Duality in N=2 minimal model holography

Recently a duality between a family of \mathcal{N}=2 supersymmetric higher spin theories on AdS3, and the 't Hooft like limit of a class of Kazama-Suzuki models (that are parametrised by N and k) was proposed. The higher spin theories can be described by a Chern-Simons theory based on the infinite-dimensional Lie algebra shs[\mu], and under the duality, \mu is to be identified with \lambda=N/(N+k+1). Here we elucidate the structure of the (quantum) asymptotic symmetry algebra sW_{\infty}[\mu] for arbitrary \mu and central charge c. In particular, we show that for each value of the central charge, there are generically four different values of \mu that describe the same sW_{\infty} algebra. Among other things this proves that the quantum symmetries on both sides of the duality agree; this equivalence does not just hold in the 't Hooft limit, but even at finite N and k.