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

TL;DR

This paper uses the R"omelsberger index on S^3 x R to establish constraints on the R-charges of free bound states in supersymmetric theories, supporting duality invariance and confirming a conjecture about R-charge bounds.

Contribution

It proves that the R"omelsberger index excludes certain singularities and constrains R-charges of bound states, confirming a weak form of Intriligator's conjecture.

Findings

Excluded singularities from 'states moving in from infinity' on S^3 x R.

Established R-charge bounds for bound states in supersymmetric theories.

Provided a proof supporting the RG-invariance of the R"omelsberger index.

Abstract

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.