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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?

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I think a good question should be self-contained. This one is not. What's a VOA, what is the relation between your question and the question you're linking to? Please provide more background. –  Marcin Kotowski Oct 15 '11 at 1:59
VOA=Vertex operator Algebra, which is like the chiral part of a CFT. The question I link to talks about embeddings of CFTs, which in particular would be morphisms between them. I wandered whether a "morphism" of CFT (as an inclusion would be) had a physical interpretation. –  Reimundo Heluani Oct 15 '11 at 2:03
A naive guess would be that morphisms of VOAs would correspond physically to domain walls coupling different CFTs. –  Andrew Neitzke Oct 20 '11 at 18:45
@Andy: thanks, that does sound like something a friend would say, but I suspect that in order to think this way you need to promote the category to a higher one and then these would be "1-morphisms" which in turn won't be the usual morphisms of VOA... and then again I'm getting further away from what I was hoping. –  Reimundo Heluani Oct 20 '11 at 23:06

3 Answers 3

up vote 9 down vote accepted

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..

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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.

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I'm sorry Urs but I was hoping for a physical interpretation of the category structure and I what I can read of your answer is a physical interpretation of isomorphisms and to some extent of monomorphisms. I am very familiar with Huang's work, but calling that a physical interpretation is some leap, most people working on CFTs won't know what an operad is, and remember that CFTs are also used in condensed matter physics, it is not only a stringy thing. More than the monoidal structure (composite systems) I am interested in basic things like associativity for composition of morphisms and such. –  Reimundo Heluani Oct 15 '11 at 21:07
Any kind of object only has an interpretation with a given definition of homomorphism. The very fact that the definition of a VOA has something to do with CFT rests in the morphisms. For instance, if we redefine the morphisms between VOAs to be linear maps of the underlying vector spaces, then VOAs would become equivalent to just vector spaces, and all the CFT structure were lost. Or, even more drastically, if you declared that there is precisely one morphism from any VOA to any other, then that would make the theory of VOAs become equivalent to the theory of the contractible space. –  Urs Schreiber Oct 17 '11 at 6:08
Moreover, without the homomorphisms defined, no universal construction of the objects exists. For instance the adjoint construction of open-closed TCFTs from open (T)CFTs in arxiv.org/abs/math/0412149 rests on the definition of morphisms between (T)CFTs. –  Urs Schreiber Oct 17 '11 at 6:13
Generally, if you pass from just the category of VOAs to that of full CFTs, you see more exmaples of the direct physical role of homomorphisms. For instance morphisms from a trivial (n+1)-dimensional theory to a nontrivial one define "twisted" n-dimensional theories, as in arxiv.org/abs/1108.0189 . In a similar fashion the very definition of rational 2d CFT is holographically given, as discussed in arxiv.org/abs/hep-th/0612306 . –  Urs Schreiber Oct 17 '11 at 6:23

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.

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Thanks André and David. Actually I was thinking on you by the "friend" in my comment above and was hoping that the answer would not have to involve higher categories. I'll accept BZ's answer just because it is an hour older and David may need the reputation to comment. –  Reimundo Heluani Nov 3 '11 at 21:24
Thanks Reimundo - the inability to comment was incredibly frustrating! –  David Ben-Zvi Nov 3 '11 at 23:21
If I understand correctly, Kapustin-Saulina are talking about a (higher) category of CFTs, not of VOAs -- in other words I think they're dealing implicitly with chiral bimodules for VOAs rather than morphisms. (In other words the natural notion of map of field theories doesn't correspond to a map of VOAs). –  David Ben-Zvi Nov 3 '11 at 23:25
The CFT's in Kapustin-Saulina are not chiral (they are when the torus has definite signature), and the sentence "the natural notion of map of field theories doesn't correspond to a map of VOAs" is definitely correct. –  André Nov 4 '11 at 8:30

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