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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.

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    $\begingroup$ Interesting higher category theory in the early-mid 90s was somewhat inspired by LQG or precursors, eg work by John Baez. $\endgroup$
    – David Roberts ♦
    Commented Mar 29, 2015 at 15:45

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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.

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One topic which started in the general area of LQG but now has taken a life of its own is the theory random tensors and group field theories. For its connections to mathematics you can have a look at the talks and courses from the program I co-organized last summer in Vienna: http://www.mat.univie.ac.at/~kratt/esi3/

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