Tadpole versus anomaly cancellation in D=4,6 compact IIB orientifolds

It is often stated in the literature concerning D = 4, 6 compact type IIB o rientifolds that tadpole cancellation conditions (i) uniquely fix the gauge group (up to Wilson lines and/or moving of branes) and (ii) are equivalent to gauge anomaly cancellation. We study the relationship between tadpole a nd anomaly cancellation conditions and qualify both statements. In general the tadpole cancellation conditions imply gauge anomaly cancellation but ar e stronger than the latter conditions in D = 4, N = 1 orientifolds. We also find that tadpole cancellation conditions in Z(N) D = 4;6 compact orientif olds do not completely fix the gauge group and we provide new solutions dif ferent from those previously reported in the literature.