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

TL;DR

This work constructs a non-unitary $Sp(N)$-invariant cubic theory of $N$ anti-commuting scalars and one commuting scalar, renormalizable in $6$ dimensions, and demonstrates IR fixed points at imaginary couplings in $6-\epsilon$. By three-loop calculations, it provides $\epsilon$-expansions for operator dimensions and the sphere free energy $\tilde{F}$, finding that the $F$-theorem holds despite non-unitarity and that the $1/N$ expansion maps to the corresponding $O(N)$ theory via $N \to -N$. The analysis identifies interacting non-unitary five-dimensional CFTs with real operator dimensions and proposes a duality to a minimal higher-spin theory in $dS_6$ with Neumann boundary conditions; for $N=2$ the fixed point exhibits $OSp(1|2)$ symmetry, suggesting a family of $OSp(1|2)$-symmetric CFTs below six dimensions that match the $q \to 0$ limit of the $q$-state Potts model. The results extend the landscape of higher-spin dS/CFT dualities to higher dimensions and link non-unitary field theories to unitary higher-spin gravity in de Sitter space.

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 supergroup $OSp(1|2)$. 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 $q \rightarrow 0$ limit of the $q$-state Potts model.

Figures (3)

• Figure 1: The zeroes of the one loop $\beta$ functions and the RG flow directions for the $OSp(1|2)$ model. The coordinates are defined via $g_1 = i\sqrt{\frac{(4\pi)^3\epsilon}{5}} x$, $g_2 = i\sqrt{\frac{(4\pi)^3\epsilon}{5}} y$, and the red dots correspond to the stable IR fixed points.
• Figure 2: Diagrams contributing to the mixing of $\sigma^2$ and $\Omega_{ij}\chi^i\chi^j$ operators to two loop order.
• Figure 3: Different Padé approximations of $\Delta_{\chi}$ for the $N=2$ model.