### String- and Fivebrane-Structures

#### Posted by Urs Schreiber

Today appeared

Christopher L. Douglas, André G. Henriques, Michael A. Hill
* Homological obstructions to string orientations*

(arXiv)

which presents a new method to check if a given manifold admits a String-structure.

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.