**Fields Institute Communications**

Volume: 54;
2008;
297 pp;
Hardcover

MSC: Primary 11; 14; 33; 81;

**Print ISBN: 978-0-8218-4484-7
Product Code: FIC/54**

List Price: $126.00

AMS Member Price: $100.80

MAA Member Price: $113.40

**Electronic ISBN: 978-1-4704-3088-7
Product Code: FIC/54.E**

List Price: $118.00

AMS Member Price: $94.40

MAA Member Price: $106.20

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# Modular Forms and String Duality

Share this page *Edited by *
*Noriko Yui; Helena Verrill; Charles F. Doran*

A co-publication of the AMS and Fields Institute

Modular forms have long played a key role in the theory of numbers,
including most famously the proof of Fermat's Last Theorem. Through
its quest to unify the spectacularly successful theories of quantum
mechanics and general relativity, string theory has long suggested
deep connections between branches of mathematics such as topology,
geometry, representation theory, and combinatorics. Less well-known
are the emerging connections between string theory and number
theory. This was indeed the subject of the workshop Modular Forms and
String Duality held at the Banff International Research Station
(BIRS), June 3–8, 2006. Mathematicians and physicists alike
converged on the Banff Station for a week of both introductory
lectures, designed to educate one another in relevant aspects of their
subjects, and research talks at the cutting edge of this rapidly
growing field.

This book is a testimony to the BIRS Workshop, and it covers a wide
range of topics at the interface of number theory and string theory,
with special emphasis on modular forms and string duality. They
include the recent advances as well as introductory expositions on
various aspects of modular forms, motives, differential equations,
conformal field theory, topological strings and Gromov–Witten
invariants, mirror symmetry, and homological mirror symmetry. The
contributions are roughly divided into three categories: arithmetic
and modular forms, geometric and differential equations, and physics
and string theory.

The book is suitable for researchers working at the interface of
number theory 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 number theory and physics.

# Table of Contents

## Modular Forms and String Duality

- Cover Cover11
- Title page iii4
- Contents v6
- Acknowledgments vii8
- Introduction ix10
- List of participants xiii14
- Schedule of workshops xv16
- Aspects of arithmetic and modular forms 120
- Motives and mirror symmetry for Calabi–Yau orbifolds 322
- String modular motives of mirrors of rigid Calabi–Yau varieties 4766
- Update on modular non-rigid Calabi–Yau threefolds 6584
- Finite index subgroups of the modular group and their modular forms 83102
- Aspects of geometric and differential equations 103122
- Apéry limits of differential equations of order 4 and 5 105124
- Hypergeometric systems in two variables, quivers, dimers and dessins d’enfants 125144
- Some properties of hypergeometric series associated with mirror symmetry 163182
- Ramanujan-type formulae for 1/𝜋: A second wind? 179198
- Aspects of physics and string theory 189208
- Meet homological mirror symmetry 191210
- Orbifold Gromov–Witten invariants and topological strings 225244
- Conformal field theory and mapping class groups 247266
- 𝑆𝐿(2,ℂ) Chern–Simons theory and the asymptotic behavior of the colored Jones polynomial 261280
- Open strings and extended mirror symmetry 279298
- Back Cover Back Cover1320