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How valuable for physics and math would be a description of a quantum field theory of non-abelian gerbes? If one could derive the instanton-part of the AGT-correspondence as a low-energy duality from such a theory would that be a sufficient prove?
I don't know much about the topic so I won't make an answer, but I remember there being references to bundle gerbes for nonassociative quantum mechanics and string theories. Richard Szabo's papers on the geometric quantization of nonassociative theories might help, though I don't know if he talks about non-abelian gerbes specifically. A search for non-abelian gerbes in physics leads me to a paper by Schweigert and Waldorf: https://arxiv.org/abs/1807.05398v1
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