Character Identities Between Affine and Virasoro Vertex Operator Algebra Modules

The affine vertex operator algebras for $\mathfrak{sl}_2$ and the Virasoro minimal models are related by Drinfeld-Sokolov reduction and by the Goddard-Kent-Olive coset construction. In this work, we propose another connection based on certain character identities between these vertex operator algebras and their modules. This relates the simple affine vertex operator algebras $L_k(\mathfrak{sl}_2)$ at admissible levels $k=-2+q/p$ to the rational $(q,3p)$-minimal models $L_\mathrm{Vir}(c_{q,3p},0)$, and also extends to the nonadmissible levels with $q=1$. Several special cases are particularly interesting. In the nonadmissible case $q=1$, the character identities extend to certain abelian intertwining algebras, specifically $\mathcal{V}^{(p)}$ and the doublet $\mathcal{A}^{(3p)}$. Specialising further to $p=2$, where $\mathcal{V}^{(2)}$ is the simple small $\mathcal{N}=4$ superconformal algebra of central charge $c=-9$, this recovers, via the 4d/2d-correspondence, a known identity between the Schur indices of the 4d $\mathcal{N}=4$ supersymmetric Yang-Mills theory for $\mathrm{SU}(2)$ and the 4d $\mathcal{N}=2$ $(3,2)$ Argyres-Douglas theory. In the boundary admissible case $q=2$, in a similar vein, we obtain an identity between the Schur indices of 4d $\mathcal{N}=2$ Argyres-Douglas theories of types $(A_1,D_{2n+1})$ and $(A_1,A_{6n})$. On the other hand, for integral levels, $p=1$, where both involved vertex operator algebras are strongly rational, our character identity induces a Galois conjugation between the representation categories $\mathrm{Rep}(L_{-2+q}(\mathfrak{sl}_2))$ and $\mathrm{Rep}(L_\mathrm{Vir}(c_{q,3},0))$. We conjecture that the characters are related by the action of certain Hecke operators. Finally, we also sketch how to extend the results of this paper to relaxed highest-weight and Whittaker modules.