Semiclassical Quantum Gravity: Obtaining Manifolds from Graphs

Understand

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.