scholarly journals DIFFERENTIAL EQUATIONS AND INTERTWINING OPERATORS

2005 ◽  
Vol 07 (03) ◽  
pp. 375-400 ◽  
Author(s):  
YI-ZHI HUANG
Keyword(s):  
Differential Equations ◽  
Vertex Operator ◽  
Operator Algebra ◽  
Vertex Operator Algebra ◽  
Tensor Category ◽  
Extension Property ◽  
Singular Points ◽  
Intertwining Operators ◽  
Natural Structure ◽  
Systems Of Differential Equations

We show that if every module W for a vertex operator algebra V = ∐n∈ℤV(n) satisfies the condition dim W/C1(W)<∞, where C1(W) is the subspace of W spanned by elements of the form u-1w for u ∈ V+ = ∐n>0 V(n) and w ∈ W, then matrix elements of products and iterates of intertwining operators satisfy certain systems of differential equations. Moreover, for prescribed singular points, there exist such systems of differential equations such that the prescribed singular points are regular. The finiteness of the fusion rules is an immediate consequence of a result used to establish the existence of such systems. Using these systems of differential equations and some additional reductivity conditions, we prove that products of intertwining operators for V satisfy the convergence and extension property needed in the tensor product theory for V-modules. Consequently, when a vertex operator algebra V satisfies all the conditions mentioned above, we obtain a natural structure of vertex tensor category (consequently braided tensor category) on the category of V-modules and a natural structure of intertwining operator algebra on the direct sum of all (inequivalent) irreducible V-modules.

scholarly journals Differential equations and logarithmic intertwining operators for strongly graded vertex algebras

2017 ◽  
Vol 19 (02) ◽  
pp. 1650009 ◽  
Author(s):  
Jinwei Yang
Keyword(s):  
Differential Equations ◽  
Category Theory ◽  
Tensor Category ◽  
Extension Property ◽  
Vertex Algebra ◽  
Intertwining Operators ◽  
Matrix Elements ◽  
Systems Of Differential Equations ◽  
Convergence And Extension ◽  
Logarithmic Intertwining Operators

We derive certain systems of differential equations for matrix elements of products and iterates of logarithmic intertwining operators among strongly graded generalized modules for a strongly graded vertex algebra under a certain finiteness condition and a condition related to the horizontal gradings. Using these systems of differential equations, we verify the convergence and extension property needed in the logarithmic tensor category theory for such strongly graded generalized modules developed by Huang, Lepowsky and Zhang.

scholarly journals Simple current extensions beyond semi-simplicity

2019 ◽  
Vol 22 (01) ◽  
pp. 1950001 ◽  
Author(s):  
Thomas Creutzig ◽  
Shashank Kanade ◽  
Andrew R. Linshaw
Keyword(s):  
Vertex Operator ◽  
Operator Algebra ◽  
Vertex Operator Algebra ◽  
Arbitrary Order ◽  
Tensor Category ◽  
Sufficient Criterion ◽  
Conformal Dimension ◽  
Half Integer ◽  
Simple Currents

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.

scholarly journals A LOGARITHMIC GENERALIZATION OF TENSOR PRODUCT THEORY FOR MODULES FOR A VERTEX OPERATOR ALGEBRA

2006 ◽  
Vol 17 (08) ◽  
pp. 975-1012 ◽  
Author(s):  
YI-ZHI HUANG ◽  
JAMES LEPOWSKY ◽  
LIN ZHANG
Keyword(s):  
Tensor Product ◽  
Vertex Operator ◽  
Operator Algebra ◽  
Vertex Operator Algebra ◽  
Vertex Algebra ◽  
Intertwining Operators

We describe a logarithmic tensor product theory for certain module categories for a "conformal vertex algebra". In this theory, which is a natural, although intricate, generalization of earlier work of Huang and Lepowsky, we do not require the module categories to be semisimple, and we accommodate modules with generalized weight spaces. The corresponding intertwining operators contain logarithms of the variables.

scholarly journals RIGIDITY AND MODULARITY OF VERTEX TENSOR CATEGORIES

2008 ◽  
Vol 10 (supp01) ◽  
pp. 871-911 ◽  
Author(s):  
YI-ZHI HUANG
Keyword(s):  
Matrix Element ◽  
Vertex Operator ◽  
Vertex Operator Algebra ◽  
Tensor Category ◽  
General Version ◽  
Natural Structure ◽  
Tensor Categories ◽  
Modular Tensor Category ◽  
Verlinde Formula ◽  
Completely Reducible

Let V be a simple vertex operator algebra satisfying the following conditions: (i) V(n)= 0 for n < 0, V(0)= ℂ1 and V′ is isomorphic to V as a V-module. (ii) Every ℕ-gradable weak V-module is completely reducible. (iii) V is C2-cofinite. (In the presence of Condition (i), Conditions (ii) and (iii) are equivalent to a single condition, namely, that every weak V-module is completely reducible.) Using the results obtained by the author in the formulation and proof of the general version of the Verlinde conjecture and in the proof of the Verlinde formula, we prove that the braided tensor category structure on the category of V-modules is rigid, balanced and nondegenerate. In particular, the category of V-modules has a natural structure of modular tensor category. We also prove that the tensor-categorical dimension of an irreducible V-module is the reciprocal of a suitable matrix element of the fusing isomorphism under a suitable basis.

Vertex algebras associated to the affine Lie algebras of abelian polynomial current algebras

2016 ◽  
Vol 27 (05) ◽  
pp. 1650046 ◽  
Author(s):  
Jinwei Yang
Keyword(s):  
Differential Equations ◽  
Lie Algebras ◽  
Tensor Category ◽  
Extension Property ◽  
Vertex Algebras ◽  
Affine Lie Algebras ◽  
Systems Of Differential Equations ◽  
Convergence And Extension ◽  
Logarithmic Intertwining Operators ◽  
Current Algebras

We construct a family of vertex algebras associated to the affine Lie algebra of polynomial current algebras of finite-dimensional abelian Lie algebras, along with their modules and logarithmic modules. These vertex algebras and their (logarithmic) modules are strongly [Formula: see text]-graded and quasi-conformal. We then show that matrix elements of products and iterates of logarithmic intertwining operators among these logarithmic modules satisfy certain systems of differential equations. Using these systems of differential equations, we verify the convergence and extension property needed in the logarithmic tensor category theory developed by Huang, Lepowsky and Zhang.

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
Keyword(s):  
General Theory ◽  
Vertex Operator ◽  
Operator Algebra ◽  
Vertex Operator Algebra ◽  
Operator Algebras ◽  
Central Charge ◽  
Tensor Category ◽  
Lie Superalgebra ◽  
Minimal Models ◽  
Intertwining Operator

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.

scholarly journals DIFFERENTIAL EQUATIONS, DUALITY AND MODULAR INVARIANCE

2005 ◽  
Vol 07 (05) ◽  
pp. 649-706 ◽  
Author(s):  
YI-ZHI HUANG
Keyword(s):  
Differential Equations ◽  
Vertex Operator ◽  
Operator Algebra ◽  
Correlation Functions ◽  
Vertex Operator Algebra ◽  
Modular Invariance ◽  
Genus Zero ◽  
Modular Invariant ◽  
The One ◽  
Genus One

We solve the problem of constructing all chiral genus-one correlation functions from chiral genus-zero correlation functions associated to a vertex operator algebra satisfying the following conditions: (i) V(n) = 0 for n < 0 and V(0) = ℂ1, (ii) every ℕ-gradable weak V-module is completely reducible and (iii) V is C2-cofinite. We establish the fundamental properties of these functions, including suitably formulated commutativity, associativity and modular invariance. The method we develop and use here is completely different from the one previously used by Zhu and others. In particular, we show that the q-traces of products of certain geometrically-modified intertwining operators satisfy modular invariant systems of differential equations which, for any fixed modular parameter, reduce to doubly-periodic systems with only regular singular points. Together with the results obtained by the author in the genus-zero case, the results of the present paper solves essentially the problem of constructing chiral genus-one weakly conformal field theories from the representations of a vertex operator algebra satisfying the conditions above.

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