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.     

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     

Twisted Logarithmic Modules of Vertex Algebras

Motivated by logarithmic conformal field theory and Gromov–Witten theory, we introduce a notion of a twisted module of a vertex algebra under an arbitrary (not necessarily semisimple) automorphism. Its main feature is that the twisted fields involve the logarithm of the formal variable. We develop the theory of such twisted modules and, in particular, derive a Borcherds identity and commutator formula for them. We investigate in detail the examples of affine and Heisenberg vertex algebras.     

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     

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.     

Representations of Vertex Operator Algebra V L + for Rank One Lattice L

We classify the irreducible modules for the fixed point vertex operator subalgebra V_L⁺ of the vertex operator algebra V_L associated to a positive definite even lattice of rank 1 under the automorphism lifted from the у isometry of L.     

Automorphisms of the Weyl Algebra

We discuss a conjecture which says that the automorphism group of the Weyl algebra in characteristic zero is canonically isomorphic to the automorphism group of the corresponding Poisson algebra of classical polynomial symbols. Several arguments in favor of this conjecture are presented, all based on the consideration of the reduction of the Weyl algebra to positive characteristic Mathematics Subject Classification (2000) 13N10, 16S32, 16H05     

Vertex Operator Solutions to the Discrete KP-Hierarchy

Vertex operators, which are disguised Darboux maps, transform solutions of the KP equation into new ones. In this paper, we show that the bi-infinite sequence obtained by Darboux transforming an arbitrary KP solution recursively forward and backwards, yields a solution to the discrete KP-hierarchy. The latter is a KP hierarchy where the continuous space $x$-variable gets replaced by a discrete $n$-variable. The fact that these sequences satisfy the discrete KP hierarchy is tantamount to certain bilinear relations connecting the consecutive KP solutions in the sequence. At the Grassmannian level, these relations are equivalent to a very simple fact, which is the nesting of the associated infinite-dimensional planes (flag). The discrete KP hierarchy can thus be viewed as a container for an entire ensemble of vertex or Darboux generated KP solutions. It turns out that many new and old systems lead to such discrete (semi-infinite) solutions, like sequences of soliton solutions, with more and more solitons, sequences of Calogero–Moser systems, having more and more particles, just to mention a few examples; this is developed in [3]. In this paper, as another example, we show that the q-KP hierarchy maps, via a kind of Fourier transform, into the discrete KP hierarchy, enabling us to write down a very large class of solutions to the q-KP hierarchy. This was also reported in a brief note [4]. Received: 27 August 1998 / Accepted: 24 November 1998     

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.     

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.     

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.     

Probabilistic Fuzzy Sets and a Related Operator Algebra

The rules of union and intersection of probabilistic fuzzy sets guided us to construct a related operator algebra. In a Hilbert space, where each fuzzy set is represented by an orthonormal vector, the union and the intersection operators generate a well-defined algebra with a unique representation. PACS NUMBER: 02.10.-v     

The Algebra of Formal Twisted Pseudodifferential Symbols and a Noncommutative Residue

Motivated by Connes–Moscovici’s notion of a twisted spectral triple, we define an algebra of formal twisted pseudodifferential symbols with respect to a twisting of the base algebra. We extend the Adler–Manin trace and the logarithmic cocycle on the algebra of pseudodifferential symbols to our twisted setting. We also give a general method to construct twisted pseudodifferential symbols on crossed product algebras.     

Bounded Subquotients of Pseudodifferential Operator Modules

Recently there have been several papers on the action of the Virasoro Lie algebra on the projective decompositions of the modules of pseudodifferential operators on the circle. We use their results to prove that a wide class of the uniserial (completely indecomposable) bounded modules of the Virasoro Lie algebra may be realized as subquotients of such modules of pseudodifferential operators. This gives easy proofs of the existence of many previously known uniserial modules, and moreover yields some hitherto undiscovered.Partially supported by NSA grant MDA 904-03-1-0004.     

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。     

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     

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.