Regge's Inferno

Abstract

We study large-spin operators in conformal field theories (CFTs) in spacetime dimensions $d>2$ by placing the theory on appropriate pp-wave backgrounds. We show that these geometries admit Heisenberg-group symmetries, and that these symmetries, combined with locality of quantum fields on such spacetimes, impose strong constraints on the asymptotic spectrum in the large-spin limit. The pp-wave backgrounds probe both the small-twist regime, corresponding to the Regge or light-cone bootstrap, and a strongly coupled regime of large twist. Finally, we demonstrate that causality (or the requirement that the energy be bounded from below) leads to a new unitarity bound in $3+1$ dimensions.

Figures (4)

• Figure 1: A cartoon of different regimes in $(2+1)$-dimensional CFTs in the $(\tau, J_z)$ plane. Numerical bootstrap techniques are effective in probing the low-twist, low-spin regime, while analytic bootstrap methods (the Lightcone bootstrap) capture the large-spin sector at small twist. The pp wave geometry developed in this work captures operators with large spin and arbitrarily large twist with $\tau/\sqrt{J_z}$ fixed at large $J_z$.
• Figure 2: Highly boosted states correspond to wavefunctions that localize near the equator of $S^2$, within a narrow strip of angular width $\delta\theta\sim J_z^{-\frac{1}{2}}$.
• Figure 3: Exact $F_{3d}(\beta)$ for the free scalar theory (black) compared with its low-temperature (red) and high-temperature (blue) asymptotics. We emphasize that this plot is meant only as an illustrative free-theory example: in interacting CFTs the interpolation between the two regimes can be more intricate, and may involve additional structure---or even phase transitions.
• Figure 4: Half-infinite slab: At $\epsilon=0$ and $\epsilon =2$, $\log Z$ develops a pole and the residue of the pole, as a function of $\beta$ may be non-analytic.