From Conformal Blocks to Path Integrals in the Vaidya Geometry

TL;DR

The paper demonstrates how conformal-block expansions in a large-$c$ CFT undergoing a Vaidya quench mirror the bulk path integral in AdS$_3$-Vaidya, by recasting the block sum as a sum over bulk worldline configurations labeled by crossing points $x_c$. It shows that off-shell bulk worldlines correspond to subdominant conformal blocks and that Lorentzian thermalization requires summing over many channels, effectively introducing complexified OPE channels and heavy operator exchanges to reproduce the gravity result. Through a monodromy calculation of the vacuum block and a bulk Witten diagram analysis, the authors establish a precise matching between a boundary integral over channels $\int dx_c$ and the bulk geodesic/path integral, including complex saddles. This work clarifies how the bulk path integral emerges from a channel-summed CFT description in dynamical, non-equilibrium settings, with implications for thermalization and non-perturbative bulk reconstruction.

Abstract

Correlators in conformal field theory are naturally organized as a sum over conformal blocks. In holographic theories, this sum must reorganize into a path integral over bulk fields and geometries. We explore how these two sums are related in the case of a point particle moving in the background of a 3d collapsing black hole. The conformal block expansion is recast as a sum over paths of the first-quantized particle moving in the bulk geometry. Off-shell worldlines of the particle correspond to subdominant contributions in the Euclidean conformal block expansion, but these same operators must be included in order to correctly reproduce complex saddles in the Lorentzian theory. During thermalization, a complex saddle dominates under certain circumstances; in this case, the CFT correlator is not given by the Virasoro identity block in any channel, but can be recovered by summing heavy operators. This effectively converts the conformal block expansion in CFT from a sum over intermediate states to a sum over channels that mimics the bulk path integral.

Figures (8)

• Figure 1.1: Schematic representation of the setup and main result. In a collapsing black hole, the boundary conformal block expansion becomes a sum over channels labelled by a boundary point $x_c$. This corresponds semiclassically to a bulk geodesic crossing the shall of infalling matter at a point $(x_c,z_c(t))$ with $z_c$ the radial coordinate. Both in the CFT and in the bulk this crossing point takes on complex values, signaling that a complex saddle point dominates the bulk path integral, and no single channel dominates in CFT.
• Figure 2.1: Monodromy path $\Gamma$ labelled by the crossing point $x_c$.
• Figure 2.2: Lines where solutions to the critical equation (\ref{['eq:cond']}) merge with the real-$x_c$ axis. The solid blue line is the lightcone of $\mathcal{O}(t_1,x)$ defined by $-(t_2-t_1)^2+x^2=0$. The dashed orange line is a mirrored light cone defined by $-(t_2+t_1)^2+x^2=0$. The dotted green line is obtained by evaluating (\ref{['eq:t2sol']}) at $x_c=y_\star$ with $y_\star$ defined as the solution to (\ref{['eq:ystar']}). The horizontal red line defines the moment of the quench $t_2=0$.
• Figure 2.3: Saddle point solutions to (\ref{['eq:cond']}) for $x=0$ and fixed $t_1<0$. As $t_2$ increases from zero the number of real solutions goes from three to five to one. Arrows indicate the direction of movement of the saddles for increasing $t_2$.
• Figure 2.4: Saddle point solutions to (\ref{['eq:cond']}) for $0<x<-t_1$ and fixed $t_1<0$. We have picked a configuration where $|x+t_1|<t_c<|x-t_1|$ however the story is similar for any possible ordering.