Mirror Symmetry (Clay Mathematics Monographs, V. 1) [Hardcover]

Kentaro Hori (Author, Editor), Sheldon Katz (Editor), Albrecht Klemm (Editor), Rahul Pandharipande (Editor), Richard Thomas (Editor), Cumrun Vafa (Editor), Ravi Vakil (Editor), Eric Zaslow (Editor)

Book Description

ISBN-10: 0821829556 | ISBN-13: 978-0821829554 | Publication Date: August 19, 2003

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. <P>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.

Editorial Reviews

Review

"This book, a product of the collective efforts of the lecturers at the School organized ... by the Clay Mathematics Institute, is a valuable contribution to the continuing intensive collaboration of physicists and mathematicians. It will be of great value to young and mature researchers in both communities interested in this fascinating modern grand unification project." ---- Yuri Manin, Max Planck Institute for Mathematics, Bonn, Germany

Product Details

• Hardcover: 929 pages
• Publisher: American Mathematical Society (August 19, 2003)
• Language: English
• ISBN-10: 0821829556
• ISBN-13: 978-0821829554
• Product Dimensions: 10.2 x 7.2 x 2.2 inches
• Shipping Weight: 4.1 pounds (View shipping rates and policies)
• Average Customer Review: 5.0 out of 5 stars  See all reviews (1 customer review)
• Amazon Best Sellers Rank: #725,068 in Books (See Top 100 in Books)

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5 of 7 people found the following review helpful:

5.0 out of 5 stars Detailed overview of the subject, May 15, 2005

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Dr. Lee D. Carlson (Baltimore, Maryland USA) - See all my reviews
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This review is from: Mirror Symmetry (Clay Mathematics Monographs, V. 1) (Hardcover)

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.