A measure on supermoduli space indeed exists and can be used to define all-loop amplitudes. The problem is that, except for small genus, it does not appear that the supermoduli spaces are split supermanifolds. Therefore there is no way to split the integration up into an integral over odd coordinates and an integral over even coordinates. The problem with higher-loop amplitudes is then a technical one: there is a definition, but we do not know how to evaluate the expression in closed form.I am aware of:
- no supersymmetric measure (like d4p in ordinary QFT) known beyond a few loops
- therefore no definition of higher-loop amplitutes
For the bosonic string, modular invariance maps the region of geometries where UV divergences would appear to points in moduli space which correspond to IR physics. The bosonic theory has a special IR divergence due to the tachyon. The other potential IR divergences come from special points in moduli space where you have degenerate handles or punctures coming together.- no proof of finiteness of higher-loop amplitutes
In the superstring, the tachyon is absent. Furthermore, in the cases where supermoduli space is split, the integration over the odd moduli contributes to the measure over even moduli. We expect IR divergences to be related to the same phenomena as in the bosonic string. In the 1-loop and 2-loop vacuum cases, these divergences vanish after summing over the spin-structures. If the supermoduli space is not split, you can't sum over spin structures until you've integrated over the supermoduli. Witten's work is an attempt to investigate these IR divergences in this general case.
It's rarely the case that one expects a perturbation series to converge, rather perturbative series are asymptotic series. Even in ordinary QM, if we consider the harmonic oscillator with a ##\lambda x^4## perturbation, the radius of convergence of the perturbation series is zero.- no proof of convergence of the perturbation series