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I find when I read a paper, Costello" The Gromov-Witten potential associated to TCFT"

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    $\begingroup$ Which paper? What other thoughts did you have? How about a little context? $\endgroup$
    – j.c.
    Jan 30, 2010 at 0:40
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    $\begingroup$ Please say more in your question. What kind of TFTs are you looking at? What paper are you reading? By Deligne-Mumford space, do you mean the space of genus g complex curves? $\endgroup$
    – S. Carnahan
    Jan 30, 2010 at 0:49
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    $\begingroup$ Please provide more information. $\endgroup$ Jan 30, 2010 at 1:28
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    $\begingroup$ -1, stone soup. Please read the mathoverflow.net/howtoask page. $\endgroup$ Jan 30, 2010 at 7:03
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    $\begingroup$ Did you by any chance make a typo when editing this? I don't even see a complete sentence in the body. $\endgroup$ Jan 30, 2010 at 20:17

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As I said in the comments, you should read AJ's answer to this question.

If you haven't read Costello's paper "Topological conformal field theories and Calabi-Yau categories", then you should definitely take a look at it, as the paper you reference is the sequel to this paper. You should also take a look at Lurie's TFTs paper.

In Costello's work and in Lurie's work, you will notice that TCFTs, by definition, involve moduli spaces of nonsingular Riemann surfaces (or nonsingular algebraic curves).

On the other hand, in order to do Gromov-Witten theory, we also need to consider moduli spaces of certain singular Riemann surfaces ("stable" Riemann surfaces). This is where Deligne-Mumford spaces come into play. So Gromov-Witten theory and TCFT are not exactly the same thing; they involve different moduli spaces. The idea of Costello (and Kontsevich) is that sometimes we can take a TCFT and extend the theory from the uncompactified moduli space to the compactified moduli space, thus getting something which is "a Gromov-Witten theory" associated to the TCFT.

One of Costello's and Kontsevich's motivations comes from mirror symmetry. The idea is that the Fukaya category of a compact symplectic manifold $X$ should give a TCFT. This is why I asked this question. Then, we should be able to extend this TCFT to the DM boundary and obtain the Gromov-Witten theory of the manifold. On the mirror side, for example the derived category of coherent sheaves on a Calabi-Yau variety $Y$ should also give a TCFT. Again, if we extend this TCFT to the DM boundary, we should get "a Gromov-Witten theory", which will not be the Gromov-Witten theory of $Y$, but it should at least be related to the Gromov-Witten theory of whatever $Y$ is mirror to.

I might be wrong about this, but I think that in some sense we have to consider compactifications of the moduli spaces, such as the Deligne-Mumford compactification (but there are other possible compactifications), because in order to obtain things like partition functions or the Gromov-Witten potential function, we must do integrals over the moduli spaces in question. But if the moduli spaces are non-compact, which they are, there may be issues in defining these integrals. So one way to get around this is to compactify.

In any case that is at least vaguely the broad picture. If you want to know more details you will have to clean up your question and make it more specific.

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  • $\begingroup$ Hi,kevin,I still have a question. Why partition function of a TCFT is the integral over the moduli space of Riemann surface (or its compactification)? By definition of partition function, we shall integrate something over the configuration space of fields, but what is the configuration space of fields for a general TCFT or just TFT? That's why I was asking this question. $\endgroup$
    – HYYY
    Jul 22, 2010 at 9:24
  • $\begingroup$ Costello's article seems hard to me. $\endgroup$
    – HYYY
    Jul 22, 2010 at 9:24
  • $\begingroup$ The space of fields is the appropriate space of maps from (nodal) Riemann surfaces to your target manifold. In other words it is $\overline{M}_{g,n}(X)$. The partition function involves integrals of certain classes over this space of fields. You can also express these integrals as integrals over $\overline{M}_{g,n}$, by pushing forward along the "forgetful map" $\overline{M}_{g,n}(X) \to \overline{M}_{g,n}$. $\endgroup$ Jul 22, 2010 at 23:52
  • $\begingroup$ Life is hard. :) $\endgroup$ Jul 22, 2010 at 23:52
  • $\begingroup$ Thanks!Kevin. but I thought Sigma model is only one kind of topological (or conformal) field theory. $\endgroup$
    – HYYY
    Jul 23, 2010 at 1:25

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