Physical interpretation to the category of CFTs

Question

This question comes from reading Andre's question where I wandered whether that question even makes sense physically. In mathematics, VOAs form a category, does this category as a whole have a physical interpretation?

Answer 1

Any kind of objects, in physics or in math, is only genuinely defined when you also define what the homomorphisms between them are supposed to be, hence when you define the category that the objects form.

Notably the invertible homomorphisms (the isomorphism) encode which objects are equivalent. There may be objects that superficially look very different and neverteless have invertible morphisms between them. Without the morphisms, you would never be able to tell that these objects are in fact equivalent. In physics this important phenomenon is often referred to as "duality". See the Gannon-Höhn Database of Vertex Operator Algebras and Modular Categories for a list of nontrivial isomorphisms in the category of VOAs.

Next, the homomorphisms which behave like injections (the monomorphisms) are needed to even define what it means to have a subobject. Physically this means: a subsystem. You can't even say (precisely) "heterotic string" or "CY compactification" or the like without a notion of monomorphism of VOAs.

Then there are the monoidal structure morphisms. VOAs form a monoidal category under tensor product. So it makes sense to ask if the tensor product of a VOA with its dual has morphisms to and from the trivial VOA. If these exist in a certain way, this encodes self-duality of the VOA. The famous physical example here is the Moonshine VOA encoding a certain bosonic string compactifications. Mathematically this is a self-dual object of rank 24 and no grade 1-subspaces in the monoidal category of VOAs, and by a famous conjecture by Frenkel, Lepowsky and Meurman it is uniquely characterized by this universal property in the category of VOAs. This means that the Monster string background exists due to universal structure in the category of VOAs.

Finally, only with the category of VOAs in hand is it possible to check for equivalences to other categories and hence discover equivalences of VOAs as a whole to other structures. The famous example of such an equivalence of most fundamental importance to the physics descriped by VOAs is Huang's theorem which says that the category of VOAs is equivalent to that of holomorphic algebras over the operad $\mathcal{HS}$ of punctured holomorphic spheres. As opposed to the definition of VOAs, an $\mathcal{HS}$-algebra is manifestly a genus-0 holomorphic 2d CFT in that it is a rule that assigns a correlator to each punctured conformal sphere, such that these correlators obey the sewing law. Starting from this Huang's student Kong derived precisley the extra structure neccessary to promote a VOA to a full 2d CFT, hence to a full string theory background (see the references here). So it's the category theory of VOAs that fully informs us about their full physical meaning in the first place.

Answer 2

My first naive guess (in line with Andy's comment but not terribly well thought out) is that no, morphisms of VOAs are not physically terribly natural - rather the more natural thing to consider is an appropriate notion of bimodule for two VOAs, which would be domain walls between the corresponding chiral CFTs (or 3d TFTs, in the rational case).

To make an analogy one dimension down, let's think that we have an associative (or if you prefer $A_\infty$) algebra, and we use it to try to define a 2d TFT --- i.e. we can always integrate it over the circle to get a vector space (take Hochschild homology or center, depending on how you think of circles) and if it's fully dualizable (f.d. semisimple in the abelian case or "smooth proper" in the dg case) we can integrate it over 2-manifolds to get numbers. However from the TFT point of view what's important is the category of modules over the algebra rather than the algebra itself -- i.e., relations between two algebras are given by bimodules (ie Morita morphisms), not necessarily by morphisms of algebras. These are exactly domain walls between 2d field theories. (This is also natural from thinking of 2d TFTs as noncommutative varieties --- only in the commutative case does it really make sense to focus on maps of algebras, since in that case you can recover the algebra from the corresponding category or TFT).

[If you want honest 2d CFTs rather than modular functors then you want a modification of the story above..]

Likewise I think (following Costello and Lurie) of a VOA as what you attach to a point in a 2d chiral CFT or modular functor (ie we're attaching vector spaces to Riemann surfaces, obtained by integrating the VOA over the surface --- conformal blocks, aka chiral homology). The coarse topological analog is an E_2 algebra. In any case what seems physically meaningful is domain walls between these modular functors (or the corresponding 3d TFT if it makes sense), and these are "chiral bimodules" for two vertex algebras: something you can put on a wall, so that on either side you have bulk operators given by your two VOAs (and in particular there's also a "boundary OPE" structure on this chiral bimodule --- topologically this would be an associative algebra object in bimodules over two E_2 algebras).

Anyway to summarize usual morphisms of vertex algebras are special cases of something more natural, which are domain walls of chiral CFTs, which give monoidal functors between the monoidal categories of left modules ("boundary conditions") for the two VOAs..

Answer 3

The category of CFT's (and related 3D TQFT) has been studied by Kapustin and Saulinas in their recent paper Topological boundary conditions in abelian Chern-Simons theory.

The quote "we obtain a classification of such theories up to isomorphism" from their abstract refers to a notion of isomorphism that is clearly higher-categorical (more precisely, it appears to be 3-categorical).

See also Surface operators in 3d Topological Field Theory and 2d Rational Conformal Field Theory by the same authors.