The String Coffee Table

The target space configurations which give rise to non-abelian 2-forms are not at all well understood. One would expect, though, that they would involve stacks of NS-5 branes on which membranes may end. The boundary of these membranes appear as strings (fundamental strings or ‘little strings’) in the world-volume theory of the NS branes, generalizing the way how open string endpoints appear as ‘quarks’ in the world-volume theory of D-branes. Just like a nonabelian 1-form couples to these ‘quarks’, i.e. to the boundary of an open string, a non-abelian 2-form should couple to the boundary of an open membrane, i.e. a to string on an NS-5 brane.

This analogy strongly suggests that there is a single Chan-Paton-like factor associated to each string living on the stack of NS-5 branes, indicating which of the $N$ branes in the stack it is associated with. This Chan-Paton factor should be the degree of freedom that the non-abelian $B$-field acts on.

Hence the higher-dimensional generalization of ordinary gauge theory should, in terms of strings, involve the steps upwards the dimensional ladder indicated in the following table:

(1-)gauge theory $\to$ 2-gauge theory

string ending on D-brane $\to$ membrane ending on NS brane

‘quark’ on D-brane $\to$ string on NS brane

nonabelian 1-form gauge field $A$ $\to$ nonabelian 2-form gauge field $B$

coupling to the boundary of a 1-brane (string) $\to$ coupling to the boundary of a 2-brane (membrane)}

Chan-Paton factor indicating which D-brane in the stack the ‘quark’ sits on $\to$ Chan-Paton-like factor indicating with NS brane in the stack the membrane boundary string sits on.

These heuristic considerations, vague as they are, receive some substantiation by the fact that indeed the contexts in which nonablian 2-forms have been argued to arise naturally are little string theories and the effective $N=(\mathrm{2,0})$ $D=5+1$ supersymmetric field theories living on these NS-5 branes.

The study of little strings and these $N=(\mathrm{2,0})$ QFTs in six dimensions has proven to be quite involved and no good understanding of the non-abelian 2-form from this target space perspective has emerged so far. Here we shall make no attempt to say anything directly about the physics of membranes attached to NS-5 branes. Instead the strategy here is to look at the worldsheet theory of strings and see from the worldsheet perspective if anything can be said about superconformal field theories that involve a non-abelian 2-form in their moduli space. Even though we will not be able to exhibit a direct correspondence between certain such SCFTs and the target space physics of membrane boundary strings on NS branes, we will be able to recognize in the formal structure of these SCFTs some of the above mentioned expected properties of such theories

This works by way of generalized Wilson lines along the string: Let ${ℰ}^{†\mu }=\frac{1}{\sqrt{2}}({\psi }^{\mu }+i{\overline{\psi }}^{\mu })$ denote the polar composition of left and right-moving worldsheet fermions $\psi ,\overline{\psi }$. It is known that an ordinary abelian $B$-field background manifests itself in the worldsheet SCFT in the appearance of a term

and that a non-abelian 1-form $A$-field background comes from a boundary state deformation with the object

, where $\mathrm{P}$ denotes path ordering along the string. (This is the classical form of this object, whose quantization will involve divergences that have to and can be dealt with either by regularization and/or by imposing the background equations of motion for $A$.)

This makes it quite compelling to construct worldsheet SCFTs involving non-abelian 2-forms $B$ by deformation with a generalized Wilson line along the string of the form

which is what we are going to do in the following. This generalized Wilson line suggests itself not only as the straightforward generalization of the two special cases mentioned above, but confirms its expected role also by the fact that, as we will show, it induces in the worldsheet supercharge (which is a generalized exterior derivative on loop space) a term that is precisely the non-abelian covariant exterior derivative on loop space which computes non-abelian surface holonomy on the worldsheet. This is precisely as it should be in light of the fact that for abelian $B$ the supercharge can be seen to include the corresponding abelian connection on loop space which computes the abelian surface holonomy ${\int }_{\mathrm{worldsheet}}B$. Given all that is known about the worldsheet theories for abelian $B$ and non-abelian $A$-field backgrounds, this shows that the above operator should be the correct worldsheet deformation operator that induces a non-abelian 2-form background.

But one point deserves further attention: Note that we did not include a trace over the path-ordered exponential. The reason for that is quite simple: If we included a trace, making the formerly group-(representation)-valued expression a scalar, this could not give rise to an exterior derivative on loop space which is covariant with respect to the given non-abelian gauge group, simply because its connection term would be locally a loop space 1-form taking scalar values instead of non-abelian Lie-algebra values. This way the relation to non-abelian surface holonomy would be completely lost and one could not expect the associated SCFT to describe any non-abelianness of the background.

There is a further indication that not taking the trace is the correct thing to do: Recall from the discussion at the beginning of this section that we should expect states of strings in a non-abelian 2-form background to have a single Chan-Paton-like degee of freedom, i.e. to carry one fundamental index of the gauge group. This implies that in particular boundary states carry such an index and hence any deformation operator acting on them must accordingly act on that index. This is precisely what an un-traced generalized Wilson line does.

However, by not taking the trace the point $\sigma =0\sim 2\pi$ on the string/loop becomes a preferred point in a sense. The object is reparameterization invariant only under those reparameterizations that leave the point $\sigma =0\sim 2\pi$ where it is. Commuting it with a general generator of $\sigma$-reparameterizations produces two ‘boundary’ terms (even though we do not really have any boundary at $\sigma =0\sim 2\pi$ on the closed string) at $\sigma =0$ and $\sigma =2\pi$ which only mutually cancel when traced over.

This might at first sight appear as a problem for our proposed way to study worldsheet theories in non-abelian 2-form backgrounds. After all, reparameterization invariance on the string must be preserved by any reasonable physical theory. But a closer look at the general theory of non-abelian surface holonomy indicates how the situation should clarify:

As soon as a non-abelian 2-form is considered and any notion of non-abelian surface-holonomy associated with it, a fundamental question that arises is at which point the non-abelian holonomy of a given surface ‘lives’. It must be associated with some point, because it really lives in a given fiber of a gauge bundle over target space, and it has to be specified in which one. This issue becomes quite rigorously clarified in the 2-group description of surface holonomy. In the 2-group description non-abelian surface holonomy is a functor that assigns 2-group elements to ‘bigons’, which are surfaces with two special points on their boundary (a ‘source’ and a ‘target’ vertex). It is a theorem that every 2-group uniquely comes from a ‘crossed module’ (a tuple of two groups, one associated with our 1-form $A$ the other with the 2-form $B$ together with a way for $A$ to act on $B$ by derivations) in such a way that when composing any two bigons, their total surface holonomy is obtained by parallel transporting (with respect to the 1-form A) group elements from the individual ‘source’ vertices to the common source vertex of the combined bigon. This theorem shows that it is inevitable to associate surface holonomy with ‘preferred’ points.

On the other hand, these points are not absolutely preferred. One can choose any other point on the boundary of the given bigon as a the source vertex. This is done by, again, parallel transporting the surface holonomy from the original source vertex to the new one. A more detailed description of this and related facts of 2-group theory is beyond the scope of this paper, but can be found elsewhere.

Indeed, it is shown there that the preferred point $\sigma =0\sim 2\pi$ is to be identified with the source vertex in 2-group theory.

That this is true can indeed be seen quite clearly from the expression given below, which we find for the non-abelian covariant loop space exterior derivative obtained by deforming the ordinary loop space exterior derivative with.

Indeed, the loop space covariant exterior derivative to be discussed in the following involves integrating the non-abelian 2-form $B$ over the string, while parallel transporting its value back from every point to $\sigma =2\pi$. As mentioned above, and as already noted by Christiaan Hofmann, this parallel transport back to a common reference point is necessary in order to obtain a well defined element of a fiber in a non-abelian gauge bundle.

It should be plausible from these comments, and is proven in detail elsewhere, that, given non-abelian differential forms $A$ and $B$, the surface holonomy computed using the covariant loop space connections discussed here coincide with those computed by 2-group theory, when the former are integrated over loops in based loop space, i.e. the space of all loops with a common point in target space.

But this should finally clarify the appearance of the preferred point $\sigma =0$ also from the point of view of worldsheet reparameterization invariance: Whenever we work with a non-abelian 2-form the worldsheet surpercharges, which are generalized exterior derivatives on loop space, must be regarded as operators on based loop space (for some given base point $x$), in terms of which, for the reasons just discussed, all computations must necessarily take place. However, the choice of this base point is arbitrary, as surface holonomies computed with differing base points are related simply by parallel transport between the two base points.

The restriction to based loop space may seem like a drastic step from the string worldsheet point of view, even though the above general considerations make it seem to be quite inevitable. On the other hand, the boundary state formalism that we will be concerned with in the following involves spherical closed string worldsheets at tree level, and these are precisely the closed curves in based loop space.

Given all this, the conceptual issues that we encounter in our worldsheet perspective approach to non-abelian 2-form field backgrounds are somewhat unusual, but are quite in accordance with both the general features expected of the target space theory as well as general facts known about non-abelian 2-gauge theory. While it of course does not completely solve the issue of superstring propagation in non-abelian 2-form backgrounds, the present approach is hoped to illuminate some crucial aspects of any complete theory that does so, and in particular of its worldsheet formulation.