Table Of ContentProgress in Mathematics
Volume 160
Series Editors
Hyman Bass
Joseph Oesterle
Alan Weinstein
Topological Field Theory,
Primitive Forms
and Related Topics
Masaki Kashiwara
Atsushi Matsuo
Kyoji Saito
Ikuo Satake
Editors
Springer Science+Business Media, LLC
Masaki Kashiwara Atsushi Matsuo
Research Institute for Department of Mathernatical Sciences
Mathernatical Sciences University of Tokyo
Kyoto University Tokyo, Japan 153
Kyoto, Japan 606-01
Kyoji Saito Ikuo Satake
Research Institute for Departrnent of Mathernatics
Mathernatical Sciences Osaka University
Kyoto University Toyonaka-city, Japan 560
Kyoto, Japan 606-01
Library of Congress Cataloging-in-Publication Data
Topological field theory, primitive forrns and related topics I Masaki
Kashiwara ... let aI.], editors.
p. cm. --(Progress in rnathernatics ; v. 160)
Includes bibJiographicaI references.
ISBN 978-1-4612-6874-1 ISBN 978-1-4612-0705-4 (eBook)
DOI 10.1007/978-1-4612-0705-4
I. Quantum field theory--Congresses. 2. Algebraic topology
-Congresses. 3. Mathematical physics--Congresses. I. Kashiwara,
Masaki, 1947- . 11. Series: Progress in rnathernatics (Boston,
Mass.) ; vol. 160.
QCI74.45.AIT684 1998
530.l4'--DC21 98-4532
CIP
AMS Subject Classification: 14NIO, 14130, 14F32, 17B67, 17Bxx, 17B81, 17B62,
17B69, 22E50, 81T40, 81RIO, IIF46, 83E30, 17B65, 13PIO, 32G20, 33D45, 33D80,
20F36,32S30,5IFI5,IIF55,05C,20F55,32S40
Printed on acid-free paper.
© 1998 Springer Science+Business Media New York
Originally published by Birkhäuser Boston in 1998
Softcover reprint ofthe hardcover 1s t edition 1998
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ISBN 978-1-4612-6874-1
Reformatted from the authors' disks by TEXniques, Inc., Boston, MA
987654321
Contents
Preface .............................................................. vii
Symposia ............................................................ ix
Degenerate Double Affine Hecke Algebra and Conformal Field Theory
TOMOYUKI ARAKAWA, TAKES HI SUZUKI,
and AKIHIRO TSUCHIYA ............................................ 1
Vertex Algebras
RICHARD E. BORCHERDS .......................................... 35
Extensions of Conformal Modules
SHUN-JEN CHENG, VICTOR G. KAC,
and MINORU WAKIMOTO .......................................... 79
String Duality and a New Description of the E6 Singularity
TOHRU EGUCHI .................................................. 131
A Mirror Theorem for Toric Complete Intersections
ALEXANDER GIVENTAL ........................................... 141
Precious Siegel Modular Forms of Genus Two
VALERI GRITSENKO .............................................. 177
Non-Abelian Conifold Transitions and N = 4 Dualities
in Three Dimensions
KENTARO HORI, HIROSI OOGURI, and CUMRUN VAFA ............ 205
GKZ Systems, Grabner Fans, and Moduli Spaces
of Calabi-Yau Hypersurfaces
SHINOBU HOSONO .......... ". ..................................... 239
Semisimple Holonomic V-Modules
MASAKI KASHIWARA ............................................. 267
K3 Surfaces, Igusa Cusp Forms, and String Theory
TOSHIYA KAWAI .................................................. 273
Hodge Strings and Elements of K. Saito's Theory
of Primitive Form
ANDREI LOSEV ................................................... 305
Summary of the Theory of Primitive Forms
ATSUSHI MATSUO ................................................ 337
vi Contents
Affine Hecke Algebras and Macdonald Polynomials
MASATOSHI NOUMI ............................................... 365
Duality for Regular Systems of Weights: A Precis
KYOJI SAITO ..................................................... 379
Flat Structure and the Prepotential for the Elliptic Root System
of Type Dil,l)
IKUO SATAKE .................................................... 427
Generalized Dynkin Diagrams and Root Systems and Their Folding
JEAN-BERNARD ZUBER .......................................... .453
Preface
This volume is the proceedings of the 38th Taniguchi Symposium
"Topological Field Theory, Primitive Forms and Related Topics" (Decem
ber 9-13, 1996, at Kansai Seminar House, Kyoto) and the RIMS Sympo
sium with the same title (December 16-19 at the Research Institute for
Mathematical Sciences, Kyoto University).
In these days, the interaction of mathematics and theoretical physics
becomes stronger and stronger. Many ideas developed in physics stimu
late mathematics. Conversely, the theories developed in mathematics have
been applied to physics. For example, the theory of primitive forms was
originally developed as a theory for the period of vanishing cycle for the
hypersurface singularities. Recent development reveals its deep connection
to the concepts such as the flat structure and the WDW equations, which
appear in the topological field theory. This resonance is a main subject of
these symposia. We hope that this volume will serve both mathematicians
and physicists.
We would like to thank the Taniguchi Foundation for its generous
financial support. We would like to thank all the participants of the sym
posia for their excellent talks and contributed papers.
Special thanks go to Kumiko Matsumura for her excellent job preparing
these symposia.
The papers in this volume are all refereed. We would like to express our
sincere gratitude to all the referees for their invaluable help to this volume.
M. Kashiwara
EDITOR IN CHIEF
Symposia
The 38th Taniguchi Symposium
Topological Field Theory, Primitive Forms and Related Topics
December 9-13, 1996, at Kansai Seminar House, Kyoto
Organizing Committee
Kyoji Saito (chief)
Masaki Kashiwara
Atsushi Matsuo
Ikuo Satake
Participants
Richard E. Borcherds (Department of Pure Mathematics, University
of Cambridge)
Tooru Eguchi (Department of Physics, University of Tokyo)
Shinobu Hosono (Department of Mathematics, Toyama University)
Masaki Kashiwara (RIMS, Kyoto University)
Toshiya Kawai (RIMS, Kyoto University)
Eduard Looijenga (Mathematics Department, University of Utrecht)
Andrei S. Losev (Department of Physics, Yale University)
Atsushi Matsuo (Graduate School of Mathematical Sciences, University
of Tokyo)
David R. Morrison (Department of Mathematics, Duke University)
Masatoshi Noumi (Department of Mathematics, Kobe University)
Hirosi Ooguri (Department of Physics, University of California at Berkeley)
Kyoji Saito (RIMS, Kyoto University)
Ikuo Satake (Department of Mathematics, Osaka University)
Peter Slodowy (Mathematisches Seminar, Universitat Hamburg)
Akihiro Tsuchiya (Graduate School of Polymathematics, Nagoya
University)
Jean-Bernard Zuber (Phys. TMorique-C.E. de Saclay)
The RIMS Symposium
Topological Field Theory and Related Topics
December 16-19, 1996,
at the Research Institute for Mathematics Sciences, Kyoto University
Organizer: Kyoji Saito
Participants: 144
Degenerate Double Affine Heeke Algebra
and Conformal Field Theory
Tomoyuki Arakawa, Takeshi Suzuki, t and Akihiro Tsuchiya
Abstract
We introduce a class of induced representations of the degener
ate double affine Hecke algebra H of g[N(C) and analyze their struc
ture mainly by means of intertwiners of H . We also construct them
from .slm(C)-modules using Knizhnik-Zamolodchikov connections in
the conformal field theory. This construction provides a natural quo
tient of induced modules, which turns out to be the unique irreducible
one under a certain condition. Some conjectural formulas are presented
for the symmetric part of these quotients.
In this paper, the representations of the degenerate double affine Hecke
algebra H are discussed from view points of the conformal field theory as
sociated to the affine Lie algebra .s[mCc). The relations between the KZ
connections of the conformal field theory and the representations of the de
generate affine Hecke algebra was first discussed by Cherednik [1]. Matsuo
[2] succeeded in clarifying the relations between the differential equations
satisfied by spherical functions and KZ-connections.
In the first part of this paper we discuss the properties of the parabolic
induced modules of H, which are induced from certain one-dimensional rep
resentations of parabolic subalgebras of H.
Secondly, we will give an explicit construction of H-modules from .s[mCC)
modules through KZ-connections. It will be shown that these H-modules aris
ing from Verma modules of .s[mCC) correspond to parabolic induced modules
of H.
In the final part of this paper, we will discuss the structure of the represen
tations of the affine Hecke algebra arising from level .e integral representations
of St.nCc).
We describe the contents of the paper more precisely:
tSupported by JSPS, the Research Fellowships for Young Scientists.
Received: October 6, 1997; revised January 15, 1997.
AMS subject classification: 17B67, 22E50, 81 T40
M. Kashiwara et al. (eds.), Topological Field Theory, Primitive Forms and Related Topics
© Springer Science+Business Media New York 1998
2 T. Arakawa, T. Suzuki, and A. Tsuchiya
In §1 we introduce basic notions about the degenerate double affine Hecke
algebra 1{. In particular, intertwiners of weight spaces of 1{ play essential
roles in the analysis of 1{-modules.
In §2, we introduce parabolic induced modules and investigate their struc
ture when the parameter is generic (Definition 2.4.1), and the results are
Proposition 2.4.3, Theorem 2.4.7, and Corollary 2.4.8:
(1) The irreducibility of the standard modules are shown and their bases
are described by intertwining operators.
(2) Decompositions of the standard modules as H-modules are obtained.
(3) The symmetric part of the standard modules are decomposed into
H,
weight spaces with respect to the action of the center of and their
basis are constructed again by using intertwiners.
In the nongeneric case, we present a sufficient condition for an induced
module to have a unique irreducible quotient (Corollary 2.5.4).
§3 is devoted to some preliminaries on affine Lie algebras, and in §4 we
realize 1{-modules as a quotient space of a tensor product of 9 = slm(C)
modules. More precisely, for g-modules A, B, we consider the space
F(A,B)
/gl
= (A® ®~l (qz;l] ® cm) ® B) (A® ®~l (qztl] ®Cm) ® B),
where qztl] ® Cm is an evaluation module of 9 and g' = [g, g] acts diag
onally on the tensor product. By combining the Knizhnik-Zamolodchikov
connection with the Cherednik-Dunkl operator, we define an action of 1{ on
F(A, B) (Theorem 4.2.2).
In §5, we construct the isomorphism between a parabolic induced module
and M(f..l, A) := F(M(f..l), M*(A)) for the highest and lowest Verma module
M(f..l) and M*(A), A and f..l being weights of 9 (Proposition 5.2.3).
Since our construction turns out to be functorial, V(f..l, A) .-
F(L(f..l), L*(,X)) gives a quotient module of M(f..l, A), where L(f..l) and L*('x)
are the irreducible quotients of M(f..l) and M*(A), respectively. We focus on
the case where A and f..l are both dominant integral weights and study V(f..l, ,X).
Finally, in §6, we focus on the symmetric part of V(f..l, A) for dominant
integral weights A, f..l, and present a description of the basis by intertwiners
(Conjecture 6.2.6) as a consequence of the character formula (84), which
is still conjectural since it is proved under the assumption that a certain
sequence of 1{-modules (coming from BGG exact sequence of g) is exact
(Conjecture 6.1.1).
