Superconformal index of higher derivative $\mathcal N=1$ multiplets in four dimensions

TL;DR

This paper computes the superconformal index for higher derivative non-unitary $ =1$ multiplets in four dimensions, testing and proposing modifications to the universal relations between index coefficients and conformal anomalies.

Contribution

It extends the analysis of superconformal index relations to higher derivative non-unitary multiplets, suggesting universal modifications applicable beyond unitary cases.

Findings

Computed superconformal indices for higher derivative multiplets

Identified deviations from previously assumed universal relations

Proposed modifications to the index-anomaly relations

Abstract

Supersymmetric partition function of $\mathcal N=1$ superconformal theories on $S^1_{\beta} \times S^3$ is related to the superconformal index receiving contributions from short representations. The leading coefficients in the small $\beta$ (high "temperature") expansion of the index were previously related to the conformal anomaly coefficients of the theory. Assumptions underlying universality of these relations were tested only for simplest low-spin unitary multiplets. Here we consider examples of higher derivative non-unitary $\mathcal N=1$ multiplets that naturally appear in the context of extended conformal supergravities and compute their superconformal index. We compare the coefficients in the small $\beta$ expansion of the index with those proposed earlier for unitary multiplets and suggest some modifications that should apply universally to all types of theories. We also comment on the structure of subleading terms and the case of $\mathcal N=4$ conformal supergravity.