Pole-skipping of scalar and vector fields in hyperbolic space: conformal blocks and holography

TL;DR

The paper identifies and cross-checks pole-skipping points for scalar and vector two-point functions in hyperbolic space using field-theoretic, holographic, and near-horizon analyses. It reveals a robust link between the leading pole-skipping data and the late-time behavior of conformal blocks (and their shadows) in four-point OTOCs, encapsulated by exponents that generalize the Lyapunov/butterfly concepts to conformal-block exchanges. The results hold universally across dimensions and are validated by multiple independent methods, including exact bulk Green's functions in Rindler-AdS and near-horizon limits. This work thus provides a unifying perspective on how pole-skipping encodes information about chaos-related dynamics and conformal-block structure in holographic CFTs, with clear avenues for extension to higher-spin and non-conformal theories.

Abstract

Motivated by the recent connection between pole-skipping phenomena of two point functions and four point out-of-time-order correlators (OTOCs), we study the pole structure of thermal two-point functions in $d$-dimensional conformal field theories (CFTs) in hyperbolic space. We derive the pole-skipping points of two-point functions of scalar and vector fields by three methods (one field theoretic and two holographic methods) and confirm that they agree. We show that the leading pole-skipping point of two point functions is related with the late time behavior of conformal blocks and shadow conformal blocks in four-point OTOCs.

Figures (2)

• Figure 1: Plot of $v$ with $\mathbf{d}=10,\ \delta_1=0.6, \ \delta_3=0.4, \ \delta_2=0.2, \ \delta_4=0$. The horizontal axis is ${ \rm Re}[v]$, and the vertical axis is ${ \rm Im}[v]$. We plot $v$ from $t=8$ to $t=12$.
• Figure 2: $\log|\mathcal{G}^\Delta(\omega_E,k)|$: logarithm of the Fourier transformed conformal two point functions of scalar fields \ref{['fourierscalar2pt']} for $d=4,\,\Delta=4.5$. The blue lines represent the zeros and the red lines represent the poles of the two point function so that the intersections between the red and blue lines are pole-skipping points, which are marked as white stars (leading points) and circles (sub-leading points). The numerical values of these points agree with the \ref{['sps']}.