String group

In topology, a branch of mathematics, a string group is an infinite-dimensional group 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 K(Z,2)\rightarrow \text{String}(n)\rightarrow \text{Spin}(n)\rightarrow 0

where K(Z, 2) is an Eilenberg–MacLane space and Spin(n) is a spin group.

The string group is an entry in the Postnikov tower for the orthogonal group:

\ldots \rightarrow {\text{String}}(n)\rightarrow {\text{Spin}}(n)\rightarrow {\text{SO}}(n)\rightarrow {\text{O}}(n)

It is preceded by the fivebrane group in the tower. It is obtained by killing the $\pi _{3}$ homotopy group for ${\text{Spin}}(n)$ , in the same way that ${\text{Spin}}(n)$ is obtained from ${\text{SO}}(n)$ by killing $\pi _{1}$ . The resulting manifold cannot be any finite-dimensional Lie group, since all finite-dimensional 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).

References

Henriques, André G.; Douglas, Christopher L.; Hill, Michael A. (2008), Homological obstructions to string orientations, arXiv:0810.2131
Wockel, Christoph; Sachse, Christoph; Nikolaus, Thomas (2011), A Smooth Model for the String Group, arXiv:1104.4288
Stolz, Stephan (1996), "A conjecture concerning positive Ricci curvature and the Witten genus", Mathematische Annalen, 304 (4): 785–800, doi:10.1007/BF01446319, ISSN 0025-5831, MR 1380455
Stolz, Stephan; Teichner, Peter (2004), "What is an elliptic object?" (PDF), Topology, geometry and quantum field theory, London Math. Soc. Lecture Note Ser., 308, Cambridge University Press, pp. 247–343, doi:10.1017/CBO9780511526398.013, MR 2079378

External links

Baez, J. (2007), Higher Gauge Theory and the String Group
string group in nLab

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