Let's say for simplicity $A$ is a smooth algebra over a field $k$ ($A$ and $k$ are discrete commutative rings but from now on we are fully derived), and we will consider the $E\_2$ algebra $HH^{\bullet}(A)$ which I'll just call $H\_A$. We know by HKR it carries $T\_A$ the tangent module in cohomological degree 1 and wedges of it in higher degrees and that the $E\_2$ structure gives $T\_A$ the normal Lie bracket structure and allows it to act on $A$.


Now let's suppose we have $H\_A$ acting on $M$ which is concentrated in degree $0$ as a $E\_2$ module. As for the homotopy groups of an $E\_2$ algebra, it seems to me that $M$ should inherit both a action of $H\_A$ normally and a degree lowering (sort of Lie bracket) action. More precisely, as $E\_2$ algebra's homotopy groups inherit Gerstenhaber structure I'm guessing that $M$ will inherit a similar thing. (One supporting evidence is that the space of maps $H\_A \otimes M \to M$ should be parametrized by 2 marked 2-discs in a 2-disc so there's an $S^1$ worth of such maps giving a degree lowering map of spectra)


Now assuming the above is correct, $M$ should have a D-module structure. (Does this seem correct?) One example of a $M$ is $A$ itself, which should inherit the standard D-module structure.


The question, assuming the above, is what class of D-modules is recoverable from this procedure?



What's the relationship between a $E_2$-Hochschild Cohomology module and a D-module?
(Incorporated David Ben-Zvi's comments below.)


An $E\_2$-$\mathrm{HH}^\bullet(A)$-module is the same as a left module over the algebra of Hochschild chains $\mathrm{HH}\_\bullet(\mathrm{HH}^\bullet(A))$.


Now assume $k$ has characteristic zero. It was shown by Tamarkin and Tsygan (see Theorem 2.7.1 in "The ring of differential operators on forms in noncommutative calculus") that this algebra is $A\_\infty$ equivalent to $D(\Omega^\bullet(A))$, differential operators on differential forms on $A$ (with the zero differential and homologically graded). However, there is no natural morphism $D(A)\rightarrow D(\Omega^\bullet(A))$.


Here is another way to compute it. Suppose $H$ is an $E\_2$-algebra. Then the $\infty$-category $\mathrm{LMod}\_H$ of left modules over $H$ (viewed as an $E\_1$-algebra) is monoidal; this uses the Dunn--Lurie additivity theorem of $E\_n$ algebras, see Corollary 5.1.2.6 in Higher Algebra. I claim that the Drinfeld center of $\mathrm{LMod}\_H$ is exactly the category of $E\_2$-$H$-modules. One way to see it is to consider the corresponding framed 2d TFT; the value on the circle is the Drinfeld center which can be computed using factorization homology of $H$.


Now, let's say $A$ is a commutative dg algebra concentrated in non-positive degrees and $X=\mathrm{Spec}\ A$ the corresponding derived affine scheme. The monoidal category $\mathrm{LMod}\_{\mathrm{HH}^\bullet(A)}$ is equivalent to the monoidal category $\mathbb{H}(A)$ defined in <https://arxiv.org/abs/1801.03752> (see Section 4.1 for the precise claim). Its Drinfeld center is computed in <https://arxiv.org/abs/1709.07867> to be $\mathfrak{D}^{der}(LX)$, a variant of the derived category of $D$-modules on the derived loop space $LX = X\times\_{X\times X} X$.


If $X$ is smooth, $LX$ is eventually coconnective; so, by Example 0.2.5 $\mathfrak{D}^{der}(LX) = \mathfrak{D}(LX)$ is the usual derived category of $D$-modules on $LX$. (Note that $\mathfrak{D}(LX) = \mathfrak{D}(T[-1] X)$ which agrees with the Tamarkin--Tsygan computation.) The latter satisfies Kashiwara's lemma, i.e. it is insensitive to the derived structure. In particular, the natural pushforward functor along constant loops $\mathfrak{D}(X)\rightarrow \mathfrak{D}(LX)$ is an equivalence.


