Higher spin Riemann and Klein surfaces

Klein surfaces are generalization of Riemann surfaces to cases of non-orientable surfaces or surfaces with boundary. They are equivalent to real algebraic curves. An m-spin structure on Riemann or Klein surface is a complex line bundle $l$ such that $l^{\otimes m}$ is the cotangent bundle. We nd all m-spin structures on surfaces of Riemann and Klein, we describe they topological invariants and moduli spaces. The proofs are based on the theory of Fuchsian groups. The talk is based on joint works with Anna Pratoussevitch.