A couple of day ago, I was lamenting to a friend about the fact that I have no idea what vertex algebroids are.
During our discussion, I came up with a guess of what a vertex algebroid ** might** be.
I'm wondering whether this guess has any merit.

Question:is the definition below equivalent to what people call "vertex algebroid"?

Let me first tell you what I know about vertex algebras:

## Vertex algebras

Let $\mathbb D=Spec(\mathbb C[[t]])$ be the formal disk, and let $\mathcal M_n$ denote the moduli space of n-tuply punctured formal discs.

(While you probably have some intuition as to what it means to puncture $\mathbb D$ once, but you might be puzzled by the idea of puncturing it more than once. Indeed, the moduli space of twice punctured formal discs doesn't have any $\mathbb C$-points, so it's not possible to exhibit any twice punctured formal disc! However, given the once punctured formal disc $\mathbb D^*=Spec( \mathbb C[[t]][t ^ {-1}])$, there's $\mathbb D^*$ many places where one can perform the second puncture. So that produces a map $\mathbb D^*\to \mathcal M_2$, showing that $\mathcal M_2$ is at least non-trivial.)

The collection of all $\mathcal M_n$ looks a lot like the little discs operads:

But if you think about it, you'll see that it's not quite an operad: it's only a partially defined operad.
The operad multiplication
$$
\mathcal M_n\times \mathcal M_{k_1}\times\ldots\times\mathcal M_{k_n}\to \mathcal M_{k_1+\ldots+k_n}
$$
is only defined on a "subset" of $\mathcal M_n\times \mathcal M_{k_1}\times\ldots\times\mathcal M_{k_n}$.
But that doesn't matter so much: once can still define the notion of an algebra over this partially defined operad, and it turns out that an algebra over $\mathcal M_\bullet$ is exactly the same thing as a vertex algebra.

## Vertex algebroids?

Fix a smooth variety $X$ and consider the moduli space of n-tuply punctured formal discs equipped with a map to $X$. Those again form a partial operad (probably it's better to call it a colored partial operad), and one can consider algebras over it.

Could it be that those algebras are equivalent to vertex algebroids over $X$?

Two-Dimensional Conformal Geometry and Vertex Operator Algebrastalks about partial operads. Those give what he calls "geometric vertex algebras". He then shows (if I remember correctly) that the category of geometric vertex algebras is equivalent to the category of vertex algebras. Note however that his punctured discs are not formal: they have non-zero spatial extent. If you let the punctured discs be formal, then you get something that is much closer to the actual definition of vertex algebras. $\endgroup$ – André Henriques Feb 9 '13 at 23:07`E infty ring spaces and E infty ring spectra, VII\S1, SLN 577, 1977. I defined the`

little convex bodies'' partial operads there for purposes of multiplicative infinite loop space theory (one point being that the little discs operads do not suspend properly). In that context Steiner came up with a better solution using honest operads, in`A canonical operad pair''. Further off, partial DGAs and their modules come up in mixed Tate motives (Kriz-May,`

Operads, algebras, modules, and motives''. $\endgroup$ – Peter May Feb 10 '13 at 1:54as themselves already are (encodings of) "somethingoids" :D $\endgroup$ – მამუკა ჯიბლაძე May 10 '16 at 8:37