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CONTEMPORARY MATHEMATICS 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 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, 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.
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