The Intermediate Vertex Subalgebras of the Lattice Vertex Operator Algebras

A notion of intermediate vertex subalgebras of lattice vertex operator algebras is introduced, as a generalization of the notion of principal subspaces. Bases and the graded dimensions of such subalgebras are given. As an application, it is shown that the characters of some modules of an intermediate vertex subalgebra between E ₇ and E ₈ lattice vertex operator algebras satisfy some modular differential equations. This result is an analogue of the result concerning the “hole” of the Deligne dimension formulas and the intermediate Lie algebra between the simple Lie algebras E ₇ and E ₈.     

Vertex Operator Algebras Associated to Admissible Representations of

The Kac-Wakimoto admissible modules for are studied from the point of view of vertex operator algebras. It is shown that the vertex operator algebra L(l,0) associated to irreducible highest weight modules at admissible level is not rational if l is not a positive integer. However, a suitable change of the Virasoro algebra makes L(l,0) a rational vertex operator algebra whose irreducible modules are exactly these admissible modules for and for which the fusion rules are calculated. It is also shown that the q-dimensions with respect to the new Virasoro algebra are modular functions. Received: 4 April 1996/Accepted: 1 August 1996     

Generalized Twisted Modules Associated to General Automorphisms of a Vertex Operator Algebra

We introduce a notion of a strongly ${\mathbb{C}^{\times}}We introduce a notion of a strongly \mathbbC^×{\mathbb{C}^{\times}}-graded, or equivalently, \mathbbC/\mathbbZ{\mathbb{C}/\mathbb{Z}}-graded generalized g-twisted V-module associated to an automorphism g, not necessarily of finite order, of a vertex operator algebra. We also introduce a notion of a strongly \mathbbC{\mathbb{C}}-graded generalized g-twisted V-module if V admits an additional \mathbbC{\mathbb{C}}-grading compatible with g. Let V=\coprod_{n ? \mathbbZ}V_(n){V=\coprod_{n\in \mathbb{Z}}V_{(n)}} be a vertex operator algebra such that V₍₀₎=\mathbbC1{V_{(0)}=\mathbb{C}\mathbf{1}} and V _(n) = 0 for n < 0 and let u be an element of V of weight 1 such that L(1)u = 0. Then the exponential of 2p?{-1}  Res_x Y(u, x){2\pi \sqrt{-1}\; {\rm Res}_{x} Y(u, x)} is an automorphism g _u of V. In this case, a strongly \mathbbC{\mathbb{C}}-graded generalized g _u -twisted V-module is constructed from a strongly \mathbbC{\mathbb{C}}-graded generalized V-module with a compatible action of g _u by modifying the vertex operator map for the generalized V-module using the exponential of the negative-power part of the vertex operator Y(u, x). In particular, we give examples of such generalized twisted modules associated to the exponentials of some screening operators on certain vertex operator algebras related to the triplet W-algebras. An important feature is that we have to work with generalized (twisted) V-modules which are doubly graded by the group \mathbbC/\mathbbZ{\mathbb{C}/\mathbb{Z}} or \mathbbC{\mathbb{C}} and by generalized eigenspaces (not just eigenspaces) for L(0), and the twisted vertex operators in general involve the logarithm of the formal variable.     

Generating Generalized Bessel Equations by Virtue of Bose Operator Algebra and Entangled State Representations

With the help of Bose operator identities and entangled state representation and based on our previous work [Phys. Lett. A 325 (2004) 188] we derive some new generalized Bessel equations which also have Bessel function as their solution. It means that for these intricate higher-order differential equations, we can get Bessel function solutions without using the expatiatory power-series expansion method.     

Interwining Operators for Vertex Representations of Toroidal Lie Algebras

The construction of the vertex representation of the toroidal Lie algebra ^[n] depends on the way of labelling the points in the dual ⁿ of the torus Tⁿ. Thus there is a built-in symmetry of the vertex representation with respect to the symmetry of Zⁿ. In conjunction with this, the energy operator L₀ gives rise to intertwining operators which reflect the symmetry of the vertex representation with respect to S¹ action on .     

Modularity in Orbifold Theory for Vertex Operator Superalgebras

This paper is about the orbifold theory for vertex operator superalgebras. Given a vertex operator superalgebra V and a finite automorphism group G of V, we show that the trace functions associated to the twisted sectors are holomorphic in the upper half plane for any commuting pairs in G under the C₂-cofinite condition. We also establish that these functions afford a representation of the full modular group if V is C₂-cofinite and g-rational for any g ∈ G.Supported by NSF grants, China NSF grant 10328102 and a Faculty research grant from the University of California at Santa Cruz     

Mirror Extensions of Vertex Operator Algebras

The mirror extensions for vertex operator algebras are studied. Two explicit examples of extensions of affine vertex operator algebras of type A are given which are not simple current extensions.     

A Characterization of Vertex Operator Algebra {L(\frac{1}{2},0)\otimes L(\frac{1}{2},0)}

We study a simple, rational and C ₂-cofinite vertex operator algebra whose weight 1 subspace is zero, the dimension of weight 2 subspace is greater than or equal to 2 and with c = c? = 1. Under some additional conditions it is shown that such a vertex operator algebra is isomorphic to ${L(\frac{1}{2},0)\otimes L(\frac{1}{2},0)}We study a simple, rational and C ₂-cofinite vertex operator algebra whose weight 1 subspace is zero, the dimension of weight 2 subspace is greater than or equal to 2 and with c = c̃ = 1. Under some additional conditions it is shown that such a vertex operator algebra is isomorphic to L(\frac12,0)?L(\frac12,0){L(\frac{1}{2},0)\otimes L(\frac{1}{2},0)}.     

Unbounded Representations of q-Deformation of Cuntz Algebra

We study a deformation of the Cuntz–Toeplitz C ^*-algebra determined by the relations ${a_i^*a_i=1+q a_ia_i^*,\, a_i^*a_j=0}$ . We define its well-behaved unbounded *-representations and classify all irreducible ones up to unitary equivalence.     

多模海森堡代数的算符恒等式

利用算符代数中的分析方法,得到了多模海森堡(Heisenberg)代数中的BCH公式和压缩算符的展开式。     

Modular Invariance for Twisted Modules over a Vertex Operator Superalgebra

The purpose of this paper is to generalize Zhu’s theorem about characters of modules over a vertex operator algebra graded by integer conformal weights, to the setting of a vertex operator superalgebra graded by rational conformal weights. To recover ${SL_2(\mathbb{Z})}$ S L 2 ( Z ) -invariance of the characters it turns out to be necessary to consider twisted modules alongside ordinary ones. It also turns out to be necessary, in describing the space of conformal blocks in the supersymmetric case, to include certain ‘odd traces’ on modules alongside traces and supertraces. We prove that the set of supertrace functions, thus supplemented, spans a finite dimensional ${SL_2(\mathbb{Z})}$ S L 2 ( Z ) -invariant space. We close the paper with several examples.     

体积算符对顶角作用的重耦矩阵

用重耦理论的图形计算法,系统推出了非微扰量子引力自旋结网圈表象体积算符对顶角作用的重耦矩阵的图形表式和记号表式     

Representations of the Weyl Algebra in Quantum Geometry

The Weyl algebra of continuous functions and exponentiated fluxes, introduced by Ashtekar, Lewandowski and others, in quantum geometry is studied. It is shown that, in the piecewise analytic category, every regular representation of having a cyclic and diffeomorphism invariant vector, is already unitarily equivalent to the fundamental representation. Additional assumptions concern the dimension of the underlying analytic manifold (at least three), the finite wide triangulizability of surfaces in it to be used for the fluxes and the naturality of the action of diffeomorphisms – but neither any domain properties of the represented Weyl operators nor the requirement that the diffeomorphisms act by pull-backs. For this, the general behaviour of C*-algebras generated by continuous functions and pull-backs of homeomorphisms, as well as the properties of stratified analytic diffeomorphisms are studied. Additionally, the paper includes also a short and direct proof of the irreducibility of .     

Braided Tensor Categories and Extensions of Vertex Operator Algebras

Let V be a vertex operator algebra satisfying suitable conditions such that in particular its module category has a natural vertex tensor category structure, and consequently, a natural braided tensor category structure. We prove that the notions of extension (i.e., enlargement) of V and of commutative associative algebra, with uniqueness of unit and with trivial twist, in the braided tensor category of V-modules are equivalent.     

Twisted Sectors for Tensor Product Vertex Operator Algebras Associated to Permutation Groups

Let V be a vertex operator algebra, and for k a positive integer, let g be a k-cycle permutation of the vertex operator algebra V ^{⊗ k} . We prove that the categories of weak, weak admissible and ordinary g-twisted modules for the tensor product vertex operator algebra V ^{⊗ k} are isomorphic to the categories of weak, weak admissible and ordinary V-modules, respectively. The main result is an explicit construction of the weak g-twisted V ^{⊗ k} -modules from weak V-modules. For an arbitrary permutation automorphism g of V ^{⊗ k} the category of weak admissible g-twisted modules for V ^{⊗ k} is semisimple and the simple objects are determined if V is rational. In addition, we extend these results to the more general setting of γg-twisted V ^{⊗ k} -modules for γ a general automorphism of V acting diagonally on V ^{⊗ k} and g a permutation automorphism of V ^{⊗ k} . Received: 20 April 2000 / Accepted: 20 January 2002     

Genus Two Partition and Correlation Functions for Fermionic Vertex Operator Superalgebras I

We define the partition and n-point correlation functions for a vertex operator superalgebra on a genus two Riemann surface formed by sewing two tori together. For the free fermion vertex operator superalgebra we obtain a closed formula for the genus two continuous orbifold partition function in terms of an infinite dimensional determinant with entries arising from torus Szegő kernels. We prove that the partition function is holomorphic in the sewing parameters on a given suitable domain and describe its modular properties. Using the bosonized formalism, a new genus two Jacobi product identity is described for the Riemann theta series. We compute and discuss the modular properties of the generating function for all n-point functions in terms of a genus two Szegő kernel determinant. We also show that the Virasoro vector one point function satisfies a genus two Ward identity.     

The Structure of Parafermion Vertex Operator Algebras: General Case

The structure of the parafermion vertex operator algebra associated to an integrable highest weight module for any affine Kac-Moody algebra is studied. In particular, a set of generators for this algebra has been determined.     

Vertex Operator Solutions of¶2d Dimensionally Reduced Gravity

We apply algebraic and vertex operator techniques to solve two dimensional reduced vacuum Einstein's equations. This leads to explicit expressions for the coefficients of metrics solutions of the vacuum equations as ratios of determinants. No quadratures are left. These formulas rely on the identification of dual pairs of vertex operators corresponding to dual metrics related by the Kramer-Neugebauer symmetry.     

Torus Chiral <Emphasis Type="Italic">n</Emphasis>-Point Functions for Free Boson and Lattice Vertex Operator Algebras

 We obtain explicit expressions for all genus one chiral n-point functions for free bosonic and lattice vertex operator algebras. We also consider the elliptic properties of these functions. Received: 6 May 2002 / Accepted: 4 October 2002 Published online: 24 January 2003 RID="*" ID="*" Partial support provided by NSF DMS-9709820 and the Committee on Research, University of California, Santa Cruz RID="⋆⋆" ID="⋆⋆" Supported by an Enterprise Ireland Basic Research Grant and the Millenium Fund, National University of Ireland, Galway Communicated by L. Takhtajan     

On the Classification of Admissible Representations of the Virasoro Algebra

For the Virasoro algebra, the problems of classifying bounded admissible representations, on the one hand, and finite-length extensions of highest weight modules, on the other hand, are wild: they are as complicated as the problem of classifying a pair of matrices. This follows from results by Martin and Piard and Feigin and Fuchs.