ebab7d0da53c51438720d60c6e2e467c - /page/wikibook/entry.mu

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String group

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In <a href="Topology" title="Topology">topology</a>, a branch of <a href="Mathematics" title="Mathematics">mathematics</a>, a string group is an infinite-dimensional group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {String} (n)}">

<mi>String</mi>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {String} (n)}</annotation>

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</math><img src="./c1202a8296407ac2858a7c3dacab0144c1b97136.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.415ex; height:2.843ex;" alt="{\displaystyle \operatorname {String} (n)}" loading="lazy"> introduced by <a href="#CITEREFStolz1996">Stolz (1996)</a> as a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3}">

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<annotation encoding="application/x-tex">{\displaystyle 3}</annotation>

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</math><img src="./991e33c6e207b12546f15bdfee8b5726eafbbb2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 3}" loading="lazy">-connected cover of a <a href="Spin_group" title="Spin group">spin group</a>. A string manifold is a <a href="Manifold" title="Manifold">manifold</a> with a lifting of its <a href="Frame_bundle" title="Frame bundle">frame bundle</a> to a string group bundle. This means that in addition to being able to define <a href="Holonomy" title="Holonomy">holonomy</a> along paths, one can also define holonomies for surfaces going between strings. There is a short <a href="Exact_sequence" title="Exact sequence">exact sequence</a> of <a href="Topological_group" title="Topological group">topological groups</a><blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\rightarrow {\displaystyle K(\mathbb {Z} ,2)}\rightarrow \operatorname {String} (n)\rightarrow \operatorname {Spin} (n)\rightarrow 0}">

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<mi>String</mi>

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<annotation encoding="application/x-tex">{\displaystyle 0\rightarrow {\displaystyle K(\mathbb {Z} ,2)}\rightarrow \operatorname {String} (n)\rightarrow \operatorname {Spin} (n)\rightarrow 0}</annotation>

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</math><img src="./640c35c0c52334f962b64764673968d41532aadb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.546ex; height:2.843ex;" alt="{\displaystyle 0\rightarrow {\displaystyle K(\mathbb {Z} ,2)}\rightarrow \operatorname {String} (n)\rightarrow \operatorname {Spin} (n)\rightarrow 0}" loading="lazy"></blockquote>where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(\mathbb {Z} ,2)}">

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<annotation encoding="application/x-tex">{\displaystyle K(\mathbb {Z} ,2)}</annotation>

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</math><img src="./63bb28635180d936dc5a1a96702fbb6354b8158e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.622ex; height:2.843ex;" alt="{\displaystyle K(\mathbb {Z} ,2)}" loading="lazy"> is an <a href="Eilenberg%E2%80%93MacLane_space" title="Eilenberg–MacLane space">Eilenberg–MacLane space</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Spin} (n)}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Spin} (n)}</annotation>

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</math><img src="./049140eca932e4de307c8bf196ed124234ae0286.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.728ex; height:2.843ex;" alt="{\displaystyle \operatorname {Spin} (n)}" loading="lazy"> is a spin group. The string group is an entry in the <a href="Whitehead_tower" class="mw-redirect" title="Whitehead tower">Whitehead tower</a> (dual to the notion of <a href="Postnikov_tower" class="mw-redirect" title="Postnikov tower">Postnikov tower</a>) for the <a href="Orthogonal_group" title="Orthogonal group">orthogonal group</a>:<blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdots \rightarrow \operatorname {Fivebrane} (n)\to \operatorname {String} (n)\rightarrow \operatorname {Spin} (n)\rightarrow \operatorname {SO} (n)\rightarrow \operatorname {O} (n)}">

<mi>Fivebrane</mi>

<mi>String</mi>

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<annotation encoding="application/x-tex">{\displaystyle \cdots \rightarrow \operatorname {Fivebrane} (n)\to \operatorname {String} (n)\rightarrow \operatorname {Spin} (n)\rightarrow \operatorname {SO} (n)\rightarrow \operatorname {O} (n)}</annotation>

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</math><img src="./3a8e723d89c27906cf920d2495b2ec36914acb7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:62.574ex; height:2.843ex;" alt="{\displaystyle \cdots \rightarrow \operatorname {Fivebrane} (n)\to \operatorname {String} (n)\rightarrow \operatorname {Spin} (n)\rightarrow \operatorname {SO} (n)\rightarrow \operatorname {O} (n)}" loading="lazy"></blockquote>It is obtained by killing the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{3}}">

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<annotation encoding="application/x-tex">{\displaystyle \pi _{3}}</annotation>

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</math><img src="./d4a078131a0c973b388cf10c0ac6d7598fe21578.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.379ex; height:2.009ex;" alt="{\displaystyle \pi _{3}}" loading="lazy"> <a href="Homotopy_group" title="Homotopy group">homotopy group</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Spin} (n)}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Spin} (n)}</annotation>

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</math><img src="./049140eca932e4de307c8bf196ed124234ae0286.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.728ex; height:2.843ex;" alt="{\displaystyle \operatorname {Spin} (n)}" loading="lazy">, in the same way that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Spin} (n)}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Spin} (n)}</annotation>

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</math><img src="./049140eca932e4de307c8bf196ed124234ae0286.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.728ex; height:2.843ex;" alt="{\displaystyle \operatorname {Spin} (n)}" loading="lazy"> is obtained from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {SO} (n)}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {SO} (n)}</annotation>

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</math><img src="./79c49648527ab4aacf6c03c15633727606cc7d22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.305ex; height:2.843ex;" alt="{\displaystyle \operatorname {SO} (n)}" loading="lazy"> by killing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{1}}">

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<annotation encoding="application/x-tex">{\displaystyle \pi _{1}}</annotation>

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</math><img src="./542cbd3dacd0a061d666ed7fc4ed7ad15b47444b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.379ex; height:2.009ex;" alt="{\displaystyle \pi _{1}}" loading="lazy">. The resulting manifold cannot be any finite-dimensional <a href="Lie_group" title="Lie group">Lie group</a>, since all finite-dimensional compact Lie groups have a non-vanishing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{3}}">

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<annotation encoding="application/x-tex">{\displaystyle \pi _{3}}</annotation>

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</math><img src="./d4a078131a0c973b388cf10c0ac6d7598fe21578.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.379ex; height:2.009ex;" alt="{\displaystyle \pi _{3}}" loading="lazy">. The fivebrane group follows, by killing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{7}}">

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<annotation encoding="application/x-tex">{\displaystyle \pi _{7}}</annotation>

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</math><img src="./4ca28200b057380aab5d9d944f7ecef70f3b72b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.379ex; height:2.009ex;" alt="{\displaystyle \pi _{7}}" loading="lazy">.

More generally, the construction of the Postnikov tower via short exact sequences starting with Eilenberg–MacLane spaces can be applied to any <a href="Lie_group" title="Lie group">Lie group</a> G, giving the string group String(G).

<div class="mw-heading mw-heading2"><h2 id="Intuition_for_the_string_group">Intuition for the string group</h2></div>

The relevance of the Eilenberg-Maclane space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(\mathbb {Z} ,2)}">

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<annotation encoding="application/x-tex">{\displaystyle K(\mathbb {Z} ,2)}</annotation>

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</math><img src="./63bb28635180d936dc5a1a96702fbb6354b8158e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.622ex; height:2.843ex;" alt="{\displaystyle K(\mathbb {Z} ,2)}" loading="lazy"> lies in the fact that there are the homotopy equivalences<blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(\mathbb {Z} ,1)\simeq U(1)\simeq B\mathbb {Z} }">

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<annotation encoding="application/x-tex">{\displaystyle K(\mathbb {Z} ,1)\simeq U(1)\simeq B\mathbb {Z} }</annotation>

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</math><img src="./6b07b405399775e474aab8d4c11dd8ebd6fafd5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.888ex; height:2.843ex;" alt="{\displaystyle K(\mathbb {Z} ,1)\simeq U(1)\simeq B\mathbb {Z} }" loading="lazy"></blockquote>for the <a href="Classifying_space" title="Classifying space">classifying space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\mathbb {Z} }">

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<annotation encoding="application/x-tex">{\displaystyle B\mathbb {Z} }</annotation>

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</math><img src="./bd707158e798951b6f2734920654d67966610712.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.314ex; height:2.176ex;" alt="{\displaystyle B\mathbb {Z} }" loading="lazy">, and the fact <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(\mathbb {Z} ,2)\simeq BU(1)}">

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<annotation encoding="application/x-tex">{\displaystyle K(\mathbb {Z} ,2)\simeq BU(1)}</annotation>

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</math><img src="./dd819ec793184d9b1b3c0a93a5fb759eb455c069.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.239ex; height:2.843ex;" alt="{\displaystyle K(\mathbb {Z} ,2)\simeq BU(1)}" loading="lazy">. Notice that because the complex spin group is a group extension<blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\to K(\mathbb {Z} ,1)\to \operatorname {Spin} ^{\mathbb {C} }(n)\to \operatorname {Spin} (n)\to 0}">

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<annotation encoding="application/x-tex">{\displaystyle 0\to K(\mathbb {Z} ,1)\to \operatorname {Spin} ^{\mathbb {C} }(n)\to \operatorname {Spin} (n)\to 0}</annotation>

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</math><img src="./30d7ff4d2d647bb03d768f2bdd2f1ec3c7e24777.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.279ex; height:3.176ex;" alt="{\displaystyle 0\to K(\mathbb {Z} ,1)\to \operatorname {Spin} ^{\mathbb {C} }(n)\to \operatorname {Spin} (n)\to 0}" loading="lazy"></blockquote>the String group can be thought of as a "higher" complex spin group extension, in the sense of <a href="Higher_group" class="mw-redirect" title="Higher group">higher group theory</a> since the space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(\mathbb {Z} ,2)}">

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<annotation encoding="application/x-tex">{\displaystyle K(\mathbb {Z} ,2)}</annotation>

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</math><img src="./63bb28635180d936dc5a1a96702fbb6354b8158e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.622ex; height:2.843ex;" alt="{\displaystyle K(\mathbb {Z} ,2)}" loading="lazy"> is an example of a higher group. It can be thought of the topological realization of the <a href="Groupoid" title="Groupoid">groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} U(1)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {B} U(1)}</annotation>

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</math><img src="./836ec6ff1d68f8f766a7c2e4d59032160260ef79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.655ex; height:2.843ex;" alt="{\displaystyle \mathbf {B} U(1)}" loading="lazy"> whose object is a single point and whose morphisms are the group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(1)}">

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<annotation encoding="application/x-tex">{\displaystyle U(1)}</annotation>

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</math><img src="./e62b00d74ee0cefb86cc052365625abff56d43e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.754ex; height:2.843ex;" alt="{\displaystyle U(1)}" loading="lazy">. Note that the homotopical degree of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(\mathbb {Z} ,2)}">

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<annotation encoding="application/x-tex">{\displaystyle K(\mathbb {Z} ,2)}</annotation>

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</math><img src="./63bb28635180d936dc5a1a96702fbb6354b8158e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.622ex; height:2.843ex;" alt="{\displaystyle K(\mathbb {Z} ,2)}" loading="lazy"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}">

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<annotation encoding="application/x-tex">{\displaystyle 2}</annotation>

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</math><img src="./901fc910c19990d0dbaaefe4726ceb1a4e217a0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}" loading="lazy">, meaning its homotopy is concentrated in degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}">

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<annotation encoding="application/x-tex">{\displaystyle 2}</annotation>

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</math><img src="./901fc910c19990d0dbaaefe4726ceb1a4e217a0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}" loading="lazy">, because it comes from the <a href="Homotopy_fiber" title="Homotopy fiber">homotopy fiber</a> of the map<blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {String} (n)\to \operatorname {Spin} (n)}">

<mi>String</mi>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {String} (n)\to \operatorname {Spin} (n)}</annotation>

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</math><img src="./233fad030c7218cc6128553121469741f3d75671.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.757ex; height:2.843ex;" alt="{\displaystyle \operatorname {String} (n)\to \operatorname {Spin} (n)}" loading="lazy"></blockquote>from the Whitehead tower whose homotopy cokernel is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(\mathbb {Z} ,3)}">

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<annotation encoding="application/x-tex">{\displaystyle K(\mathbb {Z} ,3)}</annotation>

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</math><img src="./5e755e4522e9df27562834f27fd3f693c2f8567b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.622ex; height:2.843ex;" alt="{\displaystyle K(\mathbb {Z} ,3)}" loading="lazy">. This is because the homotopy fiber lowers the degree by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

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<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

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</math><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Understanding_the_geometry">Understanding the geometry</h3></div>

The geometry of String bundles requires the understanding of multiple constructions in homotopy theory,<a href="#cite_note-1">[1]</a> but they essentially boil down to understanding what <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(\mathbb {Z} ,2)}">

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<annotation encoding="application/x-tex">{\displaystyle K(\mathbb {Z} ,2)}</annotation>

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</math><img src="./63bb28635180d936dc5a1a96702fbb6354b8158e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.622ex; height:2.843ex;" alt="{\displaystyle K(\mathbb {Z} ,2)}" loading="lazy">-bundles are, and how these higher group extensions behave. Namely, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(\mathbb {Z} ,2)}">

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<annotation encoding="application/x-tex">{\displaystyle K(\mathbb {Z} ,2)}</annotation>

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</math><img src="./63bb28635180d936dc5a1a96702fbb6354b8158e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.622ex; height:2.843ex;" alt="{\displaystyle K(\mathbb {Z} ,2)}" loading="lazy">-bundles on a space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> are represented geometrically as <a href="Bundle_gerbe" title="Bundle gerbe">bundle gerbes</a> since any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(\mathbb {Z} ,2)}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle K(\mathbb {Z} ,2)}</annotation>

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</math><img src="./63bb28635180d936dc5a1a96702fbb6354b8158e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.622ex; height:2.843ex;" alt="{\displaystyle K(\mathbb {Z} ,2)}" loading="lazy">-bundle can be realized as the homotopy fiber of a map giving a homotopy square<blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}P&\to &*\\\downarrow &&\downarrow \\M&\xrightarrow {} &K(\mathbb {Z} ,3)\end{matrix}}}">

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</mtd>

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<mtr>

<mtd>

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<mtd>

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<mtr>

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<mtd>

<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}P&\to &*\\\downarrow &&\downarrow \\M&\xrightarrow {} &K(\mathbb {Z} ,3)\end{matrix}}}</annotation>

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</math><img src="./ddaa0de9372cdf3f1fe104e1e64f3920267df79f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:17.785ex; height:9.509ex;" alt="{\displaystyle {\begin{matrix}P&\to &*\\\downarrow &&\downarrow \\M&\xrightarrow {} &K(\mathbb {Z} ,3)\end{matrix}}}" loading="lazy"></blockquote>where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(\mathbb {Z} ,3)=B(K(\mathbb {Z} ,2))}">

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<annotation encoding="application/x-tex">{\displaystyle K(\mathbb {Z} ,3)=B(K(\mathbb {Z} ,2))}</annotation>

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</math><img src="./d714e596b40a33dc0efc29aa755b939a4dbc0a3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.916ex; height:2.843ex;" alt="{\displaystyle K(\mathbb {Z} ,3)=B(K(\mathbb {Z} ,2))}" loading="lazy">. Then, a string bundle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\to M}">

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<annotation encoding="application/x-tex">{\displaystyle S\to M}</annotation>

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</math><img src="./24ad5efc5478da62b83c74039650dfac91d162dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.556ex; height:2.176ex;" alt="{\displaystyle S\to M}" loading="lazy"> must map to a spin bundle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} \to M}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {S} \to M}</annotation>

</semantics>

</math><img src="./c5f3c4246ffdfaa72d6af71e14a109ef55a6ce6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.349ex; height:2.176ex;" alt="{\displaystyle \mathbb {S} \to M}" loading="lazy"> which is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(\mathbb {Z} ,2)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K(\mathbb {Z} ,2)}</annotation>

</semantics>

</math><img src="./63bb28635180d936dc5a1a96702fbb6354b8158e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.622ex; height:2.843ex;" alt="{\displaystyle K(\mathbb {Z} ,2)}" loading="lazy">-equivariant, analogously to how spin bundles map equivariantly to the frame bundle.

<div class="mw-heading mw-heading2"><h2 id="Fivebrane_group_and_higher_groups">Fivebrane group and higher groups</h2></div>

The fivebrane group can similarly be understood<a href="#cite_note-2">[2]</a> by killing the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{7}(\operatorname {Spin} (n))\cong \pi _{7}(\operatorname {O} (n))}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi _{7}(\operatorname {Spin} (n))\cong \pi _{7}(\operatorname {O} (n))}</annotation>

</semantics>

</math><img src="./0201421274916c2bd42e4201545f2979ea1ed72c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.216ex; height:2.843ex;" alt="{\displaystyle \pi _{7}(\operatorname {Spin} (n))\cong \pi _{7}(\operatorname {O} (n))}" loading="lazy"> group of the string group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {String} (n)}">

<mi>String</mi>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {String} (n)}</annotation>

</semantics>

</math><img src="./c1202a8296407ac2858a7c3dacab0144c1b97136.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.415ex; height:2.843ex;" alt="{\displaystyle \operatorname {String} (n)}" loading="lazy"> using the Whitehead tower. It can then be understood again using an exact sequence of <a href="N-group_(category_theory)" title="N-group (category theory)">higher groups</a><blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\to K(\mathbb {Z} ,6)\to \operatorname {Fivebrane} (n)\to \operatorname {String} (n)\to 0}">

</mrow>

<mi>Fivebrane</mi>

<mi>String</mi>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0\to K(\mathbb {Z} ,6)\to \operatorname {Fivebrane} (n)\to \operatorname {String} (n)\to 0}</annotation>

</semantics>

</math><img src="./845f669d83b4b5ea32894218790b254de3f276ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.138ex; height:2.843ex;" alt="{\displaystyle 0\to K(\mathbb {Z} ,6)\to \operatorname {Fivebrane} (n)\to \operatorname {String} (n)\to 0}" loading="lazy"></blockquote>giving a presentation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Fivebrane} (n)}">

<mi>Fivebrane</mi>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Fivebrane} (n)}</annotation>

</semantics>

</math><img src="./3dd308a9624876ab87c98245c23bfe794a4b8fd3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.32ex; height:2.843ex;" alt="{\displaystyle \operatorname {Fivebrane} (n)}" loading="lazy"> it terms of an iterated extension, i.e. an extension by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(\mathbb {Z} ,6)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K(\mathbb {Z} ,6)}</annotation>

</semantics>

</math><img src="./099f92e13587c943f0cdd16c3378937aebbf5fdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.622ex; height:2.843ex;" alt="{\displaystyle K(\mathbb {Z} ,6)}" loading="lazy"> by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {String} (n)}">

<mi>String</mi>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {String} (n)}</annotation>

</semantics>

</math><img src="./c1202a8296407ac2858a7c3dacab0144c1b97136.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.415ex; height:2.843ex;" alt="{\displaystyle \operatorname {String} (n)}" loading="lazy">. Note map on the right is from the Whitehead tower, and the map on the left is the homotopy fiber.

<ul><li><a href="Gerbe" title="Gerbe">Gerbe</a></li>

<li><a href="N-group_(category_theory)" title="N-group (category theory)">N-group (category theory)</a></li>

<li><a href="Elliptic_cohomology" title="Elliptic cohomology">Elliptic cohomology</a></li>

<li>String bordism</li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFJurco2011" class="citation journal cs1">Jurco, Branislav (August 2011). "Crossed Module Bundle Gerbes; Classification, String Group and Differential Geometry". International Journal of Geometric Methods in Modern Physics. 08 (5): 1079–1095. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0510078">math/0510078</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2011IJGMM..08.1079J">2011IJGMM..08.1079J</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2FS0219887811005555">10.1142/S0219887811005555</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0219-8878">0219-8878</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:1347840">1347840</a>.</cite>

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<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFSatiSchreiberStasheff2009" class="citation journal cs1">Sati, Hisham; Schreiber, Urs; Stasheff, Jim (November 2009). "Fivebrane Structures". Reviews in Mathematical Physics. 21 (10): 1197–1240. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0805.0564">0805.0564</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2009RvMaP..21.1197S">2009RvMaP..21.1197S</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2FS0129055X09003840">10.1142/S0129055X09003840</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0129-055X">0129-055X</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:13307997">13307997</a>.</cite>

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<ul><li><cite id="CITEREFHenriquesDouglasHill2011" class="citation cs2">Henriques, André G.; Douglas, Christopher L.; Hill, Michael A. (2011), "Homological obstructions to string orientations", Int. Math. Res. Notices, 18: 4074–4088, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0810.2131">0810.2131</a>, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008arXiv0810.2131D">2008arXiv0810.2131D</a></cite></li>

<li><cite id="CITEREFWockelSachseNikolaus2013" class="citation cs2">Wockel, Christoph; Sachse, Christoph; Nikolaus, Thomas (2013), "A Smooth Model for the String Group", International Mathematics Research Notices, 2013 (16): 3678–3721, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1104.4288">1104.4288</a>, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2011arXiv1104.4288N">2011arXiv1104.4288N</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fimrn%2Frns154">10.1093/imrn/rns154</a></cite></li>

<li><cite id="CITEREFStolz1996" class="citation cs2">Stolz, Stephan (1996), "A conjecture concerning positive Ricci curvature and the Witten genus", <a href="Mathematische_Annalen" title="Mathematische Annalen">Mathematische Annalen</a>, 304 (4): 785–800, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01446319">10.1007/BF01446319</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0025-5831">0025-5831</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1380455">1380455</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:123359573">123359573</a></cite></li>

<li><cite id="CITEREFStolzTeichner2004" class="citation cs2">Stolz, Stephan; Teichner, Peter (2004), <a rel="nofollow" class="external text" href="http://math.ucr.edu/home/baez/qg-winter2007/Oxford.pdf">"What is an elliptic object?"</a> (PDF), Topology, geometry and quantum field theory, London Math. Soc. Lecture Note Ser., vol. 308, <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, pp. 247–343, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FCBO9780511526398.013">10.1017/CBO9780511526398.013</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780521540490</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2079378">2079378</a></cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite id="CITEREFBaez2007" class="citation cs2"><a href="John_Baez" class="mw-redirect" title="John Baez">Baez, J.</a> (2007), <a rel="nofollow" class="external text" href="http://math.ucr.edu/home/baez/esi/">Higher Gauge Theory and the String Group</a></cite></li>

<li><a href="https://arxiv.org/abs/math/0504123v2" class="extiw external" title="arxiv:math/0504123v2">From Loop Groups to 2-groups</a> - gives a characterization of String(n) as a <a href="2-group" title="2-group">2-group</a></li>

<li><a rel="nofollow" class="external text" href="https://ncatlab.org/nlab/show/string+group">string group</a> at the <a href="NLab" title="NLab">nLab</a></li>

<li><a rel="nofollow" class="external text" href="https://ncatlab.org/nlab/show/Whitehead+tower">Whitehead tower</a> at the <a href="NLab" title="NLab">nLab</a></li>

<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20180519183342/http://math.ucr.edu/home/baez//qg-winter2007/Oxford.pdf">What is an elliptic object?</a></li></ul></div><div><div class="zim-footer">

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