# Witten in Crete

Witten is lecturing at a conference in Crete this week and some of his transparencies are already online. He is talking about perturbative gauge theory amplitudes and the idea of interpreting them in terms of strings in twistor space. He motivates this by noting that AdS/CFT is useful for understanding gauge theories at large g^2N, but at short distances asymptotic freedom implies g^2N is small and to understand gauge theory in terms of strings you need to do so for all g^2N. He warns “I can’t promise that what I’ll explain will turn out ot be useful in a string description of QCD, but at least I’ll tell you interesting things about perturbative gauge theory!”.

For something completely different, the latest on the Landscape is that, at least this week it predicts low energy supersymmetry, maybe.

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### 23 Responses to Witten in Crete

1. Hi Peter -

yes, that’s what I am talking about – in the abelian case. In the nonabelian case things are not as simple. There it does not make sense to simply intergrate over S^1, since that amounts to comparing elements in different fibers. So in the nonabelian case some parallel transport using a 1-form has to be there in order to relate the values of the 2-form at different points of the loop.

Something along these lines was what Christiaan Hofman originally guessed, or proposed. I think the crucial point missing in Hofman’s proposal is that there is parallel trasport back and forth, as in the equation at the very bottom of this MathML enabled entry. This also explains why you don’t see the parallel transport in the abelian case.

2. Peter says:

Here’s the way I know to construct a one-form on loop space from a 2-form on the space itself. Is your construction the same?

There’s a tautological gadget called the evaluation map

ev: S^1 x Maps(S^1,M) \rightarrow M

that just takes a point on the circle parametrizing the loop to the corresponding point on M given by the loop.

Given an n-form on M, pull-back to S^1 x Maps(S^1,M) and integrate over S^1 to get an n-1 form on Maps(S^1,M)

3. what sort of geometry is a “connection 2-form” supposed to represent?

Ah, I see. That’s due to sloppy terminology on my part.

The point is that a 2-form on target space lifts to a 1-form on loop space. What I derive in that paper is that a deformation of the worldsheet supercharges by something which should describe strings in nonabelian 2-form backgrounds indeed does produce a connection 1-form on loop space which is this 2-form lifted to loop space.

This is – almost – what had been expected before, in hep-th/0207017 (see top of p. 7). But I derive a little correction to the formula given there and show that this correction is necessary for some crucial properties.

An easy heuristic way to see why target-space 2-forms correspond to loop space 1-forms is the following physical picture:

A point particle is charged under a 1-form and the 1-form is integrated over the worldline. A string is charged under a 2-form and the 2-form is integrated over the worldsheet. If you make an ADM split on the worldsheet, i.e. introduce a slicing of the ‘tube’ into ‘circles’, then you can see how ‘each point’ on these circles is similar to a point particle which is charged under the 1-form obtained by contracting B with the tangent to the circle.

Now sum this up for all points in the ‘circle’ and you get the 1-form connection on loop space.

At least this is the simple picture for the abelian case. In the nonabelian case you cannot just sum up the contributions from the different points, since you cannot compare elements of the bundle at different fibers. This is why Hofman expected the target space 1-form to play the role of parallel transporting the nonabelian 2-form values from each point of the string slice to some (arbitrary) origin on the circle. And indeed, this is what drops out from my deformation, which also demonstrates that there has to be parallel transport back and forth.

In writing this, I am thinking of superstrings. Most of the work on higher gauge theory has been in broader contexts, mostly coming from a field theoretic or a purely mathematical point of view. But in any case a 2-form YM theory must somehow be about ‘line particles’, and then the above reasoning is relevant.

Probably loop people like Baez, Girelli and Pfeiffer are thinking of parallel transport of spin network edges, instead. BTW, I think the text

Girelli & Pfeiffer: Higher gauge theory – differential forms versus integral formulation (2004)

is the most important one on 2-form gauge theory that I have seen. It clarifies an important open issue in John Baez’ paper and very nicely elucidates the geometrical visualization. But it also seems to leave the authors puzzled: Namely they derive that the whole 2-form gauge theory businenss can only be consistent when the 2-form equals minus the field strength of the 1-form. This seems to drastically constrain the number of interesting higher gauge theory Lagrangians that one can write down.

But from the string/loop space perpective this condition, as I show, is natural. It is equivalent to the 1-form connection on loop space to be flat, which again is the condition that closed strings don’t couple to the nonabelian 2-form, and they should not, since they cannot carry Chan-Paton factors.

I wasn’t aware of Girreli & Pfeiffer until after I had derived this condition myself, but I think that my derivation still helps understanding 2-form gauge theory.

I’ll look up the literature that you mentioned. Many thanks!

4. Peter says:

Hi Urs,

I looked a little bit at your “higher gauge theory” stuff, but I’m afraid my initial reaction is that of a mathematician. What’s the underlying geometrical gadget that this is supposed to be? A connection is an equivariant way of relating nearby fibers in a fiber bundle, and curvature is the holonomy about an infinitesimal loop in a bundle. I can make these ideas explicit using connection 1-forms and curvature 2-forms, but what sort of geometry is a “connection 2-form” supposed to represent?

The kind of thing I’m doing with Dirac operators is essentially what is in a recent Freed-Hopkins-Teleman paper, the closest point of contact with string theory is probably some work of Greg Landweber’s which is basically about N=2 superconformal coset models, see math.RT/0005057 on the arXiv.

5. Peter wrote:

the Dirac operator on certain loop spaces is also a crucial part of what I’ve been thinking about

Oh, interesting. I didn’t know that you have been thinking along these lines. Maybe we can discuss some stuff.

As you may have seen, I currently think that loop space differential geometry has something to say about what is called ‘higher’ gauge theory. The point is that many people have tried to more or less guess or use trial and error to find general properties of 2-form generalizations of Yang-Mills.

In a kind of reversed fashion I tried to study 2-form Yang-Mills from the worldsheet point of view, seeing how the worldsheet theory determines the target space effective field theory.

After some steps in the dark I think that I now understand what’s going on. As soon as my draft passes my advisor’s revision I’ll put it on the arXiv. Since I have learned in the past that there is nothing like shameless self-advertisement, here is the link:

Nonabelian 2-form connections from 2d BSCFT deformations

I have had discussion about this with all kinds of people by now, but I am always happy to receive more comments.

6. I don’t know much about Dixon’s work, but he gave a talk in Paris about how to use N=4 SYM to compute QCD amplitudes and how that leads to connections with all kinds of computations done in AdS/CFT.

7. Peter says:

Hi JC,

Weird but true: I just checked the SPIRES HEPNames database trying to remember who Dixon’s advisor was (Dixon and I overlapped as grad students for several years at Princeton). The example SPIRES gives for a search of this database is

find name dixon and field hep-ph and not undergrad mit

which leads precisely to Dixon’s entry.

Back when Dixon got his Ph.D.(1986), there wasn’t a whole lot of choice in the matter, you pretty much absolutely had to be working on string theory if you wanted to get a job. I’d also be curious to know why he stopped working on string theory and what he thinks of it these days.

8. JC says:

Peter,

I remember in early 90′s when string theory was in sort of a slump. From what I remember, all that work on conformal field theories and string field theory didn’t seem to make much new progress, while the mirror symmetry stuff eventually worked its course and subsequently “flatlined” shortly thereafter. (It was years before Seiberg-Witten theory and D-branes revived string theory).

It seems like even back then, many string folks were finding many rationalizations and excuses to justify working on string theory. I remember some folks were even looking at “string inspired” stuff like Lance Dixon’s work in the early 90′s that found easier ways to calculate Yang-Mills amplitudes, as an “excuse” to justify doing string theory. (It’s interesting that Witten’s present work on twistors is attempting to explain the simple looking formulas that showed up in the papers of Dixon, Kosower, et al. from that era). I always wondered what made Dixon defect from string theory back in those days, considering many of his big papers from the 80′s were all on string theory.

It seems like the sociology behind string theory is very much like being sold the “dream” of the “holy grail” of a consistent theory of quantum gravity,

In the 1970′s, the sociological circumstances in the mid 70′s which led to the mass abdication of string theory seemed to be ‘t Hooft’s renormalization results and the subsequent “rebirth” of field theory. In this case, what led to string theory’s first demise is pretty much how traditional science should work, when a better theory makes predictions that are subsequently confirmed by experimental results. Perhaps why there hasn’t been a mass abdication yet in today’s string theory, maybe has to do with the fact that the “dream” of the “holy grail” in a consistent theory of quantum gravity, is still very much alive and well in the spirits and minds of many string theory folks.

For many “mass movements” in history, whether benevolent or malevolent, there’s almost always a “dream” and/or “utopia” of some sort that is used to sustain the movement which keeps the “true believers” in line, as well as a way to get new recruits to their cause. Even once the “mass movement” becomes the “status quo”, the “dream” and/or “utopia” is still repeated over and over as propaganda. (This is what happened in many communist countries, like Soviet Russia and China, after awhile). The systems created by the “mass movements” usually start to crumble and eventually self-destruct when the propaganda of the “dream” and/or “utopia” can no longer sustain the spirits and minds of the people.

The only obvious scenario I can think of that could greatly destroy the “dream” of the “holy grail” of a consistent quantum gravity theory in the form of string theory, would be if Witten publicly abdicates and drops string theory for good.

9. Peter says:

Hi Urs,

Sure, one could attack any kind of speculative work as “wishful thinking”, but I don’t think I’m doing that. In 1984 when people were hopeful that their new speculation was about to lead to some real predictions, that was one thing. Twenty years later, after the accumulation of a lot of evidence that this idea doesn’t work, that’s something else.

My comment about sociology was meant to refer more to the question of why the more senior leaders of this field haven’t given up and moved on to something else. The question of what a young theorist should do and what the sociological pressures are for them is a bit different. String theory is by now such a huge subject that there are plenty of things for someone starting out to try and do something with, and they’re not the ones who should be expected to take leadership and change the direction of the whole field. Whether LQG or string theory or whatever, it’s very hard to start a career in this business, and no one does it because it is the easy thing to do.

The fact that string theory includes so many different approaches allows you to find one that is at least mathematically interesting and may lead somewhere. While I’m not doing string theory, the Dirac operator on certain loop spaces is also a crucial part of what I’ve been thinking about, so if string theory leads you there, that’s great.

10. Hi Sean Carroll -

since this blog does not support threaded comments my reply to Peter Woit’s comment appeared as a reply to your comment.

Yes, I fully agree with what you say. ‘Fortunate’ and ‘unfortunate’ are inappropriate adjectives in this context, anyway, their only purpose being in casual conversation. There are facts and there is right and wrong, and we need to figure it out.

P.S. BTW, thanks for all the stuff that you make available online. I have learned GR from your notes (well, and some other reading, too, of course) and have tutored a GR class using them. Really nice. But I guess you have heard that before…

11. Sure, it’s wishful thinking. No matter which person starts to think about any physics beyond the standard model, he or she will tend to wish that there is a chance that this thinking actually leads in the correct direction. No matter which approach you favor over string theory, if its about beyond-the-standard-model then it is necessarily wishful thinking.

BTW, I am a good counterexample for the ‘sociological reasons to go into string theory’. I found that it is pretty hard for someone like me to get ‘into’ the community. Experiences from talking to people indicated that it would have been much easier for me, as far as sociology is involved, to do some LQG, for instance.

Trust me or not, but I do find strings more interesting as such, and here is absolutely nobody in my vicinity who could have tried to talk me into believing that.

Maybe a curious side remark: One reason why I like strings (by no means the only or most important one, but still one reason) is due to the major role the Dirac operator plays in that theory.

But I didn’t want to get into that kind of discussion again. Sorry. From now on only technical remarks/questions. Promised!

12. Sean says:

Urs, by “unfortunately for string theorists” (referring to the observed positive vacuum energy), I presume you mean “fortunately for string theorists.” The AdS vacua we understand are all supersymmetric and not like the real world. It’s hard to understand supersymmetry breaking, and it’s hard to understand a positive vacuum energy, and it’s hard to understand moduli stabilization. The hope has to be that these difficulties are somehow related to each other, which would help explain why easy-to-understand solutions to string theory don’t look like the real world. I think that’s a perfectly plausible scenario — no reason why a correct theory has to be easy for us mere mortals to immediately master, especially in the absence of detailed experimental input.

13. Peter says:

Hi Urs,

Thanks a lot for the detailed summary of the current point of view on the “landscape”. I hope that helps Pyracantha, but if not here’s a much over-simplified version:

String/M theory supposedly is a theory of strings and maybe other objects in an 11-dimensional space-time. We see 4 of the dimensions (3 space, 1 time), what about the other 7? The initial hope was that there would be a small number of possible consistent choices of these 7 dimensions, so there would be a small number of calculations you could do and see if one of them agreed with experiment.

Lately people have started to believe that there are an astronomically large number of possible consistent choices, and these are referred to as the “landscape”. The reason for this terminology is that each such choice comes with an important number attached, the energy of the vacuum, and if one imagined mapping out all possible choices on a plane, one could imagine making a topographical map, with the energy the altitude. Zero energy choices would be at sea-level, and the whole thing would presumably have peaks and valleys, with our universe sitting at the bottom of some valley.

Anyway, that’s very roughly the idea.

The standard ideology has always been that this large number of choices is just due to the fact that one only knows an approximation to the real string/M-theory, and that if one knew the real thing one would find that all or most of these choices were inconsistent. The other part of this ideology is that there is a unique real string/M-theory, for which all these choices are just possible lowest energy states. In this scenario, maybe they are only approximately at lowest energy and the true lowest energy state is something else, or maybe there really are an extremely large number of possible lowest energy states (this may include metastable states, not at lowest energy, but separated by an energy barrier from lower energy states).

My own point of view is that this standard ideology is just wishful thinking. My guess is that there isn’t a simple unique 11-dimensional theory, but something rather complicated, involving a possibly infinite number of choices as to how to set it up. People are free to keep believing the standard ideology, since it’s hard to prove a negative, to show that what they would like to exist doesn’t.

The really strange thing that has happened in recent years is that a lot of string theorists, most prominently Susskind, have adopted the point of view that, whatever string theory is, it has an astronomically large “landscape” of equally good vacuum states, and thus equally good models of the universe. I would have thought that once someone had convinced themselves that, even if there was a unique real string theory, it would be consistent with an unimaginably large number of possible models of the universe and quite possibly be completely vacuous and unable to predict anything, they would give up on the whole idea. The idea Urs mentions, that maybe only a small number of these models is consistent with some simple facts one knows about the standard model, so you could use these to make predictions about other things, seems to me to be just more wishful thinking.

Given that you don’t know the underlying theory, and what you do know leads to an essentially infinite number of possibilities, it’s not clear that the kinds of arguments that Susskind et. al. are making are even science at all. Some of these papers are weird documents, with virtually no equations, just a lot of hand-waving arguments involving massive amounts of wishful thinking and no solid conclusions. My take on all this is that the time is long past at which a reasonable person should have given up on the whole idea and moved on to something more promising, but sociological reasons are keeping this from happening.

14. Hi Pyracantha -

in so-called perturbative quantum theories one chooses a solutiuon of the classical equations of motion, the so-called ‘background’ and then studies quantum corrections to that background order by order.

For instance in ordinary quantum field theory the background might be flat Minkwoski spacetime and in that background we can imagine photons and electrons to propagate and interact in Feynman-diagram fashion. The ‘vacuum’ background together with all these particle whizzing around would then be a full perturbative state of the theory.

(One problem is that not all aspects of the full quantum theory are captured by such a perturbative procedure.)

Now, in string theory the idea is pretty much the same, only that here the particles are not pointlike but a have a small linear extension. This seemingly simple modification has drastic consequences. While in field theory there are many possible choices of fundamental particles, their interactions, and choices of background, the consistency of string interaction very much constrains all three of these. The big open question is: How much exactly?

When people talk about the ‘string theory landscape’ they are thinking of the abstract space in which each point is one consistent perturbative string theory background, i.e. one consistent choice of particle content, particle interaction and classical spacetime that they propagate in. In principle the number and position of points in this ‘theory space’ is determined by the background equations of motion of string theory (or equivalently, if you want to hear the technical terms, by the requirement that there is a supercfonformal field theory with central charge 15 on the worldsheet of the string).

There has been some recent progress in better understanding this space – but it is still immensely ill understood. Still, the progress that has been made has appeared significant enough to some people to base some more far reaching speculation on it. That’s because a good understanding of which background solutions string theory admits is the key to be able to apply string theory to pheonomenological considerations. When a string theory background is found which is consistent with the observed particles of nature, then studying the stringy quantum corrections to it would allow to deduce what this background predicts as corrections to the currently known physics.

Peter Woit here has pointed out repeatedly that some of the speculations concerning the landscape that have been published are not at all based on results that have really been calculated.

On the other hand, the mere fact that a discussion of such a ‘theory landscape’ is possible (even though not easy) is important. It is not possible in field theory of point particles. There we also have some restrictions on the Lagrangians (i.e. the particle content and interaction) that we are allowed to consider as a consistent field theory, but they are far less severe than those found in string theory.

As has been pointed out very nicely by Jacques Distler in his weblog, the points in the landscape which are consistent with the experiments that we have made are probably very rare. In any case, none has been found so far. If there is none at all, then string theory is wrong as a theory of nature. If there is a single such point, then string theory, based on the currently known data, could make predictions about for instance new particles that could be found in future colliders. (These predictions could still be disporved by experiments, of course.). If however there are very many such points then predictions for new particles etc. would be very difficult. One might, in this case, still try to make some statistical predictions. Such statistics about properties of the ‘landscape’ are currently what some people are trying to do. But it seems fair to say that this is, while an intersting idea, quite premature.

Finally, there is the theoretical possibility that the world we live in cannot be understood as a small perturbation of some background. The success of perturbative field theory suggests otherwise, but nobody can know this for sure. So one possibility is that none of the points in the ‘landscape’ correspond to the world we live in, but some nonperturbative description of string theory is necessary to describe our world. Nonperturbative description of string theory tend to be described not by full classical backgrounds, but by asymptotical backgrounds, this means roughly that at spatial infinity the background is fixed, while ‘in between’ physics is described fully nonperturbatively. Nonperturbative discriptions of string theory are known for instance for universes which asymptotically have the geometry of what is called ‘anti-deSitter Space’. This is, roughly, the shape of a universe with a negative cosmological constant.

Now, unfortunately for string theorists, recent very exciting measurements of various cosmological parameters have shown that instead we observe a cosmological constant which is positive. This means that the particular anti-deSitter non-perturbative deswcription of string theory appears not to be applicable to describe the universe that we live in.

This is probably the main reason for the current excitement about landscape discussions. Namely people are trying to find out if in the landscape admits universes which only temporarily have a positive cosmological constant, while asymptotically this constant goes negative. If this were the case then there would still be hope that the nonperturbative string theory description which involves asymptotically anti-deSitter space could be used to describe the world we observe.

So that’s what all this landscape talk is about. Unfortunately, since there is so little known for sure about the ‘landscape’ (even though the landscape is a well defined mathematical object (space of all superconformal 2d theories with c=15) which can in principle be understood exactly), some of the discussion concerned with it recently has tended to be more philosophical than scientific.

15. Pyracantha says:

Pyracantha from “Electron Blue” here, the artist who is trying to learn math/physics in middle age. I read your site in the hope that someday I’ll understand what you and your colleagues are talking about. But I have heard one phrase many times and it intrigues me. What is the “landscape?” Could you explain it in terms that a beginner like me could understand?

16. Dick Thompson says:

AR, Here are my notes on the article “Lovely as a Tree Amplitude” by Stephen K. Blau, in the Search and Discovery section of the July 2004 Physics Today. I urge you to read the original article whenever you can, it is well-written and has a lot more detail.

It reports on a three month long workshop on collider physics at the Kavli Institute in Santa Barbara. The workeshop centered around the work of Witten and colleagues on a duality between string theory and perturbative QCD. Last December, Witten discovered this relationship between a certain string theory (B-theory) and the weak coupling domain of QCD [hep-th/0312171]. Witten, F. Cochazo, and P. Svrcek (CSW) used it to greatly simplify the expression for the leading perturbation terms in some QCD diagrams [hep-th0403047] .

In the covariant expression for an all-gluon Feynman diagram each gluon helicity is described by an overdetermined 4-vector. The extra symmetry doesn’t affect the physics, but it makes the expression complicated. Following an alternative strategy of collecting particles by helicity, S. Park and T. Taylor in 1982 proposed a formula for maximal helicity violating (MHV) diagrams. The formula was proved later; it only handles massless particles. CSW show how to apply to off-shell MHV diagrams sewn together with propagators to to yield non-MHV amplitudes. The CSW construction has not been proved, but Zvi Bern at the workshop called it convincing. It satisfies certain powerful factorization properties.

Now, the twistors. A vector can be represented by two spinors. Witten considers tree-level amplitudes in spinor language, and asks how they behave under conformal transformations. To simplify expressions and symmetries, he fourier transforms into twistor space. Then when the momenta of nonvanishing MHV amplitudes are expressed in twistor space they lie in a straight line. Other helicity combinations lie on quadratics and cubics. A feature of the duality is the instanton string configurations reproduce QCD amplitudes and they wrap around structures in twistor space, giving a winding number.

R. Roiban, M. Spradin, and A. Volovich used the Witten duality conjecture to get 5-gluon amplitudes with two + helicities and three – helicities. [hep-th/0402016]

17. Lunsford, no, I was not thinking on Regge methods. I am not fluent enough on the trajectories, so I can no bet about how deep this methodology was. Was it related to the primitive string theory?

I mentioned *Runge* methods because Brouder pointed out that the systematics to classify them was the same than Connes-Kreimer for Feymann loops.

18. Dick Thompson says:

AR, about the article. I read it at the public library. I’ll be going back there tomorrow (Monday July 12) and I’ll take some notes.

19. D R Lunsford says:

AR, you mean Regge methods of course.

20. No surprise to know that there is hidden structure in Feynman diagrams. From Bogoliugov to Kreimer there has been deep insights on it (not to mention the strange connection with Runge Kutta methods, found by Brouder). But I am unsure about if the string framework is the most adequate to look into this.

A pity the Physics Today article is not open for general public. Can you tell more about?

21. Dick Thompson says:

There’s a good article on this in the current Physics Today. Witten and coworkers can only do tree level, but others working in their wake have got plausible results for loops. The idea is to do better calculations on low energy (several TeV) soft scattering. When Witten plotted the maximum helicity violating states in twistor space, they fell on a straight line.

22. The icon in the upper left corner of the webpage is Phaistos disk, an undeciphered inscription that rivals string theory as a magnet for speculation.

Slides 1, 2, and 6 show that String theory is not more claiming to be a theory of elementary entities but a yet-weakly-foundamented area of mathematics. I could have told this to you already nine years ago. It is very much as the things topologists do to study spaces: To swept a manifold with a lower dimensional one, and to look for singularities in the evolution of the map.

This does not imply that they have lost hope on an elementary entities theory, as one can read between lines. But it is an aptitude change, and it goes back to take seriously Quantum Field Theory, no more a disposable effective theory.

23. Thomas Larsson says:

I am confused about the status of twistor-string theory. Berkovits and Witten seem to claim that the twistor string does not give you the right answer beyond tree level: “Those loop amplitudes will therefore not coincide with the loop amplitudes of pure super Yang-Mills theory. ” Still, Witten keeps working on it. Does anyone understand what the goal is?