  Sigma models on coset superspaces, such as odd dimensional superspheres, play
an important role in physics and in particular the AdS/CFT correspondence. In
this work we apply recent general results on the spectrum of coset space models
and on supergroup WZNW models to study the conformal sigma model with target
space S^{3|2}. We construct its vertex operators and provide explicit formulas
for their anomalous dimensions, at least to leading order in the sigma model
coupling. The results are used to revisit a non-perturbative duality between
the supersphere and the OSP(4|2) Gross-Neveu model that was conjectured by
Candu and Saleur. With the help of powerful all-loop results for 1/2 BPS
operators in the Gross-Neveu model we are able to recover the entire zero mode
spectrum of the sigma model at a certain finite value of the Gross-Neveu
coupling. In addition, we argue that the sigma model constraints and equations
of motion are implemented correctly in the dual Gross-Neveu description. On the
other hand, high(er) gradient operators of the sigma model are not all
accounted for. It is possible that this discrepancy is related to an
instability from high gradient operators that has previously been observed in
the context of Anderson localization.
