Quantum Gravity and Effective Topology

TL;DR

The paper tackles how to characterize quantum spacetime in regimes dominated by large fluctuations, by applying topological data analysis to coarse-grained dynamical triangulations. It introduces a coarse-graining scheme at scale $\delta$ to compute Betti numbers $\beta_k(\delta)$, yielding an effective topology fingerprint that probes the presence or absence of topological obstructions to symmetry in the classical limit. Applying this to 2D quantum gravity models, it reveals distinct behavior: Lorentzian CDT exhibits global pinchings affecting $\beta_1$ and $\beta_2$ only at large $\delta$, while Euclidean EDT shows local pinchings with bubble generation influencing $\beta_2$ even at small $\delta$, consistent with its fractal structure. The approach provides a nonlocal, diffeomorphism-invariant observable framework that can be extended to higher dimensions, offering a new angle on how quantum spacetime might recover classical symmetries at coarse scales.

Abstract

We introduce a new methodology to characterize properties of quantum spacetime in a strongly quantum-fluctuating regime, using tools from topological data analysis. Starting from a microscopic quantum geometry, generated nonperturbatively in terms of dynamical triangulations (DT), we compute the Betti numbers of a sequence of coarse-grained versions of the geometry as a function of the coarse-graining scale, yielding a characteristic ``topological finger print". We successfully implement this methodology in Lorentzian and Euclidean 2D quantum gravity, defined via lattice quantum gravity based on causal and Euclidean DT, yielding different results. Effective topology also enables us to formulate necessary conditions for the recovery of spacetime symmetries in a classical limit.

Figures (27)

• Figure 1: Shape of a typical path integral configuration in 4D CDT lattice quantum gravity, illustrating the macroscopically emergent $S^4$-topology. The curve $V_3(\tau)$ has been made into a rotational body about the horizontal time axis $\tau$.
• Figure 2: Illustrating the covering algorithm: (a) initial vertex $v_0$, together with its $\delta$-ball $B_\delta (v_0)$ (red) and $\delta$-annulus $A_\delta (v_0)$ (blue); (b) random vertex $u$ from the part of the annulus not yet covered, together with its $\delta$-ball $B_\delta (u)$ (pink); (c) after several iterations, all vertices in $A_\delta (v_0)$ have been covered; (d) algorithm continues with a random vertex $v\in {\cal S}_{new}$ whose annulus is not yet covered.
• Figure 3: Triangles can have three types of colouring, depending on the colouring of their vertices, generated during the decomposition of vertices into Voronoi cells.
• Figure 4: Voronoi decomposition into cells of a 2D CDT configuration on a torus, for $\delta =3$, using a Tutte or barycentric embedding, where each vertex is located at the barycentre of its neighbours. Dashed and solid lines represent space- and timelike edges respectively, and each black dot is a seed vertex for a cell of a given colour. Note the toroidal periodicity for opposite sides of the rectangle.
• Figure 5: Voronoi cell associated with a seed vertex (in red), with the topology of an annulus (shaded region), wrapping around a thin neck of the triangulation and shown in a three-dimensional embedding.