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In the paper Chiral Koszul duality, Gaitsgory and Francis develop a notion of a chiral algebra living on an arbitrary variety $X$. When $X=\mathbf{A}^1$ and the chiral algebra is translation invariant, this reproduces the usual notion of a vertex algebra.

I'm interested in getting a picture about what can/has been said about chiral algebras one dimension up, when $X$ is a surface. For instance,

  • The notion of vertex algebra reappears throughout physics, e.g. in conformal field theory. Does the higher dimensional analogue appear in physics naturally?
  • There is an interesting relation between loop spaces/affine Grassmannian and vertex algebras, e.g. via the WZW vertex algebra $V_k(\mathfrak{g})$. Is there any interesting relation between the higher dimensional vertex algebras and say the double loop space, which as I understand it is an even more interesting object?
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    $\begingroup$ U might want to check out the work of Faonte, Hennion and Kapranov for the Bun_{G} case. Nb that even purely local things here are derived in nature, as the authors explain in detail (punctured d disc should be derived object for d>1, get new classes in coherent cohom in dim 2d-1). In general from conversation with one of the authors above my understanding is that not much at all is known. In the physics literature I often see what seems to be something like a two dim "real" chiral algebra, where antiholomorphic coordinates are used. I don't understand at all what is going on in this case. $\endgroup$
    – user108998
    Commented Mar 1, 2021 at 15:25
  • $\begingroup$ One thing that happens in higher dimension is that the term 'chiral' becomes increasingly inaccurate.. $\endgroup$
    – user1504
    Commented Mar 1, 2021 at 16:25

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