Conformal Bootstrap Approach to O(N) Fixed Points in Five Dimensions

Abstract

Whether O(N)-invariant conformal field theory exists in five dimensions with its implication to higher-spin holography was much debated. We find an affirmative result on this question by utilizing conformal bootstrap approach. In solving for the crossing symmetry condition, we propose a new approach based on specification for the low-lying spectrum distribution. We find the traditional one-gap bootstrapping is not suited since the nontrivial fixed point expected from large-N expansion sits at deep interior (not at boundary or kink) of allowed solution region. We propose two-gap bootstrapping that specifies scaling dimension of two lowest scalar operators. The approach carves out vast region of lower scaling dimensions and universally features two tips. We find that the sought-for nontrivial fixed point now sits at one of the tips, while the Gaussian fixed point sits at the other tip. The scaling dimensions of scalar operators fit well with expectation based on large-N expansion. We also find indication that the fixed point persist for lower values of N all the way down to N=1. This suggests that interacting unitary conformal field theory exists in five dimensions for all nonzero N.

Figures (9)

• Figure 1: Position of four insertion points of the sclar operators in (\ref{['setting']}). Using conformal symmetry, we fix $\vec{x_1}, \vec{x_3}, \vec{x_4}$. This leaves two degrees of freedom for the insertion point $\vec{x_2}$ lying in the $(x^1, x^D)$-subspace. The radial distance of $\vec{x_2}$ from origin is parametrized by $z,\bar{z}$. Therefore correlation function or conformal block is function of $z,\bar{z}$.
• Figure 2: The result of one-gap numerial boostrap for $N=500$. The colored region is the values scaling dimensions consistent with the unitarity and crossing symmetry. The ultraviolet fixed point predicted by the $1/N$-expansion lies at an interior of the region.
• Figure 3: Low-lying spectrum of one-gap approach traditionally used for $d<4$ versus two-gap approach we propose in this work. Left figure illustrates typical one-gap setup in bootstrap program. Right figure depicts our input of two-gap in the scalar operator spectrum. Above the unit operator, we have an isolated scalar operator of conformal scaling dimension $\Delta_{\rm min}$. All other operators of higher scaling dimension starts at $\Delta_{\rm gap}$.
• Figure 4: Result of two-gap approach for $N=500$ and $k=15$. Yellow-colored part is the allowed region, consistent with the unitarity and the crossing symmetry of 4-point correlation function. Compared to the one-gap approach result in Figure 2, the two-gap approach carves out regions of low values above the unitarity bound. Its boundary features two cusps. The ultraviolet nontrivial fixed point is located at its lower tip, while the infrared Gaussian fixed point is located at its upper tip.
• Figure 5: Result for $\Delta_{\rm{gap}}=8.0$. Here we zoomed in around near lower tip. From leftmost, each bound stands for $N=1000, N=500, N=300, N=200, N=100$, respectively. The star marks indicate location of perturbative $\frac{1}{N}$ expansion result for each $N$. For sufficiently large $N$, star mark location gradually approaches to boundary of allowed region.