Super Riemann surfaces

A super Riemann surface is a particular kind of (1,1)-dimensional complex analytic supermanifold. From the point of view of super-manifold theory, super Riemann surfaces are interesting because they furnish the simplest examples of what have become known as non-split supermanifolds, that is, supermanifolds where the odd and even parts are genuinely intertwined, as opposed to split supermanifolds which are essentially the exterior bundles of a vector bundle over a conventional manifold. However undoubtedly the main motivation for the study of super Riemann surfaces has been their relevance to the Polyakov quantisation of the spinning string. Some of the papers on super Riemann surfaces are reviewed. Although recent work has shown all super Riemann surfaces are algebraic, some areas of difficulty remain. (author)