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Costello's theorem,"open TCFT=Calabi Yau A-infinity category",he also mentions when applied to Fukaya category we can recover Gromov-Witten theory, but I see that it needs some assumption,also even if the assumption is true,I just see that the TCFT associated to Fukaya category is the same as the one associated to Gromov-Witten invariant, BUT can we from that TCFT to RECOVER Gromov-Witten invariants? (I mean the opposite direction) Because otherwise,it is not A model.

I appreciate if there is any answer.

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This is an open question. See arxiv.org/abs/math/0509264 and arxiv.org/abs/0806.0107 –  Kevin H. Lin Jun 30 '10 at 3:10
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The answer to the question in the title is “no”. –  Harald Hanche-Olsen Jun 30 '10 at 3:37
    
If Costello's theorem is what you're interested in, section 4.2 of the Lurie(-Hopkins) article on the classification on TCFT's describes a generalization that may be better suited to deal with this. –  skupers Jun 30 '10 at 6:00
    
Kevin, do you read that paper"The Gromov-Witten potential associated to a TCFT", I have several questions? –  HYYY Aug 20 '10 at 10:27
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