Undecidability in Spacetime Geometry via the AdS/CFT Correspondence

TL;DR

This paper shows that in certain holographic models, the question of which spacetime geometry emerges from quantum gravity is undecidable, linking logical limitations to gravitational theories via the AdS/CFT correspondence.

Contribution

It demonstrates how undecidability in quantum many-body physics can be holographically transmitted to gravitational theories, revealing limits of computability in spacetime geometry selection.

Findings

Undecidability can be embedded into large-N gauge theories.

The dominant bulk saddle choice becomes undecidable.

Determining the emergent spacetime geometry can be beyond computability.

Abstract

Undecidability, a hallmark of G\"odel incompleteness theorems, has recently emerged in quantum many-body physics through the spectral gap problem. We demonstrate how this logical limitation can be holographically transmitted to a class of gravitational theories via the AdS/CFT correspondence. By embedding a translationally invariant spin Hamiltonian with undecidable gap status into a large-N gauge theory, we generate an AdS dual in which the selection of dominant bulk saddle (Poincar\'e AdS or AdS soliton) is itself undecidable. Consequently, under standard semiclassical holographic assumptions, even determining which smooth spacetime geometry emerges from quantum gravity can be beyond the limits of computability.