ar X iv : h ep - t h / 02 03 17 4 v 1 1 9 M ar 2 00 2 ON DECOMPOSING N = 2 LINE BUNDLES AS TENSOR PRODUCTS OF N = 1 LINE BUNDLES

We obtain the existence of a cohomological obstruction to expressing N = 2 line bundles as tensor products of N = 1 bundles. The motivation behind this paper is an attempt at understanding the N = 2 super KP equation via Baker functions, which are special sections of line bundles on supercurves. There has been—for some time now (cf. [DG])—an interest in extending the study of the super KP equations from the case N = 1 to the case N = 2. One possible way to do this, that could be particularly useful for understanding the geometry of these equations, would be via Baker functions: Roughly speaking, Baker functions are special unique sections with parameters of certain families of line bundles, satisfying the condition that any section of the corresponding line bundle is given by a differential operator applied to the Baker function. From the geometric point of view, their relevance stems from the fact that they allow us to reinterpret equations such as the KP (or its super analogs), as describing deformations of line bundles over curves (or supercurves). Thus, as a necessary step towards such a study of the super KP equations, one must first understand the geometry of N = 2 superline bundles, and this note is aimed towards that goal. Specifically, what we obtain here is the existence of a cohomological obstruction to expressing N = 2 line bundles as tensor products of 1991 Mathematics Subject Classification. 14M30.