Five dimensional $O(N)$-symmetric CFTs from conformal bootstrap

TL;DR

The paper investigates $O(N)$-symmetric CFTs in $d=5$ via the conformal bootstrap, emphasizing lower bounds on the current central charge $C_J$ and their relation to a proposed UV fixed point. It contrasts large-$N$ predictions for operator dimensions $\\Delta_\
$, $
$ and central charges $C_T$, $C_J$ with bootstrap bounds, and benchmarks results against analogous $d=3$ analyses to gauge finite-$N$ effects. The main finding is that, in the large-$N$ limit, the $C_J$ bounds agree with UV fixed point expectations, while for finite $N$ there are discrepancies and no clear evidence for a conformal window; varying assumptions on spin-0 gaps does not decisively reveal a saturating CFT. The work outlines possible interpretations and calls for further exploration of higher-spin data and AdS$_6$/CFT$_5$ connections, as well as cross-checks with $b psilon$-expansion resummations.

Abstract

We investigate the conformal bootstrap approach to $O(N)$ symmetric CFTs in five dimension with particular emphasis on the lower bound on the current central charge. The bound has a local minimum for all $N>1$, and in the large $N$ limit we propose that the minimum is saturated by the critical $O(N)$ vector model at the UV fixed point, the existence of which has been recently argued by Fei, Giombi, and Klebanov. The location of the minimum is generically different from the minimum of the lower bound of the energy-momentum tensor central charge when it exists for smaller $N$. To better understand the situation, we examine the lower bounds of the current central charge of $O(N)$ symmetric CFTs in three dimension to compare. We find the similar agreement in the large $N$ limit but the discrepancy for smaller $N$ with the other sectors of the conformal bootstrap.

Figures (6)

• Figure 1: The lower bounds of the current central charge $C_J ^c (\Delta _\phi)$ for $d=5$$O(N)$ symmetric CFTs with $N= 200, 500,1000$. The dots are the large $N$ predictions of the $(\Delta _\phi ,C_J )$ in (\ref{['eq:largeN-Deltaphi']}) and (\ref{['eq:largeN-CurrentCC']}).
• Figure 2: The lower bounds of the current central charge $C_J ^c (\Delta _\phi)/C_J ^{\mathrm{free}}$ for $d=5$$O(N)$ symmetric CFTs with $N= 20, 35,50, 100$. The dots are the large $N$ predictions of the $\Delta _\phi$.
• Figure 3: $C_{J,T} ^c (\Delta _\phi)/C_{J,T} ^{\mathrm{free}}$ for $d=5$$O(N)$ symmetric CFTs with $N= 2, 3,5, 10$. The solid lines are the lower bounds of the current central charge while the dashed lines are those of the energy-momentum tensor central charge with the corresponding colors.
• Figure 4: $C_J ^c (\Delta _\phi)/C_J ^{\mathrm{free}}$ for $d=5$$O(2)$ symmetric CFTs obtained by assuming $\Delta_{S,T} \ge \Delta_0$ with $\Delta _0$ running over $1.5, 1.65, 1.8, 2, 2.2, 2.4, 2.6$.
• Figure 5: $C_J ^c (\Delta _\phi)/C_J ^{\mathrm{free}}$ for $d=3$$O(N)$ symmetric CFTs with $N= 2, 3,5,9,10,20,40$. The bounds are completely general -- i.e. no assumption other than the unitarity bound is made. The dots are the large $N$ predictions of $(\Delta_\phi , C_J)$ for $N=40, 20,10$ critical vector models from the left.