Semiclassical Quantum Gravity: Obtaining Manifolds from Graphs

TL;DR

This paper develops a method to determine when a discrete graph structure can be associated with a unique manifold, specifically in the context of quantum gravity, and provides a procedure to construct such manifolds from certain graphs.

Contribution

It introduces a procedure to construct PL-manifolds from fixed-valence graphs relevant to semiclassical loop quantum gravity, addressing the inverse problem in discrete geometry.

Findings

The procedure successfully constructs manifolds from certain graphs.

Graphs not completing the procedure relate to non-manifold structures or small-scale features.

Extension of the method to more general graphs is briefly discussed.

Abstract

We address the "inverse problem" for discrete geometry, which consists in determining whether, given a discrete structure of a type that does not in general imply geometrical information or even a topology, one can associate with it a unique manifold in an appropriate sense, and constructing the manifold when it exists. This problem arises in a variety of approaches to quantum gravity that assume a discrete structure at the fundamental level; the present work is motivated by the semiclassical sector of loop quantum gravity, so we will take the discrete structure to be a graph and the manifold to be a spatial slice in spacetime. We identify a class of graphs, those whose vertices have a fixed valence, for which such a construction can be specified. We define a procedure designed to produce a cell complex from a graph and show that, for graphs with which it can be carried out to completion, the resulting cell complex is in fact a PL-manifold. Graphs of our class for which the procedure cannot be completed either do not arise as edge graphs of manifold cell decompositions, or can be seen as cell decompositions of manifolds with structure at small scales (in terms of the cell spacing). We also comment briefly on how one can extend our procedure to more general graphs.