Anomalies in the Space of Coupling Constants and Their Dynamical Applications II
Clay Cordova, Daniel S. Freed, Ho Tat Lam, Nathan Seiberg
TL;DR
The paper generalizes the notion of ’t Hooft anomalies to dependence on coupling constants in 4d gauge theories, revealing a mixed anomaly between the center’s one-form symmetry and θ-periodicity in pure Yang–Mills, and extends this to SU(N) and Sp(N) theories with fundamental matter. It provides explicit anomaly formulas for a wide class of gauge groups, including PSU(N), E6, E7, Spin variants, and Sp(N), with coefficients tied to Pontryagin-square data and center structures; the results unify phase transitions and interface dynamics under θ-shifts. The authors develop a geometric and cohomological framework to analyze anomalies for general groups, including obstruction-theory methods and a higher-group viewpoint, and show how various IR realizations (Chern–Simons theories, TQFTs, or sigma-models) can saturate the same anomaly on interfaces. They also apply the framework to 4d QCD with fundamental matter, deriving gcd-based conditions (L=gcd(N,N_f) or related gcds) that govern the presence of anomalies and constrain interface theories, thereby linking long-distance dynamics to topological terms across a broad landscape of gauge theories.
Abstract
We extend our earlier work on anomalies in the space of coupling constants to four-dimensional gauge theories. Pure Yang-Mills theory (without matter) with a simple and simply connected gauge group has a mixed anomaly between its one-form global symmetry (associated with the center) and the periodicity of the $θ$-parameter. This anomaly is at the root of many recently discovered properties of these theories, including their phase transitions and interfaces. These new anomalies can be used to extend this understanding to systems without discrete symmetries (such as time-reversal). We also study $SU(N)$ and $Sp(N)$ gauge theories with matter in the fundamental representation. Here we find a mixed anomaly between the flavor symmetry group and the $θ$-periodicity. Again, this anomaly unifies distinct recently-discovered phenomena in these theories and controls phase transitions and the dynamics on interfaces.