I think I heard there is such a theory, but I just can't find reference.So I am asking if there really has such a theory and reference if yes. Thanks firstly!
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4$\begingroup$ HYYY -- this and some of your other questions are potentially quite interesting but I think some more details and/or background are in order (e.g. what kind of answer do you expect in this particular question?) -1; to be reverted if more details are given. $\endgroup$– algoriCommented Jun 30, 2010 at 3:01
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$\begingroup$ Hi,algori,thank you! I have edited it now.I think I just need reference (or better, comment) $\endgroup$– HYYYCommented Jun 30, 2010 at 3:10
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2$\begingroup$ HYYY -- I'm sorry but it seems to me that your question has become a bit longer but still does not contain any extra information with respect to the original version. You are looking for a reference to the chain level GW theory, but you do not tell us what you expect it to be or what properties you would like it to have. I think I have a guess as to what it may be and I'd be interested to know if my guess is correct but it's your question so it's for you to set up the rules of the game i.e. to state what requirements something has to fulfill to be called a chain level GW theory. $\endgroup$– algoriCommented Jun 30, 2010 at 3:38
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1$\begingroup$ I think a chain-level GW theory should be something like a cohomological field theory (CohFT - as defined by Kontsevich-Manin), except with (co)chains C^* everywhere instead of cohomology H^*. I don't know if this is written up anywhere. My guess is no. $\endgroup$– Kevin H. LinCommented Jun 30, 2010 at 3:50
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$\begingroup$ I think Kevin Lin is correct and as far as I know, this is still a conjecture. Costello says this in his articles on Calabi-Yau $A_\infty$-categories and on the Gromov-Witten potential, but these articles were written a few years ago, so something may have happened in the mean time. $\endgroup$– skupersCommented Jun 30, 2010 at 5:54
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A theory of the type you might be talking about is stated in Sullivan's "Theorem 4" in "String topology and sigma models", http://books.google.com/books?hl=en&lr=&id=yuGic0WClQ4C&oi=fnd&pg=PA1&dq=string+topology+and+sigma+models+sullivan&ots=2Oympt_Z4g&sig=W4GOnI6Cn-7IjsJEH7x9c8-qnnY#v=onepage&q=string%20topology%20and%20sigma%20models%20sullivan&f=false. Whether it is a Theorem is somewhat unclear to me.
answered Jul 31, 2010 at 1:44