I spent most of last week commuting down to Rutgers to participate in a workshop on “Groups and Algebras in M-theory”, organized by Lisa Carbone. Lisa was a student of Hyman Bass’s here at Columbia some years back, and in recent years has been working on Kac-Moody groups and algebras over finite fields.
Much is known about one special class of Kac-Moody algebras, the so-called affine Lie algebras. These are basically Lie algebras associated to loop groups, with a central extension. The study of the representation theory of these algebras is closely connected to quantum field theory in 2d space-time dimensions, and my first talk was about this topic. For more details about this, from the point of view I was taking, see the remarkable book by Pressley and Segal called “Loop Groups”, lecture notes from 1985 by Goddard and Olive at the Erice Summer School and Srni Winter school (see Int. J. Mod. Phys. A1:303, 1986), and Witten’s paper “Quantum Field Theory, Grassmanians and Algebraic Curves” in Communications in Mathematical Physics, 113 (1988) 529-600.
An elaboration of these ideas in one direction leads to the concept of a “Vertex Operator Algebra” (first introduced by Richard Borcherds), and the study of these was pioneered by Jim Lepowsky, who also participated in the workshop, together with his ex-student and now Rutgers faculty member Yi-Zhi Huang. Several other current and ex-students of Lepowsky and Huang were also there and gave talks. For more about vertex operator algebras, see the recent short review by Lepowsky, or the materials on Huang’s web-site. A VOA is essentially the same thing that Beilinson and Drinfeld call a chiral algebra, and these have applications in the geometric Langlands program.
What Lisa is really interested in is the non-affine case, where relatively little is known. Non-affine Kac-Moody algebras and groups seem to have no known tractable realizations, and many basic questions about both the algebras and the groups, as well as their representations, remain open. In recent years several of these algebras have been conjectured to have something to do with M-theory, most notably E11, and the study of this connection has been the main focus of the work of Peter West, who gave a series of talks at the Rutgers workshop. For some more about this, see his recent papers, especially one on The Symmetry of M-theories. West’s graduate student P. P. Cook also has a weblog, and recently wrote a posting explaining a bit about this topic.
Greg Moore was at many of the talks and kept the speakers honest. He gave a fast-pace talk covering some older work, roughly the same material as in his paper with Jeff Harvey entitled Algebras, BPS States and Strings. I gave a second talk explaining a bit about my point of view on the Freed-Hopkins-Teleman theorem and its relation to representation theory and QFT.
After the talks Thursday afternoon there was a discussion section on what is going on with string theory, supersymmetry, and mathematics. No one was willing to defend work on the “Landscape” and I was surprised to find myself pretty much in agreement with quite a few people there about the way string theory has been pursued in recent years. On the whole the mathematicians are kind of bemused by the whole string theory controversy. The subject has certainly led to some very interesting and important mathematics, and they are happy to concentrate on that, although interested to hear about the controversy surrounding string theory in physics.
Thank you for Thomas Larsson for a hint to look Segal’s book. I will do it when I visit Helsinki.
The Dynkin diagrams of simply laced algebras appear also in the Jones inlusions of hyper-finite type II1 factors of von Neumann algebras (see also the article V. Jones (2003), In and around the origin of quantum groups, arXiv:math.OA/0309199) and there are reasons to believe that minimal conformal theories with simply laced quantum groups with few exceptions correspond to Jones indices M:N1 factor. There are good reasons to hope that conformal field theory structure is in some sense an inherent feature of this kind of geometry. E10 and E11 seem to represent steps in this direction.
In TGD framework the counterpart for this group is infinite-dimensional group of generalized canonical symmetries of δ M4+×CP2: the conformal structure is inherited from δ M4+. An interpretation as a Kac-Moody group obtained by localizing canonical transformations of S2×CP2 localized with respect to the radial lightlike coordinate of M4+ is in question.
Peter: “If one could find a simple symmetry principle underlying M-theory”
Stone duality for higher descent
AR – I’ve never seen any explanation even of how gauge invariance as a main principle is supposed to emerge from ST. (Not that I really care 🙂 I saw a half-hearted attempt to pull the QED Lagrangian from such and so string confuguration but it was a real stretch, so to speak.
Glad you like the blog!
If one could find a simple symmetry principle underlying M-theory, that would be very interesting, and make the whole idea much more well-defined and potentially useful. But I’m not really convinced by any of the attempts so far. Up to you to see what you think from West and other’s papers. As they say, we report, you decide….
Is this post insinuating that “The Symmetry of M-theories” could be the next superstring revolution if it is explicitly approved and explained by some superstring boss?
I just wanted to say thank you for this blog. I discovered it while Googling on the Penrose book “Road to Reality”–a book I am enjoying tremendously but am having an extremely difficult time with. I am a lay person but find this discussion incredibly fascinating and am desperately trying to learn enough math and physics to follow these discussions. It’s probably hopeless : – ). But thank you for this blog.
Matti, this is a good point. I had forgotten that the vertex operator construction of Frenkel-Kac and Segal only works for level 1 reps of simply-laced algebras (ADE). I think that Pressley-Segal has a chapter about vertex operators towards the end of their book, but they call them blips for some reason.
In this connection see this new preprint by H. Nicolai:
(..link from It’s equal but it’s different.)
It would be interesting to known whether vertex operator construction exists for non-simply laced algebras, in particular G_2. Goddard and Olive proposed such a construction for all of them except G_2. I failed to find any definite answer to this question from web.
That’s Richard Borcherds, of course. Sorry.
Robert Borcherds has done great things on Moonshine and Monsters, but he is too young to have introduced vertex operators. I think there is a difference between vertex operator algebras and more general vertex algebras, though. Maybe he had something to do with the latter.
According to Goddard’s and Olive’s IJMPA review, vertex operators were first introduced by the first generation of string theorists, refs 71,89,90,91: Fubini and Veneziano (1970), Nambu (1969), Fubini, Gordon and Veneziano (1969), Gervais (1970), although very similar things were already done by Skyrme (1961). Vertex operators were first applied to affine Kac-Moody algebras in refs 95,72,65: for SU(N) by Halpern (1975), and in the general case by Frenkel and Kac (1980) and Segal (1981).
Incidentally, the multi-dimensional Virasoro algebra, which I claim is the correct quantum constraint algebra of general-covariant theories, was first constructed as a vertex operator algebra by Eswara Rao and Moody, Vertex representations for n-toroidal Lie algebras and a generalization of the Virasoro algebras. Comm. in Math. Phys., 159 (1994) 239-264. It might be noted that Bob Moody is rather well known, e.g. as a coinventor of Kac-Moody algebras.
I didn’t really intend to make any claims about who was responsible for the idea of a VOA, just to comment on who was at the workshop. I had thought of mentioning Borcherd’s work, since several talks referred to it. To avoid confusion, I’ll rewrite the post slightly, adding a reference to Borcherds.
Such silly revisionism. Vertex algebras were invented by Richard Borcherds; its hardly convincing to claim that it is a different notion when you add a Virasoro and the word “operator” in between “vertex” and “algebra”…
(Mind you, with a Fields’ medal, he really doesnt need protecting.)