Can the energy bound $E \geq 0$ imply supersymmetry?

Abstract

We utilize the integrality conjecture to show that the torus partition function of a fermionic rational conformal theory in the Ramond-Ramond sector becomes a constant when the bound $h^R \ge \frac{c}{24}$ is satisfied, where $h^R$ denote the conformal weights of Ramond states and $c$ is the central charge. The constant-valued Ramond-Ramond partition function strongly suggests the presence of supersymmetry unless a given theory has free fermions. The lower bound $h^R \ge \frac{c}{24}$ can then be identified with the unitarity bound of $\mathcal{N}=1$ supersymmetry. We thus propose that, for rational CFTs without free fermions, $(h^R-c/24) \geq 0$ can imply supersymmetry.