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.

References

• Henriques, André G.; Douglas, Christopher L.; Hill, Michael A. (2008), Homological obstructions to string orientations
• Wockel, Christoph; Sachse, Christoph; Nikolaus, Thomas (2011), A Smooth Model for the String Group
• 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?", 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