Homological mirror symmetry and algebraic cycles springerlink. 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. In this paper we describe a homological mirror symmetry approach to classical problems in algebric geometry rationality questions and the hodge conjecture. The mirror symmetry leads the physicists to do important predictions about the rational curves on the quintic threefold, which were partially proved very late by people from algebraic geometry.
This book collects various perspectives, contributed by both mathematicians and physicists, on the bmodel and its role in mirror symmetry. The contributions in this book are based on lectures that took place during workshops with the following thematic titles. Contents preface xiii goal of the book xiii relation to physics xiv how to read the book xv acknowledgements xvii our hope xviii notation xix chapter 1. Aug, 20 this book collects various perspectives, contributed by both mathematicians and physicists, on the bmodel and its role in mirror symmetry. Algebraic geometry seems to have acquired the reputation of being esoteric, exclusive, and very abstract, with adherents who are secretly plotting to take over all the rest of mathematics. Download pdf mirror symmetry clay mathematics monographs.
The witten equation, mirror symmetry, and quantum singularity. Lectures on mirror symmetry and topological string theory. And what is the present status of research in mirror symmetry, i mean what type of problems are people working on. Tcft and when computable, provide algebraic geometry with very nontrivial new enumerative information. Homological mirror symmetry is the area of mathematics concentrated around several categorical equivalences connecting symplectic and holomorphic or algebraic geometry. This is a very specialized text focusing on the marriage of mirror symmetry a powerful concept that arises in string theory and the language of algebraic geometry. Ive been around with people whore doing donaldsonthomas theory one algebraic geometry side of mirror symmetry and personally willing to know more about homological mirror symmetry. 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.
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. Hochschild cohomology and group actions, differential weil descent and differentially large fields, minimum positive entropy of complex enriques surface automorphisms, nilpotent structures and collapsing ricciflat metrics on k3 surfaces, superstring field theory, superforms and supergeometry, picard. After covering the relevant background material, the main part of the monograph is devoted to the proof of mirror symmetry from various viewpoints. A brane tiling is a bipartite graph on a real twotorus which encodes the information of a quiver with relations. The galois group of a polynomial f with integral coefficients is a measure of the symmetry of the complex roots of f and the solvability of the equation f 0 by radicals, it is just a question of the symmetry of the roots of f. This book is the first completely comprehensive monograph on mirror symmetry, covering the original observations by the physicists through the most recent progress made to date. The relationship between tropical geometry and mirror symmetry goes back to the work of kontsevich and y. This book is an introduction to a collection of topics at the interface between theoretical physics and mathematics, referred to collectively as mirror symmetry.
The aim of the book is to provide a pedagogical introduction to the field of mirror symmetry from both a mathematical and physical perspective. One of the earliest references on this is algebraic geometry and effective lagrangians, emil j. This very incomplete list tries to provide some of the more accessible references on the material. Part of the physics motivation for mirror symmetry involves properties of the chiral ring of n2 superconformal field theories. Teleman berkeley gauge theory, mirror symmetry icm seoul, 2014 3 14. Modular forms around string theory, enumerative geometry and calabiyau varieties, physics around mirror symmetry, hodge theory in string theory. Homological mirror symmetry and tropical geometry to. T1 homological mirror symmetry and algebraic cycles.
This includes gromowwitten invariants, hodge theory, calabiyau 3folds, mirror symmetry and classification of varieties. Subjects discussed include toric varieties, hodge theory, kahler geometry, moduli of stable maps, calabiyau manifolds, quantum cohomology, gromovwitten invariants, and the mirror theorem. In algebraic geometry and theoretical physics, mirror symmetry is a relationship between geometric objects called calabiyau manifolds. Mirror symmetry is a phenomenon arising in string theory in which two very. The concept of mirror symmetry evolved in the late 1980s out of the study of superstring compacti. The inclusion of dbrane states in the equivalence has led to further conjectures involving calibrated submanifolds of the mirror pairs and new conjectural invariants of complex manifolds. Mirror symmetry began when theoretical physicists made some astonishing predictions about rational curves on quintic hypersurfaces in fourdimensional projective space. Mirror symmetry is an extremely vast subject enjoying rich mathematical and physical structures. Special issue mirror symmetry and algebraic geometry. However, mirror symmetry and algebraic geometry by cox and katz satisfies me more than the previous book because obviously its more mathematics. We give a brief survey of some of the geometry of mirror symmetry, written in 2004 for the encyclopaedia of mathematical physics.
The central ideas first appeared in the work of maxim kontsevich 1993. The geometric concepts needed to formulate various mathematical versions of mirror symmetry will be introduced along the way, in variable levels of detail and rigor. Free algebraic geometry books download ebooks online textbooks. Soibelman 2000, who applied methods of nonarchimedean geometry in particular, tropical curves to homological mirror symmetry. Paul seidel, homological mirror symmetry for the genus two curve, j. Conference room 127, graduate school of science bldg no. Algebraic geometry jump to navigation jump to search. Homological mirror symmetry and algebraic cycles university. Cox sheldon katz american mathematical society providence, rhode island.
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. Dirichlet branes and mirror symmetry share this page. This course will focus on various aspects of mirror symmetry. Cox and sheldon katz errata for the rst printing as of april 26, 2017.
What are your views about the claire voisins book on mirror symmetry. Cox, amherst college, ma and sheldon katz, oklahoma state university, stillwater. Topics discussed include duality of polytopes and cones as well as the famous quintic threefold and the toric variety of a reflexive polytope. Homological mirror symmetry and tropical geometry to appear. It is aimed at students who already have some basic knowledge in symplectic and complex geometry 18. As for the books and references, if you want to know just very little about whats going on, you may find mirror symmetry written by leading mathematicians as well as mathematical physicist useful. Probably a little bit out of date now in a few places, but hey.
If the tiling consists of hexagons, the quiver is the mckay quiver associated with a nite abelian subgroup of sl3, c. Mirror symmetry and algebraic geometry mathematical. Research in string theory over the last several decades has yielded a rich interaction with algebraic geometry. Mirror symmetry and algebraic geometry mathematical surveys.
Dec 17, 2005 we give a brief survey of some of the geometry of mirror symmetry, written in 2004 for the encyclopaedia of mathematical physics. Mirror symmetry by cumrun vafa, eric zaslow download link. Algebraic geometric and lagrangian fibrations asp ects 627 fo r k 3surfaces the situation is more complicated but in some case is analogous 18. This book aims to give a single, cohesive treatment of mirror symmetry from both the mathematical and physical viewpoint. Mathematical surveys and monographs publication year 1999. Yamada, notes on the flat structures associated with simple and simply elliptic singularities, in integrable systems and algebraic geometry, world sci. Mirror symmetry is an active topic of research in both the mathematics and physics communities, but among mathematicians, the amodel half of the story remains much betterunderstood than the bmodel. Mirror symmetry and algebraic geometry about this title. This book is a completely comprehensive monograph on mirror symmetry, covering the original observations by the physicists through the most recent progress made to date.
This book is intended for selfstudy or as a textbook for graduate students. Sheldon katz this text presents a comprehensive monograph on mirror symmetry, covering the original observations by the physicists through the most recent progress made up to date. Mirror symmetry and algebraic geometry, mathematical surveys and monographs. The book motivates the subject with an introduction to the physics wherein such questions arise, and therefore keeps things in prespective. The remaining articles cover a wide crosssection of the most significant research topics in algebraic geometry. Introduction the quintic threefold toric geometry mirror symmetry constructions hodge theory and yukawa couplings moduli spaces gromovwitten invariants quantum cohomology localization quantum differential equations the mirror theorem conclusion singular varieties physical. Mirror symmetry and algebraic geometry request pdf. Categorical aspects of algebraic geometry in mirror symmetry abstracts5 this is a joint work with kazushi ueda. The mathematics proof develops gromovwitten theory in the algebraic setting, beginning with the moduli spaces of curves and maps, and uses localization techniques to show that certain hypergeometric functions encode the gromovwitten invariants in genus zero, as is predicted by mirror symmetry. Homological mirror symmetry and tropical geometry ricardo. In 1985, the introduction of calabiyau manifolds into physics as a way to compactify tendimensional spacetime has led to exciting crossfertilization between physics and mathematics, especially with the discovery of mirror symmetry in 1989. Mirror symmetry and algebraic geometry book, 1999 worldcat. Tuesday 8 dec afternoon talks will take place in the room 110.
Dirichlet branes and mirror symmetry clay mathematics. This book presents a comprehensive monograph on mirror symmetry, covering. A list of typographical errors for the book can be found at the web site. Mirror conjecture predicts equivalence of a and b models of tcft on an algebraic calabiyau manifold to b and a models on its geometrical mirror. Hochschild cohomology and group actions, differential weil descent and differentially large fields, minimum positive entropy of complex enriques surface automorphisms, nilpotent structures and collapsing ricciflat metrics on k3 surfaces, superstring field theory, superforms and supergeometry, picard groups for tropical toric. What is meant by the proof of mirror symmetry 463 20. Free algebraic geometry books download ebooks online. The picture represents a portion of its real locus. This expository article explores the connection between the polar duality from polyhedral geometry and mirror symmetry from mathematical physics and algebraic geometry.
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