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<title>Higgs bundle</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Higgs bundle</span></span>

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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, a <b>Higgs bundle</b> is a pair <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (E,\varphi )}">

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<annotation encoding="application/x-tex">{\displaystyle (E,\varphi )}</annotation>

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</math></span><img src="./0de33cd31bfa7d0b2620c5c85d8680e5abd2e86c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.139ex; height:2.843ex;" alt="{\displaystyle (E,\varphi )}" loading="lazy"></span> consisting of a <a href="Holomorphic_vector_bundle" title="Holomorphic vector bundle">holomorphic vector bundle</a> <i>E</i> and a <a href="Higgs_field" class="mw-redirect" title="Higgs field">Higgs field</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

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

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</math></span><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"></span>, a holomorphic 1-form taking values in the bundle of endomorphisms of <i>E</i> such that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \wedge \varphi =0}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi \wedge \varphi =0}</annotation>

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</math></span><img src="./9683b137115bbc11a0923f295fa4531dfaa02765.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.884ex; height:2.676ex;" alt="{\displaystyle \varphi \wedge \varphi =0}" loading="lazy"></span>. Such pairs were introduced by <a href="Nigel_Hitchin" title="Nigel Hitchin">Nigel Hitchin</a> (<a href="#CITEREFHitchin1987">1987</a>),<sup id="cite_ref-hitchin1_1-0" class="reference"><a href="#cite_note-hitchin1-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> who named the field <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

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

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</math></span><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"></span> after <a href="Peter_Higgs" title="Peter Higgs">Peter Higgs</a> because of an analogy with Higgs bosons. The term 'Higgs bundle', and the condition <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \wedge \varphi =0}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi \wedge \varphi =0}</annotation>

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</math></span><img src="./9683b137115bbc11a0923f295fa4531dfaa02765.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.884ex; height:2.676ex;" alt="{\displaystyle \varphi \wedge \varphi =0}" loading="lazy"></span> (which is vacuous in Hitchin's original set-up on <a href="Riemann_surface" title="Riemann surface">Riemann surfaces</a>) was introduced later by <a href="Carlos_Simpson" title="Carlos Simpson">Carlos Simpson</a>.<sup id="cite_ref-simpson_2-0" class="reference"><a href="#cite_note-simpson-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p><p>A Higgs bundle can be thought of as a "simplified version" of a flat holomorphic <a href="Affine_connection" title="Affine connection">connection</a> on a holomorphic vector bundle, where the derivative is scaled to zero. The <a href="Nonabelian_Hodge_correspondence" title="Nonabelian Hodge correspondence">nonabelian Hodge correspondence</a> says that, under suitable stability conditions, the <a href="Category_(mathematics)" title="Category (mathematics)">category</a> of flat holomorphic connections on a smooth <a href="Projective_variety" title="Projective variety">projective complex algebraic variety</a>, the category of representations of the <a href="Fundamental_group" title="Fundamental group">fundamental group</a> of the variety, and the category of Higgs bundles over this variety are actually equivalent. Therefore, one can deduce results about <a href="Gauge_theory" title="Gauge theory">gauge theory</a> with flat connections by working with the simpler Higgs bundles.

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

<p>Higgs bundles were first introduced by Hitchin in 1987,<sup class=" nourlexpansion citation" id="ref_hitchin1"><a class="external autonumber external" href="https://en.wikipedia.org/wiki/Higgs_bundle#endnote_hitchin1">[1]</a></sup> for the specific case where the holomorphic vector bundle <i>E</i> is over a <a href="Compact_(mathematics)" class="mw-redirect" title="Compact (mathematics)">compact</a> <a href="Riemann_surface" title="Riemann surface">Riemann surface</a>. Further, Hitchin's paper mostly discusses the case where the vector bundle is rank 2 (that is, the fiber is a 2-dimensional vector space). The rank 2 vector bundle arises as the solution space to <a href="Hitchin's_equations" title="Hitchin's equations">Hitchin's equations</a> for a <a href="Principal_SU(2)-bundle" title="Principal SU(2)-bundle">principal SU(2)-bundle</a>.

</p><p>The theory on Riemann surfaces was generalized by Carlos Simpson to the case where the base manifold is compact and <a href="K%C3%A4hler_manifold" title="Kähler manifold">Kähler</a>. Restricting to the dimension one case recovers Hitchin's theory.

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<div class="mw-heading mw-heading2"><h2 id="Stability_of_a_Higgs_bundle">Stability of a Higgs bundle</h2></div>

<p>Of particular interest in the theory of Higgs bundles is the notion of a <b>stable</b> Higgs bundle. To do so, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

</mstyle>

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

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</p><p>In Hitchin's original discussion, a rank-1 subbundle labelled <i>L</i> is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

</mstyle>

</mrow>

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

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</math></span><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"></span>-invariant if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (L)\subset L\otimes K}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi (L)\subset L\otimes K}</annotation>

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</math></span><img src="./4482b1f0e5f6d322c645a04a6a6d025f2db2bd06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.5ex; height:2.843ex;" alt="{\displaystyle \varphi (L)\subset L\otimes K}" loading="lazy"></span> with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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</mrow>

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

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</math></span><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"></span> the canonical bundle over the Riemann surface <i>M</i>. Then a Higgs bundle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (E,\varphi )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (E,\varphi )}</annotation>

</semantics>

</math></span><img src="./0de33cd31bfa7d0b2620c5c85d8680e5abd2e86c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.139ex; height:2.843ex;" alt="{\displaystyle (E,\varphi )}" loading="lazy"></span> is <b>stable</b> if, for each <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

</mstyle>

</mrow>

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

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</math></span><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"></span> invariant subbundle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

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

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</math></span><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"></span> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

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

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</math></span><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"></span>,

<mfrac>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {deg} L<{\frac {1}{2}}\operatorname {deg} (\wedge ^{2}E),}</annotation>

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with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {deg} }">

</mstyle>

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

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</math></span><img src="./db3c84720f2f29f486b2bd1cecb4b1ccd54149f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.487ex; height:2.509ex;" alt="{\displaystyle \operatorname {deg} }" loading="lazy"></span> being the usual notion of degree for a complex vector bundle over a Riemann surface.

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<ul><li><a href="Hitchin_system" title="Hitchin system">Hitchin system</a></li></ul>

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

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</style><cite id="CITEREFHitchin1987" class="citation journal cs1">Hitchin, Nigel (1987). <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/plms/s3-55.1.59">"The self-duality equations on a Riemann surface"</a></span>. <i>Proceedings of the London Mathematical Society</i>. <b>55</b> (1): <span class="nowrap">59–</span>126. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2Fplms%2Fs3-55.1.59">10.1112/plms/s3-55.1.59</a><span class="reference-accessdate">. Retrieved <span class="nowrap">10 November</span> 2022</span>.</cite></span>

</li>

<li id="cite_note-simpson-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-simpson_2-0">^</a></b></span> <span class="reference-text"><cite id="simpson" class="citation journal cs1">Simpson, Carlos (1992). <a rel="nofollow" class="external text" href="http://www.numdam.org/item/PMIHES_1992__75__5_0.pdf">"Higgs bundles and local systems"</a> <span class="cs1-format">(PDF)</span>. <i>Publications Mathématiques de l'IHÉS</i>. <b>75</b> (1): <span class="nowrap">5–</span>95. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02699491">10.1007/BF02699491</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:56417181">56417181</a><span class="reference-accessdate">. Retrieved <span class="nowrap">10 November</span> 2022</span>.</cite></span>

</li>

</ol></div></div>

<ul><li><cite id="CITEREFCorlette1988" class="citation journal cs1">Corlette, Kevin (1988). <a rel="nofollow" class="external text" href="https://doi.org/10.4310%2Fjdg%2F1214442469">"Flat <i>G</i>-bundles with canonical metrics"</a>. <i><a href="Journal_of_Differential_Geometry" title="Journal of Differential Geometry">Journal of Differential Geometry</a></i>. <b>28</b> (3): <span class="nowrap">361–</span>382. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.4310%2Fjdg%2F1214442469">10.4310/jdg/1214442469</a></span>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0965220">0965220</a>.</cite></li>

<li><cite id="CITEREFGothenGarcía-PradaBradlow2007" class="citation cs2">Gothen, Peter B.; García-Prada, Oscar; Bradlow, Steven B. (2007), <a rel="nofollow" class="external text" href="https://www.ams.org/notices/200708/tx070800980p.pdf">"What is... a Higgs bundle?"</a> <span class="cs1-format">(PDF)</span>, <i><a href="Notices_of_the_American_Mathematical_Society" title="Notices of the American Mathematical Society">Notices of the American Mathematical Society</a></i>, <b>54</b> (8): <span class="nowrap">980–</span>981, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2343296">2343296</a></cite></li></ul>

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