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Theories of anti-commuting scalar fields are non-unitary, but they are ofinterest both in statistical mechanics and in studies of the higher spin deSitter/Conformal Field Theory correspondence. We consider an $Sp(N)$ invarianttheory of $N$ anti-commuting scalars and one commuting scalar, which has cubicinteractions and is renormalizable in 6 dimensions. For any even $N$ we find anIR stable fixed point in $6-\epsilon$ dimensions at imaginary values ofcoupling constants. Using calculations up to three loop order, we develop$\epsilon$ expansions for several operator dimensions and for the sphere freeenergy $F$. The conjectured $F$-theorem is obeyed in spite of the non-unitarityof the theory. The $1/N$ expansion in the $Sp(N)$ theory is related to that inthe corresponding $O(N)$ symmetric theory by the change of sign of $N$. Ourresults point to the existence of interacting non-unitary 5-dimensional CFTswith $Sp(N)$ symmetry, where operator dimensions are real. We conjecture thatthese CFTs are dual to the minimal higher spin theory in 6-dimensional deSitter space with Neumann future boundary conditions on the scalar field. For$N=2$ we show that the IR fixed point possesses an enhanced global symmetrygiven by the supergroup $OSp(1|2)$. This suggests the existence of $OSp(1|2)$symmetric CFTs in dimensions smaller than 6. We show that the $6-\epsilon$expansions of the scaling dimensions and sphere free energy in our $OSp(1|2)$model are the same as in the $q \rightarrow 0$ limit of the $q$-state Pottsmodel.

Critical $Sp(N)$ Models in $6-\epsilon$ Dimensions and Higher Spin dS/CFT