New Fields on Super Riemann Surfaces

A new (1, 1)-dimensional super vector bundle which exists on any super Riemann surface is described. Cross-sections of this bundle provide a new class of fields on a super Riemann surface which closely resemble holomorphic functions on a super Riemann surface, but which (in contrast to the case with holomorphic functions) form spaces which have a well defined dimension which does not change as odd moduli become non-zero. Super Riemann surfaces are (1, 1)-dimensional holomorphic complex supermanifolds which have interesting mathematical features and have been intensively studied because of their use in a very elegant and effective approach to the Polyakov quantization of the spinning string. In this approach, which has been developed by a number of authors, for instance in the works of Baranov, Manin, Frolov and Schwarz [1], Baranov and Schwarz [2], Belavin and Knizhnik [3], Rosly, Schwarz and Voronov [10] and Voronov [11], many ∗Research supported by a Royal Society University Research Fellowship