Cardy Formulae for SUSY Theories in d=4 and d=6

TL;DR

This paper derives formulas for counting BPS states and supersymmetric partition functions in 4D and 6D supersymmetric theories, linking high-temperature asymptotics to anomalies, especially the a-c combination.

Contribution

It provides new Cardy-like formulas for supersymmetric theories in four and six dimensions, connecting asymptotic state counting to anomaly coefficients.

Findings

High-temperature asymptotics in 4D N=1 theories fixed by anomalies.

The a-c anomaly combination influences the counting of BPS states.

Proposed analogous formulas for 6D (1,0) theories.

Abstract

We consider supersymmetric theories on a space with compact space-like slices. One can count BPS representations weighted by (-1)^F, or, equivalently, study supersymmetric partition functions by compactifying the time direction. A special case of this general construction corresponds to the counting of short representations of the superconformal group. We show that in four-dimensional N=1 theories the "high temperature" asymptotics of such counting problems is fixed by the anomalies of the theory. Notably, the combination a-c of the trace anomalies plays a crucial role. We also propose similar formulae for six-dimensional (1,0) theories.