Effective Field Theory for Chaotic CFTs

TL;DR

This work builds an effective field theory for chaotic two-dimensional CFTs at large central charge by promoting holomorphic and anti-holomorphic reparametrizations to soft modes that couple to external probes through bilocal vertices. The framework naturally exhibits pole skipping and yields Lorentzian propagators on Schwinger-Keldysh and higher-OTO contours, enabling computation of OTOCs that reveal a maximal Lyapunov exponent and light-speed butterfly velocity. It provides explicit calculations of 4-, 6-, and general 2k-point OTOCs, showing that higher-point functions scramble on longer timescales while maintaining the same exponential growth rate, with scrambling times scaling as t_*^{(k)} = (k-1) log c. The analysis ties conformal symmetry, SL(2,R) gauge redundancies, and an emergent entropy current to chaos in 2D CFTs and suggests avenues for real-time thermal physics, TTbar deformations, and potential extensions to higher dimensions and dissipative regimes.

Abstract

We derive an effective field theory for general chaotic two-dimensional conformal field theories with a large central charge. The theory is a specific and calculable instance of a more general framework recently proposed in [1]. We discuss the gauge symmetries of the model and how they relate to the Lyapunov behaviour of certain correlators. We calculate the out-of-time-ordered correlators diagnosing quantum chaos, as well as certain more fine-grained higher-point generalizations, using our Lorentzian effective field theory. We comment on potential future applications of the effective theory to real-time thermal physics and conformal field theory.

Figures (4)

• Figure 1: Contours in the complex $\omega$-plane, defining the retarded, advanced, and Keldysh (symmetric) correlation functions, respectively. The contour ${\cal C}^K = {\cal C}^R - {\cal C}^A$. The red crosses denote the poles of the integrand at $\omega \in \{-i,0\pm i \varepsilon,i\}$, where $\varepsilon>0$ is a small regulator that enforces consistent boundary conditions.
• Figure 2: We choose the $k$-OTO contour in the complex time plane to be such that all legs are equally separated by $\varphi$ in the imaginary direction. We often set either $\varphi=\frac{\pi}{k}$ (equal separations) or $\varphi = \delta \ll 1$ (small separation limit). The legs of the contour are labelled by indices $\alpha,\beta,\ldots=1,\ldots,2k$.
• Figure 3: Illustration of the 4- and 6-point out-of-time-ordered correlators that we consider.
• Figure 4: Illustration of the arrangement of operators in our $2k$-point observable $F_{2k}$, for the case $k=5$. It provides a generalization of the familiar 4-point OTOC, which has the property that the associated time of exponential growth (which we call as $k$-scrambling time $t_*^{(k)}$) is maximal: $t_*^{(k)} = (k-1) \log c$. The red solid lines indicate bilocal operators ${\cal B}_{(h,\bar{h})}$. The red wiggly lines schematically indicate the (maximally braided) propagation of the scrambling mode $\epsilon$.