Anomalies on ALE spaces and phases of gauge theory

A bstract We show that certain ’t Hooft anomalies not detected by standard closed four-dimensional probes can 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 ℝℙ 3 , which carries torsion and thus furnishes additional cohomological data absent on conventional backgrounds. For a theory with symmetry G 1 × 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 η -invariant associated with the ℝℙ 3 boundary. The anomaly detected in this way imposes additional constraints on asymptotically free gauge theories with fixed trivial asymptotic boundary conditions. In particular, infrared composite spectra that match anomalies on standard closed manifolds may nevertheless fail to reproduce the EH anomaly, and thus cannot by themselves furnish a complete symmetry-preserving infrared realization.