What Operator is the Superconformal Index Counting?

Given a differential operator $mathcal{D}$ with adjoint $mathcal{D}^dagger$, the analytical index of $mathcal{D}$ is usually defined by
$$text{ind }mathcal{D}=dimkermathcal{D}-dimkermathcal{D}^dagger.$$
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 $Q$. The superconformal index is the Witten index for superconformal field theories in radial quantization. My question is can I think of the superconformal index $I(beta_j)$ as counting the analytical index of some operator $mathcal{D}$? If so, is it the supercharge operator $Q$? Or is it a different operator?