String group

In topology, a branch of mathematics, a string group is an infinite-dimensional group \operatorname{String}(n) introduced by 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{\displaystyle K(\mathbb {Z} ,2)}\rightarrow \operatorname{String}(n)\rightarrow \operatorname{Spin}(n)\rightarrow 0 where K(\mathbb{Z},2) is an Eilenberg–MacLane space and \operatorname{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: \cdots\rightarrow \operatorname{Fivebrane}(n) \to \operatorname{String}(n)\rightarrow \operatorname{Spin}(n)\rightarrow \operatorname{SO}(n) \rightarrow \operatorname{O}(n) It is obtained by killing the \pi_3 homotopy group for \operatorname{Spin}(n) , in the same way that \operatorname{Spin}(n) is obtained from \operatorname{SO}(n) by killing \pi_1 . The resulting manifold cannot be any finite-dimensional Lie group, since all finite-dimensional compact Lie groups have a non-vanishing \pi_3 . The fivebrane group follows, by killing \pi_7 .…

In topology, a branch of mathematics, a string group is an infinite-dimensional group \operatorname{String}(n) introduced by 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{\displaystyle K(\mathbb {Z} ,2)}\rightarrow \operatorname{String}(n)\rightarrow \operatorname{Spin}(n)\rightarrow 0 where K(\mathbb{Z},2) is an Eilenberg–MacLane space and \operatorname{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: \cdots\rightarrow \operatorname{Fivebrane}(n) \to \operatorname{String}(n)\rightarrow \operatorname{Spin}(n)\rightarrow \operatorname{SO}(n) \rightarrow \operatorname{O}(n) It is obtained by killing the \pi_3 homotopy group for \operatorname{Spin}(n) , in the same way that \operatorname{Spin}(n) is obtained from \operatorname{SO}(n) by killing \pi_1 . The resulting manifold cannot be any finite-dimensional Lie group, since all finite-dimensional compact 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). \end{matrix} where K(\mathbb{Z},3) = B(K(\mathbb{Z},2)) . Then, a string bundle S \to M must map to a spin bundle \mathbb{S} \to M which is K(\mathbb{Z},2) -equivariant, analogously to how spin bundles map equivariantly to the frame bundle. ==References== ==External links== From Loop Groups to 2-groups - gives a characterization of String(n) as a 2-group What is an elliptic object?

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