Dirac and Majorana spinors on non-orientable Riemann surfaces (Q1093170)

It is known that Weyl spinors cannot be consistently defined on non- orientable manifolds. In the present note the authors prove that Dirac spinors may be defined on non-orientable Riemann surfaces, and that Majorana spinors cannot be defined on closed non-orientable Riemann surfaces having an odd Euler number. The latter result shows that a procedure in superstring theory proposed by \textit{A. M. Polyakov} [Phys. Lett. 103 B, 207-210 (1981); ibid., 211-213 (1981)] is inconsistent.