties over finite fields.
Gross and Siebert identified a class of singular Lagrangian torus fibrations which arise when smoothing toroidal degenerations, and which come in pairs that are related by mirror symmetry. We identify an immersed Lagrangian in each of these local models which supports a moduli space of objects that is isomorphic to the mirror space, and prove a homological mirror statement along the way.
We say a mirror pair of Calabi-Yau varieties exhibits strong arithmetic mirror symmetry if the number of points on each variety over a finite field is equivalent, modulo the order of that field. We search for strong mirror symmetry in pencils of toric hypersurfaces generated using polar dual pairs of reflexive polytopes. We characterize the pencils of elliptic curves where strong arithmetic mirror symmetry arises, and provide experimental evidence that the phenomenon generalizes to higher dimensions. We also provide experimental evidence that pencils of K3 surfaces with the same Picard-Fuchs e…
We give an example of the recent proposed mirror construction of Strominger, Yau and Zaslow in ``Mirror Symmetry is T-duality,'' hep-th/9606040. The paper first considers mirror symmetry for K3 surfaces in light of this construction. We then consider the example of mirror symmetry for Calabi-Yau threefolds of the type considered by Voisin and Borcea, of the form SxE/involution where S is a K3 surface with involution, and E is an elliptic curve. We show how dualizing a family of special Lagrangian real 3-tori does actually produce the mirrors in these examples.
One of the attractions of homological mirror symmetry is that it not only implies the previous predictions of mirror symmetry (e.g., curve counts on the quintic), but it should in some sense be `less of a coincidence' than they are and therefore easier to prove. In this survey we explain how Seidel's approach to mirror symmetry via versality at the large volume/large complex structure limit makes this idea precise.
In this paper, we partially extend recent results of Wan concerning the relationship between the zeta functions of a Calabi-Yau hypersurface and its (singular) mirror variety.
This work establishes a subtle connection between mirror symmetry for Calabi-Yau threefolds and that of curves of higher genus. The linking structure is what we call a perverse curve. We show how to obtain such from Calabi-Yau threefolds in the Batyrev mirror construction and prove that their Hodge diamonds are related by the mirror duality.
We prove Kontsevich's homological mirror symmetry conjecture for certain mirror pairs arising from Batyrev-Borisov's `dual reflexive Gorenstein cones' construction. In particular we prove HMS for all Greene-Plesser mirror pairs (i.e., Calabi-Yau hypersurfaces in quotients of weighted projective spaces). We also prove it for certain mirror Calabi-Yau complete intersections arising from Borisov's construction via dual nef partitions, and also for certain Calabi-Yau complete intersections which do not have a Calabi-Yau mirror, but instead are mirror to a Calabi-Yau subcategory of the derived cate…
We show that counting functions of covers of $\mathbb{C}^\times$ are equal to sums of integrals associated to certain `Feynman' graphs. This is an analogue of the mirror symmetry for elliptic curves by Dijkgraaf.
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.
We introduce the notion of a tropical coamoeba which gives a combinatorial description of the Fukaya category of the mirror of a toric Fano stack. We show that the polyhedral decomposition of a real n-torus into (n + 1) permutohedra gives a tropical coamoeba for the mirror of the projective space, and prove a torus-equivariant version of homological mirror symmetry for the projective space. As a corollary, we obtain homological mirror symmetry for toric orbifolds of the projective space.
In the case of toric varieties, we continue the pursuit of Kontsevich's fundamental insight, Homological Mirror Symmetry, by unifying it with the Mori program. We give a refined conjectural version of Homological Mirror Symmetry relating semi-orthogonal decompositions of the $B$-model on toric varieties to semi-orthogonal decompositions on the $A$-model on the mirror Landau-Ginzburg models. As evidence, we prove a new case of Homological Mirror Symmetry for a toric surface whose anticanonical bundle is not nef, namely a certain blow-up of $¶^2$ at three infinitesimally near points.
Homological mirror symmetry for crepant resolutions of Gorenstein toric singularities leads to a pair of conjectures on certain hypergeometric systems of PDEs. We explain these conjectures and verify them in some cases.
Ideas of Fukaya and Kontsevich-Soibelman suggest that one can use Strominger-Yau-Zaslow's geometric approach to mirror symmetry as a torus duality to construct the mirror of a symplectic manifold equipped with a Lagrangian torus fibration as a moduli space of simple objects of the Fukaya category supported on the fibres. In the absence of singular fibres, the construction of the mirror is explained in this framework, and, given a Lagrangian submanifold, a (twisted) coherent sheaf on the mirror is constructed.
We prove homological mirror symmetry for Lefschetz fibrations obtained as disconnected sums of polynomials of types A or D. The proof is based on the behavior of the Fukaya category under the addition of a polynomial of type D.
The Cox construction presents a toric variety as a quotient of affine space by a torus. The category of coherent sheaves on the corresponding stack thus has an evident description as invariants in a quotient of the category of modules over a polynomial ring. Here we give the mirror to this description, and in particular, a clean new proof of mirror symmetry for smooth toric stacks.
