The notion of vertex operator algebra arose in the vertex operator construction of the monster see flm1, b1 and flm2. The online database of vertex operator algebras and modular. This notation is a natural analog of the notion of block designs or spherical designs when the elements of the design are based on selforthogonal binary codes or. Pdf conformal field theory, vertex operator algebras and.
These algebras can be understood as stringtheoretic analogues of lie algebras and of commutative associative algebras. We discuss the classification of strongly regular vertex operator algebras voas with exactly three simple modules. Provided rankv 6 0, this action implies that dim vn 6 0 in. A vector space v for each u 2v there exists a vertex operator yu. Operator algebras and mathematical physics download. Vertex algebras are algebraic objects that encapsulate the concept of operator product expansion from twodimensional conformal field theory. A vertex operator algebra is called rational if every admissible module is a direct sum of simple admissible modules.
From this point of view, the notion of vertex operator thus appeared before the underlying concept of vertex algebra. The notion of vertex operator algebra arises naturally in the vertex operator construction of the monsterthe largest sporadic finite simple group. I will also explain how experimental mathematics is shaping the landscape. The main result is that roughly speaking the integral of a vertex operator over an ndimensional cycle is a vertex di. Lattice vertex algebras i translation invariance 83 lattice vertex algebras 85 18.
The results obtained in the study of operator algebras are phrased in algebraic terms, while the techniques used are highly analytic. In local quantum field theory one considers a hilbert space h of. 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. 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. Invariants for finite dimensional groups in vertex operator 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. Here denotes the formal exponential series and is another formal variable commuting with all. Vertex algebras and algebraic curves edward frenkel. We will study the theory of semisimple and kacmoody lie algebras, and some other in. The classical part of a vertex operator algebra gannon, terry, 2019 chapter vi.
On axiomatic approaches to vertex operator algebras and. We also prove that any vertex operator algebra equipped with a nondegenerate, virasoro preserving, bilinear form gives rise to a corresponding vertex operator coalgebra. We study a simple, rational and c 2cofinite vertex operator algebra whose weight 1 subspace is zero, the dimension of weight 2 subspace is greater than or equal to 2 and with c c. Let v be a simple vertex operator algebra and g a finite automorphism group. Metrics details download to read the full article text. The online database of vertex operator algebras and. These algebras can be understood as stringtheoretic. Zn dis0di 2 and prove that if the minimum weight of d is greater than 2 then. Mirror extensions of vertex operator algebras springerlink. The structure of parafermion vertex operator algebras. The structure of vertex operator algebras allows us to. Vertex algebras and algebraic curves edward frenkel download. These vertex operators satisfy some complicated relations, which are then used as the definition of a vertex algebra.
From the viewpoint of finite group theory, the construction of a vertex operator algebra has one advantage. Knapp, basic algebra, digital second edition east setauket, ny. Vertex algebra theory provides an effective tool to study them in a unified way. However, a suitable change of the virasoro algebra makes ll,0 a rational. 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. Free bosonic vertex operator algebras on genus two riemann. Vertex operator algebras and the monster, volume 4 1st edition. Vertex operator algebra encyclopedia of mathematics. 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 operator algebras, generalized doubles and dual pairs. Extension of vertex operator algebras by a selfdual. In this paper we describe our operadic reformulation.
We first construct a module for the corresponding affine lie algebra of level 0 from the dual space of the space of functions on the solutions space of an integrable system, by applying the formal uniformization theorem of. 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. In addition to physical applications, vertex operator algebras have proven useful in purely mathematical contexts such as monstrous moonshine and the geometric langlands correspondence. This process is experimental and the keywords may be updated as the learning algorithm improves. In the book, a mathematician encounters new algebraic structures that originated from einsteins special relativity postulate and heisenbergs uncertainty principle. Vertex operator operator algebra vertex operator algebra vertex operator algebra versus mirror extension these keywords were added by machine and not by the authors. Similarly, a vertex operator superalgebra is shortly referred to as a super voa or an svoa. Vertex operator algebras are mathematically rigorous objects corresponding to chiral algebras in conformal field theory. Twisted representations of vertex operator algebras.
In this talk, i will give an exposition of some new generalizations of this motivated proof to other families of identities, and i will relate these ideas to vertex operator algebra 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 algebras are fast becoming ubiquitous in many areas of modern mathematics, with applications to representation theory, algebraic geometry, the theory of finite groups, modular functions, topology, integrable systems, and. Operator algebras and mathematical physics download ebook. Section 10 and 11 give a few miscellaneous thoughts about gvertex algebras. Open buy once, receive and download all available ebook formats, including pdf, epub, and mobi for kindle. Description the goal of this thesis is to explicitly construct vertex operator algebras and their representations from classical integrable systems.
From another perspective, the theory of vertex operator algebras and their modules forms the algebraic foundation of conformal field theory. Then lk,0 has a vertex operator sub algebra mhk,0 generated by the cartan subalgebra h of g. 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. Pdf we will give a short reminder for vertex operator algebra notion and corresponding. Vertex operator algebras, generalized doubles and dual. Twisted representations of vertex operator algebras springerlink. Purchase vertex operator algebras and the monster, volume 4 1st edition. The commutant kg,k of mhk,0 in lk,0 is called the parafermion vertex operator algebra. An example of a vertex algebra 44 bosonfermion correspondence 50 10. Vertex operator algebras are a class of algebras underlying a number of recent constructions, results, and themes in mathematics. Apr 21, 2011 missible modules and ordinary mo dules for a vertex operator algebra. We prove the existence and the regularity of the extension by a selfdual simple current for certain regular vertex operator algebras.
Keller eth zuric h hs 2017 under construction version as of. A vertex operator superalgebra v v0 v1 with v1 0 is called a vertex operator algebra and we call it a voa for short. 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. We obtain closed formulas for the genus two partition function for the heisenberg free bosonic string and for any pair of simple heisenberg modules. In the other direction, given a commutative unital algebra with derivation t, we can construct vertex operators by the. Vertex operator algebras are a fundamental class of algebraic structures that arose in mathematics and physics in the 1980s. First we aim to provide algebraic topologists with a timely introduction to some of the algebraic ideas associated with vertex operator 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. The online database of vertex operator algebras and modular categories version 0. The physics superselection principle in vertex operator algebra. Vertex operator algebras and modular objects are intimately connected through the study of their graded dimensions of modules, also known as characters. The aim of this course is to inform students about theories of lie algebras, vertex algebras and other structures related to these theories.
Communicated by walter feit, december, 1985 abstract it is known that the adjoint representation of any kacmoody. Dimension formulae and generalised deep holes of the leech lattice vertex operator algebra. Abstractwe introduce the notion of a conformal design based on a vertex operator algebra. Vertex algebras are fast becoming ubiquitous in many areas of modern mathematics, with applications to representation theory, algebraic geometry, the theory of finite groups, modular functions, topology, integrable systems, and combinatorics. We give a natural extension of the notion of the contragredient module for a vertex operator algebra. We will call any module with linear operators diu and bilinear operators u nv satisfying relations iv above a vertex algebra, so v is a vertex algebra. We define the partition and npoint functions for a vertex operator algebra on a genus two riemann surface formed by sewing two tori together. Borcherds identity 101 representation theory of vertex algebras. Pdf we present recent progress in theory of local conformal nets which is an operator algebraic approach to study chiral. Vertex algebras, kacmoody algebras, and the monster. Intro to on axiomatic approaches to vertex operator algebras and modules of frenkel, huang, lepowsky mentions voa similarities to lie algebras and commutative associative algebras. On axiomatic approaches to vertex operator algebras and modules.
The radical of a vertex operator algebra associated to a. We first construct a module for the corresponding affine lie algebra of level 0 from the dual space of the space of functions on the solutions space of an integrable system, by applying the formal uniformization theorem of barron, huang and lepowsky. 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 operators in algebraic topology andrew baker version 24 14041998 introduction this paper is intended for two rather di. Although the study of operator algebras is usually. Vertex operator algebras, partitions, experimental. Vertex operator algebras and the monster volume 4 pure. 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. Vertex operator algebras and the monster, volume 4. 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. It is shown that the vertex operator algebra ll,0 associated to irreducible highest weight modules at admissible level \lp\over q2\ is not rational if l is not a positive integer. On regularised quantum dimensions of the singlet vertex.
Cfts vs vertex operator algebras mathematics stack exchange. Vertex operator algebras and the monster 1st edition isbn. Vertex operator algebras and conformal field theory are now known to be deeply related to many important areas of mathematics. If the vertex operator algebra is rational, and satisfies a certain cofiniteness condition, this connection is well known, while in the nonrational case the situation is far from being understood.
A holomorphic vertex operator algebra of central charge 24 with. 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. Several books elucidating the properties and theory of vertex operator algebras voa are now available but this is one of the early ones. Lattice vertex algebras iii uniqueness 95 existence and uniqueness of 96 20. Pdf introduction to vertex operator algebras imoonshine. The structure of parafermion vertex operator algebras core. 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. Vertex algebras for beginners university lecture series. Then there is a natural right gaction on the set of all inequivalent irreducible.
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. M s 0 and d 1 the full automorphism group of m d is finite. Vertex operator algebras are a mathematical approach to twodimensional chiral conformal field theory. Pdf relations for modular forms from vertex operator algebras. We investigate the invariant vertex operator subalgebras of the vertex operator algebras associated with the a, d, e series of simply laced root lattices and the. 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.
Borcherds, trinity college, cambridge cb2 1tq, england. Finally, we explicitly calculate the vertex operator coalgebra structure and unique bilinear form for the heisenberg algebra case, which corresponds to considering free bosons. Some finiteness properties of regular vertex operator. 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. Introduction to vertex operator algebras imoonshine and verdex operator algebra.
164 997 1526 1528 1546 83 1572 1196 271 1358 389 722 215 955 1390 1526 614 399 731 1360 1145 1320 1384 939 93 1138 1339 1445 1116 525 1046 629 160