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

TL;DR

This paper explores the Romelsberger index's role as an RG-invariant that can potentially uniquely identify Seiberg dualities in N=1 supersymmetric theories, revealing deep connections between mathematical conditions and physical symmetries.

Contribution

It demonstrates that the conditions for total ellipticity in the index align with symmetry and anomaly matching, supporting the index as a robust duality discriminator and revealing the fundamental nature of core symmetries.

Findings

Total ellipticity conditions match anomaly and symmetry constraints.

Index remains invariant through complex dynamical steps.

Primitive Seiberg dualities are fundamental, not incidental.

Abstract

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.