DOCSLIB.ORG

A Contemporary Study in Gauge Theory and Mathematical Physics: Holomorphic Anomaly in Gauge Theory on ALE Space & Freudenthal Gauge Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Contemporary Study in Gauge Theory and Mathematical Physics: Holomorphic Anomaly in Gauge Theory on ALE Space & Freudenthal Gauge Theory A Contemporary Study in Gauge Theory and Mathematical Physics: Holomorphic Anomaly in Gauge Theory on ALE Space & Freudenthal Gauge Theory by Sheng-Yu Darren Shih A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Bruno Zumino, Chair Professor Ori J. Ganor Professor Mariusz Wodzicki Spring 2014 A Contemporary Study in Gauge Theory and Mathematical Physics: Holomorphic Anomaly in Gauge Theory on ALE Space & Freudenthal Gauge Theory Copyright 2014 by Sheng-Yu Darren Shih 1 Abstract A Contemporary Study in Gauge Theory and Mathematical Physics: Holomorphic Anomaly in Gauge Theory on ALE Space & Freudenthal Gauge Theory by Sheng-Yu Darren Shih Doctor of Philosophy in Physics University of California, Berkeley Professor Bruno Zumino, Chair This thesis covers two distinct parts: Holomorphic Anomaly in Gauge Theory on ALE Space and Freudenthal Gauge Theory. In part I, I presented a concise review of the Seiberg-Witten curve, Nekrasov’s Ω- background, geometric engineering and the holomorphic anomaly equation followed by my published work: Holomorphic Anomaly in Gauge Theory on ALE Space, where an Ω- deformed = 2 SU(2) gauge theory on A space and its five dimension lift is studied. N 1 We find that the partition functions can be reproduced via special geometry and the holo- morphic anomaly equation. Schwinger type integral expressions for the boundary conditions at the monopole/dyon point in moduli space are inferred. The interpretation of the five- dimensional partition function as the partition function of a refined topological string on A (local P1 P1) is suggested. 1× × In part II, I give a comprehensive review of the Freudenthal Triple System, including Freudenthal’s original construction from Jordan Triple Systems and its relation to Lie alge- bra, Yang-Baxter equation, and 4d = 2 BPS black holes, where the novel Freudenthal-dual N was discovered. I also present my published work on the Freudenthal Gauge Theory, where we construct the most generic gauge theory admitting F-dual, and prove a no-go theorem that forbids coupling of a F-dual invariant gauge theory to supersymmetry. i I dedicate this work to my advisor Prof. Bruno Zumino. It is really my greatest honor to work with one of the most insightful physicist and finest gentleman in the world. ii Contents Contents ii List of Figures v List of Tables vi 1 Introduction 1 I Holomorphic Anomaly in Gauge Theory on ALE Space 3 2 Background Knowledge 4 2.1 Review of =2 SU(2)gaugetheory...................... 4 2.1.1 LowEnergyEffectiveN Theory and Prepotential . 5 2.1.2 Seiberg-WittenCurve........................... 5 2.2 Nekrasov’s Localization Formula and the Ω Background . .... 7 2.2.1 Nekrasov’s Partition Function . 7 2.2.2 TheΩBackground ............................ 8 2.3 Geometric Engineering of =2 SU(2)PureGaugeTheory . 10 2.3.1 Generalities of GeometricN Engineering . 10 2.3.2 Double Scaling Limit . 11 2.4 Review of Holomorphic Anomaly Equation . 11 2.4.1 TopologicalTwist............................. 12 2.4.2 Holomorphic Anomaly Equation . 13 2.4.3 Extended Holomorphic Anomaly Equation . 15 3 Holomorphic Anomaly in Gauge Theory in ALE Space 17 3.1 Introduction.................................... 17 3.2 Instanton counting via localization . 19 3.2.1 Generalities ................................ 19 3.2.2 A1 viaorbifold .............................. 20 3.2.3 A1 viablowup............................... 22 3.3 B-model approach to SU(2)gaugetheoryin4d . .. 23 iii 3.3.1 Generalities ................................ 23 + 3.3.2 Z2 ..................................... 26 − 3.3.3 Z2 ..................................... 28 3.4 Instanton partition functions in 5d . 30 3.4.1 Generalities ................................ 30 3.4.2 U(1).................................... 31 3.4.3 U(2).................................... 33 3.5 B-model approach to U(2)gaugetheoryin5d . .. 34 3.5.1 Generalities ................................ 34 + 3.5.2 ZSU(2) ................................... 35 − 3.5.3 ZSU(2) ................................... 36 3.6 Conclusion..................................... 37 II Freudenthal Gauge Theory 39 4 Background Knowledge 40 4.1 FreudenthalTripleSystem . 40 4.1.1 Review of Jordan Algebras of Cubic Forms . 41 4.1.2 Freudenthal’s construction . 42 4.2 Freudenthal Triple System and Lie Algebra . .. 43 4.2.1 From FTS to (Exceptional) Lie Algebra . 43 4.2.2 FromLieAlgebratoFTS ........................ 44 4.3 Yang-Baxter Equation and Okubo’s solution . .. 46 4.3.1 OverviewofYang-BaxterEquation . 46 4.3.2 Okubo’ssolution ............................. 46 4.4 5d/4d =2SupergravityandF-dual . 47 4.4.1N 5-dim =2SUGRA .......................... 48 4.4.2 DimensionN Reduction to 4-dim . 49 4.4.3 The Freudenthal Triple System and 4D BlackHoles.......... 50 4.4.4 F-dual................................... 51 5 Freudenthal Gauge Theory 53 5.1 Introduction.................................... 53 5.2 FreudenthalTripleSystems . 54 5.2.1 Definition of a Freudenthal Triple System . 55 5.2.2 FreudenthalDuality ........................... 56 5.3 FreudenthalGaugeTheory. 58 5.3.1 GlobalSymmetry............................. 58 5.3.2 GaugeSymmetry ............................. 59 5.3.3 The Minimal Action . 60 5.4 Generalization? .................................. 61 iv 5.4.1 CoupletoaVectorSpace......................... 61 5.4.2 ANo-GoTheorem ............................ 62 5.5 ConcludingRemarks ............................... 63 5.6 Appendix ..................................... 64 5.6.1 A Freudenthalduality ......................... 64 5.6.2 B Axioms of V ............................. 65 A Lie Algebra from FTS 66 B Review of Special Geometry 67 B.1 StringTheoryConstruction . 67 B.1.1 IIB/Y ................................... 68 B.1.2 IIA/Y ................................... 69 C The Yang-Baxtere Equation and the Spin Chain 70 C.1 SpinChainandIntegrableSystems . 70 C.2 FromYBEToFCR................................ 72 Bibliography 73 v List of Figures 2.1 Dual 1-cycles (γ1,γ2) of u .............................. 7 C 0 r>0 2.2 Nodal surfaces corresponding to g and g ................... 14 r D D 2.3 Nodal surface Σg fromsurgerygeometry . 14 b vi List of Tables 4.1 Grading of the Lie algebra g(K)........................... 44 4.2 Grading of the Lie algebra g fromitsrootsystem . .. 45 4.3 Freudenthal’smagicsquare . .. .. 48 4.4 5-dim =2SUGRAfieldcontents......................... 48 N A.1 ProductruleforLiebracket.. ... 66 B.1 Mirror pair (Y,Y )intypeIIA/IIBconstruction.. 68 e vii Acknowledgments I would like to thank Prof. Bruno Zumino and his former student Prof. Pe-Ming Ho, who was also my previous advisor at National Taiwan University. From them I not only learned the skill to do independent research but also the deeper philosophy underlying the beauty of theoretical physics. I’d also like to thank Prof. Ori J. Ganor for encouraging me to take the daunting challenges posed by nature. Finally, I want to thank my collaborators Daniel Krefl, Alessio Marrani, Anthony Tagliaferro, and my wife Cong-Xin Qiu to share unique insight with me and to be a great companion in our research. 1 Chapter 1 Introduction This thesis is composed of two parts: Holomorphic Anomaly in Gauge Theory on ALE Space and Freudenthal Gauge Theory. They summarize my two distinct research topics during my graduate study at UC Berkeley. In part I, we focus on the holomorphic anomaly in gauge theory in Asymptotically Locally Euclidean (ALE) spaces. Holomorphic anomaly equation [11, 12] is long known to be a powerful tool in the study of topological string moduli space and the gauge theories descended from it. It admits a generalization, called extended holomorphic anomaly equation [54, 55], which is applicable to the mysterious Ω-background originally introduced by N. Nekrasov to regularize the divergent moduli space measure of the gauge theory in R4 [71]. In this study, we take a step further towards a non-trivial spacetime; We consider four-dimensional Ω-deformed = 2 supersymmetric SU(2) gauge theory on A space and its lift to five N 1 dimensions. We find that the partition functions can be reproduced via special geometry and the holomorphic anomaly equation, whose boundary conditions at the monopole/dyon point in moduli space can be expressed by Schwinger type integrals. We also find the five- dimensional partition function bears an interpretation as the partition function of a refined 1 1 topological string on A1 (local P P ). The result is presented in our published paper [53]. In part II, we move× the topic× to an exotic algebraic system called Freudenthal Triple System, which was originally introduced by mathematician Hans Freudenthal in his study of exceptional Lie algebras [29, 30]. Such an algebraic system distinguished by a non-symmetric ternary product plays important role in both pure math and theoretical physics research. From the mathematics side, besides its original appearance in the study of exceptional Lie algebras, it’s also shown to lead to solutions to the renowned Yang-Baxter equation [74, 75], which paves the way to new insights in the study of braiding groups, and integrable systems. From the physics side, it’s known to classify the scalar manifold of the 4d = 2 Maxwell-Einstein supergravity, and governs the black hole entropy in the theory. Later,N it was discoverd that such black holes are related by a non-linear duality called Freudenthal- dual (F-dual for
Load more

Recommended publications
  • George W. Whitehead Jr
    George W. Whitehead Jr. 1918–2004 A Biographical Memoir by Haynes R. Miller ©2015 National Academy of Sciences. Any opinions expressed in this memoir are those of the author and do not necessarily reflect the views of the National Academy of Sciences. GEORGE WILLIAM WHITEHEAD JR. August 2, 1918–April 12 , 2004 Elected to the NAS, 1972 Life George William Whitehead, Jr., was born in Bloomington, Ill., on August 2, 1918. Little is known about his family or early life. Whitehead received a BA from the University of Chicago in 1937, and continued at Chicago as a graduate student. The Chicago Mathematics Department was somewhat ingrown at that time, dominated by L. E. Dickson and Gilbert Bliss and exhibiting “a certain narrowness of focus: the calculus of variations, projective differential geometry, algebra and number theory were the main topics of interest.”1 It is possible that Whitehead’s interest in topology was stimulated by Saunders Mac Lane, who By Haynes R. Miller spent the 1937–38 academic year at the University of Chicago and was then in the early stages of his shift of interest from logic and algebra to topology. Of greater importance for Whitehead was the appearance of Norman Steenrod in Chicago. Steenrod had been attracted to topology by Raymond Wilder at the University of Michigan, received a PhD under Solomon Lefschetz in 1936, and remained at Princeton as an Instructor for another three years. He then served as an Assistant Professor at the University of Chicago between 1939 and 1942 (at which point he moved to the University of Michigan).
    [Show full text]
  • Qisar-Alexander-Ollongren-Astrolinguistics.Pdf
    Astrolinguistics Alexander Ollongren Astrolinguistics Design of a Linguistic System for Interstellar Communication Based on Logic Alexander Ollongren Advanced Computer Science Leiden University Leiden The Netherlands ISBN 978-1-4614-5467-0 ISBN 978-1-4614-5468-7 (eBook) DOI 10.1007/978-1-4614-5468-7 Springer New York Heidelberg Dordrecht London Library of Congress Control Number: 2012945935 © Springer Science+Business Media New York 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, speci fi cally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on micro fi lms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied speci fi cally for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a speci fi c statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
    [Show full text]
  • IMU Secretary An: [email protected]; CC: Betreff: IMU EC CL 05/07: Vote on ICMI Terms of Reference Change Datum: Mittwoch, 24
    Appendix 10.1.1 Von: IMU Secretary An: [email protected]; CC: Betreff: IMU EC CL 05/07: vote on ICMI terms of reference change Datum: Mittwoch, 24. Januar 2007 11:42:40 Anlagen: To the IMU 2007-2010 Executive Committee Dear colleagues, We are currently experimenting with a groupware system that may help us organize the files that every EC member should know and improve the voting processes. Wolfgang Dalitz has checked the open source groupware systems and selected one that we want to try. It is more complicated than we thought and does have some deficiencies, but we see no freeware that is better. Here is our test run with a vote on a change of the ICMI terms of reference. To get to our voting system click on http://www.mathunion.org/ec-only/ To log in, you have to type your last name in the following version: ball, baouendi, deleon, groetschel, lovasz, ma, piene, procesi, vassiliev, viana Right now, everybody has the same password: pw123 You will immediately get to the summary page which contains an item "New Polls". The question to vote on is: Vote-070124: Change of ICMI terms of reference, #3, see Files->Voting->Vote- 070124 for full information and you are supposed to agree, disagree or abstain by clicking on the corresponding button. Full information about the contents of the vote is documented in the directory Voting (click on the +) where you will find a file Vote-070124.txt (click on the "txt icon" to see the contents of the file). The file is also enclosed below for your information.
    [Show full text]
  • Arxiv:1404.2323V1 [Math.AG] 8 Apr 2014 Membranes and Sheaves
    Membranes and Sheaves Nikita Nekrasov and Andrei Okounkov April 2014 Contents 1 A brief introduction 2 1.1 Overview.............................. 2 1.2 MotivationfromM-theory . 3 1.3 Planofthepaper ......................... 7 1.4 Acknowledgements ........................ 7 2 Contours of the conjectures 8 2.1 K-theorypreliminaries ...................... 8 2.2 Theindexsheaf.......................... 10 2.3 Comparison with Donaldson-Thomastheory . 15 2.4 Fields of 11-dimensional supergravity and degree zero DT counts 19 3 The DT integrand 24 3.1 Themodifiedvirtualstructuresheaf. 24 3.2 The interaction term Φ ...................... 28 arXiv:1404.2323v1 [math.AG] 8 Apr 2014 4 The index of membranes 33 4.1 Membranemoduli......................... 33 4.2 Deformationsofmembranes . 36 5 Examples 37 5.1 Reducedlocalcurves ....................... 37 5.2 Doublecurves........................... 39 5.3 Single interaction between smooth curves . 42 5.4 HigherrankDTcounts. .. .. 43 5.5 EngineeringhigherrankDTtheory . 45 1 6 Existence of square roots 49 6.1 Symmetricbundlesonsquares . 49 6.2 SquarerootsinDTtheory . 50 6.3 SquarerootsinM-theory. 52 7 Refined invariants 54 7.1 Actionsscalingthe3-form . 54 7.2 Localization for κ-trivialtori................... 56 7.3 Morsetheoryandrigidity . 60 8 Index vertex and refined vertex 61 8.1 Toric Calabi-Yau 3-folds . 61 8.2 Virtualtangentspacesatfixedpoints . 63 8.3 Therefinedvertex ........................ 66 A Appendix 70 A.1 Proofofthebalancelemma . 70 1 A brief introduction 1.1 Overview Our goal in this paper is to discuss a conjectural correspondence between enumerative geometry of curves in Calabi-Yau 5-folds Z and 1-dimensional sheaves on 3-folds X that are embedded in Z as fixed points of certain C×- actions. In both cases, the enumerative information is taken in equivariant K-theory, where the equivariance is with respect to all automorphisms of the problem.
    [Show full text]
  • Mathematicians Fleeing from Nazi Germany
    Mathematicians Fleeing from Nazi Germany Mathematicians Fleeing from Nazi Germany Individual Fates and Global Impact Reinhard Siegmund-Schultze princeton university press princeton and oxford Copyright 2009 © by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW All Rights Reserved Library of Congress Cataloging-in-Publication Data Siegmund-Schultze, R. (Reinhard) Mathematicians fleeing from Nazi Germany: individual fates and global impact / Reinhard Siegmund-Schultze. p. cm. Includes bibliographical references and index. ISBN 978-0-691-12593-0 (cloth) — ISBN 978-0-691-14041-4 (pbk.) 1. Mathematicians—Germany—History—20th century. 2. Mathematicians— United States—History—20th century. 3. Mathematicians—Germany—Biography. 4. Mathematicians—United States—Biography. 5. World War, 1939–1945— Refuges—Germany. 6. Germany—Emigration and immigration—History—1933–1945. 7. Germans—United States—History—20th century. 8. Immigrants—United States—History—20th century. 9. Mathematics—Germany—History—20th century. 10. Mathematics—United States—History—20th century. I. Title. QA27.G4S53 2008 510.09'04—dc22 2008048855 British Library Cataloging-in-Publication Data is available This book has been composed in Sabon Printed on acid-free paper. ∞ press.princeton.edu Printed in the United States of America 10 987654321 Contents List of Figures and Tables xiii Preface xvii Chapter 1 The Terms “German-Speaking Mathematician,” “Forced,” and“Voluntary Emigration” 1 Chapter 2 The Notion of “Mathematician” Plus Quantitative Figures on Persecution 13 Chapter 3 Early Emigration 30 3.1. The Push-Factor 32 3.2. The Pull-Factor 36 3.D.
    [Show full text]
  • Arxiv:1512.05388V2 [Hep-Th] 2 Jan 2016 1
    BPS/CFT CORRESPONDENCE: NON-PERTURBATIVE DYSON-SCHWINGER EQUATIONS AND qq-CHARACTERS NIKITA NEKRASOV Abstract. We study symmetries of quantum field theories involving topologically dis- tinct sectors of the field space. To exhibit these symmetries we define special gauge invariant observables, which we call the qq-characters. In the context of the BPS/CFT correspondence, using these observables, we derive an infinite set of Dyson-Schwinger- type relations. These relations imply that the supersymmetric partition functions in the presence of Ω-deformation and defects obey the Ward identities of two dimen- sional conformal field theory and its q-deformations. The details will be discussed in the companion papers. Contents 1. Introduction 4 1.1. Dyson-Schwinger equations 4 1.2. Non-perturbative Dyson-Schwinger identities 6 1.3. Organization of the presentation 8 1.4. Acknowledgements 10 2. The BPS/CFT correspondence 12 2.1. = 2 partition functions 12 2.2. DefectN operators and lower-dimensional theories 13 2.3. The Y- and X-observables 14 2.4. The physics of X-observables 14 arXiv:1512.05388v2 [hep-th] 2 Jan 2016 2.5. Hidden symmetries 17 2.6. Some notations. 17 2.7. Equivariant virtual Chern polynomials 22 3. Supersymmetric gauge theories 22 3.1. Quivers 22 3.2. Quivers with colors 22 3.3. The symmetry groups 22 3.4. The parameters of Lagrangian 24 3.5. The group H 25 3.6. Perturbative theory 26 3.7. Realizations of quiver theories 28 4. Integration over instanton moduli spaces 30 1 2 NIKITA NEKRASOV 4.1. Instanton partition function 30 4.2.
    [Show full text]
  • Public Recognition and Media Coverage of Mathematical Achievements
    Journal of Humanistic Mathematics Volume 9 | Issue 2 July 2019 Public Recognition and Media Coverage of Mathematical Achievements Juan Matías Sepulcre University of Alicante Follow this and additional works at: https://scholarship.claremont.edu/jhm Part of the Arts and Humanities Commons, and the Mathematics Commons Recommended Citation Sepulcre, J. "Public Recognition and Media Coverage of Mathematical Achievements," Journal of Humanistic Mathematics, Volume 9 Issue 2 (July 2019), pages 93-129. DOI: 10.5642/ jhummath.201902.08 . Available at: https://scholarship.claremont.edu/jhm/vol9/iss2/8 ©2019 by the authors. This work is licensed under a Creative Commons License. JHM is an open access bi-annual journal sponsored by the Claremont Center for the Mathematical Sciences and published by the Claremont Colleges Library | ISSN 2159-8118 | http://scholarship.claremont.edu/jhm/ The editorial staff of JHM works hard to make sure the scholarship disseminated in JHM is accurate and upholds professional ethical guidelines. However the views and opinions expressed in each published manuscript belong exclusively to the individual contributor(s). The publisher and the editors do not endorse or accept responsibility for them. See https://scholarship.claremont.edu/jhm/policies.html for more information. Public Recognition and Media Coverage of Mathematical Achievements Juan Matías Sepulcre Department of Mathematics, University of Alicante, Alicante, SPAIN [email protected] Synopsis This report aims to convince readers that there are clear indications that society is increasingly taking a greater interest in science and particularly in mathemat- ics, and thus society in general has come to recognise, through different awards, privileges, and distinctions, the work of many mathematicians.
    [Show full text]
  • Hans Duistermaat's Contributions to Poisson
    HANS DUISTERMAAT’S CONTRIBUTIONS TO POISSON GEOMETRY REYER SJAMAAR Abstract. Hans Duistermaat was scheduled to lecture in the 2010 School on Poisson Geometry at IMPA, but passed away suddenly. This is a record of a talk I gave at the 2010 Conference on Poisson Geometry (the week after the School) to share some of my memories of him and to give a brief assessment of his impact on the subject. Johannes Jisse (Hans) Duistermaat (1942–2010) earned his doctorate in 1968 at the University of Utrecht under the direction of Hans Freudenthal. After holding a postdoctoral position at the University of Lund, a professorship at the Univer- sity of Nijmegen, and a visiting position at the Courant Institute, he returned to Utrecht in 1975 to take over Freudenthal’s chair after the latter’s retirement. He held this chair until his own retirement in 2007. He continued to play a role in the life of the Utrecht Mathematical Institute and kept up his mathematical activities until he was struck down in March 2010 by a case of pneumonia contracted while on chemotherapy for cancer. I got to know Hans Duistermaat as an undergraduate when I took his freshman analysis course at Utrecht. At that time he taught the course from lecture notes that were in style and content close to Dieudonné’s book [23]. Although this was a smashing success with some students, I think Hans realized he had to reduce the potency so as not to leave behind quite so many of us, and over the years the lecture notes grew into his and Joop Kolk’s still very substantial undergraduate textbook [31].
    [Show full text]
  • Model Building and SUSY Breaking on D-Branes
    Model building and SUSY breaking on D-branes Dmitry Malyshev a dissertation presented to the faculty of princeton university in candidacy for the degree of doctor of philosophy recommended for acceptance by the department of physics Adviser: Herman Verlinde September 2008 c Copyright by Dmitry Malyshev, 2008. All rights reserved. Abstract In the recent years there has been an increase of the interest in applying the String theory to construct viable extensions of the Standard Model and to find stringy analogs of the known field theory models for supersymmetry breaking. In my dissertation I will focus on the constructions involving the D-branes at the Calabi-Yau singularities in type IIB string theory. The main motivations for this choice are: the decoupling of the D-brane field theory from the supergravity, the well established tools for deriving the field theories from the configurations of the D-branes, and the possibility of using the gauge-gravity correspondence in solving some non-perturbative aspects in the field theory. A construction of an extended supersymmetric Standard Model on a D3-brane at the del Pezzo 8 singularity is presented in the first part of the dissertation. In the second part we discuss the possible representations of the SUSY breaking models in String Theory and obtain the metastable SUSY breaking vacuum in a system of D-branes on the suspended pinch point singularity. The gauge mediation of this SUSY breaking model is described in details. In the simplest model the spectrum of particles in the visible sector has a split SUSY breaking, the sfermions are one order of magnitude heavier than the gauginos.
    [Show full text]
  • Nikita Nekrasov 講義録
    Nikita Nekrasov 講義録 December, 1998 講義録作成1 ：橋本幸士, 岸本 功 （京大理） 目次 第 I 部 超共形ゲージ理論と AdS 超重力理論 2 1 Introduction と motivation 3 2 Orbifold 理論 6 3 時空の状況 12 4 Orbifold から conifold への変形 17 5 まとめ 19 第 II 部 N =2ゲージ理論における状態のカウント 21 6 導入 22 7 D3-プローブ上の理論 22 8M理論への持ち上げ/引き落とし 27 9 結論 39 1この講義録は「Workshop ”ゲージ理論の力学と弦双対性” 1998 年 12 月 16 日（水）–12月 18 日（金）東京工業大学国際交流会館」における Nekrasov 氏の講義に基づくものです。第 I 部が 16 日に、第 II 部が 18 日に行われました。 1 第 I 部 超共形ゲージ理論とAdS超重力理論 (SUPER)CONFORMAL GAUGE THEORIES AND ANTI-DE-SITTER SUPERGRAVITY2 Nikita Nekrasov 1. Introduction と motivation ——哲学, D-brane と付随した場の理論, 特異な幾何からの conformal 理論 2. Orbifold 理論 ——理論の構築（：gauge 群, matter の内容, （超）対称性）, β 関数の計算, 摂動論の解析 3. 時空の状況 ——near-horizon limit, 超重力における変形 4. Orbifold から conifold への変形 ——場の理論の実現, Higgs branch の幾何, 時空解釈（conjecture） 2文献 [1, 2] に基づく。[3, 4, 5, 6, 7, 8] も見よ。 2 第 1 章 Introduction と motivation 双対性に関する動きの中でおそらく最も（そしておそらく唯一つの）重要な教訓は、面白い物理は面白い幾何に関係 しているという発想に立ち返るということである。Einstein の時代に比べての違いは、勿論量子的な物理量が古典的な 幾何を通じて表現されうるという仮定である。例えば、N =2SU(2) gauge 理論の有効結合定数（と長距離相関関数） は補佐的な楕円曲線 Eu の modular 変数 τ(u) を用いて表される： y2 =(x − u)(x2 − Λ4). (1) ここで u = hTrφ2i は真空の moduli 空間を label する。 u-plane θ(u) 4πi Eu: τ(u)= 2π + e2(u) vacuum 図 1: N =2SU(2) gauge 理論の moduli 空間。 物理から幾何、また逆に幾何から物理への変換を理解するのに必要不可欠な道具は D-brane である。D-brane に関し ての良い点は以下の二つである: (1) D-brane の上に弦が端点を持つので、いくつかの D-brane がお互いに重なるとその worldvolume 上に非可換 gauge boson が発生する。(2) D-brane は Ramond-Ramond 場の電荷を持っており、（適切な条 件の下で）弦理論における soliton として記述されうる。 (1) (2) 図 2: D-brane の二つの描像。(1) 開弦の端点としての D-brane、(2) 弦理論の soliton としての D-brane。 一般的な発想は、gauge 理論とそれに対応する soliton の背景を伝播する弦の理論を等しくするように、D-brane を用 いる、ということである。もし（IIB 型超弦理論で）N 枚の D3-brane が平坦な空間内でお互いに重なっていると、この 重なった brane は二つの記述法を持つ: 1.
    [Show full text]
  • Mini-Symposium on String Theory and M-Theory
    Mini-Symposium on String Theory and M-Theory Chair: Michael R. Douglas Duiliu-EmanuelDiaconescu JaumeGomis ChrisM.Hull AlbrechtKlemm Jos´eM.F.Labastida MarcosMarin˜ o NikitaNekrasov ChristophSchweigert AngelM.Uranga Duiliu-Emanuel Diaconescu Institute for Advanced Study, Princeton K-theory from M-theory Wednesday, 12.30 – 13.00, Room B I report on joint work with Gregory Moore and Edward Witten —“E8 Gauge Theory, and a Derivation of K-Theory from M-Theory”. It has become clear in recent years that type II string theories contain nonperturbative objects —called D-branes— which support gauge fields. From a more abstract point of view, D-brane charges are classified by K-theory classes of the space- time manifold X. Since D-branes are charged under Ramond-Ramond fields, E. Witten and then G. Moore and E. Witten proposed a K-theoretic interpretation of Ramond-Ramond fluxes themselves. This is more striking, as the RR fields are states in the perturbative closed string spectrum which a priori have no connection with vector bundles and K-theory. In the present work, we show that this proposal is consistent with M-theory. The main idea is a detailed comparison of the leading terms in the long distance partition function of M-theory and IIA string theory. The analysis relies heavily on index theory and homotopy theory techniques. It is found a precise agreement which confirms the internal consistency of M-theory, as well as the validity of the K-theoretic formalism in string theory. Jaume Gomis Caltech University D-branes in nongeometric phases of string theory Tuesday, 12.00 – 12.30, Room B Recent progress in string theory hinges upon our improved understanding of its nonperturbative states.
    [Show full text]
  • Notices of the American Mathematical
    ISSN 0002-9920 Notices of the American Mathematical Society AMERICAN MATHEMATICAL SOCIETY Graduate Studies in Mathematics Series The volumes in the GSM series are specifically designed as graduate studies texts, but are also suitable for recommended and/or supplemental course reading. With appeal to both students and professors, these texts make ideal independent study resources. The breadth and depth of the series’ coverage make it an ideal acquisition for all academic libraries that of the American Mathematical Society support mathematics programs. al January 2010 Volume 57, Number 1 Training Manual Optimal Control of Partial on Transport Differential Equations and Fluids Theory, Methods and Applications John C. Neu FROM THE GSM SERIES... Fredi Tro˝ltzsch NEW Graduate Studies Graduate Studies in Mathematics in Mathematics Volume 109 Manifolds and Differential Geometry Volume 112 ocietty American Mathematical Society Jeffrey M. Lee, Texas Tech University, Lubbock, American Mathematical Society TX Volume 107; 2009; 671 pages; Hardcover; ISBN: 978-0-8218- 4815-9; List US$89; AMS members US$71; Order code GSM/107 Differential Algebraic Topology From Stratifolds to Exotic Spheres Mapping Degree Theory Matthias Kreck, Hausdorff Research Institute for Enrique Outerelo and Jesús M. Ruiz, Mathematics, Bonn, Germany Universidad Complutense de Madrid, Spain Volume 110; 2010; approximately 215 pages; Hardcover; A co-publication of the AMS and Real Sociedad Matemática ISBN: 978-0-8218-4898-2; List US$55; AMS members US$44; Española (RSME). Order code GSM/110 Volume 108; 2009; 244 pages; Hardcover; ISBN: 978-0-8218- 4915-6; List US$62; AMS members US$50; Ricci Flow and the Sphere Theorem The Art of Order code GSM/108 Simon Brendle, Stanford University, CA Mathematics Volume 111; 2010; 176 pages; Hardcover; ISBN: 978-0-8218- page 8 Training Manual on Transport 4938-5; List US$47; AMS members US$38; and Fluids Order code GSM/111 John C.
    [Show full text]