String group

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

-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

The string group is an entry in the Whitehead tower (dual to the notion of Postnikov tower) for the orthogonal group:

⋯ → Fivebrane ⁡ ( n ) → String ⁡ ( n ) → Spin ⁡ ( n ) →

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

The resulting manifold cannot be any finite-dimensional Lie group, since all finite-dimensional compact Lie groups have a non-vanishing

The fivebrane group follows, by killing

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).

The relevance of the Eilenberg-Maclane space

lies in the fact that there are the homotopy equivalences

the String group can be thought of as a "higher" complex spin group extension, in the sense of higher group theory since the space

It can be thought of the topological realization of the groupoid

whose object is a single point and whose morphisms are the group

Note that the homotopical degree of

, meaning its homotopy is concentrated in degree

, because it comes from the homotopy fiber of the map

from the Whitehead tower whose homotopy cokernel is

This is because the homotopy fiber lowers the degree by

The geometry of String bundles requires the understanding of multiple constructions in homotopy theory,[1] but they essentially boil down to understanding what

-bundles are, and how these higher group extensions behave.

are represented geometrically as bundle gerbes since any

-bundle can be realized as the homotopy fiber of a map giving a homotopy square

must map to a spin bundle

-equivariant, analogously to how spin bundles map equivariantly to the frame bundle.

The fivebrane group can similarly be understood[2] by killing the

It can then be understood again using an exact sequence of higher groups

Note map on the right is from the Whitehead tower, and the map on the left is the homotopy fiber.