Exploring Quantum Spacetime with Topological Data Analysis

TL;DR

The paper proposes a nonperturbative quantum gravity probe based on topological data analysis by defining effective homology from coarse-grained triangulations of dynamical geometries. It develops a concrete coarse-graining pipeline (S_δ Voronoi-Delaunay) and measures Betti numbers $\langle\beta_i(\delta)\rangle$, connecting their scale dependence to the foam-like fractal structure of 2D Euclidean quantum gravity. The key finding is that $\langle\beta_2(\delta)\rangle$ exhibits pronounced short-scale topological features that diminish with $\delta$ and that the bubble distribution yields a string susceptibility $\gamma_{\mathrm{str}}$ near $-1/2$, matching universal gravity results. This establishes a quantitative framework to assess quantum spacetime topology and suggests the method can be extended to higher dimensions and Lorentzian settings, enabling new insights into quantum spacetime foam. The work leverages GUDHI and demonstrates a principled bridge between TDA techniques and nonperturbative quantum gravity observables.

Abstract

In a novel application of the tools of topological data analysis (TDA) to nonperturbative quantum gravity, we introduce a new class of observables that allows us to assess whether quantum spacetime really resembles a ``quantum foam" near the Planck scale. The key idea is to investigate the Betti numbers of coarse-grained path integral histories, regularized in terms of dynamical triangulations, as a function of the coarse-graining scale. In two dimensions our analysis exhibits the well-known fractal structure of Euclidean quantum gravity.

Figures (4)

• Figure 1: Typical configuration contributing to the Euclidean path integral (\ref{['pi2dedt']}) over 2D dynamical triangulations, for $N_2=100k$.
• Figure 2: Voronoi decomposition with resolution $\delta=8$ of a typical configuration, with $N_2=10k$. Thin black lines are those of the original triangulation $T$.
• Figure 3: Expectation value of the Betti number $\beta_2$, as a function of the resolution $\delta\in [2,53]$ and for $N_2=200k$.
• Figure 4: Expectation value of the abundance of bubbles of volume $n$, for $\delta=2$ and $N_2=50k$, together with a best fit for the string susceptibility.