CONTEMPORARY MATHEMATICS 403

Gromov-Witten Theory of Spin Curves and Orbifolds

AMS Special Session on Gromov-Witten Theory of Spin Curves and Orbifolds May 3-4, 2003 , San Francisco State University San Francisco, California

Tyler J. Jarvis Takashi Kimura Arkady Vaintrob Editors http://dx.doi.org/10.1090/conm/403

Gromov-Witten Theory of Spin Curves and Orbifolds CoNTEMPORARY MATHEMATICS

403

Gromov-Witten Theory of Spin Curves and Orbifolds

AMS Special Session on Gromov-Witten Theory of Spin Curves and Orbifolds May 3-4, 2003 San Francisco State University San Francisco, California

Tyler J. Jarvis Takashi Kimura Arkady Vaintrob Editors

American Mathematical Society Providence. Rhode Island Editorial Board Dennis DeTurck, managing editor George Andrews Carlos Berenstein Andreas Blass Abel Klein This volume contains the proceedings of an AMS Special Session entitled "Gromov- Witten theory of spin curves and orbifolds" held May 3~4, 2003, at San Francisco State University, San Francisco, CA, with support from NSF grants DMS-0105788, DMS-0204824, and DMS-0104397.

2000 Mathematics Subject Classification. Primary 14N35, 53D45.

Library of Congress Cataloging-in-Publication Data AMS Special Session on Gromov-Witten Theory of Spin Curves and Orbifolds (2003 : San Fran- cisco, Calif.) Gromov-Witten theory of spin curves and orbifolds: AMS Special Session on Gromov-Witten Theory of Spin Curves and Orbifolds, May 3~4, 2003, San Francisco, California/ Tyler J. Jarvis, Takashi Kimura, Arkady Vaintrob, editors. p. em. - (Contemporary mathematics, ISSN 0271-4132 ; 403) Includes bibliographical references. ISBN 0-8218-3534-3 (alk. paper) 1. Gromov-Witten invariants-Congresses. 2. Frobenius manifolds-Congresses. 3. Orbifolds- Congresses. 4. Singularities (Mathematics)-Congresses. 5. Homology theory-Congresses. I. Jarvis, Tyler Jamison, 1966- II. Kimura, Takashi, 1963- III. Vaintrob, Arkady, 1956- IV. Title. V. Contemporary mathematics (American Mathematical Society) ; v. 403. QA665.A47 2003 516'.07-dc22 2005057126

Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Math- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to reprint-permissiontnams. org. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) © 2006 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Copyright of individual articles may revert to the public domain 28 years after publication. Contact the AMS for copyright status of individual articles. Printed in the United States of America. § The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http: I /www. ams. org/ 10 9 8 7 6 5 4 3 2 1 11 10 09 08 07 06 Contents

Introduction vii Moduli Spaces of Curves with Effective r-Spin Structures A. POLISHCHUK 1 A Construction of Witten's Top Chern Class in K-Theory ALESSANDRO CHIODO 21 Witten's Conjecture and the Virasoro Conjecture for Genus up to Two Y.-P. LEE 31 ldempotents on the Big Phase Space XIAOBO LIU 43 Singularities with Symmetries, Orbifold Frobenius Algebras and Mirror Symmetry RALPH M. KAUFMANN 67 The Cohomology Ring of Crepant Resolutions of Orbifolds YONGBIN RUAN 117 Differential Characters on Orbifolds and String Connections I: Global Quotients ERNESTO LUPERCIO and BERNARDO URIBE 127 HKR characters and higher twisted sectors JACK MORAVA 143 Combinatorics of Binomial Decompositions of the Simplest Hodge Integrals S. V. SHADRIN 153 The Orbifold Cohomology of the Moduli of Genus-Two Curves JAMES SPENCER 167 List of Participants and Abstracts 185

v Introduction

In the past few years it has become quite clear that the theory of r-spin curves and the theory of Gromov-Witten invariants have much in common. That connec- tion has been made even more definite by the recent developments in the theory of orbicurves and orbifold stable maps, as developed by W. Chen, Y. Ruan, D. Abramovich, A. Vistoli and their collaborators. That theory has not only simpli- fied the r-spin constructions considerably, but has also shown the spin theory to be very similar to orbifold Gromov-Witten theory. The conference in San Francisco was intended to bring together researchers working on a wide variety of different aspects of these theories in the hope of further illuminating these connections. The conference was very successful and the papers here go even farther than the conference in bringing together many formerly disparate aspects of the theory of spin curves and orbifold Gromov-Witten theory. The first three papers, those of A. Polishchuk, A. Chiodo, and Y.-P. Lee, are directly connected to Witten's Conjecture for r-spin curves, which was the initial motivation for much of the interest in r-spin curve theory. Polishchuk and Chiodo's papers are closely tied to issues of the virtual class on the moduli space or r-spin curves, whereas Y.-P. Lee's paper shows that, from the axiomatic properties of that class, and using Givental's potentials with known relations of Dubrovin, Zhag, Getzler, and X. Liu, the Witten conjecture holds in genus 1 and 2. The relations developed by X. Liu play an important role in Y.-P. Lee's proof, and they also help illuminate other essential structures in general Gromov-Witten theory and other cohomological field theories. Liu's paper included here, on rela- tions in the large phase space, is even more strongly connected to r-spin theory, since the descent axiom of that theory, coupled with the theory of stable spin maps, illustrates a surprising connection between the large phase space of usuftl Gromov- Witten theory and the r-spin correlators. Another important aspect of the theory of r-spin curves and its generalizations is the relation it has to singularities, especially as a sort of Landau-Ginzberg A- model. The bridge from spin theories to singularities is through the orbifolding of Frobenius algebras (and cohomological field theories), as described in the paper of Kaufmann. Orbifold cohomology is also closely tied to singularity theory, as detailed in the various results and conjectures described in Y. Ruan's paper on crepant resolutions. And just as Ruan's work illuminates the connections between orbifold cohomology of a quotient singularity and the ordinary cohomology of a crepant resolution, so also the paper of Lupercio and Uribe shows that other cohomology theories on orbifolds are related, in particular the Beilinson-Deligne cohomology and the Cheeger-Simons cohomology of a global quotient orbifold are canonically isomorphic.

vii viii INTRODUCTION

An important part of the relationship described by Ruan between orbifold cohomology and the usual cohomology of resolutions is the fact that orbifold co- homology keeps track of not only the so-called "untwisted sector," but also that of twisted sectors, corresponding to the various fixed-point loci of isotropy groups. These appear to carry precisely the information needed to reconstruct the ordinary cohomology of the crepant resolution of the singularity, at least in many interesting cases. Spin curve theory also has a state space that is larger than one might have expected, corresponding in some way to a collection of twisted sectors. The paper of Morava examines the twisted sector constructions and shows how in a general category of orbispaces the inertia stack constructions can be used and generalized to understand and organize the higher twisted sectors. Finally, despite the power and interest of these theories, there is still a real shortage of computed examples, so the computation by S. Shadrin of Hodge in- tegrals and that of James Spencer of the orbifold cohomology of the moduli of genus-two curves (especially since this moduli space is so important in its own right) are welcome contributions. We are very pleased to make these papers available to the public in this volume, and we expect to see much more activity at the confluence of these exciting subjects in the near future.

T.J.J., T.K., and A.V. Meeting Participants and Abstracts

Gromov-Witten Theory of Spin Curves and Orbifolds Special session of the 2003 Spring Western Section Meeting of the American Mathematical Society San Francisco State University, San Francisco, California May 3-4, 2003

Plenary Lecture Arkady Vaintrob. Department of Mathematics; University of Oregon; Eugene, Oregon 97 403; USA Title: Higher spin curves and Gromov- Witten theory. Abstract No. 987-14-170: Witten conjectured and Kontsevich proved that the intersection theory on the moduli spaces of Riemann surfaces is governed by an infinite hierarchy of partial differential equations of the Korteweg-de Vries type. Later Witten generalized his original conjecture to the case of higher spin curves and the Gelfand-Dickey integrable hierarchies. In the talk we will discuss this conjecture and recent progress in this area from the point of view of the theory of Gromov-Witten invariants and co homological quantum field theories. Special Session Bernardo Uribe. Department of Mathematics, , Ann Arbor, Michigan 48109; USA Title: Gerbes over orbifolds and their TQFTs. Abstract No. 987-55-61: In this talk I will discuss my recent work (with Ernesto Lupercio) on which, following Segal, we associate to every gerbe with connection (B-field) a line bundle with connective structure over the loop orbifold, via a generalized holonomy map. I will show how this line bundle once localized to the inertia orbifold (twisted sectors) produces local coefficients (fiat line bundles) for the orbifold cohomology (these coincide with the "inner local systems" in Ruan's theory.) I will also explain how these line bundles are the coefficients that Freed-Hopkins-Teleman use in their work of Twisted equivariant K-theory. When the orbifold is a manifold we recuperate the 2-dimensional TQFT constructed by Gawedzki.

185 186 MEETING PARTICIPANTS AND ABSTRACTS

Xiaobo Liu. Department of Mathematics; University of Notre Dame; Notre Dame, Indiana 46556; USA Title: From moduli space of curves to Gromov- Witten invariants. Abstract No. 987-53-78: In this talk, I will discuss how much imformation on Gromov Witten invariants of compact symplectic manifolds (or smooth projective varieties) can (or expected to) be obtained from knowledge of tautological ring of moduli space of stable curves. Tom Graber. Department of Mathematics; University of California; Berkeley, California 94720; USA Title: Orbifold Gromov- Witten theory. Abstract No. 987-14-163: I will discuss the algebro-geometric definition of orbifold Gromov-Witten invariants, and illustrate their computability by computing some quantum cohomology rings of orbifolds which are not global quotients. Dan Abramovich. Department of Mathematics; Brown University; Providence, Rhode Island 02912; USA James Spencer. Department of Mathematics; Rice University; Houston, Texas 77005; USA Title: The stringy Chow rings of M2 and M 2. Abstract No. 987-14-151: The stringy Chow ring of M 2 with integer coefficients, and the analogous ring of M 2 .with rational coefficients are described. Some ideas regarding the latter ring with integer coefficients are discussed. Mainak Poddar. Department of Mathematics; Michigan State University; East Lansing, Michigan 48824; USA Title: The Global Mackay-Ruan Correspondence via Motivic Integration. Abstract No. 987-14-189: We show how the motivic integration methods of Kontsevich, Denef-Loeser and Looijenga can be adapted to prove the McKay-Ruan correspondence, a generalization of the cohomological McKay-Reid correspondence to complete algebraic varieties with only Gorenstein quotient singularities (that are not necessarily global quotients). This is a joint work with E. Lupercio. Tyler J Jarvis. Department of Mathematics; Brigham Young University; Provo, Utah 84602; USA Title: New cohomological field theories arising from quasi-homogeneous polynomials. Abstract No. 987-14-160: Cohomological field theories-that is, systems of cohomology classes satisfying cutting axioms like those of Gromov-Witten classes-are relatively rare. The most studied source of these is, of course, the theory of stable maps. Another source is the theory of r-spin curves. I will discuss recent joint work with Huijun Fan and Yong-Bin Ruan on new cohomological field theories defined by the choice of a quasi-homogeneous polynomial W with an isolated singularity at the origin. SPECIAL SESSION I87

Takashi Kimura. Department of Mathematics; Boston University; Boston, Massachusetts 02215; USA Title: Pointed admissible G-covers, equivariant topological field theories and analogies with the moduli space of higher spin curves. Abstract No. 987-14-192: We generalize the notion of an equivariant topological field theory associated to a finite group G through the introduction of the moduli space of pointed admissible G-covers in a joint work with T. J. Jarvis and R. Kaufmann. In this talk, we will explain its relationship to orbifold cohomology and will describe an analogy between such theories and the r-spin cohomological field theory from the moduli space of higher spin curves. Alexander Givental. Department of Mathematics; University of California; Berkeley, California 94 720; USA Title: nKdV hierarchies and An-I-singularities. Abstract No. 987-14-103: According to a conjecture of E. Witten, intersection theory on moduli spaces of complex curves with n-spin structures is governed by an approporiate tau-function of the nKdV hierarchy of integrable systems. In genus 0, the intersection theory in question is known to yield the Frobenius structure of the An-I-singularity. We study the relationship between Frobenius structures, singularities and integrable hierarchies. Our results show, in particular, that Witten's conjecture follows from the corrsponding Virasoro constraints. Yuan-Pin Lee. Department of Mathematics; ; Salt Lake City, Utah 84112-0090; USA Title: Semisimple Probenius manifolds and higher spin curves. Abstract No. 987-14-149: A project to study correlators of higher spin curves via the theory of semisimple Frobenius manifolds will be presented. Some progress related to the generalized Witteng conjecture in genus 1 and 2 will be reported. Alexander Polishchuk. Department of Mathematics; University of Oregon; Eugene, Oregon 97403; USA Title: Some remarks on Witten's top Chern class. Abstract No. 987-14-113: Generalized Witten conjecture asserts that certain intersection numbers for moduli spaces of stable curves with n-spin structures can be arranged in a generating function that will be a tau-function of the nKdV-hierarchy. I will show how to express these intersection numbers in terms of standard intersection numbers for the usual moduli spaces of curves. 188 MEETING PARTICIPANTS AND ABSTRACTS

Alessandro Chiodo. Laboratoire Dieudonne, Universite de Nice, Valrose, 06108 Nice cedex 2, FRANCE Title: Witten's top Chern class in K-theory. Abstract No. 987-14-127: We present a generalization of the top Chern class in algebraic K-theory. We use it to define Witten's top Chern class directly in the K-ring compatibly with Polishchuk-Vaintrob's definition in the Chow ring. Jack Morava. Department of Mathematics; Johns Hopkins University; Baltimore, Maryland 21218; USA Title: A homotopy-theoretical Heisenberg group. Abstract No. 987-55-183: We study the Madsen-Tillmann spectrum CP~ as a quotient of the Mahowald pro-object CP~ 00 , which is closely related to the Tate cohomology of circle actions. That theory has an associated symplectic structure, whose symmetries define the Virasoro operations on the cohomology of compactified moduli space constructed by Kontsevich and Witten. Ralph M. Kaufmann. Department of Mathematics; Oklahoma State University, Department of Mathematics, MS 401, Stillwater Oklahoma 74078; USA Title: On Gromov- Witten theory for global quotients. Abstract No. 987-14-176: In the setting of global quotients with respect to an action of a finite group G, all the ingredients of Gromov-Witten theory should be replaced by their G-equivariant counterparts. The first step being the replacement of the Frobenius algebras, or topological field theory, corresponding to the cohomology, which is to be deformed. The second structure is that of cohomological field theory which provides the link between the geometry of the deformations and their algebra. This information is encoded in the cohomology of certain moduli spaces of curves with extra structure. Lastly, the moduli spaces of maps of these curves with extra structure together with their virtual fundamental classes realizing the algebraic deformation structure on cohomology have to be studied. We will discuss several aspects of this program, of which we have completed the first step and the second in collaboration with Tyler Jarvis and Takashi Kimura. Our joint work also contains partial results on the last step. Ernesto Lupercio. Departamento de Matematicas, CINVESTAV, Apartado Postal 14-740 07000 Mexico, D.F. MEXICO Title: Inertia Orbifolds, Configuration Spaces and the Ghost Loop Space. Abstract No. 987-55-171: In this talk I will define the ghost loop orbifold £ 8 X of an orbifold X consisting of those loops that remain constant in the coarse moduli space of X. I will explain a configuration

space model for L 8 X. From this I will exhibit the relation between the Hoschild and cyclic homologies of the inertia orbifold of X (that generate the so-called twisted sectors in ) and the ordinary and

equivariant homologies of L 8 X. I will also show how this clarifies the relation between orbifold K-theory, Chen-Ruan orbifold cohomology and periodic cyclic homology. SPECIAL SESSION 189

Huijun Fan. Max Planck Institute for Mathematics in the Sciences; D-04103 Leipzig; Germany Title: A Generalization of Spin Orbifold Quantum Cohomology Arising From Quasi-Homogeneous Polynomials. Abstract No. 987-53-122: W-spin orbifold quantum cohomology is a combination and generalization of two theories, Chen-Ruan' s orbifold quantum cohomology and Jarvis-Kimura-Vaintrob's quantum cohomology theory on higher-spin curves. We show that the two theories can be combined in the orbifold category. To construct the CohFT, we study the module space of the decoupled two equations on orbifold: - - aw a 0, asi a- f = = Si ' where W is a nondegenerate quasi-homogeneous polynomial (if W(s) = ~sr, then the second equation is the r-spin equation). We prove if W is pure Neveu-Schwarz, then we can construct a CohFT. If the target symplectic manifold is a point, then we deduce the CohFT of higher spin curves. In addition, we can get the W -spin cohomological field theory. Yongbin Ruan. Department of Mathematics; University of Wisconsin; Madison, Wisconsin 53706; USA Title: Gerbe and twisted orbifold quantum cohomology. Abstract No. 987-53-104: In this talk, we will show how to use gerbes to twist orbifold quantum cohomology. Titles in This Series

403 Tyler J. Jarvis, Takashi Kimura, and Arkady Vaintrob, Editors, Gromov-Witten theory of spin curves and orbifolds, 2006 402 Zvi Arad, Mariagrazia Bianchi, Wolfgang Herfort, Patrizia Longobardi, Mercede Maj, and Carlo Scoppola, Editors, Ischia group theory 2004, 2006 401 Katrin Becker, Melanie Becker, Aaron Bertram, PaulS. Green, and Benjamin McKay, Editors, Snowbird lectures on string geometry, 2006 400 Shiferaw Berhanu, Hua Chen, Jorge Hounie, Xiaojun Huang, Sheng-Li Tan, and Stephen S.-T. Yau, Editors, Recent progress on some problems in several complex variables and partial differential equations, 2006 399 Dominique Arlettaz and Kathryn Hess, Editors, An Alpine anthology of homotopy theory, 2006 398 Jay Jorgenson and Lynne Walling, Editors, The ubiquitous heat kernel, 2006 397 Jose M. Munoz Porras, Sorin Popescu, and Rubi E. Rodriguez, Editors, The geometry of Riemann surfaces and Abelian varieties, 2006 396 Robert L. Devaney and Linda Keen, Editors, Complex dynamics: Twenty-five years after the appearance of the Mandelbrot set, 2006 395 Gary R. Jensen and Steven G. Krantz, Editors, 150 Years of Mathematics at Washington University in St. Louis, 2006 394 Rostislav Grigorchuk, Michael Mihalik, Mark Sapir, and Zoran Sunik, Editors, Topological and asymptotic aspects of group theory, 2006 393 Alec L. Matheson, Michael I. Stessin, and Richard M. Timoney, Editors, Recent advances in operator-related function theory, 2006 392 Stephen Berman, Brian Parshall, Leonard Scott, and Weiqiang Wang, Editors, Infinite-dimensional aspects of representation theory and applications, 2005 391 Jiirgen Fuchs, Jouko Mickelsson, Grigori Rozenblioum, Alexander Stolin, and Anders Westerberg, Editors, Noncommutative geometry and representation theory in mathematical physics, 2005 390 Sudhir Ghorpade, Hema Srinivasan, and Jugal Verma, Editors, Commutative algebra and , 2005 389 James Eells, Etienne Ghys, Mikhail Lyubich, Jacob Palis, and Jose Searle, Editors, Geometry and dynamics, 2005 388 Ravi Vakil, Editor, Snowbird lectures in algebraic geometry, 2005 387 Michael Entov, Yehuda Pinchover, and Michah Sageev, Editors, Geometry, spectral theory, groups, and dynamics, 2005 386 Yasuyuki Kachi, S. B. Mulay, and Pavlos Tzermias, Editors, Recent progress in arithmetic and algebraic geometry, 2005 385 Sergiy Kolyada, Yuri Manin, and Thomas Ward, Editors, Algebraic and topological dynamics, 2005 384 B. Diarra, A. Escassut, A. K. Katsaras, and L. Narici, Editors, Ultrametric functional analysis, 2005 383 Z.-C. Shi, Z. Chen, T. Tang, and D. Yu, Editors, Recent advances in adaptive computation, 2005 382 Mark Agranovsky, Lavi Karp, and David Shoikhet, Editors, Complex analysis and dynamical systems II, 2005 381 David Evans, Jeffrey J. Holt, Chris Jones, Karen Klintworth, Brian Parshall, Olivier Pfister, and Harold N. Ward, Editors, Coding theory and quantum computing, 2005 380 Andreas Blass and Yi Zhang, Editors, Logic and its applications, 2005 379 Dominic P. Clemence and Guoqing Tang, Editors, Mathematical studies in nonlinear wave propagation, 2005 TITLES IN THIS SERIES

378 Alexandre V. Borovik, Editor, Groups, languages, algorithms, 2005 377 G. L. Litvinov and V. P. Maslov, Editors, Idempotent mathematics and mathematical physics, 2005 376 Jose A. de Ia Peiia, Ernesto Vallejo, and Natig Atakishiyev, Editors, Algebraic structures and their representations, 2005 375 Joseph Lipman, Suresh Nayak, and Pramathanath Sastry, Variance and duality for cousin complexes on formal schemes, 2005 374 Alexander Barvinok, Matthias Beck, Christian Haase, Bruce Reznick, and Volkmar Welker, Editors, Integer points in polyhedra-geometry, number theory, algebra, optimization, 2005 373 0. Costin, M. D. Kruskal, and A. Macintyre, Editors, Analyzable functions and applications, 2005 372 Jose Burillo, Sean Cleary, Murray Elder, Jennifer Taback, and Enric Ventura, Editors, Geometric methods in group theory, 2005 371 Gui-Qiang Chen, George Gasper, and Joseph Jerome, Editors, Nonlinear partial differential equations and related analysis, 2005 370 Pietro Poggi-Corradini, Editor, The p-harmonic equation and recent advances in analysis, 2005 369 Jaime Gutierrez, Vladimir Shpilrain, and Jie-Tai Yu, Editors, Affine algebraic geometry, 2005 368 Sagun Chanillo, Paulo D. Cordaro, Nicholas Hanges, Jorge Hounie, and Abdelhamid Meziani, Editors, Geometric analysis of PDE and several complex variables, 2005 367 Shu-Cheng Chang, Bennett Chow, Sun-Chin Chu, and Chang-Shou Lin, Editors, Geometric evolution equations, 2005 366 Bernheim Bool3-Bavnbek, Gerd Grubb, and Krzysztof P. Wojciechowski, Editors, Spectral geometry of manifolds with boundary and decompositon of manifolds, 2005 365 Robert S. Doran and Richard V. Kadison, Editors, Operator algebras, quantization, and non-commutative geometry, 2004 364 Mark Agranovsky, Lavi Karp, David Shoikhet, and Lawrence Zalcman, Editors, Complex analysis and dynamical systems, 2004 363 Anthony To-Ming Lau and Volker Runde, Editors, Banach algebras and their applications, 2004 362 Carlos Concha, Raul Manasevich, Gunther Uhlmann, and Michael S. Vogelius, Editors, Partial differential equations and inverse problems, 2004 361 Ali Enayat and Roman Kossak, Editors, Nonstandard models of arithmetic and set theory, 2004 360 Alexei G. Myasnikov and Vladimir Shpilrain, Editors, Group theory, statistics, and cryptography, 2004 359 S. Dostoglou and P. Ehrlich, Editors, Advances in differential geometry and general relativity, 2004 358 David Burns, Christian Popescu, Jonathan Sands, and David Solomon, Editors, Stark's Conjectures: Recent work and new directions, 2004 357 John Neuberger, Editor, Variational methods: open problems, recent progress, and numerical algorithms, 2004 356 ldris Assani, Editor, Chapel Hill ergodic theory workshops, 2004

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/. This volume is a collection of articles on orbifolds, algebraic curves with higher spin structures, and related invariants of Gromov-Witten type. Orbifold Gromov-Witten theory generalizes quantum cohomology for orbifolds, whereas spin cohomological field theory is based on the moduli spaces of higher spin curves and is related by Witten's conjecture to the Gelfand-Dickey integrable hierarchies. A common feature of these two very different looking theories is the central role played by orbicurves in both of them. Insights in one theory can often yield insights into the other. This book brings together for the first time papers related to both sides of this interaction. The articles in the collection cover diverse topics, such as geometry and topology of orbifolds, cohomological field theories, orbifold Gromov-Witten theory, G-Frobenius algebra and singularities, Frobenius manifolds and Givental's quantization formalism, moduli of higher spin curves and spin cohomological field theory.

ISBN 0- 8218- 3534- 3

9 780821 835340