Let $X$ be a smooth projective variety over $\mathbf{C}$. Consider the category $\mathbf{D}_{X}^{\text{perf}}$ of perfect complexes of left $D$-modules on $X$. It is well known that there is an isomorphism, $\mathit{HH}_{*}(\mathbf{D}_{X}^{\text{perf}})\cong H^{2d-*}_{dR}(X)$, where $\mathit{HH}$ is Hochschild homology and $d$ is the complex dimension of $X$. We thus have a Chern character map taking a perfect complex of $D$-modules to an element of $\mathit{HH}_{0}\cong H^{2d}_{dR}$. Now one obtains also a Riemann–Roch type theorem in this context, as explained in D. Shklyarov - Hirzebruch–Riemann–Roch theorem for DG algebras, as well as other papers of the same author. For example in the case of the category of perfect complexes of quasi-coherent sheaves on $X$, one obtains the usual HRR formula.
Question
Is it known what explicit form this takes? In particular, what are the Chern characters explicitly and what is the corresponding Riemann–Roch formula?
Guess
Here is a guess as to how this goes for the $D$-module $\mathcal{O}$ (or, indeed, any flat vector bundle). We have a standard resolution $\mathcal{D}\otimes\bigwedge^{*} T\rightarrow \mathcal{O}$ and functoriality of the Chern character construction reduces to computing the pushforward on Hochschild homology from perfect sheaves on $X$ to perfect $\mathcal{D}$-modules. This pushforward should be the projection of $\mathit{HH}_{0}(X)=\bigoplus H^{p,p}(X)$ to $H^{d,d}\cong H^{2d}_{dR}$. Indeed it is a natural map thus either the projection or $0$, and I believe it is non-zero in the case of $\mathbf{P}^{1}$ by explicit calculation. If so we get that the $D$-module theoretic Chern character of $\mathcal{O}$ is $(-1)^{d}c_{d}(X)$. It seems thus that the Riemann–Roch type theorem in this context should be something like the Hopf index theorem. Is this true and if so has it been written down in detail somewhere?
