Genus-zero modular functors and intertwining operator algebras
Yi-Zhi Huang,Yi-Zhi Huang +1 more
TLDR
In this paper, it was shown that the notion of intertwining operator algebras of central charge is isomorphic to rational genus-zero modular functors (certain analytic partial operads) satisfying a certain generalized meromorphicity property.Abstract:
In [7] and [9], the author introduced the notion of intertwining operator algebra, a nonmeromorphic generalization of the notion of vertex operator algebra involving monodromies. The problem of constructing intertwining operator algebras from representations of suitable vertex operator algebras was solved implicitly earlier in [5]. In the present paper, we generalize the geometric and operadic formulation of the notion of vertex operator algebra given in [3, 4, 11, 12, 8] to the notion of intertwining operator algebra. We show that the category of intertwining operator algebras of central charge is isomorphic to the category of algebras over rational genus-zero modular functors (certain analytic partial operads) of central charge c satisfying a certain generalized meromorphicity property. This result is one main step in the construction of genus-zero conformal field theories from representations of vertex operator algebras announced in [7]. One byproduct of the proof of the present isomorphism theorem is a geometric construction of (framed) braid group representations from intertwining operator algebras and, in particular, from representations of suitable vertex operator algebras.read more
Citations
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Differential equations and intertwining operators
TL;DR: In this paper, it was shown that if every module W for a vertex operator algebra V = ∐n∈ℤV(n) satisfies the condition dim W/C1(W) 0 V(n), and w ∈ W, then matrix elements of products and iterates of intertwining operators satisfy certain systems of differential equations.
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Differential equations and intertwining operators
TL;DR: In this article, it was shown that if every module W for a vertex operator algebra V satisfies the condition that the dimension of W/C_1(W) is less than infinity, where C is the subspace of W spanned by elements of the form u-1}w for u in V of positive weight and w in W in W, then matrix elements of products and iterates of intertwining operators satisfy certain systems of differential equations.
Journal ArticleDOI
Cofiniteness conditions, projective covers and the logarithmic tensor product theory
TL;DR: In this article, a projective cover of irreducible V-modules in the category of grading-restricted generalized V -modules was constructed and shown to be a finite abelian category.
Journal ArticleDOI
Full field algebras
TL;DR: In this paper, the Verlinde conjecture was shown to hold for a simple vertex operator algebra V satisfying certain natural finiteness and reductivity conditions, and a non-degenerate bilinear form on the space of intertwining operators for V was constructed.
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Differential equations, duality and modular invariance
TL;DR: In this article, the authors solved the problem of constructing chiral genus-one correlation functions from a vertex operator algebra and established the fundamental properties of these functions, including suitably formulated commutativity, associativity and modular invariance.
References
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Journal ArticleDOI
Vertex algebras, Kac-Moody algebras, and the Monster.
TL;DR: An integral form is constructed for the universal enveloping algebra of any Kac-Moody algebras that can be used to define Kac's groups over finite fields, some new irreducible integrable representations, and a sort of affinization of anyKac-moody algebra.
Journal ArticleDOI
A natural representation of the Fischer-Griess Monster with the modular function J as character
TL;DR: A natural action of the Monster is obtained on V compatible with the action of [unk], thus conceptually explaining a major part of the numerical observations known as Monstrous Moonshine.
Journal ArticleDOI
A theory of tensor products for module categories for a vertex operator algebra, III
Yi-Zhi Huang,James Lepowsky +1 more
TL;DR: In this article, a tensor product theory for modules for a vertex operator algebra is developed, where the goal is to construct a "vertex tensor category" structure on the category of modules.
Book
Two-Dimensional Conformal Geometry and Vertex Operator Algebras
TL;DR: The equivalance between the algebraic and geometric formulations of vertex operator algebra is proved in this paper, where the authors introduce a geomatric notion of vertex algebra in terms of complex powers of the determinant line bundles over certain moduli spaces (parameter spaces) of spheres with punctures and local analytic co-ordinates.
Book ChapterDOI
Tensor Products of Modules for a Vertex Operator Algebra and Vertex Tensor Categories
Yi-Zhi Huang,James Lepowsky +1 more
TL;DR: In this article, a tensor product theory of classes of modules for vertex operator algebra is presented, which is based on both the formal-calculus approach to vertex algebra theory developed in [FLM2] and [FHL] and the precise geometric interpretation of the notion of vertex algebra established in [H1].