Timeline for Generator of a Fukaya category with certain properties
Current License: CC BY-SA 2.5
10 events
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| Sep 17, 2010 at 23:34 | history | edited | Kevin H. Lin | CC BY-SA 2.5 |
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| Sep 17, 2010 at 19:51 | comment | added | Daniel Pomerleano | Thanks a lot Mohammed for taking the time to clarify! That's exactly what I wanted to know. | |
| Sep 17, 2010 at 19:40 | comment | added | Mohammed Abouzaid | My point is that if you're interested in having a general theory, symplectic topology currently only produces (1) manifolds like cotangent bundles which are not compact, but where you can hope to have Fukaya categories like you originally asked for in your question (2) examples like toric varieties, where there are Lagrangian tori which are disjoint and generate, but have deformed cohomology rings. There is no general class in between that we know of an understand. The LG model for $x^3$ probably appears for toric varieties if you allow bulk deformations. | |
| Sep 17, 2010 at 19:09 | comment | added | Daniel Pomerleano | Sorry to keep pushing the point but I just must be really confused about something... consider k[[x]] with superpotential $x^3$. Then the Calabi Yau algebra associated to this is defined by a two-d vector space 1,e(e is odd) with no $m_2$ and a single $m_3(e,e,e)=1$. Am I wrong? This is what I wrote above and it seems to agree with Dyckerhoff Thm 4.7. Where is my mistake? Indeed if you change m_2 the only $A(\infty)$ algebra up to equivalence you can get should be the semisimple one. I would believe that the "geometric examples" involve "changing $m_2$" for example the mirror to $CP^n.$ | |
| Sep 17, 2010 at 18:35 | comment | added | Mohammed Abouzaid | If you allow yourself to vary $m_2$, then you're essentially allowing an arbitrary proper Calabi-Yau algebra, and there are indeed going to be some that are smooth (though I'm not the expert on this subject). The question becomes too general at this stage to probably have a good answer. The example that you have in mind for the Landau-Ginzburg model requires changing $m_2$. | |
| Sep 17, 2010 at 3:28 | comment | added | Daniel Pomerleano | How about if I give you Z/2Z grading and I allow you to vary even $m_2$ but as you were suggesting I want $HF^*(L,L)$ to be smooth as a dg-algebra and I want it to make sense for some possibly small non-zero value of the parameter t? In other words, I want it to give me a non-formal deformation of the de Rham algebra to a smooth algebra? | |
| Sep 17, 2010 at 3:06 | comment | added | Daniel Pomerleano | Hum, that last sentence sounds like the kind of explanation I'd like to understand a little bit more, but I'm currently confused by it... isn't that exactly what happens in the Landau Ginzburg models (k[[x]],x^n) n>2 the Koszul dual to that is k[e]/e^2 in an odd variable, with a single higher operation e^(n-tensor power)-->1. Your claim is maybe that this can't happen except in the case of torus? Or is this an issue of Z/2Z grading versus Z grading? | |
| Sep 17, 2010 at 1:05 | comment | added | Mohammed Abouzaid | I intentionally answered the question for $A_{\infty}$ algebra structures as opposed to cohomology because I can't think of any reasonably geometric criterion which make the products agree, while the higher products to diverge. Moreover, I highly doubt that you can construct a smooth $A_\infty$ category by taking the $A_\infty$ structure on cohomology and changing the higher products. | |
| Sep 16, 2010 at 15:20 | comment | added | Daniel Pomerleano | Hi Mohammed, It's great to have such an expert answering questions. One quick thing, I think maybe either I did not say what I meant or I am misinterpreting what you wrote, but when I say that I want the homology of HF*(X)=H*(L,L) I meant as an ordinary algebra(in other words agreeing up to m_2) there can be different higher operations. | |
| Sep 16, 2010 at 11:20 | history | answered | Mohammed Abouzaid | CC BY-SA 2.5 |