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

The R\"omelsberger 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 R\"omelsberger 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.