Old and New Fields on Super Riemann Surfaces

The “new fields” or “superconformal functions” on N = 1 super Riemann surfaces introduced recently by Rogers and Langer are shown to coincide with the Abelian differentials (plus constants), viewed as a subset of the functions on the associated N = 2 super Riemann surface. We confirm that, as originally defined, they do not form a super vector space. It has been known for some time that the globally defined holomorphic functions on a generic super Riemann surface (SRS) with odd spin structure do not form a super vector space [1]. Neither do the holomorphic superconformal tensors of weights −1 through 2 [2]. That is, one cannot uniquely express an arbitrary tensor as a linear combination of some small basis set. The proposed “basis” inevitably contains nilpotent tensors which are annihilated by some elements of the set of constant Grassmann parameters (exterior algebra) Λ. Therefore linear combinations with coefficients from Λ cannot be unique. This is a special case of the general fact that sheaf cohomology groups H i(M,F) of complex supermanifolds over Λ are Λ-modules, but these modules are not necessarily free. In comparison with the theory of Riemann surfaces used in bosonic string theory, this complicates the discussion of period matrices, Jacobian varieties, functional determinants, and other ingredients in the construction of superstring amplitudes [3].