Eric D’Hoker and D.H. Phong this past week finally posted two crucial papers with results from their work on two-loop superstring amplitudes. The first one shows gauge slice independence of the two-loop N-point function, the second shows that, for N less than 3 and for low-order terms at N less than 4, there are no two-loop corrections to the low energy effective action.
D’Hoker and Phong have been studying superstring amplitudes for nearly twenty years, and are justly proud of their recent results, which are a tour de force of careful calculation. Over the years there have been many claims made about two-loop amplitudes, but until their work, no one had managed to really sort out the gauge dependence issues and write down gauge-independent amplitudes. For some comments about some of the issues involved at genus 2 and higher, see postings by Jacques Distler here, here, and here.
I don’t think D’Hoker and Phong will be coming out with complete results for genus 3 anytime soon, so the state of the art is that there is now a finite and well-defined version of the two-loop superstring amplitudes, with the problem of higher loops still open. While claims abound about the finiteness of higher-loop amplitudes, before believing them one should first take a look at the tricky problems that D’Hoker and Phong had to overcome to get well-defined two-loop amplitudes.
Update: Jacques Distler has a new posting about multi-loop amplitudes and potential problems with the Berkovits version of the superstring (he explains in more detail the possible problems with the BRST and picture-changing operators I mentioned). For some mysterious reason Jacques neglects to refer to my posting or comments about this. I encourage those commenters who seemed convinced I didn’t know what I was talking about to now take up their arguments with him.