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Jun 27 at 11:44 comment added Reimundo Heluani What a coincidence, but 2 weeks ago I was told by Dylan Butson that a formal proof of the statement about topological factorization algebras in my comment from 2014 appears in his thesis. I would contact him, not sure what his user name here is.
Jun 24 at 12:37 comment added W.Rether @ReimundoHeluani do you know whether there's any updates about this work in Boston, Chicago and Paris? Or do you have a reference for Topological vertex algebras being the AG counterpart for E2 algebras?
Oct 31, 2014 at 9:24 comment added Reimundo Heluani Continuing: little is known for general vertex algebras or general factorization algebras in the Beilinson-Drinfeld case. Namely those which will not come from holonomic D-modules (which is the right finiteness and locally constant condition in the AG side). I've heard many times and from many that applying Chiral Homology and then RH one should obtain a factorization algebra. However, there are some subtleties involved since analitification does not preserve the chiral operations for general D-modules (quasi-coherent as O-mod).
Oct 31, 2014 at 9:20 comment added Reimundo Heluani I think much is said in the folklore but you won't find much written down. At least I know of three big groups working in Boston, Chicago and Paris trying to sort these things. In particular, the AG analog of Topological Factorization Algebra does exist (and predates the topological one) and is a Factorization algebra in Beilinson-Drinfeld. E_2 algebras are locally constant (or constructible) such algebras, their AG counterpart correspond to what are called Topological Vertex algebras. Presumably RH goes from the AG side to the top. side and this has been claimed by many to be done.
Oct 31, 2014 at 3:53 comment added Akhil Mathew @DavidBen-Zvi: Sounds like the statement I was looking for. I'll ask Sam about this further when I next see him. Thanks.
Oct 31, 2014 at 3:27 comment added David Ben-Zvi Roughly speaking, you want to consider crystals of categories where all associated D-modules (eg Homs of objects) are actually flat vector bundles, hence you can apply usual R-H. More precisely you have three sym monoidal categories, Loc_B, Loc_dR and D (local systems, flat vector bundles and D-modules). Module categories for the first two are the same via R-H. A module for D is "integrable" if it's induced from a Loc_dR-module, hence we can attach to it a Loc_B-module. (This is a garbled version of a formulation of Sam Raskin.)
Oct 31, 2014 at 3:09 comment added Akhil Mathew Actually, it looks like you're indicating it does apply in the geometric Satake setting -- so I suppose what I'm looking for is a description (or reference for) the version of Riemann-Hilbert that applies here.
Oct 31, 2014 at 2:28 comment added Akhil Mathew In particular, I'd be curious if it applies in the geometric Satake setting as well (though I'd also like to understand the E_2-structure in the affine Kac-Moody case).
Oct 31, 2014 at 2:16 comment added Akhil Mathew @DavidBen-Zvi: Could you elaborate on the condition of "integrability" which enables one to obtain the E_2-category?
Oct 31, 2014 at 2:01 comment added David Ben-Zvi In the Gaitsgory-Lurie Talbot story I think you're in the former setting --- you have a chiral category (coming from representations of Kac-Moody algebras) which happens to be integrable, so after Riemann-Hilbert you get a braided tensor category. Same in the geometric Satake setting, you have algebraic and topological chiral categories related by Riemann-Hilbert. The formula for the E_2 structure comes from trivilaizing your flat category over the disc, writing a monoidal structure using collisions of two points along a ray then describing the monodromy as those points braid in the disc.
Oct 31, 2014 at 1:52 comment added David Ben-Zvi @AkhilMathew: I don't see why one should expect to "extract" something locally constant from something crystalline. Categories are not the issue: the analog would be extracting a local system from an algebraic D-module. There's a condition we could ask for the D-module, that it be a flat vector bundle, and then (analytically) we can apply Riemann-Hilbert and get a local system. But I don't know if there's a very well behaved "integration" functor on D-modules (something like extending to infinite order differential operators) which would "extract" a local system from one.
Oct 31, 2014 at 1:45 comment added David Ben-Zvi @TheoJohnson-Freyd: unital chiral (or factorization) algebras automatically come with a de Rham (or crystal) structure.
Oct 31, 2014 at 1:42 comment added David Ben-Zvi @S.Carnahan: there's a notion of quasicoherent sheaf of categories --- locally it's given as a module category over the monoidal category of quasicoherent sheaves. You can then ask for a crystal structure (descent to de Rham space, I assume we're in characteristic zero) if you will. The definitive reference is arXiv:1306.4304
Oct 31, 2014 at 1:27 comment added Akhil Mathew @TheoJohnson-Freyd: Yes, I believe that the "crystal" datum (for a chiral category) is supposed to be the algebro-geometric analog of local (or, at least, infinitesimal) constancy.
Oct 31, 2014 at 1:24 comment added Theo Johnson-Freyd I think it's not quite right to think of a chiral algebra as an $E_2$ algebra. The latter should give examples of the former (up to some framing, perhaps), but chiral algebras should not, in general, be "locally constant". Think about a much less categorical setting of functions. Locally constant functions are very different from holomorphic functions. Or am I misunderstanding the notion of "chiral algebra" --- is there a "de Rham" that I missed somewhere?
Oct 31, 2014 at 1:16 comment added Akhil Mathew @S.Carnahan : You are right, perhaps this is part of the definition.
Oct 31, 2014 at 1:15 history edited Akhil Mathew CC BY-SA 3.0
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Oct 30, 2014 at 22:29 comment added S. Carnahan What does it mean for a crystal of categories to be quasi-coherent?
Oct 30, 2014 at 21:09 history asked Akhil Mathew CC BY-SA 3.0