The gravitational chiral anomaly of spin-1/2 field in the presence of twisted boundary conditions for ordinary field theory

We calculate the chiral anomaly in the neighbourhood of the fixed point space M_h which is constructed by the group action of a discrete symmetry h on a compact manifold M. The Feynman diagrams approach for the corresponding supersymmetric quantum mechanical system with twisted boundary conditions is used. The result we derive in this way agrees with the generalization of the ordinary index theorem (the G-index theorem) on the spin complex.