A wonderful long-promised paper by Dan Freed, Mike Hopkins and Constantin Teleman entitled Loop Groups and Twisted K-theory II has just appeared. They have advertised it in the past under various names such as “K-theory, Loop Groups and Dirac Families”, but their latest way of organizing their work seems to be to relabel the two-year old Twisted K-theory and Loop Group Representations (which recently has been updated, improved and expanded with new material) as “Loop Groups and Twisted K-theory III”. Working backwards it seems, they now advertise a “Loop Groups and Twisted K-theory I” as still to appear, hopefully in less than two years.

I don’t mean to give them a hard time about this. They are doing wonderful work, continually refining and improving on their results, and the paper is worth the wait. At the moment I don’t have time to do them justice by explaining much about their results or the conjectural relations that I see to quantum field theory, but I wrote a little bit about this a while back in another context. In the future I’ll try and find time to write some more entries about this material.

Also related to this is a new paper of Michael Atiyah and Graeme Segal called Twisted K-theory and cohomology which discusses the relation of twisted K-theory to twisted and untwisted cohomology.

Teleman has also recently made available on his web-site a preliminary version of notes from his fascinating talk at the algebraic geometry conference in Seattle this past summer, entitled Loop Groups, G-bundles on curves. He starts off with some philosophy he claims comes from lessons learned in working with moduli of bundles:

*(i) K-theory is better than cohomology
(ii) Stacks are better than spaces
(iii) Symmetry*

The first and third points I’m well aware of, and he has convinced me to spend some more time learning about stacks by his next point, which I hope may clarify some issues that confused me when I was writing my notes on Quantum Field Theory and Representation Theory. According to Teleman, the fundamental K-homology class of a classifying stack BG gives a notion of “integration over BG” in K-theory that corresponds precisely to that of taking the G-invariants of a representation. This idea has been a fundamental motivation for me for quite a while. It seems to me that one fundamental question about the path integral formulation of the standard model is “why are we looking at the space of connections and trying to integrate over it?” The K-theory philosophy gives a potential answer to this: we’re looking at the space of connections because it is the classifying space of the gauge group, and we’re integrating over it because we want to be able to pick out the invariant piece of a gauge group representation. I’ll try and write up more about this later, especially if learning some more about stacks ends up really clarifying things for me as I hope.

On a somewhat different topic, Teleman recently gave a very interesting talk at Santa Barbara entitled The Structure of 2D Semi-simple Field Theories.

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weâ€™re looking at the space of connections because it is the classifying space of the gauge group, and weâ€™re integrating over it because we want to be able to pick out the invariant piece of a gauge group representation.Hmm. I, for one, would like to see you elaborate on this. I know you wrote some of this up in your notes on the arXiv, but it’s nice to see it written up in words. I would like a summary of the idea in “physics language.” As I understand it you want to recast the ill-defined path integral formalism as some sort of sophisticated (and currently not well-understood) representation theory. Is that the gist of it? It sounds like this might make QFT more rigorous but it doesn’t seem that it would have calculational advantages. What do you think would be the physics applications if your program worked?

What do you think would be the physics applications if your program worked?Peter, I’m sure, will provide his own answer, but let me pipe up anyways. Providing a definition of the path integral in terms of the algebraic topology of a stack would almost certainly allow us to introduce new computational tools into quantum field theory. Algebraic topologists have a fascinating collection of Really Powerful Computational Gadgetry, and I for one, would love to see what they can do to QFT. It’d be nice, for instance, to compute correlation functions using simplicial resolutions.

There would probably also be conceptual advantages. Right now, the status of the path integral is a bit murky. We write down some classical action, and then we define the path integral measure in a rather ad hoc fashion, choosing some regularization. But our computations are supposed to be independent of our choice of regularization. It would be nice to interpret this by saying that each regularization is a different way of representing a fundamental class.

Or, less trivially, it would be nice to explain anomalies as obstructions to the existence of a fundamental class, just like we think of the Stieffel-Whitney class as an obstruction to the existence of an orientation. This would provide a very clean explanation for the fact that they tend to be measured by characteristic classes.

AJ, what you say sounds very nice, but it’s unclear to me how one would ever hope to compute correlation functions using algebraic topology. Probably this is because of my elementary understanding of algebraic topology. In what sort of calculation does one really capture the sort of local information we are after in QFT correlators? I’d like some example of an algebraic topology calculation where one gets back a continuous function. (I can see that a calculation might for instance tell one about a DeRham class, but one still has to figure out what is the functional form of the form, yes?) It seems to me that the question of computing correlators where perturbation theory does not apply is currently formulated as a difficult analysis problem (solving some infinite coupled set of Schwinger-Dyson equations, subject to certain constraints), and I don’t see how the proposed approach would do any better than reformulating it as a different difficult analysis problem.

Hi Anonymous,

Let me start with an analogy. You can write down the Schrodinger equation for an atom, even a simple one like the hydrogen atom, but then you have a difficult PDE to try and solve. Without exploiting the rotational symmetry group and what you know about its representations (i.e. using angular momentum operators and corresponding decompositions of states), it’s hard to get anywhere, but if you exploit what you know about representations of SU(2), all sorts of calculations become possible. Perhaps if we understood better how to reformulate certain QFTs from the ground up in terms of representation theory, we would get all sorts of new insights and possible calculational methods. Maybe some of these would involve the kind of algebraic topology AJ mentions (the Teleman idea I described directly relates representation theory and algebraic topology: you study the representation ring by looking at the K-theory of the classifying space). If you’re an incurable optimist, you might hope these new insights will give you a new idea about how to solve one of the big problems of particle theory: e.g. how is electroweak symmetry broken? how do you unify gravity and the standard model?

This is in some sense just a fancy way of saying something well-known: as Weinberg in his latest article points out, physics in the 20th century made progress by finding symmetries and exploiting them. He thinks this idea is played out and we need to change paradigms and abandon it. I don’t. I think physicists need to understand that “exploiting symmetry” = representation theory, and mathematicians know a vast amount about representation theory that physicists have never absorbed. Before abandoning hope like Weinberg, people should try and understand this kind of mathematics and see what can be done with it.

The Freed-Hopkins-Teleman stuff is relevant to 2d QFT, but precisely 2d chiral gauge theories, which are the 2d analog of the standard model in 4d. My specific hope is that if you properly sort out the 2d story, you’ll get a new idea about how to approach things in 4d. From a more physical point of view, my claim would be that, at a non-perturbative level, chiral gauge theories are still extremely ill-understood. Sure, we know how to use BRST to get a consistent perturbation expansion and to cancel anomalies, but we don’t know much about what happens non-perturbatively. There are all sorts of problems with the BRST formalism (Gribov ambiguities just for a start), my hope is that the FHT kind of stuff will allow a reformulation of BRST that makes more sense non-perturbatively, and might expose some new and interesting structure we can use to do real physics.

Anyway, at the moment, this is all a pipe-dream, but I see quite a few reasons to believe that it is a promising direction to pursue.

The fact that K-theory is “better” than homology to classify possible cycles (on which D-branes can be wrapped) has been known to the string theory community since 1998.

http://arxiv.org/abs/hep-th/9810188

I placed “better”in quotation marks because there is of course no universal adjective “better”. Some things may be described by homology, some things may be described by K-theory, and only more specific statements are meaningful.

Also, the D-brane people have been using stacks, sheaves, and other things for quite some time.

Nevertheless, I am happy to see that some mathematical insights ignited by string theory have become so powerful that even Peter Woit was forced to notice and admit the reality.

Despite what certain string theorists think, K-theory and sheaves have been around since the 1950s, stacks since around 1970. It’s just absurd to claim that they are “mathematical insights ignited by string theory”. Atiyah has been promoting the idea that K-theory is sometimes more fundamental than cohomology since the early 1960s.

AJ, what you say sounds very nice, but itâ€™s unclear to me how one would ever hope to compute correlation functions using algebraic topology. […] Iâ€™d like some example of an algebraic topology calculation where one gets back a continuous functionShort answer: Use a _really_ big space, something that bears the same resemblance to the space of connections as a sphere does to the smallest complex computing its cohomology. Let me be clear: I don’t believe the tools exist yet to treat spaces this huge. But I think there’s good reason to hope that they could be developed within the context of homotopical algebra, and I believe that algebraic topologists have been taking steps in the right direction. The elliptic cohomology theories, for instance, give us formal power series, instead of just elements of a finite-dimensional vector space.

Moreover, I do expect that, if we could reinterpret path integrals in this fashion, a lot of the usual gadgetry from algebraic topology would carry over, giving us new methods of computing. Maybe we’d run afoul of conservation of trouble, and computations would still be next to impossible. I don’t know. But it’s amusing to speculate.

I should probably emphasize that this what I’m talking about here is even more pie-in-the-sky than Peter’s suggestion, which (if I understand him correctly) could be crudely characterized as “develop an analogue of geometric quantization which works for the group of gauge transformations”. I’m focusing more on the connection between fundamental classes and integration; he’s looking more at the representation theory side. What he’s suggesting is probably actually a more promising idea — certainly closer to being not-a-fantasy. My only complaint is that I don’t know how we’re supposed to think about regularization and renormalization in that context. There should be some conceptual explanation for the fact that the choice of regularization doesn’t matter in well-defined theories.

Dear Peter, I am not talking about K-theory being universally more fundamental than homology because, as I’ve indicated, this statement is meaningless. I am talking about a very specific example where homology played (and plays) a role, which are wrapped D-branes. Apologies to Prof. Atiyah, but some general statements that a legitimate object is universally more interesting than another legitimate – and very natural – object have no testable meaning.

String theory is the only known context in which the question “is K-theory more faithful or fundamental than homology” can have a well-defined meaning.

Lubos,

“String theory is the only known context in which the question â€œis K-theory more faithful or fundamental than homologyâ€? can have a well-defined meaning.”

Your delusional mania that things only make sense in the context of string theory is getting weirder and weirder, suggesting organic brain damage. If you want to know why Teleman is saying this, read his papers, he has well-defined technical reasons for what he says, reasons which have zero to do with string theory. Similarly, Atiyah back in the early sixties had very good reasons for what he was saying (hint: there’s this thing called the Atiyah-Singer index theorem….)

Perhaps by “only known context”, Lubos means “only context known to Lubos”.

So all this talk about using representation theory to define a path integral seems to raise the question: what about scalar field theory? If there’s no gauge group, shouldn’t the representation theory all be trivial? And yet we have no nice mathematically satisfying definition of the path integral for, say, phi^4 theory. I must be missing your point.

I’m not conjecturing that all QFTs can be understood in terms of representation theory, just the fundamental one that has to do with the real world (i.e. the standard model or some extension of it). In 4d, you can rigorously define phi^4 theory, but all evidence is that you end up with a trivial theory in the continuum limit. Which may explain why you don’t see elementary scalars. Actually I think there are things you can say about scalar field theory using representation theory, but they’re pretty minimal (just as in QM for any Hamiltonian you can invoke the Heisenberg algebra, but things get more interesting when there are other symmetries around).

Point that confuses me about this, in topological field theories usually the space of physical states tends to be finite dimensional and the number of observables quite small compared to that of a generic QFT. In generic QFT there are new issues to do with renormalization that are crucial in its understanding (e.g. the difference between asymptotically free and trivial theories will only show up then). Is the statement that some infinite dimensional version of K-theory useful for understanding generic field theories, with all their infinities?

Hi Moshe,

In the kind of 2d QFT that I think Freed-Hopkins-Teleman is relevant to (basically a chiral fermion coupled to a gauge theory), there’s only a finite dim space of observables and it’s topological, when you’re looking at the gauge invariant sector. But the constructions are infinite-dimensional and involve the infinite dimensional gauge group, and infinite dimensional space of connections. Freed-Hopkins-Teleman are working in infinite dimensions. But the end result is a finite dim piece they describe in terms of finite-dimensional K-theory. The only renormalization sort of trickery involved here is the standard sort that gives the anomaly, i.e. you only need normal-ordering to get rid of infinities.

I certainly don’t know much about how to do things in 4d, where you do have to face coupling constant renormalization and infinite dimensional spaces of gauge invariant observables. I don’t believe that this kind of abstract mathematics will give insight into renormalization. Rather the opposite: that what physicists have learned about renormalization will be needed to understand exactly when it even makes sense to talk about things like representations of gauge groups in 3 or 4d, and what kinds of constructions of these actually can work.

Again, I don’t think of these ideas as a general tool that will be applicable to generic QFTs. Rather, I’d hope they will pick out a very specific QFT or set of QFTs, one’s which are very simple in representational-theoretic terms (just as certain special QM systems are special in purely involving representation theory, e.g. the quantization of a co-adjoint orbit in the compact Lie group case.) A.J. is right to say what I’m trying to understand is how to do geometric quantization in certain special infinite dim cases. In my pipe-dream this class of special cases will include the standard model. Anyway, the 2d case is something with a lot of beautiful mathematical structure, so it’s well worth sorting out for its own sake and to see what it teaches you.

Thanks Peter, one of the things that I liked about the topological string on twistor space story is the (probably well known) fact that the sheaf cohomology of the appropriate twistor space is infinite dimensional and gives you the fields of N=4 SYM, that is pretty.

BTW, the question recently raised by John Baez on the String Coffee Table seems to be closely related to the desire to understand a field theoretic analog of geometric quantization, I think.

There the question was, in a way, if we can understand something defined by a symplectic 2-form on an infinite-dimensional space in terms of a 2+n form on a finite dimensional space.

In particular, for 2D QFT it seems that it should be possible to re-express geometric quantization in terms of bundles over loop space in terms of gerbes over target space. Maybe, somehow, sort of … ðŸ™‚