[Math] Interplay between Loop Quantum Gravity and Mathematics

Question

It is known that there are many interesting connections between String Theory and modern Mathematics, with a rich feedback going on in both directions: there have been advances in mathematics thanks to research in String Theory and vice-versa. For the interested reader, some of the interactions between String Theory and Mathematics have been reviewed in

http://rsta.royalsocietypublishing.org/content/roypta/368/1914/913.full.pdf

by Atiyah, Dijkgraaf and Hitchin. However, String Theory is not the only quantum theory of gravity on the market: it has a much less popular competitor called Loop Quantum Gravity. My question is, are there any interesting interactions between mathematics and LQG? Has there been any advances in mathematics thanks to LQG research, as it has happened with ST?

Thanks.

Answer 1

Here is one area of interplay between loop quantum gravity and combinatorics:

Oriented Matroids – Combinatorial Structures Underlying Loop Quantum Gravity, Brunnemann and Rideout (2010).

We analyze combinatorial structures which play a central role in determining spectral properties of the volume operator in loop quantum gravity (LQG). We demonstrate that these structures are a special representation of the general mathematical concept of an oriented matroid. This opens up a new mathematical arena to study LQG, and to overcome certain technical and conceptual obstacles which, at present, make concrete analytical and numerical investigations of the full theory very difficult. In particular it becomes feasible to develop a systematic and unified treatment of global (connectedness) and local (embedding) properties of the graph poset underlying LQG.