Vertex Algebra Structure Research Articles

Deforming vertex algebras by vertex bialgebras

This is a continuation of a previous study initiated by the third author on nonlocal vertex bialgebras and smash product nonlocal vertex algebras. In this paper, we study a notion of right [Formula: see text]-comodule nonlocal vertex algebra for a nonlocal vertex bialgebra [Formula: see text] and give a construction of deformations of vertex algebras with a right [Formula: see text]-comodule nonlocal vertex algebra structure and a compatible [Formula: see text]-module nonlocal vertex algebra structure. We also give a construction of [Formula: see text]-coordinated quasi modules for smash product nonlocal vertex algebras. As an example, we give a family of quantum vertex algebras by deforming the vertex algebras associated to non-degenerate even lattices.

Covariant derivatives of eigenfunctions along parallel tensors over space forms and a conjecture motivated by the vertex algebraic structure

We study the covariant derivatives of an eigenfunction for the Laplace–Beltrami operator on a complete, connected Riemannian manifold with nonzero constant sectional curvature. We show that along every parallel tensor, the covariant derivative is a scalar multiple of the eigenfunction. We also show that the scalar is a polynomial depending on the eigenvalue and prove some properties. A conjecture motivated by the study of vertex algebraic structure on space forms is also announced, suggesting the existence of interesting structures in these polynomials that awaits further exploration.

Principal subspaces for the quantum affine vertex algebra in type A1(1)

By using the ideas of Feigin and Stoyanovsky and Calinescu, Lepowsky and Milas we introduce and study the principal subspaces associated with the Etingof-Kazhdan quantum affine vertex algebra of integer level $k\geqslant 1$ and type $A_1^{(1)}$. We show that the principal subspaces possess the quantum vertex algebra structure, which turns to the usual vertex algebra structure of the principal subspaces of generalized Verma and standard modules at the classical limit. Moreover, we find their topological quasi-particle bases which correspond to the sum sides of certain Rogers-Ramanujan-type identities.

Cocommutative vertex bialgebras

In this paper, the structure of cocommutative vertex bialgebras is investigated. For a general vertex bialgebra V, it is proved that the set G(V) of group-like elements is naturally an abelian semigroup, whereas the set P(V) of primitive elements is a vertex Lie algebra. For g∈G(V), denote by Vg the connected component containing g. Among the main results, it is proved that if V is a cocommutative vertex bialgebra, then V=⊕g∈G(V)Vg, where V1 is a vertex subbialgebra which is isomorphic to the vertex bialgebra VP(V) associated to the vertex Lie algebra P(V), and Vg is a V1-module for g∈G(V). In particular, this shows that every cocommutative connected vertex bialgebra V is isomorphic to VP(V) and hence establishes the equivalence between the category of cocommutative connected vertex bialgebras and the category of vertex Lie algebras. Furthermore, under the condition that G(V) is a group and lies in the center of V, it is proved that V=VP(V)⊗C[G(V)] as a coalgebra where the vertex algebra structure is explicitly determined.

Gluing vertex algebras

We relate commutative algebras in braided tensor categories to braid-reversed tensor equivalences, motivated by vertex algebra representation theory. First, for C a braided tensor category, we give a detailed account of the canonical algebra construction in the Deligne product C⊠Crev. Especially, we show that if C is semisimple but not necessarily finite or rigid, then ⨁X∈Irr(C)X′⊠X is a commutative algebra, where X′ is a representing object for the functor HomC(•⊗CX,1C) (assuming X′ exists) and the sum runs over all inequivalent simple objects of U. Conversely, let A=⨁i∈IUi⊠Vi be a simple commutative algebra in a Deligne product U⊠V with U semisimple and rigid but not necessarily finite, and V rigid but not necessarily semisimple. We show that if the unit objects 1U and 1V form a commuting pair in A in a suitable sense, then there is a braid-reversed equivalence between (sub)categories of U and V that sends Ui to Vi⁎.These results apply when U and V are braided (vertex) tensor categories of modules for simple vertex operator algebras U and V, respectively: Given τ:Irr(U)→Obj(V) such that τ(U)=V, we glue U and V along U⊠V via τ to create A=⨁X∈Irr(U)X′⊗τ(X). Then under certain conditions, τ extends to a braid-reversed equivalence between U and V if and only if A has a simple conformal vertex algebra structure that (conformally) extends U⊗V. As examples, we glue suitable Kazhdan-Lusztig categories at generic levels to construct new vertex algebras extending the tensor product of two affine vertex subalgebras, and we prove braid-reversed equivalences between certain module subcategories for affine vertex algebras and W-algebras at admissible levels.

Building Vertex Algebras from Parts

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-divisible groups, so simple currents organized into odd order abelian groups always produce vertex algebras. Furthermore, in cases most relevant to conformal field theory (i.e., when we have well-behaved contragradients and tensor products), we obtain our spaces of intertwining operators naturally, and the evenness obstruction reduces to the question of whether the contragradient bilinear form on certain order 2 currents is symmetric or skew-symmetric. We show that if we are given a simple regular VOA and integral-weight modules parametrized by a group of even units in the fusion ring, then the direct sum admits the structure of a simple regular VOA, called the simple current extension, and this structure is unique up to isomorphism.

Principal subspaces of twisted modules for certain lattice vertex operator algebras

This is the third in a series of papers studying the vertex-algebraic structure of principal subspaces of twisted modules for lattice vertex operator algebras. We focus primarily on lattices [Formula: see text] whose Gram matrix contains only non-negative entries. We develop further ideas originally presented by Calinescu, Lepowsky, and Milas to find presentations (generators and relations) of the principal subspace of a certain natural twisted module for the vertex operator algebra [Formula: see text]. We then use these presentations to construct exact sequences involving this principal subspace, which give a set of recursions satisfied by the multigraded dimension of the principal subspace and allow us to find the multigraded dimension of the principal subspace.

Quantum current algebras associated with rational R-matrix

We study quantum current algebra A(R‾) associated with the rational R-matrix of glN and we give explicit formulae for the elements of its center at the critical level. Due to Etingof–Kazhdan's construction, the level c vacuum module Vc(R‾) for the algebra A(R‾) possesses a quantum vertex algebra structure for any complex number c. We prove that any module for the quantum vertex algebra Vc(R‾) is naturally equipped with a structure of restricted A(R‾)-module of level c and vice versa.

Structure of classical (finite and affine) $\mathcal W$-algebras

First, we derive an explicit formula for the Poisson bracket of the classical finite \mathcal W -algebra \mathcal W^{\mathrm {fin}}(\mathfrak g,f) , the algebra of polynomial functions on the Slodowy slice associated to a simple Lie algebra \mathfrak g and its nilpotent element f . On the other hand, we produce an explicit set of generators and we derive an explicit formula for the Poisson vertex algebra structure of the classical affine \mathcal W -algebra \mathcal W(\mathfrak g,f) . As an immediate consequence, we obtain a Poisson algebra isomorphism between \mathcal W^{\mathrm {fin}}(\mathfrak g,f) and the Zhu algebra of \mathcal W(\mathfrak g,f) . We also study the generalized Miura map for classical \mathcal W -algebras.

Vertex-algebraic structure of principal subspaces of standard $A^{(2)}_2$-modules, I

Extending earlier work of the authors, this is the first in a series of papers devoted to the vertex-algebraic structure of principal subspaces of standard modules for twisted affine Kac–Moody algebras. In this part, we develop the necessary theory of principal subspaces for the affine Lie algebra [Formula: see text], which we expect can be extended to higher rank algebras. As a "test case", we consider the principal subspace of the basic [Formula: see text]-module and explore its structure in depth.

A motivated proof of Gordon’s identities

We generalize the “motivated proof” of the Rogers–Ramanujan identities given by Andrews and Baxter to provide an analogous “motivated proof” of Gordon’s generalization of the Rogers–Ramanujan identities. Our main purpose is to provide insight into certain vertex-algebraic structure being developed.

VERTEX-ALGEBRAIC STRUCTURE OF THE PRINCIPAL SUBSPACES OF CERTAIN $A_{1}^{(1)}$-MODULES, I: LEVEL ONE CASE

This is the first in a series of papers in which we study vertex-algebraic structure of Feigin–Stoyanovsky's principal subspaces associated to standard modules for both untwisted and twisted affine Lie algebras. A key idea is to prove suitable presentations of principal subspaces, without using bases or even "small" spanning sets of these spaces. In this paper, we prove presentations of the principal subspaces of the basic [Formula: see text]-modules. These convenient presentations were previously used in work of Capparelli–Lepowsky–Milas for the purpose of obtaining the classical Rogers–Ramanujan recursion for the graded dimensions of the principal subspaces.

Chiral de Rham complex and orbifolds

Suppose that a finite group G G acts on a smooth complex variety X X . Then this action lifts to the Chiral de Rham complex Ω X ch \Omega ^{\operatorname {ch}}_{X} of X X and to its cohomology by automorphisms of the vertex algebra structure. We define twisted sectors for Ω X ch \Omega ^{\operatorname {ch}}_{X} (and their cohomologies) as sheaves of twisted vertex algebra modules supported on the components of the fixed-point sets X g , g ∈ G X^{g}, g \in G . Each twisted sector sheaf carries a BRST differential and is quasi-isomorphic to the de Rham complex of X g X^{g} . Putting the twisted sectors together with the vacuum sector and taking G G -invariants, we recover the additive and graded structures of Chen-Ruan orbifold cohomology. Finally, we show that the orbifold elliptic genus is the partition function of the direct sum of the cohomologies of the twisted sectors.

Nonlocal vertex algebras generated by formal vertex operators

This is the first paper in a series to study vertex algebra-like objects arising from infinite-dimensional quantum groups (quantum affine algebras and Yangians). In this paper we lay the foundations for this study. For any vector space W, we study what we call quasi compatible subsets of Hom (W,W((x))) and we prove that any maximal quasi compatible subspace has a natural nonlocal (namely noncommutative) vertex algebra structure with W as a natural faithful quasi module in a certain sense, and that any quasi compatible subset generates a nonlocal vertex algebra with W as a quasi module. In particular, taking W to be a highest weight module for a quantum affine algebra we obtain a nonlocal vertex algebra with W as a quasi module. We also formulate and study a notion of quantum vertex algebra and we give general constructions of nonlocal vertex algebras, quantum vertex algebras and their modules.

Modified regular representations of affine and Virasoro algebras, VOA structure and semi-infinite cohomology

We find a counterpart of the classical fact that the regular representation R ( G ) of a simple complex group G is spanned by the matrix elements of all irreducible representations of G. Namely, the algebra of functions on the big cell G 0 ⊂ G of the Bruhat decomposition is spanned by matrix elements of big projective modules from the category O of representations of the Lie algebra g of G, and has the structure of a g ⊕ g -module. The standard regular representation R ( G ^ ) of the affine group G ^ has two commuting actions of the Lie algebra g ^ with total central charge 0, and carries the structure of a conformal field theory. The modified versions R ′ ( G ^ ) and R ′ ( G ^ 0 ) , originating from the loop version of the Bruhat decomposition, have two commuting g ^ -actions with central charges shifted by the dual Coxeter number, and acquire vertex operator algebra structures derived from their Fock space realizations, given explicitly for the case G = SL ( 2 , C ) . The quantum Drinfeld–Sokolov reduction transforms the representations of the affine Lie algebras into their W -algebra counterparts, and can be used to produce analogues of the modified regular representations. When g = sl ( 2 , C ) the corresponding W -algebra is the Virasoro algebra. We describe the Virasoro analogues of the modified regular representations, which are vertex operator algebras with the total central charge equal to 26. The special values of the total central charges in the affine and Virasoro cases lead to the non-trivial semi-infinite cohomology with coefficients in the modified regular representations. The inherited vertex algebra structure on this cohomology degenerates into a supercommutative associative superalgebra. We describe these superalgebras in the case when the central charge is generic, and identify the 0th cohomology with the Grothendieck ring of finite-dimensional G-modules. We conjecture that for the integral values of the central charge the 0th semi-infinite cohomology coincides with the Verlinde algebra and its counterpart associated with the big projective modules.

A new construction of vertex algebras and quasi-modules for vertex algebras

In this paper, a new construction of vertex algebras from more general vertex operators is given and a notion of quasimodule for vertex algebras is introduced and studied. More specifically, a notion of quasilocal subset(space) of Hom ( W , W ( ( x ) ) ) for any vector space W is introduced and studied, generalizing the notion of usual locality in the most possible way, and it is proved that on any maximal quasilocal subspace there exists a natural vertex algebra structure and that any quasilocal subset of Hom ( W , W ( ( x ) ) ) generates a vertex algebra. Furthermore, it is proved that W is a quasimodule for each of the vertex algebras generated by quasilocal subsets of Hom ( W , W ( ( x ) ) ) . A notion of Γ -vertex algebra is also introduced and studied, where Γ is a subgroup of the multiplicative group C × of nonzero complex numbers. It is proved that any maximal quasilocal subspace of Hom ( W , W ( ( x ) ) ) is naturally a Γ -vertex algebra and that any quasilocal subset of Hom ( W , W ( ( x ) ) ) generates a Γ -vertex algebra. It is also proved that a Γ -vertex algebra exactly amounts to a vertex algebra equipped with a Γ -module structure which satisfies a certain compatibility condition. Finally, two families of examples are given, involving twisted affine Lie algebras and certain quantum torus Lie algebras.

Vertex algebras and the formal loop space

We construct a certain algebro-geometric version $\mathcal{L}(X)$ of the free loop space for a complex algebraic variety X. This is an ind-scheme containing the scheme $\mathcal{L}^{0}(X)$ of formal arcs in X as studied by Kontsevich and Denef-Loeser. We describe the chiral de Rham complex of Malikov, Schechtman and Vaintrob in terms of the space of formal distributions on $\mathcal{L}(X)$ supported in $\mathcal{L}^{0}(X)$ . We also show that $\mathcal{L}(X)$ possesses a factorization structure: a certain non-linear version of a vertex algebra structure. This explains the heuristic principle that “all” linear constructions applied to the free loop space produce vertex algebras.

ON ABELIAN COSET GENERALIZED VERTEX ALGEBRAS

This paper studies the algebraic aspect of a general abelian coset theory with a work of Dong and Lepowsky as our main motivation. It is proved that the vacuum space ΩV (or the space of highest weight vectors) of a Heisenberg algebra in a general vertex operator algebra V has a natural generalized vertex algebra structure in the sense of Dong and Lepowsky and that the vacuum space ΩW of a V-module W is a natural ΩV-module. The automorphism group Aut ΩVΩV of the adjoint ΩV-module is studied and it is proved to be a central extension of a certain torsion free abelian group by C×. For certain subgroups A of Aut ΩVΩV, certain quotient algebras [Formula: see text] of ΩV are constructed. Furthermore, certain functors among the category of V-modules, the category of ΩV-modules and the category of [Formula: see text]-modules are constructed and irreducible ΩV-modules and [Formula: see text]-modules are classified in terms of irreducible V-modules. If the category of V-modules is semisimple, then it is proved that the category of [Formula: see text]-modules is semisimple.

Vertex algebras generated by Lie algebras

In this paper we introduce a notion of vertex Lie algebra U, in a way a “half” of vertex algebra structure sufficient to construct the corresponding local Lie algebra L( U) and a vertex algebra L( U). We show that we may consider U as a subset U ⊂ V ( U) which generates V( U) and that the vertex Lie algebra structure on U is induced by the vertex algebra structure on V( U). Moreover, for any vertex algebra V a given homomorphism U → V of vertex Lie algebras extends uniquely to a homomorphism V( U) → V of vertex algebras. In the second part of paper we study under what conditions on structure constants one can construct a vertex Lie algebra U by starting with a given commutator formula for fields.

Quasifinite Representations of Classical Lie Subalgebras of [formula omitted]1+∞

We show that there are precisely two, up to conjugation, anti-involutionsσ±of the algebra of differential operators on the circle preserving the principal gradation. We classify the irreducible quasifinite highest weight representations of the central extension D±of the Lie subalgebra of this algebra fixed by −σ±, and find the unitary ones. We realize them in terms of highest weight representations of the central extension of the Lie algebra of infinite matrices with finitely many non-zero diagonals over the algebra C[u]/(um+1) and its classical Lie subalgebras ofB,CandDtypes. Character formulas forpositive primitiverepresentations of D±(including all the unitary ones) are obtained. We also realize a class of primitive representations of D±in terms of free fields and establish a number of duality results between these primitive representations and finite-dimensional irreducible representations of finite-dimensional Lie groups and supergroups. We show that the vacuum moduleVcof D±carries a vertex algebra structure and establish a relationship betweenVcforc∈12Z and W-algebras.