Oriented Matroids -- Combinatorial Structures Underlying Loop Quantum Gravity

Understand

We analyze combinatorial structures which play a central role in determining spectral properties of the volume operator in loop quantum gravity (LQG).

• These structures encode geometrical information of the embedding of arbitrary valence vertices of a graph in 3-dimensional Riemannian space, and can be represented by sign strings containing relative orientations of embedded edges.
• We demonstrate that these signature factors are a special representation of the general mathematical concept of an oriented matroid.
• Moreover, we show that oriented matroids can also be used to describe the topology (connectedness) of directed graphs.