How to start learning vertex operator algebras and their representation theory
How to start learning vertex operator algebras and their representation theory
VOAs are quite hard to learn. One reason for this may be that they best require formal calculus to define. However, formal calculus is fairly intuitive (though there are definitely some subtleties), so it's not really that much of a hurdle to quickly learn. Another reason is that the definition is fairly long and notationally intimidating! This is certainly true if you define it using the Jacobi identity or without formal calculus, but there are equivalent formulations that can avoid it (like using the locality axiom instead), and these are more easily digestible. The main reason is possibly that there are no good simple examples lying around to extract intuition from. Compare this to the usual classical algebraic structures like groups, or Lie algebras, or commutative unital associative algebras, etc. where there are plenty of good simple examples to use for guidance while learning. My guess is that VOAs are hard to learn, not just because the definition is hard to comprehend, but mostly because the objects themselves are not easily constructed.
Whatever the reason, I think it's safe to say that VOAs are hard to learn. Since it's hard to learn VOAs, it's also hard to teach. This gives strong evidence for the following conjecture.
Conjecture: If you go to a talk on VOAs, the speaker will not define VOAs.
I have attended a fairly large sample of talks on VOAs and, as far as I know, this conjecture appears to be true. Hopefully, it gets disproven one day. This is a joke, of course, but my point is that the best place to learn the precise definition for VOAs is in writing, and probably a book. That said, it is still extremely helpful to look for handwavy explanations in the form of talks, YouTube videos, and discussions over strong black coffee in very small cups.
Here's another thing. Some people will tell you that you only need to know linear algebra (up to tensor products) to learn VOAs. That's kind of true, but it's going to be quite difficult. To more easily understand what's going on, you will need to know some Lie algebra representation theory. Say, Humphrey's or equivalent. You won't actually need all of it, just the representation theory and the parts before it that you need in order to understand the representation theory. This will help you a lot when learning VOAs.
And another thing. You don't need to know any (monoidal/tensor/braided/rigid/modular etc.) category theory to learn VOAs. You will, of course, need it if you want to learn how VOA representation theory can construct tensor categories.
You also don't need to learn any physics. It will definitely help you to know some very basic string theory and conformal field theory from the Yellow Book or other references (see bottom of page). It will give you great intuition, and computations are good for the soul, but I've seen plenty of people learn VOAs without knowing a significant amount of conformal field theory.
My suggestions for references learning VOAs and VOA representation theory:
Vertex algebras were defined by Borcherds in Vertex algebras, Kac-Moody algebras, and the Monster (1986). The key example was the Moonshine Module constructed by Frenkel-Lepowsky-Meurman in A natural representation of the Fischer-Griess Monster with the modular function J as character (1984). The Moonshine Module does indeed have the structure of a vertex algebra, but neither of these papers contains a proof of this fact .
Vertex operator algebras are a variant notion with additional datum and axioms defined by Frenkel-Lepowsky-Meurman in the famous Vertex operator algebras and the Monster (1988). Here, the Moonshine Module was constructed in full detail and proven to be a vertex operator algebra. Many other things are proven in this book. It also slowly illuminates the ingredients of vertex operator algebras to the reader, in a similar fashion to how they were discovered. This may be too slow for a modern reader learning VOAs for the first time, but it definitely has its place on a VOA theorist's bookshelf.
On the other hand, if you want to know what a vertex operator algebra is as quickly as possible with no examples, then Frenkel-Huang-Lepowsky's On Axiomatic Approaches to Vertex Operator Algebras and Modules (1993) is the book for you. This is a very good axiomatic treatment, especially for understanding intertwining operators. But it may be too fast for a reader who does not know of any examples.
The best book, in my opinion, is Lepowsky-Li's Introduction to Vertex Operator Algebras and Their Representations (2004). It is very precise and explains everything that it promises. It clearly explains the equivalence between different formulations of vertex algebras. It also beautifully explains how "representations" are related to "modules". One downside is that it leaves the non-trivial examples to the very end; it is essentially an axiomatic treatment for the majority of the book. So, it's best to come prepared with at least one example in your mind. It does, however, produce many examples extremely swiftly and clearly using the earlier material, which is quite nice.
Other than that, there is Kac's Vertex algebras for Beginners (1996), which gives some great physical insights. For the algebraic geometer, there is Frenkel-Benzvi's Vertex Algebras and Algebraic Curves (2001). These two books use slightly different notation than the first three books, notably the delta function. This is not much of an issue if you're careful.
In summary, to learn vertex operator algebras and their representation theory, I would suggest reading Lepowsky-Li while occasionally looking at some early chapters of Kac and Frenkel-Benzvi for some examples and a different viewpoint. If you want to learn how this fits in with the algebro-geometric story, then you should read Frenkel-Benzvi. If you want to learn how to construct the Moonshine Module, then you should read Frenkel-Lepowsky-Meurman. If you want to learn about intertwining operators (which you will need for the tensor category theory), then you should read Frenkel-Huang-Lepowsky.
Now for the tensor category theory. Assuming that you already know some category theory from Mac Lane, or equivalent, then Etingof-Gelaki-Nikshych-Ostrik's Tensor Categories (2015) is a really good book for learning what the title says. This book doesn't go into any VOA theory, but it will contain all the category-theoretic definitions that you'll need to start to understand what people mean when they say that VOA representation theory builds braided tensor and modular tensor categories.
Huang-Lepowsky-Zhang's eight-part series Logarithmic tensor category theory (known as "HLZ") explains how to construct braided tensor categories out of categories of V-modules. To fully understand this series, you will need to know a definition for VOAs that involves the Jacobi identity. This is because you'll need to do algebraic calculations with the Jacobi identity for intertwining operators.
Huang's Rigidity and modularity of vertex tensor categories proves the rigidity and modularity of the tensor categories under suitable conditions. Interestingly enough, his proof requires his results in Vertex operator algebras and the Verlinde conjecture. This is surprisingly somewhat converse to the common narrative that the Verlinde formula follows from the modular tensor category axioms.
You'll notice that Huang's modularity result appeared before HLZ. This is because there was another series A theory of tensor products for module categories worked out by Huang-Lepowsky in the 90's. You don't need to read this; HLZ is a better and more general version.
The braided tensor category construction in HLZ comes from the a vertex tensor category structure, which is roughly a category with a tensor product for each Riemann sphere with three punctures (two incoming, one outgoing). HLZ explicitly deals with only trivial local coordinates at each puncture, but vertex tensor categories as first introduced in Huang-Lepowsky's Tensor Products of Modules for a Vertex Operator Algebra and Vertex Tensor Categories can have even more structure by involving non-trivial local coordinates. The vertex tensor category theory is based on the geometric view explained in Huang's book Two-Dimensional Conformal Geometry and Vertex Operator Algebras.
My suggestions for references learning CFT:
By conformal field theory (CFT), we mean two-dimensional conformal quantum field theories. The field of study called "Conformal Field Theory" aims to study conformal field theories and related structures. There is no standard definition for a "conformal field theory", but there are various different attempts to define the object that we think of as a "conformal field theory". Different perspectives give different definitions (e.g., algebraic definitions, geometric definitions, etc.), and it becomes a mathematical problem to understand under what conditions are these various classes of objects equivalent.
The vertex-operator-algebraic study of CFT is closely related to the notion of a BPZ CFT in Belavin-Polyakov-Zamolodchikov's Infinite conformal symmetry in two dimensional quantum field theory (see also Moore-Seiberg's Classical and quantum conformal field theory). From this perspective, a chiral CFT is a (sufficiently nice) VOA and a collection of its modules, together with a choice of intertwining operators among them, satisfying certain properties (for example, a modularity condition for CFTs defined on compact Riemann surfaces on genera one or higher). A full CFT is an appropriate gluing of two chiral CFTs together, importantly to ensure that the correlation functions are single valued. Roughly speaking, the VOAs and their modules form the state space for the theory, and the intertwining operators define the fields. Some references for this include Huang's Genus-zero modular functors and intertwining operator algebras and Huang-Kong's Full field algebras and Modular invariance for conformal full field algebras. These are very abstract and do not give you a complete understanding of what a CFT is.
To develop a better intuition for CFTs, one should look at physical examples. The best place to learn this is Di Francesco-Mathieu-Sénéchal's Conformal Field Theory—this is the "standard" for learning CFT. To complement this, I would also suggest Blumenhagen-Plauschinn's Introduction to Conformal Field Theory (With Applications to String Theory), Schottenloher's A Mathematical Introduction to Conformal Field Theory, and various online notes such as the ones by Ginsparg, Qualls and Schellekens.
Another definition is that of a Segal CFT, as defined in Segal's The Definition of Conformal Field Theory. This definition is close to the mathematical notion of a topological quantum field theory. A two-dimensional topological quantum field theory (2D TQFT) is a certain kind of functor. The domain category is the category of disjoint circles as objects and cobordisms between them as the morphisms. The disjoint union plays the role of the tensor product, giving this category the structure of a (strict symmetric) braided monoidal category. The bordisms that look like a macaroni play the role of (co)evaluations, giving this monoidal category rigidity. The codomain category is the category finite-dimensional vector spaces, with the usual tensor product and duals. A 2D TQFT is a monoidal functor between these categories.
Conceptually, this says that a 2D TQFT is linear-theoretic analogy of how compact 2-manifolds with boundaries can be chopped up and glued back together, that is, a path integral. More concretely, 2D TQFTs are equivalent to Frobenius algebras. The image of the circle being the underlying vector space, the image of a pair of pants being the multiplication, the image of a cup defines the unit, and a macaroni defines the non-degenerate bilinear form.
A CFT is a complex version of a 2D TQFT. Roughly speaking, the domain category now has complex structure, and the codomain category now has Hilbert spaces. There is a program to develop a general construction of Segal CFTs from sufficiently nice vertex operator algebras.