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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJan 30th 2019

    added pointer to section 7.5 of

    diff, v11, current

    • CommentRowNumber2.
    • CommentAuthorperezl.alonso
    • CommentTimeMar 31st 2024

    Is there a reason why this article is phrased in terms of Lagrangian (as opposed to coisotropic) branes?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMar 31st 2024

    I would say: Because that’s how it’s defined. But I might not get the subtext of your question?

    • CommentRowNumber4.
    • CommentAuthorperezl.alonso
    • CommentTimeMar 31st 2024

    There is this observation going back (at least?) to Kapustin and Orlov and later used by Gukov and Witten in describing quantization via the A-model that A-branes are represented not only by Lagrangian submanifolds but more generally by coisotropic submanifolds. Shouldn’t the Fukaya category be described in terms of coisotropic subspaces if that is the relevant category for mirror symmetry?

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeApr 1st 2024
    • (edited Apr 1st 2024)

    I see; maybe somebody proposed a corresponding definition of generalized Fukaya categories?

    But the answer to your question in #2 is clearly:

    The entry on Fukaya categories speaks about Lagrangian submanifolds because these are by definition their objects – see for instance Auroux 2013, p. 22.

    Last I looked into it, many years ago, the correct definition of Fukaya categories was still felt to be elusive/unsatisfactory, due to the issue of transversality (also indicated in the Idea-section of the entry). There was the idea that a more satisfactory definition would use derived symplectic geometry, where the transversality issue is automatically dealt with. But I haven’t followed what became of this idea and what the state-of-the-art of Fukaya categories is these days.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeApr 1st 2024

    I have completed and brushed-up some of the bibitems in the list if references

    and touched the wording in the Idea-section

    diff, v14, current

    • CommentRowNumber7.
    • CommentAuthorperezl.alonso
    • CommentTimeApr 1st 2024

    I see. Sure, I suppose my question was more about how that observation fits all this in practice, whether we a. change the definition of the Fukaya category so that mirror symmetry is still phrased in terms of it(s updated version), or b. we keep the definition like that and then speak of a coisotropic extensions of the Fukaya category. One of the reasons I ask is because I’m wondering if there is a sort of clear way to see how the Fukaya category for the A-model and the category of sheaves for the B-model come about from motivic quantization of the AKSZ sigma-model.