# From Conformal Blocks to Path Integrals in the Vaidya Geometry

## TL;DR

This paper establishes a connection between conformal block sums in CFT and bulk path integrals in holography, demonstrating how off-shell particle paths encode complex saddles during black hole thermalization.

## Contribution

It reformulates conformal block expansions as bulk path integrals, revealing the role of off-shell trajectories in capturing complex saddle points in Lorentzian geometries.

## Key findings

- Off-shell worldlines correspond to subdominant conformal block contributions.
- Complex saddles dominate during thermalization, requiring summation over heavy operators.
- Conformal block sums can be interpreted as bulk path integrals in holographic duals.

## 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.