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Calabi-Yau Varieties and Mirror Symmetry
Edited by: Noriko Yui, Queen's University, Kingston, ON, Canada, and James D. Lewis, University of Alberta, Edmonton, AB, Canada
A co-publication of the AMS and Fields Institute.
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Fields Institute Communications
2003; 367 pp; hardcover
Volume: 38
ISBN-10: 0-8218-3355-3
ISBN-13: 978-0-8218-3355-1
List Price: US$122
Member Price: US$97.60
Order Code: FIC/38
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See also:

Strings 2001 - Atish Dabholkar, Sunil Mukhi and Spenta R Wadia

Mirror Symmetry - Claire Voisin

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.

The book is suitable for researchers interested in mirror symmetry and string theory.

Titles in this series are co-published with the Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).

Readership

Graduate students and research mathematicians interested in mirror symmetry and string theory.

Table of Contents

Geometric methods
  • V. V. Batyrev and E. N. Materov -- Mixed toric residues and Calabi-Yau complete intersections
  • L. Chiang and S.-s. Roan -- Crepant resolutions of \(\mathbb{C}^n/A_1(n)\) and flops of \(n\)-folders for \(n=4,5\)
  • P. L. del Angel and S. Müller-Stach -- Picard-Fuchs equations, integrable systems and higher algebraic K-theory
  • S. Hosono -- Counting BPS states via holomorphic anomaly equations
  • J. D. Lewis -- Regulators of Chow cycles on Calabi-Yau varieties
Arithmetic methods
  • P. Candelas, X. de la Ossa, and F. Rodriguez-Villegas -- Calabi-Yau manifolds over finite fields, II
  • L. Dieulefait and J. Manoharmayum -- Modularity of rigid Calabi-Yau threefolds over \(\mathbb{Q}\)
  • Y. Goto -- \(K3\) surfaces with symplectic group actions
  • T. Ito -- Birational smooth minimal models have equal Hodge numbers in all dimensions
  • B. H. Lian and S.-T. Yau -- The \(n\)th root of the mirror map
  • L. Long -- On a Shioda-Inose structure of a family of K3 surfaces
  • M. Lynker, V. Periwal, and R. Schimmrigk -- Black hole attractor varieties and complex multiplication
  • F. Rodriguez-Villegas -- Hypergeometric families of Calabi-Yau manifolds
  • R. Schimmrigk -- Aspects of conformal field theory from Calabi-Yau arithmetic
  • J. Stienstra -- Ordinary Calabi-Yau-3 crystals
  • J. Stienstra -- The ordinary limit for varieties over \(\mathbb{Z}[x_1,\ldots,x_r]\)
  • N. Yui -- Update on the modularity of Calabi-Yau varieties with appendix by Helena Verrill
  • N. Yui and J. D. Lewis -- Problems
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