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href="/nlab/show/homology">homology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/chain">chain</a>, <a class="existingWikiWord" href="/nlab/show/cycle">cycle</a>, <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/characteristic+class">characteristic class</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+characteristic+class">universal characteristic class</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> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a>/<a class="existingWikiWord" href="/nlab/show/long+exact+sequence+in+cohomology">long exact sequence in cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/twisted+%E2%88%9E-bundle">twisted ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/obstruction">obstruction</a></p> </li> </ul> <h3 id="special_and_general_types">Special and general types</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a>, <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+group+cohomology">nonabelian group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+group+cohomology">Lie group cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+cohomology">Galois cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid+cohomology">groupoid cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+groupoid+cohomology">nonabelian groupoid cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theory">cobordism cohomology theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a>, <a class="existingWikiWord" href="/nlab/show/tmf">tmf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/taf">taf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dolbeault+cohomology">Dolbeault cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/etale+cohomology">etale cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a>, <a class="existingWikiWord" href="/nlab/show/Picard+group">Picard group</a>, <a class="existingWikiWord" href="/nlab/show/Brauer+group">Brauer group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/crystalline+cohomology">crystalline cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/syntomic+cohomology">syntomic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivic+cohomology">motivic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+of+operads">cohomology of operads</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+cohomology">cyclic cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+topology">string topology</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+principal+%E2%88%9E-bundle">universal principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/groupal+model+for+universal+principal+%E2%88%9E-bundles">groupal model for universal principal ∞-bundles</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>/<a class="existingWikiWord" href="/nlab/show/gerbe">gerbe</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+constant+%E2%88%9E-stack">covering ∞-bundle</a>/<a class="existingWikiWord" href="/nlab/show/local+system">local system</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-vector+bundle">(∞,1)-vector bundle</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-vector+bundle">(∞,n)-vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/Spin+structure">Spin structure</a>, <a class="existingWikiWord" href="/nlab/show/Spin%5Ec+structure">Spin^c structure</a>, <a class="existingWikiWord" href="/nlab/show/String+structure">String structure</a>, <a class="existingWikiWord" href="/nlab/show/Fivebrane+structure">Fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+with+constant+coefficients">cohomology with constant coefficients</a> / <a class="existingWikiWord" href="/nlab/show/cohomology+with+a+local+system+of+coefficients">with a local system of coefficients</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">∞-Lie algebra cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+algebra+extensions">Lie algebra extensions</a>, <a class="existingWikiWord" href="/nlab/show/Gelfand-Fuks+cohomology">Gelfand-Fuks cohomology</a>,</li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gerstenhaber-Schack+cohomology">bialgebra cohomology</a></p> </li> </ul> <h3 id="special_notions">Special notions</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%3Fech+cohomology">?ech cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercohomology">hypercohomology</a></p> </li> </ul> <h3 id="variants">Variants</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bredon+cohomology">Bredon cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+bundle">twisted bundle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin+structure">twisted spin structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin%5Ec+structure">twisted spin^c structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+differential+c-structures">twisted differential c-structures</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+differential+string+structure">twisted differential string structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+differential+fivebrane+structure">twisted differential fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p>differential cohomology</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</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/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> </ul> </li> <li> <p><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relative+cohomology">relative cohomology</a></p> </li> </ul> <h3 id="extra_structure">Extra structure</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+structure">Hodge structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">in generalized cohomology</a></p> </li> </ul> <h3 id="operations">Operations</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+operations">cohomology operations</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cup+product">cup product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a>, <a class="existingWikiWord" href="/nlab/show/Bockstein+homomorphism">Bockstein homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration">fiber integration</a>, <a class="existingWikiWord" href="/nlab/show/transgression">transgression</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+localization">cohomology localization</a></p> </li> </ul> <h3 id="theorems">Theorems</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a>, <a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a>, <a class="existingWikiWord" href="/nlab/show/Stokes+theorem">Stokes theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+theory">Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/Hodge+theorem">Hodge theorem</a></p> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+Hodge+theory">nonabelian Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/noncommutative+Hodge+theory">noncommutative Hodge theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">Brown representability theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">hypercovering theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eckmann-Hilton+duality">Eckmann-Hilton-Fuks duality</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/cohomology+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <blockquote> <p><strong>under construction</strong></p> </blockquote> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#nonabelian_hodge_theorems'>Nonabelian Hodge theorems</a></li> <ul> <li><a href='#nonabelian_harmonic_sections'>Nonabelian harmonic sections</a></li> <li><a href='#LocalSystemsAndHiggsBundles'>Kähler case: Equivalence between Local systems and Higgs bundles</a></li> <ul> <li><a href='#relation_to_the_abelian_hodge_theorem'>Relation to the abelian Hodge theorem</a></li> <li><a href='#statement'>Statement</a></li> <li><a href='#generalizations_to_twisted_bundles'>Generalizations to twisted bundles</a></li> </ul> </ul> <li><a href='#relation_to_geometric_langlands'>Relation to geometric Langlands</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>Nonabelian Hodge theory generalizes aspects of <a class="existingWikiWord" href="/nlab/show/Hodge+theory">Hodge theory</a> from abelian cohomology (<a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a>) to <a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a>.</p> <h2 id="nonabelian_hodge_theorems">Nonabelian Hodge theorems</h2> <h3 id="nonabelian_harmonic_sections">Nonabelian harmonic sections</h3> <p>Notice or recall (for instance from <a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">generalized universal bundle</a> and <a class="existingWikiWord" href="/nlab/show/action+groupoid">action groupoid</a>) the following equivalent description of <a class="existingWikiWord" href="/nlab/show/section">section</a>s of <a class="existingWikiWord" href="/nlab/show/associated+bundle">associated bundle</a>s:</p> <p>for <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> a <a class="existingWikiWord" href="/nlab/show/group">group</a> with <a class="existingWikiWord" href="/nlab/show/action">action</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> on an object <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> witnessed by the <a class="existingWikiWord" href="/nlab/show/action+groupoid">action groupoid</a> sequence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>→</mo><mi>V</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex"> V \to V//G \to \mathbf{B}G </annotation></semantics></math></div> <p>the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/associated+bundle">associated bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">E \to X</annotation></semantics></math> to a <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>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> <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> classified by an <a class="existingWikiWord" href="/nlab/show/anafunctor">anafunctor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>←</mo><mo>≃</mo></mover><mi>Y</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">X \stackrel{\simeq}{\leftarrow} Y \to \mathbf{B}G</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>E</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>V</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>Y</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ E &\to& V//G \\ \downarrow && \downarrow \\ Y &\to& \mathbf{B}G } \,. </annotation></semantics></math></div> <p>Since this is a <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> diagram by definition, a glance at a pasting diagram of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>E</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>V</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>Y</mi></mtd> <mtd><mover><mo>→</mo><mo>=</mo></mover></mtd> <mtd><mi>Y</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && E &\to& V//G \\ & \nearrow & \downarrow && \downarrow \\ Y &\stackrel{=}{\to}& Y &\to& \mathbf{B}G } </annotation></semantics></math></div> <p>shows that <a class="existingWikiWord" href="/nlab/show/section">section</a>s</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>E</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mi>σ</mi></msup><mo>↗</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>Y</mi></mtd> <mtd><mover><mo>→</mo><mo>=</mo></mover></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && E \\ & {}^{\sigma}\nearrow & \downarrow \\ Y &\stackrel{=}{\to}& Y } </annotation></semantics></math></div> <p>are in bijection with maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>→</mo><mi>V</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">Y \to V//G</annotation></semantics></math> that make</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Y</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>V</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mo>=</mo></msup></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>Y</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Y &\to& V//G \\ \downarrow^= && \downarrow \\ Y &\to& \mathbf{B}G } </annotation></semantics></math></div> <p>commute.</p> <p>In the special case that <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> is a connected <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> and <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> a discrete group we can without restriction take <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>=</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">/</mo><mo stretchy="false">/</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Y = \hat X//\pi_1(X)</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/action+groupoid">action groupoid</a> of the <a class="existingWikiWord" href="/nlab/show/universal+cover">universal cover</a> by the <a class="existingWikiWord" href="/nlab/show/fundamental+group">homotopy group</a>, so that the classifying map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">Y \to \mathbf{B}G</annotation></semantics></math> is the same as a <a class="existingWikiWord" href="/nlab/show/group">group</a> homomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>G</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \rho : \pi_1(X) \to G \,. </annotation></semantics></math></div> <p>In that case the above says that a section of the associated bundle is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math>-equivariant map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mi>V</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \phi : \hat X \to V \,. </annotation></semantics></math></div> <p>This is the way these sections are formulated usually in the literature. The above description has the advantage that it works more generally in <a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a> for <a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>s generalized to <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>s.</p> <p>Next consider furthermore the special case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>=</mo><mi>G</mi><mo stretchy="false">/</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">V = G/K</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/coset">coset</a> <a class="existingWikiWord" href="/nlab/show/homogeneous+space">homogeneous space</a> 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> quotiented by a subgroup <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>. Then if <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> is a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> or <a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a> consider moreover a choice 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>-invariant metric on the quotient <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">G/K</annotation></semantics></math>. Also consider a <a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a> structure 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>.</p> <p>Then</p> <div class="num_defin"> <h6 id="definition">Definition</h6> <p>The <strong>energy</strong> of a <a class="existingWikiWord" href="/nlab/show/section">section</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math> of an associated <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">G/K</annotation></semantics></math>-bundle as above is the real number</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><msub><mo>∫</mo> <mi>X</mi></msub><mo stretchy="false">|</mo><mi>d</mi><mi>ϕ</mi><msup><mo stretchy="false">|</mo> <mn>2</mn></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> E(\phi) := \int_X |d \phi|^2 \,. </annotation></semantics></math></div></div> <p>Here</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math>-equivariant map describing the section as above,</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/norm">norm</a> is taken with respect to the chocen invariant <a class="existingWikiWord" href="/nlab/show/metric">metric</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">G/K</annotation></semantics></math></p> </li> <li> <p>and the <a class="existingWikiWord" href="/nlab/show/integral">integral</a> is taken with respect to the <a class="existingWikiWord" href="/nlab/show/Riemannian+metric">Riemannian metric</a> 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>.</p> </li> </ul> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>Such a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> is called <strong>harmonic</strong> if it is a <a class="existingWikiWord" href="/nlab/show/critical+point">critical point</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E(-)</annotation></semantics></math>.</p> </div> <div class="num_theorem"> <h6 id="theorem">Theorem</h6> <p><strong>(Corlette, generalizing Eells-Sampson)</strong></p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">\rho : \pi_1(X) \to G</annotation></semantics></math> is a representation with</p> <ul> <li> <p><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> a <a class="existingWikiWord" href="/nlab/show/reductive+group">reductive</a> <a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/maximal+compact+subgroup">maximal compact subgroup</a></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo stretchy="false">(</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\rho(\pi_1(X))</annotation></semantics></math> is</p> <ul> <li> <p>Zariski-dense in <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> </li> <li> <p>or its Zariski-closure is itself reductive</p> </li> </ul> </li> </ul> <p>then there exists a <em>harmonic</em> section <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> in the above sense.</p> </div> <p>This is due to (<a href="#Corlette88">Corlett 88</a>). A version of the proof is reproduced in <a href="#Simpson96">Simpson 96, p. 8</a></p> <h3 id="LocalSystemsAndHiggsBundles">Kähler case: Equivalence between Local systems and Higgs bundles</h3> <p>The <em>nonabelian Hodge theorem</em> due to (<a href="#Simpson92">Simpson 92</a>) establishes, for <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> a <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a> <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+manifold">Kähler manifold</a>, an <a class="existingWikiWord" href="/nlab/show/equivalence">equivalence</a> between (irreducible) <a class="existingWikiWord" href="/nlab/show/flat+vector+bundles">flat vector bundles</a> 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> and (stable) <a class="existingWikiWord" href="/nlab/show/Higgs+bundles">Higgs bundles</a> with vanishing <a class="existingWikiWord" href="/nlab/show/first+Chern+class">first Chern class</a>.</p> <h4 id="relation_to_the_abelian_hodge_theorem">Relation to the abelian Hodge theorem</h4> <p>The sense in which the nonabelian Hodge theorem of (<a href="#Simpson92">Simpson 92</a>) generalizes the abelian <a class="existingWikiWord" href="/nlab/show/Hodge+theorem">Hodge theorem</a> is the following (<a href="#Simpson92">Simpson 92, Introduction</a>).</p> <p>The abelian <a class="existingWikiWord" href="/nlab/show/cohomology+group">cohomology group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>ℂ</mi> <mi>disc</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^1(X,\mathbb{C}_{disc})</annotation></semantics></math> classifies <a class="existingWikiWord" href="/nlab/show/flat+vector+bundle">flat</a> <a class="existingWikiWord" href="/nlab/show/complex+line+bundles">complex line bundles</a> whose underlying <a class="existingWikiWord" href="/nlab/show/line+bundle">line bundle</a> is trivial, hence closed <a class="existingWikiWord" href="/nlab/show/differential+1-forms">differential 1-forms</a> modulo 0-forms. The abelian <a class="existingWikiWord" href="/nlab/show/Hodge+theorem">Hodge theorem</a> gives for this hence the decomposition</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>ℂ</mi> <mi>disc</mi></msub><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>H</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>𝒪</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo>⊕</mo><msup><mi>H</mi> <mn>0</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msubsup><mi>Ω</mi> <mi>X</mi> <mn>1</mn></msubsup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H^1(X,\mathbb{C}_{disc}) \simeq H^1(X, \mathcal{O}_X) \oplus H^0(X, \Omega^1_X) \,. </annotation></semantics></math></div> <p>It is this kind of relation which is generalized by the nonabelian Hodge theorem. Here one starts instead with the <a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a> set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>GL</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>ℂ</mi><msub><mo stretchy="false">)</mo> <mi>disc</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^1(X, GL_n(\mathbb{C})_{disc})</annotation></semantics></math> which classifies <a class="existingWikiWord" href="/nlab/show/flat+vector+bundle">flat</a> <a class="existingWikiWord" href="/nlab/show/rank">rank</a>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/vector+bundles">vector bundles</a> 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>, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>. The equivalence to <a class="existingWikiWord" href="/nlab/show/Higgs+bundles">Higgs bundles</a> gives now a decomposition of these structures into a <a class="existingWikiWord" href="/nlab/show/holomorphic+vector+bundle">holomorphic vector bundle</a> classified by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>GL</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msub><mi>𝒪</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^1(X, GL_n(\mathcal{O}_X))</annotation></semantics></math> and a differential 1-form with values in endomorphisms of that, subject to some conditions.</p> <h4 id="statement">Statement</h4> <p>A quick review of the theorem in (<a href="#Simpson92">Simpson 92</a>) is for instance in (<a href="#Raboso15">Raboso 15, section 1.2</a>). An elegant abstract reformulation in terms of <a class="existingWikiWord" href="/nlab/show/differential+cohesion">differential cohesion</a>/<a class="existingWikiWord" href="/nlab/show/D-geometry">D-geometry</a>, following (<a href="#Simpson96">Simpson 96</a>) is in (<a href="#Raboso15">Raboso 15, section 3.3</a>):</p> <p>Analogous to how the <a class="existingWikiWord" href="/nlab/show/de+Rham+stack">de Rham stack</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∫</mo> <mi>inf</mi></msub><mi>X</mi><mo>=</mo><msub><mi>X</mi> <mi>dR</mi></msub></mrow><annotation encoding="application/x-tex">\int_{inf} X = X_{dR}</annotation></semantics></math> of <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> is the (<a class="existingWikiWord" href="/nlab/show/homotopy+quotient">homotopy</a>) <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> of <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> by the first order <a class="existingWikiWord" href="/nlab/show/infinitesimal+neighbourhood">infinitesimal neighbourhood</a> of the <a class="existingWikiWord" href="/nlab/show/diagonal">diagonal</a> in <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 \times X</annotation></semantics></math>, so there is a space (<a class="existingWikiWord" href="/nlab/show/stack">stack</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>Dol</mi></msub></mrow><annotation encoding="application/x-tex">X_{Dol}</annotation></semantics></math> which is the formal completion of the 0-section of the <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a> of <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> (<a href="#Simpson96">Simpson 96</a>).</p> <p>Now a <a class="existingWikiWord" href="/nlab/show/flat+vector+bundle">flat vector bundle</a> 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> is essentially just a <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a> on the <a class="existingWikiWord" href="/nlab/show/de+Rham+stack">de Rham stack</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>dR</mi></msub></mrow><annotation encoding="application/x-tex">X_{dR}</annotation></semantics></math>, and a <a class="existingWikiWord" href="/nlab/show/Higgs+bundle">Higgs bundle</a> is essentially just a <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>Dol</mi></msub></mrow><annotation encoding="application/x-tex">X_{Dol}</annotation></semantics></math>. Therefore in this language the nonabelian Hodge theorem reads (for <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> a linear <a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>dR</mi></msub><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo><mo>≃</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>Dol</mi></msub><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><msup><mo stretchy="false">)</mo> <mrow><mi>ss</mi><mo>,</mo><mn>0</mn></mrow></msup><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H}(X_{dR}, \mathbf{B}G) \simeq \mathbf{H}(X_{Dol}, \mathbf{B}G)^{ss,0} \,, </annotation></semantics></math></div> <p>where the superscript on the right denotes restriction to semistable Higgs bundles with vanishing <a class="existingWikiWord" href="/nlab/show/first+Chern+class">first Chern class</a> (see <a href="#Raboso15">Raboso 15, Theorem 3.3</a>).</p> <h4 id="generalizations_to_twisted_bundles">Generalizations to twisted bundles</h4> <p>A generalization of the nonabelian Hodge theorem of (<a href="#Simpson92">Simpson 92</a>) to <a class="existingWikiWord" href="/nlab/show/twisted+bundles">twisted bundles</a> in discussed in (<a href="#Raboso15">Raboso 15</a>).</p> <h2 id="relation_to_geometric_langlands">Relation to geometric Langlands</h2> <p>Nonabelian Hodge theory is closely related to the <a class="existingWikiWord" href="/nlab/show/geometric+Langlands+correspondence">geometric Langlands correspondence</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+structure">Hodge structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/noncommutative+Hodge+theory">noncommutative Hodge theory</a></p> </li> </ul> <h2 id="references">References</h2> <ul> <li>wikipedia <a href="https://en.wikipedia.org/wiki/Nonabelian_Hodge_correspondence">nonabelian Hodge correspondence</a></li> </ul> <p>Lecture notes on nonabelian Hodge theory include:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ron+Donagi">Ron Donagi</a>, <a class="existingWikiWord" href="/nlab/show/Tony+Pantev">Tony Pantev</a>, <em>Lectures on the geometric Langlands</em></p> <p>conjecture and non-abelian Hodge theory_, 2009 (<a href="http://www.icmat.es/seminarios/langlands/school/handouts/pantev.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Alberto+Garc%C3%ADa+Raboso">Alberto García Raboso</a>, <a class="existingWikiWord" href="/nlab/show/Steven+Rayan">Steven Rayan</a>, <em>Introduction to Nonabelian Hodge Theory: flat connections, Higgs bundles, and complex variations of Hodge structure</em>, Fields Inst. Monogr. 34 (2015), 131–171 (<a href="https://arxiv.org/abs/1406.1693">arXiv</a>) (<a href="http://link.springer.com/chapter/10.1007/978-1-4939-2830-9_5">Springer</a>)</p> </li> </ul> <p>Corlette’s nonabelian Hodge theorem can be found in:</p> <ul> <li id="Corlette88">K. Corlette, <em>Flat <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>-bundles with canonical metric</em>, J. Diff Geometry 28 (1988)</li> </ul> <p>Works by <a class="existingWikiWord" href="/nlab/show/Carlos+Simpson">Carlos Simpson</a> on nonabelian Hodge theory include:</p> <ul> <li id="Simpson92"> <p><a class="existingWikiWord" href="/nlab/show/Carlos+Simpson">Carlos Simpson</a>, <em>Higgs bundles and local systems</em>, Inst. Hautes Etudes Sci. Publ. Math. (1992), no. 75, 5{95. MR 1179076 (94d:32027) (<a href="http://www.numdam.org/item?id=PMIHES_1992__75__5_0">numdam</a>)</p> </li> <li id="Simpson96"> <p><a class="existingWikiWord" href="/nlab/show/Carlos+Simpson">Carlos Simpson</a>, <em>The Hodge filtration on nonabelian cohomology</em>, Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 217{281. MR</p> <p>1492538 (99g:14028) (<a href="http://arxiv.org/abs/alg-geom/9604005">arXiv:9604005</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Carlos+Simpson">Carlos Simpson</a>, <em>Secondary Kodaira-Spencer classes and nonabelian Dolbeault cohomology</em> (<a href="http://arxiv.org/abs/alg-geom/9712020">arXiv:9712020</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Carlos+Simpson">Carlos Simpson</a>, <em>Algebraic aspects of higher nonabelian Hodge theory</em> (<a href="http://arxiv.org/abs/math/9902067">arXiv:9902067</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Carlos+Simpson">Carlos Simpson</a>, <a class="existingWikiWord" href="/nlab/show/Tony+Pantev">Tony Pantev</a>, <a class="existingWikiWord" href="/nlab/show/Ludmil+Katzarkov">Ludmil Katzarkov</a>, <em>Nonabelian mixed Hodge structures</em> (<a href="http://arxiv.org/abs/math/0006213">arXiv</a>)</p> </li> </ul> <p>The nonabelian Hodge theorem of (<a href="#Simpson92">Simpson 92</a>) is generalized to <a class="existingWikiWord" href="/nlab/show/twisted+bundles">twisted bundles</a> in:</p> <ul> <li id="Raboso15"><a class="existingWikiWord" href="/nlab/show/Alberto+Garc%C3%ADa+Raboso">Alberto García Raboso</a>, <em>A twisted nonabelian Hodge correspondence</em>, PhD thesis 2014 (<a href="http://arxiv.org/abs/1501.05872">arXiv:1501.05872</a>, <a href="http://www.math.toronto.edu/agraboso/files/TwistedNAHT_Talk_Handout.pdf">pdf slides</a>)</li> </ul> <p>Gothen’s paper starts with a survey on nonabelian Hodge correspondence,</p> <ul> <li> <p>Peter B. Gothen, <em>Higgs bundles and the real symplectic group</em>, <a href="https://arxiv.org/abs/1102.4175">arXiv:1102.4175</a></p> </li> <li> <p>C. C. Liu, S. Rayan, Y. Tanaka, <em>The Kapustin–Witten equations and nonabelian Hodge theory</em> Eur. J. Math. <strong>8</strong> (Suppl 1) 23–41 (2022) <a href="https://doi.org/10.1007/s40879-022-00538-4">doi</a></p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 1, 2023 at 12:46:32. 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