Let $X$ be a Nakajima quiver variety and $X'$ its $3d$-mirror. We consider the action of the Picard torus $\mathsf{K}=\mathrm{Pic}(X)\otimes \mathbb{C}^{\times}$ on $X'$. Assuming that $(X')^{\mathsf{K}}$ is finite, we propose a formula for the $\mathsf{K}$-character of the tangent spaces at the fixed points in terms of certain enumerative invariants of $X$ known as vertex functions.
We propose a spectral curve describing torus knots and links in the B-model. In particular, the application of the topological recursion to this curve generates all their colored HOMFLY invariants. The curve is obtained by exploiting the full Sl(2, Z) symmetry of the spectral curve of the resolved conifold, and should be regarded as the mirror of the topological D-brane associated to torus knots in the large N Gopakumar-Vafa duality. Moreover, we derive the curve as the large N limit of the matrix model computing torus knot invariants.
Using only the Fukaya category and the monodromy around large complex structure, we reconstruct the mirror map in the case of a symplectic torus. This realizes an idea described by Paul Seidel.
In this paper we discuss two major conjectures in Mirror Symmetry: Strominger-Yau-Zaslow conjecture about torus fibrations, and the homological mirror conjecture (about an equivalence of the Fukaya category of a Calabi-Yau manifold and the derived category of coherent sheaves on the dual Calabi-Yau manifold). Our point of view on the origin of torus fibrations is based on the standard differential-geometric picture of collapsing Riemannian manifolds as well as analogous considerations for Conformal Field Theories. It seems to give a description of mirror manifolds much more transparent than th…
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 this paper we give a construction of Lagrangian torus fibration for Calabi-Yau hypersurface in toric variety via the method of gradient flow. Using our construction of Lagrangian torus fibration, we are able to prove the symplectic topological version of SYZ mirror conjecture for generic Calabi-Yau hypersurface in toric variety. We will also be able to give precise formulation of SYZ mirror conjecture in general (including singular locus and duality of singular fibres).
We consider regular Calabi-Yau hypersurfaces in $N$-dimensional smooth toric varieties. On such a hypersurface in the neighborhood of the large complex structure limit point we construct a fibration over a sphere $S^{N-1}$ whose generic fibers are tori $T^{N-1}$. Also for certain one-parameter families of such hypersurfaces we show that the monodromy transformation is induced by a translation of the $T^{N-1}$ fibration by a section. Finally we construct a dual fibration and provide some evidence that it describes the mirror family.
We use the quilt formalism of Mau-Wehrheim-Woodward to give a sufficient condition for a finite collection of Lagrangian submanifolds to split-generate the Fukaya category, and deduce homological mirror symmetry for the standard 4-torus. As an application, we study Lagrangian genus two surfaces of Maslov class zero, deriving numerical restrictions on the intersections of such a surface with linear Lagrangian 2-tori in in the 4-torus.
This paper explores a refinement of homological mirror symmetry which relates exact symplectic topology to arithmetic algebraic geometry. We establish a derived equivalence of the Fukaya category of the 2-torus, relative to a basepoint, with the category of perfect complexes of coherent sheaves on the Tate curve over the "formal disc" Spec Z[[q]]. It specializes to a derived equivalence, over Z, of the Fukaya category of the punctured torus with perfect complexes on the curve y^2+xy=x^3 over Spec Z, the central fibre of the Tate curve; and, over the "punctured disc" Spec Z((q)), to an integral…
We construct pseudotoric structures (à la Tyurin) on the two-step flag variety $F\ell_{1, n-1; n}$, and explain a general relation between pseudotoric structures and special Lagrangian torus fibrations, the latter of which are important in the study of SYZ mirror symmetry. As an application, we speculate how our constructions can explain the number of terms in the superpotential of Rietsch's Landau-Ginzburg mirror.
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 obtain A- and B- series of flag manifolds for T-equivariant tangent bundles and equivariant Chern polynomial following the arguments of Lian, Liu, Yau.
We study torus actions on moduli spaces of quivers. First we give a description of the weight spaces of the induced action of the tangent space to a torus-fixed point. Then we focus on actions of tori of rank one and derive an explicit form for the attractors in the Białynicki-Birula decomposition.
Tangent spaces to Schubert varieties of type A were characterized by Lakshmibai and Seshadri. This result was extended to the other classical types by Lakshmibai. We give a uniform characterization of tangent spaces to Schubert varieties in cominuscule G/P. Our results extend beyond cominuscule G/P; they describe the tangent space to any Schubert variety in G/B at a point xB, where x is a cominuscule Weyl group element in the sense of Peterson. Our results also give partial information about the tangent space to any Schubert variety at any point. Our method is to describe the tangent spaces of…
The purpose of this paper is to exhibit a natural construction between complex geometry and symplectic geometry following the idea of mirror symmetry. Suppose we are given a family of pairs of 2-dimensional Kähler tori and stable holomorphic vector bundles on them $(\hat M_{\ep}, E_{\ep})$, \ep \in (0, 1]$, and each has structure of a Lagrangian torus fibration $π:\hat M_{\ep} \to B$ whose fibers are of diameter $O(\ep)$, and let $A_{\ep}$ be a family of hermitian Yang-Mills(HYM) connections on $E_{\ep}$. As $\ep$ goes to zero, $A_{\ep}$ will, modulo possible bubbles, converge to a connection …
In this paper we prove a mirror symmetry conjecture based on the work of Brini-Eynard-Mariño \cite{BEM} and Diaconescu-Shende-Vafa \cite{DSV}. This conjecture relates open Gromov-Witten invariants of the conifold transition of a torus knot to the topological recursion on the B-model spectral curve.
We establish a $\mathbb{Z}[[t_1,\ldots, t_n]]$-linear derived equivalence between the relative Fukaya category of the 2-torus with $n$ distinct marked points and the derived category of perfect complexes on the $n$-Tate curve. Specialising to $t_1= \ldots =t_n=0$ gives a $\mathbb{Z}$-linear derived equivalence between the Fukaya category of the $n$-punctured torus and the derived category of perfect complexes on the standard (Néron) $n$-gon. We prove that this equivalence extends to a $\mathbb{Z}$-linear derived equivalence between the wrapped Fukaya category of the $n$-punctured torus and the…
In this paper we develop a relative version of T-duality in generalized complex geometry which we propose as a manifestation of mirror symmetry. Let M be an n-dimensional smooth real manifold, V a rank n real vector bundle on M, and nabla a flat connection on V. We define the notion of a nabla-semi-flat generalized complex structure on the total space of V. We show that there is an explicit bijective correspondence between nabla-semi-flat generalized complex structures on the total space of V and nabla(dual)-semi-flat generalized complex structures on the total space of the dual of V. Similarl…
Let π: M \ra B be a Lagrangian torus fibration with singularities such that the fibers are of Maslov index zero, and unobstructed. The paper constructs a rigid analytic space M_0^\chk over the Novikov field which is a deformation of the semi-flat complex structure of the dual torus fibration over the smooth locus B_0 of π. Transition functions of M_0^\chk are obtained via A-\infty homomorphisms which capture the wall-crossing phenomenon of moduli spaces of holomorphic disks.
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.
We describe the torus fixed locus of the moduli space of stable sheaves with Hilbert polynomial $4m+1$ on the projective plane. We determine the torus representation of the tangent spaces at the fixed points, which leads to the computation of the Betti and Hodge numbers of the moduli space.
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.
Inspired by the work of Gross on topological Mirror Symmetry we construct candidate Lagrangian torus fibration models for the 105 families of smooth Fano threefolds. We prove, in the case the second Betti number is one, that the total space of each fibration is homeomorphic to the expected Fano threefold, and show that the numerical invariants coincide for all 105. Our construction relies on a notion of toric degeneration for affine manifolds with singularities, and the correspondence we obtain between polytopes and Fano manifolds is compatible with that appearing in the work of Coates-Corti-K…
This is an expository article on the Gross--Siebert approach to mirror symmetry and its interactions with the Strominger--Yau--Zaslow conjecture from a topological perspective.
Let $U$ be an affine log Calabi-Yau variety containing an open algebraic torus. We show that the naive counts of rational curves in $U$ uniquely determine a commutative associative algebra equipped with a compatible multilinear form. This proves a variant of the Frobenius structure conjecture by Gross-Hacking-Keel in mirror symmetry, and the spectrum of this algebra is supposed to give the hypothetical mirror family. Although the statement of our theorem involves only elementary algebraic geometry, our proof employs Berkovich non-archimedean analytic methods. We construct the structure constan…
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.
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.
Let $X_Σ$ be a complete toric variety. The coherent-constructible correspondence $κ$ of \cite{FLTZ} equates $\Perf_T(X_Σ)$ with a subcategory $Sh_{cc}(M_\bR;\LS)$ of constructible sheaves on a vector space $M_\bR.$ The microlocalization equivalence $μ$ of \cite{NZ,N} relates these sheaves to a subcategory $Fuk(T^*M_\bR;\LS)$ of the Fukaya category of the cotangent $T^*M_\bR$. When $X_\Si$ is nonsingular, taking the derived category yields an equivariant version of homological mirror symmetry, $DCoh_T(X_\Si)\cong DFuk(T^*M_\bR;\LS)$, which is an equivalence of triangulated tensor categories. Th…
In this article we construct Lagrangian torus fibrations for general quintic \cy hypersurfaces near the large complex limit and their mirror manifolds using gradient flow method. Then we prove the Strominger-Yau-Zaslow mirror conjecture for this class of \cy manifolds in symplectic category.
We construct a Lagrangian torus fibration on a smooth hypertoric variety and a corresponding SYZ mirror variety using $T$-duality and generating functions of open Gromov-Witten invariants. The variety is singular in general. We construct a resolution using the wall and chamber structure of the SYZ base.
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.
Let $G$ be a simple, simply connected algebraic group over the field of complex numbers. We give a necessary and a sufficient condition for a Schubert variety $X(τ)$ for which all the higher cohomologies $H^{i}(X(τ), E)$ vanish for the restriction $E$ of the tangent bundle of $G/B$ to X(τ)$. We further show that the global sections $H^{0}(X(τ), E)$ is the adjoint representation of $G$ when $G$ is simply laced.
We formulate a conjecture which describes the Fukaya category of an exact Lefschetz fibration defined by a Laurent polynomial in two variables in terms of a pair consisting of a consistent dimer model and a perfect matching on it. We prove this conjecture in some cases, and obtain homological mirror symmetry for quotient stacks of toric del Pezzo surfaces by finite subgroups of the torus as a corollary.
This paper contains an attempt to formulate rigorously and to check predictions in enumerative geometry of curves following from Mirror Symmetry. The main tool is a new notion of stable map. We give an outline of a contsruction of Gromov-Witten invariants for all algebraic projective or closed symplectic manifolds. Mirror Symmetry in the basic example of rational curves on a quintic 3-folds is reduced to certain complicated but explicit identity. The strategy of computations can be described as follows: 1) we reduce counting problems to questions concerning Chern classes on spaces of curves on…
For a particular toric variety, I explore to what extent the SYZ conjecture applied to the orbits of the torus action gives the mirror manifold, in the sense of Batyrev's mirror construction using reflexive polytopes.
These notes are aimed at mathematicians working on topics related to mirror symmetry, but are unfamiliar with the physical origins of this subject. We explain the physical concepts that enable this surprising duality to exist, using the torus as an illustrative example. Then, we develop the basic foundations of conformal field theory so that we can explain how mirror symmetry was first discovered in that context. Along the way we will uncover a deep connection between conformal field theories with (2,2) supersymmetry and Calabi-Yau manifolds. (Based on lectures given during the "Thematic Progr…
We consider effective actions of a compact torus $T^{n-1}$ on an even-dimensional smooth manifold $M^{2n}$ with isolated fixed points. We prove that under certain conditions on weights of tangent representations, the orbit space is a manifold with corners. Given that the action is Hamiltonian, the orbit space is homeomorphic to $S^{n+1} \setminus (U_1 \sqcup \ldots \sqcup U_l)$ where $S^{n+1}$ is the $(n+1)$--sphere and $U_1, \ldots, U_l$ are open domains. We apply the results to regular Hessenberg varieties and manifolds of isospectral Hermitian matrices of staircase form.
Let $G$ be a reductive complex algebraic group, $T$ a maximal torus of $G$, $B$ a Borel subgroup of $G$ containing $T$, $Φ$ the root system of $G$ w.r.t. $T$, $W$ the Weyl group of $Φ$. Denote by $\Fo = G/B$ the flag variety, by $X_w$ the Schubert subvariety of $\Fo$ associated with an element $w\in W$, and by $C_w$ the tangent cone to $X_w$ at the point $p = eB$. Then $C_w$ is a subscheme of the tangent space $T_pX_w\subseteq T_p\Fo$. Suppose $w$, $w'$ are distinct involutions in $W$. Using the so-called Kostant--Kumar polynomials, we show that if every irreducible component of $Φ$ is of type…
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 introduce algebraic structures on the polyvector fields of an algebraic torus that serve to compute multiplicities in tropical and log Gromov-Witten theory while also connecting to the mirror symmetry dual deformation theory of complex structures. Most notably these structures include a tropical quantum field theory and an $L_{\infty}$-structure. The latter is an instance of Getzler's gravity algebra, and the $l_2$-bracket is a restriction of the Schouten-Nijenhuis bracket. We explain the relationship to string topology in the appendix (thanks to Janko Latschev).
Given a ribbon graph $Γ$ with some extra structure, we define, using constructible sheaves, a dg category $CPM(Γ)$ meant to model the Fukaya category of a Riemann surface in the cell of Teichmüller space described by $Γ.$ When $Γ$ is appropriately decorated and admits a combinatorial "torus fibration with section," we construct from $Γ$ a one-dimensional algebraic stack $\widetilde{X}_Γ$ with toric components. We prove that our model is equivalent to $Perf(\widetilde{X}_Γ)$, the dg category of perfect complexes on $\widetilde{X}_Γ$.
We use Lagrangian torus fibrations on the mirror $X$ of a toric Calabi-Yau threefold $\check X$ to construct Lagrangian sections and various Lagrangian spheres on $X$. We then propose an explicit correspondence between the sections and line bundles on $\check X$ and between spheres and sheaves supported on the toric divisors of $\check X$. We conjecture that these correspondences induce an embedding of the relevant derived Fukaya category of $X$ inside the derived category of coherent sheaves on $\check X$.
This survey was written for the Current Developments in Mathematics conference, 2012, and is an updating of my article "The Strominger-Yau-Zaslow conjecture: From torus fibrations to degenerations," in the Seattle 2005 proceedings. We trace progress and thinking about the SYZ conjecture since its introduction in 1996. We begin with the original differential geometric conjecture and its refinements, and explain how it led to the algebro-geometric program developed by myself and Siebert. After explaining the overall philosophy, I explain how recent results fit into this program.
A new proof of the mirror conjecture for Fano and Calabi-Yau complete intersections in P^n is given, using only the circle action on the graph space. The proof applies to projective bundles as well, with applications to "linear" relative Calabi-Yau's and to Schubert calculus.
Let $G$ be a complex reductive group, $T$ be a maximal torus of $G$, $B$ be a Borel subgroup of $G$ containing $T$, $W$ be the Weyl group of $G$ with respect to $T$. To each element $w$ of $W$ one can associate the Schubert subvariety $X_w$ of the flag variety $G/B$, the tangent cone to $X_w$ at the identity point $p$ considered as a subcheme of the tangent space $T_p(G/B)$, and the reduced tangent cone to $X_w$ at $p$ considered as a subvariety of $T_p(G/B)$. Let $w_1$, $w_2$ be distinct involutions in $W$. We prove that if $G$ is of type $B_n$ or $C_n$, then the tangent cones corresponding t…
In the previous paper, the author defined equivariant Floer cohomology for a complete intersection in a toric variety and showed that it is isomorphic to the small quantum D-module after a mirror transformation when the first Chern class c_1(M) of the tangent bundle is nef. In this paper, even when c_1(M) is not nef, we show that the equivariant Floer cohomology reconstructs the big quantum D-module under certain conditions on the ambient toric variety. The proof is based on a mirror theorem by Coates and Givental. The reconstruction procedure here gives a generalized mirror transformation fir…
We first describe a canonical mirror partner (B-model) of the small quantum orbifold cohomology of weighted projective spaces (A-model) in the framework of differential equations: we attach to the A-model (resp. B-model) a D-module on the torus and we show that these two D-modules are isomorphic. This makes the A and B-models mirror partners and yields, in this situation, an explicit and finer version of a recent result of Iritani. Then we study, using the theory of the Kashiwara-Malgrange filtration, their degenerations at the origin and we apply our results to the construction of (classical,…
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 o…
It is known that Lagrangian torus fibers of the moment map of a toric Fano manifold $X$, equipped with flat $U(1)$-connections, are mirror to matrix factorizations of the mirror superpotential $W:\check{X}\rightarrow\bC$. Via SYZ mirror transformations, we describe how this correspondence, when $X$ is $\bP^1$ or $\bP^2$, can be explained in a geometric way.
This paper focuses on a topological version on the Strominger-Yau-Zaslow mirror symmetry conjecture. Roughly put, the SYZ conjecture suggests that mirror pairs of Calabi-Yau manifolds are related by the existence of dual special Lagrangian torus fibrations. We explore this conjecture without reference to the special Lagrangian condition. In this setting, natural questions include: does there exist a nice class of T^3-fibrations for which the dual fibration can be constructed? Do such fibrations exist on a manifold such as the quintic threefold? If so, is the dual fibration the mirror? We answe…
Let $(M,ω_M)$ be a monotone or negatively monotone symplectic manifold, or a Weinstein manifold. One can construct an "action" of $H^1(M,\mathbb{G}_m)$ on the Fukaya category (wrapped Fukaya category in the exact case) that reflects the action of $Symp^0(M,ω_M)$ on the set of Lagrangian branes. A priori this action is only analytic. The purpose of this work is to show the algebraicity of this action under some assumptions. We use this to prove a tameness result for the sheaf of Lagrangian Floer homology groups obtained by moving one of the Lagrangians via global symplectic isotopies. We also s…
We study a cylindrical Lagrangian cobordism group for Lagrangian torus fibres in symplectic manifolds which are the total spaces of smooth Lagrangian torus fibrations. We use ideas from family Floer theory and tropical geometry to obtain both obstructions to and constructions of cobordisms; in particular, we give examples of symplectic tori in which the cobordism group has no non-trivial cobordism relations between pairwise distinct fibres, and ones in which the degree zero fibre cobordism group is a divisible group. The results are independent of but motivated by mirror symmetry, and a relati…
