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nLab > Latest Changes: extended geometric quantization of 2d Chern-Simons theory

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeFeb 2nd 2013
    • (edited Feb 2nd 2013)

    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.

    • CommentRowNumber2.
    • CommentAuthorGuest
    • CommentTimeMar 15th 2023

    There are many references missing on this page, in particular I think the following excerpt is either incorrect or misleading: The higher geometric quantization of the 2-d theory yields a 2-vector space of quantum 2-states (assigned to the point n codimension 2). Under the identification of 2-vector spaces with categories of modules over an associative algebra, this space of quantum 2-states identifies (the Morita equivalence-class of) an algebra. Suitably re-interpreting traditional results about the quantization of symplectic groupods shows that this algebra is the strict deformation quantization of that Poisson manifold.

    Can you exhibit a strict deformation quantization of T*ℝT*\mathbb{R} or S 2S^2 using this?

    • Josh
    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMar 15th 2023

    Thanks for the comment. I can try to adjust that sentence when I have the time. What I did for the moment is add (in the References-section) pointer to

    and the following comment:

    Here section 5.2.2 of Nuiten (2013) essentially provides a re-casting of the quantization construction in Hawkins (2008) which gives it an invariant home and brings out the above holographic picture more explicitly.

    diff, v18, current

    • CommentRowNumber4.
    • CommentAuthorGuest
    • CommentTimeMar 20th 2023
    Thank for your reply, I think the statement is interesting but I'm still confused. When I read the paragraph my understanding is that "every Poisson manifold has a strict deformation quantization and it is obtained in this way", however Eli Hawkins paper is not about strict deformation quantization at all, it's just about constructing C*-algebras, and as far as I know it's unknown if strict deformation quantizations even exist in general .

    - Josh
    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMar 20th 2023

    Right, I had promised to look into adjusting that sentence when I have the time. Thanks for prodding me.

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