I think Witten proposed the mass gap problem as a Clay prize problem in part because of its very nonperturbative nature. He has mentioned in some articles that there is both theoretical and computational evidence that the mass gap is exponentially damped as coupling constants are taken to zero. Since perturbative expanions are basically calculating in a formal neighborhood of zero, it sounds like they would be quite useless for this problem.

Thanks for the interesting comments. Vaughan didn’t mention any sort of random walks in his talk, but given that these things already show up in classical stochastic field theories I am not surprised that some version of them also shows up in QFTs. As for the 4-dim QFTs, the sense I got from Richard was that the algebraic issues (working in the perturbative regime of formal power series) are reasonably well understood and on a fairly solid foundation, but all the analytic issues remain extremely difficult. (As an analogy, one can easily construct “global” “solutions” to the Navier-Stokes equation from analytic initial data by a formal power series expansion (e.g. using Cauchy-Kowaleski), but without any convergence result on such formal power series, this formal power series sheds no light on the global regularity problem.)

I always find it particularly fascinating when fields that at first seem unrelated turn out to have these tantalizing connections: the Jones connects with von Neumann algebras, statistical mechanics, braids etc. as you mentioned, but also with 3-dimensional topological quantum field theory. In molecular biology the Jones polynomial–and its generalisations like the HOMFLY polynomial, derivable from the SU(N) Chern-Simons–are also a potentially useful and powerful tool in classifying and identifying knotted states and topological properties of dna, polymers and enzymes; essentially ‘biological strings’. But an SU(N) Chern-Simons theory can also be related to topological string theory in the large N limit. A lot of fascinating things and connections going on here. Incidently, did Borcherd mention anything about the hard problems of actually defining QFTs on 4-manifolds, specifically the Yang-Mills gauge theory, which of course is a Clay problem?

Now it is clear why the direct use of the Hilbert space might be
difficult: We can look at the construction of the Hilbert
space using correlation functions. These correlation functions
involve higher genus Riemann surface structure, but from the
Hilbert space structure obtained from
the inner product completion, we do not see anything
about Riemann surfaces.

Anyway, to address the math question, yes, I believe that vertex operator algebras are intended as a axiomatization of a simplified part of conformal field theory. One of the simplifications is to restrict to CFT on a sphere with marked points, instead of surfaces with higher genus. Monstrous moonshine is an example CFT which is constructed with a certain amount of symmetry (that of the Leech lattice) but is revealed to have more symmetry (that of the monster group). It’s not my area either, but I think that this is a fair summary of what I have been told.

However, there is a great variety of quantum field theories in the world, so that you can easily move far away from any one set of examples and still do quantum field theory.

Richard is best known for his work in lattices and group theory, most notably in explaining the monstrous moonshine phenomenon, but in recent years he has moved to a completely different area of mathematics, namely mathematical quantum field theory (QFT)

Is monstrous moonshine really a completely different area? I was under the impression that vertex operators, infinite dimensional Grassmannians, and generalized Kac-Moody algebras arise naturally in conformal field theories, which are a baby version of QFT’s proper. Indeed, as far as I am aware Borcherd’s proof crucially relied on the no-ghost theorem from perturbative string theory.