DOCSLIB.ORG

A Family of Regular Vertex Operator Algebras with Two Generators

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Family of Regular Vertex Operator Algebras with Two Generators DOI: 10.2478/s11533-006-0045-2 Research article CEJM 5(1) 2007 1–18 A family of regular vertex operator algebras with two generators Draˇzen Adamovi´c∗ Department of Mathematics, University of Zagreb, 10 000 Zagreb, Croatia Received 7 September 2006; accepted 20 November 2006 Abstract: For every m ∈ C \{0, −2} and every nonnegative integer k we define the vertex operator 3m (super)algebra Dm,k having two generators and rank m+2 .Ifm is a positive integer then Dm,k can be realized as a subalgebra of a lattice vertex algebra. In this case, we prove that Dm,k is a regular vertex operator (super)algebra and find the number of inequivalent irreducible modules. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved. Keywords: Vertex operator algebras, vertex operator superalgebras, rationality, regularity, lattice vertex operator algebras MSC (2000): 17B69 1 Introduction In the theory of vertex operator (super)algebras, the classification and construction of rational vertex operator (super)algebras are important problems. These problems are connected with the classification of rational conformal field theories in physics. The rationality of certain familiar vertex operator (super)algebras was proved in papers [1– 3, 7, 8, 14, 20, 25]. It is natural to consider rational vertex operator (super)algebras of certain rank. In particular, in the rank one case for every positive integer k we have√ the well-known rational vertex operator (super)algebra Fk associated to the lattice kZ. These vertex operator (super)algebras are generated by two generators. In the present paper we will be concentrated on vertex operator (super)algebras of rank ∗ E-mail: [email protected] 2D.Adamovi´c / Central European Journal of Mathematics 5(1) 2007 1–18 3m ∈ C\{ − } cm = m+2 , m 0, 2 . This rank has the vertex operator algebra L(m, 0) associated to the irreducible vacuum slˆ2–module of level m and the vertex operator superalgebra Lcm associated to the vacuum module for the N = 2 superconformal algebra with central charge cm (cf.[2, 3, 11, 15–17]). In the case m = 1 these vertex operator (super)algebras ∼ ∼ are included into the family Fk, k ∈ N,sinceL(1, 0) = F2 and Lc1 = F3.Themain purpose of this article is to include L(m, 0) and Lcm into the family Dm,k, k ∈ Z≥0,of rational vertex operator (super)algebras of rank cm for arbitrary positive integer m. In fact, for every m ∈ C \{0, −2} we define the vertex operator (super)algebra Dm,k as a subalgebra of the vertex operator (super)algebra L(m, 0) ⊗ Fk (cf. Section 4). In the special case k =1,Dm,1 is in the N = 2 vertex operator superalgebra Lcm constructed ∼ by using the Kazama-Suzuki mapping (cf. [15, 19]). We also have that Dm,0 = L(m, 0) ∼ and D1,k = Fk+2. Moreover, we shall demonstrate that Dm,k has many properties similar to those of affine and N = 2 superconformal vertex algebras. When m is not a nonnegative integer, then Dm,k has infinitely many irreducible repre- sentations. Thus, it is not rational (cf. Section 4). In order to construct new examples of rational vertex operator (super)algebras we shall consider the case when m is a positive integer. Then Dm,k can be embedded into a lattice vertex algebra (cf. Section 5). In fact, we shall prove that ∼ Dm,k ⊗ F−k = L(m, 0) ⊗ F− k (mk+2) (k even), (1) ∼ 2 Dm,k ⊗ F−k = L(m, 0) ⊗ F−2k(mk+2) ⊕ L(m, m) ⊗ MF−2k(mk+2) (k odd). (2) These relations completely determine the structure of Dm,k ⊗ F−k as a weak L(m, 0)– module. In [9] the notion of a regular vertex operator algebra was introduced, i. e. rational vertex operator algebra with the property that every weak module is completely reducible. The relations (1)and(2), together with the regularity results from [9]and[21]implythat Dm,k is a simple regular vertex operator algebra if k is even, and a simple regular vertex operator superalgebra if k is odd. It was shown in [5] that regularity is equivalent to rationality and C2–cofiniteness. Therefore, vertex operator (super)algebras Dm,k are also rational and C2–cofinite. Let us here discuss the case k =2n,wheren is a positive integer. The relation (1) suggests that one can study the dual pair (Dm,2n,F−2n) directly inside L(m, 0) ⊗ F−2n(nm+1). This approach requires many deep results on the structure of the vertex operator algebra L(m, 0) and deserves to be investigated independently. Instead of this approach, we realize the vertex algebra Dm,2n ⊗F−2n inside a larger lattice vertex algebra. Then the formulas for the generators are much simpler (cf. Section 6). The similar analysis can be done when k is odd (cf. Section 7). This approach was also used in [3] for studying the fusion rules for the N = 2 vertex operator superalgebra Dm,1. Our results show that for every m ∈ N, there exists an infinite family of rational vertex operator algebras of rank cm. We believe that these algebras will have an important role in the classification of rational vertex operator algebras of this rank. As an example, in this paper we shall consider in detail the vertex operator (super)algebras of rank c4 =2. D. Adamovi´c / Central European Journal of Mathematics 5(1) 2007 1–18 3 Then our vertex operator (super)algebras Dm,k admit nice realizations. In Section 8 we show that D4,k is a Z2–orbifold model of a lattice vertex operator superalgebra under an automorphism of order two. This paper is a slightly modified version of the preprint math.QA/0111055. 2 Preliminaries In this section we recall the definition of vertex operator superalgebras their modules (cf. [12, 13, 18, 20]). We also recall the basic properties of regular vertex operator superalgebras. Let V = V0¯ ⊕ V1¯ be any Z2–graded vector space. Then any element u ∈ V0¯ (resp. u ∈ V1¯) is said to be even (resp. odd). We define |u| = 0if¯ u is even and |u| = 1if¯ u is odd. Elements in V0¯ or V1¯ are called homogeneous. Whenever |u| is written, it is understood that u is homogeneous. Definition 2.1. A vertex superalgebra is a triple (V,Y,1)whereV = V0¯ ⊕ V1¯ is a Z2– graded vector space, 1 ∈ V0¯ is a specified element called the vacuum of V ,andY is a linear map · → −1 Y ( ,z): V (End V )[[z, z ]]; −n−1 −1 a → Y (a, z)= anz ∈ (End V )[[z, z ]] n∈Z satisfying the following conditions for a, b ∈ V : (V1) |anb| = |a| + |b|. (V2) anb =0forn sufficiently large. d (V3) [D, Y (a, z)] = Y (D(a),z)= dz Y (a, z), where D ∈ End V is defined by D(a)=a−21. (V4) Y (1,z)=IV (the identity operator on V ). (V5) Y (a, z)1 ∈ (End V )[[z]] and limz→0 Y (a, z)1 = a. (V6) The following Jacobi identity holds − − −1 z1 z2 − − |a||b| −1 z2 z1 z0 δ Y (a, z1)Y (b, z2) ( 1) z0 δ − Y (b, z2)Y (a, z1) z0 z0 −1 z1 − z0 = z2 δ Y (Y (a, z0)b, z2). z2 A vertex superalgebra V is called a vertex operator superalgebra if there is a ∈ special element ω V0¯(called the Virasoro element) whose vertex operator we write −n−1 −n−2 in the form Y (ω,z)= n∈Z ωnz = n∈Z L(n)z , such that − m3−m ∈ C (V7) [L(m),L(n)] = (m n)L(m + n)+δm+n,0 12 c, c =rankV . (V8) L(−1) = D. 1 ⊕ 1 Z ¯ ⊕ ¯ ⊕ 1 (V9) V = n∈ ZV (n)isa 2 –graded so that V0 = n∈ZV (n), V1 = n∈ +ZV (n), 2 2 L(0) |V (n)= nIV |V (n),dimV (n) < ∞,andV (n)=0forn sufficiently small. 4D.Adamovi´c / Central European Journal of Mathematics 5(1) 2007 1–18 We shall sometimes refer to the vertex operator superalgebra V as quadruple (V,Y,1,ω). Remark 2.2. If in the definition of vertex (operator) superalgebra the odd subspace V1¯ = 0 we get the usual definition of vertex (operator) algebra. We will say that the vertex operator superalgebra is generated by the set S if { 1 ··· r | 1 r ∈ ∈ Z ∈ Z } V =spanC un1 unr 1 u ,...,u S, n1,...,nr ,r ≥0 . A subspace I ⊂ V is called an ideal in the vertex operator superalgebra V if anI ⊂ I for every a ∈ V and n ∈ Z. A vertex operator superalgebra V is called simple if it does not contain any proper non-zero ideal. There is a canonical automorphism σV of the vertex operator superalgebra V such that σV |V0¯ =1andσV |V1¯ = −1. Definition 2.3. Let V be a vertex operator superalgebra. A weak V –module is a pair (M,YM ), where M = M0¯ ⊕ M1¯ is a Z2–graded vector space, and YM (·,z) is a linear map −1 −n−1 YM : V → End(M)[[z, z ]],a → YM (a, z)= anz , n∈Z satisfying the following conditions for a, b ∈ V and v ∈ M: (M1) |anv| = |a| + |v| for any a ∈ V . (M2) YM (1,z)=IM . (M3) anv =0forn sufficiently large. (M4) The following Jacobi identity holds − − −1 z1 z2 − − |a||b| −1 z2 z1 z0 δ YM (a, z1)YM (b, z2) ( 1) z0 δ − YM (b, z2)YM (a, z1) z0 z0 −1 z1 − z0 = z2 δ YM (Y (a, z0)b, z2).
Load more

Recommended publications
  • 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.
    [Show full text]
  • Conformal Killing Superalgebras
    CONFORMAL SYMMETRY SUPERALGEBRAS PAUL DE MEDEIROS AND STEFAN HOLLANDS ABSTRACT. We show how the rigid conformal supersymmetries associated with a certain class of pseudo-Riemannian spin manifolds define a Lie superalgebra. The even part of this superalgebra con- tains conformal isometries and constant R-symmetries. The odd part is generated by twistor spinors valued in a particular R-symmetry representation. We prove that any manifold which admits a con- formal symmetry superalgebra of this type must generically have dimension less than seven. Moreover, in dimensions three, four, five and six, we provide the generic data from which the conformal symmetry superalgebra is prescribed. For conformally flat metrics in these dimensions, and compact R-symmetry, we identify each of the associated conformal symmetry superalgebras with one of the conformal su- peralgebras classified by Nahm. We also describe several examples for Lorentzian metrics that are not conformally flat. CONTENTS 1. Introduction 2 2. Preliminaries 3 2.1. Tensors, vectors and conformal Killing vectors 3 2.2. Clifford modules, spinors and twistor spinors 4 3. Conformal structure 6 4. Spinorial bilinear forms 6 4.1. Complex case 8 4.2. Real case 9 5. Conformal symmetry superalgebras 10 5.1. Brackets 10 5.2. Conformal invariance 11 5.3. Jacobi identities 12 5.4. Real forms 12 6. Comparison with Nahm for conformally flat metrics 14 7. Some non-trivial examples in Lorentzian signature 15 arXiv:1302.7269v2 [hep-th] 3 Jun 2013 7.1. Brinkmann waves 16 7.2. Lorentzian Einstein-Sasaki manifolds 17 7.3. Fefferman spaces 17 7.4. Direct products 18 Acknowledgments 19 References 19 Date: 9th October 2018.
    [Show full text]
  • S-Duality, the 4D Superconformal Index and 2D Topological QFT
    Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N S-duality, the 4d Superconformal Index and 2d Topological QFT Leonardo Rastelli with A. Gadde, E. Pomoni, S. Razamat and W. Yan arXiv:0910.2225,arXiv:1003.4244, and in progress Yang Institute for Theoretical Physics, Stony Brook Rutgers, November 9 2010 Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N A new paradigm for 4d =2 susy gauge theories N (Gaiotto, . ) Compactification of the (2, 0) 6d theory on a 2d surface Σ, with punctures. = =2 superconformal⇒ theories in four dimensions. N Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N A new paradigm for 4d =2 susy gauge theories N (Gaiotto, . ) Compactification of the (2, 0) 6d theory on a 2d surface Σ, with punctures. = =2 superconformal⇒ theories in four dimensions. N Space of complex structures Σ = parameter space of the 4d • theory. Moore-Seiberg groupoid of Σ = (generalized) 4d S-duality • Vast generalization of “ =4 S-duality as modular group of T 2”. N Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N A new paradigm for 4d =2 susy gauge theories N (Gaiotto, . ) Compactification of the (2, 0) 6d theory on a 2d surface Σ, with punctures. = =2 superconformal⇒ theories in four dimensions. N Space of complex structures Σ = parameter space of the 4d • theory.
    [Show full text]
  • Arxiv:1905.09269V2
    Anomaly cancellation in the topological string Kevin Costello∗ Perimeter Institute of Theoretical Physics 31 Caroline St N, Waterloo, ON N2L 2Y5, Canada Si Li† Department of Mathematical Sciences and Yau Mathematical Sciences Center, Jingzhai, Tsinghua University, Beijing 100084, China (Dated: January 7, 2020) We describe the coupling of holomorphic Chern-Simons theory at large N with Kodaira- Spencer gravity. We explain a new anomaly cancellation mechanism at all loops in pertur- bation theory for open-closed topological B-model. At one loop this anomaly cancellation is analogous to the Green-Schwarz mechanism. As an application, we introduce a type I version of Kodaira-Spencer theory in complex dimensions 3 and 5. In complex dimension 5, we show that it can only be coupled consistently at the quantum level to holomorphic Chern-Simons theory with gauge group SO(32). This is analogous to the Green-Schwarz mechanism for the physical type I string. This coupled system is conjectured to be a supersymmetric localization of type I string theory. In complex dimension 3, the required gauge group is SO(8). Contents I. Introduction 2 II. Open-closed topological B-model 3 A. The open-string sector 3 arXiv:1905.09269v2 [hep-th] 5 Jan 2020 B. The closed-string sector 6 C. Coupling closed to open strings at leading order 8 D. Kodaira-Spencer gravity in BV formalism 9 E. Green-Schwarz mechanism 13 F. The holomorphic stress-energy tensors 15 ∗Electronic address: [email protected] †Electronic address: [email protected] 2 G. Large N single trace operators 19 H.
    [Show full text]
  • 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.
    [Show full text]
  • Arxiv:2002.02662V2 [Math.RT]
    BLOCKS AND CHARACTERS OF D(2|1; ζ)-MODULES OF NON-INTEGRAL WEIGHTS CHIH-WHI CHEN, SHUN-JEN CHENG, AND LI LUO Abstract. We classify blocks in the BGG category O of modules of non-integral weights for the exceptional Lie superalgebra D(2|1; ζ). We establish various reduction methods, which connect some types of non-integral blocks of D(2|1; ζ) with integral blocks of general linear Lie superalgebras gl(1|1) and gl(2|1). We compute the characters for irreducible D(2|1; ζ)-modules of non-integral weights in O. Contents 1. Introduction 1 2. Preliminaries 3 3. Classification of blocks 10 4. Reduction methods and characters in the generic and 1-integer cases 15 5. Character formulas in 2-integer case 19 6. Primitive spectrum of D(2|1; ζ) 23 References 25 1. Introduction 1.1. Since a Killing-Cartan type classification of finite-dimensional complex simple Lie superalge- bras has been achieved by Kac [Kac77] in 1977 there has been substantial efforts by mathematicians and physicists in the study of their representation theory. The most important class of the simple arXiv:2002.02662v2 [math.RT] 20 Feb 2020 Lie superalgebras is the so-called basic classical, which includes the classical series of types ABCD. Among these basic Lie superalgebras, there are 3 exceptional ones: D(2|1; ζ), G(3) and F (3|1). 1.2. The irreducible character problem is one of the main theme in representation theory of Lie (super)algebras. While complete answers to the irreducible character problem in the BGG cat- egories for basic Lie superalgebras of types ABCD are now known (see [CLW11, CLW15, Ba17, BLW17, BW18]), the BGG category of the exceptional Lie superalgebras were not until recently.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Duality and Integrability in Topological String Theory
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by SISSA Digital Library International School of Advanced Studies Mathematical Physics Sector Academic Year 2008–09 PhD Thesis in Mathematical Physics Duality and integrability in topological string theory Candidate Andrea Brini Supervisors External referees Prof. Boris Dubrovin Prof. Marcos Marino˜ Dr. Alessandro Tanzini Dr. Tom Coates 2 Acknowledgements Over the past few years I have benefited from the interaction with many people. I would like to thank first of all my supervisors: Alessandro Tanzini, for introducing me to the subject of this thesis, for his support, and for the fun we had during countless discussions, and Boris Dubrovin, from whose insightful explanations I have learned a lot. It is moreover a privilege to have Marcos Mari˜no and Tom Coates as referees of this thesis, and I would like to thank them for their words and ac- tive support in various occasions. Special thanks are also due to Sara Pasquetti, for her help, suggestions and inexhaustible stress-energy transfer, to Giulio Bonelli for many stimulating and inspiring discussions, and to my collaborators Renzo Cava- lieri, Luca Griguolo and Domenico Seminara. I would finally like to acknowledge the many people I had discussions/correspondence with: Murad Alim, Francesco Benini, Marco Bertola, Vincent Bouchard ;-), Ugo Bruzzo, Mattia Cafasso, Hua-Lian Chang, Michele Cirafici, Tom Claeys, Andrea Collinucci, Emanuel Diaconescu, Bertrand Ey- nard, Jarah Evslin, Fabio Ferrari Ruffino, Barbara
    [Show full text]
  • 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).
    [Show full text]
  • Introduction to Conformal Field Theory and String
    SLAC-PUB-5149 December 1989 m INTRODUCTION TO CONFORMAL FIELD THEORY AND STRING THEORY* Lance J. Dixon Stanford Linear Accelerator Center Stanford University Stanford, CA 94309 ABSTRACT I give an elementary introduction to conformal field theory and its applications to string theory. I. INTRODUCTION: These lectures are meant to provide a brief introduction to conformal field -theory (CFT) and string theory for those with no prior exposure to the subjects. There are many excellent reviews already available (or almost available), and most of these go in to much more detail than I will be able to here. Those reviews con- centrating on the CFT side of the subject include refs. 1,2,3,4; those emphasizing string theory include refs. 5,6,7,8,9,10,11,12,13 I will start with a little pre-history of string theory to help motivate the sub- ject. In the 1960’s it was noticed that certain properties of the hadronic spectrum - squared masses for resonances that rose linearly with the angular momentum - resembled the excitations of a massless, relativistic string.14 Such a string is char- *Work supported in by the Department of Energy, contract DE-AC03-76SF00515. Lectures presented at the Theoretical Advanced Study Institute In Elementary Particle Physics, Boulder, Colorado, June 4-30,1989 acterized by just one energy (or length) scale,* namely the square root of the string tension T, which is the energy per unit length of a static, stretched string. For strings to describe the strong interactions fi should be of order 1 GeV. Although strings provided a qualitative understanding of much hadronic physics (and are still useful today for describing hadronic spectra 15 and fragmentation16), some features were hard to reconcile.
    [Show full text]
  • 1 the Superalgebra of the Supersymmetric Quantum Me- Chanics
    CBPF-NF-019/07 1 Representations of the 1DN-Extended Supersymmetry Algebra∗ Francesco Toppan CBPF, Rua Dr. Xavier Sigaud 150, 22290-180, Rio de Janeiro (RJ), Brazil. E-mail: [email protected] Abstract I review the present status of the classification of the irreducible representations of the alge- bra of the one-dimensional N− Extended Supersymmetry (the superalgebra of the Supersym- metric Quantum Mechanics) realized by linear derivative operators acting on a finite number of bosonic and fermionic fields. 1 The Superalgebra of the Supersymmetric Quantum Me- chanics The superalgebra of the Supersymmetric Quantum Mechanics (1DN-Extended Supersymme- try Algebra) is given by N odd generators Qi (i =1,...,N) and a single even generator H (the hamiltonian). It is defined by the (anti)-commutation relations {Qi,Qj} =2δijH, [Qi,H]=0. (1) The knowledge of its representation theory is essential for the construction of off-shell invariant actions which can arise as a dimensional reduction of higher dimensional supersymmetric theo- ries and/or can be given by 1D supersymmetric sigma-models associated to some d-dimensional target manifold (see [1] and [2]). Two main classes of (1) representations are considered in the literature: i) the non-linear realizations and ii) the linear representations. Non-linear realizations of (1) are only limited and partially understood (see [3] for recent results and a discussion). Linear representations, on the other hand, have been recently clarified and the program of their classification can be considered largely completed. In this work I will review the main results of the classification of the linear representations and point out which are the open problems.
    [Show full text]