Torsion Constraints and Super Riemann Surfaces

Super Riemann surfaces are important in superstring theories as the generalization of the bosonic world sheet. In one approach to their study, one introduces two-dimensional supergravity, subject to certain conditions on the field strengths. Another approach builds super Riemann surfaces from superconformal patching data with no mention of the constraints. We show the equivalence of these two approaches and in particular interpret the torsion constraints as integrability conditions of a certain geometrical structure.