String group

In topology, a branch of mathematics, a string group is an infinite-dimensional group formula_1 introduced by as a formula_2-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 groupsformula_3where formula_4 is an Eilenberg–MacLane space and formula_5 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:formula_6It is obtained by killing the formula_7 homotopy group for formula_5, in the same way that formula_5 is obtained from formula_10 by killing formula_11. The resulting manifold cannot be any finite-dimensional Lie group, since all finite-dimensional compact Lie groups have a non-vanishing formula_7. The fivebrane group follows, by killing formula_13.
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 formula_4 lies in the fact that there are the homotopy equivalencesformula_15for the classifying space formula_16, and the fact formula_17. Notice that because the complex spin group is a group extensionformula_18the String group can be thought of as a "higher" complex spin group extension, in the sense of higher group theory since the space formula_4 is an example of a higher group. It can be thought of the topological realization of the groupoid formula_20 whose object is a single point and whose morphisms are the group formula_21. Note that the homotopical degree of formula_4 is formula_23, meaning its homotopy is concentrated in degree formula_23, because it comes from the homotopy fiber of the mapformula_25from the Whitehead tower whose homotopy cokernel is formula_26. This is because the homotopy fiber lowers the degree by formula_27.
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 formula_28-bundles are, and how these higher group extensions behave. Namely, formula_28-bundles on a space formula_30 are represented geometrically as bundle gerbes since any formula_28-bundle can be realized as the homotopy fiber of a map giving a homotopy squareformula_32where formula_33. Then, a string bundle formula_34 must map to a spin bundle formula_35 which is formula_28-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 formula_37 group of the string group formula_38 using the Whitehead tower. It can then be understood again using an exact sequence of higher groupsformula_39giving a presentation of formula_40 it terms of an iterated extension, i.e. an extension by formula_41 by formula_38. Note map on the right is from the Whitehead tower, and the map on the left is the homotopy fiber.