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Modular Forms and String Duality

Edited by: Noriko Yui Queen’s University, Kingston, ON, Canada
Helena Verrill Louisiana State University, Baton Rouge, LA
Charles F. Doran University of Washington, Seattle, WA
A co-publication of the AMS and Fields Institute
Available Formats:
Electronic ISBN: 978-1-4704-3088-7
Product Code: FIC/54.E
List Price: $118.00 MAA Member Price:$106.20
AMS Member Price: $94.40 Click above image for expanded view Modular Forms and String Duality Edited by: Noriko Yui Queen’s University, Kingston, ON, Canada Helena Verrill Louisiana State University, Baton Rouge, LA Charles F. Doran University of Washington, Seattle, WA A co-publication of the AMS and Fields Institute Available Formats:  Electronic ISBN: 978-1-4704-3088-7 Product Code: FIC/54.E  List Price:$118.00 MAA Member Price: $106.20 AMS Member Price:$94.40
• Book Details

Fields Institute Communications
Volume: 542008; 297 pp
MSC: 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 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.

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 non-rigid 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 - Ramanujan-type 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

• Requests

Review Copy – for reviewers who would like to review an AMS book
Accessibility – to request an alternate format of an AMS title
Volume: 542008; 297 pp
MSC: 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 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.

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 non-rigid 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 - Ramanujan-type 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
Review Copy – for reviewers who would like to review an AMS book
Accessibility – to request an alternate format of an AMS title
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