ebab7d0da53c51438720d60c6e2e467c - /page/wikibook/entry.mu

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<title>Wess–Zumino–Witten model</title>

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Wess–Zumino–Witten model

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</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Wess%E2%80%93Zumino_model" title="Wess–Zumino model">Wess–Zumino model</a>.</div>

In <a href="Theoretical_physics" title="Theoretical physics">theoretical physics</a> and <a href="Mathematics" title="Mathematics">mathematics</a>, a Wess–Zumino–Witten (WZW) model, also called a Wess–Zumino–Novikov–Witten model, is a type of <a href="Two-dimensional_conformal_field_theory" title="Two-dimensional conformal field theory">two-dimensional conformal field theory</a> named after <a href="Julius_Wess" title="Julius Wess">Julius Wess</a>, <a href="Bruno_Zumino" title="Bruno Zumino">Bruno Zumino</a>, <a href="Sergei_Novikov_(mathematician)" title="Sergei Novikov (mathematician)">Sergei Novikov</a> and <a href="Edward_Witten" title="Edward Witten">Edward Witten</a>.<a href="#cite_note-1">[1]</a><a href="#cite_note-2">[2]</a><a href="#cite_note-3">[3]</a><a href="#cite_note-4">[4]</a> A WZW model is associated to a <a href="Lie_group" title="Lie group">Lie group</a> (or <a href="Supergroup_(physics)" title="Supergroup (physics)">supergroup</a>), and its symmetry algebra is the <a href="Affine_Lie_algebra" title="Affine Lie algebra">affine Lie algebra</a> built from the corresponding <a href="Lie_algebra" title="Lie algebra">Lie algebra</a> (or <a href="Lie_superalgebra" title="Lie superalgebra">Lie superalgebra</a>). By extension, the name WZW model is sometimes used for any conformal field theory whose symmetry algebra is an affine Lie algebra.<a href="#cite_note-BYB-5">[5]</a>

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

<div class="mw-heading mw-heading3"><h3 id="Definition">Definition</h3></div>

For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }">

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

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</math><img src="./9e1f558f53cda207614abdf90162266c70bc5c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }" loading="lazy"> a <a href="Riemann_surface" title="Riemann surface">Riemann surface</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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

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</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> a <a href="Lie_group" title="Lie group">Lie group</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> a (generally complex) number, let us define the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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

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</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">-WZW model on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }">

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

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</math><img src="./9e1f558f53cda207614abdf90162266c70bc5c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }" loading="lazy"> at the level <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">. The model is a <a href="Nonlinear_sigma_model" class="mw-redirect" title="Nonlinear sigma model">nonlinear sigma model</a> whose <a href="Action_(physics)" title="Action (physics)">action</a> is a functional of a field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma :\Sigma \to G}">

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<annotation encoding="application/x-tex">{\displaystyle \gamma :\Sigma \to G}</annotation>

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</math><img src="./a3b516f6fe4da7a04f0ca9c89cd6e4098ed5ba17.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.318ex; height:2.676ex;" alt="{\displaystyle \gamma :\Sigma \to G}" loading="lazy">:

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<annotation encoding="application/x-tex">{\displaystyle S_{k}(\gamma )=-{\frac {k}{8\pi }}\int _{\Sigma }d^{2}x\,{\mathcal {K}}\left(\gamma ^{-1}\partial ^{\mu }\gamma ,\gamma ^{-1}\partial _{\mu }\gamma \right)+2\pi kS^{\mathrm {W} Z}(\gamma ).}</annotation>

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</math><img src="./d49c4b3def7cbecab6fddc734cd510093330421a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:56.289ex; height:5.843ex;" alt="{\displaystyle S_{k}(\gamma )=-{\frac {k}{8\pi }}\int _{\Sigma }d^{2}x\,{\mathcal {K}}\left(\gamma ^{-1}\partial ^{\mu }\gamma ,\gamma ^{-1}\partial _{\mu }\gamma \right)+2\pi kS^{\mathrm {W} Z}(\gamma ).}" loading="lazy"></dd></dl>

Here, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }">

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

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</math><img src="./9e1f558f53cda207614abdf90162266c70bc5c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }" loading="lazy"> is equipped with a flat <a href="Euclidean_metric" class="mw-redirect" title="Euclidean metric">Euclidean metric</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{\mu }}">

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

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</math><img src="./1be3e5cf12940b60f8569a9927ad28823277177a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.458ex; height:2.843ex;" alt="{\displaystyle \partial _{\mu }}" loading="lazy"> is the <a href="Partial_derivative" title="Partial derivative">partial derivative</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {K}}}">

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

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</math><img src="./3a70fc5d5ef4fa8ce694447bef39c1aa167a68b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.771ex; height:2.176ex;" alt="{\displaystyle {\mathcal {K}}}" loading="lazy"> is the <a href="Killing_form" title="Killing form">Killing form</a> on the <a href="Lie_algebra" title="Lie algebra">Lie algebra</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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

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</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">. The Wess–Zumino term of the action is

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<annotation encoding="application/x-tex">{\displaystyle S^{\mathrm {W} Z}(\gamma )=-{\frac {1}{48\pi ^{2}}}\int _{\mathbf {B} ^{3}}d^{3}y\,\epsilon ^{ijk}{\mathcal {K}}\left(\gamma ^{-1}\partial _{i}\gamma ,\left[\gamma ^{-1}\partial _{j}\gamma ,\gamma ^{-1}\partial _{k}\gamma \right]\right).}</annotation>

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</math><img src="./00f3879460f9f931086e0ca0a76edf41fc3cd37f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:60.182ex; height:5.676ex;" alt="{\displaystyle S^{\mathrm {W} Z}(\gamma )=-{\frac {1}{48\pi ^{2}}}\int _{\mathbf {B} ^{3}}d^{3}y\,\epsilon ^{ijk}{\mathcal {K}}\left(\gamma ^{-1}\partial _{i}\gamma ,\left[\gamma ^{-1}\partial _{j}\gamma ,\gamma ^{-1}\partial _{k}\gamma \right]\right).}" loading="lazy"></dd></dl>

Here <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon ^{ijk}}">

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<annotation encoding="application/x-tex">{\displaystyle \epsilon ^{ijk}}</annotation>

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</math><img src="./30755a5e0b316fe5a4c512456e152ef93c636f44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.278ex; height:2.676ex;" alt="{\displaystyle \epsilon ^{ijk}}" loading="lazy"> is the <a href="Completely_anti-symmetric_tensor" class="mw-redirect" title="Completely anti-symmetric tensor">completely anti-symmetric tensor</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [.,.]}">

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<annotation encoding="application/x-tex">{\displaystyle [.,.]}</annotation>

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</math><img src="./740650be18c169f2d8f2d47ff04a721f726da613.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.395ex; height:2.843ex;" alt="{\displaystyle [.,.]}" loading="lazy"> is the <a href="Lie_bracket" class="mw-redirect" title="Lie bracket">Lie bracket</a>.

The Wess–Zumino term is an integral over a three-dimensional manifold <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} ^{3}}">

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

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</math><img src="./a1165a71eb37c23733ec3c2ad3938ed7a97e8347.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.955ex; height:2.676ex;" alt="{\displaystyle \mathbf {B} ^{3}}" loading="lazy"> whose boundary is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial \mathbf {B} ^{3}=\Sigma }">

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<annotation encoding="application/x-tex">{\displaystyle \partial \mathbf {B} ^{3}=\Sigma }</annotation>

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</math><img src="./6f7fa95cea15aa17c3b2e74573b99b9ffac9c648.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.05ex; height:2.676ex;" alt="{\displaystyle \partial \mathbf {B} ^{3}=\Sigma }" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Topological_properties_of_the_Wess–Zumino_term">Topological properties of the Wess–Zumino term</h3></div>

For the Wess–Zumino term to make sense, we need the field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

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

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</math><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"> to have an extension to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} ^{3}}">

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

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</math><img src="./a1165a71eb37c23733ec3c2ad3938ed7a97e8347.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.955ex; height:2.676ex;" alt="{\displaystyle \mathbf {B} ^{3}}" loading="lazy">. This requires the <a href="Homotopy_group" title="Homotopy group">homotopy group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{2}(G)}">

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<annotation encoding="application/x-tex">{\displaystyle \pi _{2}(G)}</annotation>

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</math><img src="./fe10c5b499df91223c1aaa64177ac535084e1615.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.015ex; height:2.843ex;" alt="{\displaystyle \pi _{2}(G)}" loading="lazy"> to be trivial, which is the case in particular for any compact Lie group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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

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</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">.

The extension of a given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma :\Sigma \to G}">

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<annotation encoding="application/x-tex">{\displaystyle \gamma :\Sigma \to G}</annotation>

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</math><img src="./a3b516f6fe4da7a04f0ca9c89cd6e4098ed5ba17.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.318ex; height:2.676ex;" alt="{\displaystyle \gamma :\Sigma \to G}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} ^{3}}">

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

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</math><img src="./a1165a71eb37c23733ec3c2ad3938ed7a97e8347.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.955ex; height:2.676ex;" alt="{\displaystyle \mathbf {B} ^{3}}" loading="lazy"> is in general not unique.

For the WZW model to be well-defined, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{iS_{k}(\gamma )}}">

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<annotation encoding="application/x-tex">{\displaystyle e^{iS_{k}(\gamma )}}</annotation>

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</math><img src="./3f43f8c5206e5816a5dfcf196dce7a51387dbe10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.922ex; height:2.843ex;" alt="{\displaystyle e^{iS_{k}(\gamma )}}" loading="lazy">

should not depend on the choice of the extension.

The Wess–Zumino term is invariant under small deformations of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

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

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</math><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy">, and only depends on its <a href="Homotopy_class" class="mw-redirect" title="Homotopy class">homotopy class</a>.

Possible homotopy classes are controlled by the homotopy group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{3}(G)}">

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

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</math><img src="./24dcaee34ef886ad184bc8eb9c120a7e0e17e45f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.015ex; height:2.843ex;" alt="{\displaystyle \pi _{3}(G)}" loading="lazy">.

For any compact, connected simple Lie group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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

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</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">, we have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{3}(G)=\mathbb {Z} }">

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

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</math><img src="./82ed7f998685c6129dde34c2b88cd17fa24c9ac6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.664ex; height:2.843ex;" alt="{\displaystyle \pi _{3}(G)=\mathbb {Z} }" loading="lazy">, and different extensions of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

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

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</math><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"> lead to values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{\mathrm {W} Z}(\gamma )}">

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<annotation encoding="application/x-tex">{\displaystyle S^{\mathrm {W} Z}(\gamma )}</annotation>

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</math><img src="./bd3e465af96db80e4eebcd27bbe47b2c8166e2e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.703ex; height:3.176ex;" alt="{\displaystyle S^{\mathrm {W} Z}(\gamma )}" loading="lazy"> that differ by integers. Therefore, they lead to the same value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{iS_{k}(\gamma )}}">

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<annotation encoding="application/x-tex">{\displaystyle e^{iS_{k}(\gamma )}}</annotation>

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</math><img src="./3f43f8c5206e5816a5dfcf196dce7a51387dbe10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.922ex; height:2.843ex;" alt="{\displaystyle e^{iS_{k}(\gamma )}}" loading="lazy"> provided the level obeys

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

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</math><img src="./0829895a7f6a03b7e80aac56c7df6e277fd09cdb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.249ex; height:2.176ex;" alt="{\displaystyle k\in \mathbb {Z} .}" loading="lazy"></dd></dl>

Integer values of the level also play an important role in the representation theory of the model's symmetry algebra, which is an <a href="Affine_Lie_algebra" title="Affine Lie algebra">affine Lie algebra</a>. If the level is a positive integer, the affine Lie algebra has unitary highest weight <a href="Representation_theory" title="Representation theory">representations</a> with highest <a href="Weight_(representation_theory)" title="Weight (representation theory)">weights</a> that are dominant integral. Such representations decompose into finite-dimensional subrepresentations with respect to the subalgebras spanned by each <a href="Simple_root_(root_system)" class="mw-redirect" title="Simple root (root system)">simple root</a>, the corresponding negative root and their commutator, which is a <a href="Cartan_matrix" title="Cartan matrix">Cartan generator</a>.

In the case of the noncompact simple Lie group <a href="SL2(R)" title="SL2(R)"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {SL} (2,\mathbb {R} )}">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./01c9c751fd8a5d9b5bdd3ac0a05e9eaf8f3fe57f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.429ex; height:2.843ex;" alt="{\displaystyle \mathrm {SL} (2,\mathbb {R} )}" loading="lazy"></a>,

the homotopy group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{3}(\mathrm {SL} (2,\mathbb {R} ))}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi _{3}(\mathrm {SL} (2,\mathbb {R} ))}</annotation>

</semantics>

</math><img src="./306257b1f56487708f80749bd5eebec0a15ca1db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.618ex; height:2.843ex;" alt="{\displaystyle \pi _{3}(\mathrm {SL} (2,\mathbb {R} ))}" loading="lazy"> is trivial, and the level is not constrained to be an integer.<a href="#cite_note-MO-6">[6]</a>

<div class="mw-heading mw-heading3"><h3 id="Geometrical_interpretation_of_the_Wess–Zumino_term">Geometrical interpretation of the Wess–Zumino term</h3></div>

If ea are the basis vectors for the <a href="Lie_algebra" title="Lie algebra">Lie algebra</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {K}}(e_{a},[e_{b},e_{c}])}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {K}}(e_{a},[e_{b},e_{c}])}</annotation>

</semantics>

</math><img src="./5d62bc8d07365a71cd38c8888e48b6b37dfaa946.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.176ex; height:2.843ex;" alt="{\displaystyle {\mathcal {K}}(e_{a},[e_{b},e_{c}])}" loading="lazy"> are the <a href="Structure_constants" title="Structure constants">structure constants</a> of the Lie algebra. The structure constants are completely anti-symmetric, and thus they define a <a href="Alternating_form" class="mw-redirect" title="Alternating form">3-form</a> on the <a href="Group_manifold" class="mw-redirect" title="Group manifold">group manifold</a> of G. Thus, the integrand above is just the <a href="Pullback_(differential_geometry)" title="Pullback (differential geometry)">pullback</a> of the harmonic 3-form to the ball <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} ^{3}.}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./0c0342f0c0701a2c69a220754510bb017c9084ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.602ex; height:2.676ex;" alt="{\displaystyle \mathbf {B} ^{3}.}" loading="lazy"> Denoting the harmonic 3-form by c and the pullback by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ^{*},}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma ^{*},}</annotation>

</semantics>

</math><img src="./341d41b9612f6cf49a06e867a045a1aa25e5d2a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.981ex; height:2.843ex;" alt="{\displaystyle \gamma ^{*},}" loading="lazy"> one then has

<msup>

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</mrow>

</msup>

<msub>

<msup>

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</mrow>

</msup>

</mrow>

</msub>

<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle S^{\mathrm {W} Z}(\gamma )=\int _{\mathbf {B} ^{3}}\gamma ^{*}c.}</annotation>

</semantics>

</math><img src="./867be3c46dc3d609760bb1d82c0fd7a2e4b323e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.877ex; height:5.676ex;" alt="{\displaystyle S^{\mathrm {W} Z}(\gamma )=\int _{\mathbf {B} ^{3}}\gamma ^{*}c.}" loading="lazy"></dd></dl>

This form leads directly to a topological analysis of the WZ term.

Geometrically, this term describes the <a href="Torsion_tensor" title="Torsion tensor">torsion</a> of the respective manifold.<a href="#cite_note-7">[7]</a> The presence of this torsion compels <a href="Teleparallelism" title="Teleparallelism">teleparallelism</a> of the manifold, and thus trivialization of the torsionful <a href="Affine_connection" title="Affine connection">curvature tensor</a>; and hence arrest of the renormalization flow, an <a href="Infrared_fixed_point" title="Infrared fixed point">infrared fixed point</a> of the <a href="Renormalization_group" title="Renormalization group">renormalization group</a>, a phenomenon termed geometrostasis.

<div class="mw-heading mw-heading2"><h2 id="Symmetry_algebra">Symmetry algebra</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Generalised_group_symmetry">Generalised group symmetry</h3></div>

The Wess–Zumino–Witten model is not only symmetric under global transformations by a group element in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">, but also has a much richer symmetry. This symmetry is often called the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(z)\times G({\bar {z}})}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G(z)\times G({\bar {z}})}</annotation>

</semantics>

</math><img src="./10193091e2f80006b87b8155e2fa66dc8bb19828.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.497ex; height:2.843ex;" alt="{\displaystyle G(z)\times G({\bar {z}})}" loading="lazy"> symmetry.<a href="#cite_note-8">[8]</a> Namely, given any holomorphic <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">-valued function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega (z)}">

</mstyle>

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

</semantics>

</math><img src="./e6338d81808063184f94f19751949039392332a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.576ex; height:2.843ex;" alt="{\displaystyle \Omega (z)}" loading="lazy">, and any other (completely independent of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega (z)}">

</mstyle>

</mrow>

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

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</math><img src="./e6338d81808063184f94f19751949039392332a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.576ex; height:2.843ex;" alt="{\displaystyle \Omega (z)}" loading="lazy">) antiholomorphic <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">-valued function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\Omega }}({\bar {z}})}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bar {\Omega }}({\bar {z}})}</annotation>

</semantics>

</math><img src="./c463500353c9b9e92bb43d3a66752140e785b0ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.784ex; height:3.176ex;" alt="{\displaystyle {\bar {\Omega }}({\bar {z}})}" loading="lazy">, where we have identified <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=x+iy}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z=x+iy}</annotation>

</semantics>

</math><img src="./08e90bb6b36fef59c6113eed2a08f10d77240741.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.315ex; height:2.509ex;" alt="{\displaystyle z=x+iy}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {z}}=x-iy}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bar {z}}=x-iy}</annotation>

</semantics>

</math><img src="./8f3fcfb169c13babe58e60f75cbc06c185193dfa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.523ex; height:2.509ex;" alt="{\displaystyle {\bar {z}}=x-iy}" loading="lazy"> in terms of the Euclidean space coordinates <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./5ea0abffd33a692ded22accc104515a032851dff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.519ex; height:2.009ex;" alt="{\displaystyle x,y}" loading="lazy">, the following symmetry holds:

<msub>

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<msub>

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</msub>

<msup>

<mover>

</mover>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle S_{k}(\gamma )=S_{k}(\Omega \gamma {\bar {\Omega }}^{-1})}</annotation>

</semantics>

</math><img src="./88f9a577483845c268412c3d19399f3ebdda2402.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.958ex; height:3.509ex;" alt="{\displaystyle S_{k}(\gamma )=S_{k}(\Omega \gamma {\bar {\Omega }}^{-1})}" loading="lazy"></dd></dl>

One way to prove the existence of this symmetry is through repeated application of the Polyakov–Wiegmann identity regarding products of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">-valued fields:

<msub>

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</msub>

<msup>

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</msup>

<msub>

</mrow>

</msub>

<msub>

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<msup>

</mrow>

</msup>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

<msub>

<mover>

</mover>

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<msup>

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<msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle S_{k}(\alpha \beta ^{-1})=S_{k}(\alpha )+S_{k}(\beta ^{-1})+{\frac {k}{16\pi ^{2}}}\int d^{2}x{\textrm {Tr}}(\alpha ^{-1}\partial _{\bar {z}}\alpha \beta ^{-1}\partial _{z}\beta )}</annotation>

</semantics>

</math><img src="./71b922924ed522ce6c717b101077817ae2397e4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:63.548ex; height:5.843ex;" alt="{\displaystyle S_{k}(\alpha \beta ^{-1})=S_{k}(\alpha )+S_{k}(\beta ^{-1})+{\frac {k}{16\pi ^{2}}}\int d^{2}x{\textrm {Tr}}(\alpha ^{-1}\partial _{\bar {z}}\alpha \beta ^{-1}\partial _{z}\beta )}" loading="lazy"></dd></dl>

The holomorphic and anti-holomorphic currents <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J(z)=-{\frac {1}{2}}k(\partial _{z}\gamma )\gamma ^{-1}}">

<mfrac>

</mfrac>

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<msub>

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</msub>

<msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle J(z)=-{\frac {1}{2}}k(\partial _{z}\gamma )\gamma ^{-1}}</annotation>

</semantics>

</math><img src="./7fff63c29db1c45d2eef5ea849a5101cf2771562.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.406ex; height:5.176ex;" alt="{\displaystyle J(z)=-{\frac {1}{2}}k(\partial _{z}\gamma )\gamma ^{-1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {J}}({\bar {z}})=-{\frac {1}{2}}k\gamma ^{-1}\partial _{\bar {z}}\gamma }">

<mover>

</mover>

</mrow>

<mover>

</mover>

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<mfrac>

</mfrac>

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<msup>

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</msup>

<msub>

<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\bar {J}}({\bar {z}})=-{\frac {1}{2}}k\gamma ^{-1}\partial _{\bar {z}}\gamma }</annotation>

</semantics>

</math><img src="./9b036c00d354d85ac6237169bd4d2a2020406375.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.341ex; height:5.176ex;" alt="{\displaystyle {\bar {J}}({\bar {z}})=-{\frac {1}{2}}k\gamma ^{-1}\partial _{\bar {z}}\gamma }" loading="lazy"> are the conserved currents associated with this symmetry. The singular behaviour of the products of these currents with other quantum fields determine how those fields transform under infinitesimal actions of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(z)\times G({\bar {z}})}">

<mover>

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<annotation encoding="application/x-tex">{\displaystyle G(z)\times G({\bar {z}})}</annotation>

</semantics>

</math><img src="./10193091e2f80006b87b8155e2fa66dc8bb19828.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.497ex; height:2.843ex;" alt="{\displaystyle G(z)\times G({\bar {z}})}" loading="lazy"> group.

<div class="mw-heading mw-heading3"><h3 id="Affine_Lie_algebra">Affine Lie algebra</h3></div>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

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

</semantics>

</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> be a local complex coordinate on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }">

</mstyle>

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

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</math><img src="./9e1f558f53cda207614abdf90162266c70bc5c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }" loading="lazy">,

<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle \{t^{a}\}}</annotation>

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</math><img src="./fbcc7af85569387597b06e847ba373dd0000875a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.266ex; height:2.843ex;" alt="{\displaystyle \{t^{a}\}}" loading="lazy"> an orthonormal basis (with respect to the <a href="Killing_form" title="Killing form">Killing form</a>) of the Lie algebra of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J^{a}(z)}">

<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle J^{a}(z)}</annotation>

</semantics>

</math><img src="./0252f914c3c5a7f3f64c4878d71d06bf4fc403e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.525ex; height:2.843ex;" alt="{\displaystyle J^{a}(z)}" loading="lazy"> the quantization of the field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {K}}(t^{a},\partial _{z}gg^{-1})}">

</mrow>

<msup>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {K}}(t^{a},\partial _{z}gg^{-1})}</annotation>

</semantics>

</math><img src="./60f0288d26b8c1e6ca5a9a4b7993accb55df7a7d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.359ex; height:3.176ex;" alt="{\displaystyle {\mathcal {K}}(t^{a},\partial _{z}gg^{-1})}" loading="lazy">. We have the following <a href="Operator_product_expansion" title="Operator product expansion">operator product expansion</a>:

<msup>

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</msup>

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</msup>

<mfrac>

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<mfrac>

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<mrow>

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<annotation encoding="application/x-tex">{\displaystyle J^{a}(z)J^{b}(w)={\frac {k\delta ^{ab}}{(z-w)^{2}}}+{\frac {if_{c}^{ab}J^{c}(w)}{z-w}}+{\mathcal {O}}(1),}</annotation>

</semantics>

</math><img src="./d9b223e36ee83996149122cc8670e15e161413d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:45.712ex; height:6.676ex;" alt="{\displaystyle J^{a}(z)J^{b}(w)={\frac {k\delta ^{ab}}{(z-w)^{2}}}+{\frac {if_{c}^{ab}J^{c}(w)}{z-w}}+{\mathcal {O}}(1),}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{c}^{ab}}">

</mrow>

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</msubsup>

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<annotation encoding="application/x-tex">{\displaystyle f_{c}^{ab}}</annotation>

</semantics>

</math><img src="./31299f25425314d0d94503bb6d9ba01c0766ee74.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.128ex; height:2.843ex;" alt="{\displaystyle f_{c}^{ab}}" loading="lazy"> are the coefficients such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [t^{a},t^{b}]=f_{c}^{ab}t^{c}}">

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</msubsup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [t^{a},t^{b}]=f_{c}^{ab}t^{c}}</annotation>

</semantics>

</math><img src="./93454dbc35780a77205abbd421248a7a4700f20c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.056ex; height:3.176ex;" alt="{\displaystyle [t^{a},t^{b}]=f_{c}^{ab}t^{c}}" loading="lazy">.

Equivalently, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J^{a}(z)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J^{a}(z)}</annotation>

</semantics>

</math><img src="./0252f914c3c5a7f3f64c4878d71d06bf4fc403e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.525ex; height:2.843ex;" alt="{\displaystyle J^{a}(z)}" loading="lazy"> is expanded in modes

<msup>

</mrow>

</msup>

</mrow>

</munder>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J^{a}(z)=\sum _{n\in \mathbb {Z} }J_{n}^{a}z^{-n-1},}</annotation>

</semantics>

</math><img src="./9ae0d8aadac2540fa7d92731f746cf0fb289faf8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:21.328ex; height:5.676ex;" alt="{\displaystyle J^{a}(z)=\sum _{n\in \mathbb {Z} }J_{n}^{a}z^{-n-1},}" loading="lazy"></dd></dl>

then the <a href="Current_algebra" title="Current algebra">current algebra</a> generated by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{J_{n}^{a}\}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{J_{n}^{a}\}}</annotation>

</semantics>

</math><img src="./bddae7b86913f36d51cc09aa12353cc8a88af960.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.953ex; height:2.843ex;" alt="{\displaystyle \{J_{n}^{a}\}}" loading="lazy"> is the <a href="Affine_Lie_algebra" title="Affine Lie algebra">affine Lie algebra</a> associated to the Lie algebra of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">, with a level that coincides with the level <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> of the WZW model.<a href="#cite_note-BYB-5">[5]</a>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}=\mathrm {Lie} (G)}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}=\mathrm {Lie} (G)}</annotation>

</semantics>

</math><img src="./8898b136626dc8be23c3b471f7d803b9b47606a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.038ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {g}}=\mathrm {Lie} (G)}" loading="lazy">, the notation for the affine Lie algebra is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathfrak {g}}}_{k}}">

<msub>

<mover>

</mrow>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {\mathfrak {g}}}_{k}}</annotation>

</semantics>

</math><img src="./c51811f22205fb0162d57c2af300c3f4caee228b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.26ex; height:2.843ex;" alt="{\displaystyle {\hat {\mathfrak {g}}}_{k}}" loading="lazy">.

The commutation relations of the affine Lie algebra are

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [J_{n}^{a},J_{m}^{b}]=f_{c}^{ab}J_{m+n}^{c}+kn\delta ^{ab}\delta _{n+m,0}.}</annotation>

</semantics>

</math><img src="./766ddc1931a1b60f9030d2a4f2937229b0c5cc65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.582ex; height:3.343ex;" alt="{\displaystyle [J_{n}^{a},J_{m}^{b}]=f_{c}^{ab}J_{m+n}^{c}+kn\delta ^{ab}\delta _{n+m,0}.}" loading="lazy"></dd></dl>

This affine Lie algebra is the chiral symmetry algebra associated to the left-moving currents <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {K}}(t^{a},\partial _{z}gg^{-1})}">

</mrow>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {K}}(t^{a},\partial _{z}gg^{-1})}</annotation>

</semantics>

</math><img src="./60f0288d26b8c1e6ca5a9a4b7993accb55df7a7d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.359ex; height:3.176ex;" alt="{\displaystyle {\mathcal {K}}(t^{a},\partial _{z}gg^{-1})}" loading="lazy">. A second copy of the same affine Lie algebra is associated to the right-moving currents <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {K}}(t^{a},g^{-1}\partial _{\bar {z}}g)}">

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msub>

<mover>

</mover>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {K}}(t^{a},g^{-1}\partial _{\bar {z}}g)}</annotation>

</semantics>

</math><img src="./590054c572d1ed212384a459cc76f5af3ec7de3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.544ex; height:3.176ex;" alt="{\displaystyle {\mathcal {K}}(t^{a},g^{-1}\partial _{\bar {z}}g)}" loading="lazy">. The generators <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {J}}^{a}(z)}">

<msup>

<mover>

</mover>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bar {J}}^{a}(z)}</annotation>

</semantics>

</math><img src="./9d395d1bc7a6017e376160348ba20bce6b59b944.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.822ex; height:3.176ex;" alt="{\displaystyle {\bar {J}}^{a}(z)}" loading="lazy"> of that second copy are antiholomorphic. The full symmetry algebra of the WZW model is the product of the two copies of the affine Lie algebra.

<div class="mw-heading mw-heading3"><h3 id="Sugawara_construction">Sugawara construction</h3></div>

The Sugawara construction is an embedding of the <a href="Virasoro_algebra" title="Virasoro algebra">Virasoro algebra</a> into the universal enveloping algebra of the affine Lie algebra. The existence of the embedding shows that WZW models are conformal field theories. Moreover, it leads to <a href="Knizhnik%E2%80%93Zamolodchikov_equations" title="Knizhnik–Zamolodchikov equations">Knizhnik–Zamolodchikov equations</a> for correlation functions.

The Sugawara construction is most concisely written at the level of the currents: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J^{a}(z)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J^{a}(z)}</annotation>

</semantics>

</math><img src="./0252f914c3c5a7f3f64c4878d71d06bf4fc403e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.525ex; height:2.843ex;" alt="{\displaystyle J^{a}(z)}" loading="lazy"> for the affine Lie algebra, and the <a href="Two-dimensional_conformal_field_theory#Energy–momentum_tensor" title="Two-dimensional conformal field theory">energy-momentum tensor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(z)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4749f82e035168e816434e3e3e7bf24e1c92e69b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.534ex; height:2.843ex;" alt="{\displaystyle T(z)}" loading="lazy"> for the Virasoro algebra:

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</munder>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T(z)={\frac {1}{2(k+h^{\vee })}}\sum _{a}:J^{a}J^{a}:(z),}</annotation>

</semantics>

</math><img src="./1b3abd300214987b2e8f281c0c2f25b3978d826a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.575ex; height:6.343ex;" alt="{\displaystyle T(z)={\frac {1}{2(k+h^{\vee })}}\sum _{a}:J^{a}J^{a}:(z),}" loading="lazy"></dd></dl>

where the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./cd064c6ce80ad9a8e53adebb7ad51b7635fceb0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.647ex; height:1.676ex;" alt="{\displaystyle :}" loading="lazy"> denotes normal ordering, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h^{\vee }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h^{\vee }}</annotation>

</semantics>

</math><img src="./ae135cb68f06593d8b0cdd837786b1869934c04c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.667ex; height:2.509ex;" alt="{\displaystyle h^{\vee }}" loading="lazy"> is the <a href="Coxeter_element" title="Coxeter element">dual Coxeter number</a>. By using the <a href="Operator_product_expansion" title="Operator product expansion">OPE</a> of the currents and a version of <a href="Wick's_theorem" title="Wick's theorem">Wick's theorem</a> one may deduce that the OPE of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(z)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4749f82e035168e816434e3e3e7bf24e1c92e69b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.534ex; height:2.843ex;" alt="{\displaystyle T(z)}" loading="lazy"> with itself is given by<a href="#cite_note-BYB-5">[5]</a>

<mfrac>

</mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T(y)T(z)={\frac {\frac {c}{2}}{(y-z)^{4}}}+{\frac {2T(z)}{(y-z)^{2}}}+{\frac {\partial T(z)}{y-z}}+{\mathcal {O}}(1),}</annotation>

</semantics>

</math><img src="./3e83df18b3d1f97c63ac25cdbc8f613a7443747d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:50.478ex; height:6.843ex;" alt="{\displaystyle T(y)T(z)={\frac {\frac {c}{2}}{(y-z)^{4}}}+{\frac {2T(z)}{(y-z)^{2}}}+{\frac {\partial T(z)}{y-z}}+{\mathcal {O}}(1),}" loading="lazy"></dd></dl>

which is equivalent to the Virasoro algebra's commutation relations. The central charge of the Virasoro algebra is given in terms of the level <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> of the affine Lie algebra by

<mfrac>

<mrow>

</mrow>

</mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c={\frac {k\mathrm {dim} ({\mathfrak {g}})}{k+h^{\vee }}}.}</annotation>

</semantics>

</math><img src="./2f73ab5f8defe0e40b5fc87bc57b33ff3e0d9382.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.656ex; height:6.009ex;" alt="{\displaystyle c={\frac {k\mathrm {dim} ({\mathfrak {g}})}{k+h^{\vee }}}.}" loading="lazy"></dd></dl>

At the level of the generators of the affine Lie algebra, the Sugawara construction reads

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</munder>

</mrow>

</munder>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{n\neq 0}={\frac {1}{2(k+h^{\vee })}}\sum _{a}\sum _{m\in \mathbb {Z} }J_{n-m}^{a}J_{m}^{a},}</annotation>

</semantics>

</math><img src="./30824e1bf879b50107d78008b864c53be1ad86b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:35.521ex; height:6.509ex;" alt="{\displaystyle L_{n\neq 0}={\frac {1}{2(k+h^{\vee })}}\sum _{a}\sum _{m\in \mathbb {Z} }J_{n-m}^{a}J_{m}^{a},}" loading="lazy"></dd>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

</mrow>

</munder>

</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{0}={\frac {1}{2(k+h^{\vee })}}\left(2\sum _{a}\sum _{m=1}^{\infty }J_{-m}^{a}J_{m}^{a}+J_{a}^{0}J_{a}^{0}\right).}</annotation>

</semantics>

</math><img src="./008ac5a92c6692104d6e24badd956928f9e44ee0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:45.796ex; height:7.509ex;" alt="{\displaystyle L_{0}={\frac {1}{2(k+h^{\vee })}}\left(2\sum _{a}\sum _{m=1}^{\infty }J_{-m}^{a}J_{m}^{a}+J_{a}^{0}J_{a}^{0}\right).}" loading="lazy"></dd></dl>

where the generators <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ebec334cb04f246db1139e2ca6be0b957d2ef520.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.801ex; height:2.509ex;" alt="{\displaystyle L_{n}}" loading="lazy"> of the Virasoro algebra are the modes of the energy-momentum tensor, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(z)=\sum _{n\in \mathbb {Z} }L_{n}z^{-n-2}}">

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T(z)=\sum _{n\in \mathbb {Z} }L_{n}z^{-n-2}}</annotation>

</semantics>

</math><img src="./1d49793bcfdd97258f3ff0201f278b5354f6f33c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:19.863ex; height:5.676ex;" alt="{\displaystyle T(z)=\sum _{n\in \mathbb {Z} }L_{n}z^{-n-2}}" loading="lazy">.

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

<div class="mw-heading mw-heading3"><h3 id="WZW_models_with_compact,_simply_connected_groups">WZW models with compact, simply connected groups</h3></div>

If the Lie group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> is compact and simply connected, then the WZW model is rational and diagonal: rational because the spectrum is built from a (level-dependent) finite set of irreducible representations of the affine Lie algebra called the integrable <a href="Highest_weight_representation" class="mw-redirect" title="Highest weight representation">highest weight representations</a>, and diagonal because a representation of the left-moving algebra is coupled with the same representation of the right-moving algebra.<a href="#cite_note-BYB-5">[5]</a>

For example, the spectrum of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle SU(2)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./27f8cd5de228a45abf34210c1666cd46dd87bc12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.254ex; height:2.843ex;" alt="{\displaystyle SU(2)}" loading="lazy"> WZW model at level <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {N} }</annotation>

</semantics>

</math><img src="./2a5bc4b7383031ba693b7433198ead7170954c1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.73ex; height:2.176ex;" alt="{\displaystyle k\in \mathbb {N} }" loading="lazy"> is

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</munder>

<msub>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mrow>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}_{k}=\bigoplus _{j=0,{\frac {1}{2}},1,\dots ,{\frac {k}{2}}}{\mathcal {R}}_{j}\otimes {\bar {\mathcal {R}}}_{j}\ ,}</annotation>

</semantics>

</math><img src="./ff271fd4b202761a83ea6b388ecdaa2bf9ed34cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:26.202ex; height:7.176ex;" alt="{\displaystyle {\mathcal {S}}_{k}=\bigoplus _{j=0,{\frac {1}{2}},1,\dots ,{\frac {k}{2}}}{\mathcal {R}}_{j}\otimes {\bar {\mathcal {R}}}_{j}\ ,}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {R}}_{j}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}_{j}}</annotation>

</semantics>

</math><img src="./eb6a74ae34fc1e7ffc1a13dbe2a5d8408bb35e2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.88ex; height:2.843ex;" alt="{\displaystyle {\mathcal {R}}_{j}}" loading="lazy"> is the affine highest weight representation of spin <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy">: a representation generated by a state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |v\rangle }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |v\rangle }</annotation>

</semantics>

</math><img src="./0a30e592038265dbec709de73cdb92e8cc55f6b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.679ex; height:2.843ex;" alt="{\displaystyle |v\rangle }" loading="lazy"> such that

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J_{n<0}^{a}|v\rangle =J_{0}^{-}|v\rangle =0\ ,}</annotation>

</semantics>

</math><img src="./5d5e01ba54f5a1594fb301943193dacb851da8c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.591ex; height:3.176ex;" alt="{\displaystyle J_{n<0}^{a}|v\rangle =J_{0}^{-}|v\rangle =0\ ,}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J^{-}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./02d0a4181e432dc8f77ff59cde3e7f9c2e889631.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.037ex; height:2.509ex;" alt="{\displaystyle J^{-}}" loading="lazy"> is the current that corresponds to a generator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t^{-}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./0c0ebc1bb1c2a855de8aea17c51245802c4e1258.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.35ex; height:2.509ex;" alt="{\displaystyle t^{-}}" loading="lazy"> of the Lie algebra of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle SU(2)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./27f8cd5de228a45abf34210c1666cd46dd87bc12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.254ex; height:2.843ex;" alt="{\displaystyle SU(2)}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="WZW_models_with_other_types_of_groups">WZW models with other types of groups</h3></div>

If the group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> is compact but not simply connected, the WZW model is rational but not necessarily diagonal. For example, the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle SO(3)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./16c677fee782e584fd417726201ce27c567f1e11.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.244ex; height:2.843ex;" alt="{\displaystyle SO(3)}" loading="lazy"> WZW model exists for even integer levels <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in 2\mathbb {N} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./602b457c28588f94d4368efbf3ac46c0d93ad994.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.892ex; height:2.176ex;" alt="{\displaystyle k\in 2\mathbb {N} }" loading="lazy">, and its spectrum is a non-diagonal combination of finitely many integrable highest weight representations.<a href="#cite_note-BYB-5">[5]</a>

If the group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> is not compact, the WZW model is non-rational. Moreover, its spectrum may include non highest weight representations. For example, the spectrum of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle SL(2,\mathbb {R} )}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2c45257f58499779a1f5994061fe78b18bbd8ab1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.766ex; height:2.843ex;" alt="{\displaystyle SL(2,\mathbb {R} )}" loading="lazy"> WZW model is built from highest weight representations, plus their images under the spectral flow automorphisms of the affine Lie algebra.<a href="#cite_note-MO-6">[6]</a>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> is a <a href="Supergroup_(physics)" title="Supergroup (physics)">supergroup</a>, the spectrum may involve representations that do not factorize as tensor products of representations of the left- and right-moving symmetry algebras. This occurs for example in the case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=GL(1|1)}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4a474325bd9947b3e13e15cb30e6c859e58471be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.116ex; height:2.843ex;" alt="{\displaystyle G=GL(1|1)}" loading="lazy">,<a href="#cite_note-9">[9]</a>

and also in more complicated supergroups such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=PSU(1,1|2)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G=PSU(1,1|2)}</annotation>

</semantics>

</math><img src="./9e33daeead7fb66685052f58199cd2f9926be664.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.93ex; height:2.843ex;" alt="{\displaystyle G=PSU(1,1|2)}" loading="lazy">.<a href="#cite_note-GQS-10">[10]</a>

Non-factorizable representations are responsible for the fact that the corresponding WZW models are <a href="Logarithmic_conformal_field_theory" title="Logarithmic conformal field theory">logarithmic conformal field theories</a>.

<div class="mw-heading mw-heading3"><h3 id="Other_theories_based_on_affine_Lie_algebras">Other theories based on affine Lie algebras</h3></div>

The known conformal field theories based on affine Lie algebras are not limited to WZW models.

For example, in the case of the affine Lie algebra of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle SU(2)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./27f8cd5de228a45abf34210c1666cd46dd87bc12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.254ex; height:2.843ex;" alt="{\displaystyle SU(2)}" loading="lazy"> WZW model, modular invariant torus partition functions obey an ADE classification, where the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle SU(2)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./27f8cd5de228a45abf34210c1666cd46dd87bc12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.254ex; height:2.843ex;" alt="{\displaystyle SU(2)}" loading="lazy"> WZW model accounts for the A series only.<a href="#cite_note-11">[11]</a> The D series corresponds to the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle SO(3)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./16c677fee782e584fd417726201ce27c567f1e11.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.244ex; height:2.843ex;" alt="{\displaystyle SO(3)}" loading="lazy"> WZW model, and the E series does not correspond to any WZW model.

Another example is the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{3}^{+}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./904324d13e3a73954ea2b3ddb4ba2b084822a40e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.614ex; height:3.176ex;" alt="{\displaystyle H_{3}^{+}}" loading="lazy"> model. This model is based on the same symmetry algebra as the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle SL(2,\mathbb {R} )}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2c45257f58499779a1f5994061fe78b18bbd8ab1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.766ex; height:2.843ex;" alt="{\displaystyle SL(2,\mathbb {R} )}" loading="lazy"> WZW model, to which it is related by Wick rotation. However, the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{3}^{+}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./904324d13e3a73954ea2b3ddb4ba2b084822a40e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.614ex; height:3.176ex;" alt="{\displaystyle H_{3}^{+}}" loading="lazy"> is not strictly speaking a WZW model, as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{3}^{+}=SL(2,\mathbb {C} )/SU(2)}">

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{3}^{+}=SL(2,\mathbb {C} )/SU(2)}</annotation>

</semantics>

</math><img src="./199499db762ddb13e500af5ae3f5a992402b37da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.895ex; height:3.176ex;" alt="{\displaystyle H_{3}^{+}=SL(2,\mathbb {C} )/SU(2)}" loading="lazy"> is not a group, but a coset.<a href="#cite_note-12">[12]</a>

<div class="mw-heading mw-heading2"><h2 id="Fields_and_correlation_functions">Fields and correlation functions</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Fields">Fields</h3></div>

Given a simple <a href="Lie_algebra_representation" title="Lie algebra representation">representation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> of the Lie algebra of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">, an affine primary field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi ^{\rho }(z)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi ^{\rho }(z)}</annotation>

</semantics>

</math><img src="./035f07a9cd1a9f6c7d2dfaa97345acf60a7c48e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.658ex; height:2.843ex;" alt="{\displaystyle \Phi ^{\rho }(z)}" loading="lazy"> is a field that takes values in the representation space of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy">, such that

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J^{a}(y)\Phi ^{\rho }(z)=-{\frac {\rho (t^{a})\Phi ^{\rho }(z)}{y-z}}+O(1)\ .}</annotation>

</semantics>

</math><img src="./99043ae54646ad7e291fa55569328085c8c00956.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:36.416ex; height:6.176ex;" alt="{\displaystyle J^{a}(y)\Phi ^{\rho }(z)=-{\frac {\rho (t^{a})\Phi ^{\rho }(z)}{y-z}}+O(1)\ .}" loading="lazy"></dd></dl>

An affine primary field is also a <a href="Primary_field" title="Primary field">primary field</a> for the Virasoro algebra that results from the Sugawara construction. The conformal dimension of the affine primary field is given in terms of the quadratic Casimir <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{2}(\rho )}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./1a721bd9618343958e7f7e4f4056c0c638d07e9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.727ex; height:2.843ex;" alt="{\displaystyle C_{2}(\rho )}" loading="lazy"> of the representation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> (i.e. the eigenvalue of the quadratic <a href="Casimir_element" title="Casimir element">Casimir element</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{ab}t^{a}t^{b}}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K_{ab}t^{a}t^{b}}</annotation>

</semantics>

</math><img src="./249be242649294c70b0ac1e9ddde0d5fb2e324f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.499ex; height:3.009ex;" alt="{\displaystyle K_{ab}t^{a}t^{b}}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{ab}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./3c99d9ba072964ac131b0e9d972710672e91e852.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.78ex; height:2.509ex;" alt="{\displaystyle K_{ab}}" loading="lazy"> is the inverse of the matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {K}}(t^{a},t^{b})}">

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {K}}(t^{a},t^{b})}</annotation>

</semantics>

</math><img src="./b9dcd4a6e879674100f90b8e342240f7c9019f0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.333ex; height:3.176ex;" alt="{\displaystyle {\mathcal {K}}(t^{a},t^{b})}" loading="lazy"> of the Killing form) by

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta _{\rho }={\frac {C_{2}(\rho )}{2(k+h^{\vee })}}\ .}</annotation>

</semantics>

</math><img src="./0abd5f007d867e629b1f8fb283c67a9df1530800.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.871ex; height:6.509ex;" alt="{\displaystyle \Delta _{\rho }={\frac {C_{2}(\rho )}{2(k+h^{\vee })}}\ .}" loading="lazy"></dd></dl>

For example, in the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle SU(2)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./27f8cd5de228a45abf34210c1666cd46dd87bc12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.254ex; height:2.843ex;" alt="{\displaystyle SU(2)}" loading="lazy"> WZW model, the conformal dimension of a primary field of <a href="Spin_(physics)" title="Spin (physics)">spin</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy"> is

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta _{j}={\frac {j(j+1)}{k+2}}\ .}</annotation>

</semantics>

</math><img src="./904fed0fbc73dbcab618598a519cd2f4022da357.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.736ex; height:6.009ex;" alt="{\displaystyle \Delta _{j}={\frac {j(j+1)}{k+2}}\ .}" loading="lazy"></dd></dl>

By the state-field correspondence, affine primary fields correspond to affine primary states, which are the highest weight states of <a href="Highest_weight_representation" class="mw-redirect" title="Highest weight representation">highest weight representations</a> of the affine Lie algebra.

<div class="mw-heading mw-heading3"><h3 id="Correlation_functions">Correlation functions</h3></div>

If the group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> is compact, the spectrum of the WZW model is made of highest weight representations, and all correlation functions can be deduced from correlation functions of affine primary fields via <a href="Ward_identities" class="mw-redirect" title="Ward identities">Ward identities</a>.

If the Riemann surface <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./9e1f558f53cda207614abdf90162266c70bc5c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }" loading="lazy"> is the Riemann sphere, correlation functions of affine primary fields obey <a href="Knizhnik%E2%80%93Zamolodchikov_equations" title="Knizhnik–Zamolodchikov equations">Knizhnik–Zamolodchikov equations</a>. On Riemann surfaces of higher genus, correlation functions obey Knizhnik–Zamolodchikov–Bernard equations, which involve derivatives not only of the fields' positions, but also of the surface's moduli.<a href="#cite_note-13">[13]</a>

<div class="mw-heading mw-heading2"><h2 id="Gauged_WZW_models">Gauged WZW models</h2></div>

Given a Lie subgroup <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\subset G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H\subset G}</annotation>

</semantics>

</math><img src="./15eb570a68da46b0458ff0ead693a82467ab8ddd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.989ex; height:2.176ex;" alt="{\displaystyle H\subset G}" loading="lazy">, the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G/H}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./21e7e9d6e3072ec8dd48200d755847154ea5d35c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.053ex; height:2.843ex;" alt="{\displaystyle G/H}" loading="lazy"> gauged WZW model (or coset model) is a nonlinear sigma model whose target space is the quotient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G/H}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./21e7e9d6e3072ec8dd48200d755847154ea5d35c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.053ex; height:2.843ex;" alt="{\displaystyle G/H}" loading="lazy"> for the <a href="Adjoint_action" class="mw-redirect" title="Adjoint action">adjoint action</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">. This gauged WZW model is a conformal field theory, whose symmetry algebra is a quotient of the two affine Lie algebras of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> WZW models, and whose central charge is the difference of their central charges.

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

The WZW model whose Lie group is the <a href="Universal_cover" class="mw-redirect" title="Universal cover">universal cover</a> of the group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {SL} (2,\mathbb {R} )}">

</mrow>

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</mrow>

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

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</math><img src="./01c9c751fd8a5d9b5bdd3ac0a05e9eaf8f3fe57f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.429ex; height:2.843ex;" alt="{\displaystyle \mathrm {SL} (2,\mathbb {R} )}" loading="lazy"> has been used by <a href="Juan_Maldacena" title="Juan Maldacena">Juan Maldacena</a> and <a href="Hirosi_Ooguri" title="Hirosi Ooguri">Hirosi Ooguri</a> to describe bosonic <a href="String_theory" title="String theory">string theory</a> on the three-dimensional <a href="Anti-de_Sitter_space" title="Anti-de Sitter space">anti-de Sitter space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AdS_{3}}">

<msub>

</mrow>

</msub>

</mstyle>

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

</semantics>

</math><img src="./883dda50cacc5a349d8a102d9492a82cd6dd37de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.438ex; height:2.509ex;" alt="{\displaystyle AdS_{3}}" loading="lazy">.<a href="#cite_note-MO-6">[6]</a> Superstrings on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AdS_{3}\times S^{3}}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

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

</semantics>

</math><img src="./e7ed5e9a01c6d1ee8a031788e86c1694bbba0520.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.854ex; height:3.009ex;" alt="{\displaystyle AdS_{3}\times S^{3}}" loading="lazy"> are described by the WZW model on the supergroup <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PSU(1,1|2)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle PSU(1,1|2)}</annotation>

</semantics>

</math><img src="./ff7edeaf7cec9ea8e613f97434da64690e280020.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.005ex; height:2.843ex;" alt="{\displaystyle PSU(1,1|2)}" loading="lazy">, or a deformation thereof if Ramond-Ramond flux is turned on.<a href="#cite_note-14">[14]</a><a href="#cite_note-GQS-10">[10]</a>

WZW models and their deformations have been proposed for describing

the plateau transition in the integer <a href="Quantum_Hall_effect" title="Quantum Hall effect">quantum Hall effect</a>.<a href="#cite_note-15">[15]</a>

The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle SL(2,\mathbb {R} )/U(1)}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle SL(2,\mathbb {R} )/U(1)}</annotation>

</semantics>

</math><img src="./d274daccddd8cc9a633b387b66120c9c45c09290.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.683ex; height:2.843ex;" alt="{\displaystyle SL(2,\mathbb {R} )/U(1)}" loading="lazy"> gauged WZW model has an interpretation in <a href="String_theory" title="String theory">string theory</a> as <a href="Edward_Witten" title="Edward Witten">Witten</a>'s two-dimensional Euclidean black hole.<a href="#cite_note-Witten1991-16">[16]</a>

The same model also describes certain two-dimensional statistical systems at criticality, such as the critical antiferromagnetic <a href="Potts_model" title="Potts model">Potts model</a>.<a href="#cite_note-17">[17]</a>

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

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<li id="cite_note-17"><a href="#cite_ref-17">^</a> N. Robertson, J. Jacobsen, H. Saleur, "Conformally invariant boundary conditions in the antiferromagnetic Potts model and the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle SL(2,\mathbb {R} )/U(1)}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle SL(2,\mathbb {R} )/U(1)}</annotation>

</semantics>

</math><img src="./d274daccddd8cc9a633b387b66120c9c45c09290.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.683ex; height:2.843ex;" alt="{\displaystyle SL(2,\mathbb {R} )/U(1)}" loading="lazy"> sigma model", <a rel="nofollow" class="external text" href="https://arxiv.org/abs/1906.07565">arXiv:1906.07565</a>

</li>

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

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</style><div id="Quantum_field_theories202" style="font-size:114%;margin:0 4em"><a href="Quantum_field_theory" title="Quantum field theory">Quantum field theories</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theories</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Algebraic_quantum_field_theory" title="Algebraic quantum field theory">Algebraic QFT</a></li>

<li><a href="Axiomatic_quantum_field_theory" title="Axiomatic quantum field theory">Axiomatic QFT</a></li>

<li><a href="Conformal_field_theory" title="Conformal field theory">Conformal field theory</a></li>

<li><a href="Lattice_field_theory" title="Lattice field theory">Lattice field theory</a></li>

<li><a href="Noncommutative_quantum_field_theory" title="Noncommutative quantum field theory">Noncommutative QFT</a></li>

<li><a href="Gauge_theory" title="Gauge theory">Gauge theory</a></li>

<li><a href="Quantum_field_theory_in_curved_spacetime" title="Quantum field theory in curved spacetime">QFT in curved spacetime</a></li>

<li><a href="String_theory" title="String theory">String theory</a></li>

<li><a href="Supergravity" title="Supergravity">Supergravity</a></li>

<li><a href="Thermal_quantum_field_theory" title="Thermal quantum field theory">Thermal QFT</a></li>

<li><a href="Topological_quantum_field_theory" title="Topological quantum field theory">Topological QFT</a></li>

<li><a href="Two-dimensional_conformal_field_theory" title="Two-dimensional conformal field theory">Two-dimensional conformal field theory</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Models</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Regular</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Born%E2%80%93Infeld_model" title="Born–Infeld model">Born–Infeld</a></li>

<li><a href="Euler%E2%80%93Heisenberg_Lagrangian" title="Euler–Heisenberg Lagrangian">Euler–Heisenberg</a></li>

<li><a href="Ginzburg%E2%80%93Landau_theory" title="Ginzburg–Landau theory">Ginzburg–Landau</a></li>

<li><a href="Non-linear_sigma_model" title="Non-linear sigma model">Non-linear sigma</a></li>

<li><a href="Proca_action" title="Proca action">Proca</a></li>

<li><a href="Quantum_electrodynamics" title="Quantum electrodynamics">Quantum electrodynamics</a></li>

<li><a href="Quantum_chromodynamics" title="Quantum chromodynamics">Quantum chromodynamics</a></li>

<li><a href="Quartic_interaction" title="Quartic interaction">Quartic interaction</a></li>

<li><a href="Scalar_electrodynamics" title="Scalar electrodynamics">Scalar electrodynamics</a></li>

<li><a href="Scalar_chromodynamics" title="Scalar chromodynamics">Scalar chromodynamics</a></li>

<li><a href="Soler_model" title="Soler model">Soler</a></li>

<li><a href="Yang%E2%80%93Mills_theory" title="Yang–Mills theory">Yang–Mills</a></li>

<li><a href="Yang%E2%80%93Mills%E2%80%93Higgs_equations" title="Yang–Mills–Higgs equations">Yang–Mills–Higgs</a></li>

<li><a href="Yukawa_interaction" class="mw-redirect" title="Yukawa interaction">Yukawa</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Low dimensional</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Two-dimensional_Yang%E2%80%93Mills_theory" title="Two-dimensional Yang–Mills theory">2D Yang–Mills</a></li>

<li><a href="Bullough%E2%80%93Dodd_model" title="Bullough–Dodd model">Bullough–Dodd</a></li>

<li><a href="Gross%E2%80%93Neveu_model" title="Gross–Neveu model">Gross–Neveu</a></li>

<li><a href="Schwinger_model" title="Schwinger model">Schwinger</a></li>

<li><a href="Sine-Gordon_equation" title="Sine-Gordon equation">Sine-Gordon</a></li>

<li><a href="Thirring_model" title="Thirring model">Thirring</a></li>

<li><a href="Thirring%E2%80%93Wess_model" title="Thirring–Wess model">Thirring–Wess</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Conformal</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Massless_free_scalar_bosons_in_two_dimensions" title="Massless free scalar bosons in two dimensions">2D free massless scalar</a></li>

<li><a href="Liouville_field_theory" title="Liouville field theory">Liouville</a></li>

<li><a href="Minimal_model_(physics)" title="Minimal model (physics)">Minimal</a></li>

<li><a href="Polyakov_action" title="Polyakov action">Polyakov</a></li>

</ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Supersymmetric</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="N_%3D_1_supersymmetric_Yang%E2%80%93Mills_theory" title="N = 1 supersymmetric Yang–Mills theory">N = 1 super Yang–Mills</a></li>

<li><a href="Seiberg%E2%80%93Witten_theory" title="Seiberg–Witten theory">Seiberg–Witten</a></li>

<li><a href="Super_QCD" title="Super QCD">Super QCD</a></li>

<li><a href="Wess%E2%80%93Zumino_model" title="Wess–Zumino model">Wess–Zumino</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Superconformal</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="N_%3D_4_supersymmetric_Yang%E2%80%93Mills_theory" title="N = 4 supersymmetric Yang–Mills theory">N = 4 super Yang–Mills</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Supergravity</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Higher-dimensional_supergravity" title="Higher-dimensional supergravity">Higher dimensional</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Topological</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Chern%E2%80%93Simons_theory" title="Chern–Simons theory">Chern–Simons</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Particle theory</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Chiral_model" title="Chiral model">Chiral</a></li>

<li><a href="Fermi's_interaction" title="Fermi's interaction">Fermi</a></li>

<li><a href="Nambu%E2%80%93Jona-Lasinio_model" title="Nambu–Jona-Lasinio model">Nambu–Jona-Lasinio</a></li>

<li><a href="Next-to-Minimal_Supersymmetric_Standard_Model" title="Next-to-Minimal Supersymmetric Standard Model">NMSSM</a></li>

<li><a href="Standard_Model" title="Standard Model">Standard Model</a></li>

<li><a href="Stueckelberg_action" title="Stueckelberg action">Stueckelberg</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Casimir_effect" title="Casimir effect">Casimir effect</a></li>

<li><a href="Cosmic_string" title="Cosmic string">Cosmic string</a></li>

<li><a href="History_of_quantum_field_theory" title="History of quantum field theory">History</a></li>

<li><a href="Loop_quantum_gravity" title="Loop quantum gravity">Loop quantum gravity</a></li>

<li><a href="Loop_quantum_cosmology" title="Loop quantum cosmology">Loop quantum cosmology</a></li>

<li><a href="On_shell_and_off_shell" title="On shell and off shell">On shell and off shell</a></li>

<li><a href="Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li>

<li><a href="Quantum_dynamics" title="Quantum dynamics">Quantum dynamics</a></li>

<li><a href="Quantum_foam" title="Quantum foam">Quantum foam</a></li>

<li><a href="Quantum_fluctuation" title="Quantum fluctuation">Quantum fluctuations</a>

<ul><li>links</li></ul></li>

<li><a href="Quantum_gravity" title="Quantum gravity">Quantum gravity</a>

<ul><li>links</li></ul></li>

<li><a href="Quantum_hadrodynamics" title="Quantum hadrodynamics">Quantum hadrodynamics</a></li>

<li><a href="Quantum_hydrodynamics" title="Quantum hydrodynamics">Quantum hydrodynamics</a></li>

<li><a href="Quantum_information" title="Quantum information">Quantum information</a></li>

<li><a href="Quantum_information_science" title="Quantum information science">Quantum information science</a>

<ul><li>links</li></ul></li>

<li><a href="Quantum_logic" title="Quantum logic">Quantum logic</a></li>

<li><a href="Quantum_thermodynamics" title="Quantum thermodynamics">Quantum thermodynamics</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>See also: Template:Quantum mechanics topics</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="String_theory190" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2" style="text-align:center;"><div id="String_theory190" style="font-size:114%;margin:0 4em"><a href="String_theory" title="String theory">String theory</a></div></th></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Background</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="String_(physics)" title="String (physics)">Strings</a></li>

<li><a href="Cosmic_string" title="Cosmic string">Cosmic strings</a></li>

<li><a href="History_of_string_theory" title="History of string theory">History of string theory</a>

<ul><li><a href="First_superstring_revolution" class="mw-redirect" title="First superstring revolution">First superstring revolution</a></li>

<li><a href="Second_superstring_revolution" class="mw-redirect" title="Second superstring revolution">Second superstring revolution</a></li></ul></li>

<li><a href="String_theory_landscape" title="String theory landscape">String theory landscape</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Theory</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Nambu%E2%80%93Goto_action" title="Nambu–Goto action">Nambu–Goto action</a></li>

<li><a href="Polyakov_action" title="Polyakov action">Polyakov action</a></li>

<li><a href="Bosonic_string_theory" title="Bosonic string theory">Bosonic string theory</a></li>

<li><a href="Superstring_theory" title="Superstring theory">Superstring theory</a>

<ul><li><a href="Type_I_string_theory" title="Type I string theory">Type I string</a></li>

<li><a href="Type_II_string_theory" title="Type II string theory">Type II string</a>

<ul><li><a href="Type_II_string_theory" title="Type II string theory">Type IIA string</a></li>

<li><a href="Type_II_string_theory" title="Type II string theory">Type IIB string</a></li></ul></li>

<li><a href="Heterotic_string_theory" title="Heterotic string theory">Heterotic string</a></li></ul></li>

<li><a href="N%3D2_superstring" class="mw-redirect" title="N=2 superstring">N=2 superstring</a></li>

<li><a href="F-theory" title="F-theory">F-theory</a></li>

<li><a href="String_field_theory" title="String field theory">String field theory</a></li>

<li><a href="Matrix_string_theory" title="Matrix string theory">Matrix string theory</a></li>

<li><a href="Non-critical_string_theory" title="Non-critical string theory">Non-critical string theory</a></li>

<li><a href="Non-linear_sigma_model" title="Non-linear sigma model">Non-linear sigma model</a></li>

<li><a href="Tachyon_condensation" title="Tachyon condensation">Tachyon condensation</a></li>

<li><a href="RNS_formalism" title="RNS formalism">RNS formalism</a></li>

<li><a href="GS_formalism" title="GS formalism">GS formalism</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="String_duality" title="String duality">String duality</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="T-duality" title="T-duality">T-duality</a></li>

<li><a href="S-duality" title="S-duality">S-duality</a></li>

<li><a href="U-duality" title="U-duality">U-duality</a></li>

<li><a href="Montonen%E2%80%93Olive_duality" title="Montonen–Olive duality">Montonen–Olive duality</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Particles and fields</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

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

<li><a href="Dilaton" title="Dilaton">Dilaton</a></li>

<li><a href="Tachyon" title="Tachyon">Tachyon</a></li>

<li><a href="Ramond%E2%80%93Ramond_field" title="Ramond–Ramond field">Ramond–Ramond field</a></li>

<li><a href="Kalb%E2%80%93Ramond_field" title="Kalb–Ramond field">Kalb–Ramond field</a></li>

<li><a href="Magnetic_monopole" title="Magnetic monopole">Magnetic monopole</a></li>

<li><a href="Dual_graviton" title="Dual graviton">Dual graviton</a></li>

<li><a href="Dual_photon" title="Dual photon">Dual photon</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="Brane" title="Brane">Branes</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="D-brane" title="D-brane">D-brane</a></li>

<li><a href="NS5-brane" title="NS5-brane">NS5-brane</a></li>

<li><a href="M2-brane" title="M2-brane">M2-brane</a></li>

<li><a href="M5-brane" title="M5-brane">M5-brane</a></li>

<li><a href="S-brane" title="S-brane">S-brane</a></li>

<li><a href="Black_brane" title="Black brane">Black brane</a></li>

<li><a href="Black_hole" title="Black hole">Black holes</a></li>

<li><a href="Black_string" class="mw-redirect" title="Black string">Black string</a></li>

<li><a href="Brane_cosmology" title="Brane cosmology">Brane cosmology</a></li>

<li><a href="Quiver_diagram" title="Quiver diagram">Quiver diagram</a></li>

<li><a href="Hanany%E2%80%93Witten_transition" title="Hanany–Witten transition">Hanany–Witten transition</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="Conformal_field_theory" title="Conformal field theory">Conformal field theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Virasoro_algebra" title="Virasoro algebra">Virasoro algebra</a></li>

<li><a href="Mirror_symmetry_(string_theory)" title="Mirror symmetry (string theory)">Mirror symmetry</a></li>

<li><a href="Conformal_anomaly" title="Conformal anomaly">Conformal anomaly</a></li>

<li><a href="Conformal_symmetry" title="Conformal symmetry">Conformal algebra</a></li>

<li><a href="Superconformal_algebra" title="Superconformal algebra">Superconformal algebra</a></li>

<li><a href="Vertex_operator_algebra" title="Vertex operator algebra">Vertex operator algebra</a></li>

<li><a href="Loop_algebra" title="Loop algebra">Loop algebra</a></li>

<li><a href="Kac%E2%80%93Moody_algebra" title="Kac–Moody algebra">Kac–Moody algebra</a></li>

</ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="Gauge_theory" title="Gauge theory">Gauge theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Anomaly_(physics)" title="Anomaly (physics)">Anomalies</a></li>

<li><a href="Instanton" title="Instanton">Instantons</a></li>

<li><a href="Chern%E2%80%93Simons_form" title="Chern–Simons form">Chern–Simons form</a></li>

<li><a href="Bogomol'nyi%E2%80%93Prasad%E2%80%93Sommerfield_bound" title="Bogomol'nyi–Prasad–Sommerfield bound">Bogomol'nyi–Prasad–Sommerfield bound</a></li>

<li><a href="Exceptional_Lie_group" class="mw-redirect" title="Exceptional Lie group">Exceptional Lie groups</a> (<a href="G2_(mathematics)" title="G2 (mathematics)">G2</a>, <a href="F4_(mathematics)" title="F4 (mathematics)">F4</a>, <a href="E6_(mathematics)" title="E6 (mathematics)">E6</a>, <a href="E7_(mathematics)" title="E7 (mathematics)">E7</a>, <a href="E8_(mathematics)" title="E8 (mathematics)">E8</a>)</li>

<li><a href="ADE_classification" title="ADE classification">ADE classification</a></li>

<li><a href="Dirac_string" title="Dirac string">Dirac string</a></li>

<li><a href="P-form_electrodynamics" title="P-form electrodynamics">p-form electrodynamics</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Geometry</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

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

<li><a href="Kaluza%E2%80%93Klein_theory" title="Kaluza–Klein theory">Kaluza–Klein theory</a></li>

<li><a href="Compactification_(physics)" title="Compactification (physics)">Compactification</a></li>

<li><a href="Why_10_dimensions" class="mw-redirect" title="Why 10 dimensions">Why 10 dimensions</a>?</li>

<li><a href="K%C3%A4hler_manifold" title="Kähler manifold">Kähler manifold</a></li>

<li><a href="Ricci-flat_manifold" title="Ricci-flat manifold">Ricci-flat manifold</a>

<ul><li><a href="Calabi%E2%80%93Yau_manifold" title="Calabi–Yau manifold">Calabi–Yau manifold</a></li>

<li><a href="Hyperk%C3%A4hler_manifold" title="Hyperkähler manifold">Hyperkähler manifold</a>

<ul><li><a href="K3_surface" title="K3 surface">K3 surface</a></li></ul></li>

<li><a href="G2_manifold" title="G2 manifold">G2 manifold</a></li>

<li><a href="Spin(7)-manifold" title="Spin(7)-manifold">Spin(7)-manifold</a></li></ul></li>

<li><a href="Generalized_complex_structure" title="Generalized complex structure">Generalized complex manifold</a></li>

<li><a href="Orbifold" title="Orbifold">Orbifold</a></li>

<li><a href="Conifold" title="Conifold">Conifold</a></li>

<li><a href="Orientifold" title="Orientifold">Orientifold</a></li>

<li><a href="Moduli_space" title="Moduli space">Moduli space</a></li>

<li><a href="Ho%C5%99ava%E2%80%93Witten_theory" title="Hořava–Witten theory">Hořava–Witten theory</a></li>

<li><a href="K-theory_(physics)" title="K-theory (physics)">K-theory (physics)</a></li>

<li><a href="Twisted_K-theory" title="Twisted K-theory">Twisted K-theory</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="Supersymmetry" title="Supersymmetry">Supersymmetry</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

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

<li><a href="Eleven-dimensional_supergravity" title="Eleven-dimensional supergravity">Eleven-dimensional supergravity</a></li>

<li><a href="Type_I_supergravity" title="Type I supergravity">Type I supergravity</a></li>

<li><a href="Type_IIA_supergravity" title="Type IIA supergravity">Type IIA supergravity</a></li>

<li><a href="Type_IIB_supergravity" title="Type IIB supergravity">Type IIB supergravity</a></li>

<li><a href="Superspace" title="Superspace">Superspace</a></li>

<li><a href="Lie_superalgebra" title="Lie superalgebra">Lie superalgebra</a></li>

<li><a href="Lie_supergroup" class="mw-redirect" title="Lie supergroup">Lie supergroup</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="Holography" title="Holography">Holography</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Holographic_principle" title="Holographic principle">Holographic principle</a></li>

<li><a href="AdS/CFT_correspondence" title="AdS/CFT correspondence">AdS/CFT correspondence</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="M-theory" title="M-theory">M-theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Matrix_theory_(physics)" title="Matrix theory (physics)">Matrix theory</a></li>

<li><a href="Introduction_to_M-theory" title="Introduction to M-theory">Introduction to M-theory</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">String theorists</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Mina_Aganagi%C4%87" title="Mina Aganagić">Aganagić</a></li>

<li><a href="Nima_Arkani-Hamed" title="Nima Arkani-Hamed">Arkani-Hamed</a></li>

<li><a href="Michael_Atiyah" title="Michael Atiyah">Atiyah</a></li>

<li><a href="Tom_Banks_(physicist)" title="Tom Banks (physicist)">Banks</a></li>

<li><a href="David_Berenstein" title="David Berenstein">Berenstein</a></li>

<li><a href="Raphael_Bousso" title="Raphael Bousso">Bousso</a></li>

<li><a href="Thomas_Curtright" title="Thomas Curtright">Curtright</a></li>

<li><a href="Robbert_Dijkgraaf" title="Robbert Dijkgraaf">Dijkgraaf</a></li>

<li><a href="Jacques_Distler" title="Jacques Distler">Distler</a></li>

<li><a href="Michael_R._Douglas" title="Michael R. Douglas">Douglas</a></li>

<li><a href="Gia_Dvali" class="mw-redirect" title="Gia Dvali">Dvali</a></li>

<li><a href="Sergio_Ferrara" title="Sergio Ferrara">Ferrara</a></li>

<li><a href="Willy_Fischler" title="Willy Fischler">Fischler</a></li>

<li><a href="Daniel_Friedan" title="Daniel Friedan">Friedan</a></li>

<li><a href="Sylvester_James_Gates" title="Sylvester James Gates">Gates</a></li>

<li><a href="Ferdinando_Gliozzi" title="Ferdinando Gliozzi">Gliozzi</a></li>

<li><a href="Rajesh_Gopakumar" title="Rajesh Gopakumar">Gopakumar</a></li>

<li><a href="Michael_Green_(physicist)" title="Michael Green (physicist)">Green</a></li>

<li><a href="Brian_Greene" title="Brian Greene">Greene</a></li>

<li><a href="David_Gross" title="David Gross">Gross</a></li>

<li><a href="Steven_Gubser" title="Steven Gubser">Gubser</a></li>

<li><a href="Sergei_Gukov" title="Sergei Gukov">Gukov</a></li>

<li><a href="Andrew_J._Hanson" title="Andrew J. Hanson">Hanson</a></li>

<li><a href="Jeffrey_A._Harvey" title="Jeffrey A. Harvey">Harvey</a></li>

<li><a href="Gerard_'t_Hooft" title="Gerard 't Hooft">'t Hooft</a></li>

<li><a href="Petr_Ho%C5%99ava_(theorist)" class="mw-redirect" title="Petr Hořava (theorist)">Hořava</a></li>

<li><a href="Gary_Gibbons" title="Gary Gibbons">Gibbons</a></li>

<li><a href="Shamit_Kachru" title="Shamit Kachru">Kachru</a></li>

<li><a href="Renata_Kallosh" title="Renata Kallosh">Kallosh</a></li>

<li><a href="Theodor_Kaluza" title="Theodor Kaluza">Kaluza</a></li>

<li><a href="Anton_Kapustin" title="Anton Kapustin">Kapustin</a></li>

<li><a href="Igor_Klebanov" title="Igor Klebanov">Klebanov</a></li>

<li><a href="Vadim_Knizhnik" title="Vadim Knizhnik">Knizhnik</a></li>

<li><a href="Maxim_Kontsevich" title="Maxim Kontsevich">Kontsevich</a></li>

<li><a href="Oskar_Klein" title="Oskar Klein">Klein</a></li>

<li><a href="Andrei_Linde" title="Andrei Linde">Linde</a></li>

<li><a href="Juan_Mart%C3%ADn_Maldacena" class="mw-redirect" title="Juan Martín Maldacena">Maldacena</a></li>

<li><a href="Stanley_Mandelstam" title="Stanley Mandelstam">Mandelstam</a></li>

<li><a href="Donald_Marolf" title="Donald Marolf">Marolf</a></li>

<li><a href="Emil_Martinec" title="Emil Martinec">Martinec</a></li>

<li><a href="Shiraz_Minwalla" title="Shiraz Minwalla">Minwalla</a></li>

<li><a href="Greg_Moore_(physicist)" title="Greg Moore (physicist)">Moore</a></li>

<li><a href="Sunil_Mukhi" title="Sunil Mukhi">Mukhi</a></li>

<li><a href="Robert_Myers_(physicist)" title="Robert Myers (physicist)">Myers</a></li>

<li><a href="Dimitri_Nanopoulos" title="Dimitri Nanopoulos">Nanopoulos</a></li>

<li><a href="Hora%C8%9Biu_N%C4%83stase" title="Horațiu Năstase">Năstase</a></li>

<li><a href="Nikita_Nekrasov" title="Nikita Nekrasov">Nekrasov</a></li>

<li><a href="Andr%C3%A9_Neveu" title="André Neveu">Neveu</a></li>

<li><a href="Holger_Bech_Nielsen" title="Holger Bech Nielsen">Nielsen</a></li>

<li><a href="Peter_van_Nieuwenhuizen" title="Peter van Nieuwenhuizen">van Nieuwenhuizen</a></li>

<li><a href="Sergei_Novikov_(mathematician)" title="Sergei Novikov (mathematician)">Novikov</a></li>

<li><a href="David_Olive" title="David Olive">Olive</a></li>

<li><a href="Hirosi_Ooguri" title="Hirosi Ooguri">Ooguri</a></li>

<li><a href="Burt_Ovrut" title="Burt Ovrut">Ovrut</a></li>

<li><a href="Joseph_Polchinski" title="Joseph Polchinski">Polchinski</a></li>

<li><a href="Alexander_Markovich_Polyakov" title="Alexander Markovich Polyakov">Polyakov</a></li>

<li><a href="Arvind_Rajaraman" title="Arvind Rajaraman">Rajaraman</a></li>

<li><a href="Pierre_Ramond" title="Pierre Ramond">Ramond</a></li>

<li><a href="Lisa_Randall" title="Lisa Randall">Randall</a></li>

<li><a href="Seifallah_Randjbar-Daemi" title="Seifallah Randjbar-Daemi">Randjbar-Daemi</a></li>

<li><a href="Martin_Ro%C4%8Dek" title="Martin Roček">Roček</a></li>

<li><a href="Augusto_Sagnotti" title="Augusto Sagnotti">Sagnotti</a></li>

<li><a href="Jo%C3%ABl_Scherk" title="Joël Scherk">Scherk</a></li>

<li><a href="John_Henry_Schwarz" title="John Henry Schwarz">Schwarz</a></li>

<li><a href="Nathan_Seiberg" title="Nathan Seiberg">Seiberg</a></li>

<li><a href="Stephen_Shenker" title="Stephen Shenker">Shenker</a></li>

<li><a href="Warren_Siegel" title="Warren Siegel">Siegel</a></li>

<li><a href="Eva_Silverstein" title="Eva Silverstein">Silverstein</a></li>

<li><a href="Matthias_Staudacher" title="Matthias Staudacher">Staudacher</a></li>

<li><a href="Paul_Steinhardt" title="Paul Steinhardt">Steinhardt</a></li>

<li><a href="Andrew_Strominger" title="Andrew Strominger">Strominger</a></li>

<li><a href="Raman_Sundrum" title="Raman Sundrum">Sundrum</a></li>

<li><a href="Leonard_Susskind" title="Leonard Susskind">Susskind</a></li>

<li><a href="Paul_Townsend" title="Paul Townsend">Townsend</a></li>

<li><a href="Sandip_Trivedi" title="Sandip Trivedi">Trivedi</a></li>

<li><a href="Neil_Turok" title="Neil Turok">Turok</a></li>

<li><a href="Gabriele_Veneziano" title="Gabriele Veneziano">Veneziano</a></li>

<li><a href="Erik_Verlinde" title="Erik Verlinde">Verlinde</a></li>

<li><a href="Herman_Verlinde" title="Herman Verlinde">Verlinde</a></li>

<li><a href="Edward_Witten" title="Edward Witten">Witten</a></li>

<li><a href="Tamiaki_Yoneya" title="Tamiaki Yoneya">Yoneya</a></li>

<li><a href="Alexander_Zamolodchikov" title="Alexander Zamolodchikov">Zamolodchikov</a></li>

<li><a href="Alexei_Zamolodchikov" title="Alexei Zamolodchikov">Zamolodchikov</a></li>

<li><a href="Eric_Zaslow" title="Eric Zaslow">Zaslow</a></li>

<li><a href="Bruno_Zumino" title="Bruno Zumino">Zumino</a></li>

<li><a href="Barton_Zwiebach" title="Barton Zwiebach">Zwiebach</a></li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2024-07-19" href="https://en.wikipedia.org/wiki/?title=Wess%E2%80%93Zumino%E2%80%93Witten_model&oldid=1235453796">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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