Introduction to Vertex Operator Superalgebras and Their Modules Mathematics and Its Applications
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Introduction to Supersymmetry(1)
Introduction to Supersymmetry(1) J.N. Tavares Dep. Matem¶aticaPura, Faculdade de Ci^encias,U. Porto, 4000 Porto TQFT Club 1Esta ¶euma vers~aoprovis¶oria,incompleta, para uso exclusivo nas sess~oesde trabalho do TQFT club CONTENTS 1 Contents 1 Supersymmetry in Quantum Mechanics 2 1.1 The Supersymmetric Oscillator . 2 1.2 Witten Index . 4 1.3 A fundamental example: The Laplacian on forms . 7 1.4 Witten's proof of Morse Inequalities . 8 2 Supergeometry and Supersymmetry 13 2.1 Field Theory. A quick review . 13 2.2 SuperEuclidean Space . 17 2.3 Reality Conditions . 18 2.4 Supersmooth functions . 18 2.5 Supermanifolds . 21 2.6 Lie Superalgebras . 21 2.7 Super Lie groups . 26 2.8 Rigid Superspace . 27 2.9 Covariant Derivatives . 30 3 APPENDIX. Cli®ord Algebras and Spin Groups 31 3.1 Cli®ord Algebras . 31 Motivation. Cli®ord maps . 31 Cli®ord Algebras . 33 Involutions in V .................................. 35 Representations . 36 3.2 Pin and Spin groups . 43 3.3 Spin Representations . 47 3.4 U(2), spinors and almost complex structures . 49 3.5 Spinc(4)...................................... 50 Chiral Operator. Self Duality . 51 2 1 Supersymmetry in Quantum Mechanics 1.1 The Supersymmetric Oscillator i As we will see later the \hermitian supercharges" Q®, in the N extended SuperPoincar¶eLie Algebra obey the anticommutation relations: i j m ij fQ®;Q¯g = 2(γ C)®¯± Pm (1.1) m where ®; ¯ are \spinor" indices, i; j 2 f1; ¢ ¢ ¢ ;Ng \internal" indices and (γ C)®¯ a bilinear form in the spinor indices ®; ¯. When specialized to 0-space dimensions ((1+0)-spacetime), then since P0 = H, relations (1.1) take the form (with a little change in notations): fQi;Qjg = 2±ij H (1.2) with N \Hermitian charges" Qi; i = 1; ¢ ¢ ¢ ;N. -
Inönü–Wigner Contraction and D = 2 + 1 Supergravity
Eur. Phys. J. C (2017) 77:48 DOI 10.1140/epjc/s10052-017-4615-1 Regular Article - Theoretical Physics Inönü–Wigner contraction and D = 2 + 1 supergravity P. K. Concha1,2,a, O. Fierro3,b, E. K. Rodríguez1,2,c 1 Departamento de Ciencias, Facultad de Artes Liberales, Universidad Adolfo Ibáñez, Av. Padre Hurtado 750, Viña del Mar, Chile 2 Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile, Casilla 567, Valdivia, Chile 3 Departamento de Matemática y Física Aplicadas, Universidad Católica de la Santísima Concepción, Alonso de Rivera 2850, Concepción, Chile Received: 25 November 2016 / Accepted: 5 January 2017 / Published online: 25 January 2017 © The Author(s) 2017. This article is published with open access at Springerlink.com Abstract We present a generalization of the standard cannot always be obtained by rescaling the gauge fields and Inönü–Wigner contraction by rescaling not only the gener- considering some limit as in the (anti)commutation relations. ators of a Lie superalgebra but also the arbitrary constants In particular it is well known that, in the presence of the exotic appearing in the components of the invariant tensor. The Lagrangian, the Poincaré limit cannot be applied to a (p, q) procedure presented here allows one to obtain explicitly the AdS CS supergravity [7]. This difficulty can be overcome Chern–Simons supergravity action of a contracted superal- extending the osp (2, p) ⊗ osp (2, q) superalgebra by intro- gebra. In particular we show that the Poincaré limit can be ducing the automorphism generators so (p) and so (q) [9]. performed to a D = 2 + 1 (p, q) AdS Chern–Simons super- In such a case, the IW contraction can be applied and repro- gravity in presence of the exotic form. -
1. INTRODUCTION 1.1. Introductory Remarks on Supersymmetry. 1.2
1. INTRODUCTION 1.1. Introductory remarks on supersymmetry. 1.2. Classical mechanics, the electromagnetic, and gravitational fields. 1.3. Principles of quantum mechanics. 1.4. Symmetries and projective unitary representations. 1.5. Poincar´esymmetry and particle classification. 1.6. Vector bundles and wave equations. The Maxwell, Dirac, and Weyl - equations. 1.7. Bosons and fermions. 1.8. Supersymmetry as the symmetry of Z2{graded geometry. 1.1. Introductory remarks on supersymmetry. The subject of supersymme- try (SUSY) is a part of the theory of elementary particles and their interactions and the still unfinished quest of obtaining a unified view of all the elementary forces in a manner compatible with quantum theory and general relativity. Supersymme- try was discovered in the early 1970's, and in the intervening years has become a major component of theoretical physics. Its novel mathematical features have led to a deeper understanding of the geometrical structure of spacetime, a theme to which great thinkers like Riemann, Poincar´e,Einstein, Weyl, and many others have contributed. Symmetry has always played a fundamental role in quantum theory: rotational symmetry in the theory of spin, Poincar´esymmetry in the classification of elemen- tary particles, and permutation symmetry in the treatment of systems of identical particles. Supersymmetry is a new kind of symmetry which was discovered by the physicists in the early 1970's. However, it is different from all other discoveries in physics in the sense that there has been no experimental evidence supporting it so far. Nevertheless an enormous effort has been expended by many physicists in developing it because of its many unique features and also because of its beauty and coherence1. -
Of Operator Algebras Vern I
proceedings of the american mathematical society Volume 92, Number 2, October 1984 COMPLETELY BOUNDED HOMOMORPHISMS OF OPERATOR ALGEBRAS VERN I. PAULSEN1 ABSTRACT. Let A be a unital operator algebra. We prove that if p is a completely bounded, unital homomorphism of A into the algebra of bounded operators on a Hubert space, then there exists a similarity S, with ||S-1|| • ||S|| = ||p||cb, such that S_1p(-)S is a completely contractive homomorphism. We also show how Rota's theorem on operators similar to contractions and the result of Sz.-Nagy and Foias on the similarity of p-dilations to contractions can be deduced from this result. 1. Introduction. In [6] we proved that a homomorphism p of an operator algebra is similar to a completely contractive homomorphism if and only if p is completely bounded. It was known that if S is such a similarity, then ||5|| • ||5_11| > ||/9||cb- However, at the time we were unable to determine if one could choose the similarity such that ||5|| • US'-1!! = ||p||cb- When the operator algebra is a C*- algebra then Haagerup had shown [3] that such a similarity could be chosen. The purpose of the present note is to prove that for a general operator algebra, there exists a similarity S such that ||5|| • ||5_1|| = ||p||cb- Completely contractive homomorphisms are central to the study of the repre- sentation theory of operator algebras, since they are precisely the homomorphisms that can be dilated to a ^representation on some larger Hilbert space of any C*- algebra which contains the operator algebra. -
Supergravity Backgrounds and Symmetry Superalgebras
Turkish Journal of Physics Turk J Phys (2016) 40: 113 { 126 http://journals.tubitak.gov.tr/physics/ ⃝c TUB¨ ITAK_ Review Article doi:10.3906/fiz-1510-8 Supergravity backgrounds and symmetry superalgebras Umit¨ ERTEM∗ School of Mathematics, College of Science and Engineering, The University of Edinburgh, Edinburgh, United Kingdom Received: 12.10.2015 • Accepted/Published Online: 08.12.2015 • Final Version: 27.04.2016 Abstract: We consider the bosonic sectors of supergravity theories in ten and eleven dimensions corresponding to the low energy limits of string theories and M-theory. The solutions of supergravity field equations are known as supergravity backgrounds and the number of preserved supersymmetries in those backgrounds are determined by Killing spinors. We provide some examples of supergravity backgrounds that preserve different fractions of supersymmetry. An important invariant for the characterization of supergravity backgrounds is their Killing superalgebras, which are constructed out of Killing vectors and Killing spinors of the background. After constructing Killing superalgebras of some special supergravity backgrounds, we discuss the possibilities of the extensions of these superalgebras to include the higher degree hidden symmetries of the background. Key words: Supergravity backgrounds, Killing spinors, Killing superalgebras 1. Introduction The unification of fundamental forces of nature is one of the biggest aims in modern theoretical physics. The most promising approaches for that aim include the ten-dimensional supersymmetric string theories and their eleven-dimensional unification called M-theory. There are five different string theories in ten dimensions: type I, type IIA and IIB, and heterotic E8 × E8 and SO(32) theories. However, some dualities called T-duality, S-duality, and U-duality between strong coupling and weak coupling limits of these theories can be defined and these dualities can give rise to one unified M-theory in eleven dimensions [1, 2, 3, 4]. -
Robot and Multibody Dynamics
Robot and Multibody Dynamics Abhinandan Jain Robot and Multibody Dynamics Analysis and Algorithms 123 Abhinandan Jain Ph.D. Jet Propulsion Laboratory 4800 Oak Grove Drive Pasadena, California 91109 USA [email protected] ISBN 978-1-4419-7266-8 e-ISBN 978-1-4419-7267-5 DOI 10.1007/978-1-4419-7267-5 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010938443 c Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) In memory of Guillermo Rodriguez, an exceptional scholar and a gentleman. To my parents, and to my wife, Karen. Preface “It is a profoundly erroneous truism, repeated by copybooks and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. -
From String Theory and Moonshine to Vertex Algebras
Preample From string theory and Moonshine to vertex algebras Bong H. Lian Department of Mathematics Brandeis University [email protected] Harvard University, May 22, 2020 Dedicated to the memory of John Horton Conway December 26, 1937 – April 11, 2020. Preample Acknowledgements: Speaker’s collaborators on the theory of vertex algebras: Andy Linshaw (Denver University) Bailin Song (University of Science and Technology of China) Gregg Zuckerman (Yale University) For their helpful input to this lecture, special thanks to An Huang (Brandeis University) Tsung-Ju Lee (Harvard CMSA) Andy Linshaw (Denver University) Preample Disclaimers: This lecture includes a brief survey of the period prior to and soon after the creation of the theory of vertex algebras, and makes no claim of completeness – the survey is intended to highlight developments that reflect the speaker’s own views (and biases) about the subject. As a short survey of early history, it will inevitably miss many of the more recent important or even towering results. Egs. geometric Langlands, braided tensor categories, conformal nets, applications to mirror symmetry, deformations of VAs, .... Emphases are placed on the mutually beneficial cross-influences between physics and vertex algebras in their concurrent early developments, and the lecture is aimed for a general audience. Preample Outline 1 Early History 1970s – 90s: two parallel universes 2 A fruitful perspective: vertex algebras as higher commutative algebras 3 Classification: cousins of the Moonshine VOA 4 Speculations The String Theory Universe 1968: Veneziano proposed a model (using the Euler beta function) to explain the ‘st-channel crossing’ symmetry in 4-meson scattering, and the Regge trajectory (an angular momentum vs binding energy plot for the Coulumb potential). -
Representations of Centrally Extended Lie Superalgebra $\Mathfrak {Psl}(2
Representations of centrally extended Lie superalgebra psl(2|2) Takuya Matsumoto and Alexander Molev Abstract The symmetries provided by representations of the centrally extended Lie su- peralgebra psl(2|2) are known to play an important role in the spin chain models originated in the planar anti-de Sitter/conformal field theory correspondence and one- dimensional Hubbard model. We give a complete description of finite-dimensional irreducible representations of this superalgebra thus extending the work of Beisert which deals with a generic family of representations. Our description includes a new class of modules with degenerate eigenvalues of the central elements. Moreover, we construct explicit bases in all irreducible representations by applying the techniques of Mickelsson–Zhelobenko algebras. arXiv:1405.3420v2 [math.RT] 15 Sep 2014 Institute for Theoretical Physics and Spinoza Institute Utrecht University, Leuvenlaan 4, 3854 CE Utrecht, The Netherlands [email protected] School of Mathematics and Statistics University of Sydney, NSW 2006, Australia [email protected] 1 1 Introduction As discovered by Beisert [1, 2, 3], certain spin chain models originated in the planar anti-de Sitter/conformal field theory (AdS/CFT) correspondence admit hidden symmetries pro- vided by the action of the Yangian Y(g) associated with the centrally extended Lie super- algebra g = psl(2|2) ⋉C3. This is a semi-direct product of the simple Lie superalgebra psl(2|2) of type A(1, 1) and the abelian Lie algebra C3 spanned by elements C, K and P which are central in g. Due to the results of [6], psl(2|2) is distinguished among the basic classical Lie superalgebras by the existence of a three-dimensional central extension. -
Classification of Simple Linearly Compact N-Lie Superalgebras
Classification of simple linearly compact n-Lie superalgebras The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Cantarini, Nicoletta, and Victor G. Kac. “Classification of Simple Linearly Compact n-Lie Superalgebras.” Communications in Mathematical Physics 298.3 (2010): 833–853. Web. As Published http://dx.doi.org/10.1007/s00220-010-1049-0 Publisher Springer-Verlag Version Author's final manuscript Citable link http://hdl.handle.net/1721.1/71610 Terms of Use Creative Commons Attribution-Noncommercial-Share Alike 3.0 Detailed Terms http://creativecommons.org/licenses/by-nc-sa/3.0/ Classification of simple linearly compact n-Lie superalgebras Nicoletta Cantarini∗ Victor G. Kac∗∗ Abstract We classify simple linearly compact n-Lie superalgebras with n> 2 over a field F of charac- teristic 0. The classification is based on a bijective correspondence between non-abelian n-Lie Z n−1 superalgebras and transitive -graded Lie superalgebras of the form L = ⊕j=−1Lj, where dim Ln−1 = 1, L−1 and Ln−1 generate L, and [Lj ,Ln−j−1] = 0 for all j, thereby reducing it to the known classification of simple linearly compact Lie superalgebras and their Z-gradings. The list consists of four examples, one of them being the n+1-dimensional vector product n-Lie algebra, and the remaining three infinite-dimensional n-Lie algebras. Introduction Given an integer n ≥ 2, an n-Lie algebra g is a vector space over a field F, endowed with an n-ary anti-commutative product n Λ g → g , a1 ∧ . -
Scientific Report for the Year 2000
The Erwin Schr¨odinger International Boltzmanngasse 9 ESI Institute for Mathematical Physics A-1090 Wien, Austria Scientific Report for the Year 2000 Vienna, ESI-Report 2000 March 1, 2001 Supported by Federal Ministry of Education, Science, and Culture, Austria ESI–Report 2000 ERWIN SCHRODINGER¨ INTERNATIONAL INSTITUTE OF MATHEMATICAL PHYSICS, SCIENTIFIC REPORT FOR THE YEAR 2000 ESI, Boltzmanngasse 9, A-1090 Wien, Austria March 1, 2001 Honorary President: Walter Thirring, Tel. +43-1-4277-51516. President: Jakob Yngvason: +43-1-4277-51506. [email protected] Director: Peter W. Michor: +43-1-3172047-16. [email protected] Director: Klaus Schmidt: +43-1-3172047-14. [email protected] Administration: Ulrike Fischer, Eva Kissler, Ursula Sagmeister: +43-1-3172047-12, [email protected] Computer group: Andreas Cap, Gerald Teschl, Hermann Schichl. International Scientific Advisory board: Jean-Pierre Bourguignon (IHES), Giovanni Gallavotti (Roma), Krzysztof Gawedzki (IHES), Vaughan F.R. Jones (Berkeley), Viktor Kac (MIT), Elliott Lieb (Princeton), Harald Grosse (Vienna), Harald Niederreiter (Vienna), ESI preprints are available via ‘anonymous ftp’ or ‘gopher’: FTP.ESI.AC.AT and via the URL: http://www.esi.ac.at. Table of contents General remarks . 2 Winter School in Geometry and Physics . 2 Wolfgang Pauli und die Physik des 20. Jahrhunderts . 3 Summer Session Seminar Sophus Lie . 3 PROGRAMS IN 2000 . 4 Duality, String Theory, and M-theory . 4 Confinement . 5 Representation theory . 7 Algebraic Groups, Invariant Theory, and Applications . 7 Quantum Measurement and Information . 9 CONTINUATION OF PROGRAMS FROM 1999 and earlier . 10 List of Preprints in 2000 . 13 List of seminars and colloquia outside of conferences . -
Operator Algebras: an Informal Overview 3
OPERATOR ALGEBRAS: AN INFORMAL OVERVIEW FERNANDO LLEDO´ Contents 1. Introduction 1 2. Operator algebras 2 2.1. What are operator algebras? 2 2.2. Differences and analogies between C*- and von Neumann algebras 3 2.3. Relevance of operator algebras 5 3. Different ways to think about operator algebras 6 3.1. Operator algebras as non-commutative spaces 6 3.2. Operator algebras as a natural universe for spectral theory 6 3.3. Von Neumann algebras as symmetry algebras 7 4. Some classical results 8 4.1. Operator algebras in functional analysis 8 4.2. Operator algebras in harmonic analysis 10 4.3. Operator algebras in quantum physics 11 References 13 Abstract. In this article we give a short and informal overview of some aspects of the theory of C*- and von Neumann algebras. We also mention some classical results and applications of these families of operator algebras. 1. Introduction arXiv:0901.0232v1 [math.OA] 2 Jan 2009 Any introduction to the theory of operator algebras, a subject that has deep interrelations with many mathematical and physical disciplines, will miss out important elements of the theory, and this introduction is no ex- ception. The purpose of this article is to give a brief and informal overview on C*- and von Neumann algebras which are the main actors of this summer school. We will also mention some of the classical results in the theory of operator algebras that have been crucial for the development of several areas in mathematics and mathematical physics. Being an overview we can not provide details. Precise definitions, statements and examples can be found in [1] and references cited therein. -
Kac-Moody Algebras and Applications
Kac-Moody Algebras and Applications Owen Barrett December 24, 2014 Abstract This article is an introduction to the theory of Kac-Moody algebras: their genesis, their con- struction, basic theorems concerning them, and some of their applications. We first record some of the classical theory, since the Kac-Moody construction generalizes the theory of simple finite-dimensional Lie algebras in a closely analogous way. We then introduce the construction and properties of Kac-Moody algebras with an eye to drawing natural connec- tions to the classical theory. Last, we discuss some physical applications of Kac-Moody alge- bras,includingtheSugawaraandVirosorocosetconstructions,whicharebasictoconformal field theory. Contents 1 Introduction2 2 Finite-dimensional Lie algebras3 2.1 Nilpotency.....................................3 2.2 Solvability.....................................4 2.3 Semisimplicity...................................4 2.4 Root systems....................................5 2.4.1 Symmetries................................5 2.4.2 The Weyl group..............................6 2.4.3 Bases...................................6 2.4.4 The Cartan matrix.............................7 2.4.5 Irreducibility...............................7 2.4.6 Complex root systems...........................7 2.5 The structure of semisimple Lie algebras.....................7 2.5.1 Cartan subalgebras............................8 2.5.2 Decomposition of g ............................8 2.5.3 Existence and uniqueness.........................8 2.6 Linear representations of complex semisimple Lie algebras...........9 2.6.1 Weights and primitive elements.....................9 2.7 Irreducible modules with a highest weight....................9 2.8 Finite-dimensional modules............................ 10 3 Kac-Moody algebras 10 3.1 Basic definitions.................................. 10 3.1.1 Construction of the auxiliary Lie algebra................. 11 3.1.2 Construction of the Kac-Moody algebra................. 12 3.1.3 Root space of the Kac-Moody algebra g(A) ..............