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The R\"omelsberger index on S^3 x R serves as a powerful test for conjectureddualities, relying on the claim that this object is an RG-invariant. In thiswork we support this claim by showing that the singularities suggested byWitten of "states moving in from infinity" are excluded on S^3 x R. Inaddition, we provide an application of the R\"omelsberger index, in the form ofa constraint on the RG flow of supersymmetric theories. The constraint, whichapplies for asymptotically free theories with unbroken supersymmetry andnon-anomalous R-symmetry, is the following: if the R-charges of the chiralmultiplets in the UV theory are 0<q_i<2 and the IR theory can be described as afree 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 weakversion of a conjecture proposed by Intriligator. We mention some applicationsof this result.

Constraints on the R-charges of Free Bound States from the R\"omelsberger Index