  We study the connected components of the space of higher spin bundles on
hyperbolic Klein surfaces. A Klein surface is a generalisation of a Riemann
surface to the case of non-orientable surfaces or surfaces with boundary. The
category of Klein surfaces is isomorphic to the category of real algebraic
curves. An m-spin bundle on a Klein surface is a complex line bundle whose m-th
tensor power is the cotangent bundle. The spaces of higher spin bundles on
Klein surfaces are important because of their applications in singularity
theory and real algebraic geometry, in particular for the study of real forms
of Gorenstein quasi-homogeneous surface singularities. In this paper we
describe all connected components of the space of higher spin bundles on
hyperbolic Klein surfaces in terms of their topological invariants and prove
that any connected component is homeomorphic to a quotient of a Euclidean space
by a discrete group. We also discuss applications to real forms of
Brieskorn-Pham singularities.
