scholarly journals INTERTWINING OPERATOR SUPERALGEBRAS AND VERTEX TENSOR CATEGORIES FOR SUPERCONFORMAL ALGEBRAS, I

2002 ◽  
Vol 04 (02) ◽  
pp. 327-355 ◽  
Author(s):  
YI-ZHI HUANG ◽  
ANTUN MILAS

We apply the general theory of tensor products of modules for a vertex operator algebra (developed by Lepowsky and the first author) and the general theory of intertwining operator algebras (developed by the first author) to the case of the N=1 superconformal minimal models and related models in superconformal field theory. We show that for the category of modules for a vertex operator algebra containing a subalgebra isomorphic to a tensor product of rational vertex operator superalgebras associated to the N =1 Neveu–Schwarz Lie superalgebra, the intertwining operators among the modules have the associativity property, the category has a natural structure of vertex tensor category, and a number of related results hold. We obtain, as a corollary and special case, a construction of a braided tensor category structure on the category of finite direct sums of minimal modules of central charge [Formula: see text] for the N = 1 Neveu–Schwarz Lie superalgebra for any fixed integers p, q larger than 1 such that p - q ∈ 2ℤ and (p - q)/2 and q relatively prime to each other.

1998 ◽  
Vol 09 (07) ◽  
pp. 845-863 ◽  
Author(s):  
YI-ZHI HUANG

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 [Formula: see text] 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.


2018 ◽  
Vol 168 (2) ◽  
pp. 261-285 ◽  
Author(s):  
CHING HUNG LAM ◽  
HIROKI SHIMAKURA

AbstractIn this paper, we study orbifold constructions associated with the Leech lattice vertex operator algebra. As an application, we prove that the structure of a strongly regular holomorphic vertex operator algebra of central charge 24 is uniquely determined by its weight one Lie algebra if the Lie algebra has the type A3,43A1,2, A4,52, D4,12A2,6, A6,7, A7,4A1,13, D5,8A1,2 or D6,5A1,12 by using the reverse orbifold construction. Our result also provides alternative constructions of these vertex operator algebras (except for the case A6,7) from the Leech lattice vertex operator algebra.


Author(s):  
Bin Gui

Abstract Complete unitarity is a natural condition on a CFT-type regular vertex operator algebra (VOA), which ensures that its modular tensor category (MTC) is unitary. In this paper we show that any CFT-type unitary (conformal) extension $U$ of a completely unitary VOA $V$ is completely unitary. Our method is to relate $U$ with a Q-system $A_U$ in the $C^*$-tensor category $\textrm{Rep}^{\textrm{u}}(V)$ of unitary $V$-modules. We also update the main result of [ 30] to the unitary cases by showing that the tensor category $\textrm{Rep}^{\textrm{u}}(U)$ of unitary $U$-modules is equivalent to the tensor category $\textrm{Rep}^{\textrm{u}}(A_U)$ of unitary $A_U$-modules as unitary MTCs. As an application, we obtain infinitely many new (regular and) completely unitary VOAs including all CFT-type $c<1$ unitary VOAs. We also show that the latter are in one-to-one correspondence with the (irreducible) conformal nets of the same central charge $c$, the classification of which is given by [ 29].


2012 ◽  
Vol 23 (10) ◽  
pp. 1250106 ◽  
Author(s):  
DONNY HURLEY ◽  
MICHAEL P. TUITE

We consider all genus zero and genus one correlation functions for the Virasoro vacuum descendants of a vertex operator algebra. These are described in terms of explicit generating functions that can be combinatorially expressed in terms of graph theory related to derangements in the genus zero case and to partial permutations in the genus one case.


2019 ◽  
Vol 30 (10) ◽  
pp. 1950048 ◽  
Author(s):  
Michael Penn ◽  
Christopher Sadowski ◽  
Gautam Webb

This is the third in a series of papers studying the vertex-algebraic structure of principal subspaces of twisted modules for lattice vertex operator algebras. We focus primarily on lattices [Formula: see text] whose Gram matrix contains only non-negative entries. We develop further ideas originally presented by Calinescu, Lepowsky, and Milas to find presentations (generators and relations) of the principal subspace of a certain natural twisted module for the vertex operator algebra [Formula: see text]. We then use these presentations to construct exact sequences involving this principal subspace, which give a set of recursions satisfied by the multigraded dimension of the principal subspace and allow us to find the multigraded dimension of the principal subspace.


2019 ◽  
Vol 30 (11) ◽  
pp. 1950054
Author(s):  
Guobo Chen ◽  
Dejia Cheng ◽  
Jianzhi Han ◽  
Yucai Su

The classification of irreducible weak modules over the Virasoro vertex operator algebra [Formula: see text] is obtained in this paper. As one of the main results, we also classify all irreducible weak modules over the simple Virasoro vertex operator algebras [Formula: see text] for [Formula: see text] [Formula: see text], where [Formula: see text] are coprime positive integers.


2019 ◽  
Vol 22 (01) ◽  
pp. 1950001 ◽  
Author(s):  
Thomas Creutzig ◽  
Shashank Kanade ◽  
Andrew R. Linshaw

Let [Formula: see text] be a simple vertex operator algebra (VOA) and consider a representation category of [Formula: see text] that is a vertex tensor category in the sense of Huang–Lepowsky. In particular, this category is a braided tensor category. Let [Formula: see text] be an object in this category that is a simple current of order two of either integer or half-integer conformal dimension. We prove that [Formula: see text] is either a VOA or a super VOA. If the representation category of [Formula: see text] is in addition ribbon, then the categorical dimension of [Formula: see text] decides this parity question. Combining with Carnahan’s work, we extend this result to simple currents of arbitrary order. Our next result is a simple sufficient criterion for lifting indecomposable objects that only depends on conformal dimensions. Several examples of simple current extensions that are [Formula: see text]-cofinite and non-rational are then given and induced modules listed.


2014 ◽  
Vol 156 (2) ◽  
pp. 343-361 ◽  
Author(s):  
HIROKI SHIMAKURA

AbstractIn this paper, we prove that the full automorphism group of the ${\mathbb Z}_2$-orbifold of the Barnes–Wall lattice vertex operator algebra of central charge 32 has the shape 227.E6(2). In order to identify the group structure, we introduce a graph structure on the Griess algebra and show that it is a rank 3 graph associated to E6(2).


2006 ◽  
Vol 08 (02) ◽  
pp. 247-307 ◽  
Author(s):  
B. DOYON ◽  
J. LEPOWSKY ◽  
A. MILAS

Using general principles in the theory of vertex operator algebras and their twisted modules, we obtain a bosonic, twisted construction of a certain central extension of a Lie algebra of differential operators on the circle, for an arbitrary twisting automorphism. The construction involves the Bernoulli polynomials in a fundamental way. We develop new identities and principles in the theory of vertex operator algebras and their twisted modules, and explain the construction by applying general results, including an identity that we call modified weak associativity, to the Heisenberg vertex operator algebra. This paper gives proofs and further explanations of results announced earlier. It is a generalization to twisted vertex operators of work announced by the second author some time ago, and includes as a special case the proof of the main results of that work.


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