Bounding the Space of Holographic CFTs with Chaos

TL;DR

The paper investigates how the chaos bound $\lambda_L\le{2\pi\over \beta}$ constrains holographic CFTs and their AdS duals, showing that stress-tensor exchange alone enforces $\lambda_L=2\pi/\beta$ in typical Einstein-like holographic CFTs, while finite higher-spin towers in 2d CFTs lead to bound violations and acausality. It demonstrates that infinite higher-spin towers, as in $W_{\infty}[\lambda]$ theories (Vasiliev-like), yield $\lambda_L=0$ and are compatible with unitarity only in a restricted parameter range ($\lambda\le 2$), suggesting a tensionless-string interpretation of 3d Vasiliev theory. The results rule out unitary, sparse 2d CFTs with finite higher-spin towers (e.g., $SL(N)$ gravities) as duals to consistent bulk theories, while showing that disorderless higher-spin sectors can be causal when infinitely extended. Across dimensions, the work connects CFT OPE data, Regge behavior, and bulk dynamics to delineate the space of holographic theories, and points to tension between chaos, higher spins, and unitarity as a path toward selecting viable AdS/CFT pairs. The findings imply that chaos can serve as a diagnostic for bulk locality and string-like UV completions, and motivate further study of Reggeized OTO correlators in diverse CFTs and their bulk duals.

Abstract

Thermal states of quantum systems with many degrees of freedom are subject to a bound on the rate of onset of chaos, including a bound on the Lyapunov exponent, $λ_L\leq 2π/β$. We harness this bound to constrain the space of putative holographic CFTs and their would-be dual theories of AdS gravity. First, by studying out-of-time-order four-point functions, we discuss how $λ_L=2π/β$ in ordinary two-dimensional holographic CFTs is related to properties of the OPE at strong coupling. We then rule out the existence of unitary, sparse two-dimensional CFTs with large central charge and a set of higher spin currents of bounded spin; this implies the inconsistency of weakly coupled AdS$_3$ higher spin gravities without infinite towers of gauge fields, such as the $SL(N)$ theories. This fits naturally with the structure of higher-dimensional gravity, where finite towers of higher spin fields lead to acausality. On the other hand, unitary CFTs with classical $W_{\infty}[λ]$ symmetry, dual to 3D Vasiliev or hs[$λ$] higher spin gravities, do not violate the chaos bound, instead exhibiting no chaos: $λ_L=0$. Independently, we show that such theories violate unitarity for $|λ|>2$. These results encourage a tensionless string theory interpretation of the 3D Vasiliev theory. We also perform some CFT calculations of chaos in Rindler space in various dimensions.

Figures (7)

• Figure 1: A weakly coupled theory of higher spin gravity in AdS$_3$ may be viewed as matter coupled to a $G\times G$ Chern-Simons theory for some Lie algebra $G$. The boundary gravitons of $G$ generate an asymptotic ${\cal W}$-symmetry, ${\cal W}_G$. When rank($G$) is finite, such theories are inconsistent.
• Figure 2: On the left, the analytic continuation of $z$ onto the second sheet, passing clockwise around $z=1$. On the right, the canonical Shenker:2014cwa diagram of the out-of-time-order arrangement of operators along the Euclidean time circle. $W(t)$ undergoes Lorentzian time evolution orthogonal to the circle. We have placed the operators diametrically opposite from one another in pairs, as in \ref{['e26']}, with the two pairs separated by an angle $\theta$. In terms of the imaginary time $\tau$, $\theta = {\pi\over 2}-{2\pi\over\beta}\tau$.
• Figure 3: Functions of the form \ref{['testf']} violate the chaos bound in portions of the half-strip $0\leq \theta\leq\pi$. Here we take $n=2$ (and $\epsilon<0$), suppressing coefficients. The modulus of $f$ grows with time in the lower (red) sub-strip.
• Figure 4: The decomposition of the connected piece of $\langle VVWW\rangle$ into $s$-channel conformal blocks, to leading order in $1/c$. The first term stands for the exchanges of single-trace operators ${\cal O}$. In a typical holographic CFT, $L=\infty$, but the Regge limit corresponds to an exchange of effective spin $L_{\rm eff}=2$.
• Figure 5: At ${\cal O}(1/c)$, a ${\cal W}$-algebra vacuum block (left side) branches into a sum of global blocks for simple current exchanges. All composite operator exchanges are suppressed by higher powers of $1/c$.