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Feb 24 at 3:22 comment added Connor Malin @Pulcinella That's reasonable! I'm obviously not a mathematical physicist, but the question intrigued me, and even this much was actually new to me. Hopefully it is helpful to some others.
Feb 24 at 1:42 comment added Pulcinella I think this isn't quite a satisfying answer, especially given that already a lot of this is in the body of the question. I imagine nowadays there's a cleaner way of expressing an answer, which is part of the point of my question. For instance: in the paper there's not a clear explanation of why comparing these two $P_2$ structures gives a h-adic quantisation that e.g. would make it clear how to generalise to other operads, at least $\mathbf{E}_n$, it's not clear how it relates to Kontsevich's Swiss cheese quantisation, etc.
Feb 23 at 19:01 history answered Connor Malin CC BY-SA 4.0