Showing papers on "Mirror symmetry published in 2009"
TL;DR: In this article, the effects of all D-instantons in Type II string compactifications on Calabi-Yau three-folds were derived by recasting the previously known A-type D2-instanton corrections in the language of contact geometry, covariantizing the result under electromagnetic duality, and using mirror symmetry.
Abstract: Finding the exact, quantum corrected metric on the hypermultiplet moduli space in Type II string compactifications on Calabi-Yau threefolds still poses an outstanding open problem. We address this issue by relating the quaternionic-Kahler metric on the hypermultiplet moduli space to the complex contact geometry on its twistor space. In this framework, Euclidean D-brane instanton contributions are captured by contact transformations between different locally flat patches. We derive those by recasting the previously known A-type D2-instanton corrections in the language of contact geometry, covariantizing the result under electro-magnetic duality, and using mirror symmetry. As a result, we are able to express the effects of all D-instantons in Type II compactifications concisely as a sum of dilogarithm functions. We conclude with some comments on the relation to microscopic degeneracies of four-dimensional BPS black holes and to the wall-crossing formula of Kontsevich and Soibelman, and on the form of the yet unknown NS5-brane instanton contributions.
306 citations
TL;DR: In this article, an integral structure in orbifold quantum cohomology associated to the K-group and the Γ ˆ-class was introduced. But the integral structure was not shown to match with the natural integral structure for the Landau-Ginzburg model under mirror symmetry.
296 citations
TL;DR: In this paper, the authors developed a mathematical theory of the topological vertex of Calabi-Yau three-manifolds, a theory that was originally proposed by Aganagic, A-Klemm, M-Marino and C-Vafa.
Abstract: We have developed a mathematical theory of the topological vertex—a theory that was originally proposed by M Aganagic, A Klemm, M Marino and C Vafa on effectively computing Gromov–Witten invariants of smooth toric Calabi–Yau threefolds derived from duality between open string theory of smooth Calabi–Yau threefolds and Chern–Simons theory on three-manifolds.
179 citations
Book•
04 Nov 2009
TL;DR: In this article, the authors present the new ideas coming out of the interactions of string theory and algebraic geometry in a coherent logical context, including the Strominger-Yau-Zaslow conjecture, the theory of stability conditions on triangulated categories, and a physical basis for the McKay correspondence.
Abstract: Research in string theory over the last several decades has yielded a rich interaction with algebraic geometry. In 1985, the introduction of Calabi-Yau manifolds into physics as a way to compactify ten-dimensional space-time has led to exciting cross-fertilization between physics and mathematics, especially with the discovery of mirror symmetry in 1989. A new string revolution in the mid-1990s brought the notion of branes to the forefront. As foreseen by Kontsevich, these turned out to have mathematical counterparts in the derived category of coherent sheaves on an algebraic variety and the Fukaya category of a symplectic manifold. This has led to exciting new work, including the Strominger-Yau-Zaslow conjecture, which used the theory of branes to propose a geometric basis for mirror symmetry, the theory of stability conditions on triangulated categories, and a physical basis for the McKay correspondence. These developments have led to a great deal of new mathematical work. One difficulty in understanding all aspects of this work is that it requires being able to speak two different languages, the language of string theory and the language of algebraic geometry. The 2002 Clay School on Geometry and String Theory set out to bridge this gap, and this monograph builds on the expository lectures given there to provide an up-to-date discussion including subsequent developments. A natural sequel to the first Clay monograph on Mirror Symmetry, it presents the new ideas coming out of the interactions of string theory and algebraic geometry in a coherent logical context. We hope it will allow students and researchers who are familiar with the language of one of the two fields to gain acquaintance with the language of the other. The book first introduces the notion of Dirichlet brane in the context of topological quantum field theories, and then reviews the basics of string theory. After showing how notions of branes arose in string theory, it turns to an introduction to the algebraic geometry, sheaf theory, and homological algebra needed to define and work with derived categories. The physical existence conditions for branes are then discussed and compared in the context of mirror symmetry, culminating in Bridgeland's definition of stability structures, and its applications to the McKay correspondence and quantum geometry. The book continues with detailed treatments of the Strominger-Yau-Zaslow conjecture, Calabi-Yau metrics and homological mirror symmetry, and discusses more recent physical developments. This book is suitable for graduate students and researchers with either a physics or mathematics background, who are interested in the interface between string theory and algebraic geometry.
150 citations
Journal Article•
TL;DR: The authors investigated whether vertical mirror symmetry acts as a cue in figure-ground segregation and found that symmetric shapes were easier to detect than asymmetric ones across the different noise levels, indicating that vertical mirror symmetry is indeed used as cue in perceptual grouping.
Abstract: The goal of our study is a better understanding of the role of vertical mirror symmetry in perceptual grouping. With a simple psychophysical task and a set of controlled stimuli, we investigated whether vertical mirror symmetry acts as a cue in figure-ground segregation. We asked participants to indicate which of two sequentially presented Gabor arrays contained a visual shape. The shape was defined by a subset of Gabor elements positioned along the outline of an unfamiliar shape. By adding orientation noise to these Gabor elements, the shape percept became less salient. Across the different noise levels, symmetric shapes were easier to detect than asymmetric ones. This finding indicates that vertical mirror symmetry is indeed used as a cue in perceptual grouping.
132 citations
TL;DR: In this article, the dependence of genus zero half-twisted correlators on linear sigma models with a (2, 2) locus was studied and the results of the dependence on the correlators and parameters were shown.
Abstract: We study half-twisted linear sigma models relevant to (0,2) compactifications of the heterotic string. Focusing on theories with a (2,2) locus, we examine the linear model parameter space and the dependence of genus zero half-twisted correlators on these parameters. We show that in a class of theories the correlators and parameters separate into A and B types, present techniques to compute the dependence, and apply these to some examples. These results should bear on the mathematics of (0,2) mirror symmetry and the physics of the moduli space and Yukawa couplings in heterotic compactifications.
82 citations
TL;DR: In this paper, the foundations of homological mirror symmetry for manifolds of general type are outlined and both physics and Categorical prospectives are considered, and the results of the analysis are presented.
Abstract: In this paper we outline the foundations of Homological Mirror Symmetry for manifolds of general type. Both Physics and Categorical prospectives are considered.
75 citations
Posted Content•
TL;DR: In this article, it was shown that certain superpotential and Kahler potential couplings of N = 1 supersymmetric compactifications with branes or bundles can be computed from Hodge theory and mirror symmetry.
Abstract: We show that certain superpotential and Kahler potential couplings of N=1 supersymmetric compactifications with branes or bundles can be computed from Hodge theory and mirror symmetry. This applies to F-theory on a Calabi-Yau four-fold and three-fold compactifications of type II and heterotic strings with branes. The heterotic case includes a class of bundles on elliptic manifolds constructed by Friedmann, Morgan and Witten. Mirror symmetry of the four-fold computes non-perturbative corrections to mirror symmetry on the three-folds, including D-instanton corrections. We also propose a physical interpretation for the observation by Warner that relates the deformation spaces of certain matrix factorizations and the periods of non-compact 4-folds that are ALE fibrations.
72 citations
TL;DR: In this paper, the authors studied the behavior of Ricci-flat Kahler metrics on a projective Calabi-Yau manifold when the Kahler classes degenerate to the boundary of the ample cone.
Abstract: We study the behavior of families of Ricci-flat Kahler metrics on a projective Calabi- Yau manifold when the Kahler classes degenerate to the boundary of the ample cone. We prove that if the limit class is big and nef the Ricci-flat metrics converge smoothly on compact sets outside a subvariety to a limit incomplete Ricci-flat metric. The limit can also be understood from algebraic geometry. Einstein metrics, namely metrics with constant Ricci curvature, have been an important subject of study in the field of differential geometry since the early days. The solution of the Calabi Conjecture given by Yau (Y1) in 1976 provided a very powerful existence theorem for Kahler-Einstein metrics with negative or zero Ricci curvature (the negative case was also done independently by Aubin (Au)). This produced a number of nonho- mogeneous examples of Ricci-flat manifolds. These spaces have been named Calabi-Yau manifolds by the physicists in the eighties, and have been thoroughly studied in several different areas of mathematics and physics. Prompted by the physical intuition of mirror symmetry, mathematicians have studied the ways in which Calabi-Yau manifolds can de- generate when they are moving in families. In general both the complex and symplectic (Kahler) structures are changing, and the behavior is not well understood. In this paper we will consider the case when the complex structure is fixed, and so we will be looking at a single compact projective Calabi-Yau manifold. The Kahler class is then allowed to vary inside the ample cone. As long as the class stays inside the cone, the corresponding Ricci-flat metrics vary smoothly, but they will degenerate when the class approaches the boundary of the cone. We will try to understand this degeneration process and see what the limiting space looks like. To introduce our results, let us fix some notation first. LetX be a compact projective Calabi-Yau manifold, of complex dimensionn. This is by definition a projective manifold
72 citations
Posted Content•
TL;DR: In this paper, a cohomological LG/CY correspondence between the cohomology of finite quotients of Calabi-Yau hypersurfaces inside a weighted projective space and the Fan-Jarvis-Ruan-Witten state space of the associated Landau-Ginzburg singularity theory was proved.
Abstract: We prove the classical mirror symmetry conjecture for the mirror pairs constructed by Berglund, H\"ubsch, and Krawitz. Our main tool is a cohomological LG/CY correspondence which provides a degree-preserving isomorphism between the cohomology of finite quotients of Calabi-Yau hypersurfaces inside a weighted projective space and the Fan-Jarvis-Ruan-Witten state space of the associated Landau-Ginzburg singularity theory.
67 citations
TL;DR: In this article, the authors use toric geometry to study open string mirror symmetry on compact Calabi-Yau manifolds, and derive a canonical hypergeometric system of differential equations, whose solutions determine the open/closed string mirror maps and the partition functions for spheres and discs.
Abstract: We use toric geometry to study open string mirror symmetry on compact Calabi-Yau manifolds. For a mirror pair of toric branes on a mirror pair of toric hypersurfaces we derive a canonical hypergeometric system of differential equations, whose solutions determine the open/closed string mirror maps and the partition functions for spheres and discs. We define a linear sigma model for the brane geometry and describe a correspondence between dual toric polyhedra and toric brane geometries. The method is applied to study examples with obstructed and classically unobstructed brane moduli at various points in the deformation space. Computing the instanton expansion at large volume in the flat coordinates on the open/closed deformation space we obtain predictions for enumerative invariants.
Posted Content•
TL;DR: In this paper, the Bouchard-Mari conjecture for the framed one-legged topological vertex was shown to hold for the local mirror symmetry with one $D$-brane.
Abstract: We prove the Bouchard-Mari\~no Conjecture for the framed one-legged topological vertex by deriving the Eynard-Orantin type recursion relations from the cut-and-join equation satisfied by the relevant triple Hodge integrals. This establishes a version of local mirror symmetry for the local $C^3$ geometry with one $D$-brane.
TL;DR: In this paper, the quilt formalism of Mau-Wehrheim-Woodward was used 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.
Abstract: 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.
TL;DR: Using the twistor approach to hypermultiplet moduli spaces, the authors derived the worldsheet, D(-1), and D1-instanton contributions to the generalized mirror map, relating Type IIA and Type IIB string theory compactified on generic mirror Calabi-Yau threefolds.
Abstract: Using the twistor approach to hypermultiplet moduli spaces, we derive the worldsheet, D(-1), and D1-instanton contributions to the generalized mirror map, relating Type IIA and Type IIB string theory compactified on generic mirror Calabi-Yau threefolds. For this purpose, we provide a novel description of the twistor space underlying the Type IIB hypermultiplet moduli space where the SL(2, )-action is found to be free from quantum corrections. The extent to which instanton effects may resolve the perturbative singularities of the moduli space metric is discussed.
TL;DR: In this article, the authors use toric geometry to study open string mirror symmetry on compact Calabi-Yau manifolds, and derive a canonical hypergeometric system of differential equations, whose solutions determine the open/closed string mirror maps and the partition functions for spheres and discs.
Abstract: We use toric geometry to study open string mirror symmetry on compact Calabi-Yau manifolds. For a mirror pair of toric branes on a mirror pair of toric hypersurfaces we derive a canonical hypergeometric system of differential equations, whose solutions determine the open/closed string mirror maps and the partition functions for spheres and discs. We define a linear sigma model for the brane geometry and describe a correspondence between dual toric polyhedra and toric brane geometries. The method is applied to study examples with obstructed and classically unobstructed brane moduli at various points in the deformation space. Computing the instanton expansion at large volume in the flat coordinates on the open/closed deformation space we obtain predictions for enumerative invariants.
TL;DR: In this paper, the authors considered the suspension operation on Lefschetz fibrations, which takes p(x) to p (x)-y^2 time, and proved part of homological mirror symmetry for the total spaces of canonical bundles over toric del Pezzo surfaces.
Abstract: We consider the suspension operation on Lefschetz fibrations, which takes p(x) to p(x)-y^2. This leaves the Fukaya category of the fibration invariant, and changes the category of the fibre (or more precisely, the subcategory consisting of a basis of vanishing cycles) in a specific way. As an application, we prove part of Homological Mirror Symmetry for the total spaces of canonical bundles over toric del Pezzo surfaces.
TL;DR: In this article, the authors derived the Picard-Fuchs equations satisfied by the superpotential of B-branes for holomorphic maps of worldsheets with low Euler characteristics.
Abstract: This work is concerned with branes and differential equations for oneparameter Calabi–Yau hypersurfaces in weighted projective spaces. For a certain class of B-branes, we derive the inhomogeneous Picard–Fuchs equations satisfied by the brane superpotential. In this way, we arrive at a prediction for the real BPS invariants for holomorphic maps of worldsheets with low Euler characteristics, ending on the mirror A-branes.
CERN1
TL;DR: In this paper, normal functions capturing D-brane superpotentials on several one-and two-parameter Calabi-Yau hypersurfaces and complete intersections in weighted pro-ective space were studied.
Abstract: We study normal functions capturing D-brane superpotentials on several one- and two-parameter Calabi-Yau hypersurfaces and complete intersections in weighted pro- jective space. We calculate in the B-model and interpret the results using mirror symmetry in the large volume regime, albeit without identifying the precise A-model geometry in all cases. We identify new classes of extensions of Picard-Fuchs equations, as well as a novel type of topology changing phase transition involving quantum D-branes. A 4-d domain wall which is obtained in one region of closed string moduli space from wrapping a four-chain interpolating between two Lagrangian submanifolds is, for other values of the parameters, represented by a disk ending on a single Lagrangian.
CERN1
TL;DR: In this article, normal functions capturing D-brane superpotentials on several one-and two-parameter Calabi-Yau hypersurfaces and complete intersections in weighted projective space were studied.
Abstract: We study normal functions capturing D-brane superpotentials on several one- and two-parameter Calabi-Yau hypersurfaces and complete intersections in weighted projective space. We calculate in the B-model and interpret the results using mirror symmetry in the large volume regime, albeit without identifying the precise A-model geometry in all cases. We identify new classes of extensions of Picard-Fuchs equations, as well as a novel type of topology changing phase transition involving quantum D-branes. A 4-d domain wall which is obtained in one region of closed string moduli space from wrapping a four-chain interpolating between two Lagrangian submanifolds is, for other values of the parameters, represented by a disk ending on a single Lagrangian.
Posted Content•
TL;DR: For three-partition triple Hodge integrals related to the topological vertex, the authors derived Eynard-Orantin type recursion relations from the cut-and-join equation and established a version of local mirror symmetry for the local $C^3$ geometry with three D-branes.
Abstract: 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.
Book•
01 Jan 2009
TL;DR: A survey of Calabi-Yau manifolds can be found in this article, where the authors also discuss the existence of Faddeev knots and field homomorphisms.
Abstract: Special Lagrangian fibrations, wall-crossing, and mirror symmetry Denis Auroux Sphere theorems in geometry Simon Brendle and Richard Schoen Geometric Langlands and non-Abelian Hodge theory Ron Donagi and Tony Pantev Developments around positive sectional curvature Karsten Grove Einstein metrics, four-manifolds, and conformally Kahler geometry Claude LeBrun Existence of Faddeev knots Fengbo Hang, Fanghua Lin, and Yisong Yang Milnor K2 and field homomorphisms Fedor Bogomolov and Yuri Tschinkel Arakelov inequalities Eckart Viehweg A survey of Calabi-Yau manifolds Shing-Tung Yau.
TL;DR: In this paper, a new duality of symmetry, geometric duality, was introduced to generate large families of gauge theories, with and without Chern-Simons terms, that all flow to the same conformal field theory in the infrared.
Abstract: We clarify how mirror symmetry acts on 3d theories with = 2,3 or 4 supersymmetries and non-abelian Chern-Simons terms and then construct many new examples. We identify a new duality, geometric duality, that allows us to generate large families of gauge theories, with and without Chern-Simons term, that all flow to the same conformal field theory in the infrared. In particular, we find an interesting duality of dualities: a pair of theories related via mirror symmetry can be mapped, via geometric duality, into a pair of gauge theories related by Seiberg duality. This network of dualities can be understood as the simple result that all of these theories are different realizations of one and the same system in M-theory.
TL;DR: In this article, the authors investigate the cosmological properties of an "induced gravity" brane scenario in the absence of mirror symmetry with respect to the brane and find that brane evolution can proceed along one of four distinct branches.
Abstract: We investigate the cosmological properties of an "induced gravity" brane scenario in the absence of mirror symmetry with respect to the brane. We find that brane evolution can proceed along one of four distinct branches. By contrast, when mirror symmetry is imposed, only two branches exist, one of which represents the self-accelerating brane, while the other is the so-called normal branch. This model incorporates many of the well-known possibilities of brane cosmology including phantom acceleration (w < -1), self-acceleration, transient acceleration, quiescent singularities, and cosmic mimicry. Significantly, the absence of mirror symmetry also provides an interesting way of inducing a sufficiently small cosmological constant on the brane. A small (positive) Lambda-term in this case is induced by a small asymmetry in the values of bulk fundamental constants on the two sides of the brane.
TL;DR: In this article, the authors investigate the cosmological properties of an ''induced gravity'' brane scenario in the absence of mirror symmetry with respect to the brane and find that brane evolution can proceed along one of four distinct branches.
Abstract: We investigate the cosmological properties of an `induced gravity' brane scenario in the absence of mirror symmetry with respect to the brane. We find that brane evolution can proceed along one of four distinct branches. By contrast, when mirror symmetry is imposed, only two branches exist, one of which represents the self-accelerating brane, while the other is the so-called normal branch. This model incorporates many of the well-known possibilities of brane cosmology including phantom acceleration (w < ?1), self-acceleration, transient acceleration, quiescent singularities, and cosmic mimicry. Significantly, the absence of mirror symmetry also provides an interesting way of inducing a sufficiently small cosmological constant on the brane. A small (positive) ?-term in this case is induced by a small asymmetry in the values of bulk fundamental constants on the two sides of the brane.
TL;DR: In this article, the flux superpotential in F-theory compactifications with N = 1 supersymmetry was derived for the Calabi-Yau fourfold model.
Abstract: In four-dimensional F-theory compactifications with N=1 supersymmetry the fields describing the dynamics of space-time filling 7-branes are part of the complex structure moduli space of the internal Calabi-Yau fourfold. We explicitly compute the flux superpotential in F-theory depending on all complex structure moduli, including the 7-brane deformations and the field corresponding to the dilaton-axion. Since fluxes on the 7-branes induce 5-brane charge, a local limit allows to effectively match the F-theory results to a D5-brane in a non-compact Calabi-Yau threefold with threeform fluxes. We analyze the classical and instanton contributions to the F-theory superpotential using mirror symmetry for Calabi-Yau fourfolds. The F-theory compactifications under consideration also admit heterotic dual descriptions and we discuss the identification of the moduli in this non-perturbative duality.
TL;DR: Using semiclassics, the origin of the symmetry-induced interference effects that contribute to weak localization corrections and universal conductance fluctuations are identified and a universal dependence on an asymmetry parameter gamma_{asym} is found.
Abstract: We apply the semiclassical theory of transport to quantum dots with exact and approximate spatial symmetries; left-right mirror symmetry, up-down mirror symmetry, inversion symmetry, or fourfold symmetry. In this work—the first of a pair of articles—we consider (a) perfectly symmetric dots and (b) nearly symmetric dots in which the symmetry is broken by the dot's internal dynamics. The second article addresses symmetry-breaking by displacement of the leads. Using semiclassics, we identify the origin of the symmetry-induced interference effects that contribute to weak localization corrections and universal conductance fluctuations. For perfect spatial symmetry, we recover results previously found using the random-matrix theory conjecture. We then go on to show how the results are affected by asymmetries in the dot, magnetic fields, and decoherence. In particular, the symmetry-asymmetry crossover is found to be described by a universal dependence on an asymmetry parameter gamma_asym. However, the form of this parameter is very different depending on how the dot is deformed away from spatial symmetry. Symmetry-induced interference effects are completely destroyed when the dot's boundary is globally deformed by less than an electron wavelength. In contrast, these effects are only reduced by a finite amount when a part of the dot's boundary smaller than a lead-width is deformed an arbitrarily large distance.
TL;DR: In this paper, the authors outline some applications of homological mirror symmetry to classical problems in algebraic geometry, such as rationality of algebraic varieties and the study of algebras.
Abstract: In this chapter we outline some applications of Homological Mirror Symmetry to classical problems in Algebraic Geometry, like rationality of algebraic varieties and the study of algebraic cycles. Several examples are studied in detail.
Book Chapter•
10 Feb 2009TL;DR: In this paper, the authors briefly review various aspects of the SYZ approach to mirror symmetry for non-Calabi-Yau varieties, focusing in particular on Lagrangian fibrations and wall-crossing phenomena in Floer homology.
Abstract: In this survey paper, we briefly review various aspects of the SYZ approach to mirror symmetry for non-Calabi-Yau varieties, focusing in particular on Lagrangian fibrations and wall-crossing phenomena in Floer homology. Various examples are presented, some of them new.
TL;DR: In this article, the authors verify the Landau-Ginzburg Mirror Symmetry Conjecture for Arnol'd's list of unimodal and bimodal quasi-homogeneous singularities with G the maximal diagonal symmetry group.
Abstract: We verify the Landau-Ginzburg Mirror Symmetry Conjecture for Arnol'd's list of unimodal and bimodal quasi-homogeneous singularities with G the maximal diagonal symmetry group, and include a discussion of eight axioms which facilitate the computation of FJRW-rings.
TL;DR: In this paper, the authors analyzed the N = 4 superconformal interactions of the gauge and matter superfields and the spontaneous breakdown of the super-formal symmetry in the D =3, N =4 supersymmetry.
Abstract: The mirror map in the D=3, N=4 supersymmetry connects the left and right SU(2) automorphism groups and also the superfield representations of the corresponding N=4 supermultiplets. The mirror N=4 harmonic superspaces use the harmonics of two SU(2) groups and two types of the Grassmann analyticity. The irreducible left and right N=4 supermultiplets are defined in these harmonic superspaces. We analyze the N=4 superconformal interactions of the gauge and matter superfields and the spontaneous breakdown of the superconformal symmetry. The most interesting superconformal action possesses the mirror symmetry and contains two nonlinear terms of the abelian left and right gauge superfields, and also the mixing N=4 BF interaction which yields the topological masses of the gauge fields and the nontrivial interaction of the scalar and pseudoscalar fields. The minimal interactions of the left and right N=4 hypermultiplets can be included to this abelian gauge theory. We consider also the nonlinear N=4 gauge superfield interactions.