The n-Category Café

Recall that the sequence of higher connected covers $$O(n) \leftarrow S O(n) \leftarrow Spin(n)$$ of the orthogonal group $O(n)$ in the next step is called $String(n)$ $$O(n) \leftarrow S O(n) \leftarrow Spin(n) \leftarrow String(n)$$ because just as the lift of the structure group of the tangent bundle of an oriented manifold $X$ from $S O(n)$ to $Spin(n)$ is required in order for quantum super-particles (“spinors”) to be consistent on $X$, the further lift of the structure group to $String(n)$ is required in order for quantum super-strings (heterotic superstrings, to be precise) to be consistent.

The above article starts with the observation that even though in principle the existence of such a lift – a “String structure” – on $X$ is detectable from the fact that the TMF-cohomology of $X$ exhibits Poincaré duality, this does in practice not lend itself to computations. The point of the article is to demonstrate that nevertheless there is a shadow of this Poncaré duality which is accessible and useful to determine existence or not of String-structures: namely certain patterns on the Steenrod algebra.

Somewhat in passing the authors mention that one can go further, to the next higher connected cover of $O(n)$ sitting above $String(n)$ and have similar considerations for that case:

in joint work with Hisham Sati and Jim Stasheff, we had coined the term $Fivebrane(n)$ for the next higher connected cover $$O(n) \leftarrow S O(n) \leftarrow Spin(n) \leftarrow String(n) \leftarrow Fivebrane(n)$$ and fivebrane structure for the corresponding lift of the structure group.

Of course from pure algebraic topology there is no big deal in principle in passing to ever higher connected covers of $O(n)$, reflected by the fact that all of them have the generic name $O\langle m\rangle$ for some $m$. It was Hisham Sati who recognized the physical meaning of this next higher lift as that required for super NS 5-branes to be consistent on a manifold, hence the term $Fivebrane(n)$.

We had announced part of this story in

H. Sati, U. Schreiber, J. Stasheff
$L_\infty$-connections
(arXiv, blog)

(which in its bulk deals with setting up the rational approximation to differential cocycles classifying things like String 2- and Fivebrane 6-bundles with connection)

and then introduced Fivebrane structures as such in

H. Sati, U. Schreiber, J. Stasheff
Fivebrane Structures
(arXiv, blog).

Apart from discussing the physical context such as the relation to the magnetic dual string theory, the magnetic dual Green-Schwarz mechanism, and the worldvolume anomaly of the super NS 5-brane, this states the fact that the obstruction to lifting a String-structure to a Fivebrane structure is the fractional second Pontryagin class $\frac{1}{6} p_2$.

A complete discussion of the differential nonabelian cocycles classifying String 2- and Fivebrane 6-bundles with connection, as well as their obstructing Chern-Simons 3- and 7-bundles with characteristic classes those fractional Pontryagin classes $\frac{1}{2}p_1$ and $\frac{1}{6}p_2$ is underway. I hope to present some of this, which is joint work with Hisham Sati, Zoran Škoda and Danny Stevenson, at Higher Structure 2008 in Lausanne in a few weeks in a talk.

Meanwhile Hisham Sati has further studied the String-theoretic underpinning of these questions:

Hisham Sati
$O P^2$ bundles in M-theory
(arXiv)

This builds on the observation that in 11-dimensional supergravity, where ultimately all these structures come from, the exceptional Lie group $F_4$ arises in a way that, as Hisham discusses, allows a geometric interpretation in terms of 27-dimensional $O P^2$-bundles over 11-dimensional base space, where $O P^2 = F_4/Spin(9)$ is the Cayley plane.

In section 3.3.2 from p. 23 on he discusses the higher structures (Spin, String, Fivebrane) from this perspective. The fact that $O P^2$ itself is not Fivebrane is used here, which also appears in table 1 of the above article by Douglas, Henriques and Hill.