### Three ways to loop flatness

#### Posted by Urs Schreiber

Today I was contacted by Luiz Agostinho Ferreira who kindly called my attention to the paper

Orlando Alvarez & Luiz. A. Ferreira & J. Sánchez Guillén : A New Approach to Integrable Theories in any Dimension (1998)

that regrettably I was not aware of before but which discusses matters that are relevant for my recent work hep-th/0407122 on strings in nonabelian 2-form backgrounds.

The above paper is motivated from generalizing the method of *Lax pairs* from 1+1 to higher dimensions. Integrable field theories in 1+1 dimensions admit the construction of a certain connection on 1+1d spacetime which is *flat* iff the equations of motion hold, so that all holonomies of that connection around, say, ${S}^{1}$ space, are preserved in time and hence are conserved charges.

The idea is to generalize this procedure to $d=1+p$ dimensional field theories by constructing $p$-holonomies for $p>1$ which associate group elements with a $p$-dimensional manifold, so that when the relevant $p$-curvature vanishes these are again homotopy invariants of these surfaces.

For $p=2$ this amounts to the study of connections on loop space, which is precisely what I was concerned with in the above mentioned paper (as discussed here before: I, II, III).

Ferreira et al. work on parameterized but *based* loop space, where all the loops have a given point in target space at parameter $\sigma =\mathrm{0,2}\pi $ in common. This differs a little from the setup needed for string theory, where there is no requirement about this point. So the based loop space in that paper is a subspace of the full loop space relevant in string theory. But as far as I can see this does not restrict any of the constructions and statements made by Ferreira et al., they all generalize easily to unbased loop space.

That said, the first thing to observe is that the general form of the loop space connection used by Ferreira et al. (see their equation (5.1)) is precisely that which dropped out from the boundary state deformation which I used (my equation (1.2) and (3.15)), namely (in the notation adapted from Hofman’s paper but modified as in my equation (3.12))

Here $d$ is the exterior derivative on loop space $A$ is a 1-form and $B$ a 2-form, both taking values in a nonablelian Lie-algebra, on target space and the integral is that over the pull-back of $B$ over the loop with $A$-holonomies used to parallel tranport everything to the point $\sigma =0$.

I argued in my paper that local gauge transformation on loop space translate into sensible 2-level gauge transformations on target space only when $B=-{F}_{A}$, in which case it turns out that the above connection is flat, which again makes sense since it implies that toroidal worldsheets don’t see the nonabelian background.

But $B=-{F}_{A}$ is not the only condition which implies *flatness* of the connection on loop space. Alvarez, Ferreira and Guillén don’t need to mind consistency of any boundary state deformations, of course, and they find two further sufficient conditions for flatness.

With the above notation there is a simple calculus for dealing with differential forms on loop space which are multi-path-ordered integrals (studied intensively by Chen) over the loops. Using this one finds for the curvature

Only the first term is understandable from the purely target space perspective, since this is just the ‘pullback’ (on one index) of the $B$-field strength to the loop up to these $A$-holonomies and integration.

The other terms have no direct target space analog. In particular, the field strength ${F}_{A}$ of the $A$-field shows up all the time, due to the action of the exterior derivative on the $A$-holonomies.

Alvarez, Ferreira & Guillé demand that

in order for these terms to vanish.

This is natural from the point of view of integrable systems, but not so natural when we want to think of ${F}_{A}$ really as the field strength of a physical background gauge field, like in string theory, instead of as a auxiliary connection coming from a Lax-pair method. The authors show that several known integrable systems have ‘Lax pairs’ which satisfy this condition.

That it would help to have ${F}_{A}=0$ was also discussed by Christiaan Hofman in the above mentioned paper, who also discussed constraining it to zero.

(For the connection with $B=-{F}_{A}$ that I was discussing, however, we can have ${F}_{A}\ne 0$ and there is full cancellation of all the bothersome terms.)

So with ${F}_{A}=0$ two of the terms in the above expression for the curvature vanish. Alvarez, Ferreira & Guillén now also set what one might call ‘target space curvature of $B$’ equal to zero:

With this requirement, all that remains is the exterior product of the 1-form ${\oint}_{A}(B)$ with itself. If this were an abelian 1-form, then its exterior square would vanish automatically. So that’s one condition on $B$ (together with ${F}_{A}=0={d}_{A}B$) which would give a flat connection (their equation (3.16)).

The second choice discussed by these authors is obtained by demanding not only ${d}_{A}B=0$, but even stronger that $B$ is covariantly constant (their equation (3.18)). Then, as they show and as can easily be seen, $({\oint}_{A}(B){)}^{2}=0$ follows automatically.

So this are two sets of conditions on $A$ and $B$ which yield a flat metric on loop space without having $B=-{F}_{A}$ but with instead having ${F}_{A}=0$ and constancy constraints on $B$.

It is annyoing that I wasn’t aware of this work before and I will need to cite it in a revised version of my paper. But I also think that the insights in both papers are to a good part complementary. The form of the loop space connection in both cases is found to be the same, but different conditions on the vanishing of its curvature arise for applications in integrable systems as opposed to applications for the configuration space of relativistic superstring.

(Please note that I have only read this paper today and not all of it, so I have to apologize for any inaccuracies in the above summary. I’ll appreciate corrections and comments. More on how to use the String Coffee Table can be found here.)

Posted at July 22, 2004 5:46 PM UTC
## Re: Three ways to loop flatness

Ha! I mentioned that paper to you in my

mail of 9th July. ;-)