Anomalies of 4d Spin_G 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 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= \mathbb{CP}². 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 ${\cal N}=2$ 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 hG^∨_gauge copies of a $\mathbb{CP}¹$ non-linear sigma model, including for the w₂w₃ anomalies when formulated with Spin_SU(2)global(4) structure.