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<![CDATA[//><!]]> </script> <link href="https://ncatlab.org/nlab/atom_with_headlines" rel="alternate" title="Atom with headlines" type="application/atom+xml" /> <link href="https://ncatlab.org/nlab/atom_with_content" rel="alternate" title="Atom with full content" type="application/atom+xml" /> <script type="text/javascript"> document.observe("dom:loaded", function() { generateThmNumbers(); }); </script> </head> <body> <div id="Container"> <div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> holonomy </h1> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='differential_cohomology'>Differential cohomology</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/differential+cohomology'>differential cohomology</a></strong></p> <h2 id='ingredients'>Ingredients</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/cohomology'>cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/differential+geometry'>differential geometry</a></p> </li> </ul> <h2 id='connections_on_bundles'>Connections on bundles</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/connection+on+a+bundle'>connection on a bundle</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/parallel+transport'>parallel transport</a>, <a class='existingWikiWord' href='/nlab/show/holonomy'>holonomy</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/curvature'>curvature</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/curvature+characteristic+form'>curvature characteristic form</a>, <a class='existingWikiWord' href='/nlab/show/Chern+character'>Chern character</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/Chern-Weil+theory'>Chern-Weil theory</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/Chern-Weil+homomorphism'>Chern-Weil homomorphism</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/secondary+characteristic+class'>secondary characteristic class</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/differential+characteristic+class'>differential characteristic class</a></p> </li> </ul> </li> </ul> <h2 id='higher_abelian_differential_cohomology'>Higher abelian differential cohomology</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/differential+function+complex'>differential function complex</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/orientation+in+differential+cohomology'>differential orientation</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/ordinary+differential+cohomology'>ordinary differential cohomology</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/fiber+integration+in+ordinary+differential+cohomology'>differential Thom class</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/Cheeger-Simons+differential+character'>differential characters</a>,</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/Deligne+cohomology'>Deligne cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/circle+n-bundle+with+connection'>circle n-bundle with connection</a>,</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/bundle+gerbe'>bundle gerbe with connection</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/differential+K-theory'>differential K-theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/differential+elliptic+cohomology'>differential elliptic cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/differential+cobordism+cohomology'>differential cobordism cohomology</a></p> </li> </ul> <h2 id='higher_nonabelian_differential_cohomology'>Higher nonabelian differential cohomology</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/principal+2-bundle'>principal 2-bundle</a>, <a class='existingWikiWord' href='/nlab/show/principal+infinity-bundle'>principal ∞-bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/connection+on+a+2-bundle'>connection on a 2-bundle</a>, <a class='existingWikiWord' href='/nlab/show/connection+on+a+smooth+principal+infinity-bundle'>connection on an ∞-bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/Chern-Weil+theory+in+Smooth%E2%88%9EGrpd'>Chern-Weil theory in Smooth∞Grpd</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/infinity-Lie+algebra+cohomology'>∞-Lie algebra cohomology</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/Chern-Simons+element'>∞-Chern-Simons theory</a></p> </li> </ul> <h2 id='fiber_integration'>Fiber integration</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/higher+parallel+transport'>higher holonomy</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/fiber+integration+in+differential+cohomology'>fiber integration in differential cohomology</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/fiber+integration+in+ordinary+differential+cohomology'>fiber integration in ordinary differential cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/fiber+integration+in+differential+K-theory'>fiber integration in differential K-theory</a></p> </li> </ul> </li> </ul> <h2 id='application_to_gauge_theory'>Application to gauge theory</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/gauge+theory'>gauge theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/gauge+theory'>gauge field</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/electromagnetic+field'>electromagnetic field</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/Yang-Mills+field'>Yang-Mills field</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/Kalb-Ramond+field'>Kalb-Ramond field</a>/<a class='existingWikiWord' href='/nlab/show/Kalb-Ramond+field'>B-field</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/RR+field'>RR-field</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/supergravity+C-field'>supergravity C-field</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/D%27Auria-Fr%C3%A9-Regge+formulation+of+supergravity'>supergravity</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/quantum+anomaly'>quantum anomaly</a></p> </li> </ul> <div> <p> <a href='/nlab/edit/differential+cohomology+-+contents'>Edit this sidebar</a> </p> </div></div> </div> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#properties'>Properties</a></li><li><a href='#applications'>Applications</a></li><li><a href='#higher_holonomy'>Higher holonomy</a></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a></li></ul></div> <h2 id='idea'>Idea</h2> <p>Given <a class='existingWikiWord' href='/nlab/show/connection+on+a+bundle'>connection on a bundle</a> <math class='maruku-mathml' display='inline' id='mathml_e323b8df50671ef4bb7cc48a4d4695604ba0b8bd_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∇</mo></mrow><annotation encoding='application/x-tex'>\nabla</annotation></semantics></math> over a <a class='existingWikiWord' href='/nlab/show/space'>space</a> <math class='maruku-mathml' display='inline' id='mathml_e323b8df50671ef4bb7cc48a4d4695604ba0b8bd_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, its <a class='existingWikiWord' href='/nlab/show/parallel+transport'>parallel transport</a> around some loop <math class='maruku-mathml' display='inline' id='mathml_e323b8df50671ef4bb7cc48a4d4695604ba0b8bd_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>γ</mi><mo>:</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>\gamma : [0,1] \to X</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_e323b8df50671ef4bb7cc48a4d4695604ba0b8bd_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>γ</mi><mo stretchy='false'>(</mo><mn>0</mn><mo stretchy='false'>)</mo><mo>=</mo><mi>γ</mi><mo stretchy='false'>(</mo><mn>1</mn><mo stretchy='false'>)</mo><mo>=</mo><msub><mi>x</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>\gamma(0) = \gamma(1) = x_0</annotation></semantics></math> yields an element</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e323b8df50671ef4bb7cc48a4d4695604ba0b8bd_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>hol</mi> <mo>∇</mo></msub><mo stretchy='false'>(</mo><mi>γ</mi><mo stretchy='false'>)</mo><mo>∈</mo><mi>G</mi></mrow><annotation encoding='application/x-tex'> hol_\nabla(\gamma) \in G </annotation></semantics></math></div> <p>in the <a class='existingWikiWord' href='/nlab/show/automorphism'>automorphism group</a> of the <a class='existingWikiWord' href='/nlab/show/fiber'>fiber</a> <math class='maruku-mathml' display='inline' id='mathml_e323b8df50671ef4bb7cc48a4d4695604ba0b8bd_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>P</mi> <mrow><msub><mi>x</mi> <mn>0</mn></msub></mrow></msub></mrow><annotation encoding='application/x-tex'>P_{x_0}</annotation></semantics></math> of the bundle. This is the <strong>holonomy</strong> of <math class='maruku-mathml' display='inline' id='mathml_e323b8df50671ef4bb7cc48a4d4695604ba0b8bd_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∇</mo></mrow><annotation encoding='application/x-tex'>\nabla</annotation></semantics></math> around <math class='maruku-mathml' display='inline' id='mathml_e323b8df50671ef4bb7cc48a4d4695604ba0b8bd_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>γ</mi></mrow><annotation encoding='application/x-tex'>\gamma</annotation></semantics></math>.</p> <p>Fixing a connection <math class='maruku-mathml' display='inline' id='mathml_e323b8df50671ef4bb7cc48a4d4695604ba0b8bd_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∇</mo></mrow><annotation encoding='application/x-tex'>\nabla</annotation></semantics></math> and a point <math class='maruku-mathml' display='inline' id='mathml_e323b8df50671ef4bb7cc48a4d4695604ba0b8bd_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>x \in X</annotation></semantics></math> the collection of all elements <math class='maruku-mathml' display='inline' id='mathml_e323b8df50671ef4bb7cc48a4d4695604ba0b8bd_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>hol</mi> <mo>∇</mo></msub><mo stretchy='false'>(</mo><mi>γ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>hol_\nabla(\gamma)</annotation></semantics></math> for all loops <math class='maruku-mathml' display='inline' id='mathml_e323b8df50671ef4bb7cc48a4d4695604ba0b8bd_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>γ</mi></mrow><annotation encoding='application/x-tex'>\gamma</annotation></semantics></math> based at <math class='maruku-mathml' display='inline' id='mathml_e323b8df50671ef4bb7cc48a4d4695604ba0b8bd_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> forms a subgroup of <math class='maruku-mathml' display='inline' id='mathml_e323b8df50671ef4bb7cc48a4d4695604ba0b8bd_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>, called the <a class='existingWikiWord' href='/nlab/show/holonomy+group'>holonomy group</a>.</p> <p>If the <a class='existingWikiWord' href='/nlab/show/Levi-Civita+connection'>Levi-Civita connection</a> on a <a class='existingWikiWord' href='/nlab/show/Riemannian+manifold'>Riemannian manifold</a> <math class='maruku-mathml' display='inline' id='mathml_e323b8df50671ef4bb7cc48a4d4695604ba0b8bd_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>g</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(X,g)</annotation></semantics></math> has a holonomy group that is a proper subgroup of the <a class='existingWikiWord' href='/nlab/show/special+orthogonal+group'>special orthogonal group</a> one says that <math class='maruku-mathml' display='inline' id='mathml_e323b8df50671ef4bb7cc48a4d4695604ba0b8bd_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>g</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(X,g)</annotation></semantics></math> is a manifold with <a class='existingWikiWord' href='/nlab/show/special+holonomy'>special holonomy</a>. (More precise would be: “with special holonomy group for the Levi-Civita connection”.)</p> <h2 id='properties'>Properties</h2> <p>Proposition. If on a smooth principal bundle <math class='maruku-mathml' display='inline' id='mathml_e323b8df50671ef4bb7cc48a4d4695604ba0b8bd_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>P\to X</annotation></semantics></math> there is a connection <math class='maruku-mathml' display='inline' id='mathml_e323b8df50671ef4bb7cc48a4d4695604ba0b8bd_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∇</mo></mrow><annotation encoding='application/x-tex'>\nabla</annotation></semantics></math> whose holonomy group is <math class='maruku-mathml' display='inline' id='mathml_e323b8df50671ef4bb7cc48a4d4695604ba0b8bd_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> then the structure group can be reduced to <math class='maruku-mathml' display='inline' id='mathml_e323b8df50671ef4bb7cc48a4d4695604ba0b8bd_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>.</p> <p>(…)</p> <div class='un_theorem'> <h6 id='theorem'>Theorem</h6> <p><strong>(Ambrose-Singer)</strong> <a class='existingWikiWord' href='/nlab/show/holonomy'>Ambrose-Singer theorem</a>: the <a class='existingWikiWord' href='/nlab/show/Lie+algebra'>Lie algebra</a> of the holonomy group of a <a class='existingWikiWord' href='/nlab/show/connection+on+a+bundle'>connection on a bundle</a> <math class='maruku-mathml' display='inline' id='mathml_e323b8df50671ef4bb7cc48a4d4695604ba0b8bd_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∇</mo></mrow><annotation encoding='application/x-tex'>\nabla</annotation></semantics></math> on <math class='maruku-mathml' display='inline' id='mathml_e323b8df50671ef4bb7cc48a4d4695604ba0b8bd_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> at a point <math class='maruku-mathml' display='inline' id='mathml_e323b8df50671ef4bb7cc48a4d4695604ba0b8bd_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>x \in X</annotation></semantics></math> is spanned by the <a class='existingWikiWord' href='/nlab/show/parallel+transport'>parallel transport</a> <math class='maruku-mathml' display='inline' id='mathml_e323b8df50671ef4bb7cc48a4d4695604ba0b8bd_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Ad</mi> <mrow><msub><mi>tra</mi> <mo>∇</mo></msub><mo stretchy='false'>(</mo><mi>γ</mi><mo stretchy='false'>)</mo></mrow></msub><mo stretchy='false'>(</mo><msub><mi>F</mi> <mi>A</mi></msub><mo stretchy='false'>(</mo><mi>v</mi><mo>∨</mo><mi>w</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ad_{tra_\nabla(\gamma)}(F_A(v \vee w))</annotation></semantics></math> of the <a class='existingWikiWord' href='/nlab/show/curvature'>curvature</a> <math class='maruku-mathml' display='inline' id='mathml_e323b8df50671ef4bb7cc48a4d4695604ba0b8bd_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>F</mi> <mi>A</mi></msub></mrow><annotation encoding='application/x-tex'>F_A</annotation></semantics></math> evaluated on any <math class='maruku-mathml' display='inline' id='mathml_e323b8df50671ef4bb7cc48a4d4695604ba0b8bd_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>v</mi><mo>∨</mo><mi>w</mi><mo>∈</mo><msup><mo>∧</mo> <mn>2</mn></msup><msub><mi>T</mi> <mi>y</mi></msub><mi>X</mi></mrow><annotation encoding='application/x-tex'>v \vee w \in \wedge^2 T_y X</annotation></semantics></math> at <math class='maruku-mathml' display='inline' id='mathml_e323b8df50671ef4bb7cc48a4d4695604ba0b8bd_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>y \in X</annotation></semantics></math> along any path <math class='maruku-mathml' display='inline' id='mathml_e323b8df50671ef4bb7cc48a4d4695604ba0b8bd_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>γ</mi></mrow><annotation encoding='application/x-tex'>\gamma</annotation></semantics></math> from <math class='maruku-mathml' display='inline' id='mathml_e323b8df50671ef4bb7cc48a4d4695604ba0b8bd_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>→</mo><mi>y</mi></mrow><annotation encoding='application/x-tex'>x \to y</annotation></semantics></math>.</p> </div> <p>We may think of <math class='maruku-mathml' display='inline' id='mathml_e323b8df50671ef4bb7cc48a4d4695604ba0b8bd_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Id</mi><mo>+</mo><msub><mo lspace='0em' rspace='thinmathspace'>Ad</mo> <mrow><msub><mi>tra</mi> <mo>∇</mo></msub><mo stretchy='false'>(</mo><mi>γ</mi><mo stretchy='false'>)</mo></mrow></msub><mo stretchy='false'>(</mo><msub><mi>F</mi> <mi>A</mi></msub><mo stretchy='false'>(</mo><mi>ϕ</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Id + \Ad_{tra_\nabla(\gamma)}(F_A(\phi))</annotation></semantics></math> as being the holonomy around the loop obtained by</p> <ol> <li> <p>going along <math class='maruku-mathml' display='inline' id='mathml_e323b8df50671ef4bb7cc48a4d4695604ba0b8bd_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>γ</mi></mrow><annotation encoding='application/x-tex'>\gamma</annotation></semantics></math> from <math class='maruku-mathml' display='inline' id='mathml_e323b8df50671ef4bb7cc48a4d4695604ba0b8bd_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_e323b8df50671ef4bb7cc48a4d4695604ba0b8bd_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi></mrow><annotation encoding='application/x-tex'>y</annotation></semantics></math></p> </li> <li> <p>going around the <a class='existingWikiWord' href='/nlab/show/infinitesimal+object'>infinitesimal</a> parallelogram spanned by <math class='maruku-mathml' display='inline' id='mathml_e323b8df50671ef4bb7cc48a4d4695604ba0b8bd_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>v</mi></mrow><annotation encoding='application/x-tex'>v</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_e323b8df50671ef4bb7cc48a4d4695604ba0b8bd_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>w</mi></mrow><annotation encoding='application/x-tex'>w</annotation></semantics></math>;</p> </li> <li> <p>coming back to <math class='maruku-mathml' display='inline' id='mathml_e323b8df50671ef4bb7cc48a4d4695604ba0b8bd_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> along the reverse path <math class='maruku-mathml' display='inline' id='mathml_e323b8df50671ef4bb7cc48a4d4695604ba0b8bd_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>γ</mi></mrow><annotation encoding='application/x-tex'>\gamma</annotation></semantics></math>.</p> </li> </ol> <h2 id='applications'>Applications</h2> <p>(…)</p> <h2 id='higher_holonomy'>Higher holonomy</h2> <p>The <a class='existingWikiWord' href='/nlab/show/higher+parallel+transport'>higher holonomy</a> (see there) of <a class='existingWikiWord' href='/nlab/show/circle+n-bundle+with+connection'>circle n-bundles with connection</a> is given by <a class='existingWikiWord' href='/nlab/show/fiber+integration+in+ordinary+differential+cohomology'>fiber integration in ordinary differential cohomology</a>.</p> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/connection+on+a+bundle'>connection on a bundle</a>, <a class='existingWikiWord' href='/nlab/show/connection+on+a+2-bundle'>connection on a 2-bundle</a>, <a class='existingWikiWord' href='/nlab/show/connection+on+a+smooth+principal+infinity-bundle'>connection on an infinity-bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/parallel+transport'>parallel transport</a>, <a class='existingWikiWord' href='/nlab/show/higher+parallel+transport'>higher parallel transport</a></p> </li> <li> <p><strong>holonomy</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/holonomy+group'>holonomy group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/special+holonomy'>special holonomy</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/Wilson+loop'>Wilson loop</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/nonabelian+Stokes+theorem'>nonabelian Stokes theorem</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/fiber+integration+in+differential+cohomology'>fiber integration in differential cohomology</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/fiber+integration+in+ordinary+differential+cohomology'>fiber integration in ordinary differential cohomology</a></li> </ul> </li> </ul> <h2 id='references'>References</h2> <p>With an eye towards application in <a class='existingWikiWord' href='/nlab/show/mathematical+physics'>mathematical physics</a>:</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/Mikio+Nakahara'>Mikio Nakahara</a>, Section 10.2 of: <em><a class='existingWikiWord' href='/nlab/show/Geometry%2C+Topology+and+Physics'>Geometry, Topology and Physics</a></em>, IOP 2003 (<a href='https://doi.org/10.1201/9781315275826'>doi:10.1201/9781315275826</a>, <a href='http://alpha.sinp.msu.ru/~panov/LibBooks/GRAV/(Graduate_Student_Series_in_Physics)Mikio_Nakahara-Geometry,_Topology_and_Physics,_Second_Edition_(Graduate_Student_Series_in_Physics)-Institute_of_Physics_Publishing(2003).pdf'>pdf</a>)</li> </ul> <p> </p> <p> </p> </div>  </div>  </body> </html>