Scandinavian but not Abelian
Posted by urs
Stepping out of the propeller plane on Karlstad airport, I found myself surrounded by pine forests and in an atmosphere quite unlike that on larger airports – but what I did not find was my luggage.
Apart from the obvious inconveniences this meant that a couple of papers on non-abelian 2-form fields which I had brought with me were spending the night in Copenhagen, instead of attending the conference ‘NCG and rep theory in math-phys’ with me.
Not that there weren’t plenty of other things to think about, like Schweigert’s talk on how modular tensor categories and Frobenius algebras know about open strings, as well as many very mathematical talks with categories here and functors there
but after I had given my talk on Loop space methods in string theory it turned out that several people were interested in nonabelian 2-form gauge theories, and on my way back to the hotel I had a very interesting conversation with Martin Cederwall about precisely the lost hep-th/0206130, hep-th/0207017, hep-th/0312112 which I had intended to pull out of my hat on precisely such an occasion.
But maybe I was lucky after all, because when on the next day at lunch I talked about gauge invariances in 2-form theories with Jens Fjelstad, we had to reproduce the essential formulas by ourselves on a sheet of scrap paper, instead of just looking them up, and somehow this triggered the right neurons for me, and after a nap that evening I got up and saw the light.
[Update 07/15/04: The issue discussed below can now be found discussed in hep-th/0407122.]
The point is that the 2-form on target space gives rise to an ordinary 1-form connection on loop space, of course, and that I think that I know precisely how this 1-form connection looks like, because I can derive it from boundary state deformations.
In a somewhat schematical and loose fashion we can write
(1)
following the notation in Hofman’s paper, but including a second factor of the -holonomy, as I have discussed before.
Using this connection and the ordinary formula for its gauge transformations, one can check that global gauge transformations on loop space correspond to the ordinary 1-form gauge transformations
(2)
(3)
on target space, while local gauge transformations on loop space give rise to
(4)
(5)
up to some correction terms which don’t have a target space analogue. I have given a little more detailed discussion of this on sci.physics.strings.
As with any riddle, after having written this down it looks pretty obvious, but at least I haven’t seen this clearly before.
The question now seems to be: Can we even expect to be able to write down a theory of point particles that is local in target space and respects the above gauge symmetries. What happens to the correction terms?
Rather I’d suspect something like an OSFT which has the true loop space 2-form gauge invariance, but whose level truncated effective field theory breaks some of it. But I don’t know.
When I mentioned to Martic Cederwall that we should maybe consider YM on loop space using the field strength he remarked that this would be a theory local in loop space, while ordinary OSFT is non-local in loop space (because the 3 ‘loops’ (or rather open intervals) involved in an interaction are not small deformations of each other and hence do not correspond to nearby points in loop space).
Well, so I don’t know what all this means. But as far as I can see nobody else does either, at least nobody understands it completely. Amitabha Lahiri kindly made me aware of a couple of paper he has on attempts to construct field theories with some reasonable 2-form gauge invariances. I will try to have a look at these papers and see if the Lagrangians considered there might be understood in terms of the loop space connection
(6)
Posted at July 9, 2004 8:28 AM UTC
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Re: Scandinavian but not Abelian
Hi -
In a private email (before noticing the new SCT entry), I said…
I haven’t mastered the loop space formulation, but the idea seems really natural to me. I am sure that you are correct and people will take notice of this and your deformation stuff. An obvious question, can the process be iterated? What would be the loop space of loop space? :) Would you get non-abelian 3-form stuff? :)
You responded…
Good question. My advisor asked me the same question today at lunch. My answer
was that, yes, I think this iterates. Naively at least I can imagine the loop
space over loop space, for instance (and I seem to recall John Baez having
mentioned something like that before). This would be “torus space”, the space
of all maps from the torus into target space. It should be relevant for the
supermembrane, which indeed couples to the 3-form that appears in 11d
supergravity. There should then also be nonabelian 3-forms and so on.
But - wait - we should be discussing this at the String Coffee Table. It could
need some activity. :-)
Ok! Ok! :)
Another obvious question…
Is it possible to generalize “loop” space to “string” space that includes both open and closed strings? If you could construct a “string space of string space”, then you could talk about general maps from seemingly general 2d manifolds (branes?) into target space.
Just curious, but would even the loop space of loop space admit things more general than tori? I can almost picture a Klein bottle among other things, e.g. closed strings that twist around.
[snip of some stuff about Pohlmeyer invariants I don’t think you want me reproducing in public ;)]
I also said (in regard to the lousy communication skills of most string theorists)
I know! I see you being assimilated! :) Your notes are becoming less and less comprehensible ;)
to which you replied
Ok. So we should start a project: Rephrase everything that I think can be said
about strings in loop space formulation in a way that is understandable for
non-experts in strings. I think there is a very good chance that this is
possible. Many aspects of string theory look surprisingly natural in loop
space formulation. For instance isn’t it kind of remarkable that the concept
of the spacefilling D9 brane translates in loop space language just to the
constant 0-form on loop space? What are called ‘boundary states’ in string
theory are really just the constant 0-form acted on by some unitary operators
on loop space.
This sounds like a good idea :) Maybe we can work on this together and put something on the arxives. Then again, knowing how notoriously slow I am at writing things up, you might want to go ahead on your own. I’d be happy to at least make suggestions :) We started to do this for your string theory seminar, where I was able to write down a pretty sleek coordinate independent version of the Polyakov action. Doing so made the relation to BF-YM theory kind of obvious to me. Knowing that there are probably infinite many ways to write down an action that reduced to Nambu-Goto “on shell” caused me to lose interest :)
This loop space idea is pretty neat though. Of course, I would suggest presenting the discrete version first, which would be much simpler :) Contraction “integrals” of continuum indices become summations, which more closely resembles the usual contraction of indices.
Just a thought…
I also said
One of the things that inspired me to learn some elementary string theory was a statement you made (several times in fact) in response to complaints that string theory does not predict anything. You said that the fact that string theory does not predict anything should not be considered necessarily a bad thing because Newton’s gravitational theory doesn’t predict anything either. Not without specifying initial conditions. Finding the right vacua in string theory is like trying to find the right initial configuration of planets in the solar system. Once this is done, you can make many predictions about the subsequent orbits of the planets. This made a lot of sense to me.
You replied
Nice to hear. I get kind of frustrated hearing [people] repeat [their] claim ‘no predictions’ no matter what. In precisely the same sense field theory as such does not make any predictions. I mean, just givben the statement: “The world is described by field theoy.” does not allow you to predict anything. You instead need to pick a particular Lagrangian. String theory has the advantage that all these Lagrangians are not just there to be picked but are part of a larger framework which in principle should tell you what t choose. So in principle string theory is more predictive than field theory.
I continued…
On the other hand, I am not quite convinced that the situation is as easy as that. Before Newton could write down his gravitational theory, he first had to invent his calculus. Once you have the calculus, you need some basic physical laws, like F = dp/dt. Once you have the physical laws, THEN you can construct a physical model, e.g. the solar system.
to which you replied
I am not sure about your destinction between laws and models. But one should
certainly agree with what everybody is saying, namely that the full picture of
string theory has not yet emerged.
In this case of Newton’s gravitational theory, the “law” is the law of gravitation. The “model” is the specific placement of planets. The model is governed by the law. I don’t know how this analogy quite translates for strings.
I continued
I get the feeling that string theory has not even passed the “calculus” phase yet, i.e. the “calculus of strings” is still being developed.
To which you replied
BTW, I think Martinec used to call conformal field theory the “calculus of
string”.
Neat :)
I continued
On another note, I have actually been working hard on the discrete stuff. I don’t have a lot to show for it though :) It might be a waste of time, but I have been LaTeX’ifying Robb’s 21 postulates for axiomatizing Minkowski space. Then I will LaTeX’ify Zeeman’s postulates and finally Penrose’s. I hope to compare and contrast them. I will then try to distill out the “meaning” of what they are trying to say. I will then try to use this “meaning” to motivate the discrete approaches of Sorkin, D&MH, and our notes. Well, that is the plan anyway :)
to which you replied
Robb has 21 axioms? So many?
Yes. Although all of them rely only on the relations “before” and “after”, it does seem unnecessarily convoluted. By the way, I forgot to mention Mundy in my list. Mundy basically reformulated what Robb did in a much more concise manner using the relation of being “light-like” separated as opposed to Robb’s “time-like” relation. Another motivation for :aTeX’ifying everthing is to help me verify that Mundy actually does reproduce everything Robb does.
Gotta run!
Eric
Re: Scandinavian but not Abelian
Hi -
In a private email (before noticing the new SCT entry), I said…
You responded…
Ok! Ok! :)
Another obvious question…
Is it possible to generalize “loop” space to “string” space that includes both open and closed strings? If you could construct a “string space of string space”, then you could talk about general maps from seemingly general 2d manifolds (branes?) into target space.
Just curious, but would even the loop space of loop space admit things more general than tori? I can almost picture a Klein bottle among other things, e.g. closed strings that twist around.
[snip of some stuff about Pohlmeyer invariants I don’t think you want me reproducing in public ;)]
I also said (in regard to the lousy communication skills of most string theorists)
to which you replied
This sounds like a good idea :) Maybe we can work on this together and put something on the arxives. Then again, knowing how notoriously slow I am at writing things up, you might want to go ahead on your own. I’d be happy to at least make suggestions :) We started to do this for your string theory seminar, where I was able to write down a pretty sleek coordinate independent version of the Polyakov action. Doing so made the relation to BF-YM theory kind of obvious to me. Knowing that there are probably infinite many ways to write down an action that reduced to Nambu-Goto “on shell” caused me to lose interest :)
This loop space idea is pretty neat though. Of course, I would suggest presenting the discrete version first, which would be much simpler :) Contraction “integrals” of continuum indices become summations, which more closely resembles the usual contraction of indices.
Just a thought…
I also said
You replied
I continued…
to which you replied
In this case of Newton’s gravitational theory, the “law” is the law of gravitation. The “model” is the specific placement of planets. The model is governed by the law. I don’t know how this analogy quite translates for strings.
I continued
To which you replied
Neat :)
I continued
to which you replied
Yes. Although all of them rely only on the relations “before” and “after”, it does seem unnecessarily convoluted. By the way, I forgot to mention Mundy in my list. Mundy basically reformulated what Robb did in a much more concise manner using the relation of being “light-like” separated as opposed to Robb’s “time-like” relation. Another motivation for :aTeX’ifying everthing is to help me verify that Mundy actually does reproduce everything Robb does.
Gotta run!
Eric