### Re: Self-dual strings and M5-brane anomalies

#### Posted by Urs Schreiber

[*This is a followup to Luboš’s blog entry Self-dual strings and M5-brane anomalies. Let’ see if it is possible to have an inter-blog discussion.*]

When I read Berman’s & Harvey’s hep-th/0408198 a while ago I learned a bit more about how difficult it really is to understand the situation with $Q_5 \gt 1$ coincident 5-branes.

Important for me at that point was the reference to

X. Bekaert & M. Henneaux & A. Sevrin: Chiral forms and their deformation (2000)

which demonstrates that it is impossible to have a *local* deformation of an abelian theory of self-dual 2-forms to a non-abelian one.

This result is perhaps reminiscent of the insight of

F. Girelli & H. Pfeiffer: Higher gauge theory - differential versus integral formulation (2004)

that the ‘naive’ (depending on what you still consider naive) local extension of the YM Lagrangian to nonabelian 2-forms constituting a local connection for a strict structure 2-group is really equivalent to ordinary gauge theory again and does not yield the expected extension.

The same problem from another point of view is that one can write down the dynamical fields that are expected to describe the coincident 5-branes by deriving data describing a nonabelian gerbe from M5-brane anomaly cancellation as in

P. Aschieri & B. Jurčo: Gerbes, M5-Brane Anomalies and $E_8$ Gauge Theory (2004) ,

but then it is not known (yet) how to construct gauge invariant quantities, holonomies and ‘Lagrangian-like’ objects from that data.

Though there are some first hints, I believe. I think I can show that the nonabelian gerbes (without connection) considered by Aschieri & Jurčo are equivalent to the data describing 2-transitions in 2-bundles (without connection), which were very recently introduced in

T. Bartels: Categorified gauge theory: 2-Bundles (2004) .

That’s good, because the 2-categoric context of 2-bundles should allows us to simply categorify the objects that we are looking for in the ordinary incarnation to get their nonabelian 2-form version.

In particular, it should be possible to have a 2-morphisms from the 2-groupoid of 2-paths in the base space of a 2-bundle to the structure 2-group and thus get a holonomy of non-abelian 2-forms. It remains to be seen if the non-abelian cocycle data introduced that way is still equivalent to Aschieri&Jurč’s nonabelian gerbes with connection, but the success in the case without connection suggests that this must be true.

The good thing is that if we pick the base 2-space of our 2-bundle to be $B$ with point space

being ‘spacetime’ (i.e. the 5-brane worldvolume) and the arrow space

the space of based loops over $\mathcal{M}$ (i.e. the configuration space of closed strings in the 5-brane) then such a 2-connection gives rise to a connection on the loop space $B^2$ (as the entire 2-bundle gives rise to an ordinary bundle over $\Omega$\mathcal{M}) and we know some things about how to get holonomies from such nonabelian path-space connections and how they can in principle be more general than the local strict 2-group holonomies considered by Girelli&Pfeiffer (though it remains to be better understood exactly how they are more general).

In fact, this leads me to the **cubic scaling** of degrees of freedom that Lubš talked about in his post.

So if I understand correctly the usual asymptotic scaling with $n^2$, where $n$ is the dimension of the Cartan sub-algebra of the gauge group) is simply an indication of the fact that fields in the adjoint rep and in particular the gauge bosons themselves have to make up $n \times n$ matrices and thus appear in bunches of $n^2$.

So if objects in theories on $n$ $Q_5$ branes scale faster than $n^2$ this might indicate that the gauge connection in these theories requires for its specification more than a $\mathrm{Lie}(\mathrm{SU}(n))$-valued differential form. (What Luboš addresses as fields carrying ‘three indices’).

But that’s exactly what is the case in general for the 2-connections and the connections on loop space! Bekaert,Henneaux&Sevrin in their paper mention the famous old result by Teitelboim (Phys. Lett. **167B** (1986) 63) that no ‘straightforward’ (as they say) non-abelian extension of the 2-form field exists, which is based on the assumption that the holonomy of a 2-form gauge field $B$ along a string worldsheet $\Sigma$ is

even in the nonabelian case.

But the boundary state deformation considerations that I gave in hep-th/0407122 as well as the relation to the 2-group connections shows that (what Alvarez et. al had, in a special case, considered before and what was also expressed by Hofmann) that this fails because parallel transport of the non-abelian $B$ to a reference fiber has to be included, generalizing the above to

This form of a path space connection evades Teitelboim’s no-go theorem if the 2-form $B$ and the 1-form $A$ satisfy a certain condition, dubbed *r-flatness* by Alvarez, which can be shown to be equivalent to the exchange law arising in a sesqui-connection (a 2-connection into a sesqui-group).

There is some discussion necessary concerning the uniqueness of this form, but in any case it shows that the gauge fields expected to describe this non-abelian parallel transport of strings involves *more* than one $\mathrm{Lie}(\mathrm{SU}(n))$-valued differential form and hence *more* than $n^2$ ‘fields of data, namely (at least) two such forms. This holds locally and indeed the general results on nonabelian gerbes say that there must be even more forms involved.

Now this does not yet prove the $n^3$ scaling, but this is kind of suggestive to be part of the explanation why the scaling is $\gt n^2$.

Posted at October 22, 2004 7:43 PM UTC
## Re: Re: Self-dual strings and M5-brane anomalies

Hi, Urs,

I guess you have read this literature a lot more thoroughly than I ever did, but the last time I looked at it, I remember seeing an N^3 popping out in an unexpected place: if you look at Hofman’s paper on nonabelian 2-forms, as part of the data of his connection on loop space, he uses a field with 3 indices that gets interpreted as determining an algebra structure on the N-dimensional “Chan-Paton” vector space. Roughly, this field looks like it describes the way that the fivebranes are knitted together by condensation of “pant” diagrams, which would allow strings on the M5-brane worldvolume to change their Chan-Paton indices as they interact with one another. Is it related to what you are discussing above?