/ 96 11 03 4 v 1 6 N ov 1 99 6 Superstrings in Higher Order Extensions of

The work proposes a general background of the theory of field interactions and strings in spaces with higher order anisotropy. Our approach proceeds by developing the concept of higher order anisotropic superspace which unifies the logical and mathematical aspects of modern Kaluza– Klein theories and generalized Lagrange and Finsler geometry and leads to modelling of physical processes on higher order fiber bundles provided with nonlinear and distingushed connections and metric structures. The view adopted here is that a general field theory should incorporate all possible anisotropic and stochastic manifestations of classical and quantum interactions and, in consequence, a corresponding modification of basic principles and mathematical methods in formulation of physical theories. The presentation is divided into two parts. The first five sections cover the higher order anisotropic superspaces. We focus on the geometry of distinguished by nonlinear connection vector superbundles, consider different supersymmetric extensions of Finsler and Lagrange spaces and analyze the structure of basic geometric objects on such superspaces. The remaining five sections are devoted to the theory of higher order anisotropic superstrings. In the framework of supersymmetric nonlinear sigma models in Finser extended backgrounds we prove that the low– energy dynamics of such strings contains motion equations for locally anisotropic field interactions. Our work is to be compared with important previous variants of extension of Finsler geometry and gravity (see, for instance, [7, 76, 74, 14]). There are substantial differences, because we rely on modeling of higher order anisotropic interactions on superbundle spaces and do not propose