The AdS$_3$ Propagator and the Fate of Locality

TL;DR

This paper investigates the bulk propagator in AdS$_3$ using conformal symmetry techniques, revealing UV/IR mixing and a breakdown of locality at short distances due to quantum gravitational effects.

Contribution

It extends conformal block methods to compute the AdS$_3$ propagator and analyzes how quantum gravity effects lead to locality breakdown at small scales.

Findings

Propagator matches gravitational perturbation theory at large central charge.

UV/IR mixing effects appear in the perturbative expansion.

Bulk locality breaks down at a new short-distance scale due to quantum effects.

Abstract

We recently used Virasoro symmetry considerations to propose an exact formula for a bulk proto-field $\phi$ in AdS$_3$. In this paper we study the propagator $\langle \phi \phi \rangle$. We show that many techniques from the study of conformal blocks can be generalized to compute it, including the semiclassical monodromy method and both forms of the Zamolodchikov recursion relations. When the results from recursion are expanded at large central charge, they match gravitational perturbation theory for a free scalar field coupled to gravity in our chosen gauge. We find that although the propagator is finite and well-defined at long distances, its perturbative expansion in $G_N = \frac{3}{2c}$ exhibits UV/IR mixing effects. If we nevertheless interpret $\langle \phi \phi \rangle$ as a probe of bulk locality, then when $G_N m_\phi \ll 1$ locality breaks down at the new short-distance scale $\sigma_* \sim \sqrt[4]{G_N R_{AdS}^3}$. For $\phi$ with very large bulk mass, or at small central charge, bulk locality fails at the AdS length scale. In all cases, locality `breakdown' manifests as singularities or branch cuts at spacelike separation arising from non-perturbative quantum gravitational effects.