Anomalies of 4d SpinG theories

We consider 't Hooft anomalies of four-dimensional gauge theories whose fermion matter content admits Spin(G)(4) generalized spin structure, with G either gauged or a global symmetry. We discuss methods to directly compute w(2) boolean OR w(3) 't Hooft anomalies involving Stiefel-Whitney classes of gauge and flavor symmetry bundles that such theories can have on non-spin manifolds, e.g. M-4 = CP2. Such anomalies have been discussed for SU(2) gauge theory with adjoint fermions, where they were shown to give an effect that was originally found in the Donaldson-Witten topological twist of N = 2 SYM theory. We directly compute these anomalies for a variety of theories, including general G gauge theories with adjoint fermions, SU(2) gauge theory with fermions in general representations, and Spin(N) gauge theories with fundamental matter. We discuss aspects of matching these and other 't Hooft anomalies in the IR phase where global symmetries are spontaneously broken, in particular for general G(gauge) theory with N-f adjoint Weyl fermions. For example, in the case of N-f = 2 we discuss anomaly matching in the IR phase consisting of h(Ggauge)(boolean OR) copies of a CP1 non-linear sigma model, including for the w(2)w(3) anomalies when formulated with Spin(SU(2)global)(4) structure.