Coupling a vertex algebra to a large center

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.

We prove a dimension formula for the weight-1 subspace of a vertex operator algebra $V^{\operatorname{orb}(g)}$ obtained by orbifolding a strongly rational, holomorphic vertex operator algebra $V$ of central charge 24 with a finite-order automorphism $g$. Based on an upper bound derived from this formula we introduce the notion of a generalised deep hole in $\operatorname{Aut}(V)$. Then we show that the orbifold construction defines a bijection between the generalised deep holes of the Leech lattice vertex operator algebra $V_\Lambda$ with non-trivial fixed-point Lie subalgebra and the strongly rational, holomorphic vertex operator algebras of central charge 24 with non-vanishing weight-1 space. This provides the first uniform construction of these vertex operator algebras and naturally generalises the correspondence between the deep holes of the Leech lattice $\Lambda$ and the 23 Niemeier lattices with non-vanishing root system found by Conway, Parker and Sloane.

Vertex algebras generated by Lie algebras

Starting with any R-matrix with spectral parameter, obeying the Yang-Baxter equation and a unitarity condition, we construct the corresponding infinite dimensional quantum group U_{R} in term of a deformed oscillators algebra A_R. The realization we present is an infinite series, very similar to a vertex operator. Then, considering the integrable hierarchy naturally associated to A_{R}, we show that U_{R} provides its integrals of motion. The construction can be applied to any infinite dimensional quantum group, e.g. Yangians or elliptic quantum groups. Taking as an example the R-matrix of Y(N), the Yangian based on gl(N), we recover by this construction the nonlinear Schrodinger equation and its Y(N) symmetry.

The quantum symmetry of rational conformal field theories

A vertex algebra with an action of a group $G$ comes with a notion of $g$-twisted modules, forming a $G$-crossed braided tensor category. For a Lie group $G$, one might instead wish for a notion of $(\textrm{d}+A)$-twisted modules for any $\mathfrak{g}$-connection on the formal punctured disk. For connections with a regular singularity, this reduces to $g$-twisted modules, where $g$ is the monodromy around the puncture. The case of an irregular singularity is much richer and involved, and we are not aware that it has appeared in vertex algebra language. The present article is intended to spark such a treatment, by providing a list of expectations and an explicit worked-through example with interesting applications. Concretely, we consider the vertex super algebra of symplectic fermions through its associated Clifford algebra and study its twisted module with respect to irregular $\mathfrak{s}\mathfrak{l}_{2}$-connections. We first determine the category of representations, depending on the formal type of the connection. Then we prove that a Sugawara-type construction gives a Virasoro action and we prove that as Virasoro modules our representations are direct sums of Whittaker modules. Conformal field theory with irregular singularities resp. wild ramification appear in the context of geometric Langlands correspondence, in particular work by Witten [51], and more generally in higher-dimensional context. Our original motivation in [24] comes from semiclassical limits of the generalized quantum Langlands kernel, which fibres over the space of connections (similar to the affine Lie algebra at critical level) and as such gives a family of deformations of the Feigin–Tipunin algebra. Our present article now describes, in the smallest case, the fibres and their representation categories over irregular connections.

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.

Ribbon tensor structure on the full representation categories of the singlet vertex algebras

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 center of the algebra of continuous functions on the quantum group \(\mathrm{SU}_q(2)\) is determined as well as centers of other related algebras. Several other results concerning this quantum group are given with direct proofs based on concrete realization of these algebras as algebras of operators on a Hilbert space.

The two major approaches to chiral conformal field theory - one based on operator algebras and one based on vertex algebras - both lead to representation categories which are tensor categories and, in the case of rational chiral conformal field theories, more specifically modular tensor cat- egories. In this Arbeitsgemeinschaft, we have studied algebraic structures related to tensor categories arising in conformal field theory. The notion of a module category over this tensor category is central in the construction of a full local conformal field theory in various frameworks.

Recent work on neutrino oscillations suggests that the three generations of fermions in the standard model are related by representations of the finite group A(4), the group of symmetries of the tetrahedron. Motivated by this, we explore models which extend the EPRL model for quantum gravity by coupling it to a bosonic quantum field of representations of A(4). This coupling is possible because the representation category of A(4) is a module category over the representation categories used to construct the EPRL model. The vertex operators which interchange vacua in the resulting quantum field theory reproduce the bosons and fermions of the standard model, up to issues of symmetry breaking which we do not resolve. We are led to the hypothesis that physical particles in nature represent vacuum changing operators on a sea of invisible excitations which are only observable in the A(4) representation labels which govern the horizontal symmetry revealed in neutrino oscillations. The quantum field theory of the A(4) representations is just the dual model on the extended lattice of the Lie group E6, as explained by the quantum McKay correspondence of Frenkel, Jing and Wang. The coupled model can be thought of as string field theory, but propagating on a discretized quantum spacetime rather than a classical manifold.

Introduction Poisson Lie groups Quantized universal enveloping algebras Quantized algebras of functions Quantum Weyl group and the universal quantum $R$-matrix Bibliography.

Following a recent proposal of Richard Borcherds to regard fusion as the ringlike tensor product of modules of a quantum ring, a generalization of rings and vertex algebras, we define fusion as a certain quotient of the (vector space) tensor product of representations of the symmetry algebra [Formula: see text]. We prove that this tensor product is associative and symmetric up to equivalence. We also determine explicitly the action of [Formula: see text] on it, under which the central extension is preserved. Having defined fusion in this way, determining the fusion rules is then the algebraic problem of decomposing the tensor product into irreducible representations. We demonstrate how to solve this for the case of the WZW and the minimal models and recover thereby the well-known fusion rules. The action of the symmetry algebra on the tensor product is given in terms of a comultiplication. We calculate the R matrix of this comultiplication and find that it is triangular. This seems to shed some new light on the possible rôle of the quantum group in conformal field theory.

Quantum group and quantum symmetry