# What are advantages of chiral algebras over vertex algebras?

In Chiral Algebras, Beilinson and Drinfeld introduced the notion of chiral algebra, which based on the pseudo-tensor category of D-modules. On the other hand, There is already a notion of vertex algebras based on formal power series. Chiral algebras on curves are essentially conformal vertex algebras.

My question is: what are advantages of chiral algebras over vertex algebras other than that the definition is kind of shorter?

1. It is not necessarily true that chiral algebras are essentially conformal vertex algebras, as chiral algebras are allowed to vary over the curve in a way that vertex algebras are not. For instance, on $\mathbb{A}^1$, only translation-invariant chiral algebras give rise to vertex algebras. You can see this looking just at commutative chiral algebras ($\mathcal{D}$-schemes): Take the jet scheme of a nontrivial fibration over $\mathbb{A}^1.$ However, in my (limited) personal experience, this extra generality is not used in an essential way in most applications of chiral algebras.