CONTEMPORARY MATHEMATICS 401

Snowbird Lectures on String Geometry

Proceedings of an AMS-IMS-SIAM Joint Summer Research Conference on String Geometry June 5-ll, 2004 Snowbird, Utah

Kotrin Becker Melanie Becker Aaron Bertram Paul S. Green Benjamin McKoy Editors http://dx.doi.org/10.1090/conm/401

Snowbird Lectures on String Geometry Snowbird Lectures on String Geometry

Proceedings of an AMS-IMS-SIAM Joint Summer Research Conference on String Geometry June 5-1 1,2004 Snowbird, Utah

Katrin Becker Melanie Becker Aaron Bertram Paul S, Green Benjamin McKay Editors

American Mathematical Society Providence, Rhode Island Editorial Board Dennis DeTurck, managing editor George Andrews Carlos Berenstein Andreas Blass Abel Klein The 2004 AMS-IMS-SIAM Joint Research Conference on String Geometry was held in Snowbird, Utah, June 5-11, 2004, with support from the National Science Foundation, grant DMS-9973450.

2000 Mathematics Subject Classification. Primary 14D22, 14F05, 14N10, 14N35, 14Q05, 18E30, 53C80, 81Q30, 81Q60, 81Q70, 81 T13, 81 T30, 81 T45, 81 T60.

Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Library of Congress Cataloging-in-Publication Data AMS-IMS-SIAM Joint Summer Research Conference on String Geometry (2004) Snowbird lectures on string geometry : proceedings of the AMS-IMS-SIAM Joint Summer Research Conference on String Geometry, June 5-11, 2004 / Katrin Becker... [et a!.], editors. p. em. -(Contemporary mathematics; 401) Includes bibliographical references. ISBN 0-8218-3663-3 (acid-free paper) 1. Geometry, Algebraic-Congresses. 2. Moduli theory-Congresses. 3. Gromov-Witten invariants-Congresses. 4. Calabi-Yau manifolds-Congresses. 5. String theory-Congresses. I. Becker, Katrin, 1967- II. Title. III. Contemporary mathematics (American Mathematical Society) ; v. 401. QA564.A5275 2004 516.3'.5-dc22 2005058917

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Preface vii

List of participants XI D-branes, IT-stability and 0-stability PAULS. ASPINWALL 1 Geometric transitions, non-Kahler geometries and string vacua KATRIN BECKER, MELANIE BECKER, KESHAV DASGUPTA AND RADU TATAR 15 On the K-theory of smooth toric DM stacks LEV A. BORISOV AND R. PAUL HORJA 21 Grornov-Witten, Gopakumar-Vafa, and Donaldson-Thomas invariants of Calabi-Yau threefolds SHELDON KATZ 43 Flux compactification geometries and de Sitter vacua in M-theory AXEL KRAUSE 53 N = 4 super-Yang-Mills theory and integrable spin chains RADU ROIBAN 69 Yang-Mills amplitudes from twistor string theory RADU ROIBAN, MARCUS SPRADLIN AND ANASTASIA VOLOVICH 81 Notes on correlation functions in (0, 2) theories ERIC SHARPE 93

v Preface

The interaction and cross-fertilization of mathematics and physics is ubiquitous in the history of both disciplines. However, the recent developments of string theory have led to some relatively new areas of common interest among mathematicians and physicists, and it was to explore some of these that the Snowbird conference of June 2004 was convened. Although not all papers presented were available for inclusion in this volume, the papers included provide a reasonably comprehen- sive sampling of the potential for fruitful interaction between mathematicians and physicists that exists as a result of string theory.

Much of the recent wave of interaction has been, on the mathematical side, in the area of algebraic geometry. There are two principal reasons for this: world sheets in string theory (at least in the Euclidean domain) are Riemann surfaces, and Calabi-Yau manifolds have been primary candidate vacua since the appearance of the paper of Candelas, Horowitz, Strominger and Witten on that subject. These two strands come together in the computation of instanton corrections in string theory, and lead to enumeration problems that are of interest to mathematicians and physicists alike. Perhaps the most obvious manifestation of this are the Gromov- Witten invariants, which are defined in terms of such enumerations and demonstrate the commonality of interest by their very name. The paper of Katz provides a nice review of the Gromov-Witten invariants and the related Gopakumar-Vafa and Donaldson-Thomas invariants. It goes on to present specific conjectural formulae relating the three sets of invariants and assembles evidence for the validity of the conjectures.

While it has been known for decades that some physically motivated construc- tions can be reformulated in terms of such mathematical concepts as sheaves, mod- uli and cohomology, the level of mathematical sophistication involved has increased markedly since the advent of string theory, especially in connection with the theory of D-branes. The concept IT-stability for BPS-brane configurations, with which Aspinwall's paper is concerned, is a case in point. Although motivated by the physical concept of stability against decay, it is defined in terms of the derived category of coherent sheaves on a Calabi-Yau manifold, but depends also on the complexified Kahler structure, regarded as a point in the moduli space of the mirror manifold. Aspinwall studies IT-stability for BPS-brane configurations supported on a Del Pezzo surface embedded in a Calabi-Yau manifold near the "wall" in the Kahler structure moduli space that corresponds to collapse of the Del Pezzo sur- face. In that regime, he is able to relate IT-stability to the more computable notion of 0-stability for quiver representations.

vii viii PREFACE

For some time now, there has been a consensus that toric varieties provide the appropriate context for the study of mirror symmetry from the mathematical point of view. The paper of Borisov and Horja (presented by Horja at the Snowbird conference) is concerned with Toric Deligne-Mumford stacks, which generalize toric varieties, and coherent sheaves over them, which generalize D-brane configurations. Mirror symmetry also generalizes to this context, and the authors promise to apply their results in that direction in a future paper. Sharpe's paper is concerned with (0, 2) mirror symmetry, which generalizes ordinary ((2, 2)) mirror symmetry to relate pairs of Calabi-Yau manifolds, each equipped with a holomorphic vector bundle. The manifolds involved need not be a mirror pair in the ordinary sense, except in the special case that the vector bundles are the holomorphic tangent bundles, in which case (0, 2) mirror symmetry reproduces the usual mirror symmetry. The paper computes quantum correlation functions in this context, expressed as integrals over a compactified moduli space of world sheet instantons, obtained from gauged linear sigma models. Although Calabi-Yau manifolds are, by definition, Kahler manifolds, it has long been known that there are examples of complex threefolds that admit no Kahler structure but satisfy the condition of vanishing canonical bundle that distinguishes Calabi-Yau manifolds from other Kahler manifolds. Although it has been some years since non-Kahler manifolds made their first appearance in in the physics literature in the mid 1980's, there has recently been a marked increase in the level of interest in this area among both mathematicians and physicists. From the string- theoretic point of view, the motivation to study non-Kahler manifolds is provided by the inclusion of fluxes. Such flux compactifications are of interest, as the inclusion of fluxes (i.e. non-vanishing tensor fields) in the background geometry lead to a potential for the moduli fields. String theorists hope that this will lead to a solution to the moduli space problems, so that predictions for the coupling constants for the standard model of elementary particles can be made in the future. From the mathematical point of view, such new background geometries are of interest, as many theorems of algebraic geometry need to be revisited in the context of non- Kahler manifolds with torsion. The papers of Krause and Tatar are examples of this relatively new trend. At the time of the Snowbird conference, the revival of Twistor theory in the context of String theory as Twistor-String theory was just getting underway, ex- ploiting ideas proposed by Witten. The paper of Roiban, Spradlin and Volovich, presented at the conference by Spradlin, follows Witten's prescription to compute scattering amplitudes for gluons in Yang-Mills theory. One of the features of the computation is that it involves integration over moduli spaces of instantons whose target is a complex supermanifold for which a super analogue of the Calabi-Yau condition holds, although the underlying manifold is not Calabi-Yau. Since the discovery of the AdS/CFT correspondence a number of interesting new results for large N strongly coupled Yang-Mills theories have been obtained in terms of the weakly coupled supergravity dual. Many times, there is an interesting mathemati- cal structure behind such gauge theories, which has lead string theorists to believe they might be exactly solvable. This aspect has been emphasized in the paper by R.Roiban, who in his talk at the conference pointed out the existence of a relation between N = 4 Yang-Mills theory and an integrable spin chain. PREFACE ix

To summarize, many new an interesting relations between mathematics and string theory have appeared in recent times and some of these aspects have been discussed at this conference. It has been more than a year since the Snowbird con- ference on String Geometry took place and the interaction between both disciplines continues growing. Both mathematicians and physicists have greatly benefited from this and hope that in the future advances in physics will lead to even greater ad- vances in mathematics and vice versa. List of Participants

Following is a list of the speakers at the Paul Horja, Toric Deligne-Mumford Snowbird conference with the titles of stacks and homological mirror their talks. The starred titles symmetry* correspond to papers that appear in Sheldon Katz, GW, GV, DT Invariants this volume. We have provided preprint on Calabi-Yau 3-folds* references for as many as possible of the others. Axel Krause, Flux Compactification Geometries and de Sitter in Heterotic Paul Aspinwall, D-branes on del Pezzo M-Theory* surfaces* Radu Roiban, On spin chains and field Robert Bryant, Geometry of G2 and theories* Spin(7) structures1 Eric Sharpe, Notes on correlation Philip Candelas, Arithmetic of functions in (0,2) theories* Calabi-Yau manifolds2 Marcus Spradlin, Yang-Mills Emanuel Diaconescu, Extremal Amplitudes from String Theory in transitions in Gromov-Witten theory3 Twistor Space* Ron Donagi, The particle spectrum of Andrew Strominger, Black hole 9 heterotic compactifications4 attractors and the topological string Michael Douglas, Open problems in Radu Tatar, Geometric Transitions, Calabi-Yau compactification of lib Flops and Non-Kahler Manifolds* string theory5 Daniel Freed, The C-field in M-theory6 Tamar Friedmann, GUTs and dualities from singular G 2-geometry7 Sergei Gukov, Perturbative Yang-Mills theory and moduli spaces of curves8

1 math.DG/0305124 2 hep-th/0012233, hep-th/0402133 3 hep-th/0302076 4 hep-th/0405014 5 hep-th/0303194, hep-th/0307049, math.CV /0402326, hep-th/0404116 6 hep-th/0409135 7 hep-th/0203256, hep-th/0211269 shep-th/0404085 9 hep-th/0405146

xi Titles in This Series

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 Rub! 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 Seade, 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 Kolya"da, 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 378 Alexandre V. Borovik, Editor, Groups, languages, algorithms, 2005 377 G. L. Litvinov and V. P. Maslov, Editors, Idempotent mathematics and mathematical physics, 2005 TITLES IN THIS SERIES

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 BooB-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 355 William Abikoff and Andrew Haas, Editors, In the tradition of Ahlfors and Bers, III, 2004

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstorej. The interaction and cross-fertilization of mathematics and physics is ubiquitous in the history of both disciplines. In particular, the recent developments of string theory have led to some relatively new areas of common interest among mathematicians and physicists, some of which are explored in the papers in this volume. These papers provide a reason- ably comprehensive sampling of the potential for fruitful interaction between mathemati- cians and physicists that exists as a result of string theory.

ISBN 0-8218-3663-3