Generalized rationality and a 'Jacobi identity' for intertwining operator algebras

We prove a generalized rationality property and a new identity that we call the 'Jacobi identity' for intertwining operator algebras. Most of the main properties of genus-zero conformal field theories, including the main properties of vertex operator algebras, modules, intertwining operators, Verlinde algebras, and fusing and braiding matrices, are incorporated into this identity. Together with associativity and commutativity for intertwining operators proved by the author in [H4] and [H6], the results of the present paper solve completely the problem of finding a natural purely algebraic structure on the direct sum of all inequivalent irreducible modules for a suitable vertex operator algebra. Two equivalent definitions of intertwining operator algebra in terms of this Jacobi identity are given.     

Twisted modules for quantum vertex algebras

We study twisted modules for (weak) quantum vertex algebras and we give a conceptual construction of (weak) quantum vertex algebras and their twisted modules. As an application we construct and classify irreducible twisted modules for a certain family of quantum vertex algebras.     

False theta functions and companions to Capparelli's identities

Capparelli conjectured two modular identities for partitions whose parts satisfy certain gap conditions, which were motivated by the calculation of characters for the standard modules of certain affine Lie algebras and by vertex operator theory. These identities were subsequently proved and refined by Andrews, who related them to Jacobi theta functions, and also by Alladi–Andrews–Gordon, Capparelli and Tamba–Xie. In this paper we prove two new companions to Capparelli's identities, where the evaluations are expressed in terms of Jacobi theta functions and false theta functions.     

Vertex-algebraic structure of the principal subspaces of level one modules for the untwisted affine Lie algebras of types A,D,E

Generalizing some of our earlier work, we prove natural presentations of the principal subspaces of the level one standard modules for the untwisted affine Lie algebras of types A, D and E, and also of certain related spaces. As a consequence, we obtain a canonical complete set of recursions (q-difference equations) for the (multi-)graded dimensions of these spaces, and we derive their graded dimensions. Our methods are based on intertwining operators in vertex operator algebra theory.     

Deformations of Lie algebras using σ-derivations

In this article we develop an approach to deformations of the Witt and Virasoro algebras based on σ-derivations. We show that σ-twisted Jacobi type identity holds for generators of such deformations. For the σ-twisted generalization of Lie algebras modeled by this construction, we develop a theory of central extensions. We show that our approach can be used to construct new deformations of Lie algebras and their central extensions, which in particular include naturally the q-deformations of the Witt and Virasoro algebras associated to q-difference operators, providing also corresponding q-deformed Jacobi identities.     

A theory of tensor products for modulé categories for a vertex operator algebra, IV

This is the fourth part in a series of papers developing a tensor product theory of modules for a vertex operator algebra. In this paper, we establish the associativity of P(z)-tensor products for nonzero complex numbers z constructed in Part III of the present series under suitable conditions. The associativity isomorphisms constructed in this paper are analogous to associativity isomorphisms for vector space tensor products in the sense that they relate the tensor products of three elements in three modules taken in different ways. The main new feature is that they are controlled by the decompositions of certain spheres with four punctures into spheres with three punctures using a sewing operation. We also show that under certain conditions, the existence of the associativity isomorphisms is equivalent to the associativity (or (nonmeromorphic) operator product expansion in the language of physicists) for the intertwining operators (or chiral vertex operators). Thus the associativity of tensor products provides a means to establish the (nonmeromorphic) operator product expansion.     

Twisted modules over vertex algebras on algebraic curves

We extend the geometric approach to vertex algebras developed by the first author to twisted modules, allowing us to treat orbifold models in conformal field theory. Let V be a vertex algebra, H a finite group of automorphisms of V, and C an algebraic curve such that H⊂Aut(C). We show that a suitable collection of twisted V-modules gives rise to a section of a certain sheaf on the quotient X=C/H. We introduce the notion of conformal blocks for twisted modules, and analyze them in the case of the Heisenberg and affine Kac-Moody vertex algebras. We also give a chiral algebra interpretation of twisted modules.     

Representations of centrally-extended Lie algebras over differential operators and vertex algebras

We construct irreducible modules of centrally-extended classical Lie algebras over left ideals of the algebra of differential operators on the circle, through certain irreducible modules of centrally-extended classical Lie algebras of infinite matrices with finite number of nonzero entries. The structures of vertex algebras associated with the vacuum representations of these algebras are determined. Moreover, we prove that under certain conditions, the highest-weight irreducible modules of centrally-extended classical Lie algebras of infinite matrices with finite number of nonzero entries naturally give rise to the irreducible modules of the simple quotients of these vertex algebras. From vertex algebra and its representation point of view, our results with positive integral central charge are high-order differential operator analogues of the well-known WZW models in conformal field theory associated with affine Kac-Moody algebras. Indeed, when the left ideals are the algebra of differential operators, our Lie algebras do contain affine Kac-Moody algebras as subalgebras and our results restricted on them are exactly the representation contents in WZW models. Similar results with negative central charge are also obtained.     

构造相应于有限维非退化可解李代数的顶点代数

设g是带有非退化不变对称双线性型的有限维可解李代数, 该文首先应用g的仿射李代数{\heiti $\hat{g}$}的表示理论,构造出一类水平为l的限制$\hat{g}$ -模$V_{\hat{g}}(l,0)$.然后应用顶点算子的局部理论在hom$(V_{\hat{g}}(l,0),V_{\hat{g}}(l,0)((x)))$中 找到一类顶点代数$L_{V_{\hat{g}}(l,0)}$.建立了$L_{V_{\hat{g}}(l,0)}$到 $V_{\hat{g}}(l,0)$的映射,最后证明了这类映射是顶点代数同构.     

Representations of a class of lattice type vertex algebras

In this paper we study the representation theory for certain “half lattice vertex algebras.” In particular we construct a large class of irreducible modules for these vertex algebras. We also discuss how the representation theory of these vertex algebras are related to the representation theory of some associative algebras.     

Lattice construction of logarithmic modules for certain vertex algebras

A general method for constructing logarithmic modules in vertex operator algebra theory is presented. By utilizing this approach, we give explicit vertex operator construction of certain indecomposable and logarithmic modules for the triplet vertex algebra W(p){\mathcal{W}(p)} and for other subalgebras of lattice vertex algebras and their N = 1 super extensions. We analyze in detail indecomposable modules obtained in this way, giving further evidence for the conjectural equivalence between the category of W(p){\mathcal{W}(p)}-modules and the category of modules for the restricted quantum group [`(U)]_q(sl₂){\overline{\mathcal{U}}_q(sl_2)} , q = e ^{π i/p} . We also construct logarithmic representations for a certain affine vertex operator algebra at admissible level realized in Adamović (J. Pure Appl. Algebra 196:119–134, 2005). In this way we prove the existence of the logarithmic representations predicted in Gaberdiel (Int. J. Modern Phys. A 18, 4593–4638, 2003). Our approach enlightens related logarithmic intertwining operators among indecomposable modules, which we also construct in the paper.     

Axiomatic Aspects of N = 2 Vertex Superalgebras with Odd Formal Variables

We formulate the notion of “N = 2 vertex superalgebra with two odd formal variables” using a Jacobi identity with odd formal variables in which an N = 2 superconformal shift is incorporated into the usual Jacobi identity for a vertex superalgebra. It is shown that as a consequence of these axioms, the N = 2 vertex superalgebra is naturally a representation of the Lie superalgebra isomorphic to the three-dimensional algebra of superderivations with basis consisting of the usual conformal operator and the two N = 2 superconformal operators. In addition, this superconformal shift in the Jacobi identity dictates the form of the odd formal variable components of the vertex operators, and allows one to easily derive the useful formulas in the theory. The notion of N = 2 Neveu–Schwarz vertex operator superalgebra with two odd formal variables is introduced, and consequences of this notion are derived. In particular, we develop the duality properties which are necessary for a rigorous treatment of the correspondence with the underlying supergeometry. Various other formulations of the notion of N = 2 (Neveu–Schwarz) vertex (operator) superalgebra appearing in the mathematics and physics literature are discussed, and several mistakes in the literature are noted and corrected.     

Vertex algebras associated to modified regular representations of the Virasoro algebra

We give an abstract construction, based on the Belavin–Polyakov–Zamolodchikov equations, of a family of vertex algebras with conformal elements of rank 26 associated to the modified regular representations of the Virasoro algebra. The vertex operators are obtained from the products of intertwining operators for a pair of Virasoro algebras. We explicitly determine the structure coefficients that yield the axioms of vertex algebras. In the process of our construction, we obtain new hypergeometric identities.     

Associating quantum vertex algebras to deformed Heisenberg Lie algebras

We associate quantum vertex algebras and their ϕ-coordinated quasi modules to certain deformed Heisenberg algebras.     

Local conformal nets arising from framed vertex operator algebras

We apply an idea of framed vertex operator algebras to a construction of local conformal nets of (injective type III₁) factors on the circle corresponding to various lattice vertex operator algebras and their twisted orbifolds. In particular, we give a local conformal net corresponding to the moonshine vertex operator algebras of Frenkel-Lepowsky-Meurman. Its central charge is 24, it has a trivial representation theory in the sense that the vacuum sector is the only irreducible DHR sector, its vacuum character is the modular invariant J-function and its automorphism group (the gauge group) is the Monster group. We use our previous tools such as α-induction and complete rationality to study extensions of local conformal nets.     

On the concepts of intertwining operator and tensor product module in vertex operator algebra theory

We produce counterexamples to show that in the definition of the notion of intertwining operator for modules for a vertex operator algebra, the commutator formula cannot in general be used as a replacement axiom for the Jacobi identity. We further give a sufficient condition for the commutator formula to imply the Jacobi identity in this definition. Using these results we illuminate the crucial role of the condition called the “compatibility condition” in the construction of the tensor product module in vertex operator algebra theory, as carried out in work of Huang and Lepowsky. In particular, we prove by means of suitable counterexamples that the compatibility condition was indeed needed in this theory.     

Homotopy algebras and the inverse of the normalization functor

In this paper, we investigate multiplicative properties of the classical Dold-Kan correspondence. The inverse of the normalization functor maps commutative differential graded algebras to E_∞-algebras. We prove that it in fact sends algebras over arbitrary differential graded E_∞-operads to E_∞-algebras in simplicial modules and is part of a Quillen adjunction. More generally, this inverse maps homotopy algebras to weak homotopy algebras. We prove the corresponding dual results for algebras under the conormalization, and for coalgebra structures under the normalization resp. the inverse of the conormalization.     

On certain vertex algebras and their modules associated with vertex algebroids

We study the family of vertex algebras associated with vertex algebroids, constructed by Gorbounov, Malikov, and Schechtman. As the main result, we classify all the (graded) simple modules for such vertex algebras and we show that the equivalence classes of graded simple modules one-to-one correspond to the equivalence classes of simple modules for the Lie algebroids associated with the vertex algebroids. To achieve our goal, we construct and exploit a Lie algebra from a given vertex algebroid.