ag.algebraic geometry - Why is a partition function of a Topological Conformal Field Theory related to Deligne-Mumford space

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.