We describe an isomorphism of categories conjectured by Kontsevich. If $M$ and $\widetilde{M}$ are mirror pairs then the conjectural equivalence is between the derived category of coherent sheaves on $M$ and a suitable version of Fukaya's category of Lagrangian submanifolds on $\widetilde{M}.$ We prove this equivalence when $M$ is an elliptic curve and $\widetilde{M}$ is its dual curve, exhibiting the dictionary in detail.
The paper contains a proof that elliptic genus of a Calabi-Yau manifold is a Jacobi form, finds in which dimensions the elliptic genus is determined by the Hodge numbers and shows that elliptic genera of a Calabi-Yau hypersurface in a toric variety and its mirror coincide up to sign. The proof of the mirror property is based on the extension of elliptic genus to Calabi-Yau hypersurfaces in toric varieties with Gorenstein singularities.
The main result of the present paper concerns finiteness properties of Floer theoretic invariants on affine log Calabi-Yau varieties $X$. Namely, we show that: (a) the degree zero symplectic cohomology $SH^0(X)$ is finitely generated and is a filtered deformation of a certain algebra defined combinatorially in terms of a compactifying divisor $\mathbf{D}.$ (b) For any Lagrangian branes $L_0, L_1$, the wrapped Floer groups $WF^*(L_0,L_1)$ are finitely generated modules over $SH^0(X).$ We then describe applications of this result to mirror symmetry, the first of which is an ``automatic generatio…
We find that for many Calabi-Yau threefolds with elliptic or genus one fibrations mirror symmetry factorizes between the fiber and the base of the fibration. In the simplest examples, the generic CY elliptic fibration over any toric base surface $B$ that supports an elliptic Calabi-Yau threefold has a mirror that is an elliptic fibration over a dual toric base surface $\tilde{B}$ that is related through toric geometry to the line bundle $-6K_B$. The Kreuzer-Skarke database includes all these examples and gives a wide range of other more complicated constructions where mirror symmetry also fact…
The elliptic quasimap potential function is explicitly calculated for Calabi-Yau complete intersections in projective spaces by Kim and Lho. We extend this result to local Calabi-Yau varieties. Using this as well as the wall crossing formula by Ciocan-Fontanine and Kim, we can calculate the elliptic Gromov-Witten potential function.
We study mirror symmetry of families of elliptic K3 surfaces with elliptic fibers of type $E_6,~E_7$ and $E_8$. We consider a moduli space $\mathsf{T}$ of the mirror K3 surfaces enhanced with the choice of differential forms. We show that coordinates on $\mathsf{T}$ are given by the ring of quasi modular forms in two variables, with modular groups adapted to the fiber type. We furthermore introduce an algebraic group $\mathsf{G}$ which acts on $\mathsf{T}$ from the right and construct its Lie algebra $\mathrm{Lie}(\mathsf{G})$. We prove that the extended Lie algebra generated by $\mathrm{Lie}(…
We develop the quantum Kodaira-Spencer theory on the elliptic curve and establish the corresponding higher genus B-model. We show that the partition functions of the higher genus B-model on the elliptic curve are almost holomorphic modular forms, which can be identified with partition functions of descendant Gromov-Witten invariants on the mirror elliptic curve. This gives the first compact Calabi-Yau example where mirror symmetry is established at all genera.
In this paper we calculate the elliptic genus of certain complete intersections in products of projective spaces. We show that it is equal to the elliptic genus of the Landau-Ginzburg models that are, according to Hori and Vafa, mirror partners of these complete intersections. This provides additional evidence of the validity of their construction.
For a stopped Liouville manifold arising from a Liouville sector, we construct a symplectic analogue of the formal neighborhood of the stop on the level of Fukaya categories. This geometric construction is performed via Floer-theoretic methods by allowing wrappings in the negative direction. On the other hand, inspired by homological mirror symmetry for pairs, where the mirror is the formal neighborhood of a divisor in an ambient projective variety, there is a different approach by taking a `categorical formal completion' introduced by Efimov. Our main results establishes equivalence of these …
A simple elliptic singularity of type $E_N^{(1,1)}$ ($N=6,7,8$) can be described in terms of a marginal deformation of an invertible polynomial $W$. In the papers \cite{KS} and \cite{MR} the authors proved a mirror symmetry statement for some particular choices of $W$ and used it to prove quasi-modularity of Gromov-Witten invariants for certain elliptic orbifold $\mathbb{P}^1$s. However, the choice of the polynomial $W$ and its marginal deformation $ϕ_μ$ are not unique. In this paper, we investigate the global mirror symmetry phenomenon for the one-parameter family $W+σϕ_μ$. In each case the m…
We prove the Doran-Harder-Thompson conjecture in the case of elliptic curves by using ideas from SYZ mirror symmetry. The conjecture claims that when a Calabi-Yau manifold $X$ degenerates to a union of two quasi-Fano manifolds (Tyurin degeneration), a mirror Calabi-Yau manifold of $X$ can be constructed by gluing the two mirror Landau-Ginzburg models of the quasi-Fano manifolds. The two crucial ideas in our proof are to obtain a complex structure by gluing the underlying affine manifolds and to construct the theta functions from the Landau-Ginzburg superpotentials.
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.
In this paper we outline a setup for Homological Mirror Symmetry for manifolds of general type. Both Physics and Categorical perspectives are considered.
We propose and prove a mirror theorem for the elliptic quasimap invariants for smooth Calabi-Yau complete intersections in projective spaces. The theorem combined with the wall-crossing formula appeared in paper (arXiv:1308.6377) implies mirror theorems of Zinger and Popa for the elliptic Gromov-Witten invariants for those varieties. This paper and the wall-crossing formula provide a unified framework for the mirror theory of rational and elliptic Gromov-Witten invariants.
Fibrewise T-duality (Fourier-Mukai transform) for D-branes on an elliptic Calabi-Yau three-fold $X$ is seen to have an expected adiabatic form for its induced cohomology operation only when an appropriately twisted operation resp. twisted charge is defined. Some differences with the case of $K3$ as well as connections with the spectral cover construction for bundles on $X$ are pointed out. In the context of mirror symmetry Kontsevich's association of line bundle twists (resp. a certain 'diagonal' operation) with monodromies (esp. the conifold monodromy) is made explicit and checked for two exa…
We compute the $g=1, n=1$ B-model Gromov-Witten invariant of an elliptic curve E directly from the derived category D(E). More precisely, we carry out the computation of the categorical Gromov-Witten invariant defined by Costello using as target a cyclic $A_\infty$ model of D(E) described by Polishchuk. This is the first non-trivial computation of a positive genus categorical Gromov-Witten invariant, and the result agrees with the prediction of mirror symmetry: it matches the classical (non-categorical) Gromov-Witten invariants of a symplectic 2-torus computed by Dijkgraaf.
We study categorical primitive forms for Calabi--Yau $A_\infty$ categories with semi-simple Hochschild cohomology. We classify these primitive forms in terms of certain grading operators on the Hochschild homology. We use this result to prove that, if the Fukaya category ${\sf Fuk}(M)$ of a symplectic manifold $M$ has semi-simple Hochschild cohomology, then its genus zero Gromov--Witten invariants may be recovered from the $A_\infty$-category ${\sf Fuk}(M)$ together with the closed-open map. An immediate corollary of this is that in the semi-simple case, homological mirror symmetry implies enu…
We consider a pair of quiver varieties (X;X') related by 3d mirror symmetry, where X =T*Gr(k,n) is the cotangent bundle of the Grassmannian of k-planes of n-dimensional space. We give formulas for the elliptic stable envelopes on both sides. We show an existence of an equivariant elliptic cohomology class on X $\times$ X' (the Mother function) whose restrictions to X and X' are the elliptic stable envelopes of those varieties. This implies, that the restriction matrices of the elliptic stable envelopes for X and X' are equal after transposition and identification of the equivariant parameters …
An algebro-geometric setting for the study of the Painlevé VI equation is introduced. Hamiltonian form of the equation is realized on a twisted relative cotangent bundle to the universal elliptic curve with labelled points of order two. Relations with the theory of elliptic functions and the quantum cohomology of projective plane are discussed.
We prove a tropical mirror symmetry theorem for descendant Gromov-Witten invariants of the elliptic curve, generalizing the tropical mirror symmetry theorem for Hurwitz numbers of the elliptic curve, Theorem 2.20 in [Böhm J., Bringmann K., Buchholz A., Markwig H., J. Reine Angew. Math. 732 (2017), 211-246, arXiv:1309.5893]. For the case of the elliptic curve, the tropical version of mirror symmetry holds on a fine level and easily implies the equality of the generating series of descendant Gromov-Witten invariants of the elliptic curve to Feynman integrals. To prove tropical mirror symmetry fo…
A conjecture expressing genus 1 Gromov-Witten invariants in mirror-theoretic terms of semi-simple Frobenius structures and complex oscillating integrals is formulated. The proof of the conjecture is given for torus-equivariant Gromov - Witten invariants of compact Kähler manifolds with isolated fixed points and for concave bundle spaces over such manifolds. Several results on genus 0 Gromov - Witten theory include: a non-linear Serre duality theorem, its application to the genus 0 mirror conjecture, a mirror theorem for concave bundle spaces over toric manifolds generalizing a recent result of…
We show homological mirror symmetry results relating coherent analytic sheaves on some complex elliptic surfaces and objects of certain Fukaya categories. We first define the notion of a non-algebraic Landau-Ginzburg model on $\mathbb{R} \times S^1$ and its associated Fukaya category, and show that non-Kähler sur-faces obtained by performing two logarithmic transformations to the product of the projective plane and an elliptic curve have non-algebraic Landau-Ginzburg models as their mirror spaces; this class of surface includes the classical Hopf surface $S^1\times S^3$ and other elliptic prim…
Mirror symmetry relates Gromov-Witten invariants of an elliptic curve with certain integrals over Feynman graphs. We prove a tropical generalization of mirror symmetry for elliptic curves, i.e., a statement relating certain labeled Gromov-Witten invariants of a tropical elliptic curve to more refined Feynman integrals. This result easily implies the tropical analogue of the mirror symmetry statement mentioned above and, using the necessary Correspondence Theorem, also the mirror symmetry statement itself. In this way, our tropical generalization leads to an alternative proof of mirror symmetry…
We consider a mirror symmetry of simple elliptic singularities. In particular, we construct isomorphisms of Frobenius manifolds among the one from the Gromov--Witten theory of a weighted projective line, the one from the theory of primitive forms for a universal unfolding of a simple elliptic singularity and the one from the invariant theory for an elliptic Weyl group. As a consequence, we give a geometric interpretation of the Fourier coefficients of an eta product considered by K. Saito.
We apply the homological mirror symmetry for elliptic curves to the study of indefinite theta series. We prove that every such series corresponding to a quadratic form of signature (1,1) can be expressed in terms of theta series associated with split quadratic forms and the usual theta series. We also show that indefinite theta series corresponding to univalued Massey products between line bundles on elliptic curve are modular.
Let $W\in \mathbb{C}[x_1,\cdots,x_N]$ be an invertible polynomial with an isolated singularity at origin, and let $G\subset {\sf SL}_N\cap (\mathbb{C}^*)^N$ be a finite diagonal and special linear symmetry group of $W$. In this paper, we use the category ${\sf MF}_G(W)$ of $G$-equivariant matrix factorizations and its associated VSHS to construct a $G$-equivariant version of Saito's theory of primitive forms. We prove there exists a canonical categorical primitive form of ${\sf MF}_G(W)$ characterized by $G_W^{\sf max}$-equivariance. Conjecturally, this $G$-equivariant Saito theory is equivale…
In this thesis we discuss various classical problems in enumerative geometry. We are focused on ideas and methods which can be used explicitly for practical computations. Our approach is based on studying the limits of elliptic stable envelopes with shifted equivariant or Kahler variables from elliptic cohomology to K-theory. We prove that for a variety X we can obtain K-theoretic stable envelopes for the variety X^G of the G-fixed points of X, where G is a cyclic group acting on X preserving the symplectic form. We formalize the notion of symplectic duality, also known as 3-dimensional mirror…
The construction introduced by Gross, Hacking and Keel allows one to construct a formal mirror family to a pair $(S,D)$ where $S$ is a smooth rational projective surface and $D$ a certain type of Weil divisor supporting an ample or anti-ample class. In that paper they proved two convergence results. Firstly that if the intersection matrix of $D$ is not negative semi-definite then the family they construct lifts to an algebraic family. Secondly they prove that if the intersection matrix is negative definite then their construction lifts along certain analytic strata on the base, and then over a…
This note is a continuation of our paper with E. Zaslow "Categorical mirror symmetry: the elliptic curve", math.AG/9801119. We compare some triple Massey products on elliptic curve with the corresponding Fukaya products on the symplectic torus and recover the classical identity due to Kronecker. We also compute some triple Fukaya products such that the corresponding Massey products are not correctly defined.
Perverse schobers are categorical analogs of perverse sheaves. Examples arise from varieties admitting flops, determined by diagrams of derived categories of coherent sheaves associated to the flop: in this paper we construct mirror partners to such schobers, determined by diagrams of Fukaya categories with stops, for examples in dimensions 2 and 3. Interpreting these schobers as supported on loci in mirror moduli spaces, we prove homological mirror symmetry equivalences between them. Our construction uses the coherent-constructible correspondence and a recent result of Ganatra-Pardon-Shende t…
We consider Takahashi's categorical interpretation of the Berglund-Hubsch mirror symmetry conjecture for invertible polynomials in the case of chain polynomials. Our strategy is based on a stronger claim that the relevant categories satisfy a recursion of directed $A_{\infty}$-categories, which may be of independent interest. We give a full proof of this claim on the B-side. On the A-side we give a detailed sketch of an argument, which falls short of a full proof because of certain missing foundational results in Fukaya-Seidel categories, most notably a generation statement.
We argue how boundary B-type Landau-Ginzburg models based on matrix factorizations can be used to compute exact superpotentials for intersecting D-brane configurations on compact Calabi-Yau spaces. In this paper, we consider the dependence of open-string, boundary changing correlators on bulk moduli. This determines, via mirror symmetry, non-trivial disk instanton corrections in the A-model. As crucial ingredient we propose a differential equation that involves matrix analogs of Saito's higher residue pairings. As example, we compute from this for the elliptic curve certain quantum products m_…
We establish a relationship between mirror symmetry for K3 surfaces and Arnold's strange duality for K3 surfaces. We compute various examples of mirror families. Among them the mirror moduli family for the moduli space of degree 2n polarized K3 surfaces. It turns out to be related to the moduli space of elliptic curves with level n.
6d superconformal field theories (SCFTs) are the SCFTs in the highest possible dimension. They can be geometrically engineered in F-theory by compactifying on non-compact elliptic Calabi-Yau manifolds. In this paper we focus on the class of SCFTs whose base geometry is determined by $-2$ curves intersecting according to ADE Dynkin diagrams and derive the corresponding mirror Calabi-Yau manifold. The mirror geometry is uniquely determined in terms of the mirror curve which has also an interpretation in terms of the Seiberg-Witten curve of the four-dimensional theory arising from torus compactif…
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 under…
In this paper, we will study the connections between the mirror symmetry of K3 surfaces and the geometry of the Legendre family of elliptic curves. We will prove that the mirror map of the Dwork family is equal to the period map of the Legendre family. This result provides an interesting explanation to the modularities of counting functions for K3 surfaces from the mirror symmetry point of view. We will also discuss the relations between the arithmetic geometry of smooth fibers of the Fermat pencil (Dwork family) and that of the smooth fibers of the Legendre family, e.g. Shioda-Inose structure…
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.
The derived category of an algebraic variety might be a source of a myriad of new (categorical) symmetries. Some are predicted by homological mirror symmetry, to be obtained from the fundamental group of the space of complex structures of its mirror partner. These finally lead to differential equations. We expositorily unravel a part of this conjectural master plan for a class of toric varieties.
We use a relative Fourier-Mukai transform on elliptic K3 surfaces $X$ to describe mirror symmetry. The action of this Fourier-Mukai transform on the cohomology ring of $X$ reproduces relative T-duality and provides an infinitesimal isometry of the moduli space of algebraic structures on $X$ which, in view of the triviality of the quantum cohomology of K3 surfaces, can be interpreted as mirror symmetry.
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 study the sunset Feynman integral defined as the scalar two-point self-energy at two-loop order in a two dimensional space-time. We firstly compute the Feynman integral, for arbitrary internal masses, in terms of the regulator of a class in the motivic cohomology of a 1-parameter family of open elliptic curves. Using an Hodge theoretic (B-model) approach, we show that the integral is given by a sum of elliptic dilogarithms evaluated at the divisors determined by the punctures. Secondly we associate to the sunset elliptic curve a local non-compact Calabi-Yau 3-fold, obtained as a limit of el…
This paper studies $3d$ $\mathcal{N}=4$ supersymmetric gauge theories on an elliptic curve, with the aim to provide a physical realisation of recent constructions in equivariant elliptic cohomology of symplectic resolutions. We first study the Berry connection for supersymmetric ground states in the presence of mass parameters and flat connections for flavour symmetries, which results in a natural construction of the equivariant elliptic cohomology variety of the Higgs branch. We then investigate supersymmetric boundary conditions and show from an analysis of boundary 't Hooft anomalies that t…
We study the structure of a modified Fukaya category ${\frak F}(X)$ associated with a K3 surface $X$, and prove that whenever $X$ is an elliptic K3 surface with a section, the derived category of $\fF(X)$ is equivalent to a subcategory of the derived category ${\bold D}(\hat X)$ of coherent sheaves on the mirror K3 surface $\hat X$.
Assuming the standard framework of mirror symmetry, a conjecture is formulated describing how the diffeomorphism group of a Calabi-Yau manifold Y should act by families of Fourier-Mukai transforms over the complex moduli space of the mirror X. The conjecture generalizes a proposal of Kontsevich relating monodromy transformations and self-equivalences. Supporting evidence is given in the case of elliptic curves, lattice-polarized K3 surfaces and Calabi-Yau threefolds. A relation to the global Torelli problem is discussed.
We construct an $A_{\infty}$-structure on the Ext-groups of hermitian holomorphic vector bundles on a compact complex manifold. We propose a generalization of the homological mirror conjecture due to Kontsevich. Namely, we conjecture that for mirror dual Calabi-Yau manifolds $M$ and $X$ there exists an $A_{\infty}$-functor from Fukaya's symplectic $A_{\infty}$-category of $M$ to the $A_{\infty}$-derived category of $X$ which is a homotopy equivalence on morphisms. We verify the part of this conjecture concering triple products for elliptic curves.
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.
We prove that the moduli space of the pseudo holomorphic curves in the A-model on a symplectic torus is homeomorphic to a moduli space of Feynman diagrams in the configuration space of the morphisms in the B-model on the corresponding elliptic curve. These moduli spaces determine the $A_{\infty}$ structure of the both models.
Following the idea of Aganagic--Okounkov \cite{AOelliptic}, we study vertex functions for hypertoric varieties, defined by $K$-theoretic counting of quasimaps from $\mathbb{P}^1$. We prove the 3d mirror symmetry statement that the two sets of $q$-difference equations of a 3d hypertoric mirror pairs are equivalent to each other, with Kähler and equivariant parameters exchanged, and the opposite choice of polarization. Vertex functions of a 3d mirror pair, as solutions to the $q$-difference equations, satisfying particular asymptotic conditions, are related by the elliptic stable envelopes. Vari…
Aganagic and Okounkov proved that the elliptic stable envelope provides the pole cancellation matrix for the enumerative invariants of quiver varieties known as vertex functions. This transforms a basis of a system of $q$-difference equations holomorphic in $\boldsymbol{z}$ with poles in $\boldsymbol{a}$ to a basis of solutions holomorphic in $\boldsymbol{a}$ with poles in $\boldsymbol{z}$. The resulting functions are expected to be the vertex functions of the 3d mirror dual variety. In this paper, we prove that for the cotangent bundle of the full flag variety, the functions obtained in this …
For the Fermat Calabi-Yau threefold and the theory of stability conditions [Bri07], there have been two mathematical aims given by physical reasoning. One is that we should define stability conditions by central charges of quintic periods [Hos04,Kon12,KonSoi13], which extend the Gamma class [KKP,Iri09,Iri11]. The other is that for well-motivated stability conditions on a derived Fukaya-type category, each stable object should be a Lagrangian [ThoYau]. We answer affirmatively to these aims with the simplest homological mirror symmetry (HMS for short) of the Fermat Calabi-Yau threefold [Oka09,Fu…
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…
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…
