DOCSLIB.ORG

Research Collection

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Research Collection Doctoral Thesis Non-Abelian Orbifold Theory and Twisted Modules for Vertex Operator Algebras Author(s): Gemünden, Thomas Publication Date: 2020 Permanent Link: https://doi.org/10.3929/ethz-b-000446462 Rights / License: In Copyright - Non-Commercial Use Permitted This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use. ETH Library diss. eth no. 26903 NON-ABELIANORBIFOLDTHEORYANDTWISTEDMODULES FORVERTEXOPERATORALGEBRAS A thesis submitted to attain the degree of doctor of sciences of eth zurich¨ (Dr. sc. ETH Z¨urich) presented by thomas gemunden¨ Master of Mathematics, University of Cambridge Master of Arts, University of Cambridge born on 05.02.1994 citizen of Germany accepted on the recommendation of Prof. Dr. G. Felder, Referent Prof. Dr. C. A. Keller, Korreferent Prof. Dr. N. Scheithauer, Korreferent 2020 Thomas Gem¨unden, Non-Abelian Orbifold Theory and Twisted Modules for Vertex Operator Algebras Abstract The topic of this thesis is the theory of holomorphic extensions of orbifolds of holomorphic vertex operator algebras. We develop an orbifold theory for non-abelian groups of the form Zq o Zp and give an explicit formula for the characters of the holomorphic extensions. Additionally, we show how groups of the form Zq o Zp can be realised as automorphism groups of lattice vertex operator algebras. Furthermore, we construct the action of automorphisms on twisted modules for lattice vertex operator algebras and give explicit expressions for the corresponding graded trace functions. Using these results, we conduct a survey of orbifolds by cyclic groups and groups of the form Zq o Zp of lattice vertex operator algebras associated to extremal lattices in dimensions 48 and 72. This is the first systematic survey of holomorphic vertex operator algebras at higher central charge, as well as the first systematic survey of non-abelian orbifold theories. We construct around 200 new holomorphic vertex operator algebras at central charge c = 48 and c = 72. These include those with the lowest number of low-weight states currently known among holomorphic vertex operator algebras of the respective central charges. These allow us to prove that there exist holomorphic vertex operator algebras that cannot be constructed as abelian orbifolds of lattice vertex operator algebras. Finally, we investigate the notion of `large-central charge limit' of a family of vertex operator algebras. We define a vertex algebra that can be considered such a large-central charge limit for a family of permutation orbifolds and give a sufficient condition for its existence. 3 Zusammenfassung Das Thema dieser Abhandlung ist die Theorie der holomorphen Erweiterungen der Orbifaltigkeiten von holomorphen Vertexoperatoralgebren. Wir entwickeln eine Theorie der Orbifaltigkeiten f¨urnicht-abelsche Gruppen der Form Zq o Zp und bestimmen einen expliziten Ausdruck f¨urdie Charaktere der holomorphen Erweiterungen. Zus¨atzlich, zeigen wir, wie Gruppen der Form Zq o Zp als Automorphismusgruppen von Gittervertexoperatoralgebren realisiert werden k¨onnen. Weiterhin konstruieren wir die Wirkung von Automorphismen auf die verdrehten Moduln von Gitter- vertexoperatoralgebren und bestimmen die dazugeh¨origengradierten Spurfuntionen. Mithilfe dieser Resultate unternehmen wir eine Untersuchung der Orbifaltigkeiten von zyklischen Gruppen und Gruppen der Form Zq o Zp f¨urGittervertexoperatoralgebren von extremalen Gittern in den Dimensionen 48 und 72. Dies ist die erste systematische Untersuchung von holomorphen Vertexoper- atoralgebren gr¨ossererzentraler Ladung, sowie die erste systematische Untersuchung von nicht-abelschen Orbifaltigkeiten. Wir konstruieren etwa 200 neue holomorphe Vertexoperatoralgebren der zentralen Ladungen 48 und 72. Darunter sind die holomorphen Vertexoperatoralgebren mit der jeweils kleinsten bekannten Anzahl von leichten Zust¨andenunter holomorphen Vertexoperatoralgebren ihrer zentralen Ladung. Diese erlauben uns zu beweisen, dass holomorphe Vertexoperatoralgebren existieren, die nicht als abelsche Orbifaltigkeiten von Gittervertexoperatoralgebren konstruiert werden k¨onnen. Zuletzt untersuchen wir das Konzept des `Grenzwertes zur grossen zentralen Ladung` einer Familie von Vertexoperatoralgebren. Wir definieren eine Vertexalgebra, die als Grenzwert zur grossen zentralen Ladung einer Familie von Permutationsorbifaltigkeiten angesehen werden kann und geben eine hinreichende Bedingung f¨urihre Existenz. 4 Acknowledgements I would like to express my gratitude to Prof. Dr. Christoph Keller for his excellent supervision and his patience and encouragement throughout a remarkably eventful doctorate. I would like to thank Prof. Dr. Giovanni Felder for valuable discussions and advise, helpful comments on this text and finally for taking me in as his student. I am grateful to Prof. Dr. Nils Scheithauer for invaluable advise and helpful comments on this text. I would like to thank Prof. Dr. Gerald H¨ohnfor his kind hospitality during my stay in Manhattan. Finally, I would like to thank my colleague Simon Brun, for his fantastic work as a group organiser. My work is supported by the Swiss National Science Foundation Project Grant 175494. 5 Contents Introduction 8 1 Preliminaries and Definitions 13 1.1 Formal Calculus . 13 1.2 Vertex Operator Algebras . 13 1.3 Automorphisms of Vertex Operator Algebras . 14 1.4 Modules and Twisted Modules . 14 1.5 Regularity and Rationality . 16 1.6 Holomorphic Vertex Operator Algebras . 17 1.7 Homomorphisms, Schur's Lemma and Action of Automorphisms on Twisted Modules . 17 1.8 Characters, Graded Trace Functions and Modular Invariance . 18 1.9 Intertwining operators, Fusion and Vertex Tensor Categories . 19 2 Orbifold Theory 22 2.1 Holomorphic Extensions of Vertex Operator Algebras . 22 2.2 Cyclic Orbifolds . 25 2.3 Non-Abelian Orbifolds . 27 2.3.1 Groups with Periodic Cohomology . 28 2.3.2 Semidirect Products of Cyclic Groups Zq oφ Zp .................... 29 2.3.3 Characters of Holomorphic Extensions for Zq oφ Zp ................. 30 3 Lattice Vertex Operator Algebras 33 3.1 Lattices . 33 3.1.1 Integral Lattices . 33 3.1.2 Lattice theta-functions and Modularity . 34 3.1.3 Lattice Automorphisms . 35 3.1.4 Smith Normal Form and Lattice Quotients . 36 3.2 Construction of Lattice Vertex Operator Algebras . 37 3.3 Automorphisms of Lattice Vertex Operator Algebras . 39 3.4 Graded Trace Functions for Lattice Vertex Operator Algebras . 40 3.5 Finite Automorphism Groups for Lattice Vertex Operator Algebras . 41 3.5.1 Lifting Semidirect Products of Cyclic Groups Zq oφ Zp ................ 43 4 Twisted Modules for Lattice Vertex Operator Algebras 47 4.1 Construction of Twisted Modules . 47 4.2 The Vacuum Representation . 49 4.3 A Basis for VL(^g)......................................... 52 4.4 Action of Automorphisms on Twisted Modules . 53 4.5 Graded Trace Functions for Twisted Modules . 53 4.6 Examples: Orbifolds of VE8 ................................... 55 4.6.1 An S3-Orbifold . 55 4.6.2 A Z2 × Z2-Orbifold . 55 4.6.3 A Z3 × Z3-Orbifold . 56 5 Cyclic Orbifolds of Lattice Vertex Operator Algebras at c = 48 and c = 72 57 5.1 Computational Aspects | Characters of Holomorphic Extensions for Cyclic Groups . 57 5.1.1 Eta-Quotients and Modular Transformations of Lattice Theta-Functions . 57 5.1.2 Computing Characters of Cyclic Orbifolds . 60 5.2 The Orbifold Algorithm . 61 5.3 Cyclic Orbifolds of Extremal Lattices in d =48........................ 62 5.3.1 Cyclic Orbifolds for the Lattice P48m .......................... 63 5.3.2 Cyclic Orbifolds for the Lattice P48n .......................... 63 6 5.3.3 Cyclic Orbifolds for the Lattice P48p .......................... 65 5.3.4 Cyclic Orbifolds for the Lattice P48q .......................... 65 5.4 Cyclic Orbifolds for the Lattice Γ72 ............................... 66 6 Non-Abelian Orbifolds of Lattice Vertex Operator Algebras at c = 48 and c = 72 68 6.1 Zq oφ Zp-Orbifolds for d =48.................................. 68 6.1.1 Zq oφ Zp-Orbifolds for the lattice P48p ......................... 68 6.1.2 Zq oφ Zp-Orbifolds for the lattice P48q ......................... 68 6.1.3 Zq oφ Zp-Orbifolds for the lattice P48m ......................... 69 6.1.4 Zq oφ Zp-Orbifolds for the lattice P48n ......................... 69 6.2 Zq oφ Zp-Orbifolds for the Lattice Γ72 ............................. 70 6.3 Bounds for Abelian Orbifolds . 70 7 The Large-N-Limit for Permutation Orbifolds 72 7.1 Oligomorphic Families of Permutation Groups . 72 7.1.1 Unitary Vertex Operator Algebras . 72 7.1.2 Oligomorphic Orbifolds . 73 7.1.3 Space of states . 74 7.1.4 Structure Constants . 75 7.1.5 An Example: SN ..................................... 77 7.2 The Limit of GL(N; q)-Orbifolds . 78 Bibliography 81 7 Preface The topic of this text are certain algebraic structures, called vertex operator algebras. They can be used to construct full realisations of conformal field theories, special models in particle physics that play a central role in string theory. In informal terms, our goal will be to construct new vertex operator algebras from old ones and their symmetries and hence construct new conformal field theories. The theory of vertex operator algebras is a relatively new field of study with deep roots in different areas of mathematics and physics that appear entirely unrelated at first sight. First introduced in the 1980s by Richard Borcherds [Bor86] vertex operator algebras played a crucial role in his proof of the \monstrous moonshine"
Recommended publications
  • Modular Invariance of Characters of Vertex Operator Algebras

    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.
  • From String Theory and Moonshine to Vertex Algebras

    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).
  • M5-Branes, D4-Branes and Quantum 5D Super-Yang-Mills

    M5-Branes, D4-Branes and Quantum 5D Super-Yang-Mills

    CERN-PH-TH/2010-294 KCL-MTH-10-17 M5-Branes, D4-Branes and Quantum 5D super-Yang-Mills N. Lambert a,∗,† , C. Papageorgakis b,‡ and M. Schmidt-Sommerfeld a,§ aTheory Division, CERN 1211 Geneva 23, Switzerland bDepartment of Mathematics, King’s College London The Strand, London WC2R 2LS, UK Abstract We revisit the relation of the six-dimensional (2, 0) M5-brane Conformal Field Theory compactified on S1 to 5D maximally supersymmetric Yang-Mills Gauge Theory. We show that in the broken phase 5D super-Yang-Mills contains a arXiv:1012.2882v3 [hep-th] 22 Feb 2011 spectrum of soliton states that can be identified with the complete Kaluza-Klein modes of an M2-brane ending on the M5-branes. This provides evidence that the (2, 0) theory on S1 is equivalent to 5D super-Yang-Mills with no additional UV degrees of freedom, suggesting that the latter is in fact a well-defined quantum theory and possibly finite. ∗On leave of absence from King’s College London. †E-mail address: [email protected] ‡E-mail address: [email protected] §E-mail address: [email protected] 1 Introduction Multiple M5-branes are believed to be described at low energies by a novel, interacting, strongly coupled, 6D CFT with (2, 0) supersymmetry. Very little is known about such a theory and it is not expected to have a Lagrangian description. According to the type IIA/M-theory duality it arises as the strong-coupling, UV fixed-point of multiple D4-branes whose dynamics are obtained from open string theory.
  • Pulling Yourself up by Your Bootstraps in Quantum Field Theory

    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).
  • Round Table Talk: Conversation with Nathan Seiberg

    Round Table Talk: Conversation with Nathan Seiberg

    Round Table Talk: Conversation with Nathan Seiberg Nathan Seiberg Professor, the School of Natural Sciences, The Institute for Advanced Study Hirosi Ooguri Kavli IPMU Principal Investigator Yuji Tachikawa Kavli IPMU Professor Ooguri: Over the past few decades, there have been remarkable developments in quantum eld theory and string theory, and you have made signicant contributions to them. There are many ideas and techniques that have been named Hirosi Ooguri Nathan Seiberg Yuji Tachikawa after you, such as the Seiberg duality in 4d N=1 theories, the two of you, the Director, the rest of about supersymmetry. You started Seiberg-Witten solutions to 4d N=2 the faculty and postdocs, and the to work on supersymmetry almost theories, the Seiberg-Witten map administrative staff have gone out immediately or maybe a year after of noncommutative gauge theories, of their way to help me and to make you went to the Institute, is that right? the Seiberg bound in the Liouville the visit successful and productive – Seiberg: Almost immediately. I theory, the Moore-Seiberg equations it is quite amazing. I don’t remember remember studying supersymmetry in conformal eld theory, the Afeck- being treated like this, so I’m very during the 1982/83 Christmas break. Dine-Seiberg superpotential, the thankful and embarrassed. Ooguri: So, you changed the direction Intriligator-Seiberg-Shih metastable Ooguri: Thank you for your kind of your research completely after supersymmetry breaking, and many words. arriving the Institute. I understand more. Each one of them has marked You received your Ph.D. at the that, at the Weizmann, you were important steps in our progress.
  • 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 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.
  • Electric-Magnetic Duality, Matrices, & Emergent Spacetime

    Electric-Magnetic Duality, Matrices, & Emergent Spacetime

    Electric-Magnetic Duality, Matrices, & Emergent Spacetime Shyamoli Chaudhuri1 1312 Oak Drive Blacksburg, VA 24060 Abstract This is a rough transcript of talks given at the Workshop on Groups & Algebras in M Theory at Rutgers University, May 31–Jun 04, 2005. We review the basic motivation for a pre-geometric formulation of nonperturbative String/M theory, and for an underlying eleven- dimensional electric-magnetic duality, based on our current understanding of the String/M Duality Web. We explain the concept of an emerging spacetime geometry in the large N limit of a U(N) flavor matrix Lagrangian, distinguishing our proposal from generic proposals for quantum geometry, and explaining why it can incorporate curved spacetime backgrounds. We assess the significance of the extended symmetry algebra of the matrix Lagrangian, raising the question of whether our goal should be a duality covariant, or merely duality invariant, Lagrangian. We explain the conjectured isomorphism between the O(1/N) corrections in any given large N scaling limit of the matrix Lagrangian, and the corresponding α′ corrections in a string effective Lagrangian describing some weak-coupling limit of the String/M Duality Web. arXiv:hep-th/0507116v4 23 Aug 2005 1Based on talks given at the Workshop on Groups & Algebras in M Theory, Rutgers University, May 31–Jun 04, 2005. Email: [email protected] 1 Introduction Understanding the symmetry principles and the fundamental degrees of freedom in terms of which nonperturbative String/M theory is formulated is a problem of outstanding importance in theoretical high energy physics. The Rutgers Mathematics workshop on Groups & Algebras in M Theory this summer devoted part of its schedule to an assessment of the significance of Lorentzian Kac-Moody algebras to recent conjectures for the symmetry algebra of String/M theory.
  • Chiral Algebras and Partition Functions

    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.
  • A View from the Bridge Natalie Paquette

    A View from the Bridge Natalie Paquette

    INFERENCE / Vol. 3, No. 4 A View from the Bridge Natalie Paquette tring theory is a quantum theory of gravity.1 Albert example, supersymmetric theories require particles to Einstein’s theory of general relativity emerges natu- come in pairs. For every bosonic particle there is a fermi- rally from its equations.2 The result is consistent in onic superpartner. Sthe sense that its calculations do not diverge to infinity. Supersymmetric field theory has a disheartening String theory may well be the only consistent quantum impediment. Suppose that a supersymmetric quantum theory of gravity. If true, this would be a considerable field theory is defined on a generic curved manifold. The virtue. Whether it is true or not, string theory is indis- Euclidean metric of Newtonian physics and the Lorentz putably the source of profound ideas in mathematics.3 metric of special relativity are replaced by the manifold’s This is distinctly odd. A line of influence has always run own metric. Supercharges correspond to conserved Killing from mathematics to physics. When Einstein struggled spinors. Solutions to the Killing spinor equations are plen- to express general relativity, he found the tools that he tiful in a flat space, but the equations become extremely needed had been created sixty years before by Bernhard restrictive on curved manifolds. They are so restrictive Riemann. The example is typical. Mathematicians discov- that they have, in general, no solutions. Promoting a flat ered group theory long before physicists began using it. In supersymmetric field theory to a generic curved mani- the case of string theory, it is often the other way around.
  • Introduction to Vertex Operator Algebras I 1 Introduction

    Introduction to Vertex Operator Algebras I 1 Introduction

    数理解析研究所講究録 904 巻 1995 年 1-25 1 Introduction to vertex operator algebras I Chongying Dong1 Department of Mathematics, University of California, Santa Cruz, CA 95064 1 Introduction The theory of vertex (operator) algebras has developed rapidly in the last few years. These rich algebraic structures provide the proper formulation for the moonshine module construction for the Monster group ([BI-B2], [FLMI], [FLM3]) and also give a lot of new insight into the representation theory of the Virasoro algebra and affine Kac-Moody algebras (see for instance [DL3], [DMZ], [FZ], [W]). The modern notion of chiral algebra in conformal field theory [BPZ] in physics essentially corresponds to the mathematical notion of vertex operator algebra; see e.g. [MS]. This is the first part of three consecutive lectures by Huang, Li and myself. In this part we are mainly concerned with the definitions of vertex operator algebras, twisted modules and examples. The second part by Li is about the duality and local systems and the third part by Huang is devoted to the contragradient modules and geometric interpretations of vertex operator algebras. (We refer the reader to Li and Huang’s lecture notes for the related topics.) So many exciting topics are not covered in these three lectures. The book [FHL] is an excellent introduction to the subject. There are also existing papers [H1], [Ge] and [P] which review the axiomatic definition of vertex operator algebras, geometric interpretation of vertex operator algebras, the connection with conformal field theory, Borcherds algebras and the monster Lie algebra. Most work on vertex operator algebras has been concentrated on the concrete exam- ples of vertex operator algebras and the representation theory.
  • Mathematisches Forschungsinstitut Oberwolfach Geometry, Quantum

    Mathematisches Forschungsinstitut Oberwolfach Geometry, Quantum

    Mathematisches Forschungsinstitut Oberwolfach Report No. 25/2010 DOI: 10.4171/OWR/2010/25 Geometry, Quantum Fields, and Strings: Categorial Aspects Organised by Peter Bouwknegt, Canberra Dan Freed, Austin Christoph Schweigert, Hamburg June 6th – June 12th, 2010 Abstract. Currently, in the interaction between string theory, quantum field theory and topology, there is an increased use of category-theoretic methods. Independent developments (e.g. the categorificiation of knot invariants, bun- dle gerbes and topological field theories on extended cobordism categories) have put higher categories in the focus. The workshop has brought together researchers working on diverse prob- lems in which categorical ideas play a significant role. Mathematics Subject Classification (2000): 81T, in particular 81T45 and 81T13. Introduction by the Organisers The workshop Geometry, Quantum Fields, and Strings: Categorial Aspects, organ- ised by Peter Bouwknegt (Australian National University, Canberra), Dan Freed (University of Texas, Austin), and Christoph Schweigert (University of Hamburg) was held June 6th–June 12th, 2010. The meeting was attended by 52 participants from all continents. 18 talks of one hour each were contributed to the workshop. Moreover, young researchers were offered the possibility to present short contributions. On Monday and Wednesday evening a total of 11 short talks were delivered. We would like to stress the high quality and level of interest of these contributions. The two sessions have received much attention and have led to much additional scientific discussion about the work of younger participants. For this reason, these contributions are covered in these proceedings as well. Another special event was a panel discussion on Tuesday evening on the topic “Whither the interaction of Geometry-QFT-String?”.
  • Maximally Supersymmetric String Theories in D&lt; 10

    Maximally Supersymmetric String Theories in D< 10

    FERMILAB-PUB-95/099-T NSF-ITP-94-37 hep-th/9505054 Maximally Supersymmetric String Theories in D<10 Shyamoli Chaudhuri∗ Institute for Theoretical Physics University of California Santa Barbara, CA 93106-4030 George Hockney and Joseph D. Lykken† Fermi National Accelerator Laboratory P.O. Box 500 Batavia, IL 60510 Abstract The existence of maximally supersymmetric solutions to heterotic string the- ory that are not toroidal compactifications of the ten-dimensional superstring is established. We construct an exact fermionic realization of an N=1 super- symmetric string theory in D=8 with non-simply-laced gauge group Sp(20). Toroidal compactification to six and four dimensions gives maximally extended supersymmetric theories with reduced rank (4, 12) and (6, 14) respectively. arXiv:hep-th/9505054v1 10 May 1995 ∗[email protected] †[email protected]; [email protected] Finiteness is a robust property of the perturbative amplitudes of the known su- perstring theories. N=4 supersymmetric Yang-Mills theory is known to be finite in four dimensions [1], and there is growing evidence that the theory exhibits an ex- tension of Olive-Montonen strong-weak coupling duality known as S-duality[2] [3]. A generalization of the Olive-Montonen duality of N=4 theories has also been identi- fied in N=1 supersymmetric Yang-Mills theory [4]. In string theory, conjectures for S-duality have mostly been explored in the context of toroidal compactifications of the ten-dimensional heterotic string to spacetime dimensions D<10 [5]. It would be helpful to have insight into the generic moduli space, and the generic duality group, of such maximally supersymmetric string theories.