Borcherds, trinity college, cambridge cb2 1tq, england. 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 algebra of operators isomorphic to the affine lie algebra. Save up to 80% by choosing the etextbook option for isbn. A marking determines add 1 sublattice in the associated lattices and a vf in the associated voas. Lie algebras, vertex algebras and automorphic forms 1 introduction. 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. Now, we come to see that the simple current extensions of vertex operator algebras have an incredible signi. The notion of a vertex operator algebra is due to r. Vertex operator algebras with central charges 12 and 687 v. It also admits a unique canonically twisted module, and the action of the conway group naturally extends.
The latter notion thus turns out to constitute abridgebetween the study ofmodularfunctions and the representations of sporadic simple groups. Frenkel and yongchang zhu, vertex operator algebras associated to representations of affine and virasoro algebras, duke math. As one of mysterious properties of the monster simple group, j. Bosonfermion correspondence of type b and twisted vertex. The monster vertex algebra and the moonshine conjec ture. Vertex operator algebras and the monster, volume 4 1st. Associative subalgebras of griess algebra and related topics, in. The construction uses representations of vertex operator algebras and leads to combinatorial identities of rogersramanujantype. The moonshine representation of the monster constructed by frenkel and others also has products like the ones constructed for kacmoody algebras, one of which extends the griess product on the 196884dimensional piece to the whole representation. Various equivalent definitions of the notion of vertex algebra and of the variant notion of vertex operator algebra are given below. Bibliographic record and links to related information available from the library of congress catalog information from electronic data provided by.
Coset vertex operator algebras and w algebras of a type. The proof that the moonshine module is a vertex operator algebra and the monster group acts as its automorphism group was given in flm2. Operator algebra resources this page contains lists of some resources useful to mathematicians working in the subject of operator algebras. Decomposition of the lattice vertex operator algebra v2al. Local conformal nets arising from framed vertex operator. Title introduction to vertex operator algebras imoonshine. Introduction to vertex operator algebras imoonshine and verdex operator algebra. A english translation is in preparation, here the introduction ps. Lie algebras and their vertex operator representations. From the representation theory of vertex operator algebras to. The monster is in fact the full symmetry group of this special vertex operator algebra, just as the mathieu finite group cf. Triply even binary codes and their application to framed. Vertex algebras, kacmoody algebras, and the monster pnas.
Vertex operator algebras and the monster by frenkel, igor. Our aim in this paper is to study such counterparts in. Abstract vertex operator algebras are mathematically rigorous objects corresponding to chiral algebras in conformal. Notably the character of the moonshine module is also a modular function, namely j. Its full automorphism group is the monster simple group.
Modular invariance of characters of vertex operator algebras yongchang zhu introduction in contrast with the nite dimensional case, one of the distinguished features in the theory of in nite dimensional lie algebras is the modular invariance of the characters of certain representations. I also construct an integral form for the universal enveloping algebra of any kacmoody algebra that can be used to define kacmoody groups over. A natural construction of borcherds fake baby monster lie algebra. Acted on by the monster, it admits in fact a structure of vertex algebra, known as the monster vertex algebra. Vertex algebras, kacmoody algebras, and the monster ncbi. Conformal field theory, vertex operator algebras and. Vertex operators in algebraic topology andrew baker version 24 14041998 introduction this paper is intended for two rather di. By this reason, the theory of simple current extensions is one of my central subject. Pdf modular invariance of characters of vertex operator. Vertex operator algebras and the monster sciencedirect. Mckay observed that there exists an interesting connection between the 2aconjugacy class of the monster and the extended e8 dynkin diagram. 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.
Buy vertex operator algebras and the monster volume 4 pure and applied mathematics volume 4 on free shipping on qualified orders. These rich algebraic structures provide the proper formulation forthe moonshine module construction for the monster group bib2, flmi. Scheithauer introduction to the subject and overview over the following talks. The axioms of vertex operator algebras are an axiomatization of opes. Rims workshop research on finite groups and their representations, vertex operator algebras, and algebraic combinatorics organizer. Vertex algebras, kacmoody algebras, and the monster. Originally, vertex operators arose in string theory. Bosonfermion correspondence of type b and twisted vertex algebras iana i. Vertex operator algebras and the monster volume 4 pure and applied mathematics volume 4 1st edition. Please use this link to email me if you find any link that does not work or have suggestions for information to be included here. 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. Vertex operator algebras and the monster issn book 4 and millions of other books are available for amazon kindle.
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. The super vertex operator algebra we consider admits a natural characterization, in direct analogy with that conjectured to hold for the moonshine module vertex operator algebra. Communicated by walter feit, december, 1985 abstract it is known that the adjoint representation of any kacmoody. Keller eth zuric h hs 2017 under construction version as of. Similarly, a vertex operator superalgebra is shortly referred to as a super voa or an svoa. 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 algebra was. Vertex operator algebras and differential mathematical beauty. 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. Griess algebras and conformal vectors in vertex operator. Haisheng li and yizhi huang in the conference on monster and vertex operator algebras at the research. The monster vertex algebra or moonshine module is a vertex algebra acted on by the monster group that was constructed by igor frenkel, james lepowsky, and arne meurman. Table of contents for vertex operator algebras and the monster igor frenkel, james lepowsky, arne meurman. The remaining part constructs the monster finite simple group as the automorphism group of a very special vertex operator algebra, called the moonshine module.
In fact, the theory of vertex operator algebras and their representations can be thought of as an algebraic foundation of a great number of constructions in these theories. Ternary codes and vertex operator algebras sciencedirect. We give an explicit description of the coset decomposition of the lat. Some modules over lie algebras related to the virasoro algebra chen, guobo, han, jianzhi, and su, yucai, journal of the mathematical society of japan, 2020. Some finiteness properties of regular vertex operator algebras. Description integer partition identities such as the rogersramanujan identities have deep relations with the representation theory of vertex operator algebras, among many other fields of mathematics and physics. In mathematics, a vertex operator algebra voa is an algebraic structure that plays an important role in twodimensional conformal field theory and string theory. In collaboration with james lepowsky and arne meurman, he constructed the monster vertex algebra, a vertex algebra which provides a representation of the monster group around 1990, as a member of the school of mathematics at the institute for advanced study, frenkel worked on the mathematical theory of knots, hoping to develop a theory in which the knot would be seen as a. Igor frenkel, james lepowsky and arne meurman, vertex operator algebras and the monster. A vertex operator superalgebra v v0 v1 with v1 0 is called a vertex operator algebra and we call it a voa for short. Pdf introduction to vertex operator algebras imoonshine. If the monster vertex operator algebra is unique and for each of the 69 candidates there exists a unique vertex opera tor algebra, schellekens result implies that. Vertex operator algebra encyclopedia of mathematics. Mcrae a dissertation submitted to the graduate schoolnew brunswick rutgers, the st.
Igor frenkel, james lepowsky, arne meurman, vertex operator algebras and the monster, pure and applied mathematics 4, academic press, new york 1989. Aug 25, 2017 vertex operator algebras are a mathematical approach to twodimensional chiral conformal field theory. The moonshine vertex operator algebra v constructed by frenkellepowskymeurman 7 is one of the most important examples of vertex operator algebras voas. The notion of vertex operator algebra arose in the vertex operator construction of the monster see flm1, b1 and flm2. Zhus algebra, c 2 algebras and c 2cofiniteness of parafermion vertex operator algebras. We give a natural extension of the notion of the contragredient module for a vertex operator algebra. 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. First we aim to provide algebraic topologists with a timely introduction to some of the algebraic ideas associated with vertex operator algebras.
I will also explain how experimental mathematics is shaping the landscape. Vertex operator algebra and mckays e observation on the. 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 bosonfermion correspondence of type b plays similarly. Vertex operator algebras and the monster book, 1988. Vertex operator algebras and the monster, volume 4 1st edition. Vertex operator algebras, the verlinde conjecture, and. On vertex operator algebras as sl2modules dong, chongying lin, zongzhu mason, geoffrey. Some results on the representation theory of vertex operator. In addition, it is not surprising to see the notion of vertex algebra emergingfrom string theory, as the worldsheetof a string exhibits the symmetry of a twodimensional 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 aspects of conformai. The monster vertex algebra or moonshine module was constructed by frenkel, lepowsky, and meurman in 1988 8 in. James lepowsky, and arne meurman, vertex operator algebras and the monster, pure and applied mathematics, vol.
Ams transactions of the american mathematical society. This algebraic structure was later found in bo and called the vertex algebra. Vertex operator algebras and differential geometry jian zhou mathematical beauty is more than its own reward. In addition to physical applications, vertex operator algebras have proven useful in purely mathematical contexts such as monstrous moonshine and the geometric langlands correspondence. In our structure theory of quantum vertex algebras, these conditions on s are never used. Vertex operator algebras and the monster volume 4 pure.
This vertex operator had been considered by physicists for other purposes. On vertex operator algebras, their representations, and. Vertex operator algebra and mckays e8 observation on the monster hiroshi yamauchi graduate school of mathematical sciences, university of tokyo 381 komaba, meguroku, tokyo, 1538914 japan. A theory of simple current extensions of vertex operator. Vertex operator algebras and the monster, pure and applied math. Apr 12, 2005 1820 and 22 on the tensor product theory, intertwining operator algebras, and the construction of genuszero chiral conformal field theories.
Introduction to vertex algebras, borcherds algebras, and the. Twodimensional conformal quantum field theory cft has inspired an immense amount of mathematics and has interacted with mathematics. Framed vertex operator algebras, codes and the moonshine module. Rationality of admissible affine vertex algebras in the category o arakawa, tomoyuki, duke mathematical journal, 2016. Richard borcherds, vertex algebras, kacmoody algebras, and the monster, pnas may 1, 1986 83 10 30683071 doi. Self dual vertex operator superalgebras with shadows of large minimal weight.
The weight 2 subspace v 2 of v has a structure of commutative nonassociative algebra. A spectacular example is provide by the monster vertex operator algebra, v, whose. Borcherds and arose in the algebraicization of structures. Vertex operator algebras and the monster 9780122670657. Reduced operator algebras of tracepreserving quantum. Purchase vertex operator algebras and the monster, volume 4 1st edition. I also construct an inte gral form for the universal enveloping algebra of any kacmoody algebra that. Note that in the present paper, we give a definition of a braided and a quantum vertex algebra v, not requiring that the braiding s satisfies the shift condition, unitarity, and the quantum yangbaxter equation, and replacing the hexagon relation by associativity of v. Vertex operator algebras and the monster by igor frenkel. Read generalized kacmoody lie algebras, free lie algebras and the structure of the monster lie algebra, journal of pure and applied algebra on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. The main technical result used is the associativity for intertwining operators proved in refs. We announce the construction of an explicit basis for all integrable highest weight modules over the lie algebra a 1 1.
Many examples of commutative gvertex algebras come from quantum. More surprisingly was the appearance of vertex algebras in various other mathematical theories such as finite group theory, lie theory, modular functions, and string theory. Borcherds used it to prove the monstrous moonshine conjectures, by applying the goddardthorn theorem of string theory to construct the monster lie algebra, an infinitedimensional generalized kacmoody algebra acted. Dirac the purpose of the series of the lectures is to introduce some applications of. Vertex operator algebras and representations of affine lie. Vertex operator algebras and the monster igor frenkel. Vertex operator algebra and mckays e observation on the monster. In this sense there is reason to expect that vertex operator algebras will play a key role, on one way or another, in a rigorous construction of interacting qfts in four dimensions. We give an analogue for vertex operator algebras and superalgebras of the notion of endomorphism ring of a vector space by means of a notion of local system of vertex operators for a super. Much work on vertex operator algebras has been concentrated on the concrete exam ples of vertex operator algebras and the representation. There are a large number of other vertex algebras with these properties.
The mirror extensions for vertex operator algebras are studied. Vertex operator algebras and the monster 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. Vertex operator algebras, partitions, experimental. Oct 30, 2009 triply even binary codes and their application to framed vertex operator algebras akihiro munemasa1 1graduate school of information sciences tohoku university sendai, 9808579 japan.
Gerald h ohn kansas state university lattices and vertex operator algebras. Mirror extensions of vertex operator algebras springerlink. Formal series a formal series is an in nite series in some indeterminate z where convergence is not considered, i. Two explicit examples of extensions of affine vertex operator algebras of type a are given which are not simple current extensions. Lie algebras and their vertex operator representations article in journal of algebra 442 october 20 with 33 reads how we measure reads. Scheithauer in this seminar we give an introduction to the theory of vertex algebras. 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. Anguelova abstract the bosonfermion correspondence of type a is an isomorphism between two super vertex algebras and so has singularities in the operator product expansions only at z w.
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