5d to 3d compactifications and discrete anomalies

Abstract

Much insight into the dynamics of quantum field theories can be gained by studying the relationship between field theories in different dimensions. An interesting observation is that when two theories are related by dimensional reduction on a compact surface, their ’t Hooft anomalies corresponding to continuous symmetries are also related: the anomaly polynomial of the lower-dimensional theory can be obtained by integrating that of the higher-dimensional one on the compact surface. Naturally, this relation only holds if both theories are even dimensional. This raises the question of whether similar relations can also hold for the case of anomalies in discrete symmetries, which might be true even in odd dimensions. The natural generalization to discrete symmetries is that the anomaly theories, associated with the lower and higher dimensional theories, would be related by reduction on the compact surface. We explore this idea for compactifications of 5d superconformal field theories (SCFTs) to 3d on Riemann surfaces with global-symmetry fluxes. In this context, it can be used both as a check for these compactification constructions and for discovering new anomalies in the 5d SCFTs. This opens the way to applying the same idea of dimensional reduction of the anomaly theory to more general types of compactifications.

Discrete global symmetries: gauging and twisted compactification

“…In section 3, we investigate the effects of discrete gauging on the mixed 't Hooft anomalies between the Z n 1-form symmetry and the 0-form flavor symmetry in the theory of class S of type A 2n−1 with four twisted trivial punctures and four untwisted minimal punctures. The method that we adopt to examine such 't Hooft anomalies is the one discussed in [50,51] applied to the 3d mirror theory wreathed by various subgroups of the permutation group S 4 . The case of n = 1 of this class S theory is a close cousin of SU(2) SQCD with 4 flavors, whereby if the 1-form symmetry is gauged, the resulting theory is precisely the aforementioned SQCD.…”