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Graphs and Patterns in Mathematics and Theoretical Physics, Volume 73 http://dx.doi.org/10.1090/pspum/073 Graphs and Patterns in Mathematics and Theoretical Physics This page intentionally left blank Proceedings of Symposia in PURE MATHEMATICS Volume 73 Graphs and Patterns in Mathematics and Theoretical Physics Proceedings of the Conference on Graphs and Patterns in Mathematics and Theoretical Physics Dedicated to Dennis Sullivan's 60th birthday June 14-21, 2001 Stony Brook University, Stony Brook, New York Mikhail Lyubich Leon Takhtajan Editors Proceedings of the conference on Graphs and Patterns in Mathematics and Theoretical Physics held at Stony Brook University, Stony Brook, New York, June 14-21, 2001. 2000 Mathematics Subject Classification. Primary 81Txx, 57-XX 18-XX 53Dxx 55-XX 37-XX 17Bxx. Library of Congress Cataloging-in-Publication Data Stony Brook Conference on Graphs and Patterns in Mathematics and Theoretical Physics (2001 : Stony Brook University) Graphs and Patterns in mathematics and theoretical physics : proceedings of the Stony Brook Conference on Graphs and Patterns in Mathematics and Theoretical Physics, June 14-21, 2001, Stony Brook University, Stony Brook, NY / Mikhail Lyubich, Leon Takhtajan, editors. p. cm. — (Proceedings of symposia in pure mathematics ; v. 73) Includes bibliographical references. ISBN 0-8218-3666-8 (alk. paper) 1. Graph Theory. 2. Mathematics-Graphic methods. 3. Physics-Graphic methods. 4. Man• ifolds (Mathematics). I. Lyubich, Mikhail, 1959- II. Takhtadzhyan, L. A. (Leon Armenovich) III. Title. IV. Series. QA166.S79 2001 511/.5-dc22 2004062363 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 [email protected]. 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.) © 2005 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://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 10 09 08 07 06 05 This Volume is Dedicated to Dennis Sullivan's 60th Birthday This page intentionally left blank Contents Preface ix Dennis Sullivan-A short history xiii Dennis Sullivan's List of publications xv Sigma models and string topology 1 DENNIS SULLIVAN Feynman diagrams Feynman diagrams for pedestrians and mathematicians 15 MICHAEL POLYAK Structures in Feynman graphs: Hopf algebras and symmetries 43 DIRK KREIMER Algebraic structures Notes on universal algebra 81 ALEXANDER A. VORONOV The ring of differential operators on forms in noncommutative calculus 105 DMITRI TAMARKIN and BORIS TSYGAN Twisted chiral de Rham algebras on P1 133 VASSILY GORBOUNOV, FYODOR MALIKOV, and VADIM SCHECHTMAN Manifolds: invariants and mirror symmetry Invariants of tangles with flat connections in their complements 151 RINAT KASHAEV and NIKOLAI RESHETIKHIN Tree-level invariants of three-manifolds, Massey products and the Johnson homomorphism 173 STAVROS GAROUFALIDIS and JEROME LEVINE Multivalued Morse theory, asymptotic analysis and mirror symmetry 205 KENJI FUKAYA viii CONTENTS Combinatorial aspects of dynamics Some applications of combinatorial differential topology 281 ROBIN FORMAN Extensions, quotients and generalized pseudo-Anosov maps 315 ANDRE DE CARVALHO Unimodal maps and hierarchical models 339 MICHAEL YAMPOLSKY Physics Quantum geometry in action: big bang and black holes 361 ABHAY ASHTEKAR Supersymmetry, supergravity, superspace and BRST symmetry in a simple model 381 PETER VAN NIEUWENHUIZEN This page intentionally left blank Preface This volume offers a variety of papers based on the talks delivered at the Stony Brook Conference (June 2001) "Graphs and Patterns in Mathematics and Theo• retical Physics" dedicated to Dennis Sullivan's 60th birthday. At the conference, whose scientific content was suggested by Sullivan, an attempt was made to overcome conceptual barriers between experts in various branches of mathematics and theoretical physics who encounter graphs in their research. To achieve this goal, the conference was largely based on mini-courses and survey lectures directed to experts in other areas as well as graduate students. The above idea is reflected in this volume as well: along with research papers, it contains a number of surveys aimed to introduce a reader to the corresponding fields. The volume opens with an article by Dennis Sullivan "Sigma models and string topology". It describes a background algebraic structure for the sigma model based on algebraic topology and transversality. The background allows one to write the quantum master equation of Batalin and Vilkovisky. It is shown that the Gromov moduli space of all J-holomorphic curves gives a solution to this equation, thus providing a mathematical foundation of the quantum field theory aspects of the Gromov-Witten theory. The other contributions are organized into five sections: Feynman Diagrams, Algebraic Structures, Manifolds: Invariants and Mirror Symmetry, Combinatorial Aspects of Dynamics, and Physics. This classification is rather conventional as there is a strong interplay between ideas and methodology presented in different contributions which have a common origin in topology and quantum physics. The articles in the volume are ordered in such a way that survey-style papers are followed (within a given section) by more special research contributions. Feynman diagrams This section opens with a survey "Feynman diagrams for pedestrians and math• ematicians" by M. Polyak. It introduces the reader to the machinery of Feynman diagrams (starting with a finite dimensional model), and then explains how it can be combined with the Chern-Simons theory to produce "quantum" invariants of knots and three-manifolds. The next paper, "Structures of Feynman graphs: Hopf algebras and symme• tries" by D. Kreimer describes algebraic structures on the space of Feynman dia• grams that puts the classical renormalization procedure on a regular basis. Algebraic structures The survey "Notes on universal algebra" by A. Voronov introduces the reader to Hochschild cohomology and deformation quantization, operad theory, and graph ix x PREFACE homology. In particular, it contains a sketch of Kontsevich's proof of the celebrated Formality Theorem, as well as ideas of its proof by Cattaneo and Felder motivated by quantum field theory (based, again, on the machinery of Feynman diagrams). The paper "The ring of differential operators on forms in non-commutative calculus" by D. Tamarkin and B. Tsygan gives a non-commutative generalization of the Hochschild-Kostant-Rosenberg Theorem that identifies the Hochschild co- homology of the commutative algebra of smooth functions on a manifold M with the algebra of poly vector fields on M. In particular, it gives an extension of the Formality Theorem to the case of an arbitrary associative algebra. In the paper "Twisted chiral de Rham algebras on P1" by V. Gorbounov, F. Malikov and V. Schechtman, the authors continue their exploration of chiral de Rham complexes, namely, they construct a twisted chiral de Rham complex of vertex algebras over P1. Manifolds: invariants and mirror symmetry In the paper "Invariants of tangles with flat connections" by R. Kashaev and N. Reshetikhin, the authors define invariants of tangles with flat connections in their complements extending the classical work of Reshetikhin and Turaev. In the next paper, "Tree-level invariants of three-manifolds, Massey products and the Johnson homomorphism" by S. Garoufalidis and J. Levine, the tree-level part of the finite type invariants for 3-manifolds is expressed in terms of classical algebraic topology. In the paper "Multivalued Morse theory, asymptotic analysis, and mirror sym• metry", K. Fukaya proposes a new geometric approach to the homological mirror symmetry conjecture for Calabi-Yau manifolds represented as dual torus fibrations with the same base. Combinatoral aspects of dynamics In the first paper of this section, "Some applications of combinatorial differ• ential topology", R. Forman develops Morse theory for cell complexes based on the dynamics of discrete vector fields, and discusses its applications in topology, geometry, and other fields. The paper "Extensions, quotients and generalized pseudo-Anosov maps" by A. de Carvalho describes the interplay between one- and two-dimensional dynam• ics (on graphs and on surfaces). This leads the author to a notion of a generalized pseudo-Anosov map
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