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 ?