The String Coffee Table

Thoughts on n-cube Scaling on 5-Branes

Posted by Urs Schreiber

Andrew Neitzke and Luboš Motl recently made me aware of the importance of identifying objects in 2-gauge theory that carry three ‘group indices’.

One expects that a stack of 5-branes is characterized by

- the number $n$ of coincident 5-branes

- a condensate $\{m_{ijk}\}$, $i,j,k \in \{1,2,\dots,n\}$ of BPS membranes with precisely three disconnected boundary components attached to the $i$th, the $j$th and the $k$th 5-brane.

With respect to the 2-gauge theory living on the stack of 5-branes it is clear that the parameter $n$ specifies the gauge group, like in $\mathrm{SU}(n)$.

The big question is: What in the 2-gauge theory is described by $m_{ijk}$?

Intuitively one would expect $m_{ijk}$ to measure the probability for a membrane ending on the $i$the 5-brane to coalesce with another membrane ending on the $j$the 5-brane and turning into a membrane ending on the $k$th 5-brane, roughly. This suggests that $m_{ijk}$ somehow controls the product of Lie algebra elements, maybe.

Aware of that, Andrew Neitzke identified the map $M$ in equation (21) of Christiaan Hofman’s paper hep-th/0207017 as an apparently natural candidate for an incarnation of $m_{ijk}$.

There are three obvious questions:

- Can we check if this makes sense?

- Are there other candidates?

- Are these other candidates possibly just different aspects of the same thing?

In the following I want to briefly sketch some thoughts on these questions:

First of all I would like to make the point that while equation (15) in Christiaan Hofman’s paper is very suggestive and certainly a great idea, it does not really follow (as far as I can see at least) from the considerations that he gives in (1)-(14). For one, from (14) it would follow that there should be terms proportional to $F_A$ in (15). Another difficulty is that the 2-form $B$ is ‘shuffled’ inside the multi-integral, including its $\mathfr{g}$-factor, which is not what happens if $\oint(B)$ is applied from the left on the multi-integral, as one might expect from a connection 1-form.

So if we are to physically interpret the map $M$ we first need a strict derivation of equation (15) from some loop space reasoning. Christiaan’s Hochschild complex considerations certainly show that (15) is a good idea, but how does it really arise?

The key observation is probably the shuffling property of the $B$-term. It implies that this term does not come from any multiplication from the left, but arises from a derivation inside the multi-integral, the same way that the $A$ term arises. This could then also explain why the $F_A$ terms don’t appear: They could cancel against one part of the $B$ term.

Thinking about this for a moment seems to admit only a single solution:

Recall that in the context of superstrings the natural differential operator on loop space is not the loop space exterior derivative $\mathbf{d}$, but the polar combination of the worldsheet supercharges, which reads

where $\iota_K$ is the operator of inner multiplication with the loop space vector $K^{(\mu,\sigma)} = X^{\prime\mu}(\sigma)$ ($X$ is the loop and $X^\prime$ its $\sigma$-derivative). $T$ is the string’s tension.

This operator was first considered in the second half of Witten’s ‘SUSY and Morse theory’ paper and is nowadays familiar from boundary state formalism. A boundary state for some gauge field background is simply an inhomogenous differential form on loop space of the form

Usually the $B$ term is absent here, but as I have tried to argue in hep-th/0407122 including it (which is very natural) immedietaly gives us local nonabelian connections on loop space induced by the nonabelian 2-form $B$ that have well-defined surface holonomy and are equivalent to local 2-connection in the theory of 2-groups.

So this is in a sense the generalization of the ordinary Wilson line for $A$ along the loop/string. The ordinary Wilson line is what Christiaan Hofman convincingly argued to have its place in between the factors of the multi-integrals. Now lets take his approach and the above one together and generalize in the obvious way: This leads us to consider multi-integrals of target space $(p_i+1)$-forms $\omega_i$ of the form

This is very close to what Christiaan Hofman does, it essentially just replaces the ordinary Wilson line by its supersymmetric version. The interesting point is that acting with the modified exterior derivative $\mathbf{d}_K$ which comes from the string’s supercharges on such a multi-form produces precisely the terms that Christiaan Hofman postulates in his equation (15):

where $R$ denotes some additional terms that don’t look like multi-integrals of the above form. These terms drop out however if we scale $T \to \infty$ with keeping $T B$ constant.

So this construction has a nice side and a surprising side: The nice thing is that we can very naturally derive the terms in Hofman’s equation (15), the surprising thing is that we can do so precisely only by taking the unexpected limit of large tension. Maybe this is a sign that the above needs to be improved, maybe it is a sign of some effect. (E.g. it could be that the above applies to membranes that stretch between a stack of 5-branes and some other attachment point, thus inducing large tension on their boundary strings.)

In any case, this demonstrates that it is possible in principle to derive the multi-derivations that the map $M$ which we are interested in is part of from a consistent scheme of loop space differential geometry, even one which has the right physical objects like $\mathbf{d}_K$ and $W(\sigma,\sigma^\prime)$ appearing.

Incidentally, this suggest that the $A$ and $B$ appearing here are not nessecarily the same that would also appear in an honest connection on loop space, which would be represented by a loop space 1-form of the form $\oint_{A^\prime} (B^\prime)$ and give rise to a covariant derivative

the way I have discussed before. So it seems that we really end up with two sets of a (1+2) form. This is actually nice, because the definition of a nonabelian gerbe also involves two such pairs!

But there is still a problem with idenitfying the map $M$ with a version of the $m_{ijk}$: In all of the above formulas the product in $\mathfr{g}$ is really implicit already in the multi-integrals. After all, these integals are valued in the enveloping algebra of $\mathfr{g}$ and not in some tensor product. What $M$ really does is just implementing the wedge product on the scalar coefficients. Otherwise it seems hard to give the above objects a sensible interpretation.

So it seems that if we want to identify the $m_{ijk}$ with anything determining the group product, we must to so for all such products, not just inside the map $M$. That is, we have to somehow generalize the product in $\mathfr{g}$ in general.

There seems to be no room for such a step in the theory of nonabelian gerbes. That’s why Christiaan Hofman fixes the freedom contained in the definition of the product before discussing gerbes in his paper.

But we know that 2-bundles for strict structure 2-groups are equivalent to nonabelian gerbes. On the other hand, 2-bundles are more general than that. In particular, the structure 2-group of a 2-bundle is in general weak and/or coherent (which is essentially the same) instead of strict.

But a coherent 2-group is a 2-group in which, lo and behold, the group product operation is more flexible than ordinarily! In particular, the group product here need not be associative. Instead, there is a natural transformation called the associator which tells you how the group product fails to be associative.

This is precisely the kind of degree of freedom that we are looking for. Now there is a very interesting result for coherent 2-groups:

In

J. Baez & A. Lauda: Higher Dimensional Algebra V: 2-Groups (2004)

it is proved in section 8.3 that every coherent 2-group is specified up to equivalence by the following data:

- a group $G$

- an abelian group $H$ and an action of $G$ on $H$ by automorphisms

- an element $[a]$ of the cohomology group $H^3(G,H)$ .

Incidentally, this $[a]$ specifies the associator.

This looks rather similar to the data mentioned above wich specifies the stack of 5-branes. The group $G$ would by $SU(n)$ or something. Since $H$ is abelian it does not contain a whole lot of information. But then there is the object $[a]$, which, indeed, carries three ‘group indices’.

So it seems that coherent 2-groups might provide just the right kind of degrees of freedom to account for those on a stack of 5-branes, including the $n^3$ scaling.

To check this conjecture in more detail one would have to define the notion of 2-connection in a 2-bundle which has a coherent 2-group as structure group. This has not been done yet. But we are getting closer I think.

My apologies if the above was too speculative for anyone’s taste. I don’t have any hard results here, but I think the above observations are intersting enough to warrant thinking about them if one is interested in the $n^3$-puzzle.

Posted at October 27, 2004 8:53 PM UTC