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.
where K(Z, 2) is an Eilenberg–MacLane space and Spin(n) is a spin group.
It is preceded by the fivebrane group in the tower. It is obtained by killing the homotopy group for , in the same way that is obtained from by killing . The resulting manifold cannot be any finite-dimensional Lie group, since all finite-dimensional 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).
- Henriques, André G.; Douglas, Christopher L.; Hill, Michael A. (2008), Homological obstructions to string orientations, arXiv:
- Wockel, Christoph; Sachse, Christoph; Nikolaus, Thomas (2011), A Smooth Model for the String Group, arXiv:
- Stolz, Stephan (1996), "A conjecture concerning positive Ricci curvature and the Witten genus", Mathematische Annalen, 304 (4): 785–800, ISSN 0025-5831, MR 1380455, doi:10.1007/BF01446319
- 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, MR 2079378, doi:10.1017/CBO9780511526398.013