Vertex operators in algebraic topology andrew baker version 24 14041998 introduction. 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. 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. Descargar vertex algebras and algebraic curves en pdf. Borcherds, trinity college, cambridge cb2 1tq, england.
Section 10 and 11 give a few miscellaneous thoughts about gvertex algebras. Let v be a simple vertex operator algebra satisfying the following conditions. Several books elucidating the properties and theory of vertex operator algebras voa are now available but this is one of the early ones. Similarly, a vertex operator superalgebra is shortly referred to as a super voa or an svoa. In functional analysis, an operator algebra is an algebra of continuous linear operators on a topological vector space with the multiplication given by the composition of mappings the results obtained in the study of operator algebras are phrased in algebraic terms, while the techniques used are highly analytic. Pdf on apr 28, 2015, sebastiano carpi and others published from vertex operator algebras to conformal nets and back find, read and cite all the research you need on researchgate. In the banach algebra literature, an operator algebra is just a closed subalgebra not necessarily selfadjoint of bh. The first part of the book presents a new mathematical theory of vertex operator algebras, the algebraic counterpart of twodimensional holomorphic conformal. We prove the verlinde conjecture in the following general form. Although the study of operator algebras is usually. He received his phd in mathematics from the university of zagreb.
Sometimes both meanings exist for the same qualifier, as in the sentence. On the other hand, from the physical point of view, a vertex operator algebra looks like a commutative associative algebra with identity because roughly speaking, a vertex operator algebra is. On the concepts of intertwining operator and tensor product module in vertex operator algebra theory yizhi huang, james lepowsky, haisheng li and lin zhang 1 mar 2006 journal of pure and applied algebra, vol. Vertex operator algebras in mathematics and physics. This book presents the proceedings from the workshop, vertex operator algebras in mathematics and physics, held at the fields institute. Vertex operator algebras in mathematics and physics ebook. The volume serves as an introduction to algebraic aspects of conformal field theory, which in the past 15 years revealed a variety of unusual mathematical notions. Vertex operator algebras and the monster igor frenkel. The main result is that roughly speaking the integral of a vertex operator over an ndimensional cycle is a vertex di.
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 notion of vertex operator algebra is the mathematical counterpart of the notion of operator algebra, or chiral algebra, in conformai field theory, as formalized in bpz, and, many of the algebraic considerations in this chapter were introduced and, exploited in more physical language and, settings in the vast physics literature. Vertex operator algebras and the monster igor frenkel, james lepowsky, arne meurman this work is motivated by and develops connections between several branches of mathematics and physicsthe theories of lie algebras, finite groups and modular functions in mathematics, and string theory in physics. These vertex operators satisfy some complicated relations, which are then used as the definition of a vertex algebra. Vertex operator calculus mathematical aspects of string theory. In addition to physical applications, vertex operator algebras have proven useful in purely mathematical contexts such as monstrous moonshine and the geometric langlands correspondence. K theory has revolutionized the study of operator algebras in the last few years. 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. 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 socalled moonshine monstrous con. Zhus algebra, c 2algebras and c 2cofiniteness of parafermion vertex operator algebras. In this paper we give several sufficient conditions under which a rational vertex operator algebra is regular. Pdf introduction to vertex operator algebras imoonshine. We now show how one such representation can be constructed. Three pivotal insights that are developed in the rst chapter are.
The notion of vertex operator algebra arose in the vertex operator construction of the monster see flm1, b1 and flm2. An example of a vertex algebra 44 bosonfermion correspondence 50 10. The term vertex operator in mathematics refers mainly to certain operators in a generalized sense of the term used in physics to describe interactions of physical states at a vertex in string theory and its precursor, dual resonance theory. The traditional definition of vertex operator algebra voa is long and tends to be. This construction provides an equivalence of categories between the category of vertex operator algebras and the category of vertex operator coalgebras. A uniform algebra is a subalgebra of the space ct of all continuous functions on a compact set t. The definition of vertex algebra with either minimal or maximal axioms is equivalent to borcherds definition in. Vertex operator algebras are a fundamental class of algebraic structures that arose in mathematics and physics in the 1980s. Introduction to vertex operator algebras imoonshine and verdex operator algebra article pdf available may 1995 with 75 reads how we measure reads. The duality between vertex operator algebras and coalgebras, modules and co modules 233 269 279 289 303 315 keith hubbard 339 some developments in vertex operator algebra theory, old and new james lepowsky 355 twisted modules and quasimodules for vertex operator algebras haisheng li 389 chiral algebras and partition functions.
Subsequently this led them to the discovery of what. The remaining part constructs the monster finite simple group as the automorphism group of a very special vertex operator algebra, called the moonshine module. 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. Vertex algebra theory provides an effective tool to study them in a unified way.
Vertex operator algebra encyclopedia of mathematics. In order to present the results on vertex operator algebras we need to extend the formal calculus introduced in chapter 2. In mathematics, a vertex operator algebra voa is an algebraic structure that plays an important role in twodimensional conformal field theory and string theory. If f is a subset of a c algebra a, the subc algebra generated by f, denoted by cf, is the smallest subc algebra containing f. On the concepts of intertwining operator and tensor product module in vertex operator algebra theory, with j. In other words, the original example of a vertex algebra was the vertex algebra of an even lattice, and the definition of a vertex algebra was an axiomatization of this example.
Vertex algebras, kacmoody algebras, and the monster. Descargar ktheory for operator algebras en pdf libros. Lattice vertex algebras iii uniqueness 95 existence and uniqueness of 96 20. From this point of view, the notion of vertex operator thus appeared before the underlying concept of vertex algebra. 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. Nearly all the axioms of vertex operator coalgebras are proven directly from the corresponding axiom of a vertex operator algebra.
A vertex operator superalgebra v v0 v1 with v1 0 is called a vertex operator algebra and we call it a voa for short. Vertex operator calculus mathematical aspects of string. Zentralblatt the book gives a sound introduction in the theory of vertex algebras emphasizing in particular the construction of families of examples by means of a kind of representations developped by the. It became increasingly important that vertex algebras are useful not only in the representation theory of infinitedimensional lie algebras, where they are by now ubiquitous, but also in other fields, such as algebraic geometry, theory of finite groups, modular. Complete definitions of the notions of vertex algebra and vertex operator algebra, using commuting formal variables, and are given below.
As well as the book of frenkel, lepowsky and meurman 5, the paper of dong 2 and the memoir of. For each, the coefficient of in the expansion of the vertex operator in powers of is a welldefined linear operator on, and the main point is that the operators, and span a lie. Keller eth zuric h hs 2017 under construction version as of. Commutative algebra is the study of commutative rings, which are commutative algebras over the integers. Contemporary mathematics 442 lie algebras, vertex operator. Drazen adamovic is full professor of mathematics at the university of zagreb, croatia. While formal in its approach, and using notation that can be very difficult to read, this book nevertheless gives the reader keen insights into the theory, this coming from the summaries and motivations that occur at the beginning of every chapter. Vertex algebras were first introduced as a tool used in the description of the algebraic structure of certain quantum field theories. A regular vertex operator algebra is a vertex operator algebra such that any weak module without grading is a direct sum of ordinary irreducible modules. Vertex operator algebras or vertex algebras, for short are algebras with a productoperation parameterized by points in the complex plane vertex operator algebras equipped with an action of the virasoro algebra encode the local genus0 behaviour of 2dimensional conformal field theories. A set of matrices can be a representation of a group of symmetry operators if there is a matrix corresponding to each of the symmetry operators and if the matrices obey the same multiplication table as the symmetry operators. Vertex algebra theory provides an effective tool to. Internet archive we consider exceptional vertex operator algebras and vertex operator superalgebras with the property that particular casimir vectors constructed from the primary vectors of lowest conformal weight are virasoro descendents of the vacuum.
It is called subc algebra, if it is also normclosed. The norm closure of a sub algebra is a subc algebra from continuity of the algebraic operations. In the book, a mathematician encounters new algebraic structures that originated from einsteins special relativity postulate and heisenbergs uncertainty principle. Vertex operator algebras are mathematically rigorous objects corresponding to chiral algebras in conformal field theory. This is a revised and expanded edition of kacs original introduction to algebraic aspects of conformal field theory, which was published by the ams in 1996. Vertex operator algebras and vertex operator coalgebras 2.
Examples of regular vertex operator algebras 218 8. With an article, it means an instance of some abstract structure, like a lie algebra, an associative algebra, or a vertex operator algebra. The duality between vertex operator algebras and coalgebras. Vertex operator algebras and the monster, volume 4 1st. Communicated by walter feit, december, 1985 abstract it is known that. Here denotes the formal exponential series and is another formal variable commuting with all.
A vector space v for each u 2v there exists a vertex operator yu. This work is motivated by and develops connections between several branches of mathematics and physicsthe theories of lie algebras, finite groups and modular functions in mathematics, and string theory in physics. Elliptic functions and 2point functions 204 modularinvariance and rational vertex operator algebras 6. Operator algebra and matrix algebra are similar to each other. Delta functions and expansions of formal variables.
Affine, vertex and walgebras drazen adamovic springer. Meurman, vertex operator algebras and the monster, pure and appl. Purchase vertex operator algebras and the monster, volume 4 1st edition. Vertex operator algebras and the monster overdrive. 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.