Introduction to vertex operator algebras and their. In addition to physical applications, vertex operator algebras have proven useful in purely mathematical contexts such as monstrous moonshine and the geometric langlands correspondence. 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. An example of a vertex algebra 44 bosonfermion correspondence 50 10. When we work with kacmoody algebras that so not have all real roots of norm 2, we can also construct a space v and operators u n. In theory of vertex operator algebra, one considers. Introduction to vertex operator algebras imoonshine and verdex operator algebra article pdf available may 1995 with 75 reads how we measure reads. Introduction to vertex operator algebras and their representations. The online database of vertex operator algebras and modular. Borcherds, trinity college, cambridge cb2 1tq, england. The online database of vertex operator algebras and modular categories version 0. Contemporary mathematics 442 lie algebras, vertex operator.
In the example of a trivial vertex algebra, there is the statement. The first part of the book presents a new mathematical theory of vertex operator algebras, the algebraic counterpart of twodimensional holomorphic conformal. Notably the character of the moonshine module is also a modular function, namely j. Vertex operator algebras, the verlinde conjecture, and. Localityvertex algebrasthe heisenberg vertex algebra from cft to va although kacs book vertex operators for beginners is not available online, the. Vertex algebras and algebraic curves, by edward frenkel and david. Generalized vertex algebras and relative vertex operators. The main result is that roughly speaking the integral of a vertex operator over an ndimensional cycle is a vertex di. Affine, vertex and walgebras drazen adamovic springer.
The online database of vertex operator algebras and. Vertex operator algebra in mathematics, a vertex operator algebra voa is an algebraic structure that plays an important role in conformal field theory and string theory. The authors make the first steps toward reformulating the theory of vertex algebras in a way that is suitable for algebraicgeometric applications. 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 and the monster, volume 4 1st edition. In theory of vertex operator algebra, one considers a vector space of states without an inner product and. Vertex algebras for beginners university lecture series.
Originally, vertex operators arose in string theory. Vertex algebra theory provides an effective tool to. Descargar vertex algebras and algebraic curves en pdf. A vertex operator superalgebra v v0 v1 with v1 0 is called a vertex operator algebra and we call it a voa for short.
In this article, we show that a framed vertex operator algebra v satisfying the conditions. 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. Vertex operator algebras in mathematics and physics ebook. A physicist will find familiar notions presented in a more rigorous and systematic way, possibly. In the introduction of their book 12, frenkel, lepowsky and meurman conjectured that v\ can be characterized by the following three conditions. In the book, a mathematician encounters new algebraic structures that originated from einsteins special relativity postulate and heisenbergs uncertainty principle. This excellent book is written in a selfcontained manner with detailed proofs.
Introduction to vertex operator algebras i 1 introduction. In a certain way, vertex operators are the algebraic counterparts of field operators investigated. So many exciting topics are not covered in these three lectures. A theory of tensor products for module categories for a vertex operator algebra, iv, j. The monograph is written in a n accessible and selfcontained manner, with detailed proofs and with many examples interwoven through the axiomatic treatment. 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. Vertex algebras, kacmoody algebras, and the monster. They are transcribed from the book of victor kac and you are probably an ipsock of a. The duality between vertex operator algebras and coalgebras. Nearly all the axioms of vertex operator coalgebras are proven directly from the corresponding axiom of a vertex operator algebra. Kac, pierluigi moseneder frajria, paolo papi and ozren perse, kostants pair of lie type and conformal embeddings. From this point of view, the notion of vertex operator thus appeared before the underlying concept of vertex algebra. Jan 29, 2016 vertex operator algebra in mathematics, a vertex operator algebra voa is an algebraic structure that plays an important role in conformal field theory and string theory. Here denotes the formal exponential series and is another formal variable commuting with all.
Finally, the most general of the three notions, that of abelian intertwining algebra, also illuminates the theory of intertwining operator for certain classes of vertex operator algebras. Drazen adamovic is full professor of mathematics at the university of zagreb, croatia. In order to present the results on vertex operator algebras we need to extend the formal calculus introduced in chapter 2. Section 10 and 11 give a few miscellaneous thoughts about gvertex algebras. 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. Vertex operator algebras and the monster igor frenkel. Vertex operator algebras and the monster by igor frenkel. 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. 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. Moonshine and vertex operator algebras, rims kokyuroku 904, rims, kyoto, japan, 1995, 5177. 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.
The remaining part constructs the monster finite simple group as the automorphism group of a very special vertex operator algebra, called the moonshine module. Vertex operators in algebraic topology andrew baker version 24 14041998 introduction this paper is intended for two rather di. Let v be a simple vertex operator algebra satisfying the following conditions. On the concepts of intertwining operator and tensor product module in vertex operator algebra theory, with j. Vertex operator algebras and the monster, volume 4 1st. In mathematics, a vertex operator algebra voa is an algebraic structure that plays an important role in twodimensional conformal field theory and string theory. The level one zhu algebra for the heisenberg vertex operator algebra. This book presents the proceedings from the workshop, vertex operator algebras in mathematics and physics, held at the fields institute. 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. First we aim to provide algebraic topologists with a timely introduction to some of the algebraic ideas associated with vertex operator algebras.
Vertex operator algebras are a fundamental class of algebraic structures that arose in mathematics and physics in the 1980s. The leitmotif of these notes is the idea of a vertex operator algebra voa and the relationship. A characterization of the moonshine vertex operator algebra by means of virasoro frames ching hung lam y and hiroshi yamauchi z abstract. He received his phd in mathematics from the university of zagreb. In this paper we give several sufficient conditions under which a rational vertex operator algebra is regular.
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 in mathematics and physics. A vector space v for each u 2v there exists a vertex operator yu. The book provides a detailed study of most basic families of vertex operator algebras and their representation theory. 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. 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. Vertex algebra theory provides an effective tool to study them in a unified way. These vertex operators satisfy some complicated relations, which are then used as the definition of a vertex algebra. There is a slightly weaker notion of a vertex algebra. Vertex operator algebra encyclopedia of mathematics.
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. 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. The moonshine vertex operator algebra v\ constructed by frenkellepowskymeurman 12 is one of the most important examples of vertex operator algebra voa. Part of the progress in mathematics book series pm, volume 227. 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. Communicated by walter feit, december, 1985 abstract it is known that the adjoint representation of any kacmoody. Similarly, a vertex operator superalgebra is shortly referred to as a super voa or an svoa.
Pdf introduction to vertex operator algebras imoonshine. Huangs work extended the already fascinating connections with vertex operator algebras even further, outlining rigorously the role of twodimensional conformal geometry and the sewing equation in their formulations. 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. This book is an introduction to the theory of vertex algebras with a particular emphasis on the relationship between vertex algebras and the geometry of moduli spaces of algebraic curves. Examples of regular vertex operator algebras 218 8. On vertex operator algebras, their representations, and. This construction provides an equivalence of categories between the category of vertex operator algebras and the category of vertex operator coalgebras. Elliptic functions and 2point functions 204 modularinvariance and rational vertex operator algebras 6. From this point of view, the notion of vertex operator.
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