The BPS Cardy conjecture for three-dimensional N=2 superconformal field theories

Let $a_n$ denote the coefficients in the superconformal index of a three-dimensional superconformal field theory with $\mathcal{N}=2$ supersymmetry, with

$$\mathcal{I}(q):=\operatorname{Tr}_{\mathcal{H}_{S^2}}\left[(-1)^Fq^{R/2+J_3}\right]=Z(S^2\times_qS^1)=\sum_na_nq^n.$$

The coefficients are the degeneracies of supersymmetric, or BPS, states, and the Cardy asymptotic is

$$a_n\sim\exp\left(2\pi\sqrt{\frac{1}{6}c_{\text{eff}}n}\right)\qquad\text{as }n\to\infty.$$

BPS Cardy conjecture. In every three-dimensional superconformal field theory with $\mathcal{N}=2$ supersymmetry, the spectrum of supersymmetric (BPS) states obeys this Cardy asymptotic; equivalently, the superconformal index, or the $S^2\times_qS^1$ partition function, enjoys it.

This behavior has been observed in examples related to the 3d–3d correspondence, but the paper presents it as a general property whose validity for all three-dimensional $\mathcal{N}=2$ SCFTs remains open. Non-BPS states in a three-dimensional CFT have faster growth, of order $\log a_n\sim n^{2/3}$, whereas the conjectured BPS growth is of order $n^{1/2}$.