Anomalies on ALE spaces and phases of gauge theory

We show that certain 't~Hooft anomalies that evade detection on commonly used closed four-dimensional manifolds become visible when a quantum field theory is placed on asymptotically locally Euclidean (ALE) spaces. As a concrete example, we use the Eguchi-Hanson (EH) space, whose defining features are its nontrivial second cohomology generated by the self-intersecting two-sphere and its asymptotic boundary ${\mathbb{R}\mathbb{P}}^3$, which carries torsion and thus furnishes additional cohomological data absent on conventional backgrounds. For a theory with symmetry $G_1\times G_2$, we turn on background flux for $G_1$ and probe potential anomalies by performing a global $G_2$ transformation; the resulting anomaly is captured by a five-dimensional mapping torus. The anomaly receives contributions from the four-dimensional characteristic classes on EH space as well as from the $\eta$-invariant associated with the ${\mathbb{R}\mathbb{P}}^3$ boundary. The anomaly uncovered in this way leads to new constraints on asymptotically free gauge theories. In particular, infrared composite spectra that successfully match anomalies on standard manifolds may nevertheless fail to reproduce the EH anomaly, and can therefore be excluded as the complete infrared realization of the symmetries.