Quantum Chaos in Liouville CFT

TL;DR

This paper investigates quantum chaos and scrambling in Liouville conformal field theory, revealing maximal Lyapunov exponent behavior and the transition of dominant conformal blocks over time, with implications for holography.

Contribution

It provides a detailed analysis of chaos in Liouville CFT, deriving explicit four-point functions, and connecting path integral and conformal block approaches to understand scrambling.

Findings

Liouville theory exhibits a maximal Lyapunov exponent.

Scrambling causes a shift from heavy to light primary dominance.

Explicit form of the Euclidean four-point function derived.

Abstract

Fast scrambling is a distinctive feature of quantum gravity, which by means of holography is closely tied to the behaviour of large$-c$ conformal field theories. We study this phenomenon in the context of semiclassical Liouville theory, providing both insights into the mechanism of scrambling in CFTs and into the structure of Liouville theory, finding that it exhibits a maximal Lyapunov exponent despite not featuring the identity in its spectrum. However, as we show, the states contributing to the relevant correlation function can be thought of as dressed scramblons. At a technical level we we first use the path integral picture in order to derive the Euclidean four-point function in an explicit compact form. Next, we demonstrate its equivalence to a conformal block expansion, revealing an explicit but non-local map between path integral saddles and conformal blocks. By analytically continuing both expressions to Lorentzian times, we obtain two equivalent formulations of the OTOC, which we use to study the onset of chaos in Liouville theory. We take advantage of the compact form in order to extract a Lyapunov exponent and a scrambling time. From the conformal block expansion formulation of the OTOC we learn that scrambling shifts the dominance of conformal blocks from heavy primaries at early times to the lightest primary at late times. Finally, we discuss our results in the context of holography.