The classical part of a vertex operator algebra gannon, terry, 2019. The algebraic theory of vertex operator algebras provides deep conceptual understanding of many but not yet all of the phenomena of monstrous moonshine see cn and at the same time establishes a. Generalized vertex algebras and relative vertex operators. Vertex operator algebras and the monster sciencedirect. We give a natural extension of the notion of the contragredient module for a vertex operator algebra. Room 420, research institute for mathematical sciences, kyoto university program january 5 tue 14. This book is intended for graduate students specialising in algebra, differential geometry, operator theory, and theoretical physics, and for researchers in. As the primary component of the subject of noncommutative topol ogy, k theory has opened vast new vistas within the structure theory of c algebras, as well as leading to profound and unexpected applications of opera tor algebras to problems in geometry and topology. Lattice vertex algebras iii uniqueness 95 existence and uniqueness of 96 20. Rationality of admissible affine vertex algebras in the category o arakawa, tomoyuki, duke mathematical journal, 2016. The papers on operator theory deal with the study of certain types of operator algebras. The first part of the book presents a new mathematical theory of vertex operator algebras, the algebraic. The book focusses on applications to the theory of fredholm and toeplitz operators, so it is useful if you want to do some operator theory.
Set theory and operator algebras 3 recently showing that the structure of ultrapowers and relative commutants of c algebras and ii 1 factors can depend on the choice of the ultra lter 29, 24, 26. This area has its origins in twodimensional conformal quantum field theory, and has had important applications to areas of mathematics as far a field as the theory of finite groups and the invariants of knots and of threemanifolds, as. Commutative banach algebras israel gelfand,2 giving rise to c algebras. The notion of vertex operator algebra arose in the vertex operator construction of the monster see flm1, b1 and flm2. Vertex operator algebra encyclopedia of mathematics. The first part of the book presents a new mathematical theory of vertex operator algebras, the algebraic counterpart of twodimensional holomorphic conformal quantum field theory. Pdf introduction to vertex operator algebras imoonshine. One of main issues here is to give criteria if a vertex operator algebra is not rational. The leitmotif of these notes is the idea of a vertex operator algebra voa and the. Vertex operator algebras, number theory, and related topics. The theory of vertex operator algebras has developed rapidly in the last few years. These algebras and their representations are deeply related to many directions in mathematics and physics, in particular, the representation theory of the fischergriess monster simple finite group and the connection with the phenomena of monstrous moonshine cf. The authors provide a clear account of the introductory portions of this important and.
Some results on the representation theory of vertex operator. Since the paper is from outside operator algebras computer sciencequantum computing and they actually work on. Everyday low prices and free delivery on eligible orders. Scheithauer introduction to the subject and overview over the following talks. This allows the reader to recognize the affinity between operator algebras and measure theory on locally compact spaces. The book provides a detailed study of most basic families of vertex operator algebras and their representation theory. Now i want to find an alternative book for vol 2, because i once heard that the content in this book is somewhat out of date and the theories are developed in a rather slow pace. Set theory and operator algebras 3 recently showing that the structure of ultrapowers and relative commutants of calgebras and ii 1 factors can depend on the choice of the ultra lter 29, 24, 26. Lie algebras, vertex operator algebras and their applications. It assumes a basic knowledge in functional analysis but no prior acquaintance with operator theory is. Commutative banach algebras israel gelfand,2 giving rise to calgebras. Some results on the representation theory of vertex. In mathematics, a vertex operator algebra voa is an algebraic structure that plays an important role in twodimensional conformal field theory and string theory.
This essentially selfcontained monograph develops the basic axiomatic theory of vertex operator algebras and their modules and intertwining operators, following a fundamental analogy with lie algebra theory. Vertex operator algebras are a mathematical approach to twodimensional chiral conformal field theory. The categoryv is organized into what we call the minimalk. Borcherds identity 101 representation theory of vertex algebras. Descargar ktheory for operator algebras en pdf libros. This book may be used by the beginners as an entry point to the modern theory of vertex algebras, and by more experienced readers as a guide to advanced studies in this beautiful and exciting field. Introduction to vertex operator algebras and their representations. The concept of vertex algebra happens to be a powerful tool and initially introduced by borcherds to this purpose in the proof bo of the so. They are mathematically precise counterparts of what are known in physics as chiral algebras, and in particular, they are intimately related to string theory and conformal field theory. Purchase calgebras and operator theory 1st edition. By using this extension we prove that for regular vertex operator algebras, zhusc 2finiteness condition holds, fusion rules for any three irreducible modules are finite and the vertex operator algebras themselves are finitely generated. A modeltheoretic logic suitable for study of calgebras and ii 1 factors, adapted from 10, was developed in 27.
In local quantum field theory one considers a hilbert space h of. The c algebras of compact operators play an important role in calgebra theory, and they are widely used in mathematical physics and quantum mechanics. In chapter 3 we study the properties of positive maps as well as construction of positive maps under certain conditions. These rich algebraic struc tures provide the proper formulation for the moonshine module. Theory of operator algebras i operator algebras and noncommulative geometry v by m. It consists of papers based on many of the talks given at the conference by leading experts in the algebraic, geometric, and physical aspects of. Scheithauer in this seminar we give an introduction to the theory of vertex algebras. In addition to physical applications, vertex operator algebras have proven useful in purely mathematical contexts such as monstrous moonshine and the geometric langlands correspondence the related notion of vertex algebra was. Pdf in twodimensional conformal field theory cft the building blocks are given by chiral cfts, i. The virasoro algebra kaplansky, irving, communications in mathematical physics, 1982. It is a field that has great importance for other areas of mathematics and physics, such as algebraic topology, differential geometry, and quantum mechanics. Description integer partition identities such as the rogersramanujan identities have deep relations with the representation theory of vertex operator algebras, among many other fields of mathematics and physics.
This study was continued in the paper fkrw in the framework of vertex algebra theory. Therefore, the primary goal is to provide required. Accessible to the nonspecialist, this first part of a threevolume treatise provides a clear, carefully written survey that emphasizes the theory s analytical and topological aspects. Introduction to vertex operator algebras i 1 introduction. Introduction to vertex operator algebras ii haisheng li department of mathematics university of california santa cruz, ca 95064 1 introduction this is the second of three lectures on introduction to vertex operator algebras. Contemporary mathematics 442 lie algebras, vertex operator. Originally, vertex operators arose in string theory. Vertex operator algebras and the monster 1st edition isbn. Operator theory and operator algebras are concerned with the study of linear operators, usually on vector spaces whose elements are functions. Wilson on their sixtieth birthdays, may 1721, 2005, north carolina state university, raleigh, north carolina yizhi huang, kailash misra, editors. Very technical sections are clearly labeled and there are extensive comments by the author, a good historical background and excercises. Operator theoryoperator algebras department of mathematics. They are used to describe certain types of interactions, between di. Vertex algebras, kacmoody algebras, and the monster.
In these talks, i will introduce the definition of vertex operator algebra with motivation from the segal picture of conformal field theory and discuss examples coming from the virasoro algebra, affine lie algebras, and lattices. The treatment of group c algebras is particularly good as it is in ken davidsons book r. Buy theory of operator algebras i operator algebras and noncommulative geometry v on free shipping on qualified orders. Aug 25, 2017 vertex operator algebras are a mathematical approach to twodimensional chiral conformal field theory.
A vertex operator algebra voa consists of a z graded vector space v l k. The subject is analysis, but because the vector spaces are usually infinite dimensional, the subject has a nice blend of techniques from other areas of mathematics, ranging from algebra to topology to dynamical systems. The remaining part constructs the monster finite simple group as the automorphism group of a very special vertex operator algebra, called the moonshine module. Buy on axiomatic approaches to vertex operator algebras and modules memoirs of the ams memoirs of the american mathematical society by frenkel, igor b. Vertex operator algebras and associative algebras sciencedirect. Theory of operator algebras i operator algebras and non. From this point of view, the notion of vertex operator. This work and fundamentals of the theory of operator algebras.
In some vertexoperatoralgebraic settings, the product sides arise. Nov 27, 2008 erties for vertex operator algebras e. K theory has revolutionized the study of operator algebras in the last few years. In our paper kr we began a systematic study of representations of the universal central extension of the lie algebra of differential operators on the circle.
The paper used in this book is acidfree and falls within the guidelines established to. This volume also contains a detailed introduction to the theory of quantum groups. Inspired by a recent work of frenkelzhu, we study a class of pre vertex operator algebras voa associated to the selfdual lie algebras. Among the topics investigated in the thesis are locally convex topologies, the extension and characterisation of ultraweakly continuous linear functionals on bh, and conditions that ensure the constructive existence. Purchase vertex operator algebras and the monster, volume 4 1st edition. Chongying dong and geoffrey mason work in the area of vertex operator algebras. On axiomatic approaches to vertex operator algebras and. In this lecture, we shall continue professor dongs lecture to present more fundamental properties of vertex operator. On the classification of simple vertex operator algebras.
Subject mathematics, vertex operator algebras, partitions mathematics extent 1 online resource viii, 105 p. It consists of papers based on many of the talks given at the conference by leading experts in the algebraic, geometric, and physical aspects of vertex operator algebra theory. Vertex algebra associated to abelian current lie algebras. For any even lattice r the operators u n on v satisfy the following relations. Lattice vertex algebras i translation invariance 83 lattice vertex algebras 85 18. Vertex operator algebras are a fundamental class of algebraic structures that arose in mathematics and physics in the 1980s. For a brief summary of this book, see the text of the talk given by e. A modeltheoretic logic suitable for study of c algebras and ii 1 factors, adapted from 10, was developed in 27. Vertex operator algebras can be viewed as complex analogues of both lie algebras and associative algebras. Vertex operator algebras in mathematics and physics. On axiomatic approaches to vertex operator algebras and modules.
Lie algebras, vertex operator algebras, and related topics. Operator algebras and unbounded selfadjoint operators author. Some finiteness properties of regular vertex operator algebras. These functors exhibit a bijection between the simple modules in each category. The theory we develop in chapter 1 and 2 is mostly covered in courses on operator algebras. Operator algebras and unbounded selfadjoint operators. Vertex operator algebras and conformal field theory are now known to be deeply related to many important areas of mathematics. Vertex operator algebras and the monster, volume 4 1st edition. Among the topics investigated in the thesis are locally convex topologies, the extension and characterisation of ultraweakly continuous linear functionals on bh, and conditions that ensure the constructive existence of the adjoint of a bounded linear operator on h. Keller eth zuric h hs 2017 under construction version as of. Pdf exceptional vertex operator algebras and the virasoro. This barcode number lets you verify that youre getting exactly the right version or edition of a book. Vertex operator algebras and the monster, volume 4 1st.
Research on finite groups and their representations, vertex operator algebras, and algebraic combinatorics organizer. Vertex algebras generated by lie algebras mirko prime communicated by ca. This book presents the proceedings from the workshop, vertex operator algebras in mathematics and physics, held at the fields institute. Abelian calgebras 744 compact operators 747 s8j book by r. In addition to physical applications, vertex operator algebras have proven useful in purely mathematical contexts such as monstrous moonshine and the geometric langlands correspondence. So many exciting topics are not covered in these three lectures. The books unifying theme is the banach space duality for operator algebras. The calgebras of compact operators play an important role in calgebra theory, and they are widely used in mathematical physics. We address this by focusing on the analogue of the categorical verlinde formula and its relation to the modular smatrix.
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