Table Of ContentModular Form s
and Strin g Dualit y
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http://dx.doi.org/10.1090/fic/054
FIELDS INSTITUT E
COMMUNICATIONS
THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES
Modular Form s
and Strin g Dualit y
Noriko Yui
Helena Verrill
Charles F. Dora n
Editors
American Mathematical Society
Providence, Rhode Island
The Fields Institute
for Research in Mathematical Sciences r i r r p\r
Toronto, Ontario r 1 C L D 5
The Fields Institut e
for Research in Mathematical Science s
The Fields Institute is a center for mathematical research, located in Toronto, Canada .
Our missio n i s to provid e a supportiv e an d stimulatin g environmen t fo r mathematic s
research, innovatio n an d education . Th e Institute i s supported b y the Ontario Ministr y
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Council of Canada, and seven Ontario universities (Carleton, McMaster, Ottawa, Toronto,
Waterloo, Western Ontario, and York). I n addition there are several affiliated universitie s
and corporate sponsors in both Canada and the United States .
Fields Institut e Editoria l Board : Car l R . Rieh m (Managin g Editor) , Barbar a Le e
Keyfitz (Directo r o f the Institute) , Juri s Stepran s (Deput y Director) , Jame s G . Arthu r
(Toronto), Kennet h R . Davidso n (Waterloo) , Lis a Jeffre y (Toronto) , Thoma s G . Salis -
bury (York) , Noriko Yui (Queen's) .
2000 Mathematics Subject Classification. Primar y HFxx , 14Gxx , 14J32, 14N35, 33Cxx,
81T30.
Library o f Congress Cataloging-in-Publicatio n Dat a
Modular forms and string duality / Noriko Yui, Helena Verrill, Charles F. Dor an, editors.
p. cm. — (Fields Institute Communications, ISSN 1069-5265 ; 54)
Proceedings of a workshop held at the Banff International Research Station, June 3-8, 2006.
Includes bibliographical references.
ISBN 978-0-8218-4484-7 (alk. paper)
1. Forms , Modular—Congresses . 2 . Dualit y (Mathematics)—Congresses . 3 . Mirro r
symmetry—Congresses. 4 . Number theory—Congresses. 5 . String theory—Congresses. 6 . Parti-
cles (Nuclear physics)—Congresses. I . Yui, Noriko. II . Verrill, Helena. III . Doran, Charles F.,
1971-
QA243.M695 200 8
512.7'3—dc22 200802817 3
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10 9 8 7 6 5 4 3 2 1 1 3 12 11 10 09 08
Contents
Acknowledgments vi i
Introduction i x
List of Participants xii i
Schedule of Workshops x v
Aspects of Arithmetic and Modular Form s
Motives and Mirror Symmetry for Calabi-Yau Orbifolds 3
SHABNAM KADI R and NORIK O YU I
String Modular Motives of Mirrors of Rigid Calabi-Yau Varieties 4 7
SAVAN KHAREL, MONIK A LYNKE R and ROL F SCHIMMRIG K
Update on Modular Non-Rigid Calabi-Yau Threefolds 6 5
EDWARD LE E
Finite Index Subgroups of the Modular Group and Their
Modular Forms 8 3
LING LON G
Aspects of Geometric and Differential Equation s
Apery Limits of Differential Equations of Order 4 and 5 10 5
GERT ALMKVIST , DUC O VA N STRATEN an d WADI M ZUDILI N
Hypergeometric Systems in Two Variables, Quivers, Dimers
and Dessins d'Enfants 12 5
JAN STIENSTR A
Some Properties of Hypergeometric Series Associated with
Mirror Symmetry 16 3
DON ZAGIE R and ALEKSE Y ZINGE R
Ramanujan-Type Formulae for 1/n: A Second Wind? 17 9
WADIM ZUDILI N
VI Contents
Aspects o f Physics and Strin g Theor y
Meet Homological Mirror Symmetry 19 1
MATTHEW ROBER T BALLAR D
Orbifold Gromov-Witten Invariants and Topological Strings 22 5
VINCENT BOUCHAR D
Conformal Field Theory and Mapping Class Groups 24 7
TERRY GANNO N
SL(2,C) Chern-Simon s Theory and the Asymptotic Behavior of
the Colored Jones Polynomial 26 1
SERGEI GUKO V and HITOSH I MURAKAM I
Open Strings and Extended Mirror Symmetry 279
JOHANNES WALCHE R
Acknowledgments
The editors wish to express their appreciation to all the contributors for prepar-
ing their manuscripts for the Fields Communication Series , which required extr a
effort presenting not only current developments but also the history of the subjects
treated in their articles.
All papers in this volume were refereed very rigorously. We are deeply grateful
to all our referees for their time-consuming effort an d discipline in evaluating the
articles.
All papers were copy-edited by Arthur Greenspoon of Mathematical Reviews.
The editors and the Fields Institute are grateful for his help smoothing out, English
and mathematical presentations.
The worksho p wa s supporte d b y th e Banf f Internationa l Researc h Statio n
(BIRS) throug h th e five-day workshop s program. W e thank BIR S for thei r fi-
nancial support. W e enjoyed the excellent support o f the staff a t BIRS, and ar e
grateful for their hospitality. In addition, some young participants from the United
States were supported in part by Mathematical Sciences Research Institute (MSRI)
Berkeley.
Last but not least, we thank Debbie Iscoe of the Fields Institute for her help
reformatting articles and assembling this volume for publication.
Noriko Yui, Helena Verrill and Charles Dor an
June 2008
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Introduction
Modular forms have long played a key role in the theory of numbers, including
most famousl y th e proof of Fermat's Last Theorem . Throug h its quest to unif y
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. Les s well-
known are the emerging connections between string theory and number theory - the
subject of the workshop Modular Forms and String Duality held at the Banff Inter-
national Research Station (BIRS), June 3-8, 2006. Mathematicians and physicists
alike converged on the Banff Station for a week of introductory lectures, designed
to educate one another in relevant aspects of their subjects, and of research talks
at the cutting edge of this rapidly growing field.
The workshop was organized by Charles F. Doran, Helena Verrill and Noriko
Yui. Th e workshop was a huge success. Altogethe r thirty-seven mathematician s
and physicists converged at the BIRS for the five day workshop. Twenty-si x one
hour talks were presented. Som e were introductory lecture s by mathematician s
designed to prepare physicists in modular forms, quasimodular forms, modularity of
Galois representations, and toric geometry. At the same time, introductory lectures
by physicists were intended to educate mathematicians on some aspects of mirror
symmetry and string theory in connection with number theory. These introductory
lectures were scheduled in the mornings of the early days of the workshop. Research
talks were scheduled in the afternoons and later days. They covered recent advances
on various aspects of modular forms, differential equations, conformal field theory,
topological strings and Gromov-Witten invariants, holomorphic anomaly equations,
motives, mirror symmetry, homological mirror symmetry, construction of Calabi-
Yau manifolds, among others.
Summary o f scientific an d other objective s
Physical duality symmetries relate special limits of the various consistent string
theories (Types I, II, Heterotic string and their cousins, including M-theory and F-
theory) one to another. Th e comparison of the mathematical descriptions of these
theories often reveal s quite deep and unexpected mathematica l conjectures. Th e
best know n string duality to mathematicians, Typ e IIA/II B duality , als o called
mirror symmetry , ha s inspired many new developments in algebraic and arith-
metic geometry, number theory, toric geometry, Riemann surface theory, and in-
finite dimensional Lie algebras. Othe r string dualities such as Heterotic/Type I I
duality an d F-Theory/Heterotic strin g duality have also, more recently, led to a
series of mathematical conjectures, many involving elliptic curves, K3 surfaces, and
modular forms. Modular forms and quasi-modular forms play a central role in mir-
ror symmetry, in particular as generating functions counting the number of curves
on Calabi-Yau manifolds and describing Gromov-Witten invariants. Thi s has led
ix
