  We study the vacuum structure of spin-3 higher-spin supergravity in AdS_3
spacetime. The theory can be written as a Chern-Simons theory based on the Lie
superalgebra sl(3|2). We find three distinct AdS_3 vacua, AdS^(1), AdS^(2) and
AdS^(p), each corresponding to one embedding of the osp(1|2) subalgebra into
the sl(3|2) algebra. We explicitly construct the RG flows from AdS^(1) to
AdS^(p) and from AdS^(2) to AdS^(p), which identifies AdS^(p) as an IR vacuum
and AdS^(1), AdS^(2) as two different UV vacua. Thus a duality is found between
the two UV theories in the sense that the two theories, each with a chemical
potential turned on, flow to the same IR theory. Moreover, we identify a
similar structure in the Hamiltonian reductions of the 2d Wess-Zumino-Witten
(WZW) model with sl(3|2)-valued currents by matching the chiral symmetries
there with the asymptotic symmetries of the three different embeddings. Our
computation gives an RG interpretation of (certain types of) the Hamiltonian
reductions. In addition, it gives a hint of a duality between the 3d
higher-spin supergravity and some conformally extended super-Toda theory as
suggested by Mansfield and Spence for the bosonic case.
