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

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

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

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

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

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

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, Bibcode:2008arXiv0810.2131D
• Wockel, Christoph; Sachse, Christoph; Nikolaus, Thomas (2011), A Smooth Model for the String Group, arXiv:1104.4288, Bibcode:2011arXiv1104.4288N
• 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
• Whitehead tower in nLab