Filling the gaps with PCO's

Ashoke Sen, Edward Witten
2015 Journal of High Energy Physics  
Superstring perturbation theory is traditionally carried out by using picturechanging operators (PCO's) to integrate over odd moduli. Naively the PCO's can be inserted anywhere on a string worldsheet, but actually a constraint must be placed on PCO insertions to avoid spurious singularities. Accordingly, it has been long known that the simplest version of the PCO procedure is valid only locally on the moduli space of Riemann surfaces, and that a correct PCO-based algorithm to compute scattering
more » ... compute scattering amplitudes must be based on piecing together local descriptions. Recently, "vertical integration" was proposed as a relatively simple method to do this. Here, we spell out in detail what vertical integration means if carried out systematically. This involves a hierarchical procedure with corrections of high order. One might anticipate such a structure from the viewpoint of super Riemann surfaces. a zero-mode if it is allowed to have simple poles at the positions of PCO's [3] . But that fact is really not important for the present paper. 2 The simplest way to explain what one means by "reasonable behavior" is to say that the bad set is an orbifold of complex codimension 1 or real codimension 2 even if one compactifies the moduli space of Riemann surfaces by allowing the usual degenerations.
doi:10.1007/jhep09(2015)004 fatcat:mo7k6v7qwrer7dfixr2vcjejdy