String group
# String group

In topology, a branch of mathematics, a string group is an infinite-dimensional group $\operatorname {String} (n)$ introduced by Stolz (1996) as a $3$-connected cover of a spin group. A string manifold is a manifold with a lifting of its frame bundle to a string group bundle. This means that in addition to being able to define holonomy along paths, one can also define holonomies for surfaces going between strings. There is a short exact sequence of topological groups

   $0\rightarrow {\displaystyle K(\mathbb {Z} ,2)}\rightarrow \operatorname {String} (n)\rightarrow \operatorname {Spin} (n)\rightarrow 0$

where $K(\mathbb {Z} ,2)$ is an Eilenberg–MacLane space and $\operatorname {Spin} (n)$ is a spin group. The string group is an entry in the Whitehead tower (dual to the notion of Postnikov tower) for the orthogonal group:

   $\cdots \rightarrow \operatorname {Fivebrane} (n)\to \operatorname {String} (n)\rightarrow \operatorname {Spin} (n)\rightarrow \operatorname {SO} (n)\rightarrow \operatorname {O} (n)$

It is obtained by killing the $\pi _{3}$ homotopy group for $\operatorname {Spin} (n)$, in the same way that $\operatorname {Spin} (n)$ is obtained from $\operatorname {SO} (n)$ by killing $\pi _{1}$. The resulting manifold cannot be any finite-dimensional Lie group, since all finite-dimensional compact Lie groups have a non-vanishing $\pi _{3}$. The fivebrane group follows, by killing $\pi _{7}$.

More generally, the construction of the Postnikov tower via short exact sequences starting with Eilenberg–MacLane spaces can be applied to any Lie group G, giving the string group String(G).

## Intuition for the string group

The relevance of the Eilenberg-Maclane space $K(\mathbb {Z} ,2)$ lies in the fact that there are the homotopy equivalences

   $K(\mathbb {Z} ,1)\simeq U(1)\simeq B\mathbb {Z}$

for the classifying space $B\mathbb {Z}$, and the fact $K(\mathbb {Z} ,2)\simeq BU(1)$. Notice that because the complex spin group is a group extension

   $0\to K(\mathbb {Z} ,1)\to \operatorname {Spin} ^{\mathbb {C} }(n)\to \operatorname {Spin} (n)\to 0$

the String group can be thought of as a "higher" complex spin group extension, in the sense of higher group theory since the space $K(\mathbb {Z} ,2)$ is an example of a higher group. It can be thought of the topological realization of the groupoid $\mathbf {B} U(1)$ whose object is a single point and whose morphisms are the group $U(1)$. Note that the homotopical degree of $K(\mathbb {Z} ,2)$ is $2$, meaning its homotopy is concentrated in degree $2$, because it comes from the homotopy fiber of the map

   $\operatorname {String} (n)\to \operatorname {Spin} (n)$

from the Whitehead tower whose homotopy cokernel is $K(\mathbb {Z} ,3)$. This is because the homotopy fiber lowers the degree by $1$.

### Understanding the geometry

The geometry of String bundles requires the understanding of multiple constructions in homotopy theory, but they essentially boil down to understanding what $K(\mathbb {Z} ,2)$-bundles are, and how these higher group extensions behave. Namely, $K(\mathbb {Z} ,2)$-bundles on a space $M$ are represented geometrically as bundle gerbes since any $K(\mathbb {Z} ,2)$-bundle can be realized as the homotopy fiber of a map giving a homotopy square

   ${\begin{matrix}P&\to &*\\\downarrow &&\downarrow \\M&\xrightarrow &K(\mathbb {Z} ,3)\end{matrix}}$

where $K(\mathbb {Z} ,3)=B(K(\mathbb {Z} ,2))$. Then, a string bundle $S\to M$ must map to a spin bundle $\mathbb {S} \to M$ which is $K(\mathbb {Z} ,2)$-equivariant, analogously to how spin bundles map equivariantly to the frame bundle.

## Fivebrane group and higher groups

The fivebrane group can similarly be understood by killing the $\pi _{7}(\operatorname {Spin} (n))\cong \pi _{7}(\operatorname {O} (n))$ group of the string group $\operatorname {String} (n)$ using the Whitehead tower. It can then be understood again using an exact sequence of higher groups

   $0\to K(\mathbb {Z} ,6)\to \operatorname {Fivebrane} (n)\to \operatorname {String} (n)\to 0$

giving a presentation of $\operatorname {Fivebrane} (n)$ it terms of an iterated extension, i.e. an extension by $K(\mathbb {Z} ,6)$ by $\operatorname {String} (n)$. Note map on the right is from the Whitehead tower, and the map on the left is the homotopy fiber.

