Quantum geometry of the Dirac fermions. Dimensional reduction of the spinor functional

The bosonic path integral formalism is developed for Dirac fermions interacting with a nonabelian gauge field in the D-dimensional Euclidean space-time. The representation for the effective action and correlation functions of interacting fermions as sums over all bosonic paths on the complex projective space CP2d-1, d=[D/2]-1 is derived where all spinor structure is absorbed by the one-dimensional Wess-Zumino term. 4 refs