A chiral algebra on a smooth curve $X$, in the sense of Beilinson-Drinfeld, is a right $D\_{X}$-module with a *chiral bracket*, which is a map $\mathcal{V}^{\boxtimes 2}(\infty\Delta)\rightarrow \Delta\_{\*}\mathcal{V}$ satisfying certain conditions making it look like a Lie bracket. In fact $\mathcal{V}$ is a Lie algebra with respect to a certain *pseudo-tensor* structure on $D\_{X}$-modules. A pseudo-tensor structure is what one obtains when one forgets the tensor product but remembers the associated multilinear maps. Multi-linear maps from a collection $\{\mathcal{V}\_{i\in I}\}$ to $\mathcal{W}$ are defined to be $\mathrm{Hom}\_{D\_{X^{I}}}(\boxtimes\_{i\in I}\mathcal{V}\_{i}(\infty\Delta\_{\mathrm{big}}),\Delta\_{\*}\mathcal{W})$. One thus obtains a complex of vector spaces $C^{\*}\_{CE}(\mathcal{V})$ via the appropriate construction of Chevalley-Eilenberg cochains, which makes sense in a pseudo-tensor category. Notably this formalism *does not* produce CE chains.


If $X$ is taken to a be a formal disc $D$, then we obtain a vertex algebra. I believe in this case the above construction produces the vertex algebra cohomology (with coefficients in the adjoint module) studied in the work of Bakalov, de Sole, Heluani and Kac ([arXiv link](https://arxiv.org/abs/1806.08754)). Is this true? If not what is the precise relation?


Now if $\mathcal{V}$ is a chiral algebra, then BD construct its *factorization homology*. This is constructed as de Rham homology of the associated $D$-module on the Ran space of $X$. In chiral terms I believe it is gotten by taking (the colimit over finite sets $I$) of the Chevalley Cousins complexes on $X^{I}$, cf. these notes by Rozenblyum ([on Gaitsgory's webpage](http://people.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Apr20-27-May4(Chiral).pdf)). 


This can also be performed on a formal disc $D$\*, in which case one obtains what one might call Vertex *Homology*. Has this been studied anywhere? Is there a relation with the work of BdSHK cited above, in particular does the BdSHK complex act on it?


For example, if I'm not mistaken we compute $H^{\mathrm{vtx}}\_{0}(V)=V/\operatorname{Res}(V)$, where $\operatorname{Res}(V)$ is the image of all the residues $\operatorname{res}\_{z=0}(v(z))$ of the fields, which has a very similar flavour to the complex of BdSH. 


\*Edit, in accordance with David Ben-Zvi's comments it seems that vertex algebra homology as defined in this question would be better defined over a punctured disc $D^{\*}$. 



Factorization and vertex algebra cohomology

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> If $X$ is taken to a be a formal disc $D$, then we obtain a vertex algebra. I believe in this case the above construction produces the vertex algebra cohomology (with coefficients in the adjoint module) studied in the work of Bakalov, de Sole, Heluani and Kac. Is this true? If not what is the precise relation?
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This is not exactly so. If you take $X=\mathbb{A}^1$ and your D-modules are translation equivariant and you consider the translation equivariant CE complex you will get the same complex that we studied. 


Chiral algebras on the disc are **not** the same thing as vertex algebras. If you consider chiral algebras equivariant with respect to the action of the group $Aut\_\mathcal{O}$ of changes of coordinates then they are the same as *quasi-conformal* vertex algebras (notation of Frenkel Ben-Zvi). The $Aut\_\mathcal{O}$-equivariant complex as you describe on the punctured disk will be equivalent to ours in the quasi-conformal setting. 



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> This can also be performed on a formal disc $D^\*$, in which case one obtains what one might call Vertex Homology. Has this been studied anywhere? Is there a relation with the work of BdSHK cited above, in particular does the BdSHK complex act on it?
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Again if you want something that should be called "vertex homology" then you should consider either $Aut\_{\mathcal O}$ equivariant objects on the disk or translation equivariant objects on the line. 


There are a few statements in this answer that are plain wrong cause I had in mind translation invariants sections on the disk. I'll strike them through. 


~~Factorization homology of a unital chiral algebra vanishes on an open proper subset of a curve.~~ Factorization homology of the restriction of a chiral algebra to an affine proper subset of a curve vanishes. This is the content of Lemma 4.3.4 in Beilinson and Drinfeld's book. 


~~I also believe that David's comments about Factorization homology on the disk are wrong. There's a big difference between the topological setting and the algebraic setting and between $E\_2$ algebras and the chiral/factorization algebras of Beilinson and Drinfeld. In particular, the chiral homology of the disk with coefficient on any unital chiral algebra vanishes. This is because chiral homology is not just the CE complex of a Lie algebra on the Ran space, but also it's de Rham complex. In particular as you point out the degree zero homology of this complex is $V/V\_{(-1)}V = 0$ and this is because we have the vacuum vector to play around. Therefore the image of $\mathbb{1}$ in chiral homology vanishes and then by the last comment before the mentioned Lemma 4.3.4 the whole complex vanishes.~~ 


Chiral homology of a curve is really a global object and you cannot detect any of it by looking at formal disks (see however 2.4.12 in BD in agreement with David's comments). The complex studied in BdSHK is purely a local object, or rather a translation equivariant complex on the line. I am not aware of any direct connection between them. 


There are however a couple of instances where you can compute the global chiral homology by local considerations: the case of $X=\mathbb{P}^1$ and the case of an elliptic curve. Both cases because you have global coordinates. In degree $0$ this is the work of Zhu way before chiral homology was defined. In the elliptic curve case we have a preprint with J. van Ekeren describing the nodal curve limit of the first chiral homology group and we are supposed to post a continuation describing the general elliptic curve soon. 


**Edit:** To reflect David's comments. Indeed by $U\subset X$ open they really mean an open affine, so that the example of a disconnected curve is not allowed. The point needed is the vanishing of the de Rham cohomology. They use this lemma heavily in proving the quasi-isomorphism of the complex with supports in 4.4.2--4.4.3. 


~~Commutative vertex algebras are not a counter-example to this statement as the functor of coinvariants on the disk is canonically zero as long as your algebras are unital. Here the very naive notion of coinvariants is clear, the Cousin complex on $D^2$ is already enough to compute the homology in degree $0$ and this group is $V/V\_{(-1)}V$ (think of homologies with support at the origin).~~


As for the $E\_2$ algebras business: here is where things get sloppy. As far as I know the folklore is that $E\_2$ algebras are essentially the same as topological vertex algebras. With this I have no objection, however notice that the latter is a vertex algebra together with a BRST differential, and that this equivalence is inherently derived. Since the BRST cohomology of a topological vertex algebra is typically stupid (it lies in conformal weight zero and is a commutative algebra) then it's no wonder that the chiral homology viewed as a complex with this extra BRST differential is silly. However the complex without taking this differential into account is still very interesting, you capture irreducible modules and Jacobi elliptic functions on genus $1$ (at least in rational cases) and so forth. 


The second point is more subtle: I have seen in many places the analogy drawn between Factorization homology in the topological world and chiral homology in the algebraic setting. I may be wrong and very much outdated in the literature that I can cope with, but I have not seen anything written to prove an actual dictionary. There is no formal way of constructing a factorization algebra (in the topological meaning) coming from a factorization algebra in the algebraic world. Folklore expected that if one takes the complex that computes chiral homology on the Ran space of $X$ *á la BD* and one takes an analitification of this and then sections with compact support, this is something that looks like a Factorization algebra in the topological setting. As far as I know the problem is that the process of analytification (or tensoring with the analytic de Rham complex) does not commute with tensor products and this breaks down being a Lie/Commutative algebra. 


I don't claim that having a topological intuition is wrong, just that at least when trying to translate that intuition into an actual theorem in BD's setting the situation is not so transparent even in the simplest of cases (for example the zeoro-th Hochshild Homology of higher Zhu algebras equals zero-th chiral homology of vertex algebras, this is a theorem that requires a lot of theory and a bunch of further assumptions that have no topological counterparts). 


