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

Gromov-Witten Theory of Spin Curves and Orbifolds

http://dx.doi.org/10.1090/conm/403

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

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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, University of Michigan, 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

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.

I87

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; University of Utah; 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 L8 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 string theory) and the ordinary and equivariant homologies of L8 X. I will also show how this clarifies the relation between orbifold K-theory, Chen-Ruan orbifold cohomology and periodic cyclic homology.

SPECIAL SESSION

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.

189

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 f = 0, asi = a- ' 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 algebraic geometry, 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