## July 12, 2004

### Not Abelian - but Flat

#### Posted by urs

[Update 07/15/04: The issue discussed below can now be found discussed in hep-th/0407122.]

Over the weekend I had some time to think about what I had written about nonabelian 2-form connections here at the String Coffee Table as well as over on sci.physics.research.

The result are some refinements on my draft Nonabelian 2-form connections from 2d BSCFT deformations, which is beginning to approach a final form.

Using boundary state deformation theory I had read off the loop space connection induced by a nonabelian 2-form background from the deformed worldsheet supercharges. This is a 1-form connection on loop space, which is essentially what Christiaan Hofman writes as

(1)$d+\mathrm{iT}{\oint }_{A}\left(B\right)\phantom{\rule{thinmathspace}{0ex}},$

with the only difference that I derive that (at least in the context that I am working in) one has to define this a little differently, as explained before.

Then one can study gauge transformations of this 1-form connection in the usual way and try to read off what this implies for the target space fields $A$ and $B$.

It can be checked that global gauge transformations on loop space give rise to the ordinary target space gauge transformations obtained by adjoining unitary group elements.

The problem was that for loop-space-local gauge transformations one obtains the expected terms - plus plenty of other terms that don’t have a target space analogue.

As Christiaan Hofman and Hendryk Pfeiffer kindly told me, in one way or another this is a well known problem. What I think is new is the particular string/loop space perspective that I am using. This can be used for the following simple physical argument, which, as I now realize, is the heuristic version of the key for making further progress:

A non-abelian 2-form background must obviously couple to open strings only, since there are no Chan-Paton factors for closed strings (even though the abelian 2-form of course does couple to the closed string). This means that whatever connection we find, it should not couple to closed strings. But noting that a torus worldsheet in target space is nothing but a closed path in loop space, this tells us that we should be looking for loop space connections which are flat. That’s because this implies that the surface holonomy associated with every closed target space surface, which is the same as the line holonomy of the respective closed path in loop space, is always the trivial element $g=1$. The nontrivial physics is then induced only on open strings via the deformed boundary state.

So let’s look for configurations of $A$ and $B$ fields that make $\left(d+i{\oint }_{A}\left(B\right){\right)}^{2}=0$. There are probably several ways to derive these, but I have found that the boundary state deformation theory allows a very simple criteria: The boundary state deformation operator should not depend on worldsheet fermions.

This is precisely the case when the sum of $A$-field strength with the 2-form field vanishes:

(2)${F}_{A}+B=0\phantom{\rule{thinmathspace}{0ex}}.$

One can explicitly check that in this case the obnoxious corrections terms cancel each other and one finds that in this case that infinitesimal local gauge transformations in loop space of the form

(3)$U\left(X\right)=1-i{\oint }_{A}\left(\Lambda \right)+\cdots$

do indeed induce the expected

(4)$A↦A+\Lambda$
(5)$B↦B-{d}_{A}\Lambda$

(to first order in $\Lambda$), and one can also explicitly check that in this case the loop space connection is flat.

Now, to my delight, after I had convinced myself of the above relations I opened up

Florian Girelli & Hendryk Pfeiffer: Higher gauge theory - differential versus integral formulation (2004)

and found that the authors of that paper, by completely different reasoning (based on 2-groups introduced by John Baez), find precisely the same condition (their (3.25))!

Cool. So the result is not new (but not very old, either), but I still think that it is kind of interesting to see how the abstract category-theory reasoning meets string theory and in particular the loop space deformation theory that I have been working on.

To summarize: Nonabelian 2-form connections can consistently be defined only when ${F}_{A}+B=0$. This makes physical sense, because it is precisely the condition which says that the connection on loop space is flat, so that closed strings do not couple to the nonablian background, and they shouldn’t. The corresponding deformed boundary state $\mathrm{P}epx\left(\int A\cdot {X}^{\prime }\right)\mid \alpha 〉$ should describe open strings in the nonabelian 1- and 2-form background.

What do you all think?

Posted at July 12, 2004 6:09 PM UTC

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