Holographic One-Point Function and Geodesics in SdS$_3$

Abstract

Grinberg and Maldacena showed that heavy thermal one-point functions in AdS/CFT can encode complex geodesics reaching a black hole singularity. We study the de Sitter analogue in three-dimensional Schwarzschild--de Sitter space, restricting to the finite cyclic quotients $\mathrm{dS}_3/\mathbb Z_q$. We define a one-point function through a differentiate dictionary and show that an analogous result holds for the Bunch--Davies-prepared integral. Using the exact light field Green function, the bulk integral reduces exactly to a convergent one-dimensional transform, whose large-mass limit is controlled by the near-defect region and governed by a complex saddle. Its action reproduces a complex geodesic length from future infinity to the conical defect, whose real part is the renormalized boundary-to-horizon proper time and whose imaginary part is the horizon-to-defect distance. The imaginary part appears in the one-point coefficient normalized in the Lorentzian source basis; in the Euclidean basis the same saddle contributes only the real part, the two being related by a fixed connection factor.