A view of the bulk from the worldline

TL;DR

The paper develops a worldline quantum-mechanics approach to AdS/CFT that reorganizes bulk dynamics through a two-parameter perturbative expansion in $G_N$ and the inverse mass $m^{-1}$, offering a complementary perspective to Witten diagrams. It shows that leading gravitational exchanges reproduce global conformal blocks via geodesic Witten diagrams, and, in AdS$_3$, yield the Virasoro block with systematic quantum corrections that exponentiate to encode higher-loop information. The framework is built by detailing gauge-fixed worldline path integrals, boundary conditions at infinity, and a vertex-operator formalism for coupling to gravity, then applied to four-point functions to demonstrate block structure, exponentiation, and higher-order saddles. The authors discuss extensions to finite temperature, higher-point blocks, OPE blocks, Lorentzian kinematics, and connections to Chern-Simons formulations and conformal bootstrap, highlighting the potential of bulk worldline methods to illuminate quantum gravity and the bulk/boundary dictionary.

Abstract

A new method to compute correlation functions in AdS$_{d+1}$ in general dimension is introduced, considering a particle quantised in the worldline formalism of quantum field theory, coupled to bulk fields, in particular gravity, quantised in the standard manner. This gives a systematic two-parameter perturbative expansion organised by small parameters $G_N$ and the inverse mass of the particle, complementary to the usual Witten diagram expansion. This connects closely to CFT language, with the geodesic Witten diagram representation of global conformal blocks emerging naturally, and for two dimensions gives a bulk representation of the Virasoro block, with a systematic method for computing quantum corrections. The global conformal block and other contributions are shown to exponentiate in correlation functions in any dimension, corresponding to pieces of Witten diagrams at arbitrary loop order.

Figures (2)

• Figure 1: A two-loop, two graviton diagram, of order $m G_N^2$. The dotted lines indicate the locations of the unperturbed geodesics for the two particles, with an independent quantum mechanics living on each of these. This diagram contributes to $\langle \langle\mathcal{V}_h\mathcal{V}_h\rangle_{WL1}\langle\mathcal{V}_h\rangle_{WL2}\rangle_\text{gravity}$, with a one-loop contribution to the two-point function of the graviton vertex operator on the first worldline, the tree-level one-point function of the vertex operator on the second, and a gravity three-point function of $h$ to tie them all together.
• Figure 2: The two diagrams appearing at tree-level, two-graviton order, $m^3G_N^2$.