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.

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)

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!

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.

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.

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.

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.

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.

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…