General considerations of matter coupling with the self-dual connection

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.