Introduction despite being the focus of a great deal of attention, kontsevichs homological mirror symmetry conjecture kon95 has been fully proved in only a handful of cases. The eufunded hms project will conduct research to highlight new aspects of the relationship between homological mirror symmetry and hodge. This is a writeup of the authors talk in the conference algebraic geometry in east asia 2016 held at the university of tokyo in january 2016. We prove homological mirror symmetry for a smooth ddimensional calabiyau hypersurface in projective space, for any d 2 for example, d 3 is the quintic threefold. A homogeneous degree 5 polynomial equation in 5 variables determinesa quintic 3fold in cp4. The central ideas first appeared in the work of maxim kontsevich 1993. Homological mirror symmetry group mathematical institute. Nick sheridan has worked on proving homological mirror symmetry in various fundamental cases, and on deriving consequences for symplectic topology with smith, and for gromovwitten theory with ganatra and perutz. Generalized homological mirror symmetry and cubics citeseerx. Given a punctured riemann surface with a pairofpants decomposition, we compute itswrapped fukaya category in a suitable model by reconstructing it from those of various pairsof pants. Mirror symmetry states that to every calabiyau manifold \x\ with complex structure and symplectic symplectic structure there is another dual manifold \x\vee \, so that the properties of \x\ associated to the complex structure e. Pdf homological mirror symmetry and algebraic cycles. Lectures on homological mirror symmetry, nick sheridan. For every calabiyau 3manifold xthere exists a mirror partner x with a symplectic form.
Our team simons collaboration on homological mirror symmetry. Homological mirror symmetry ias school of mathematics. In this paper, we propose a way to avoid this problem, and discuss the homological mirror symmetry. Homological mirror symmetry for open riemann surfaces from. Katzarkov has proposed a generalization of kontsevichs mirror symmetry conjecture, covering some varieties of general type. Homological algebra of mirror symmetry springerlink. Kontsevich formulation of homological mirror symmetry. Homological mirror symmetry for curves of higher genus. Tduality and homological mirror symmetry for toric varieties bohan fanga, chiuchu melissa liub. Soibelman 2000, who applied methods of nonarchimedean geometry in particular, tropical curves to homological mirror symmetry. Homological mirror symmetry is a mathematical conjecture made by maxim kontsevich. Author links open overlay panel bohan fang a chiuchu melissa liu b david treumann a eric zaslow a. We consider some classical examples from a new point of. On the homological mirror symmetry conjecture for pairs of pants arxiv.
Kontsevich in his icm talk in zuric h in 1994 ko can be formulated as follows. This question is related to this other one intuition for sduality. Jun 14, 2014 the first in a series of lectures by nick sheridan veblen research instructor at ias, princeton on homological mirror symmetry. Mirror symmetry functor on objects over sectors in the space of dolbeault forms. Pdf homological mirror symmetry for punctured spheres. Homological mirror symmetry for hypersurface cusp singularities, selecta mathematica, 2017, pp. 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 mirror symmetry was originally discovered by physicists. Mirror symmetry 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 zeroes. Smith, homological mirror symmetry for the fourtorus, duke math. Homological mirror symmetry is the area of mathematics concentrated around several categorical equivalences connecting symplectic and holomorphic or algebraic geometry.
Tduality and homological mirror symmetry for toric. Homological mirror symmetry for the pair of pants denis auroux by institute for advanced study. The first in a series of lectures by nick sheridan veblen research instructor at ias, princeton on homological mirror symmetry. Y calabiyauc1 0mirrorpair dbcohx dfy dfx dbcohy cohxcategoryofcoherentsheavesonx complexmd. 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 progress in. Deformation theory, homological algebra, and mirror symmetry kenji fukaya download bok. Homological mirror symmetry for elliptic curves april 25, 20 we prove homological mirror symmetry for elliptic curves. It seeks a systematic mathematical explanation for a phenomenon called mirror symmetry first observed by physicists studying string theory.
It seeks a systematic mathematical explanation for a phenomenon called. An introduction to homological mirror symmetry and the case of. Tduality and homological mirror symmetry for toric varieties. It has been extended to fano cases by considering landauginzburg models as mirrors to fano varieties 14. So okada, homological mirror symmetry of fermat polynomials arxiv. It was proposed by katzarkov as a generalization of origin. Homological mirror symmetry for the quartic surface. Speculations about homological mirror symmetry for affine. Together with abouzaid, ganatra, and iritani, he has introduced a new approach to the gamma conjecture using syz fibrations. Abouzaid, homological mirrror symmetry without corrections. In combination with the subsequent work of mikhalkin on the. We prove the homological mirror conjecture for toric del pezzo surfaces.
Here is a list with references that give complete proofs of homological mirror symmetry on certain types of spaces. Homological geometry and mirror symmetry alexander b. Homological mirror symmetry, the study of dualities of certain quantum field. Manifolds with mirrorsymmetric hodge tables are called geometrical mirrors. Namely, for any calabiyau manifold the hodge diamond is unchanged by a rotation by. The author proves kontsevichs form of the mirror symmetry conjecture for on the symplectic geometry side a quartic surface in \\mathbbc p3\. This is done by applying the methods seidel developed for quartic surfaces to the much easier onedimensional case. We speculate that after certain nonlinear twist the fukaya category becomes equivalent to the category of holonomic modules over a quantized algebra of functions. We prove a version of this conjecture in the simplest example, relating the fukaya category of a genus two curve to the category of landauginzburg branes on a certain singular rational surface. 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. Yukawa couplings and numbers of rational curves on the quintic.
The name comes from the symmetry among hodge numbers. The pieces are glued together in the sense that the restrictions of the wrappedfloer complexes from two adjacent pairs of pants to their. This paper is devoted to homological mirror symmetry conjecture for curves of higher genus. Citeseerx document details isaac councill, lee giles, pradeep teregowda. This thesis is concerned with kontsevichs homological mirror symmetry conjecture. 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. We discuss the possible relationship of homological mirror symmetry with deformation quantization. Deformation theory, homological algebra, and mirror symmetry. The homological mirror symmetry conjecture, as stated by m.
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. Abstractwe discuss an approach to studying fano manifolds based on homological mirror. Homological mirror symmetry and tropical geometry ricardo. The relationship between tropical geometry and mirror symmetry goes back to the work of kontsevich and y. Calabiyau manifolds, mirror symmetry, and ftheory part i duration. Hodge numbers of a nonsingular quintic are known to be. Introduction to homological mirror symmetry springerlink. T1 homological mirror symmetry and algebraic cycles. Download in pdf, epub, and mobi format for read it on your kindle device, pc, phones or tablets. Homological mirror symmetry, deformation quantization and. This talk will provide an introduction to the relevant concepts and illustrate the statement on one simple example.
Homological mirror symmetry for punctured spheres article pdf available in journal of the american mathematical society 264. Im interested in the physical intuition of the langlands program, therefore i need to understand what physicists think about homological mirror symmetry. This was later extended, again by kontsevich 23, to the noncalabiyau setting as well. Nov 05, 2015 simons collaboration on homological mirror symmetry 887 views 1. This paper investigates the structure of the fukayaseidel category for the mirror potentials. We give a survey on the series of papers 16, 2325 where the author and his collaborators daniel pomerleano and kazushi ueda show how stromingeryauzaslow syz transforms can be applied to understand the geometry of kontsevichs homological mirror. In this case, the mirror object is a regular function on an algebraic torus. Volume 229, issue 3, 15 february 2012, pages 18731911. Simons collaboration on homological mirror symmetry u.
As an example of work done in the reading groups, participants made notable progress on the geometric understanding of gammaintegral structures this is still work in. Homological mirror symmetry for the genus two curve. In algebraic geometry and theoretical physics, mirror symmetry is a relationship between geometric objects called calabiyau manifolds. On the homological mirror symmetry conjecture for pairs of.
Please contact us for feedback and comments about this page. Mirror symmetry translates the dimension number of the p, qth differential form h p,q for the original manifold into h np,q of that for the counter pair manifold. N2 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. From a class of calabiyau dg algebras to frobenius manifolds via primitive forms takahashi, atsushi, 2019. Picardfuchs equation and canonical coordinates for the quintic mirror family. This lecture was given on november 4, 20, and the video can be. The present volume, intended to be a monograph, covers mirror symmetry from the homological and torus. Methods and structures november 711, 2016 agenda all talks will take place in wolfensohn hall monday, 117. In contrast, homological mirror symmetry for these is comparatively well understoodsee for instance 5, 15. Homological mirror symmetry, hodge theory, and symplectic.