Universal two-parameter even spin [formula omitted]-algebra

We introduce the notion of ``local system of $\Bbb{Z}_{T}$-twisted vertex operators'' on a $\Bbb{Z}_{2}$-graded vector space $M$, generalizing the notion of local system of vertex operators [Li]. First, we prove that any local system of $\Bbb{Z}_{T}$-twisted vertex operators on $M$ has a vertex superalgebra structure with an automorphism $\sigma$ of order $T$ with $M$ as a $\sigma$-twisted module. Then we prove that for a vertex (operator) superalgebra $V$ with an automorphism $\sigma$ of order $T$, giving a $\sigma$-twisted $V$-module $M$ is equivalent to giving a vertex (operator) superalgebra homomorphism from $V$ to some local system of $\Bbb{Z}_{T}$-twisted vertex operators on $M$. As applications, we study the twisted modules for vertex operator (super)algebras associated to some well-known infinite-dimensional Lie (super)algebras and we prove the complete reducibility of $\Bbb{Z}_{T}$-twisted modules for vertex operator algebras associated to standard modules for an affine Lie algebra.

The varieties of semi-conformal vectors of affine vertex operator algebras

In this work, we introduce Urod algebras associated to simply laced Lie algebras as well as the concept of translation of W-algebras. Both results are achieved by showing that the quantum Hamiltonian reduction commutes with tensoring with integrable representations; that is, for V and L an affine vertex algebra and an integrable affine vertex algebra associated with $\mathfrak {g}$ , we have the vertex algebra isomorphism $H_{DS,f}^{0}(V\otimes L)\cong H_{DS,f}^{0}(V)\otimes L$ , where in the left-hand-side the Drinfeld–Sokolov reduction is taken with respect to the diagonal action of $\widehat {\mathfrak {g}}$ on $V{\otimes } L$ . The proof is based on some new construction of automorphisms of vertex algebras, which may be of independent interest. As corollaries, we get fusion categories of modules of many exceptional W-algebras, and we can construct various corner vertex algebras. A major motivation for this work is that Urod algebras of type A provide a representation theoretic interpretation of the celebrated Nakajima–Yoshioka blowup equations for the moduli space of framed torsion free sheaves on $\mathbb {CP}^{2}$ of an arbitrary rank.

Elements of the annihilating ideal of a standard module

Twisted representations of vertex operator algebras associated to affine Lie algebras

We continue the study of the vertex operator algebra L(k, 0) associated to a type affine Lie algebra at admissible one-third integer levels, , initiated in Axtell and Lee [3]. We show there are only finitely many irreducible L(k, 0)-modules from the category 𝒪. The proof relies on knowledge of an explicit formula for the singular vectors. After obtaining this formula, we are able to prove there are only finitely many irreducible A(L(k, 0))-modules from the category 𝒪. The main result then follows from the bijective correspondence in A(V)-theory.

Inspired by a recent work of Frenkel-Zhu, we study a class of (pre-)vertex operator algebras (voa) associated to the self-dual Lie algebras. Based on a few elementary structural results we propose thatV, the category of Z+-graded prevoasV in whichV[0] is one-dimensional, is a proper setting in which to study and classify simple objects. The categoryV is organized into what we call the minimalkth types. We introduce a functor Γ—which we call the Frenkel-Lepowsky-Meurman functor—that attaches to each object inV a Lie algebra. This is a key idea which leads us to a (relative) classification of thesimple minimal first type. We then study the set of all Virasoro structures on a fixed minimal first typeV, and show that they are in turn classified by the orbits of the automorphism group Aut(Γ(V)) in cent(Γ(V)). Many new examples of voas are given. Finally, we introduce a generalized Kac-Casimir operator and give a simple proof of the irreducibility of the prolongation modules over the affine Lie algebras.

We show that the Constant r-term Krichever-Novikov-type called by Nuyts and platten are derivation algebras of certain k-graded commutative algebra parametrized by Laurent polynomials, which implies that the algebras have natural central extensions. The affinizations associated with these commutative algebras are also introduced. More-Over, we give vertex operator representations of corresponding affine Kac-Moody algebras. By our vertex operator approach, we construct related k-graded Super-Virasoro algebras and their representations.

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.

This is the first in a series of papers which describe the action of an affine Lie algebra with central charge $n$ on the moduli space of $U(n)$-instantons on a four manifold $X$. This generalises work of Nakajima, who considered the case when $X$ is an ALE space. In particular, this describes the combinatorial complexity of the moduli space as being precisely that of representation theory, and thus will lead to a description of the Betti numbers of moduli space as dimensions of weight spaces. This Lie algebra acts on the space of conformal blocks (i\.e\., the cohomology of a determinant line bundle on the moduli space) generalising the ``insertion'' and ``deletion'' operations of conformal field theory, and indeed on any cohomology theory. In the particular case of $U(1)$-instantons, which is essentially the subject of this present paper, the construction produces the basic representation after Frenkel-Kac. Then the well known quadratic nature of $ch_2$, $$ch_2 = \frac{1}{2} c_1\cdot c_1 - c_2 $$ becomes precisely the formula for the eigenvalue of the degree operator, i\.e\. the well known quadratic behaviour of affine Lie algebras.

After giving some definitions for vertex operator SUPERalgebras and their modules, we construct an associative algebra corresponding to any vertex operator superalgebra, such that the representations of the vertex operator algebra are in one-to-one correspondence with those of the corresponding associative algebra. A way is presented to decribe the fusion rules for the vertex operator superalgebras via modules of the associative algebra. The above are generalizations of Zhu's constructions for vertex operator algebras. Then we deal in detail with vertex operator superalgebras corresponding to Neveu-Schwarz algebras, super affine Kac-Moody algebras, and free fermions. We use the machinery established above to find the rationality conditions, classify the representations and compute the fusion rules. In the appendix, we present explicit formulas for singular vectors and defining relations for the integrable representations of super affine algebras. These formulas are not only crucial for the theory of the corresponding vertex operator superalgebras and their representations, but also of independent interest.

Twisted tensor products of quantum affine vertex algebras and coproducts

In this article, we shall describe the center of the universal affine vertex superalgebra $V^{\kappa _{c}}(\mathfrak{g})$ associated with $\mathfrak{g}=\mathfrak{s}\mathfrak{l}_{2|1}, \mathfrak{g}\mathfrak{l}_{2|1}$ at the critical level $\kappa _{c}$ and prove the conjecture of A. Molev and E. Ragoucy [24] in this case. The center $\mathfrak{z}(V^{\kappa _{c}}(\mathfrak{s}\mathfrak{l}_{2|1}))$ turns out to be isomorphic to the large level limit $\ell \rightarrow \infty $ of a vertex subalgebra, called the parafermion vertex algebra $K^{\ell } (\mathfrak{s}\mathfrak{l}_{2})$, of the affine vertex algebra $V^\ell (\mathfrak{s}\mathfrak{l}_{2})$. The key ingredient of the proof is to understand the principal $\mathcal{W}$-superalgebra $\mathcal{W}^{\kappa _{c}}(\mathfrak{s}\mathfrak{l}_{2|1})$ at the critical level. It relates the center $\mathfrak{z}(V^{\kappa _{c}}(\mathfrak{s}\mathfrak{l}_{2|1}))$ to $V^\infty (\mathfrak{s}\mathfrak{l}_{2})$ via the Kazama–Suzuki duality while it has a surprising coincidence with $V^{\kappa _{c}}(\mathfrak{g}\mathfrak{l}_{1|1})$, whose center has been recently described. Moreover, the centers $\mathfrak{z}(V^{\kappa _{c}}(\mathfrak{s}\mathfrak{l}_{2|1}))$ and $\mathfrak{z}(\mathcal{W}^{\kappa _{c}}(\mathfrak{s}\mathfrak{l}_{2|1}))$ are proven to coincide as a byproduct. A general conjecture is proposed, which describes the center $\mathfrak{z}(V^{\kappa _{c}}(\mathfrak{s}\mathfrak{l}_{n|m}))$ with $n>m$ as a large level limit of “the dual side”, that is, the parafermion-type subalgebras of $\mathcal{W}$-algebras $\mathcal{W}^\ell (\mathfrak{s}\mathfrak{l}_{n}, \mathbb{O}_{[n-m,1^{m}]})$ associated with hook-type partitions $[n-m,1^{m}]$, known also as vertex algebras at the corner.

For each simple Lie algebra g consider the corresponding affine vertex algebra V_{crit}(g) at the critical level. The center of this vertex algebra is a commutative associative algebra whose structure was described by a remarkable theorem of Feigin and Frenkel about two decades ago. However, only recently simple formulas for the generators of the center were found for the Lie algebras of type A following Talalaev's discovery of explicit higher Gaudin Hamiltonians. We give explicit formulas for generators of the centers of the affine vertex algebras V_{crit}(g) associated with the simple Lie algebras g of types B, C and D. The construction relies on the Schur-Weyl duality involving the Brauer algebra, and the generators are expressed as weighted traces over tensor spaces and, equivalently, as traces over the spaces of singular vectors for the action of the Lie algebra sl_2 in the context of Howe duality. This leads to explicit constructions of commutative subalgebras of the universal enveloping algebras U(g[t]) and U(g), and to higher order Hamiltonians in the Gaudin model associated with each Lie algebra g. We also introduce analogues of the Bethe subalgebras of the Yangians Y(g) and show that their graded images coincide with the respective commutative subalgebras of U(g[t]).

The monster sporadic group is the automorphism group of a central charge $c=24$ vertex operator algebra (VOA) or meromorphic conformal field theory (CFT). In addition to its $c=24$ stress tensor $T(z)$, this theory contains many other conformal vectors of smaller central charge; for example, it admits $48$ commuting $c=\frac12$ conformal vectors whose sum is $T(z)$. Such decompositions of the stress tensor allow one to construct new CFTs from the monster CFT in a manner analogous to the Goddard-Kent-Olive (GKO) coset method for affine Lie algebras. We use this procedure to produce evidence for the existence of a number of CFTs with sporadic symmetry groups and employ a variety of techniques, including Hecke operators, modular linear differential equations, and Rademacher sums, to compute the characters of these CFTs. Our examples include (extensions of) nine of the sporadic groups appearing as subquotients of the monster, as well as the simple groups ${}^2{E}_6(2)$ and ${F}_4(2)$ of Lie type. Many of these examples are naturally associated to McKay's $\widehat{E_8}$ correspondence, and we use the structure of Norton's monstralizer pairs more generally to organize our presentation.