  It has been shown for low-spin fields that the use of only the self-dual part
of the connection as basic variable does not lead to extra conditions or
inconsistencies. We study whether this is true for more general chiral action.
We generalize the chiral gravitational action, and assume that half-integer
spin fields are coupled with torsion linearly. The equation for torsion is
solved and substituted back into the generalized chiral action, giving
four-fermion contact terms. If these contact terms are complex, the imaginary
part will give rise to extra conditions for the gravitational and matter field
equations. We study the four-fermion contact terms taking spin-1/2 and spin-3/2
fields as examples.
