Bounds on triangle anomalies in (3+1)D

How many charged degrees of freedom are necessary to accommodate a certain amount of 't Hooft anomaly? Using the conformal bootstrap for the four-point function of flavor current multiplets, we show that in all $(3+1)\mathrm{D}$ superconformal field theories the 't Hooft anomaly of a continuous flavor symmetry is bounded from above by the $3/2$ power of the current two-point function coefficient, which can be thought of as a measure for the amount of charged degrees of freedom. We check our bounds against free fields and supersymmetric quantum chromodynamics in the conformal window.