From the Representation Theory of Vertex Operator Algebras to Modular Tensor Categories in Conformal Field Theory
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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. -
Modular Invariance of Characters of Vertex Operator Algebras
JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 9, Number 1, January 1996 MODULAR INVARIANCE OF CHARACTERS OF VERTEX OPERATOR ALGEBRAS YONGCHANG ZHU Introduction In contrast with the finite dimensional case, one of the distinguished features in the theory of infinite dimensional Lie algebras is the modular invariance of the characters of certain representations. It is known [Fr], [KP] that for a given affine Lie algebra, the linear space spanned by the characters of the integrable highest weight modules with a fixed level is invariant under the usual action of the modular group SL2(Z). The similar result for the minimal series of the Virasoro algebra is observed in [Ca] and [IZ]. In both cases one uses the explicit character formulas to prove the modular invariance. The character formula for the affine Lie algebra is computed in [K], and the character formula for the Virasoro algebra is essentially contained in [FF]; see [R] for an explicit computation. This mysterious connection between the infinite dimensional Lie algebras and the modular group can be explained by the two dimensional conformal field theory. The highest weight modules of affine Lie algebras and the Virasoro algebra give rise to conformal field theories. In particular, the conformal field theories associated to the integrable highest modules and minimal series are rational. The characters of these modules are understood to be the holomorphic parts of the partition functions on the torus for the corresponding conformal field theories. From this point of view, the role of the modular group SL2(Z)ismanifest. In the study of conformal field theory, physicists arrived at the notion of chi- ral algebras (see e.g. -
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. -
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). -
Pulling Yourself up by Your Bootstraps in Quantum Field Theory
Pulling Yourself Up by Your Bootstraps in Quantum Field Theory Leonardo Rastelli Yang Institute for Theoretical Physics Stony Brook University ICTP and SISSA Trieste, April 3 2019 A. Sommerfeld Center, Munich January 30 2019 Quantum Field Theory in Fundamental Physics Local quantum fields ' (x) f i g x = (t; ~x), with t = time, ~x = space The language of particle physics: for each particle species, a field Quantum Field Theory for Collective Behavior Modelling N degrees of freedom in statistical mechanics. Example: Ising! model 1 (uniaxial ferromagnet) σi = 1, spin at lattice site i ± P Energy H = J σiσj − (ij) Near Tc, field theory description: magnetization '(~x) σ(~x) , ∼ h i Z h i H = d3x ~ ' ~ ' + m2'2 + λ '4 + ::: r · r 2 m T Tc ∼ − Z h i H = d3x ~ ' ~ ' + m2'2 + λ '4 + ::: r · r The dots stand for higher-order \operators": '6, (~ ' ~ ')'2, '8, etc. r · r They are irrelevant for the large-distance physics at T T . ∼ c Crude rule of thumb: an operator is irrelevant if its scaling weight [ ] > 3 (3 d, dimension of space). O O ≡ Basic assignments: ['] = 1 d 1 and [~x] = 1 = [~ ] = 1. 2 ≡ 2 − − ) r So ['2] = 1, ['4] = 2, [~ ' ~ '] =3, while ['8] = 4 etc. r · r First hint of universality: critical exponents do not depend on details. E.g., C T T −α, ' (T T )β for T < T , etc. T ∼ j − cj h i ∼ c − c QFT \Theory of fluctuating fields” (Duh!) ≡ Traditionally, QFT is formulated as a theory of local \quantum fields”: Z H['(x)] Y − Z = d'(x) e g x In particle physics, x spacetime and g = ~ (quantum) 2 In statistical mechanics, x space and g = T (thermal). -
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. -
COSETS of AFFINE VERTEX ALGEBRAS INSIDE LARGER STRUCTURES Vertex Algebra Are a Fundamental Class of Algebraic Structures That Ar
COSETS OF AFFINE VERTEX ALGEBRAS INSIDE LARGER STRUCTURES THOMAS CREUTZIG AND ANDREW R. LINSHAW ABSTRACT. Given a finite-dimensional reductive Lie algebra g equipped with a nondegen- erate, invariant, symmetric bilinear form B, let Vk(g;B) denote the universal affine vertex algebra associated to g and B at level k. Let Ak be a vertex (super)algebra admitting a homomorphism Vk(g;B) !Ak. Under some technical conditions on Ak, we characterize the commutant Ck = Com(Vk(g;B); Ak) for generic values of k. We establish the strong 0 0 0 0 finite generation of Ck in the following cases: Ak = Vk(g ;B ), Ak = Vk−l(g ;B ) ⊗ F, and 0 0 00 00 0 00 Ak = Vk−l(g ;B ) ⊗ Vl(g ;B ). Here g and g are finite-dimensional Lie (super)algebras containing g, equipped with nondegenerate, invariant, (super)symmetric bilinear forms B0 and B00 which extend B, l 2 C is fixed, and F is a free field algebra admitting a homomor- phism Vl(g;B) !F. Our approach is essentially constructive and leads to minimal strong finite generating sets for many interesting examples. 1. INTRODUCTION Vertex algebra are a fundamental class of algebraic structures that arose out of confor- mal field theory and have applications in a diverse range of subjects. The coset or commu- tant construction is a standard way to construct new vertex algebras from old ones. Given a vertex algebra V and a subalgebra A ⊂ V, Com(A; V) is the subalgebra of V which com- mutes with A. This was introduced by Frenkel and Zhu in [FZ], generalizing earlier constructions in representation theory [KP] and physics [GKO], where it was used to con- struct the unitary discrete series representations of the Virasoro algebra. -
Intertwining Operator Superalgebras and Vertex Tensor Categories for Superconformal Algebras, Ii
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 354, Number 1, Pages 363{385 S 0002-9947(01)02869-0 Article electronically published on August 21, 2001 INTERTWINING OPERATOR SUPERALGEBRAS AND VERTEX TENSOR CATEGORIES FOR SUPERCONFORMAL ALGEBRAS, II YI-ZHI HUANG AND ANTUN MILAS Abstract. We construct the intertwining operator superalgebras and vertex tensor categories for the N = 2 superconformal unitary minimal models and other related models. 0. Introduction It has been known that the N = 2 Neveu-Schwarz superalgebra is one of the most important algebraic objects realized in superstring theory. The N =2su- perconformal field theories constructed from its discrete unitary representations of central charge c<3 are among the so-called \minimal models." In the physics liter- ature, there have been many conjectural connections among Calabi-Yau manifolds, Landau-Ginzburg models and these N = 2 unitary minimal models. In fact, the physical construction of mirror manifolds [GP] used the conjectured relations [Ge1] [Ge2] between certain particular Calabi-Yau manifolds and certain N =2super- conformal field theories (Gepner models) constructed from unitary minimal models (see [Gr] for a survey). To establish these conjectures as mathematical theorems, it is necessary to construct the N = 2 unitary minimal models mathematically and to study their structures in detail. In the present paper, we apply the theory of intertwining operator algebras developed by the first author in [H3], [H5] and [H6] and the tensor product theory for modules for a vertex operator algebra developed by Lepowsky and the first author in [HL1]{[HL6], [HL8] and [H1] to construct the intertwining operator algebras and vertex tensor categories for N = 2 superconformal unitary minimal models. -
A Spatial Operator Algebra
International Journal of Robotics Research vol. 10, pp. 371-381, Aug. 1991 A Spatial Op erator Algebra for Manipulator Mo deling and Control 1 2 1 G. Ro driguez , K. Kreutz{Delgado , and A. Jain 2 1 AMES Department, R{011 Jet Propulsion Lab oratory Univ. of California, San Diego California Institute of Technology La Jolla, CA 92093 4800 Oak Grove Drive Pasadena, CA 91109 Abstract A recently develop ed spatial op erator algebra for manipulator mo deling, control and tra- jectory design is discussed. The elements of this algebra are linear op erators whose domain and range spaces consist of forces, moments, velo cities, and accelerations. The e ect of these op erators is equivalent to a spatial recursion along the span of a manipulator. Inversion of op erators can be eciently obtained via techniques of recursive ltering and smo othing. The op erator algebra provides a high-level framework for describing the dynamic and kinematic b ehavior of a manipu- lator and for control and tra jectory design algorithms. The interpretation of expressions within the algebraic framework leads to enhanced conceptual and physical understanding of manipulator dynamics and kinematics. Furthermore, implementable recursive algorithms can be immediately derived from the abstract op erator expressions by insp ection. Thus, the transition from an abstract problem formulation and solution to the detailed mechanization of sp eci c algorithms is greatly simpli ed. 1 Intro duction: A Spatial Op erator Algebra A new approach to the mo deling and analysis of systems of rigid b o dies interacting among them- selves and their environment has recently b een develop ed in Ro driguez 1987a and Ro driguez and Kreutz-Delgado 1992b . -
Chiral Algebras and Partition Functions
Provided by the author(s) and NUI Galway in accordance with publisher policies. Please cite the published version when available. Title Partition functions and chiral algebras. Author(s) Tuite, Michael P. Publication Date 2007-05 Publication Mason, G., Tuite, M.P. (2007) Partition functions and chiral Information algebras Lie algebras, vertex operator algebras and their applications, CONTEMPORARY 442 Publisher American Mathematical Society Link to publisher's http://www.ams.org/bookstore-getitem/item=conm-442 version Item record http://hdl.handle.net/10379/4848 Downloaded 2021-09-29T23:55:39Z Some rights reserved. For more information, please see the item record link above. Chiral Algebras and Partition Functions Geoffrey Mason∗ Department of Mathematics, University of California Santa Cruz, CA 95064, U.S.A. MichaelP.Tuite† Department of Mathematical Physics, National University of Ireland, Galway, Ireland. In Honor of Jim Lepowsky and Robert Wilson Abstract We discuss recent work of the authors concerning correlation functions and partition functions for free bosons/fermions and the b-c or ghost system. We compare and contrast the nature of the 1-point functions at genus 1, and explain how one may understand the free boson partition function at genus 2 via vertex operators and sewing complex tori. 1 Introduction This paper is based on the talk given by one of the authors at the North Carolina State Conference honoring Jim Lepowsky and Robert Wilson. The paper concerns the idea of partition functions in the theory of chiral algebras. The genus 1 partition function of a vertex operator algebra - a.k.a. the graded dimension - has been studied extensively, but the case when either the genus is greater than 1 or else the chiral algebra is not a vertex operator algebra ∗Partial support provided by NSF DMS-0245225 and the Committee on Research, University of California, Santa Cruz †Supported by the Millenium Fund, National University of Ireland, Galway 1 has received little attention from mathematicians thus far. -
Clifford Algebra Analogue of the Hopf–Koszul–Samelson Theorem
Advances in Mathematics AI1608 advances in mathematics 125, 275350 (1997) article no. AI971608 Clifford Algebra Analogue of the HopfKoszulSamelson Theorem, the \-Decomposition C(g)=End V\ C(P), and the g-Module Structure of Ãg Bertram Kostant* Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Received October 18, 1996 1. INTRODUCTION 1.1. Let g be a complex semisimple Lie algebra and let h/b be, respectively, a Cartan subalgebra and a Borel subalgebra of g. Let n=dim g and l=dim h so that n=l+2r where r=dim gÂb. Let \ be one- half the sum of the roots of h in b. The irreducible representation ?\ : g Ä End V\ of highest weight \ is dis- tinguished among all finite-dimensional irreducible representations of g for r a number of reasons. One has dim V\=2 and, via the BorelWeil theorem, Proj V\ is the ambient variety for the minimal projective embed- ding of the flag manifold associated to g. In addition, like the adjoint representation, ?\ admits a uniform construction for all g. Indeed let SO(g) be defined with respect to the Killing form Bg on g. Then if Spin ad: g Ä End S is the composite of the adjoint representation ad: g Ä Lie SO(g) with the spin representation Spin: Lie SO(g) Ä End S, it is shown in [9] that Spin ad is primary of type ?\ . Given the well-known relation between SS and exterior algebras, one immediately deduces that the g-module structure of the exterior algebra Ãg, with respect to the derivation exten- sion, %, of the adjoint representation, is given by l Ãg=2 V\V\.(a) The equation (a) is only skimming the surface of a much richer structure.