While classically this procedure is always possible, quantum mechanically the U ( 1 ) symmetries may be complex structures and the B model exists when the first Chern classes of associated bundles sum to zero whereas the A model exists when the difference of the Chern classes is zero.
22.
A yet more complicated, yet more accurate and geometrically enlightening, approach is to understand that the gauge covariant derivative is ( exactly ) the same thing as the exterior covariant derivative on a principal fibre bundle of the gauge theory; and, for the case of spinors, the associated bundle would be a spin bundle of the spin structure.
23.
Another special class of fiber bundles, called "'principal bundles "', are bundles on whose fibers a free and transitive representation ? of " G " on a vector space " V ", a vector bundle with ? ( " G " ) ?" Aut ( " V " ) as a structure group may be constructed, known as the associated bundle.
