Constraints on the R-charges of free bound states from the Römelsberger index

Abstract

The Römelsberger index on S 3 × $$ \mathbb{R} $$ 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 × $$ \mathbb{R} $$ . 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 q i ∈ (0, 2) and the IR theory can be described as a free theory of chiral bound states, then the R-charges of these bound states, $$ {\tilde{q}}_j $$ , are constrained such that $$ {\tilde{q}}_j\in \left(0,2\right) $$ . We thus provide a proof of a weak version of a conjecture proposed by Intriligator. We mention some applications of this result.