We consider ’t Hooft anomalies of four-dimensional gauge theories whose fermion matter content admits SpinG(4) generalized spin structure, with G either gauged or a global symmetry. We discuss methods to directly compute w2 ∪ w3 ’t Hooft anomalies involving Stiefel-Whitney classes of gauge and flavor symmetry bundles that such theories can have on non-spin manifolds, e.g. M4 = 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 \( \mathcal{N} \) = 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 Ggauge theory with Nf adjoint Weyl fermions. For example, in the case of Nf = 2 we discuss anomaly matching in the IR phase consisting of \( {h}_{G_{\textrm{gauge}}}^{\vee } \) copies of a ℂℙ1 non-linear sigma model, including for the w2w3 anomalies when formulated with \( {\textrm{Spin}}_{\textrm{SU}{(2)}_{\textrm{global}}}(4) \) structure.
