TITLE: Gromov-Witten theory and compactifications of the moduli of curves
QUESTION [13 upvotes]: Why, from a string theory perspective, is it natural to consider the Deligne-Mumford (resp. Kontsevich) compactification of the moduli of curves (resp. maps [from curves to a target space X]) rather than some other compactification? 
In any case, what other compactifications of the moduli of curves have been studied? Similarly, what other compactifications of the moduli of maps have been studied? Do any of these other compactifications lead to an interesting "Gromov-Witten theory"?

REPLY [4 votes]: I would prefer to leave this as a comment to Tony Pantev's answer, but I don't have enough reputation to do so.  Anyway, as Tony mentions the Laumon compactification provides a semismall resolution of singularities of the Drinfeld compactification.  However, to the best of my knowledge, the Laumon compactification is special to the case where G = SL_n, whereas the Drinfeld compactification can be defined for any reductive algebraic group.  In Kuznetsov's paper, he says in the introduction that he would like to study the resolutions of the Drinfeld compactification for groups besides SL_n in a future paper.  As far as I know, no such paper was ever written.  Does anyone know if Kuznetsov ever wrote such a paper and, if not, if anyone else has ever worked on this?
One partial solution to this question seems like it might be buried in the paper by Braverman, Finkelberg, Gaitsgory, and Mirkovic where they compute the IC sheaves of the Drinfeld compactification.  This is related because, assuming a semismall resolution of singularities existed, its cohomology would compute the intersection cohomology of the Drinfeld compactification.  However, given their use of finite-dimensional Zastava spaces to model the singularities of the Drinfeld compactification (and hence to compute the IC sheaf), it does not seem to me that a semismall resolution can be found in their paper (although maybe this means that if a semismall resolution existed it should also provide a resolution of each of the Zastava spaces?).
If appropriate, I would be happy to start this as its own topic.