Constraints on the R-charges of Free Bound States from the Römelsberger Index

TL;DR

This work demonstrates that the Römelsberger index on $S^3\times\mathbb{R}$ is RG-invariant by arguing away singularities from states moving in from infinity, and uses this framework to derive a constraint on IR R-charges: if UV chiral multiplets have $0<q_i<2$ and the IR is a free theory of bound chiral states, then all IR R-charges satisfy $0<\tilde{q}_j<2$. The analysis combines a mode decomposition on $S^3$ leading to localized Landau-level-like eigenfunctions with a generating-function calculation of the index via the plethystic exponential, yielding results like $I(t,y)=\Gamma(t^q; ty, t/y)$ for $q\in(0,2]$. A key semi-classical argument shows that turning on a superpotential does not create or annihilate low-∆ states, supporting RG-invariance, while the IR R-charge constraint provides a weak version of Intriligator's conjecture and clarifies IR phases in models such as the $SU(2)$ theory with a $3/2$-representation chiral field. Collectively, these results reinforce the index as a robust, topological invariant for studying non-perturbative dynamics and IR phases in supersymmetric theories, with potential applications to dualities and phase structure analyses.

Abstract

The Römelsberger index on S^3 x R serves as a powerful test for conjectured dualities, relying on the claim that this object is an RG-invariant. In this work we support this claim by showing that the singularities suggested by Witten of "states moving in from infinity" are excluded on S^3 x R. In addition, we provide an application of the Römelsberger index, in the form of a constraint on the RG flow of supersymmetric theories. The constraint, which applies for asymptotically free theories with unbroken supersymmetry and non-anomalous R-symmetry, is the following: if the R-charges of the chiral multiplets in the UV theory are 0<q_i<2 and the IR theory can be described as a free theory of chiral bound states, then the R-charges of these bound states, ~q_j, are constrained such that 0<~q_j<2. We thus provide a proof of a weak version of a conjecture proposed by Intriligator. We mention some applications of this result.