## The context

In a beautiful paper, Malikov-Schechtman-Vaintrob defined a canonical sheaf of vertex algebras equipped with a differential on any manifold $X$ (either in the $C^\infty$, complex analytic or algebraic context). They called it the **chiral de Rham complex** (it is called this way because the ordinary de Rham complex embed into the chiral de Rham complex, and this embedding is a quasi-isomorphism), and denoted it $\Omega^{ch}_X$.

They also proved in the complex analytic setting that $\Omega^{ch}_X$ carries the structure of a conformal vertex algebra. Moreover, if $X$ is Calabi-Yau (in the weak sens: $X$ admits a global holomorphic volume form) then $\Omega^{ch}_X$ admits the structure of a topological vertex algebra (such are structures are in 1-1 correspondance with $N=2$ superconformal vertex algebra structures, aren't they?).

In another paper (also very nice), Ben-Zvi-Heluani-Szczesny proved that in the $C^\infty$ context, we have that:

- if $X$ is Riemannian then $\Omega^{ch}_X$ admits a $N=1$ superconformal vertex algebra structure.
- if the metric is Kähler and Ricci-flat then $\Omega^{ch}_X$ inherits a $N=2$ superconformal structure.

## The question(s)

My question is then

What is the relation between those $N=2$ superconfromal structures when $X$ is Calabi-Yau.

From what I understand, when $X$ is kähler the complex analytic chiral de Rham complex embbed into the $C^\infty$ chiral de Rham complex, and the $N=1$ superconformal structure of Ben-Zvi-Heluani-Szczesny restricts to the conformal structure of Malikov-Schechtman-Vaintrob.

But it seems that the $N=2$ superconformal structure of Ben-Zvi-Heluani-Szczesny does not restrict to the one of Malikov-Schechtman-Vaintrob in the case when $X$ is Calabi-Yau unless the metric is flat.

Does anybody understand what is going on there?

In yet another paper Heluani contructs **yet another $N=2$ superconformal structure** on any kähler manifold $X$, which commutes with the one constructed by Ben-Zvi-Heluani-Szczesny when $X$ is Calabi-Yau.

Is this new $N=2$ superconformal structure related to the one constructed by Malikov-Schechtman-Vaintrob ? If not, then do the three $N=2$ structures commute ?