arXiv subjects
Publications and source records attributed to Xuanzhong Dai.
We prove that the simple affine vertex algebra $L_k(\mathfrak{sl}_2)$ is classically free at the boundary admissible level, which confirms a conjecture of Andrews, van Ekeren, and Heluani.
The Feigin-Semikhatov duality asserts that the Heisenberg cosets of the subregular $W$-algebra of $\mathfrak{sl}_n$ at level $k$ and the one of the principal $W$-superalgebra of $\mathfrak{sl}_{n|1}$ at level $\ell$ coincide when the levels satisfy the Feigin-Frenkel relation $(k+n)(\ell+n-1)=1$. A similar duality holds between the subregular $W$-algebra of $\mathfrak{so}_{2n+1}$ and the principal $W$-superalgebra of $\mathfrak{osp}_{2|2n}$. We study these dualities in the critical/large level limit. We describe the centerless subregular $W$-algebra at the critical level as an orbifold of the large level limit of the principal $W$-superalgebra times a lattice VOA. Our construction yields a functor between certain categories of the two involved vertex algebras. We show that in this set-up one in fact gets block-wise equivalences of categories. Studying the principal block of the large level limit of the principal $W$-superalgebra then gives us the structure of the principal blocks of the subregular $W$-algebras in the category of weight modules (which is much larger than the more common category of lower bounded modules).
We study the representations of some simple affine vertex algebras at non-admissible level arising from rank one 4D SCFTs. In particular, we classify the irreducible highest weight modules of $L_{-2}(G_2)$ and $L_{-2}(B_3)$. It is known by the works of Adamović and Perše that these vertex algebras can be conformally embedded into $L_{-2}(D_4)$. We also compute the associated variety of $L_{-2}(G_2)$, and show that it is the orbifold of the associated variety of $L_{-2}(D_4)$ by the symmetric group of degree 3 which is the Dynkin diagram automorphism group of $D_4$. This provides a new interesting example of associated variety satisfying a number of conjectures in the context of orbifold vertex algebras.
Chiral de Rham complex introduced by Malikov et al. in 1998, is a sheaf of vertex algebras on any complex analytic manifold or non-singular algebraic variety. Starting from the vertex algebra of global sections of chiral de Rham complex on the upper half plane, we consider the subspace of $Γ$-invariant sections that are meromorphic at the cusps. The space is again a vertex operator algebra, with a linear basis consisting of lifting formulas of meromorphic modular forms. We will describe two types of lifting formulas, and generalize the Rankin-Cohen bracket to the meromorphic modular forms. As an application, we will show that the vertex algebras constructed by congruence subgroups are simple.
For any congruence subgroup $Γ$, we study the vertex operator algebra $Ω^{ch}(\mathbb H,Γ)$ constructed from the $Γ$-invariant global sections of the chiral de Rham complex on the upper half plane, which are holomorphic at all the cusps. We introduce an $SL(2,\mathbb R)$-invariant filtration on the global sections and show that the $Γ$-invariants on the graded algebra is isomorphic to certain copies of modular forms. We also give an explicit formula for the lifting of modular forms to $Ω^{ch}(\mathbb H,Γ)$ and compute the character formula of $Ω^{ch}(\mathbb H,Γ)$. Furthermore, we show that the vertex algebra structure modifies the Rankin-Cohen bracket, and the modified bracket becomes non-zero between constant modular forms involving the Eisenstein series.
In this paper we study the vertex operator algebra $\mathscr D^{\text{ch}}(\mathbb H,Γ)$ constructed from the fixed points of the chiral differential operators on the upper half plane which is holomorphic at all the cusps, under the action of the congruence subgroup $Γ$. To this end, we introduce an $SL(2,\mathbb R)$-invariant filtration labeled by partition pairs and study its successive quotient. We show that the successive quotient under the cuspidal condition is isomorphic to the space of modular forms. And we also give a description of the structure of $\mathscr D^{\text{ch}}(\mathbb H,Γ)$ and compute its character.
We introduce the concept of linear topological modules over vertex algebras and apply it to representations of $β-γ$ system and affine Kac-Moody algebras.
In this paper, we construct a representation of loop group and derive the formula of the corresponding representation of the affine Kac-Moody algebra with level 1. And we also provide a concrete realization of Whittaker functionals in the dual representation.