We consider several topologically twisted Chern-Simons-matter theories and propose boundary VOAs whose module categories should model the category of line operators of the 3d bulk. Our main examples come from the topological $A$ and $B$ twists of the exotic $\mathcal{N}=4$ Chern-Simons-matter theories of Gaiotto-Witten, but we show that there is a topological "$A$-twist" for a much larger class of $\mathcal{N}\neq4$ theories. We illustrate a particular example of this new class of theories that admits the $p=2$ singlet VOA $\mathfrak{M}(2)$ on its boundary and comment on its relation to the $\psi \to \infty$ limit of the Gaiotto-Rap{\v c}{\'a}k corner VOA $Y_{1,1,0}[\psi]$.

Niklas Garner

We show that the representation theory for the toroidal extended affine Lie algebras is controlled by a vertex operator algebra which is a tensor product of four VOAs: a sub-VOA of the hyperbolic lattice VOA, two affine VOAs and a Virasoro VOA. A tensor product of irreducible modules for these VOAs admits the structure of an irreducible module for the toroidal extended affine Lie algebra. We also show that for N=12, the sub-VOA of the hyperbolic lattice VOA becomes an exceptional irreducible module for the extended affine Lie algebra of rank 0.

Yuly Billig

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 controls the representation theory of gdiv is a tensor product of an affine VOA Vaff(c) at level c, a sub-VOA of a hyperbolic lattice VOA, affine sl(N) VOA and a Virasoro VOA at level cvir with the following condition on the central charges: 2(N+1) + rank Vaff(c) + cvir = 26.

Yuly Billig

In this article, we determine quantum dimensions and fusion rules for the orbifold code VOA $ V^\tau_{L_{C \times D}}$. As an application, we also construct certain $3$-local subgroups inside the automorphism group of the VOA $V^\sharp$, where $V^\sharp$ is a holomorphic VOA obtained by the $\mathbb{Z}_3$-orbifold construction on the Leech lattice VOA.

Hsian-Yang Chen Ching Hung Lam

We use Kazhdan-Lusztig tensoring to, first, describe annihilating ideals of highest weight modules over an affine Lie algebra in terms of the corresponding VOA and, second, to classify tilting functors, an affine analogue of projective functors known in the case of a simple Lie algebra. For the sake of completeness, the classification of annihilating ideals is borrowed from our previous work, q-alg/9711011; the part on tilting functors is new.

Igor B. Frenkel Feodor Malikov

We discuss the quantization of the $\widehat{\mathfrak{sl}}_2$ coset vertex operator algebra $\mathcal{W}D(2,1;\alpha)$ using the bosonization technique. We show that after quantization there exist three families of commuting integrals of motion coming from three copies of the quantum toroidal algebra associated to ${\mathfrak{gl}}_2$.

B. Feigin M. Jimbo E. Mukhin

In this article we study a VOA with two Miyamoto involutions generating $S_3$. In \cite{M3}, Miyamoto showed that a VOA generated by two conformal vectors whose Miyamoto involutions generate an automorphism group isomorphic to $S_3$ is isomorphic to one of the four candidates he listed. We construct one of them and prove that our VOA is actually the same as $\mathrm{VA}(e,f)$ studied by Miyamoto. We also show that there is an embedding into the moonshine VOA. Using our VOA, we can define the 3A-triality in the Monster.

Shinya Sakuma Hiroshi Yamauchi

It is one of the remarkable results of vertex operator algebras (VOAs) that the graded traces (one-point correlation functions) of holomorphic VOAs are modular functions. This paper explores the question of which modular functions arise as the graded traces of holomorphic VOAs. For VOAs of small central charge, i.e., $c\le 24$, and a non-zero weight-one subspace, we find that the only conditions imposed on the modular fuctions are those that arise easily out of our condition that the VOAs be of CFT type, that is that they have no negative-weight subspaces and their zero-weight subspace is generated by the vacuum vector.

Katherine L. Hurley

We identify vertex operator algebras (VOAs) of a class of Argyres-Douglas (AD) matters with two types of non-abelian flavor symmetries. They are the $W$ algebra defined using nilpotent orbit with partition $[q^m,1^s]$. Gauging above AD matters, we can find VOAs for more general $\mathcal{N}=2$ SCFTs engineered from 6d $(2,0)$ theories. For example, the VOA for general $(A_{N-1}, A_{k-1})$ theory is found as the coset of a collection of above $W$ algebras. Various new interesting properties of 2d VOAs such as level-rank duality, conformal embedding, collapsing levels, coset constructions for known VOAs can be derived from 4d theory.

Dan Xie Wenbin Yan

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.

B. Feigin M. Jimbo T. Miwa