  We discuss supersymmetric surface defects in compactifications of six
dimensional minimal conformal matter of type SU(3) and SO(8) to four
dimensions. The relevant field theories in four dimensions are N=1 quiver gauge
theories with SU(3) and SU(4) gauge groups respectively. The defects are
engineered by giving space-time dependent vacuum expectation values to baryonic
operators. We find evidence that in the case of SU(3) minimal conformal matter
the defects carry SU(2) flavor symmetry which is not a symmetry of the four
dimensional model. The simplest case of a model in this class is SU(3) SQCD
with nine flavors and thus the results suggest that this admits natural surface
defects with SU(2) flavor symmetry. We analyze the defects using the
superconformal index and derive analytic difference operators introducing the
defects into the index computation. The duality properties of the four
dimensional theories imply that the index of the models is a kernel function
for such difference operators. In turn, checking the kernel property
constitutes an independent check of the dualities and the dictionary between
six dimensional compactifications and four dimensional models.
