The center of H(Y)

Let $Y$ be a derived geometric stack defined over an algebraically closed
field of characteristic zero and satisfying three technical conditions:
eventual coconnectivity, perfection and local finite presentation. % For such
$Y$, we consider the DG category $IndCoh(Y \times_{Y_{dR}} Y)$ of ind-coherent
sheaves on the formal completion of the diagonal of $Y$, equipped with the
convolution monoidal structure. Within $IndCoh(Y \times_{Y_{dR}} Y)$, we single
out a monoidal full subcategory $\mathbb H(Y)$, defined as follows: an object
of $IndCoh(Y \times_{Y_{dR}} Y)$ belongs to $\mathbb H(Y)$ if its pullback
along the diagonal $\Delta: Y \to Y \times_{Y_{dR}} Y$ is quasi-coherent.
Such DG category $\mathbb H(Y)$ will play an important role in several future
applications. For instance, let $G$ be a connected reductive group, with
Langlands dual $\check{G}$, and ${LocSys}_{\check{G}}$ the stack of
$\check{G}$-local systems on a smooth complete curve. Then $\mathbb
H({LocSys}_{\check{G}})$ is expected to act on both sides of the geometric
Langlands correspondence compatibly with the conjectural equivalence
$IndCoh_{N}(LocSys_{\check{G}}) \to D-mod(Bun_G)$.
In this paper, we identify the Drinfeld center of $\mathbb H(Y)$ with
"$D-mod(LY)$", a certain version of the DG category of left D-modules on the
loop stack $LY := Y \times_{Y \times Y} Y$. Contrarily to ordinary (left)
D-modules, "$D-mod(LY)$" is sensitive to the derived structure of $LY$. By
construction, "$D-mod(LY)$" is the DG category of modules for a monad
$U^Q(\mathbb T_{LY})$ acting on $QCoh(LY)$. In turn, the monad $U^Q(\mathbb
T_{LY})$ is obtained from the universal envelope of the Lie algebroid $\mathbb
T_{LY}$ by a renormalization procedure that uses the PBW filtration.