Moshe's recent questions on formalizing quantum field theory and lattices as a definition of field theory remind me of something I occasionally idly wonder about, and maybe this site can tell me the answer. Are there any mathematicians working on defining quantum field theory by beginning with a rigorous definition of CFTs, and working from there?
The reason I ask is that I think this is how most of us in physics think about quantum field theory (that is, in a Wilsonian way): to define a QFT, you start with a UV fixed point, and deform it with some relevant operator. So if you had a general theory of CFTs, you would know how to understand how CFTs respond to external sources for operators, and getting a more general QFT would "just" mean turning on a spatially homogeneous source for some operator and seeing how it responds.
The other object we study is "effective field theory," which you might imagine in this language is a CFT that serves as the IR fixed point, together with some notion of equivalence class of irrelevant operators deforming "up" away from that point (being agnostic about whether you ever reach a UV fixed point).
Very (extremely) naively, I would suspect mathematicians might have better luck studying the space of CFTs rather than trying to begin with all QFTs. And you might imagine this approach would be well-suited for questions physicists might care about (like, say, whether there is an "a-theorem" or something similar, analogous to the c-theorem in 2 dimensions, characterizing RG flows as irreversible).
Axiomatic/algebraic/constructive field theory seems to worry about all kinds of field theory at once, and other mathematicians seem to be trying to dig up interesting structure in perturbation theory, which I'm not sure will ever lead to progress in nonperturbatively understanding field theory. I know there are some mathematicians who work on CFTs. (I found this MathOverflow question that has a lot of links to work by mathematicians on CFTs, for instance.) But I wonder if any have tried to work on CFTs as a route to understanding QFT more generally.
Answer
CFT consists for most mathematicians - who are interested in this topic - currently of the study of vertex operator algebras, see this question on math overflow:
You can find a little bit more about the topic and the work of several mathematicians here:
As you can see from the answers on mathoverflow, vertex algebras were not invented for the study of CFT, and that they form an axiomatic abstraction of operator algebra products was noted only later.
A personal and very subjective note: One should not underestimate the amount of theoretical physics that is necessary to understand what a QFT is to physicists. Most mathematicians that encounter physics for the first time since highschool through some QFT framework seem to be quite taken aback by the high intrance fee they'd have to pay to understand this. This is my own, personal explanation for the observtion that most mathematicians study the formal machinery only, in order to use it to prove some new mathematical theorems, but only very rarely in order to better understand what physicists do. Although you'll find quite a lot of work by quite a lot of rather famous mathematicians if you follow the links above, AFAIK there is none doing the work you describe.
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