The Romelsberger Index, Berkooz Deconfinement, and Infinite Families of Seiberg Duals

Romelsberger's index has been argued to be an RG-invariant and, therefore, Seiberg-duality-invariant object that counts protected operators in the IR SCFT of an N=1 theory. These claims have so far passed all tests. In fact, it remains possible that this index is a perfect discriminant of duality. The investigation presented here bolsters such optimism. It is shown that the conditions of total ellipticity, which are needed for the mathematical manifestation of duality, are equivalent to the conditions ensuring non-anomalous gauge and flavor symmetries and the matching of (most) 't Hooft anomalies. Further insights are gained from an analysis of recent results by Craig, et al. It is shown that a non-perturbative resolution of an apparent mismatch of global symmetries is automatically accounted for in the index. It is then shown that through an intricate series of dynamical steps, the index not only remains fixed, but the only integral relation needed is the one that gives the "primitive" Seiberg dualities, perhaps hinting that the symmetry at the core is fundamental rather than incidental.