Homological mirror symmetry for elliptic curves april 25, 20 we prove homological mirror symmetry for elliptic curves. In this paper we outline the foundations of homological mirror symmetry for manifolds of general type. Mirror symmetry ms was discovered several years ago in string theory as a duality between families of 3dimensional calabiyau manifolds more precisely, complex algebraic manifolds possessing holomorphic volume elements without zeros. Pdf homological mirror symmetry and algebraic cycles. The objects, representing dbranes, are the lagrangian submanifolds in the amodel and the complexes of. Pdf homological mirror symmetry and torus fibrations yan. The simons collaboration on homological mirror symmetry brings together a group of leading mathematicians working towards the goal of proving homological mirror symmetry hms in full generality, and fully exploring its applications mirror symmetry, which emerged in the late 1980s as an unexpected physical duality between quantum field theories, has been a major source of. Aninvitationtohomologicalmirrorsymmetry denis auroux harvard university imsa inaugural conference, miami, september 6, 2019 partially supported by nsf and by the. Each session will be devoted to a fairly broad topic that should be split among two or more responsible speakers.
Simons collaboration on homological mirror symmetry u. Calabiyau manifolds, mirror symmetry, and ftheory part i duration. The homological mirror symmetry conjecture of maxim kontsevich states that the derived category of coherent sheaves on one calabiyau manifold is equivalent in a certain sense to the fukaya category of its mirror. Partially supported by grant inaid for scientific research no 09304008 and no. Seminar homological mirror symmetry heidelberg university. Homological geometry and mirror symmetry alexander b. In algebraic geometry and theoretical physics, mirror symmetry is a relationship between geometric objects called calabiyau manifolds.
Today, i want to survey developments in another generalization, known as heterotic mirror symmetry or 0,2 mirror symmetry. Homological mirror symmetry for calabiyau hypersurfaces. Wintersemester 201920 seminar on homological mirror symmetry talks general idea. Floer homology and mirror symmetry i kenji fukaya 0. Givental and others published homological geometry and mirror symmetry find, read and. Abouzaid, homological mirrror symmetry without corrections.
Kontsevich introduced his homological mirror symmetry. The present volume, intended to be a monograph, covers mirror symmetry from the homological and torus. In particular, homological mirror symmetry hms is a relationship between symplectic and algebraic geometry. This relation is by no means straightforward, and its exploration has been the driving force behind several recent developments in. Homological mirror symmetry for curves of higher genus. We prove a form of homological mirror symmetry for the genus two curve on the symplectic or amodel side. The technical restrictions are largely foundational in nature.
Homological mirror symmetry group mathematical institute. Homological mirror symmetry and topological recursion, miami, january 27february 1, 2020 organizers. Pdf homological mirror symmetry and torus fibrations. Context i our main result is to prove homological mirror symmetry for a smooth hypersurface in cpn 1 of degree n. Homological mirror symmetry for the genus two curve. In 1994, maxim kontsevich 68 made his fundamental homological mirror symmetry conjecture, a profound e. Homological algebra of mirror symmetry maxim kontsevich maxplanckinstitut fu. This relation is by no means straightforward, and its exploration has been the driving force behind several recent developments in algebraic geometry. Heterotic mirror symmetry, and quantum sheaf cohomology. Abstractwe discuss an approach to studying fano manifolds based on homological mirror.
Homological mirror symmetry and torus fibrations request pdf. Yau gave an overview of the results obtained by the various groups in the collaboration aimed at proving and categorizing the syz mirror symmetry, which constructs mirror calabiyau varieties by taking dual torus. Homological mirror symmetry for the genus two curve arxiv. In the onedimensional homological mirror symmetry hms, the amodel on a symplectic torus corresponds to the bmodel on an elliptic curve.
Here is a list with references that give complete proofs of homological mirror symmetry on certain types of spaces. Nov 05, 2015 simons collaboration on homological mirror symmetry 887 views 1. Homological mirror symmetry in dimension one bernd kreu. In terms of homological mirror symmetry formulated by kontsevich kon95, x is called to be a mirror of xif d. Smith, homological mirror symmetry for the fourtorus, duke math. This is done by applying the methods seidel developed for quartic surfaces to the much easier onedimensional case. I for n 5 this is the quintic threefold also considered by noharaueda using my results. In this paper, we propose a way to avoid this problem, and discuss the homological mirror symmetry.
Please contact us for feedback and comments about this page. The first in a series of lectures by nick sheridan veblen research instructor at ias, princeton on homological mirror symmetry. Calabiyau hypersurfaces in projective space outline 1 gromovwitten invariants 2 mirror symmetry 1. Simons collaboration on homological mirror symmetry denis auroux harvard university homological mirror symmetry september 6, 2019 114. I kontsevich introduced his homological mirror symmetry. As one would expect, significant advances were made towards the core conjectures of homological mirror symmetry. It seeks a systematic mathematical explanation for a phenomenon called mirror symmetry first observed by physicists studying string theory. Simons collaboration on homological mirror symmetry 887 views 1. Lectures on homological mirror symmetry, nick sheridan. Homological mirror symmetry ias school of mathematics. This was the second annual meeting of the simons collaboration on homological mirror symmetry.
Orlov after ideas of kontsevich, seidel, hori, vafa. This paper investigates the structure of the fukayaseidel category for the mirror potentials. An introduction to homological mirror symmetry and the. Kontsevich formulation of homological mirror symmetry. I for n 4 this is the quartic k3 reproduces the result of seidel. Zaslow pz of a weak version of kontsevichs homological mirror symmetry. This paper is devoted to homological mirror symmetry conjecture for curves of higher genus. We hope that this volume is a natural sequel to mirror symmetry, 242, written by hori, katz, klemm, pandharipande, thomas, vafa, vakil and zaslow, which was a product of the.
Simons collaboration on homological mirror symmetry 2017. I know basic complex geometry, kahler manifolds, symplectic manifolds in the geometric side and also reading some material for my course on syz conjecture. Mirror symmetry was discovered several years ago in string theory as a dual ity between families of 3dimensional calabiyau manifolds more precisely, com. Tduality and homological mirror symmetry for toric varieties bohan fanga, chiuchu melissa liub. Symplectic geometry, floer homology, mirror symmetry, dbrane. This book is suitable for graduate students and researchers with either a physics or mathematics background, who are interested in the interface between string. Apr 09, 2018 the proposal that its always possible to use a torus fibration to move from one side of the mirror to the other is called the syz conjecture, after its originators. I am taking a reading course in mirror symmetry, which will focus on the syz side. Dirichlet branes and mirror symmetry clay mathematics. Aurouxs notes for a course on mirror symmetry at berkeley. Supersymmetric sigma model on t2 and mirror symmetry 307. Mathematicians explore mirror link between two geometric. Kontsevichs homological mirror symmetry 1994, which relates the fukaya category of a symplectic manifold to the derived category of coherent sheaves of a mirror space, and the stromingeryauzaslow syz conjecture 1996, which gives a geometric underpinning for the construction of mirror spaces.
Furthermore we discover an enumerative meaning of the inverse mirror map for elliptic curve quotients. Discovered accidentally in a computer experiment, such mirror 3folds very soon took their place in various string models of the 10dimensional universe. Pdf remarks on the homological mirror symmetry for tori. Homological mirror symmetry for calabiyau hypersurfaces in. The simons collaboration on homological mirror symmetry brings together a group of leading mathematicians working towards the goal of proving homological mirror symmetry hms in full generality, and fully exploring its applications. The appearance of semiorthgonal decompositions in algebraic and symplectic geometry can, therefore, be viewed as an additional structure one should attach to the homological mirror symmetry conjecture. Homological mirror symmetry for hypersurface cusp singularities ailsa keating 0 0 cambridge, uk we study versions of homological mirror symmetry for hypersurface cusp singularities and the three hypersurface simple elliptic singularities. In 1991, the number of degreed rational curves on the quintic threefold was unknown, for d 4. Manifolds with mirrorsymmetric hodge tables are called geometrical mirrors. Introduction homological mirror symmetry hms relates algebraic and symplectic geometry through their associated categorical structures. Pdf homological mirror symmetry in dimension one semantic.
This expository article is an attempt to illustrate the power of kontsevichs homological mirror. Tduality and homological mirror symmetry for toric varieties. Hodge structures, quantum cohomology, and mirror symmetry. Math 277 topics in differential geometry fall 2009. Mirror symmetry ms was discovered several years ago in string theory as a duality between families of 3dimensional calabiyau manifolds more precisely, complex algebraic manifolds possessing holomorphic. W e close b y out lining the c urrent state of the. Yau gave an overview of the results obtained by the various groups in the collaboration aimed at proving and categorizing the syz mirror symmetry, which constructs mirror calabiyau varieties by taking dual torus fibrations. Pdf homological mirror symmetry for manifolds of general. Emerging developments and applications march 17, 2017 agenda all talks will take place in simonyi hall monday, march. So okada, homological mirror symmetry of fermat polynomials arxiv.
Proving it has become one of the foundational questions in mirror symmetry along with the homological mirror symmetry conjecture, proposed by maxim kontsevich in 1994. Homological mirror symmetry hms is the study of the relations between three types of mathematical objects. The proposal that its always possible to use a torus fibration to move from one side of the mirror to the other is called the syz conjecture, after its originators. Both physics and categorical prospectives are considered. We consider some classical examples from a new point of. Complex geometry is a beautiful and welldeveloped branch in mathematics which behaves much neater than real geometry. This equivalence provides a precise mathematical formulation of mirror symmetry in topological string theory. Math 277 section 3 topics in differential geometry fall 2009 d.
Homological algebra of mirror symmetry springerlink. It was proposed by katzarkov as a generalization of original kontsevich. This thesis is concerned with kontsevichs homological mirror symmetry conjecture. Homological mirror symmetry for hypersurface cusp singularities sel. Purpose and background of the research mirror symmetry is a conjecture proposed in. The book continues with detailed treatments of the stromingeryauzaslow conjecture, calabiyau metrics and homological mirror symmetry, and discusses more recent physical developments. Introduction despite being the focus of a great deal of attention, kontsevichs homo. Homological mirror symmetry, the study of dualities of certain quantum field. Homological mirror symmetry, the study of dualities of certain quantum field theories in a mathematically rigorous form, has developed into a flourishing subject on its own. This lecture was given on november 4, 20, and the video can be.
An exercise in mirror symmetry imperial college london. I for n 3 this is the elliptic curve related to the result of polishchukzaslow. Homological mirror symmetry, the study of dualities of certain quantum field theories in a mathematically rigorous form, has developed into a flourishing. Motivated by the study of homological mirror symmetry, a gauge theory analogue of the collapsing of ricciflat kahler metrics was conjectured by fukaya conjecture 5. Concerning mirror symmetry, batyrev 6 and batyrevborisov 7 gave very general constructions of mirror pairs of calabiyau manifolds occurring as complete intersections in toric varieties. Homological mirror symmetry, the study of dualities of certain quantum field theories in a mathematically rigorous form, has developed into a flourishing subject on its own over the past years. I am interested in learning mirror symmetry, both from the syz and homological point of view. The term refers to a situation where two calabiyau manifolds look very different geometrically but are nevertheless equivalent when employed as extra dimensions of string theory.
As an application, the construction is applied to spheres with three orbifold points to produce their quantumcorrected mirrors and derive homological mirror symmetry. Homological mirror symmetry for fano surfaces denis auroux joint work with l. Homological mirror symmetry is a mathematical conjecture made by maxim kontsevich. Jun 14, 2014 the first in a series of lectures by nick sheridan veblen research instructor at ias, princeton on homological mirror symmetry. The name comes from the symmetry among hodge numbers. Find materials for this course in the pages linked along the left. Calabiyau hypersurfaces in projective space curvecounting on the quintic threefold number of lines on a quintic threefold. The objects, representing dbranes, are the lagrangian submanifolds in the amodel and the complexes of the coherent sheaves in the bmodel. Homological mirror symmetry for fano surfaces uc berkeley math. Pantev this event is partially supported by the simons collaboration on homological mirror symmetry, with the assistance of the university of miami department of mathematics. Deformation theory, homological algebra, and mirror symmetry.
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