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You may recall that we have been thinking about extended geometric pre-quantization a good bit. So far we haven’t been sure how exactly to generalize the notion of polarization to n-plectic moduli $\infty$-stacks in general and hence only had some tentative steps towards genuine extended geometric quantization. (Chris Rogers has a discussion of 2-plectic geometric quantization of the topological WZW term in preparation, that certainly goes in the right direction).
Now, it occurred to me that for the extended geometric quantization of 2d Chern-Simons theory, the answer is actually already in the literature – in disguise and hence not fully recognized. Exploring this example should help to understand how the general case will work.
So I started making some notes that contain the story to the degree that I currently understand it, in the hope that this will be further developed. This is now at
extended geometric quantization of 2d Chern-Simons theory.
There is an introduction and overview there, which should give the main idea.
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