Super-Riemann surfaces, loop measure, etc…

Abstract

An explicit expression for the loop measure over supermoduli space is derived for a specific choice of local coordinates. Geometrical interpretation of these local coordinates over supermoduli space allows one to construct a matrix of superperiods and suggests the generalization of theta functions. A non-analytic factor over supermoduli space turns out to be det ( T R − T L ) −5 where T R ( T L) is a matrix of superperiods corresponding to the right (left) geometry.