Moving NS Punctures on Super Spheres

One of the subtleties that has made superstring perturbation theory intricate at high string loop order is the fact that as shown by Donagi and Witten, supermoduli space is not holomorphically projected, nor is it holomorphically split. In recent years, Sen (further refined by Sen and Witten) has introduced the notion of vertical integration in moduli space. This enables one to build BRST-invariant and well-defined amplitudes by adding certain correction terms to the contributions associated to the traditional “delta function” gauge fixing for the worldsheet gravitino on local patches. The Sen and Witten approach is made possible due to there being no obstruction to a smooth splitting of supermoduli space, but it may not necessarily be the most convenient or natural solution to the problem. In particular, this approach does not determine what these corrections terms actually are from the outset. Instead, it shows that such correction terms in principle exist, and when included make all perturbative amplitudes well-defined. There may be situations however where one would like to instead have a well-defined and fully determined path integral at arbitrary string loop order from the outset. In this paper, I initiate an alternative (differential-geometric) approach that implements the fact that a smooth gauge slice for supermoduli space always exists. As a warmup, I focus specifically on super Riemann surfaces with the topology of a sphere in heterotic string theory, incorporating the corresponding super curvature locally, and introduce a new well-defined smooth gauge fixing that leads to a globally defined path integral measure that translates arbitrary fixed ($-1$) picture NS vertex operators (or handle operators) (that may or may not be offshell) to integrated (0) picture. I also provide some comments on the extension to arbitrary super Riemann surfaces.