Supersymmetric Vertex Algebras

We define and study the structure of SUSY Lie conformal and vertex algebras. This leads to effective rules for computations with superfields. Supported in part by NSF grants DMS-0201017 and DMS-0501395.     

Modules-at-Infinity for Quantum Vertex Algebras

This is a sequel to [Li4] and [Li5] in a series to study vertex algebra-like structures arising from various algebras such as quantum affine algebras and Yangians. In this paper, we study two versions of the double Yangian , denoted by DY _q (sl ₂) and with q a nonzero complex number. For each nonzero complex number q, we construct a quantum vertex algebra V _q and prove that every DY _q (sl ₂)-module is naturally a V _q -module. We also show that -modules are what we call V _q -modules-at-infinity. To achieve this goal, we study what we call -local subsets and quasi-local subsets of for any vector space W, and we prove that any -local subset generates a (weak) quantum vertex algebra and that any quasi-local subset generates a vertex algebra with W as a (left) quasi module-at-infinity. Using this result we associate the Lie algebra of pseudo-differential operators on the circle with vertex algebras in terms of quasi modules-at-infinity.     

Vertex Operators for Boundary Algebras

We construct embeddings of boundary algebras into ZF algebras . Since it is known that these algebras are the relevant ones for the study of quantum integrable systems (with boundaries for and without for ), this connection allows to make the link between different approaches of the systems with boundaries. The construction uses the well-bred vertex operators built recently, and is classified by reflection matrices. It relies only on the existence of an R-matrix obeying a unitarity condition, and as such can be applied to any infinite dimensional quantum group.     

Rationality of Bershadsky-Polyakov Vertex Algebras

We prove the conjecture of Kac-Wakimoto on the rationality of exceptional W-algebras for the first non-trivial series, namely, for the Bershadsky-Polyakov vertex algebras ${W_3^{(2)}}$ W 3 ( 2 ) at level k =  p/2?3 with ${p = 3, 5, 7, 9, \dots}$ p = 3 , 5 , 7 , 9 , ? . This gives new examples of rational conformal field theories.     

P-Twisted Affine Lie Algebras and Their Associated Vertex Algebras

For any finite-dimensional semisimple Lie algebra g, a Z₊-graded vertex algebra is construsted on the vacuum representation V_k(\hat{g}[ θ]) of \hat{g}[θ], which is a one-dimentional central extension of θ-invariant subspace on the loop algebra Lg=g\otimes C((t^1/p)).     

P-Twisted Affine Lie Algebras and Their Associated Vertex Algebras

For any finite-dimensional semisimple Lie algebra g, a Z＋-graded vertex algebra is construsted on the vacuum representation Vk（g[θ]of g[θ]）,which is a one-dimentionM central extension of 8-invariant subspace on the loop algebra Lg=g C（（t^1/p））.     

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.     

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.     

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 Cohomology Theory of Grading-Restricted Vertex Algebras

We introduce a cohomology theory of grading-restricted vertex algebras. To construct the correct cohomologies, we consider linear maps from tensor powers of a grading-restricted vertex algebra to “rational functions valued in the algebraic completion of a module for the algebra,” instead of linear maps from tensor powers of the algebra to a module for the algebra. One subtle complication arising from such functions is that we have to carefully address the issue of convergence when we compose these linear maps with vertex operators. In particular, for each ${n \in \mathbb{N}}$ , we have an inverse system ${\{H^{n}_{m}(V, W)\}_{m \in \mathbb{Z}_{+}}}$ of nth cohomologies and an additional nth cohomology ${H_{\infty}^{n}(V, W)}$ of a grading-restricted vertex algebra V with coefficients in a V-module W such that ${H_{\infty}^{n}(V, W)}$ is isomorphic to the inverse limit of the inverse system ${\{H^{n}_{m}(V, W)\}_{m\in \mathbb{Z}_{+}}}$ . In the case of n = 2, there is an additional second cohomology denoted by ${H^{2}_{\frac{1}{2}}(V, W)}$ which will be shown in a sequel to the present paper to correspond to what we call square-zero extensions of V and to first order deformations of V when W = V.     

Lagrangian Approach to Sheaves of Vertex Algebras

We explain how sheaves of vertex algebras are related to mathematical structures encoded by a class of Lagrangians. The exposition is focused on two examples: the WZW model and the (1,1)-supersymmetric σ-model. We conclude by showing how to construct a family of vertex algebras with base the Barannikov-Kontsevich moduli space thus furnishing the B-model moduli for Witten’s half-twisted model. Partially supported by the National Science Foundation.     

Quantum Kac– Algebras and Vertex Representations

We introduce an affinization of the quantum Kac–Moody algebra associated to a symmetric generalized Cartan matrix. Based on the affinization, we construct a representation of the quantum Kac–Moody algebra by vertex operators from bosonic fields. We also obtain a combinatorial indentity about Hall–Littlewood polynomials.     

Open-String Vertex Algebras,Tensor Categories and Operads

We introduce notions of open-string vertex algebra, conformal open-string vertex algebra and variants of these notions. These are open-string-theoretic, noncommutative generalizations of the notions of vertex algebra and of conformal vertex algebra. Given an open-string vertex algebra, we show that there exists a vertex algebra, which we call the meromorphic center, inside the original algebra such that the original algebra yields a module and also an intertwining operator for the meromorphic center. This result gives us a general method for constructing open-string vertex algebras. Besides obvious examples obtained from associative algebras and vertex (super)algebras, we give a nontrivial example constructed from the minimal model of central charge We establish an equivalence between the associative algebras in the braided tensor category of modules for a suitable vertex operator algebra and the grading-restricted conformal open-string vertex algebras containing a vertex operator algebra isomorphic to the given vertex operator algebra. We also give a geometric and operadic formulation of the notion of grading-restricted conformal open-string vertex algebra, we prove two isomorphism theorems, and in particular, we show that such an algebra gives a projective algebra over what we call the Swiss-cheese partial operad.Acknowledgement. We would like to thank Jürgen Fuchs and Christoph Schweigert for helpful discussions and comments. We are also grateful to Jim Lepowsky for comments. The research of Y.-Z. H. is supported in part by NSF grant DMS-0070800.     

The q-Analogous Vertex Operator and Quantum Affine Algebras

The extended quantum affine algebras of q-analogous vertex operator have been constructed in this Jet ter, and the similar algebra structure of q-analogous fermionic vertex operator is also analysed.     

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     

First and Second Cohomologies of Grading-Restricted Vertex Algebras

Let V be a grading-restricted vertex algebra and W a V-module. We show that for any ${m\in \mathbb{Z}_{+}}$ , the first cohomology ${H^{1}_{m}(V, W)}$ of V with coefficients in W introduced by the author is linearly isomorphic to the space of derivations from V to W. In particular, ${H^{1}_{m}(V, W)}$ for ${m\in \mathbb{N}}$ are equal (and can be denoted using the same notation H ¹(V, W)). We also show that the second cohomology ${H^{2}_{\frac{1}{2}}(V, W)}$ of V with coefficients in W introduced by the author corresponds bijectively to the set of equivalence classes of square-zero extensions of V by W. In the case that W = V, we show that the second cohomology ${H^{2}_{\frac{1}{2}}(V, V)}$ corresponds bijectively to the set of equivalence classes of first order deformations of V.     

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.     

Affine Vertex Operator Algebras and Modular Linear Differential Equations

In this paper, we list all affine vertex operator algebras of positive integral levels whose dimensions of spaces of characters are at most 5 and show that a basis of the space of characters of each affine vertex operator algebra in the list gives a fundamental system of solutions of a modular linear differential equation. Further, we determine the dimensions of the spaces of characters of affine vertex operator algebras whose numbers of inequivalent simple modules are not exceeding 20.