Electronic ISBN:  9781470430887 
Product Code:  FIC/54.E 
List Price:  $118.00 
MAA Member Price:  $106.20 
AMS Member Price:  $94.40 

Book DetailsFields Institute CommunicationsVolume: 54; 2008; 297 ppMSC: Primary 11; 14; 33; 81;
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 wellknown 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.ReadershipGraduate students and research mathematicians interested in number theory and physics.

Table of Contents

Aspects of arithmetic and modular forms

Shabnam Kadir and Noriko Yui  Motives and mirror symmetry for Calabi–Yau orbifolds

Savan Kharel, Monika Lynker and Rolf Schimmrigk  String modular motives of mirrors of rigid Calabi–Yau varieties

Edward Lee  Update on modular nonrigid Calabi–Yau threefolds

Ling Long  Finite index subgroups of the modular group and their modular forms

Aspects of geometric and differential equations

Gert Almkvist, Duco van Straten and Wadim Zudilin  Apéry limits of differential equations of order 4 and 5

Jan Stienstra  Hypergeometric systems in two variables, quivers, dimers and dessins d’enfants

Don Zagier and Aleksey Zinger  Some properties of hypergeometric series associated with mirror symmetry

Wadim Zudilin  Ramanujantype formulae for $1/\pi $: A second wind?

Aspects of physics and string theory

Matthew Ballard  Meet homological mirror symmetry

Vincent Bouchard  Orbifold Gromov–Witten invariants and topological strings

Terry Gannon  Conformal field theory and mapping class groups

Sergi Gukov and Hitoshi Murakami  $SL(2,\mathbb {C})$ Chern–Simons theory and the asymptotic behavior of the colored Jones polynomial

Johannes Walcher  Open strings and extended mirror symmetry


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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 wellknown 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.
Graduate students and research mathematicians interested in number theory and physics.

Aspects of arithmetic and modular forms

Shabnam Kadir and Noriko Yui  Motives and mirror symmetry for Calabi–Yau orbifolds

Savan Kharel, Monika Lynker and Rolf Schimmrigk  String modular motives of mirrors of rigid Calabi–Yau varieties

Edward Lee  Update on modular nonrigid Calabi–Yau threefolds

Ling Long  Finite index subgroups of the modular group and their modular forms

Aspects of geometric and differential equations

Gert Almkvist, Duco van Straten and Wadim Zudilin  Apéry limits of differential equations of order 4 and 5

Jan Stienstra  Hypergeometric systems in two variables, quivers, dimers and dessins d’enfants

Don Zagier and Aleksey Zinger  Some properties of hypergeometric series associated with mirror symmetry

Wadim Zudilin  Ramanujantype formulae for $1/\pi $: A second wind?

Aspects of physics and string theory

Matthew Ballard  Meet homological mirror symmetry

Vincent Bouchard  Orbifold Gromov–Witten invariants and topological strings

Terry Gannon  Conformal field theory and mapping class groups

Sergi Gukov and Hitoshi Murakami  $SL(2,\mathbb {C})$ Chern–Simons theory and the asymptotic behavior of the colored Jones polynomial

Johannes Walcher  Open strings and extended mirror symmetry