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 ?


1 Answer 1


Ooops, I should start reading this site. In some sense all of these N=2 structures are the same. Unfortunately we now understand the situation well better than we did then.

To simplify the answer you may think of the $C^\infty$ Chiral de Rham (CDR) as the tensor product of a holomorphic CDR with a anti-holomorphic one (both commuting, and I presume this is the embedding you mentioned in the post). Then [MSV] constructed two commuting copies of N=2: one in each sector. The N=2 structure of [BZHS] would be their difference.

The unfortunate misleading comment in [BZHS] about the N=2 structure there agreeing with [MSV]'s only with flat metrics is due to the following. The fields in [BZHS] are written in general coordinate systems that's why they depend on a choice of a global holomorphic volume form in the CY (N=2) case. This dependence enters there as derivatives of $\sqrt{\det g}$. The fields in [MSV] are written in the coordinates where the volume form is taken to be constant and so this term does not appear. One of the advantages of having the fields in general coordinates was to treat the N=4 case which is the main point of [BZHS].

To compare the two N=2 structures of [MSV] (holomorphic + anti-holomorphic) with the two in the second paper you mentioned (I'll call that [H]) the situation is subtler. In short: I claim that there is a much more natural embedding of holomorphic and anti-holomorphic CDR into the smooth one than the obvious one, and this has to do with different topological twistings.

Having the fields in general coordinates makes it clear that the two different N=2 structures in [H] correspond to the complex and symplectic structures of the Kahler manifold (or more generally to the two generalized complex structures defining the Kahler structure). In order to compare the two N=2 structures of [MSV] with the two in [H] one needs to embed the (anti)holomorphic CDR into the smooth one in a different way which essentially amounts to the orthogonal identifications (here I'm using the standard Kahler structure on $\mathbb{C}$).

$dz \rightarrow dz + \partial_{\bar z}, \qquad \partial_{z} \rightarrow d\bar{z} + \partial_z$

This is the content of Remark 8 of this article which in turn is an infinite dimensional manifestation of the fact that the p,q decomposition of generalized Kahler manifolds is not the Dolbeaut decomposition, but rather this orthogonal transformation of it as explained in Gualtieri's article. With this different embedding then the two structures coincide (but this is non-trivial since the Kahler form enters the embedding).

Perhaps is just a form of advertizing, but CDR (at least for the purpose of these computations) should be viewed as a Courant-algebroid version of the (N=1 super) affine vertex algebra associated to a Lie algebra with an invariant non-degenerate form (,). The computations in that article with Zabzine mentioned above might look longer due to the generality of the situation (in the Generalized CY situation dilatons are unavoidable) but are coordinate free and explicitly invariant, perhaps taking a look at that article will clarify better your questions than this post. I hope this helped a little.


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