# on superstring perturbation theory

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I'd like to know what is the current state of knowledge regarding this topic (I'm aware of some papers by Witten, Berkovits, D'Hoker and Nekrasov, plus an old review by Nelson)

What I'd like to know is, in particular:

i. what is the role of Deligne-Mumford compactification (and of supermoduli spaces) in relation to IR and UV behavior (both in the bosonic and super string; we are actually integrating over the compatcification of moduli space everytime, correct?)

ii. what are the advantages of the different formalisms, e.g. GS, RNS or Berkovits (also, how do spin structures enrich the game?)

iii. one requires modular invariance for one-loop amplitudes (and this is related to UV behavior, right?), and then gets the different types of superstrings; what happens at higher genus/loop order? are there other constraints, or these are enough?

sorry if I messed things up, feel free to correct me

Witten has written a series of papers on superstring perturbation theory in the NSR formulation over the last couple of years which addresses several issues that arose in the late 80's and subsequently in the work of D' Hoker and Phong. In my experience, his papers are the best place start learning something from scratch as he writes very well. How I wish I had his clarity of thought! If I find the time, I will take a look at his papers and comment here. His paper with Donagi shows that the super moduli space is not split for genus $>5$ as one would have liked. I think it might mess up the equivalence of the Green-Schwarz and the NSR strings.
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