On decomposing \(N=2\) line bundles as tensor products of \(N=1\) line bundles (Q1868230)

If \(N\) denotes the number of odd generators of a super Grassmann algebra, the authors find a cohomology class which measures the obstruction to expressing a line bundle over an \(N = 2\) super Riemann surface associated to an \(N = 1\) supercurve \(M\) as a tensor product of line bundles over \(M\) (and the dual supercurve \(\widehat{M}\) ). Some interesting remarks on the significance of this result are also presented.