  Theories of anti-commuting scalar fields are non-unitary, but they are of
interest both in statistical mechanics and in studies of the higher spin de
Sitter/Conformal Field Theory correspondence. We consider an $Sp(N)$ invariant
theory of $N$ anti-commuting scalars and one commuting scalar, which has cubic
interactions and is renormalizable in 6 dimensions. For any even $N$ we find an
IR stable fixed point in $6-\epsilon$ dimensions at imaginary values of
coupling constants. Using calculations up to three loop order, we develop
$\epsilon$ expansions for several operator dimensions and for the sphere free
energy $F$. The conjectured $F$-theorem is obeyed in spite of the non-unitarity
of the theory. The $1/N$ expansion in the $Sp(N)$ theory is related to that in
the corresponding $O(N)$ symmetric theory by the change of sign of $N$. Our
results point to the existence of interacting non-unitary 5-dimensional CFTs
with $Sp(N)$ symmetry, where operator dimensions are real. We conjecture that
these CFTs are dual to the minimal higher spin theory in 6-dimensional de
Sitter space with Neumann future boundary conditions on the scalar field. For
$N=2$ we show that the IR fixed point possesses an enhanced global symmetry
given 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 Potts
model.
