Dan Freed recently gave the Andrejewski Lectures at the Max Planck Institute for Mathematics in the Sciences in Leipzig, and has put the slides from his first lecture on-line. These give a beautiful overview of his work with Hopkins and Teleman relating loop group representations and equivariant K-theory, and explain one aspect of the relation to topological quantum field theory. His second and third lectures aren’t available on-line. The second was supposed to cover the way they use Dirac operators, which is explained in their papers. The third lecture was evidently about the relation to Chern-Simons, which isn’t in their papers so far, and which I’d be quite curious to know more about.
This fall, Dan will be giving a graduate course on Loop Groups and Algebraic Topology, which should be quite interesting.
I have a question/suggestion. Everything these days seems to involve twisting, and apparently there are different kinds of twists (e.g. Witten’s new paper has a different kind of twist from his previous ones). Could we get a general intro to the twisting concept? Are twists the new representations?
Dick,
Sorry, but I don’t immediately see any connection between the two kinds of twisting. Maybe there is one if you think about it the right way though…
Amusing to see you have a new pet project. As it happens, the subject at hand (namely G/G WZW models) was explored and left by string theorists more than 10 years ago. The main statements mathematicians are making have long been known. See old papers by string theorists for the spectrum of the Dirac/BRST operator, the relation with the fusion rules, and the relation to Chern-Simons on S^1, amongst many other things.
M.,
It’s not a new pet project, it’s an old pet project, see http://www.arxiv.org/abs/hep-th/0206135
Besides my own speculations about the significance of Freed-Hopkins-Teleman for physics (which have nothing to do with string theory), there’s a huge amount of interesting new ideas in their papers that are not in the string theory literature.
Some comments here may have been accidentally deleted. If so, my apologies and please repost them if you can.
As much as I admire mathematical work of this high quality, as a physicist I have a different completely autonomous access to the problems connected with the Verlinde algebra. For me the issue is related to an extension of the Nelson-Symanzik duality in the new context of chiral QFT (i.e. in the context of a more involved spin-statistics connection due to the appearance of braid group statistics). This is a kind of symmetry which is particularily interesting for 2-dim. QFT at a finite temperature. It is almost trivial if you are allowed to represent your QFT in terms of Feynman-Kac representation, it is just the symmetry between space and Euclidean time in a temperature state. A more general and rigorous setting of N-S duality can be found in a recent paper (C. Gerard and C. Jaekel, mat-ph/0403048). This is of course related to the issue of Euclideanization (B.S. hep-th/0603118), one of the most subtle issues in QFT (unfortunately it has been banalized beyond recognition, for critical remarks in this direction see K-H. Rehren, hep-th/0411086) which in the standard context is inexorably linked with the names Osterwalder Schrader (and also Symanzik, Nelson and Guerra).
But in the chiral context with braid group statistics it takes on a completely new form and the relevant Euclideanization is that of the modular localization approach (B.S. hep-th/0504206 (published in AOP) and hep-th/0603118). It uses the amazing power of the modular theory of operator algebras (adjusted to the localization concepts of AQFT) and reproduces among other things the Verlinde relation as a special case of an extended concept of temperature duality (a kind of self-duality in which the Euclideanization leads to a QFT which is of the same noncommutative kind as the original one, which the old Nelson-Symanzik duality within the O-S Euclideanization setting cannot provide) of a full thermal theory (and not just of the zero-point (partition) function). The conformal symmetry of the underlying QFT permits to make a complex extension of temperature to the tau parameter. The two field theories live (in the sense of physical localization) on circles, the connecting torus is nothing else than the old connecting Bargman-Hall-Wightman analyticity region (albeit in a more sophisticated veil) which has no direct operator interpretation. Since the charge transportation around the circle mixes the charge superselection sectors (F-R-S, Rev. Math. Phys., Special Issue, (1992) 113), the Euclideanization involves the statistics character matrix S defined by Rehren. Although in all concrete model calculations Rehren’s S turned out to be the same as Verlinde’s S, the derivation by modular Euclideanization provides the missing structural argument why they are always the same.
My remarks could be misinterpreted as lamenting about recognition of work I have been involved with, but this would be much too superficial; what makes me really sad is that tremendous schism which caused young particle physicists to have lost their own rich conceptual past and which presently could be of help to do the fruitfull kind of mathematical physics with physics at the helm of moderation; but what I see is that physicists take their ideas and orders from mathematicians, the less agreeable aspect of the otherwise positive legacy of the Atiyah-Witten era.
I could continue this list. Take the categorical aspect of the old Moore-Seiberg work and all the subsequent papers in that frame of mind (most recently the Fuchs-Runkel-Schweigert etc. work). It is a good illustration of the present Zeitgeist that papers which develop these issues from an entirely physical spacetime perspective (as e.g. the paper of Rehren and myself on Einstein Causality and Artin Braids which appeared at the same time and several subsequent papers in this conceptual line) are hardly noticed outside a small community which has not suffered from that grand rupture in particle physics. Or take the recent works on n-categories (see the blogs in the string coffee table). The oldest and physically best motivated work on such issue (it shows you how hard the AQFTists worked in order to find a structural entrance to gauge theories) by John Roberts is not even mentioned!
The situation reminds me of a newspaper report I red a long time ago in the US. It said that a catchup factory had to close down because the taste of their product was too natural i.e. too close to real tomatos (not the plastic stuff which you find often in supermarkets in Europe and the US); most of the costumers already got to like the type which is mixed with pineapple and a lot of other less healthy things. Physicists only accept mathematical products if the underlying concepts have a certain distance from physics!
Reading the recent blogs on Dan Freed’s work, I also learned to appreciate the subtle (but I still think unintended) humor of Lubos Motl. He is of course right in pointing out the metaporical link between those combining and splitting Euclidean waterhoses which Dan Freed and string theory share (I seriously hope that Freed does not require do interpret them the same way). Lubos Motl plays the (in modern times often missing) role a jester of the string community, a kind of supreme Lord of Misrule. If they have any sense of humor they would support his tenure at Harvard. His role of a jester also works perfectly with repect to the climate change issue. By carrying rightwing propaganda to its extremes, he creates a relaxed cabaret atmosphere which has the opposite effect from what it purpots to do, a perfect Lord of Misrule!
Peter
This morning there was a post of Motl in this series of posts. My last remaks refer to this post and without is some part of my comments on string theory& Dan Freeds mathematical work runs into the void. Peter, did you loose your humor? There was no peronal attack in that one which could have annoyed you. Maybe we can ask him to re-import it.
Bert,
I certainly did not delete any comment of Lubos’s here intentionally. If any recent comments in this thread disappeared, I don’t know how it happened.