String group explained

In topology, a branch of mathematics, a string group is an infinite-dimensional group

\operatorname{String}(n) introduced by 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 → {\displaystyle K( Z , 2 )} → \operatorname{String}(n) → \operatorname{Spin}(n) → 0

where K( 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:

… → \operatorname{Fivebrane}(n) \to \operatorname{String}(n) → \operatorname{Spin}(n) → \operatorname{SO}(n) → \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( Z , 2 ) lies in the fact that there are the homotopy equivalences

K( Z , 1 ) \simeq U( 1 ) \simeq B Z

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

0 \to K( Z , 1 ) \to \operatorname{Spin} ^{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( Z , 2 ) is an example of a higher group. It can be thought of the topological realization of the groupoid B U( 1 ) whose object is a single point and whose morphisms are the group U( 1 ) . Note that the homotopical degree of K( 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( 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,^[1] but they essentially boil down to understanding what

K( Z , 2 ) -bundles are, and how these higher group extensions behave. Namely, K( Z , 2 ) -bundles on a space M are represented geometrically as bundle gerbes since any K( 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( Z , 3 ) \end{matrix}

where K( Z , 3 ) = B(K( Z , 2 )) . Then, a string bundle S \to M must map to a spin bundle S \to M which is K( 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^[2] 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( 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( 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.

See also

• Gerbe
• N-group (category theory)
• Elliptic cohomology
• String bordism

External links

• • From Loop Groups to 2-groups - gives a characterization of String(n) as a 2-group
    • What is an elliptic object?

Notes and References

1. Jurco. Branislav. August 2011. Crossed Module Bundle Gerbes; Classification, String Group and Differential Geometry. International Journal of Geometric Methods in Modern Physics. 08. 5. 1079–1095. 10.1142/S0219887811005555. 0219-8878. math/0510078. 2011IJGMM..08.1079J . 1347840.
2. Sati. Hisham. Schreiber. Urs. Stasheff. Jim. November 2009. Fivebrane Structures. Reviews in Mathematical Physics. 21. 10. 1197–1240. 10.1142/S0129055X09003840. 0805.0564. 2009RvMaP..21.1197S . 13307997. 0129-055X.