We identify the algebra of matrix elements of big projective modules in category O with the regular functions on the big Bruhat cell of G. Analogous extensions of the regular representations of the affine Lie and Virasoro algebras yield vertex operator algebras, equipped with two commuting actions with special values of the total central charge. In the generic case, we identify the structure of these VOAs, compute their semi-infinite cohomology. Its superalgebra structure encodes the fusion rules for the associated tensor categories. Full details are provided for the affine sl(2) and Virasoro algebras, and various generalizations and extensions are discussed.
In several examples it has been observed that a module category of a vertex operator algebra (VOA) is equivalent to a category of representations of some quantum group. The present article is concerned with developing such a duality in the case of the Virasoro VOA at generic central charge; arguably the most rudimentary of all VOAs, yet structurally complicated. We do not address the category of all modules of the generic Virasoro VOA, but we consider the infinitely many modules from the first row of the Kac table. Building on an explicit quantum group method of Coulomb gas integrals, we give …
In this paper we study the first cohomologies for the following three examples of vertex operator algebras: (i) the simple affine VOA associated to a simple Lie algebra with positive integral level; (ii) the Virasoro VOA corresponding to minimal models; (iii) the lattice VOA associated to a positive definite even lattice. We prove that in all these cases, the first cohomology $H^1(V, W)$ are given by the zero-mode derivations when $W$ is any $V$-module with an $\N$-grading (not necessarily by the operator $L(0)$). This agrees with the conjecture made by Yi-Zhi Huang and the author in 2018. For…
We construct vertex operator representations for the full (N+1)-toroidal Lie algebra g. We associate with g a toroidal vertex operator algebra, which is a tensor product of an affine VOA, a sub-VOA of a hyperbolic lattice VOA, affine sl(N) VOA and a twisted Heisenberg-Virasoro VOA. The modules for the toroidal VOA are also modules for the toroidal Lie algebra g. We also construct irreducible modules for an important subalgebra gdiv of the toroidal Lie algebra that corresponds to the divergence free vector fields. This subalgebra carries a non-degenerate invariant bilinear form. The VOA that co…
In this paper, we show that $σ$-involutions associated to extendable c=4/5 Virasoro vectors generate a 3-transposition group in the automorphism group of a vertex operator algebra (VOA). Several explicit examples related to lattice VOA are also discussed in details. In particular, we show that the automorphism group of the VOA $V_{K_{12}}^{\hatν}$ associated to the Coxeter Todd lattice $K_{12}$ contains a subgroup isomorphic to ${}^+Ω^{-}(8,3)$.
We give an abstract construction, based on the Belavin-Polyakov-Zamolodchikov equations, of a family of vertex operator algebras of rank $26$ associated to the modified regular representations of the Virasoro algebra. The vertex operators are obtained from the tensor products of intertwining operators for a pair of Virasoro algebras. We explicitly determine the structure coefficients that yield the axioms of VOAs. In the process of our construction, we obtain new hypergeometric identities.
The derivation algebras, automorphism groups and second cohomology groups of the generalized loop Schrödinger-Virasoro algebras are completely determined in this paper.
The purpose of this paper is to introduce the cohomology of various algebras over an operad of moduli spaces including the cohomology of conformal field theories (CFT's) and vertex operator algebras (VOA's). This cohomology theory produces a number of invariants of CFT's and VOA's, one of which is the space of their infinitesimal deformations.
The energy bounds condition for intertwining operators of unitary rational vertex operator algebras (VOAs) was studied, first by A.Wassermann for type $A$ affine VOAs, and later by T.Loke for $c<1$ Virasoro VOAs, and by V.Toledano-Laredo for type $D$ affine VOAs. In this paper, we extend their results to affine VOAs of type $B$, $C$, and $G_2$. As a consequence, the modular tensor categories of these unitary vertex operator algebras are unitary.
The first cohomology group of a generalized loop Virasoro algebra with coefficients in the tensor product of its adjoint module is shown to be trivial. The result is applied to prove that Lie bialgebra structures on generalized loop Virasoro algebras are coboundary triangular. We then generalize the results to generalized map Virasoro algebras.
The purpose of this paper is to study Virasoro extensions of the q-deformed Witt Hom-Lie superalgebra. Moreover, we provide the cohomology and deformations of the Ramond Hom-superalgebra and special Ramond Hom-superalgebra.
In this paper, we study the structure theory of a class of not-finitely graded Lie algebras related to generalized Virasoro algebras. In particular,the derivation algebras, the automorphism groups and the second cohomology groups of these Lie algebras are determined.
We describe Zhu recursion for a vertex operator algebra (VOA) on a general genus Riemann surface in the Schottky uniformization where $n$-point correlation functions are written as linear combinations of $(n-1)$-point functions with universal coefficients. These coefficients are identified with specific geometric structures on the Riemann surface. We apply Zhu recursion to the Heisenberg VOA and determine all its correlation functions. For a general VOA, Zhu recursion with respect to the Virasoro vector is shown to lead to conformal Ward identities expressed in terms of derivatives with respec…
We discuss the classification of strongly regular vertex operator algebras (VOAs) with exactly three simple modules whose character vector satisfies a monic modular linear differential equation with irreducible monodromy. Our Main Theorem provides a classification of all such VOAs in the form of one infinite family of affine VOAs, one individual affine algebra and two Virasoro algebras, together with a family of eleven exceptional character vectors and associated data that we call the $U$-series. We prove that there are at least $15$ VOAs in the $U$-series occurring as commutants in a Schellek…
In this paper, we introduce two kinds of Lie conformal algebras associated with the loop Schrödinger-Virasoro Lie algebra and the extended loop Schrödinger-Virasoro Lie algebra, respectively. The conformal derivations, the second cohomology groups of these two conformal algebras are completely determined. And nontrivial free conformal modules of rank one and Z-graded free intermediate series modules over these two conformal algebras are also classified in the present paper
In 1993, Lian-Zuckerman constructed two cohomology operations on the BRST complex of a conformal vertex algebra with central charge 26. They gave explicit generators and relations for the cohomology algebra equipped with these operations in the case of the c=1 model. In this paper, we describe another such example, namely, the semi-infinite Weil complex of the Virasoro algebra. The semi-infinite Weil complex of a tame Z-graded Lie algebra was defined in 1991 by Feigin-Frenkel, and they computed the linear structure of its cohomology in the case of the Virasoro algebra. We build on this result …
The main aim of this article is to prove the one-dimensionality of the third algebraic cohomology of the Virasoro algebra with values in the adjoint module. We announced this result in a previous publication with only a sketch of the proof. The detailed proof is provided in the present article. We also show that the third algebraic cohomology of the Witt and the Virasoro algebra with values in the trivial module is one-dimensional. We consider purely algebraic cohomology, i.e. our results are independent of any topology chosen. The vanishing of the third algebraic cohomology of the Witt algebr…
We build a bridge between two algebraic structures in SCFT: a VOA in the Schur sector of 4d $\mathcal{N}=2$ theories and an associative algebra in the Higgs sector of 3d $\mathcal{N}=4$. The natural setting is a 4d $\mathcal{N}=2$ SCFT placed on $S^3\times S^1$: by sending the radius of $S^1$ to zero, we recover the 3d $\mathcal{N}=4$ theory, and the corresponding VOA on the torus degenerates to the associative algebra on the circle. We prove that: 1) the Higgs branch operators remain in the cohomology; 2) all the Schur operators of the non-Higgs type are lifted by line operators wrapped on th…
We define a quasiclassical limit of the Lian-Zuckerman homotopy BV algebra (quasiclassical LZ algebra) on the subcomplex, corresponding to "light modes", i.e. the elements of zero conformal weight, of the semi-infinite (BRST) cohomology complex of the Virasoro algebra associated with vertex operator algebra (VOA) with a formal parameter. We also construct a certain deformation of the BRST differential parametrized by a constant two-component tensor, such that it leads to the deformation of the $A_{\infty}$-subalgebra of the quasiclassical LZ algebra. Altogether this gives a functor the categor…
A proof of the vanishing of the third cohomology group of the Witt algebra with values in the adjoint module is given. Moreover, we provide a sketch of the proof of the one-dimensionality of the third cohomology group of the Virasoro algebra with values in the adjoint module. The proofs given in the present article are completely algebraic and independent of any underlying topology. They are a generalization of the ones provided by Schlichenmaier, who proved the vanishing of the second cohomology group of the Witt and the Virasoro algebra by using purely algebraic methods. In the case of the t…
By elementary and direct calculations the vanishing of the (algebraic) second Lie algebra cohomology of the Witt and the Virasoro algebra with values in the adjoint module is shown. This yields infinitesimal and formal rigidity or these algebras. The first (and up to now only) proof of this important result was given 1989 by Fialowski in an unpublished note. It is based on cumbersome calculations. Compared to the original proof the presented one is quite elegant and considerably simpler.
We study a vertex operator algebra (VOA) V related to the M(3,p) Virasoro minimal series. This VOA reduces in the simplest case p=4 to the level two integrable vacuum module of $\hat{sl}_2$. On V there is an action of a commutative current a(z), which is an analog of the current e(z) of $\hat{sl}_2$. Our main concern is the subspace W generated by this action from the highest weight vector of V. Using the Fourier components of a(z), we present a monomial basis of W and a semi-infinite monomial basis of V. We also give a Gordon type formula for their characters.
In this paper, Lie bialgebra structures on the extended Schrodinger-Virasoro Lie algebra are classified. It is obtained that all the Lie bialgebra structures on L are triangular coboundary. As a by-product, it is derived that the first cohomology group is trivial.
In this paper, we study the structure theory of a class of not-finitely graded Lie algebras related to generalized Heisenberg-Virasoro algebras. In particular, the derivation algebras, the automorphism groups and the second cohomology groups of these Lie algebras are determined.
It is typical for a semi-infinite cohomology complex associated with a graded Lie algebra to occur as a vertex operator (or chiral) superalgebra where all the standard operators of cohomology theory, in particular the differential, are modes of vertex operators (fields). Although vertex operator superalgebras -with the inherent Virasoro action- are regarded as part of Conformal Field Theory (CFT), a VOSA may exhibit a square-zero operator (often, but not always, the semi-infinite cohomology differential) for which the Virasoro algebra acts trivially in the cohomology. Capable of shedding its C…
Conformal algebra is an axiomatic description of the operator product expansion of chiral fields in conformal field theory. On the other hand, it is an adequate tool for the study of infinite-dimensional Lie algebras satisfying the locality property. The main examples of such Lie algebras are those ``based'' on the punctured complex plane, like the Virasoro algebra and loop algebras. In the present paper we develop a cohomology theory of conformal algebras with coefficients in an arbitrary module. It possesses standard properties of cohomology theories; for example, it describes extensions and…
We consider the algebraic structure of $\mathbb{N}$-graded vertex operator algebras with conformal grading $V=\oplus_{n\geq 0} V_n$ and $\dim V_0\geq 1$. We prove several results along the lines that the vertex operators $Y(a, z)$ for $a$ in a Levi factor of the Leibniz algebra $V_1$ generate an affine Kac-Moody subVOA. If $V$ arises as a shift of a self-dual VOA of CFT-type, we show that $V_0$ has a `de Rham structure' with many of the properties of the de Rham cohomology of a complex connected manifold equipped with Poincaré duality.
We consider the M(2,3) Minimal Liouville gravity, whose states in the gravity sector are represented by irreducible modules of the Virasoro algebra. We present a recursive construction for BRST cohomology classes. This construction is based on using an explicit form of singular vectors in irreducible modules of the Virasoro algebra. We construct an algebra of operators acting on the BRST cohomology space. The operator algebra of physical states is established by use of these operators.
We study properties of a C_2-cofinite vertex operator algebra of CFT type. If it is also rational and V'\cong V, then the rigidity of the tensor category of modules has been proved by Huang. When we treat an irrational C_2-cofinite VOA, the rigidity is too strong, because it is almost equivalent to be rational as we see. We introduce a natural weaker condition "semi-rigidity". Under this condition, we prove the following results. For a projective cover P of a V-module V and a finitely generated V-module M, the projective cover of M is a direct summand of the tensor product P\boxtimes M defined…
Given a collection of modules of a vertex algebra parametrized by an abelian group, together with one dimensional spaces of composable intertwining operators, we assign a canonical element of the cohomology of an Eilenberg-Mac Lane space. This element describes the obstruction to locality, as the vanishing of this element is equivalent to the existence of a vertex algebra structure with multiplication given by our intertwining operators, and given existence, the structure is unique up to isomorphism. The homological obstruction reduces to an "evenness" problem that naturally vanishes for 2-div…
Given a simple, simply laced, complex Lie algebra $\bfg$ corresponding to the Lie group $G$, let $\bfnp$ be the subalgebra generated by the positive roots. In this paper we construct a BV-algebra $\fA[\bfg]$ whose underlying graded commutative algebra is given by the cohomology, with respect to $\bfnp$, of the algebra of regular functions on $G$ with values in $\mywedge (\bfnp\backslash\bfg)$. We conjecture that $\fA[\bfg]$ describes the algebra of {\it all} physical (i.e., BRST invariant) operators of the noncritical $\cW[\bfg]$ string. The conjecture is verified in the two explicitly known c…
The noncritical $D=4$ $W_3$ string is a model of $W_3$ gravity coupled to two free scalar fields. In this paper we discuss its BRST quantization in direct analogy with that of the $D=2$ (Virasoro) string. In particular, we calculate the physical spectrum as a problem in BRST cohomology. The corresponding operator cohomology forms a BV-algebra. We model this BV-algebra on that of the polyderivations of a commutative ring on six variables with a quadratic constraint, or, equivalently, on the BV-algebra of (polynomial) polyvector fields on the base affine space of $SL(3,C)$. In this paper we atte…
