By the Riemann-Hilbert correspondence, there is an equivalence between
(1) $\mathcal{D}\operatorname{-mod}(X)$ , the (derived) category of holonomic D-modules on a complex variety X, and
(2) $D^b_c(X)$ , the (derived) category of constructible sheaves on X.
There is a "naive" t-structure we can put on both categories. In $\mathcal{D}\operatorname{-mod}(X)$ , we can look at a t-structure whose heart $\mathcal{D}\operatorname{-mod}^\heartsuit$ is a complex (of D-modules) concentrated in degree 0. In $D^b_c(X)$ , we can look at the naive t-structure whose heart $D^{b \heartsuit}_c$ is a complex (of constructible sheaves) concentrated in degree 0.
It's known that if we transfer the naive t-structure on $\mathcal{D}\operatorname{-mod}(X)$ to $D^b_c(X)$ (using the equivalence above), $\mathcal{D}\operatorname{-mod}^\heartsuit$ is identified with "perverse sheaves" on X.
My question is:
If we map $D^{b\heartsuit}_c$ to the category of D-modules using the Riemann-Hilbert correspondence, what subcategory of $\mathcal{D}\operatorname{-mod}$ do we get? Does this have a well-known name?
More generally, is there some geometric/nice description of what the naive t-structure on $D^b_c$ becomes on $\mathcal{D}{\operatorname{-mod}}$ ?
