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?


2 Answers 2


Some comments:

  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.

  2. The chiral algebra formalism replaces the power series manipulations of vertex algebras with algebraic geometry. Personally I find the algebra-geometric explanations more conceptual, but this really is a matter of taste. For most calculations at the level of vector spaces, one can translate between the two formalisms.

  3. Most importantly, chiral algebras are equivalent to factorization algebras, i.e. D-modules on the Ran space satisfying a factorization condition. Firstly, this more abstract perspective allows for very neat geometric constructions of factorization algebras. As an example, for an affine Lie algebra one can give both the vacuum module and the integrable vacuum module a factorization algebra structure (and thus chiral and vertex algebra structures) by interpreting them in terms of the Beilinson-Drinfeld Grassmannian (which is a factorization space.) The real power of this interpretation comes when you want to work one categorical level higher. Then you have a notion of factorization category, which has no vertex analogue. If you want to work with such categories, you need to make sure that all your constructions work factorizably and it's easier to do so if you are working in the factorization setting from the start.


The main advantage of chiral algebras over vertex algebras is that they admit "very functorial" definitions, and this helps more general concepts and constructions appear naturally. The usual examples involve factorization spaces like the Beilinson-Drinfeld Grassmannian, and applications to the Geometric Langlands program. Another example is the concept of chiral homology, which can be viewed as derived version of a coinvariant construction, but has taken on a new life as factorization homology in the theory of extended TQFTs.

I disagree with the claim that the definition of "chiral algebra" is shorter than that of "vertex algebra". You really need to build up a substantial body of theory even to describe the chiral pseudo-tensor structure on D-modules. Vertex algebras are fundamentally vector spaces with a funny looking multiplication structure.

If you are trying to decide whether you want to invest more time in learning about chiral algebras versus vertex algebras, it may be worth your time to ask which notion has yielded substantial results that interest you. Chiral algebras have been available to the public for about 20 years (since Gaitsgory's IAS notes), and vertex algebras have been around for about 30. That is, neither notion is particularly new.


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