Hyperbolic Geometry of Superstring Perturbation Theory

Seyed Faroogh Moosavian, Roji Pius

Published 2017-03-30Version 1

The hyperbolic structure of perturbative superstring theory in RNS framework is explored, with the goal of providing a systematic method for explicitly evaluating both the on-shell and the off-shell amplitudes in closed superstring theory with arbitrary number of external states and loops. The set of local coordinates around the punctures on the Riemann surface satisfying the off-shell factorization requirement needed for defining the off-shell amplitudes is specified using the metric on the Riemann surface having $-1$ constant curvature. An explicit procedure for distributing picture changing operators on the Riemann surface over the whole moduli space in a way that satisfies the off-shell factorization requirement and at the same time avoids the spurious singularities is described. The superstring amplitude is expressed as a sum of real integrals over certain covering spaces of the moduli space. These covering spaces are identified as explicit domains inside the Teichm\"uller space of the Riemann surfaces parameterized using the Fenchel-Nielsen coordinates.