Critical Sp(N ) models in 6 − ϵ dimensions and higher spin dS/CFT

Abstract

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 − ϵ dimensions at imaginary values of coupling constants. Using calculations up to three loop order, we develop ϵ 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 super-group OSp(1|2). This suggests the existence of OSp(1|2) symmetric CFTs in dimensions smaller than 6. We show that the 6 − ϵ expansions of the scaling dimensions and sphere free energy in our OSp(1|2) model are the same as in the q → 0 limit of the q-state Potts model.