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