The n-Category Café

Thanks for the question. I just have half a second, no time to really reply right now – but you can essentially find the answer here:

K. Lechner, M. Tonin, Worldvolume and target space anomalies in the $D=10$ super–fivebrane sigma–model.

More later.

5 am in a hotel in Dublin. While she is in the bathroom, I have another 1.5 seconds to reply to the above question:

yes, the (super-)5-brane is a gadget pretty much like the (super-)string, just 5+1 instead of 1+1 dimensional.

In 10 dimensions, the critical number of dimensions for the string, the 5-brane is the magnetic dual to the ordinary (“electric”) string and there are various reasons to expect that heterotic string theory, which arises as the effective target space theory (the second quantization) of the heterotic superstring has a dual formulation with the superstring replaced by the super 5-brane.

On the target space one such reason is the fact that there is a well known dual formulation of the target space theory: the ordinary target space theory of the heterotic string is a higher gauge theory involving a 2-bundle/1-gerbe: the (twisted) Kalb-Ramond field, which is the “electric” background field for the string.

The dual formulation is a higher gauge theory involving a 6-bundle/5-gerbe: the magnetic dual of the Kalb-Ramond field, which is the “magnetic” background field for the fivebrane.

This dual formulation of the heterotic background theory is originally due to Chamseddine, as far as I am aware.

To this dual target space theory one expects a corresponding $\sigma$-model for a super-5-brane from which the former arises as the effective background theory.

This super 5-brane worldsheet theory is the one discussed in the article I mentioned. As discussed there, it is in principle clear how and that this works, even though one technical subtlety concerning the interpretation of a prefactor of 1/2 remains.

But the upshot is that from the target space theory alone one can deduce, as I discussed here that heterotic string theory, in order to cancel the anomalies induced by the fermions it contains (gravitino, dilatino and gaugino) needs to contain a specific electric string and magnetic fivebrane current.

The electric string current, measured by a 4-form, had famously also been derived from the point of view of the worldsheet theory of the String. Originally by Killingback (review here). From this point of view it arises as a higher generalization of the condition that the target space for the super particle (the spinning particle) has to have Spin structure: the target of the heterotic superstring even has to have what is called a “String structure”.

Similarly one expects that the $\sigma$-model for the super 5-brane is consistent only if a certain structure on the target trivializes which carries an 8-form curvature. Such a thing is a 7-bundle or 6-gerbe with connection, or degree 8 differential cocycle. And indeed, as the article I linked to demonstrates, that’s what one finds.

In our article we discuss this “Fivebrane” structure as a generalization of the formerly known String structure.

If we disregard the gauge bundle for a moment (regarding the case that it is trivial, for instance), we have:

$X$ is Spin if $X$ is orientable and $w_2(X)$ vanishes.

$X$ is String if X is $Spin$ and $\frac{1}{2}p_1(X)$ vanishes.

$X$ is Fivebrane if X is $String$ and $\frac{1}{6}p_2(X)$ vanishes.

(These fractional Pontrjagin classes are reviewed/explained in section 4.4.)

If the gauge bundle is nontrivial the situation becomes a little more intricate. In that case the structure that obstructs the existence of the Fivebrane structure has a curvature 8-form which is a linear combination of the second Pontrjagin characteristic class of the Spin bundle and the fourth Chern class of the gauge bundle and of a couple of characteristic forms which are wedge products of nontrivial characteristic forms.

We discuss various special cases in which these decomposable characteristic forms vanish anyway. But more generally the question is:

what is the integral singular cohomology class of a 7-bundle/6-gerbe (whatever you call it) which has a curvature 8-form being the sum of given characteristic forms.

I think that, using the construction by Brylinski-McLaughlin, one can show that the integral class depends only on the indecomposable characteristic forms appearing in the curvature $n+1$-form. The other contributions suspend to 0 and hence do not contribute in that construction to the underlying topology, but just to the connection put on it.

This will be discussed in more detail elsewhere.

Ah, so even the old experts misspell ‘principal bundle’!

As Allen Knutson realized, a ‘principle bundle’ is one with moral fiber.

Later, James Dolan started using ‘moral fiber’ to mean ‘homotopy fiber’ — since in homotopy theory, the ‘morally correct’ version of the fiber is the homotopy fiber.

Putting this all together, I think we should use ‘principle bundle’ to denote the homotopy-theoretic analogue of a principal bundle. Its moral fibers will be $A_\infty$ spaces.