Anomalies of 4d $Spi{n}_G$ Theories

Abstract

We consider 't Hooft anomalies of four-dimensional gauge theories whose fermion matter content admits $Spi{n}_G(4)$ generalized spin structure, with $G$ either gauged or a global symmetry. We discuss methods to directly compute $w_2\cup w_3$ 't Hooft anomalies involving Stiefel-Whitney classes of gauge and flavor symmetry bundles that such theories can have on non-spin manifolds, e.g. $M_4={\mathbb{CP}}^2$. 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 $\mathcal{N}=2$ 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 $G_{\mathrm{gauge}}$ 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 $h_{G_{\mathrm{gauge}}}^{\vee }$ copies of a ${\mathbb{CP}}^1$ non-linear sigma model, including for the $w_2{w}_3$ anomalies when formulated with $Spi{n}_{SU(2{)}_{\mathrm{global}}}(4)$ structure.

48 pages plus Appendix