### 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

following the notation in Hofman’s paper, but including a second factor of the $A$-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

on target space, while local *gauge* transformations on loop space give rise to

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 $(d+{\oint}_{A}(B){)}^{2}$ 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

## 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