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started Calabi-Yau category
what about vertical categorification of Frobenius algebras? they should be closely related to the notion of modular tensor category (if not exactly the same thing)
fRight, so the notion of Calabi-Yau $A_\infty$-category is one aspect of vertical categorification. It yields “$(\infty,1)$-Frobenius algebroids” if you wish.
But then next should come “2-Frobenius algebroids” which should be “modular bicategories” such that those with a single object are modular tensor categories.
There must be literature on this, but I am not sure where the best place to go is.
inserted the full definition of $A_\infty$ CY-categories at Calabi-Yau category.
created a very short stub for Calabi-Yau variety, essentially to address the links pointing there from Calabi-Yau category.
Thanks!
“Calabi-Yau $A_\infty$-categories classify TCFTs. This is where they get there name from, because these TCFTs in turn may be constructed from sigma-models whose targets ar Calabi-Yau spaces.”
but is this true?? I would have bet this was some fields medal result! :) I would write “because nontrivial examples of TCFTs may be constructed in a natural way from.. but it could be me having missed a proof of every TCFT coming from a sigma-model :(
Oh, I didn’t mean to say that every one arises this way. Just that the motivating ones that made people invent the name “Calabi-Yau category” arose this way.
If you could clarify that sentence in the entry for me I’d be grateful. have to run now…
done.
Calabi-Yau A ∞-categories classify TCFTs. This is where they get there name from, because these TCFTs in turn may be constructed from sigma-models whose targets ar Calabi-Yau spaces.” but is this true??
Well it is, but really TCFT is of not crucial here. This came from community (Bondal, Orlov, Kontsevich…) which from late 1980-s studied (following early ideas of Beilinson) replacing varieties by their derived categories of coherent sheaves. Thus the derived category of coherent sheaves on Calabi Yau has some properties and generalizing those we come to Calabi Yau (triangulated or A-infinity) categories. Not each of them comes from a TCFT, there are Calabi Yau derived categories which are “noncommutative” and do not come from a TCFT on a commutative space, Calabi Yau or whatever. TCFTs are important for general motivation of the importance of the field but much further then the iommediate connection between derived categories of coherent sheaves on varieties and varieties themselves. So I would write THIS as a reason and TCFT as a benefit.
Included Zoran’ remark in the Idea section at Calabi-Yau category and edited the Properties section.
By the way, as Costello emphasizes a lot: the derived category of coherent sheaves is not the Calabi-Yau category that encodes the corresponding TCFT, but an $A_\infty$-refinement of it is. Apparently this $A_\infty$-refinemnet is not entirely clarified. See the discussion on page 35 here.
Costello emphasizes a lot: the derived category of coherent sheaves is not the Calabi-Yau category that encodes the corresponding TCFT, but an A ∞-refinement of it is
You have to be methodologically honest. In A infinity you do not have Hilbert space. You get it from it for free if you want with lots of work. So, you chose to equate the A infinity data with the physical theory, what means you just care if you have sufficient data, not the traditional setting of Hilbert space explicitly. But then, it is now known that for a quasiprojective smooth variety each derived category has a unique enhancement. So there is no difference from the point of view of data if you do A infinity or you do triangulated, as long as you work with varieties. This was not known at the time Costello’s article was written and it is known now due Orlov and Lunts who emphasise its philosophical importance for physics.
You have to be methodologically honest. In A infinity you do not have Hilbert space.
Don’t understand what you mean by this. If you mean the space of states associated to th circle by the TCFT: this is the Hochschild homology of the corresponding $A_\infty$-category.
But then, it is now known that for a quasiprojective smooth variety each derived category has a unique enhancement. So there is no difference from the point of view of data if you do A infinity or you do triangulated, as long as you work with varieties. This was not known at the time Costello’s article was written and it is known now due Orlov and Lunts who emphasise its philosophical importance for physics.
Ah, do you have the precise reference?
this is the Hochschild homology of the corresponding A ∞-category
Of course. But it is not given. It is derived from the data. Similarly the Ainfinity category itself is derived from the derived category by Lunts-Orlov. For the reference look in nlab under enhanced triangulated category. I discussed this in several of our discussions on derived vs. A-infinity. It is important I think.
Now I see there was only arxiv version there, and the final version is a bit optimised in J. AMS. This reference is under Valery Lunts. I will add the link to the journal in a bit.
Okay, thanks, Zoran. So you are saying that the open problem that Kevin Costello mentions on page 35 here has been solved?
Yes, I think so.
Okay, could you add a remark to this extent in the Examples-section at Calabi-Yau category?
But notice that the problem that Costello mentions is not the existence of the $A_\infty$-structure, but the existence of the cyclically invariant form that gives the CY structure. It is clear that this follows from the result you have in mind?
It is not clear to me, but the constructions in the paper of Lunts and Orlov are flexible enough to allow various extensions. Namely there are 3 versions of equivalences of enhancements considered, rouighly how natural/functorial the whole equivalence is. So with the strongest version it is probably possible, but I do not know.
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