Symplectic Geometry And Mirror Symmetry

Author : Kenji Fukaya
Publisher : World Scientific
Page : 510 pages
File Size : 40,6 Mb
Release : 2001
Category : Mathematics
ISBN : 9789810247140

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Symplectic Geometry and Mirror Symmetry by Kenji Fukaya Pdf

In 1993, M Kontsevich proposed a conceptual framework for explaining the phenomenon of mirror symmetry. Mirror symmetry had been discovered by physicists in string theory as a duality between families of three-dimensional Calabi-Yau manifolds. Kontsevich's proposal uses Fukaya's construction of the Aì-category of Lagrangian submanifolds on the symplectic side and the derived category of coherent sheaves on the complex side. The theory of mirror symmetry was further enhanced by physicists in the language of D-branes and also by Strominger-Yau-Zaslow in the geometric set-up of (special) Lagrangian torus fibrations. It rapidly expanded its scope across from geometry, topology, algebra to physics.In this volume, leading experts in the field explore recent developments in relation to homological mirror symmetry, Floer theory, D-branes and Gromov-Witten invariants. Kontsevich-Soibelman describe their solution to the mirror conjecture on the abelian variety based on the deformation theory of Aì-categories, and Ohta describes recent work on the Lagrangian intersection Floer theory by Fukaya-Oh-Ohta-Ono which takes an important step towards a rigorous construction of the Aì-category. There follow a number of contributions on the homological mirror symmetry, D-branes and the Gromov-Witten invariants, e.g. Getzler shows how the Toda conjecture follows from recent work of Givental, Okounkov and Pandharipande. This volume provides a timely presentation of the important developments of recent years in this rapidly growing field.

Author : Kenji Fukaya,Yong Geun Oh,K Ono,Gang Tian
Publisher : World Scientific
Page : 510 pages
File Size : 43,8 Mb
Release : 2001-11-19
Category : Mathematics
ISBN : 9789814490405

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Symplectic Geometry And Mirror Symmetry - Proceedings Of The 4th Kias Annual International Conference by Kenji Fukaya,Yong Geun Oh,K Ono,Gang Tian Pdf

In 1993, M Kontsevich proposed a conceptual framework for explaining the phenomenon of mirror symmetry. Mirror symmetry had been discovered by physicists in string theory as a duality between families of three-dimensional Calabi-Yau manifolds. Kontsevich's proposal uses Fukaya's construction of the A∞-category of Lagrangian submanifolds on the symplectic side and the derived category of coherent sheaves on the complex side. The theory of mirror symmetry was further enhanced by physicists in the language of D-branes and also by Strominger-Yau-Zaslow in the geometric set-up of (special) Lagrangian torus fibrations. It rapidly expanded its scope across from geometry, topology, algebra to physics.In this volume, leading experts in the field explore recent developments in relation to homological mirror symmetry, Floer theory, D-branes and Gromov-Witten invariants. Kontsevich-Soibelman describe their solution to the mirror conjecture on the abelian variety based on the deformation theory of A∞-categories, and Ohta describes recent work on the Lagrangian intersection Floer theory by Fukaya-Oh-Ohta-Ono which takes an important step towards a rigorous construction of the A∞-category. There follow a number of contributions on the homological mirror symmetry, D-branes and the Gromov-Witten invariants, e.g. Getzler shows how the Toda conjecture follows from recent work of Givental, Okounkov and Pandharipande. This volume provides a timely presentation of the important developments of recent years in this rapidly growing field.

Author : Eric D'Hoker,Duong H. Phong,Shing-Tung Yau
Publisher : American Mathematical Soc.
Page : 394 pages
File Size : 55,5 Mb
Release : 2002
Category : Duality (Nuclear physics)
ISBN : 9780821833353

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Mirror Symmetry IV by Eric D'Hoker,Duong H. Phong,Shing-Tung Yau Pdf

This book presents contributions of participants of a workshop held at the Centre de Recherches Mathematiques (CRM), University of Montreal. It can be viewed as a sequel to Mirror Symmetry I (1998), Mirror Symmetry II (1996), and Mirror Symmetry III (1999), copublished by the AMS and International Press. The volume presents a broad survey of many of the noteworthy developments that have taken place in string theory, geometry, and duality since the mid 1990s. Some of the topics emphasized include the following: Integrable models and supersymmetric gauge theories; theory of M- and D-branes and noncommutative geometry; duality between strings and gauge theories; and elliptic genera and automorphic forms. Several introductory articles present an overview of the geometric and physical aspects of mirror symmetry and of corresponding developments in symplectic geometry. The book provides an efficient way for a very broad audience of mathematicians and physicists to explore the frontiers of research into this rapidly expanding area.

Author : David A. Cox,Sheldon Katz
Publisher : American Mathematical Soc.
Page : 469 pages
File Size : 48,9 Mb
Release : 1999
Category : Mathematics
ISBN : 9780821821275

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Mirror Symmetry and Algebraic Geometry by David A. Cox,Sheldon Katz Pdf

Mathematicians wanting to get into the field ... will find a very well written and encyclopaedic account of the mathematics which was needed in, and was developed from, what now might be termed classical mirror symmetry. --Bulletin of the LMS The book is highly recommended for everyone who wants to learn about the fascinating recent interplay between physics and mathematics. --Mathematical Reviews Mirror symmetry began when theoretical physicists made some astonishing predictions about rational curves on quintic hypersurfaces in four-dimensional projective space. Understanding the mathematics behind these predictions has been a substantial challenge. 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. Subjects discussed include toric varieties, Hodge theory, Kahler geometry, moduli of stable maps, Calabi-Yau manifolds, quantum cohomology, Gromov-Witten invariants, and the mirror theorem.

Author : Korea Institute for Advanced Study,0 K. Fukaya
Publisher : Unknown
Page : 498 pages
File Size : 48,5 Mb
Release : 2001
Category : Electronic
ISBN : OCLC:248722339

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Symplectic Geometry and Mirror Symmetry by Korea Institute for Advanced Study,0 K. Fukaya Pdf

Author : Mark Gross
Publisher : American Mathematical Soc.
Page : 338 pages
File Size : 53,5 Mb
Release : 2011-01-20
Category : Mathematics
ISBN : 9780821852323

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Tropical Geometry and Mirror Symmetry by Mark Gross Pdf

Tropical geometry provides an explanation for the remarkable power of mirror symmetry to connect complex and symplectic geometry. The main theme of this book is the interplay between tropical geometry and mirror symmetry, culminating in a description of the recent work of Gross and Siebert using log geometry to understand how the tropical world relates the A- and B-models in mirror symmetry. The text starts with a detailed introduction to the notions of tropical curves and manifolds, and then gives a thorough description of both sides of mirror symmetry for projective space, bringing together material which so far can only be found scattered throughout the literature. Next follows an introduction to the log geometry of Fontaine-Illusie and Kato, as needed for Nishinou and Siebert's proof of Mikhalkin's tropical curve counting formulas. This latter proof is given in the fourth chapter. The fifth chapter considers the mirror, B-model side, giving recent results of the author showing how tropical geometry can be used to evaluate the oscillatory integrals appearing. The final chapter surveys reconstruction results of the author and Siebert for ``integral tropical manifolds.'' A complete version of the argument is given in two dimensions.

Author : Anton Kapustin,Maximilian Kreuzer
Publisher : Springer Science & Business Media
Page : 281 pages
File Size : 43,5 Mb
Release : 2009
Category : Mathematics
ISBN : 9783540680291

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Homological Mirror Symmetry by Anton Kapustin,Maximilian Kreuzer Pdf

An ideal reference on the mathematical aspects of quantum field theory, this volume provides a set of lectures and reviews that both introduce and representatively review the state-of-the art in the field from different perspectives.

Author : Anonim
Publisher : American Mathematical Soc.
Page : 698 pages
File Size : 52,7 Mb
Release : 2009
Category : Mathematics
ISBN : 9780821838488

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Dirichlet Branes and Mirror Symmetry by Anonim Pdf

Research in string theory has generated a rich interaction with algebraic geometry, with exciting work that includes the Strominger-Yau-Zaslow conjecture. This monograph builds on lectures at the 2002 Clay School on Geometry and String Theory that sought to bridge the gap between the languages of string theory and algebraic geometry.

Author : Tyler J. Jarvis,Nathan Priddis
Publisher : American Mathematical Society
Page : 203 pages
File Size : 50,5 Mb
Release : 2021-02-26
Category : Mathematics
ISBN : 9781470457006

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Singularities, Mirror Symmetry, and the Gauged Linear Sigma Model by Tyler J. Jarvis,Nathan Priddis Pdf

This volume contains the proceedings of the workshop Crossing the Walls in Enumerative Geometry, held in May 2018 at Snowbird, Utah. It features a collection of both expository and research articles about mirror symmetry, quantized singularity theory (FJRW theory), and the gauged linear sigma model. Most of the expository works are based on introductory lecture series given at the workshop and provide an approachable introduction for graduate students to some fundamental topics in mirror symmetry and singularity theory, including quasimaps, localization, the gauged linear sigma model (GLSM), virtual classes, cosection localization, $p$-fields, and Saito's primitive forms. These articles help readers bridge the gap from the standard graduate curriculum in algebraic geometry to exciting cutting-edge research in the field. The volume also contains several research articles by leading researchers, showcasing new developments in the field.

Author : Ricardo Castano-Bernard,Fabrizio Catanese,Maxim Kontsevich,Tony Pantev,Yan Soibelman,Ilia Zharkov
Publisher : Springer
Page : 445 pages
File Size : 49,7 Mb
Release : 2014-10-07
Category : Mathematics
ISBN : 9783319065144

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Homological Mirror Symmetry and Tropical Geometry by Ricardo Castano-Bernard,Fabrizio Catanese,Maxim Kontsevich,Tony Pantev,Yan Soibelman,Ilia Zharkov Pdf

The relationship between Tropical Geometry and Mirror Symmetry goes back to the work of Kontsevich and Y. Soibelman (2000), who applied methods of non-archimedean geometry (in particular, tropical curves) to Homological Mirror Symmetry. In combination with the subsequent work of Mikhalkin on the “tropical” approach to Gromov-Witten theory and the work of Gross and Siebert, Tropical Geometry has now become a powerful tool. Homological Mirror Symmetry is the area of mathematics concentrated around several categorical equivalences connecting symplectic and holomorphic (or algebraic) geometry. The central ideas first appeared in the work of Maxim Kontsevich (1993). Roughly speaking, the subject can be approached in two ways: either one uses Lagrangian torus fibrations of Calabi-Yau manifolds (the so-called Strominger-Yau-Zaslow picture, further developed by Kontsevich and Soibelman) or one uses Lefschetz fibrations of symplectic manifolds (suggested by Kontsevich and further developed by Seidel). Tropical Geometry studies piecewise-linear objects which appear as “degenerations” of the corresponding algebro-geometric objects.

Author : Paul Seidel
Publisher : American Mathematical Soc.
Page : 236 pages
File Size : 42,8 Mb
Release : 2015-06-26
Category : Mirror symmetry
ISBN : 9781470410971

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Homological Mirror Symmetry for the Quartic Surface by Paul Seidel Pdf

The author proves Kontsevich's form of the mirror symmetry conjecture for (on the symplectic geometry side) a quartic surface in C .

Author : Ricardo Castaño-Bernard,Yan S. Soibelman,Ilia Zharkov
Publisher : American Mathematical Soc.
Page : 184 pages
File Size : 51,5 Mb
Release : 2010
Category : Science
ISBN : 9780821858516

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Mirror Symmetry and Tropical Geometry by Ricardo Castaño-Bernard,Yan S. Soibelman,Ilia Zharkov Pdf

This volume contains contributions from the NSF-CBMS Conference on Tropical Geometry and Mirror Symmetry, which was held from December 13-17, 2008 at Kansas State University in Manhattan, Kansas. It gives an excellent picture of numerous connections of mirror symmetry with other areas of mathematics (especially with algebraic and symplectic geometry) as well as with other areas of mathematical physics. The techniques and methods used by the authors of the volume are at the frontier of this very active area of research.

Author : Kentaro Hori
Publisher : American Mathematical Soc.
Page : 954 pages
File Size : 46,5 Mb
Release : 2003
Category : Mathematics
ISBN : 9780821829554

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Mirror Symmetry by Kentaro Hori Pdf

This thorough and detailed exposition is the result of an intensive month-long course on mirror symmetry sponsored by the Clay Mathematics Institute. It develops mirror symmetry from both mathematical and physical perspectives with the aim of furthering interaction between the two fields. The material will be particularly useful for mathematicians and physicists who wish to advance their understanding across both disciplines. Mirror symmetry is a phenomenon arising in string theory in which two very different manifolds give rise to equivalent physics. Such a correspondence has significant mathematical consequences, the most familiar of which involves the enumeration of holomorphic curves inside complex manifolds by solving differential equations obtained from a ``mirror'' geometry. The inclusion of D-brane states in the equivalence has led to further conjectures involving calibrated submanifolds of the mirror pairs and new (conjectural) invariants of complex manifolds: the Gopakumar-Vafa invariants. This book gives a single, cohesive treatment of mirror symmetry. Parts 1 and 2 develop the necessary mathematical and physical background from ``scratch''. The treatment is focused, developing only the material most necessary for the task. In Parts 3 and 4 the physical and mathematical proofs of mirror symmetry are given. From the physics side, this means demonstrating that two different physical theories give isomorphic physics. Each physical theory can be described geometrically, and thus mirror symmetry gives rise to a ``pairing'' of geometries. The proof involves applying $R\leftrightarrow 1/R$ circle duality to the phases of the fields in the gauged linear sigma model. The mathematics proof develops Gromov-Witten 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 Gromov-Witten invariants in genus zero, as is predicted by mirror symmetry. Part 5 is devoted to advanced topi This one-of-a-kind book is suitable for graduate students and research mathematicians interested in mathematics and mathematical and theoretical physics.

Author : Shing-Tung Yau
Publisher : American Mathematical Soc.
Page : 460 pages
File Size : 40,9 Mb
Release : 1998
Category : Conformal invariants
ISBN : 9780821827437

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Mirror Symmetry I by Shing-Tung Yau Pdf

Vol. 1 represents a new ed. of papers which were originally published in Essays on mirror manifolds (1992); supplemented by the additional volume: Mirror symmetry 2 which presents papers by both physicists and mathematicians. Mirror symmetry 1 (the 1st volume) constitutes the proceedings of the Mathematical Sciences Research Institute Workshop of 1991.

Author : Kodŭng Kwahagwŏn (Korea). International Conference,Kenji Fukaya
Publisher : World Scientific
Page : 940 pages
File Size : 45,5 Mb
Release : 2001
Category : Mirror symmetry
ISBN : 9812799826

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Symplectic Geometry and Mirror Symmetry by Kodŭng Kwahagwŏn (Korea). International Conference,Kenji Fukaya Pdf

In 1993, M. Kontsevich proposed a conceptual framework for explaining the phenomenon of mirror symmetry. Mirror symmetry had been discovered by physicists in string theory as a duality between families of three-dimensional Calabi–Yau manifolds. Kontsevich's proposal uses Fukaya's construction of the A∞-category of Lagrangian submanifolds on the symplectic side and the derived category of coherent sheaves on the complex side. The theory of mirror symmetry was further enhanced by physicists in the language of D-branes and also by Strominger–Yau–Zaslow in the geometric set-up of (special) Lagrangian torus fibrations. It rapidly expanded its scope across from geometry, topology, algebra to physics. In this volume, leading experts in the field explore recent developments in relation to homological mirror symmetry, Floer theory, D-branes and Gromov–Witten invariants. Kontsevich-Soibelman describe their solution to the mirror conjecture on the abelian variety based on the deformation theory of A∞-categories, and Ohta describes recent work on the Lagrangian intersection Floer theory by Fukaya–Oh–Ohta–Ono which takes an important step towards a rigorous construction of the A∞-category. There follow a number of contributions on the homological mirror symmetry, D-branes and the Gromov–Witten invariants, e.g. Getzler shows how the Toda conjecture follows from recent work of Givental, Okounkov and Pandharipande. This volume provides a timely presentation of the important developments of recent years in this rapidly growing field.