{{Short description|Infinite-dimensional group in topology}} In topology, a branch of 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 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<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 and <math>\operatorname{Spin}(n)</math> 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:<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 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, 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–MacLane spaces can be applied to any 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 equivalences<blockquote><math>K(\mathbb{Z},1) \simeq U(1) \simeq B\mathbb{Z}</math></blockquote>for the 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<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 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 <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 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 gerbes 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<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 *N-group (category theory) *Elliptic cohomology *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 | 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 | 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 - gives a characterization of String(n) as a 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