Higher spin 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 structure on a Klein surface is a complex line bundle whose $m$-th tensor power is the cotangent bundle. We describe all $m$-spin structures on Klein surfaces of genus greater than one and determine the conditions for their existence. In particular we compute the number of $m$-spin structures on a Klein surface in terms of its natural topological invariants.