Spinors in Weyl Geometry
  We consider the wave equation for spinors in ${\cal D}$-dimensional Weyl geometry. By appropriately coupling the Weyl vector $\phi _{\mu}$ as well as the spin connection $\omega _{\mu a b } $ to the spinor field, conformal invariance can be maintained. The one loop effective action generated by the coupling of the spinor field to an external gravitational field is computed in two dimensions. It is found to be identical to the effective action for the case of a scalar field propagating in two dimensions. 
A.H. Fariborz and D.G.C. McKeon