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. Conformal field theory and vertex operator algebras runkel, ingo, 2019. The online database of vertex operator algebras and. Vertex operator algebras and the monster by frenkel, igor. Top global course special lectures 2 vertex operator algebras and integrable systems lecture 1 boris feigin kyoto university distinguished visiting professor landau institute for theoretical. Vertex operator algebras and the monster 9780122670657. 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 in mathematics and physics ebook. Communicated by walter feit, december, 1985 abstract it is known that. In addition to physical applications, vertex operator algebras have proven useful in purely mathematical contexts such as monstrous moonshine and the geometric langlands correspondence. Vertex operator algebras and 3d n 4 gauge theories arxiv. The duality between vertex operator algebras and coalgebras, modules and comodules keith hubbard abstract. Vertex operator algebras and integrable systems by shrjing chen thesis director. Vertex operator algebras, number theory and related topics.
Vertex operator algebras and operads yizhi huang and james lepowsky this paper is dedicated to the memory of lawrence corwin, january 20, 1943 march 19, 1992. It will be useful for graduate students and active researchers interested in the theory of vertex operator algebras and their applications. Pdf document information annals of mathematics fine hall washington road princeton university princeton, nj 08544, usa phone. 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. We introduce two ktheories, one for vector bundles whose. The authors consider unitary simple vertex operator algebras whose vertex operators satisfy certain energy bounds and a strong form of locality and call them strongly local. Vertex operator algebras and the monster ebook, 1988. We now show how one such representation can be constructed. Let v be a simple vertex operator algebra satisfying the following conditions. Prepared for submission to jhep vertex operator algebras and 3d n 4 gauge theories kevin costello, 1davide gaiotto 1perimeter institute for theoretical physics, waterloo, ontario, canada n2l 2y5 abstract. 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 operator algebras or vertex algebras, for short are algebras with a productoperation parameterized by points in the complex plane. Framed vertex operator algebras, codes and moonshine module 409 2sets.
We discuss properties and examples of discrete groups in connec tion with their operator algebras and. 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. This book presents the proceedings from the workshop, vertex operator algebras in mathematics and physics, held at the fields institute. Rationality of virasoro vertex operator algebras weiqiang wang department of mathematics massachusetts institute of technology 1 introduction vertex operator algebras voa were introduced by borcherds b as an axiomatic description of the holomorphic part of a conformal. Framed vertex operator algebras, codes and the moonshine. Geometric interpretation of vertex operator algebras.
On vertex operator algebras, their representations, and. Frenkel and benzvi, vertex algebras and algebraic curves, available online through the library. Introduction to vertex operator algebras imoonshine and verdex operator algebra article pdf available may 1995 with 75 reads how we measure reads. The first part of the book presents a new mathematical theory of vertex operator algebras, the algebraic counterpart of twodimensional holomorphic conformal. Abstract vertex operator algebras are mathematically rigorous objects corresponding to chiral algebras in conformal. 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 solid foundation for the algebraic. It is intended as a pedagogical companion for the beginner, an introduction to some of the main ideas in this area of analysis, a compendium of problems i. From vertex operator algebras to conformal nets and back.
Pdf simple currents and extensions of vertex operator. In mathematics, a vertex operator algebra voa is an algebraic structure that plays an important role in twodimensional conformal field theory and string theory. We will discuss the gaps in our knowledge at the end of chapter 4, but we hope this thesis will be a valuable resource for readers who. Ktheory associated to vertex operator algebras chongying dong1, kefeng liu2, xiaonan ma3, jian zhou4 abstract. 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. The algebra of bounded operators on a hilbert space h is.
Descargar vertex algebras and algebraic curves en pdf. We address this by focusing on the analogue of the categorical verlinde formula and its relation to the modular smatrix. A dirichlet operator algebra is a nonselfadjoint operator algebra \mathcala. Vertex operator algebras equipped with an action of the virasoro algebra encode the local genus0 behaviour of 2dimensional conformal field theories. The notion of vertex operator algebra arises naturally in frenkel, lepowsky, and meurmans vertex operator construction ofthe monstersporadic group1. Representations of vertex operator algebras and superalgebras by weiqiang wang submitted to the department of mathematics on may 5, 1995, in partial fulfillment of the requirements for the degree of doctor of philosophy in mathematics abstract representations of vertex operator algebras and superalgebras are studied. The current set of notes is an activityoriented companion to the study of linear functional analysis and operator algebras. A marking determines add 1 sublattice in the associated lattices and a vf in the associated voas. Using this we explain and generalize some recent work of s. The original vertex operator algebras are equivalent to conformal vertex algebra in modern texts.
We construct an equivalence between the categories of vertex operator algebras and vertex operator coalgebras. Preface to the encyclopaedia subseries on operator algebras and non commutative. Vertex operator algebras and the zeta function nasaads. Description the goal of this thesis is to explicitly construct vertex operator algebras and their representations from classical integrable systems. All these examples can be understood by lifting the fourdimensional theories to six dimensional scfts compacti ed on a riemann surface. My collaboration with geo rey schellekens list three descriptions of schellekens list. Introduction to vertex operator algebras and their.
The structure of parafermion vertex operator algebras. Simple currents and extensions of vertex operator algebras. Themonsteris in fact the symmetry group of a special vertex operator algebra, the moonshine module see the introduction in ref. See also kac, vertex algebras for beginners and chapter 10 of schottenloher. Purchase vertex operator algebras and the monster, volume 4 1st edition.
We give an explicit description of the coset decomposition of the lat. Buy introduction to vertex operator algebras and their representations progress in mathematics on free shipping on qualified orders. Twisted representations of vertex operator algebras. In the end, we fell short of completely understanding borcherdss construction of the lattice vertex operator algebra with some details still eluding us. We announce a new type of jacobi identity for vertex operator algebras, incorporating values of the riemann zeta function at negative integers. 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. Early examples can be found in fourdimensional, topologically twisted n 4 super yang mills 1 and in deformed fourdimensional n 2 gauge theory 2,3. We introduce two mirror constructions of vertex operator algebras as. We then investigate to what degree weak modules, generalized modules and ordinary modules carry corre.
The involution is the usual adjoint mapping and the norm is the usual operator norm. The authors make the first steps toward reformulating the theory of vertex algebras in a way that is suitable for algebraicgeometric applications. 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. First we aim to provide algebraic topologists with a timely introduction to some of the algebraic ideas associated with vertex operator algebras. Save up to 80% by choosing the etextbook option for isbn. Vertex operator algebras, number theory, and related topics a conference in honor of geo rey mason sacramento, june 2018 gerald h ohn kansas state university lattices and vertex operator algebras.
The notion of vertex operator algebra arose in the vertex operator construction of the monster see flm1, b1 and flm2. Communicated by walter feit, december, 1985 abstract it is known that the adjoint representation of any kacmoody. Conformal field theory, vertex operator algebras and. Although the study of operator algebras is usually. The online database of vertex operator algebras and modular categories version 0. Pdf introduction to vertex operator algebras imoonshine. Williams, equivalence and isomorphism for groupoid. Keller eth zuric h hs 2017 under construction version as of. On vertex operator algebras, their representations, and corresponding twodimensional conformal geometry a thesis submitted to the mathematics department of the university of notre dame in partial ful. Vertex operator algebras and the monster, volume 4 1st. Vertex algebras, kacmoody algebras, and the monster. Borcherds, trinity college, cambridge cb2 1tq, england. 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. Vertex operator algebras and the monster igor frenkel.
Blochs relating values of the zeta function with the commutators of certain operators and lie algebras of differential operators. Vertex operator algebras, the verlinde conjecture, and. Operator algebra and matrix algebra are similar to each other. On any hilbert space, the identity operator is denoted by i, without pre cising the associated space.
Sergei lukyanov and yizhi huang the goal of this thesis is to explicitly construct vertex operator algebras and their representations from classical integrable systems. Then bh is an algebra with respect to addition, multiplication with scalars, and composition of operators, it is a. Pure and applied mathematics vertex operator algebras. We use cookies to make interactions with our website easy and meaningful, to better understand the use of our services, and to tailor advertising.
1317 1310 541 21 726 602 1282 244 1160 609 216 775 932 1565 205 1080 1400 841 695 861 961 773 797 1573 632 1146 1521 354 556 1220 206 576 1254 346 475 571 1475 395 295 844 364 561 806 1457 437 829 768 1258 642 1027