Character identities between affine and Virasoro vertex operator algebra modules

Abstract The affine vertex operator algebras for $$\mathfrak {sl}_2$$ 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 $$L_k(\mathfrak {sl}_2)$$ L k ( sl 2 ) at admissible levels $$k=-2+q/p$$ k = - 2 + q / p to the rational ( q , 3 p )-minimal models $$L_\textrm{Vir}(c_{q,3p},0)$$ L Vir ( c q , 3 p , 0 ) , and also extends to the nonadmissible levels with $$q=1$$ q = 1 . Several special cases are particularly interesting. In the nonadmissible case $$q=1$$ q = 1 , the character identities extend to certain abelian intertwining algebras, specifically $$\mathcal {V}^{(p)}$$ V ( p ) and the doublet $$\smash {\mathcal {A}^{(3p)}}$$ A ( 3 p ) . Specialising further to $$p=2$$ p = 2 , where $$\smash {\mathcal {V}^{(2)}}$$ V ( 2 ) is the simple small $$\mathcal {N}=4$$ N = 4 superconformal algebra of central charge $$\smash {c=-9}$$