What Operator is the Superconformal Index Counting?

Question

Given a differential operator $\mathcal{D}$ with adjoint $\mathcal{D}^\dagger$, the index of $\mathcal{D}$ is usually defined (mathematically) by
$$\text{ind }\mathcal{D}=\dim\ker\mathcal{D}-\dim\ker\mathcal{D}^\dagger.$$
Alternatively, we can talk about the superconformal index
$$I(\beta_j) = \mbox{Tr}_{\mathcal{H}}(-1)^F e^{-\gamma\{Q,Q^\dagger\}}e^{-\sum_{j}\beta_j t_j},$$
where $F$ is the fermion number, $Q$ is the supercharge, and $t_j$'s are generators of the Cartan subalgebra of the superconformal and flavor symmetry algebra that commute with $Q$. In short, the superconformal index is the Witten index for superconformal field theories in radial quantization. My question is what operator $\mathcal{D}$ is the superconformal index $I(\beta_j)$ counting? Is it the supercharge operator $Q$? Or is it a different operator?