Conserved Topological Defects in Non-Embedded Graphs in Quantum Gravity

TL;DR

This paper investigates conserved topological quantities in non-embedded graphs within quantum gravity, revealing new invariants related to topological defects across different graph evolution rules.

Contribution

It identifies conserved quantities in non-embedded graphs under various dynamics, extending previous work on embedded graphs and topological invariants.

Findings

Conserved quantities expressed as topological defects.

Complete set of invariants for graphs dual to 2D complexes.

Expected invariants for graphs dual to 3D complexes.

Abstract

We follow up on previous work which found that commonly used graph evolution moves lead to conserved quantities that can be expressed in terms of the braiding of the graph in its embedding space. We study non-embedded graphs under three distinct sets of dynamical rules and find non-trivial conserved quantities that can be expressed in terms of topological defects in the dual geometry. For graphs dual to 2-dimensional simplicial complexes we identify all the conserved quantities of the evolution. We also indicate expected results for graphs dual to 3-dimensional simplicial complexes.