Calculations of the chiral condensate (RX) on the lattice using stagge
red fermions and the Lanczos algorithm are presented. Four gauge field
s are considered: the quenched non-Abelian field, an Abelian-projected
field, and monopole and photon fields further decomposed from the Abe
lian field. Abelian projection is performed in the maximal Abelian gau
ge and in the Polyakov gauge. The results show that monopoles in the m
aximal Abelian gauge largely reproduce the chiral condensate values of
the full non-Abelian theory, in both SU(2) and SU(3) color.