Mirror Symmetry (Clay Mathematics Monographs, V. 1)

Book Description

This thorough and detailed exposition is the result of an intensive month-long course sponsored by the Clay Mathematics Institute. It develops mirror symmetry from both mathematical and physical perspectives. The material will be particularly useful for those wishing to advance their understanding by exploring mirror symmetry at the interface of mathematics and physics.

This one-of-a-kind volume offers the first comprehensive exposition on this increasingly active area of study. It is carefully written by leading experts who explain the main concepts without assuming too much prerequisite knowledge. The book is an excellent resource for graduate students and research mathematicians interested in mathematical and theoretical physics.

Customer Reviews:

Detailed overview of the subject.......2005-05-16

Mirror symmetry has become an established branch of mathematics and mathematical physics, and research in the subject has resulted in brilliant developments. This sizable book contains essentially some (polished) lecture notes of a seminar series in mirror symmetry that was given in the spring of 2000. This reviewer only studied Part 5 of the book, entitled "Advanced Topics" and so only that part will be reviewed here. In addition, space constraints then dictate only a small portion of this part can be reviewed. Needless to say, any reader who intends to tackle this book will need a substantial background in modern mathematics and advanced physics, and a sizable commitment in time. The time spent is well worth it though, as both the mathematics and physics behind mirror symmetry has to rank as one of the most fascinating research topics in the last two decades.

In the chapter entitled "Topological Strings" the authors consider the functional integration of worldsheet geometries. This project involves essentially the integration over the complex structures of Riemann surfaces. Referring to this procedure as "quantum gravity", they do not address it in-depth, but instead focus on the coupling of topological sigma models to worldsheet gravity, which is called `topological string theory' in the literature. The authors first consider the case where the target is a Kahler manifold whose first Chern class is zero, since for this case the quantum cohomology ring is less easy to obtain, i.e. it can obtain contributions from holomorphic maps of any degree. Even for the case where there is no coupling to gravity, the degree 0 contribution is related to the classical intersection number. The contributions from higher degree result in the deformation of the classical cohomology ring into the quantum cohomology ring. The authors then ask whether there are any other correlators that will give nontrivial (non-zero) invariants in genus 0. Posing this question leads to the WDVV equation and the genus 0 topological string partition function. The n-point correlation functions of topological strings can then be defined as the nth partial derivatives of this function. For higher genus cases, the correlators are all zero, but the authors show the connection between the higher genus partition function and holomorphic anomalies. The case of three-dimensional Calabi-Yau manifolds is special, if one concentrates on the integration over the complex structures of the worldsheet. When the complex dimension of this moduli space is 3(g-1) then there are isolated points where holomorphic maps exist. Defining a topological string theory for Calabi-Yau threefolds is straightforward, as the author shows, and proceeds analogously to the case of topological field theory. A measure is defined on the moduli space of Riemann surfaces of genus g that cancels the axial charge anomaly. A genus g (>1) topological string amplitude, which is a section of a bundle over the moduli space of Calabi-Yau manifolds, is then obtained from this procedure. Modulo the presence of holomorphic anomalies, the authors show that the definition of topological string amplitudes is consistent with the topological symmetry. The origin of these holomorphic anomalies is discussed in fair detail by the authors, having their origin in the boundaries of the moduli space.

The rigorous mathematical formulation of mirror symmetry is of course of great interest to mathematicians. Because of its origin in string theory and quantum field theory, mirror symmetry has not yet received this kind of rigor. Chapters 37 and 38 of this book discuss some of the approaches that attempt to put mirror symmetry on a more rigorous foundation. One of these involves the use of `derived categories,' an approach that was recommended by the mathematician Maxim Kontsevich. The discussion in these chapters takes place in the context of D-branes, and Kontsevich conjectures that mirror symmetry is the equivalence of two categories: the derived category of coherent sheaves, and the category of Lagrangian submanifolds with flat U(1) connections. Specifically the equivalence entails the equivalence between the bounded derived category of coherent sheaves or `B-cycles' and the category of A-cycles with compositions defined in terms of holomorphic maps from disks. This latter category is derived from the Fukaya A-infinity category, as is shown by the authors. They discuss in detail this category, being essentially a generalization of a differential, graded algebra, especially how to obtain the compositions. In chapter 37, the authors give an explicit example of the equivalence of these categories for the case of the elliptic curve. The elliptic curve is interesting in this regard in that it is its own mirror, i.e. the complex parameter is mapped to the complexified Kahler parameter by the mirror map.

The derived category has sometimes been a stumbling block to those who want to understand the Kontsevich conjecture. The authors do not attempt to give the reader the needed insight into this kind of category, but merely take it to be a collection of all holomorphic bundles and coherent sheaves. Sheaves in this category can be subtracted from each other using a map between them. Physically, this subtraction corresponds to the annihilation of branes and anti-branes via a tachyon. Derived categories though are straightforward to think about if one views them from the standpoint of algebraic topology. Derived categories are rich enough to include notions of localization and triangulated objects (i.e. "complexes") and maps (i.e. morphisms) between these objects. This is a kind of "homology" but what is of main interest are homotopies between the morphisms. The class of homotopic morphisms between two complexes forms an abelian group and one can then obtain a category consisting of complexes as objects and classes of homotopic morphisms as morphisms. A cohomology functor can then be defined on this category, along with graded objects and differentials between them. The homotopic category can be given a "triangulation" and morphisms in this category that give rise to isomorphisms in cohomology are given special status, called `quasimorphisms.' The localization of this category with respect to quasimorphisms is called a derived category.

The New Ambidextrous Universe: Symmetry and Asymmetry from Mirror Reflections to Superstrings: Third Revised Edition

Average customer rating:

Wonderful, but somewhat out-of-date (only at the end).
Review from a non-scientific perspective
Frames superstrings and twistors
IS SUPERSTRING THEORY A RECREATIONAL MATHEMATICAL THEOLOGY?

The New Ambidextrous Universe: Symmetry and Asymmetry from Mirror Reflections to Superstrings: Third Revised Edition
Martin Gardner
Manufacturer: Dover Publications
ProductGroup: Book
Binding: Paperback

Genetics | Evolution | Science | Subjects | Books

General | Science | Subjects | Books

General | Physics | Science | Subjects | Books

Look Inside Science Books | Trip | Specialty Stores | Books
Similar Items:

Are Universes Thicker Than Blackberries?: Discourses on Godel, Magic Hexagrams, Little Red Riding Hood, and Other Mathematical and Pseudoscientific Topics
The Whys of a Philosophical Scrivener
The Force of Symmetry
The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems
Gardner's Whys & Wherefores

ASIN: 0486442446

Book Description

This newly updated edition of a well-known work explores a pair of modern science's most fundamental discoveries: the asymmetric DNA helix and the overthrow of parity (left-right symmetry) in particle physics. Absorbing and thought-provoking, The New Ambidextrous Universe was written by Martin Gardner, one of Dover's most popular authors,.

Customer Reviews:

Wonderful, but somewhat out-of-date (only at the end)........2002-01-16

I think that THE NEW AMBIDEXTROUS UNIVERSE (1990) is a wonderful book on symmetry and asymmetry in the worlds of everyday life, chemistry, physics, and unification theories. Everything in this book is noteworthy, and also up-to-date except for the last few chapters.
It is a very good updating of the previous (1978) edition, which concluded with many open questions in elementary particle physics that were resolved (and new questions raised) in 1978 - 1989.
It is high time for this book to be updated if Mr. Gardner can manage it (he is rather elderly; born in 1914), and a publisher will take a new edition. Books like this are gueling to revise and update.

Review from a non-scientific perspective.......2002-01-14

I'm not going to say that I understand all of this. Most of it is way over my head, but after reading it, I can say that I understand more now than I did before. I'm planning on attacking it again in a couple years. Overall, however, Gardner does a good job of bring complicated scientific theory down to a plain English level by using diagrams and analogies.

Frames superstrings and twistors.......2000-04-29

Every decade Gardner updates this book. The five new chapters in the 1990 edition, including material on twistors and superstrings, are well worth the price. What Gardner does best is frame the new theories within a historical perspective. For example, he says it is impossible not to compare string theory with Lord Kelvin's (W. Thompson) 1958 theory of vortex strings. Vortex string theory was fashionable for at least fifty years. Gardner shows the vortex string theory and the superstring theory to be kissing cousins: Lord Kelvin used perfect fluid to refer to the superstring quantum vacuum -- both referring to the same sub-space area. String theory speaks of vibrating frequencies of energy while vortex rings were also vibrating frequencies that gave the atoms different properties. Instead of quantum foam with jittering virtual particles, vortex theory had vortex sponges with billions of vortex motions whirling in all directions.

IS SUPERSTRING THEORY A RECREATIONAL MATHEMATICAL THEOLOGY?.......2000-03-30

Gardner's account of Roger Penrose's twistor theory is short and excellent. Physicists have gotten tangled up trying to speak of deeper down events which are hidden from view due to their sub-Planck length size (10 to the minus 33rd power of a centimeter). Here it is pointed out that "on a sufficiently small scale the concept of a space-time point evaporates in the complex space of twistor theory." Twistor theory, like superstring theory, was merely trying to formulate how the submicroscopic particles come into being. Both theories consist of math and lack any experimental verification. To repeat, the author discusses these obtuse theories in a way that frames their overall direction of thought. Gardner appears to agree with Howard George who calls superstring theory a "recreational mathematical theology." The bottom line -- both twistor and string theory are philosophy -- not physics.

Mirror Symmetry and Algebraic Geometry (Mathematical Surveys & Monographs)
Average customer rating:

Excellent overview of mirror symmetry

Mirror Symmetry and Algebraic Geometry (Mathematical Surveys & Monographs)
David A. Cox
Manufacturer: Amer Mathematical Society
ProductGroup: Book
Binding: Paperback

General | Science | Subjects | Books
General | Mathematics | Science | Subjects | Books
Algebraic Geometry | Geometry & Topology | Mathematics | Science | Subjects | Books
Applied | Mathematics | Science | Subjects | Books | Biomathematics | Computer Mathematics | Differential Equations | Engineering | Game Theory | General | Graph Theory | Linear Programming | Probability & Statistics | Vector Analysis
All Titles | Qualifying Textbooks - Fall 2007 | Stores | Books
Science | Qualifying Textbooks - Fall 2007 | Stores | Books
Similar Items:

Mirror Symmetry (Clay Mathematics Monographs, V. 1)

Enumerative Geometry and String Theory

Compact Complex Surfaces (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics)

Calabi-Yau Manifolds and Related Geometries

Introduction to Toric Varieties. (AM-131)

ASIN: 082182127X

Book Description

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 the first 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, Kähler geometry, moduli of stable maps, Calabi-Yau manifolds, quantum cohomology, Gromov-Witten invariants, and the mirror theorem.

Customer Reviews:

Excellent overview of mirror symmetry.......2001-03-18

This book is one of the few monographs on mirror symmetry that is not a collection of articles written by specialists. It attempts to put mirror symmetry on a mathematically rigorous foundation and does so to a large degree. The book opens with a review of the motivations for mirror symmetry in quantum field theory and superstring theory. The content of this chapter is straightforward reading for physicists/string theorists but mathematicians may have trouble with the physical reasoning employed. The chapter explains the motivation for the mathematical constructions performed in the rest of the book. The author does a good job of presenting the mathematics in a form that is as rigorous as possible. The predictions made by physicists to quantities in in algebraic geometry are too interesting from a mathematical standpoint to let lay couched in the formalism of path integrals. The book gives many examples of mirror symmetry constructions that are rigorous mathematically, most of these involving toric varieties. A general methodology for finding the mirror of a given Calabi-Yau manifold is still unknown according to the author. By far the best chapter in the book is the one on quantum cohomology, as this tool has so many applications in algebraic and symplectic geometry. One is always impressed on the originality and breadth of ideas that have been employed in this subject. There are several topics in mirror symmetry that are not discussed in the book, but even with these omissions, this is a fine addition to the literature on the subject. One question that immediately arises when thinking about mirror symmetry is there is anything that is interesting in the case of Calabi-Yau manifolds over finite fields. The special case that comes to mind is for elliptic curves. Are mirrors of Calabi-Yau manifolds easier to find in the finite field case and does the mirror have a group operation related to the one on the original manifold (elliptic curve)? These questions are not addressed in this book, but answering them may have important ramifications for applications of mirror symmetry to the field of cryptography for example.

Calabi-Yau Manifolds and Related Geometries

Average customer rating:

Calabi-Yau Manifolds and Related Geometries
Mark Gross , Daniel Huybrechts , and Dominic Joyce
Manufacturer: Springer
ProductGroup: Book
Binding: Paperback

General | Science | Subjects | Books

General | Mathematics | Science | Subjects | Books

General | Physics | Science | Subjects | Books

Mathematical Physics | Physics | Science | Subjects | Books

All Amazon Upgrade | Amazon Upgrade | Stores | Books

Professional & Technical | Amazon Upgrade | Stores | Books

Science | Amazon Upgrade | Stores | Books

All Titles | Qualifying Textbooks - Fall 2007 | Stores | Books

Professional | Qualifying Textbooks - Fall 2007 | Stores | Books

Science | Qualifying Textbooks - Fall 2007 | Stores | Books
Similar Items:

Complex Geometry: An Introduction (Universitext)
Enumerative Geometry and String Theory
Compact Manifolds with Special Holonomy (Oxford Mathematical Monographs)
Calabi Yau Manifolds: A Bestiary for Physicists
Mirror Symmetry (Clay Mathematics Monographs, V. 1)

ASIN: 3540440593

Book Description

This book is an expanded version of lectures given at a summer school on symplectic geometry in Nordfjordeid, Norway, in June 2001. The unifying feature of the book is an emphasis on Calabi-Yau manifolds. The first part discusses holonomy groups and calibrated submanifolds, focusing on special Lagrangian submanifolds and the SYZ conjecture. The second studies Calabi-Yau manifolds and mirror symmetry, using algebraic geometry. The final part describes compact hyperkahler manifolds, which have a geometric structure very closely related to Calabi-Yau manifolds.
The book is an introduction to a very active field of research, on the boundary between mathematics and physics. It is aimed at graduate students and researchers in geometry and string theory and intended as an introductory text, requiring only limited background knowledge. Proofs or sketches are given for many important results. Moreover, exercises are provided.

Kaleidoscopes, Hubcaps, & Mirrors: Symmetry & Transformations (Connected Mathematics Series)

Average customer rating:

Kaleidoscopes, Hubcaps, & Mirrors: Symmetry & Transformations (Connected Mathematics Series)
Glenda Lappan
Manufacturer: Dale Seymour Publications
ProductGroup: Book
Binding: Paperback

General | Science & Technology | Teens | Subjects | Books

Mathematics | Science & Technology | Teens | Subjects | Books

jp-unknown3 | Specialty Stores | Books
Similar Items:

Thinking With Mathematical Models: Representing Relationships
Moving Straight Ahead: Linear Relationships (Connected Mathematics Series)
Say It With Symbols
Samples & Populations: Data & Statistics
Looking for Pythagoras: The Pythagorean Theorem (Connected Mathematics Series)

ASIN: 1572321881

The Ambidextrous Universe: Mirror Asymmetry and Time-Reversed Worlds

Average customer rating:

Pilates for the Mind
DON'T BUY THIS BOOK!

The Ambidextrous Universe: Mirror Asymmetry and Time-Reversed Worlds
Martin Gardner
Manufacturer: Scribner
ProductGroup: Book
Binding: Paperback

General | History & Philosophy | Science | Subjects | Books
ASIN: 068415790X

Customer Reviews:

Pilates for the Mind.......2004-10-02

I read this book 25 years ago and still remember it as a mind-expanding experience. I am a person with slight skills in the maths and sciences, but Gardner took me farther than I've ever gone. Easy to understand and, if I recall correctly, very funny.

DON'T BUY THIS BOOK!.......2002-01-16

This is the 1978 edition of a book that was updated and re-published in 1990! It is rather silly that anyone is even offering the old edition for sale.
The paperback (pub in 1991) is available and for sale at a lower price!

Actually, the 1990 edition is somewhat behind the times in its sections on elementary particle physics, etc., and it is past time for this book to be updated again.

Kaleidoscopes, Hubcaps, and Mirrors: Symmetry and Transformations
Average customer rating:
Kaleidoscopes, Hubcaps, and Mirrors: Symmetry and Transformations

Manufacturer: Prentice Hall
ProductGroup: Book
Binding: Paperback
ASIN: 0131808044

Calabi-Yau Varieties and Mirror Symmetry (Fields Institute Communications, V. 38.)
Average customer rating:
Calabi-Yau Varieties and Mirror Symmetry (Fields Institute Communications, V. 38.)

Manufacturer: American Mathematical Society
ProductGroup: Book
Binding: Hardcover

General | Science | Subjects | Books
General | Mathematics | Science | Subjects | Books
Algebraic Geometry | Geometry & Topology | Mathematics | Science | Subjects | Books
Algebraic Geometry | Geometry & Topology | Mathematics | Professional Science | Professional & Technical | Subjects | Books
ASIN: 0821833553

Book Description

The idea of mirror symmetry originated in physics, but in recent years, the field of mirror symmetry has exploded onto the mathematical scene. It has inspired many new developments in algebraic and arithmetic geometry, toric geometry, the theory of Riemann surfaces, and infinite-dimensional Lie algebras among others.

The developments in physics stimulated the interest of mathematicians in Calabi-Yau varieties. This led to the realization that the time is ripe for mathematicians, armed with many concrete examples and alerted by the mirror symmetry phenomenon, to focus on Calabi-Yau varieties and to test for these special varieties some of the great outstanding conjectures, e.g., the modularity conjecture for Calabi-Yau threefolds defined over the rationals, the Bloch-Beilinson conjectures, regulator maps of higher algebraic cycles, Picard-Fuchs differential equations, GKZ hypergeometric systems, and others.

The articles in this volume report on current developments. The papers are divided roughly into two categories: geometric methods and arithmetic methods. One of the significant outcomes of the workshop is that we are finally beginning to understand the mirror symmetry phenomenon from the arithmetic point of view, namely, in terms of zeta-functions and L-series of mirror pairs of Calabi-Yau threefolds.

Book Description

The investigation of discrete symmetries is a fascinating subject which has been central to the agenda of physics research for 50 years, and has been the target of many experiments, ongoing and in preparation, all over the world. This book approaches the subject from a somewhat less traditional angle: while being self-contained and suitable to the reader who wants to acquire a solid knowledge of the topic, it puts more emphasis on the experimental aspects of the field, trying to provide a wider picture than usual and to convey the intellectual challenge of experimental physics. The book includes the related connection to phenomenology, a purpose for which the precision experiments in this field - often rather elegant and requiring a good amount of ingenuity - are very well suited. The book discusses discrete symmetries (parity, charge conjugation, time reversal, and of course CP symmetry) in microscopic (atomic, nuclear, and particle) physics, and includes the detailed description of some key or representative experiments. The book discusses their principles and challenges more than the historical development. The main past achievements and the most recent developments are both included. The level goes from introductory to advanced. While mainly addressed to graduate students, the book can also be useful to undergraduates (by skipping some of the more advanced sections, and utilizing the brief introductions to some topics in the appendices), and to young researchers looking for a wider modern overview of the issues related to CP symmetry.

An Introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces (Theoretical and Mathematical Physics)

Average customer rating:

An Introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces (Theoretical and Mathematical Physics)
Martin Schlichenmaier
Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover

General | Earth Sciences | Science | Subjects | Books

General | Science | Subjects | Books

General | Mathematics | Science | Subjects | Books

Mathematical Physics | Physics | Science | Subjects | Books

Explorations in Mathematical Physics: The Concepts Behind an Elegant Language
Mathematical Methods for Engineers and Scientists 3: Fourier Analysis, Partial Differential Equations and Variational Methods
Lost Causes in and beyond Physics

ASIN: 3540711740

Book Description

This book gives an introduction to modern geometry. Starting from an elementary level the author develops deep geometrical concepts, playing an important role nowadays in contemporary theoretical physics. He presents various techniques and viewpoints, thereby showing the relations between the alternative approaches.

At the end of each chapter suggestions for further reading are given to allow the reader to study the touched topics in greater detail.

This second edition of the book contains two additional more advanced geometric techniques: (1) The modern language and modern view of Algebraic Geometry and (2) Mirror Symmetry.

The book grew out of lecture courses. The presentation style is therefore similar to a lecture. Graduate students of theoretical and mathematical physics will appreciate this book as textbook. Students of mathematics who are looking for a short introduction to the various aspects of modern geometry and their interplay will also find it useful. Researchers will esteem the book as reliable reference.

Books:

Books Index

Books Home

Recommended Books