Twisted differential operator, chiral differential operator, $???$ (continue the sequence) Let $X$ be a smooth variety.

*

*One can define the notion of a sheaf of twisted differential operators (TDO) on $X$. They "quantise" functions on $T^*X$. Examples include the usual sheaf of differential operators $\mathcal{D}$. TDOs are classified by the complex of sheaves
$$K_1\ =\ \Omega_X^1\ \longrightarrow\ \Omega^{2,cl}_X$$
meaning that TDOs are parametrised by the first hypercohomology $H^1$ of this complex, and for each TDO its automorphisms are $H^0$. You should maybe think of the obstruction $ob_X$ to the existence of TDOs as living in $H^2$, but it's just that $ob_X=0$ for all $X$ (because of the existence of $\mathcal{D}$).

*One can also define chiral differential operators (CDO). They "quantise" functions on $\text{Arc}(T^*X)=\text{Map}(\text{disk},T^*X)$, and are even vertex algebras. They are classified by (see 5.3 of linked article)
$$K_2\ =\ \Omega_X^2\ \longrightarrow\ \Omega^{3,cl}_X$$
in the same way as in 1, except that now the obstruction $ob_X=ch_2(T_X)$ does not vanish, so CDOs do not necessarily exist.

*?


My question is whether this pattern continues, i.e. are there interesting geometric objects classified as above by complexes of sheaves $K_n$, and whether e.g. they are related to iterated loop spaces in the same way that CDOs are related to loop spaces. The obvious guess is of course $K_n=\Omega^n_X\to \Omega_{X}^{n+1,cl}$.
 A: $\newcommand{\co}{\mathcal{O}} \newcommand{\H}{\mathrm{H}} \newcommand{\dR}{\mathrm{dR}} \newcommand{\GG}{\mathbf{G}}$
Let $k = \mathbf{C}$, let $X$ be a (smooth) variety over $k$, and let $X_\dR$ be its de Rham space (so that the cohomology of $\Gamma(X_\dR; \co_{X_\dR})$ is $\H^\ast_\dR(X/k)$. There is a canonical map $f: X \to X_\dR$, and the global sections of the fiber of the map $\co_{X_\dR} \to f_\ast \co_X$ is $\H^\ast(X; \Omega^{\bullet\geq 1}_{X/k})$. In other words, the map $\co_{X_\dR} \to f_\ast \co_X$ corresponds to the projection $\Omega^{\bullet}_{X/k} \to \co_X$ (whose fiber is $\Omega^{\bullet\geq 1}_{X/k}$, with $\Omega^i_{X/k}$ placed in homological degree $-i$). From this point of view (some of the indices may be off, let me know if you find mistakes):

*

*The group $\H^2(X; \Omega^{\bullet\geq 1}_{X/k})$ may be understood as $\pi_0$ of the space of $\GG_a$-gerbes on $X_\dR$ which pull back to the trivial $\GG_a$-gerbe along $f$. Note that $\H^2(X; \Omega^{\bullet\geq 1}_{X/k}) = \H^1(X; K_1)$ in your notation. (The relation to TDOs comes from an identification of $\GG_a$-gerbes on $X_\dR$ which pull back to the trivial $\GG_a$-gerbe along $f$ with $\GG_m$-gerbes on $X_\dR$ which pull back to the trivial $\GG_m$-gerbe along $f$, using the exponential on the formal completion of $\GG_a$. Because $\mathrm{QCoh}(X_\dR)$ describes D-modules on $X$ (using the finite-type assumption on $X$), $\GG_m$-gerbes on $X_\dR$ correspond to twistings of $\mathrm{DMod}(X)$. See Gaitsgory-Rozenblyum's https://arxiv.org/abs/1111.2087 for more on this.)


*Similarly, $\H^3(X; \Omega^{\bullet\geq 2}_{X/k}) = \H^1(X; K_2)$ in your notation, and this can be understood as $\pi_0$ of the space of $\GG_a$-$2$-gerbes on $X_\dR$ which pull back to the trivial $\GG_a$-$2$-gerbe along $f$.


*In general (if I haven't snuck in a $\pm 1$ mistake), if we define the space of $\GG_a$-$n$-gerbes on $Y$ to be the space of maps $Y \to B^{n+1} \GG_a$, then  $\H^{n+1}(X; \Omega^{\bullet\geq n}_{X/k}) = \H^1(X; K_n)$ in your notation, and this can be understood as $\pi_0$ of the space of $\GG_a$-$n$-gerbes on $X_\dR$ which pull back to the trivial $\GG_a$-$n$-gerbe along $f$.
For this to be a satisfying answer, we should relate $\GG_a$-$2$-gerbes on $X_\dR$ to chiral differential operators . This should be given by an algebraic version of transgression (which, in topology, would be a map $\H^\ast(Y) \to \H^{\ast-1}(LY)$ where $Y$ is an oriented manifold). I don't know a reference for this in the algebraic setting (but I would like to know of one if it exists!), but if you are interested I can try to sketch the idea. I think $K_n$ wouldn't be related to iterated free loop spaces, but rather to some algebraic analogue of $\mathrm{Map}(S^n, Y)$.
