# String group

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{{Short description|Infinite-dimensional group in topology}}
In [topology](/source/topology), a branch of [mathematics](/source/mathematics), a '''string group''' is an infinite-dimensional group <math>\operatorname{String}(n)</math> introduced by {{harvtxt|Stolz|1996}} as a [<math>3</math>-connected](/source/Homotopical_connectivity) cover of a [spin group](/source/spin_group). A '''string manifold''' is a [manifold](/source/manifold) with a lifting of its [frame bundle](/source/frame_bundle) to a string group bundle. This means that in addition to being able to define [holonomy](/source/holonomy) along paths, one can also define holonomies for surfaces going between strings. There is a short [exact sequence](/source/exact_sequence) of [topological group](/source/topological_group)s<blockquote><math>0\rightarrow{\displaystyle K(\mathbb {Z} ,2)}\rightarrow \operatorname{String}(n)\rightarrow \operatorname{Spin}(n)\rightarrow 0</math></blockquote>where <math>K(\mathbb{Z},2)</math> is an [Eilenberg–MacLane space](/source/Eilenberg%E2%80%93MacLane_space) and <math>\operatorname{Spin}(n)</math> is a spin group. The string group is an entry in the [Whitehead tower](/source/Whitehead_tower) (dual to the notion of [Postnikov tower](/source/Postnikov_tower)) for the [orthogonal group](/source/orthogonal_group):<blockquote><math>\cdots\rightarrow \operatorname{Fivebrane}(n) \to \operatorname{String}(n)\rightarrow \operatorname{Spin}(n)\rightarrow \operatorname{SO}(n) \rightarrow \operatorname{O}(n) </math></blockquote>It is obtained by killing the <math>\pi_3</math> [homotopy group](/source/homotopy_group) for <math>\operatorname{Spin}(n)</math>, in the same way that <math>\operatorname{Spin}(n)</math> is obtained from <math>\operatorname{SO}(n)</math> by killing <math>\pi_1</math>. The resulting manifold cannot be any finite-dimensional [Lie group](/source/Lie_group), since all finite-dimensional compact Lie groups have a non-vanishing <math>\pi_3</math>. The fivebrane group follows, by killing <math>\pi_7</math>.

More generally, the construction of the Postnikov tower via short exact sequences starting with  Eilenberg&ndash;MacLane spaces can be applied to any [Lie group](/source/Lie_group) ''G'', giving the string group ''String''(''G'').

== Intuition for the string group ==
The relevance of the Eilenberg–MacLane space <math>K(\mathbb{Z},2)</math> lies in the fact that there are the [homotopy equivalence](/source/homotopy_equivalence)s<blockquote><math>K(\mathbb{Z},1) \simeq U(1) \simeq B\mathbb{Z}</math></blockquote>for the [classifying space](/source/classifying_space) <math>B\mathbb{Z}</math>, and the fact <math>K(\mathbb{Z},2) \simeq BU(1)</math>. Notice that because the complex spin group is a [group extension](/source/group_extension)<blockquote><math>0\to K(\mathbb{Z},1) \to \operatorname{Spin}^\mathbb{C}(n) \to \operatorname{Spin}(n) \to 0</math></blockquote>the String group can be thought of as a "higher" complex spin group extension, in the sense of [higher group theory](/source/Higher_group) since the space <math>K(\mathbb{Z},2)</math> is an example of a higher group. It can be thought of the topological realization of the [groupoid](/source/groupoid) <math>\mathbf{B}U(1)</math> whose object is a single point and whose morphisms are the group <math>U(1)</math>. Note that the homotopical degree of <math>K(\mathbb{Z},2)</math> is <math>2 </math>, meaning its homotopy is concentrated in degree <math>2 </math>, because it comes from the [homotopy fiber](/source/homotopy_fiber) of the map<blockquote><math>\operatorname{String}(n) \to \operatorname{Spin}(n) </math></blockquote>from the Whitehead tower whose homotopy cokernel is <math>K(\mathbb{Z},3) </math>. This is because the homotopy fiber lowers the degree by <math>1 </math>.

=== Understanding the geometry ===
The geometry of String bundles requires the understanding of multiple constructions in homotopy theory,<ref>{{Cite journal|last=Jurco|first=Branislav|date=August 2011|title=Crossed Module Bundle Gerbes; Classification, String Group and Differential Geometry|journal=International Journal of Geometric Methods in Modern Physics|volume=08|issue=5|pages=1079–1095|doi=10.1142/S0219887811005555|issn=0219-8878|arxiv=math/0510078|bibcode=2011IJGMM..08.1079J |s2cid=1347840}}</ref> but they essentially boil down to understanding what <math>K(\mathbb{Z},2) </math>-bundles are, and how these higher group extensions behave. Namely, <math>K(\mathbb{Z},2) </math>-bundles on a space <math>M </math> are represented geometrically as [bundle gerbe](/source/bundle_gerbe)s since any <math>K(\mathbb{Z},2) </math>-bundle can be realized as the homotopy fiber of a map giving a homotopy square<blockquote><math>\begin{matrix}
P & \to & * \\
\downarrow & & \downarrow \\
M & \xrightarrow{} & K(\mathbb{Z},3)
\end{matrix} </math></blockquote>where <math>K(\mathbb{Z},3) = B(K(\mathbb{Z},2)) </math>. Then, a string bundle <math>S \to M </math> must map to a spin bundle <math>\mathbb{S} \to M </math> which is <math>K(\mathbb{Z},2) </math>-equivariant, analogously to how spin bundles map equivariantly to the frame bundle.

== Fivebrane group and higher groups ==
The fivebrane group can similarly be understood<ref>{{Cite journal|last1=Sati|first1=Hisham|last2=Schreiber|first2=Urs|last3=Stasheff|first3=Jim|date=November 2009|title=Fivebrane Structures|journal=Reviews in Mathematical Physics|volume=21|issue=10|pages=1197–1240|doi=10.1142/S0129055X09003840|arxiv=0805.0564|bibcode=2009RvMaP..21.1197S |s2cid=13307997|issn=0129-055X}}</ref> by killing the <math>\pi_7(\operatorname{Spin}(n)) \cong \pi_7(\operatorname{O}(n)) </math> group of the string group <math>\operatorname{String}(n) </math> using the Whitehead tower. It can then be understood again using an exact sequence of [higher groups](/source/N-group_(category_theory))<blockquote><math>0 \to K(\mathbb{Z},6) \to \operatorname{Fivebrane}(n) \to \operatorname{String}(n) \to 0 </math></blockquote>giving a presentation of <math>\operatorname{Fivebrane}(n) </math> it terms of an iterated extension, i.e. an extension by <math>K(\mathbb{Z},6) </math> by <math>\operatorname{String}(n) </math>. Note map on the right is from the Whitehead tower, and the map on the left is the homotopy fiber.

== See also ==

* [Gerbe](/source/Gerbe)
*[N-group (category theory)](/source/N-group_(category_theory))
*[Elliptic cohomology](/source/Elliptic_cohomology)
*[String bordism](/source/String_bordism)

==References==
{{reflist}}
*{{Citation | last1=Henriques | first1=André G. | last2=Douglas | first2=Christopher L. | last3=Hill | first3=Michael A. | title=Homological obstructions to string orientations | journal=Int. Math. Res. Notices | arxiv=0810.2131| year=2011| volume=18 | pages=4074–4088 | bibcode=2008arXiv0810.2131D }}
*{{Citation | last1=Wockel | first1=Christoph | last2=Sachse | first2=Christoph | last3=Nikolaus | first3=Thomas | title=A Smooth Model for the String Group  | journal=International Mathematics Research Notices | arxiv=1104.4288| year=2013| volume=2013 | issue=16 | pages=3678–3721 | doi=10.1093/imrn/rns154 | bibcode=2011arXiv1104.4288N}}
*{{Citation | last1=Stolz | first1=Stephan | title=A conjecture concerning positive Ricci curvature and the Witten genus | doi=10.1007/BF01446319 | mr=1380455 | year=1996 | journal=[Mathematische Annalen](/source/Mathematische_Annalen) | issn=0025-5831 | volume=304 | issue=4 | pages=785–800| s2cid=123359573 }}
*{{Citation | last1=Stolz | first1=Stephan | last2=Teichner | first2=Peter | title=Topology, geometry and quantum field theory | publisher=[Cambridge University Press](/source/Cambridge_University_Press) | series=London Math. Soc. Lecture Note Ser. | doi=10.1017/CBO9780511526398.013 | mr=2079378 | year=2004 | volume=308 | chapter=What is an elliptic object? | pages=247–343 | isbn=9780521540490 |chapter-url=http://math.ucr.edu/home/baez/qg-winter2007/Oxford.pdf}}

==External links==
*{{citation|first=J.|last=Baez|authorlink=John Baez|title=Higher Gauge Theory and the String Group|url=http://math.ucr.edu/home/baez/esi/|year=2007 }}
*[From Loop Groups to 2-groups](/source/arxiv%3Amath%2F0504123v2) - gives a characterization of String(n) as a [2-group](/source/2-group)
*{{nlab|id=string+group|title=string group}}
*{{nlab|id=Whitehead+tower|title=Whitehead tower}}
*[https://web.archive.org/web/20180519183342/http://math.ucr.edu/home/baez//qg-winter2007/Oxford.pdf What is an elliptic object?]

Category:Group theory
Category:Differential geometry
Category:String theory
Category:Homotopy theory

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Adapted from the Wikipedia article [String group](https://en.wikipedia.org/wiki/String_group) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/String_group?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.