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float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="differential_cohomology">Differential cohomology</h4> <div class="hide"><div> <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/differential+orientation">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/differential+Thom+class">differential Thom class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+characters">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+%E2%88%9E-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+an+%E2%88%9E-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/%E2%88%9E-Lie+algebra+cohomology">∞-Lie algebra cohomology</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Simons+theory">∞-Chern-Simons theory</a></p> </li> </ul> <h2 id="fiber_integration">Fiber integration</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+holonomy">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+field">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/B-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-Fre+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> </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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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+group">automorphism group</a> of the <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math> around <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi></mrow><annotation encoding="application/x-tex">\gamma</annotation></semantics></math>.</p> <p>Fixing a connection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math> and a point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi></mrow><annotation encoding="application/x-tex">\gamma</annotation></semantics></math> based at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> forms a subgroup of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math> whose holonomy group is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/Ambrose-Singer+theorem">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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> at a point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi></mrow><annotation encoding="application/x-tex">\gamma</annotation></semantics></math> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi></mrow><annotation encoding="application/x-tex">\gamma</annotation></semantics></math> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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+space">infinitesimal</a> parallelogram spanned by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi></mrow><annotation encoding="application/x-tex">w</annotation></semantics></math>;</p> </li> <li> <p>coming back to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> along the reverse path <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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+holonomy">higher holonomy</a> (see there) of <a class="existingWikiWord" href="/nlab/show/circle+n-bundles+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+an+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> </body></html> </div> <div class="revisedby"> <p> Last revised on October 19, 2020 at 09:50:25. 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