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Sigma model

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In <a href="Physics" title="Physics">physics</a>, a sigma model is a <a href="Field_theory_(physics)" class="mw-redirect" title="Field theory (physics)">field theory</a> that describes the field as a <a href="Point_particle" title="Point particle">point particle</a> confined to move on a fixed manifold. This manifold can be taken to be any <a href="Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a>, although it is most commonly taken to be either a <a href="Lie_group" title="Lie group">Lie group</a> or a <a href="Symmetric_space" title="Symmetric space">symmetric space</a>. The model may or may not be quantized. An example of the non-quantized version is the <a href="Skyrme_model" class="mw-redirect" title="Skyrme model">Skyrme model</a>; it cannot be quantized due to non-linearities of power greater than 4. In general, sigma models admit (classical) <a href="Topological_soliton" class="mw-redirect" title="Topological soliton">topological soliton</a> solutions, for example, the <a href="Skyrmion" title="Skyrmion">skyrmion</a> for the Skyrme model. When the sigma field is coupled to a gauge field, the resulting model is described by <a href="Ginzburg%E2%80%93Landau_theory" title="Ginzburg–Landau theory">Ginzburg–Landau theory</a>. This article is primarily devoted to the <a href="Classical_field_theory" title="Classical field theory">classical field theory</a> of the sigma model; the corresponding quantized theory is presented in the article titled "<a href="Non-linear_sigma_model" title="Non-linear sigma model">non-linear sigma model</a>".

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

The name has roots in particle physics, where a sigma model describes the interactions of <a href="Pion" title="Pion">pions</a>. Unfortunately, the "sigma meson" is not described by the sigma-model, but only a component of it.<a href="#cite_note-1">[1]</a>

The sigma model was introduced by <a href="#CITEREFGell-MannLévy1960">Gell-Mann & Lévy (1960</a>, section 5); the name σ-model comes from a field in their model corresponding to a spinless meson called σ, a <a href="Scalar_meson" title="Scalar meson">scalar meson</a> introduced earlier by <a href="Julian_Schwinger" title="Julian Schwinger">Julian Schwinger</a>.<a href="#cite_note-2">[2]</a> The model served as the dominant prototype of <a href="Spontaneous_symmetry_breaking" title="Spontaneous symmetry breaking">spontaneous symmetry breaking</a> of O(4) down to O(3): the three axial generators broken are the simplest manifestation of <a href="Chiral_symmetry_breaking" title="Chiral symmetry breaking">chiral symmetry breaking</a>, the surviving unbroken O(3) representing <a href="Isospin" title="Isospin">isospin</a>.

In conventional <a href="Particle_physics" title="Particle physics">particle physics</a> settings, the field is generally taken to be <a href="SU(N)" class="mw-redirect" title="SU(N)">SU(N)</a>, or the vector subspace of quotient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (SU(N)_{L}\times SU(N)_{R})/SU(N)}">

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<annotation encoding="application/x-tex">{\displaystyle (SU(N)_{L}\times SU(N)_{R})/SU(N)}</annotation>

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</math><img src="./ee612392c3caa3c863fab92de0c79a6cca925497.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.107ex; height:2.843ex;" alt="{\displaystyle (SU(N)_{L}\times SU(N)_{R})/SU(N)}" loading="lazy"> of the product of left and right chiral fields. In <a href="Condensed_matter" class="mw-redirect" title="Condensed matter">condensed matter</a> theories, the field is taken to be <a href="Orthogonal_group" title="Orthogonal group">O(N)</a>. For the <a href="Rotation_group" class="mw-redirect" title="Rotation group">rotation group</a> O(3), the sigma model describes the <a href="Isotropic" class="mw-redirect" title="Isotropic">isotropic</a> <a href="Ferromagnet" class="mw-redirect" title="Ferromagnet">ferromagnet</a>; more generally, the O(N) model shows up in the <a href="Quantum_Hall_effect" title="Quantum Hall effect">quantum Hall effect</a>, <a href="Superfluid" class="mw-redirect" title="Superfluid">superfluid</a> <a href="Helium-3" title="Helium-3">Helium-3</a> and <a href="Spin_chain" title="Spin chain">spin chains</a>.

In <a href="Supergravity" title="Supergravity">supergravity</a> models, the field is taken to be a <a href="Symmetric_space" title="Symmetric space">symmetric space</a>. Since symmetric spaces are defined in terms of their <a href="Involution_(mathematics)" title="Involution (mathematics)">involution</a>, their tangent space naturally splits into even and odd parity subspaces. This splitting helps propel the <a href="Dimensional_reduction" title="Dimensional reduction">dimensional reduction</a> of <a href="Kaluza%E2%80%93Klein" class="mw-redirect" title="Kaluza–Klein">Kaluza–Klein</a> theories.

In its most basic form, the sigma model can be taken as being purely the <a href="Kinetic_energy" title="Kinetic energy">kinetic energy</a> of a point particle; as a field, this is just the <a href="Dirichlet_energy" title="Dirichlet energy">Dirichlet energy</a> in Euclidean space.

In two spatial dimensions, the O(3) model is <a href="Completely_integrable" class="mw-redirect" title="Completely integrable">completely integrable</a>.

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

The <a href="Lagrangian_density" class="mw-redirect" title="Lagrangian density">Lagrangian density</a> of the sigma model can be written in a variety of different ways, each suitable to a particular type of application. The simplest, most generic definition writes the Lagrangian as the metric trace of the pullback of the metric tensor on a <a href="Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a>. For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi :M\to \Phi }">

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

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</math><img src="./a77379175059df79ffa6c9c757e31ae91e5b5766.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.057ex; height:2.509ex;" alt="{\displaystyle \phi :M\to \Phi }" loading="lazy"> a <a href="Field_(physics)" title="Field (physics)">field</a> over a <a href="Spacetime" title="Spacetime">spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">, this may be written as

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}={\frac {1}{2}}\sum _{i=1}^{n}\sum _{j=1}^{n}g_{ij}(\phi )\;\partial ^{\mu }\phi _{i}\partial _{\mu }\phi _{j}}</annotation>

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</math><img src="./cf8b548a37ce44134ac761a712a892a820e289f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:30.503ex; height:7.176ex;" alt="{\displaystyle {\mathcal {L}}={\frac {1}{2}}\sum _{i=1}^{n}\sum _{j=1}^{n}g_{ij}(\phi )\;\partial ^{\mu }\phi _{i}\partial _{\mu }\phi _{j}}" loading="lazy"></dd></dl>

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

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<annotation encoding="application/x-tex">{\displaystyle g_{ij}(\phi )}</annotation>

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</math><img src="./3627d71891449a92d55533da04f4d83005bec30e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.781ex; height:3.009ex;" alt="{\displaystyle g_{ij}(\phi )}" loading="lazy"> is the <a href="Metric_tensor" title="Metric tensor">metric tensor</a> on the field space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi \in \Phi }">

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<annotation encoding="application/x-tex">{\displaystyle \phi \in \Phi }</annotation>

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</math><img src="./ebacc2aa8186316fc76398fdc12f5a3c47a964b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.904ex; height:2.509ex;" alt="{\displaystyle \phi \in \Phi }" loading="lazy">, and the <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"> are the derivatives on the underlying <a href="Spacetime_manifold" class="mw-redirect" title="Spacetime manifold">spacetime manifold</a>.

This expression can be unpacked a bit. The field space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }">

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

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</math><img src="./aed80a2011a3912b028ba32a52dfa57165455f24.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 \Phi }" loading="lazy"> can be chosen to be any <a href="Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a>. Historically, this is the "sigma" of the sigma model; the historically-appropriate symbol <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="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy"> is avoided here to prevent clashes with many other common usages of <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="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy"> in geometry. Riemannian manifolds always come with a metric tensor <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="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy">. Given an <a href="Atlas_(topology)" title="Atlas (topology)">atlas of charts</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }">

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

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</math><img src="./aed80a2011a3912b028ba32a52dfa57165455f24.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 \Phi }" loading="lazy">, the field space can always be <a href="Local_trivialization" class="mw-redirect" title="Local trivialization">locally trivialized</a>, in that given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\subset \Phi }">

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

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</math><img src="./afc5f4f8038c661afd127b1fec8f78b240d22a5b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.559ex; height:2.176ex;" alt="{\displaystyle U\subset \Phi }" loading="lazy"> in the atlas, one may write a map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\to \mathbb {R} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle U\to \mathbb {R} ^{n}}</annotation>

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</math><img src="./f400295637439de6e8f3acab12c306b180aa817f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.293ex; height:2.343ex;" alt="{\displaystyle U\to \mathbb {R} ^{n}}" loading="lazy"> giving explicit local coordinates <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi =(\phi ^{1},\cdots ,\phi ^{n})}">

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<annotation encoding="application/x-tex">{\displaystyle \phi =(\phi ^{1},\cdots ,\phi ^{n})}</annotation>

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</math><img src="./d59074c2e7ee610dba76c25dc0016e789292f010.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.515ex; height:3.176ex;" alt="{\displaystyle \phi =(\phi ^{1},\cdots ,\phi ^{n})}" loading="lazy"> on that patch. The metric tensor on that patch is a matrix having components <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{ij}(\phi ).}">

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<annotation encoding="application/x-tex">{\displaystyle g_{ij}(\phi ).}</annotation>

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</math><img src="./26cc7bddaaca3aafd6620a9e7460a87628f8799e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.428ex; height:3.009ex;" alt="{\displaystyle g_{ij}(\phi ).}" loading="lazy">

The base manifold <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> must be a <a href="Differentiable_manifold" title="Differentiable manifold">differentiable manifold</a>; by convention, it is either <a href="Minkowski_space" title="Minkowski space">Minkowski space</a> in <a href="Particle_physics" title="Particle physics">particle physics</a> applications, flat two-dimensional <a href="Euclidean_space" title="Euclidean space">Euclidean space</a> for <a href="Condensed_matter" class="mw-redirect" title="Condensed matter">condensed matter</a> applications, or a <a href="Riemann_surface" title="Riemann surface">Riemann surface</a>, the <a href="Worldsheet" title="Worldsheet">worldsheet</a> in <a href="String_theory" title="String theory">string theory</a>. The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{\mu }\phi =\partial \phi /\partial x^{\mu }}">

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

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</math><img src="./052d0d4e4a25218947db77d913d6578c294bc27f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.679ex; height:3.009ex;" alt="{\displaystyle \partial _{\mu }\phi =\partial \phi /\partial x^{\mu }}" loading="lazy"> is just the plain-old <a href="Covariant_derivative" title="Covariant derivative">covariant derivative</a> on the base spacetime manifold <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M.}">

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

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</math><img src="./b865c33e30eb83000cd6387517c66dbbf3c3df9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.089ex; height:2.176ex;" alt="{\displaystyle M.}" loading="lazy"> When <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> is flat, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{\mu }\phi =\nabla \phi }">

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

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</math><img src="./a45d3c86441bc10afa4790c5bd6778fbc38345aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.263ex; height:2.843ex;" alt="{\displaystyle \partial _{\mu }\phi =\nabla \phi }" loading="lazy"> is just the ordinary <a href="Gradient" title="Gradient">gradient</a> of a scalar function (as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">

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

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</math><img src="./72b1f30316670aee6270a28334bdf4f5072cdde4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" loading="lazy"> is a scalar field, from the point of view of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> itself.) In more precise language, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{\mu }\phi }">

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

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</math><img src="./6ccdcb812d70b2fd2b3d8619c2b5caf1e4a40c1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.843ex; height:2.843ex;" alt="{\displaystyle \partial _{\mu }\phi }" loading="lazy"> is a <a href="Section_(fiber_bundle)" title="Section (fiber bundle)">section</a> of the <a href="Jet_bundle" title="Jet bundle">jet bundle</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\times \Phi }">

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<annotation encoding="application/x-tex">{\displaystyle M\times \Phi }</annotation>

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</math><img src="./97f683003972ef878bfeddbbce40c86cbb17e320.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.961ex; height:2.176ex;" alt="{\displaystyle M\times \Phi }" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Example:_O(n)_non-linear_sigma_model">Example: O(n) non-linear sigma model</h3></div>

Taking <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{ij}=\delta _{ij}}">

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<annotation encoding="application/x-tex">{\displaystyle g_{ij}=\delta _{ij}}</annotation>

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</math><img src="./d3704c69da4961f99cd8d8e9abaf9c9f5f607bb9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.194ex; height:3.009ex;" alt="{\displaystyle g_{ij}=\delta _{ij}}" loading="lazy"> the <a href="Kronecker_delta" title="Kronecker delta">Kronecker delta</a>, i.e. the scalar <a href="Dot_product" title="Dot product">dot product</a> in Euclidean space, one gets the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n)}">

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

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</math><img src="./34109fe397fdcff370079185bfdb65826cb5565a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.977ex; height:2.843ex;" alt="{\displaystyle O(n)}" loading="lazy"> non-linear sigma model. That is, write <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi ={\hat {u}}}">

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<annotation encoding="application/x-tex">{\displaystyle \phi ={\hat {u}}}</annotation>

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</math><img src="./5c88b728cefb7fd5fab0310ceee55e06c76442db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.814ex; height:2.509ex;" alt="{\displaystyle \phi ={\hat {u}}}" loading="lazy"> to be the <a href="Unit_vector" title="Unit vector">unit vector</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

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

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</math><img src="./c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy">, so that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {u}}\cdot {\hat {u}}=1}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {u}}\cdot {\hat {u}}=1}</annotation>

</semantics>

</math><img src="./6fab0a2e429cdfb2056d03bcf584261a15570aab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.599ex; height:2.176ex;" alt="{\displaystyle {\hat {u}}\cdot {\hat {u}}=1}" loading="lazy">, with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdot }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ba2c023bad1bd39ed49080f729cbf26bc448c9ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.439ex; margin-bottom: -0.61ex; width:0.647ex; height:1.176ex;" alt="{\displaystyle \cdot }" loading="lazy"> the ordinary Euclidean dot product. Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {u}}\in S^{n-1}}">

<mover>

</mover>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {u}}\in S^{n-1}}</annotation>

</semantics>

</math><img src="./c4ed0dc8a10f3e3d08a5b8d4de189a15b32d359b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.011ex; height:2.676ex;" alt="{\displaystyle {\hat {u}}\in S^{n-1}}" loading="lazy"> the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n-1)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./df88c6333caaf6471cf277f24b802ff9931b133e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.207ex; height:2.843ex;" alt="{\displaystyle (n-1)}" loading="lazy">-<a href="Sphere" title="Sphere">sphere</a>, the <a href="Isometries" class="mw-redirect" title="Isometries">isometries</a> of which are the <a href="Rotation_group" class="mw-redirect" title="Rotation group">rotation group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./34109fe397fdcff370079185bfdb65826cb5565a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.977ex; height:2.843ex;" alt="{\displaystyle O(n)}" loading="lazy">. The Lagrangian can then be written as

</mrow>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mover>

</mover>

</mrow>

<msub>

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</msub>

<mover>

</mover>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}={\frac {1}{2}}\nabla _{\mu }{\hat {u}}\cdot \nabla _{\mu }{\hat {u}}}</annotation>

</semantics>

</math><img src="./bbc1702f34a0b6f5082b2d96803a471d50265193.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.358ex; height:5.176ex;" alt="{\displaystyle {\mathcal {L}}={\frac {1}{2}}\nabla _{\mu }{\hat {u}}\cdot \nabla _{\mu }{\hat {u}}}" loading="lazy"></dd></dl>

For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=3}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./1c5a5a42ced00df920fad4ab2d4acdb960a4105b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=3}" loading="lazy">, this is the <a href="Continuum_limit" title="Continuum limit">continuum limit</a> of the <a href="Isotropic" class="mw-redirect" title="Isotropic">isotropic</a> <a href="Ferromagnet" class="mw-redirect" title="Ferromagnet">ferromagnet</a> on a lattice, i.e. of the <a href="Classical_Heisenberg_model" title="Classical Heisenberg model">classical Heisenberg model</a>. For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=2}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a02c8bd752d2cc859747ca1f3a508281bdbc3b34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=2}" loading="lazy">, this is the continuum limit of the <a href="Classical_XY_model" title="Classical XY model">classical XY model</a>. See also the <a href="N-vector_model" title="N-vector model">n-vector model</a> and the <a href="Potts_model" title="Potts model">Potts model</a> for reviews of the <a href="Lattice_model_(physics)" title="Lattice model (physics)">lattice model</a> equivalents. The continuum limit is taken by writing

<msub>

</mrow>

</msub>

<mover>

</mover>

</mrow>

<mfrac>

<mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

</mfrac>

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<annotation encoding="application/x-tex">{\displaystyle \delta _{h}[{\hat {u}}](i,j)={\frac {{\hat {u}}_{i}-{\hat {u}}_{j}}{h}}}</annotation>

</semantics>

</math><img src="./7c406714ab628c3b773cc987ace5b40830df7627.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.582ex; height:5.843ex;" alt="{\displaystyle \delta _{h}[{\hat {u}}](i,j)={\frac {{\hat {u}}_{i}-{\hat {u}}_{j}}{h}}}" loading="lazy"></dd></dl>

as the <a href="Finite_difference" title="Finite difference">finite difference</a> on neighboring lattice locations <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i,j.}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./5cf94d0a36b46a8917d7e18017d59c7ff200cedb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.441ex; height:2.509ex;" alt="{\displaystyle i,j.}" loading="lazy"> Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{h}[{\hat {u}}]\to \partial _{\mu }{\hat {u}}}">

<msub>

</mrow>

</msub>

<mover>

</mover>

</mrow>

<msub>

</mrow>

</msub>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta _{h}[{\hat {u}}]\to \partial _{\mu }{\hat {u}}}</annotation>

</semantics>

</math><img src="./bb3f27e7accd36def318948f690c92bd5de441a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.236ex; height:3.009ex;" alt="{\displaystyle \delta _{h}[{\hat {u}}]\to \partial _{\mu }{\hat {u}}}" loading="lazy"> in the limit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\to 0}">

</mstyle>

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

</semantics>

</math><img src="./3f15c1e6800f6e3fca9fec99720fc67a13780215.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.116ex; height:2.176ex;" alt="{\displaystyle h\to 0}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {u}}_{i}\cdot {\hat {u}}_{j}\to \partial _{\mu }{\hat {u}}\cdot \partial _{\mu }{\hat {u}}}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mover>

</mover>

</mrow>

<msub>

</mrow>

</msub>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {u}}_{i}\cdot {\hat {u}}_{j}\to \partial _{\mu }{\hat {u}}\cdot \partial _{\mu }{\hat {u}}}</annotation>

</semantics>

</math><img src="./cdf0537a33424da8b9f5f114379ceeb80e1d850a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.916ex; height:2.843ex;" alt="{\displaystyle {\hat {u}}_{i}\cdot {\hat {u}}_{j}\to \partial _{\mu }{\hat {u}}\cdot \partial _{\mu }{\hat {u}}}" loading="lazy"> after dropping the constant terms <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {u}}_{i}\cdot {\hat {u}}_{i}=1}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {u}}_{i}\cdot {\hat {u}}_{i}=1}</annotation>

</semantics>

</math><img src="./b64b8ad85d8f76ccdd2cec48b73f2aeaa2b2a8c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.199ex; height:2.509ex;" alt="{\displaystyle {\hat {u}}_{i}\cdot {\hat {u}}_{i}=1}" loading="lazy"> (the "bulk magnetization").

<div class="mw-heading mw-heading2"><h2 id="In_geometric_notation">In geometric notation</h2></div>

The sigma model can also be written in a more fully geometric notation, as a <a href="Fiber_bundle" title="Fiber bundle">fiber bundle</a> with fibers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }">

</mstyle>

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

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</math><img src="./aed80a2011a3912b028ba32a52dfa57165455f24.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 \Phi }" loading="lazy"> over a <a href="Differentiable_manifold" title="Differentiable manifold">differentiable manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">. Given a <a href="Section_(fiber_bundle)" title="Section (fiber bundle)">section</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi :M\to \Phi }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a77379175059df79ffa6c9c757e31ae91e5b5766.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.057ex; height:2.509ex;" alt="{\displaystyle \phi :M\to \Phi }" loading="lazy">, fix a point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in M.}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./127645d1eb7572c38f34e5b855f5d789404f6d93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.259ex; height:2.176ex;" alt="{\displaystyle x\in M.}" loading="lazy"> The <a href="Pushforward_(differential)" title="Pushforward (differential)">pushforward</a> at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> is a map of tangent bundles

<msub>

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</msub>

<msub>

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</msub>

<msub>

</mrow>

</msub>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} _{x}\phi :T_{x}M\to T_{\phi (x)}\Phi \quad }</annotation>

</semantics>

</math><img src="./b4e87c77c6604aa3b7007ebe7c18d276523f6ff3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:23.164ex; height:3.009ex;" alt="{\displaystyle \mathrm {d} _{x}\phi :T_{x}M\to T_{\phi (x)}\Phi \quad }" loading="lazy"> taking <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad \partial _{\mu }\mapsto {\frac {\partial \phi ^{i}}{\partial x^{\mu }}}\partial _{i}}">

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \quad \partial _{\mu }\mapsto {\frac {\partial \phi ^{i}}{\partial x^{\mu }}}\partial _{i}}</annotation>

</semantics>

</math><img src="./14ab132ee1d0555e60f58e3c8616abbeb6b78263.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.136ex; height:5.843ex;" alt="{\displaystyle \quad \partial _{\mu }\mapsto {\frac {\partial \phi ^{i}}{\partial x^{\mu }}}\partial _{i}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{\mu }=\partial /\partial x^{\mu }}">

<msub>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d254e4a2f61c99020e632761d22fafc4f076eef8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.908ex; height:3.009ex;" alt="{\displaystyle \partial _{\mu }=\partial /\partial x^{\mu }}" loading="lazy"> is taken to be a local orthonormal <a href="Basis_(mathematics)" class="mw-redirect" title="Basis (mathematics)">vector space basis</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle TM}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ea000afb5769206ddd5fd43f458430d04422ddeb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.078ex; height:2.176ex;" alt="{\displaystyle TM}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{i}=\partial /\partial q^{i}}">

<msub>

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</msub>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

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

</semantics>

</math><img src="./48e36c26de79f09c5f9c3564dbc66a7e3f0b9b77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.81ex; height:3.176ex;" alt="{\displaystyle \partial _{i}=\partial /\partial q^{i}}" loading="lazy"> the vector space basis on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\Phi }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./8fde55e2780b39398a53007c5b20668b48204ba0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.314ex; height:2.176ex;" alt="{\displaystyle T\Phi }" loading="lazy">. The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} \phi }">

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \phi }</annotation>

</semantics>

</math><img src="./eee4bd0f85e672337902fc76e0854573d36ae34c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.678ex; height:2.509ex;" alt="{\displaystyle \mathrm {d} \phi }" loading="lazy"> is a <a href="Differential_form" title="Differential form">differential form</a>. The sigma model <a href="Action_(physics)" title="Action (physics)">action</a> is then just the conventional <a href="Hodge_star" class="mw-redirect" title="Hodge star">inner product</a> on vector-valued k-forms

</mrow>

<mfrac>

</mfrac>

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<msub>

</mrow>

</msub>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}={\frac {1}{2}}\int _{M}\mathrm {d} \phi \wedge {\star \mathrm {d} \phi }}</annotation>

</semantics>

</math><img src="./0b6ed234bdc1a8e736a75a96fc41a8f47ddb31bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.716ex; height:5.676ex;" alt="{\displaystyle {\mathcal {S}}={\frac {1}{2}}\int _{M}\mathrm {d} \phi \wedge {\star \mathrm {d} \phi }}" loading="lazy"></dd></dl>

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

</mstyle>

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

</semantics>

</math><img src="./1caa4004cb216ef2930bb12fe805a76870caed94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \wedge }" loading="lazy"> is the <a href="Wedge_product" class="mw-redirect" title="Wedge product">wedge product</a>, and the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \star }">

</mstyle>

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

</semantics>

</math><img src="./bd316a21eeb5079a850f223b1d096a06bfa788c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.035ex; margin-bottom: -0.206ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \star }" loading="lazy"> is the <a href="Hodge_star" class="mw-redirect" title="Hodge star">Hodge star</a>. This is an inner product in two different ways. In the first way, given any two differentiable forms <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ,\beta }">

</mstyle>

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

</semantics>

</math><img src="./e4b46b57cfa0011b643037751809904d915c1b48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.854ex; height:2.509ex;" alt="{\displaystyle \alpha ,\beta }" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

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

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">, the Hodge dual defines an invariant inner product on the space of differential forms, commonly written as

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \langle \!\langle \alpha ,\beta \rangle \!\rangle \ =\ \int _{M}\alpha \wedge \star \beta }</annotation>

</semantics>

</math><img src="./de2c6c474d15f07f5789c9c9af75ed2055cd66fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.548ex; height:5.676ex;" alt="{\displaystyle \langle \!\langle \alpha ,\beta \rangle \!\rangle \ =\ \int _{M}\alpha \wedge \star \beta }" loading="lazy"></dd></dl>

The above is an inner product on the space of square-integrable forms, conventionally taken to be the <a href="Sobolev_space" title="Sobolev space">Sobolev space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}.}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle L^{2}.}</annotation>

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</math><img src="./9175d3b6bb8c26e6acb3fab128b34a294491c876.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.284ex; height:2.676ex;" alt="{\displaystyle L^{2}.}" loading="lazy"> In this way, one may write

</mrow>

<mfrac>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}={\frac {1}{2}}\langle \!\langle \mathrm {d} \phi ,\mathrm {d} \phi \rangle \!\rangle }</annotation>

</semantics>

</math><img src="./5ec7664062b6e83b4088532f80a2f4fd21512239.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.824ex; height:5.176ex;" alt="{\displaystyle {\mathcal {S}}={\frac {1}{2}}\langle \!\langle \mathrm {d} \phi ,\mathrm {d} \phi \rangle \!\rangle }" loading="lazy">.</dd></dl>

This makes it explicit and plainly evident that the sigma model is just the <a href="Kinetic_energy" title="Kinetic energy">kinetic energy</a> of a point particle. From the point of view of the manifold <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">, the field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./72b1f30316670aee6270a28334bdf4f5072cdde4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" loading="lazy"> is a scalar, and so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} \phi }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \phi }</annotation>

</semantics>

</math><img src="./eee4bd0f85e672337902fc76e0854573d36ae34c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.678ex; height:2.509ex;" alt="{\displaystyle \mathrm {d} \phi }" loading="lazy"> can be recognized just the ordinary <a href="Gradient" title="Gradient">gradient</a> of a scalar function. The Hodge star is merely a fancy device for keeping track of the <a href="Volume_form" title="Volume form">volume form</a> when integrating on curved spacetime. In the case that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> is flat, it can be completely ignored, and so the action is

</mrow>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}={\frac {1}{2}}\int _{M}\Vert \nabla \phi \Vert ^{2}d^{m}x}</annotation>

</semantics>

</math><img src="./3c404fc6f5c01904ab967196ce37ee663dae02da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.538ex; height:5.676ex;" alt="{\displaystyle {\mathcal {S}}={\frac {1}{2}}\int _{M}\Vert \nabla \phi \Vert ^{2}d^{m}x}" loading="lazy">,</dd></dl>

which is the <a href="Dirichlet_energy" title="Dirichlet energy">Dirichlet energy</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./72b1f30316670aee6270a28334bdf4f5072cdde4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" loading="lazy">. Classical extrema of the action (the solutions to the <a href="Lagrange_equation" class="mw-redirect" title="Lagrange equation">Lagrange equations</a>) are then those field configurations that minimize the Dirichlet energy of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./72b1f30316670aee6270a28334bdf4f5072cdde4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" loading="lazy">. Another way to convert this expression into a more easily-recognizable form is to observe that, for a scalar function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:M\to \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:M\to \mathbb {R} }</annotation>

</semantics>

</math><img src="./3f7c129dfb3446a2251aabc1cb3eba0b13ee7b21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.95ex; height:2.509ex;" alt="{\displaystyle f:M\to \mathbb {R} }" loading="lazy"> one has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} {\star }f=0}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} {\star }f=0}</annotation>

</semantics>

</math><img src="./0d061f5f69f56338e36a3956bdcbdbe0c9c4dd8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.994ex; height:2.509ex;" alt="{\displaystyle \mathrm {d} {\star }f=0}" loading="lazy"> and so one may also write

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}={\frac {1}{2}}\langle \!\langle \phi ,\Delta \phi \rangle \!\rangle }</annotation>

</semantics>

</math><img src="./530cb341409925f396de00536a68f99c2eaf1660.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.174ex; height:5.176ex;" alt="{\displaystyle {\mathcal {S}}={\frac {1}{2}}\langle \!\langle \phi ,\Delta \phi \rangle \!\rangle }" loading="lazy"></dd></dl>

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

</mstyle>

</mrow>

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

</semantics>

</math><img src="./32769037c408874e1890f77554c65f39c523ebe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }" loading="lazy"> is the <a href="Laplace%E2%80%93Beltrami_operator" title="Laplace–Beltrami operator">Laplace–Beltrami operator</a>, i.e., the ordinary <a href="Laplacian" class="mw-redirect" title="Laplacian">Laplacian</a> when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> is flat.

That there is another, second inner product in play simply requires not forgetting that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} \phi }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \phi }</annotation>

</semantics>

</math><img src="./eee4bd0f85e672337902fc76e0854573d36ae34c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.678ex; height:2.509ex;" alt="{\displaystyle \mathrm {d} \phi }" loading="lazy"> is a vector from the point of view of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./aed80a2011a3912b028ba32a52dfa57165455f24.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 \Phi }" loading="lazy"> itself. That is, given any two vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v,w\in T\Phi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v,w\in T\Phi }</annotation>

</semantics>

</math><img src="./32360be5c98c3e60b3c418d76fdd0769200e2b2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.981ex; height:2.509ex;" alt="{\displaystyle v,w\in T\Phi }" loading="lazy">, the Riemannian metric <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{ij}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e7c1130c3dec178129b287a3672c72f88e773832.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.586ex; height:2.343ex;" alt="{\displaystyle g_{ij}}" loading="lazy"> defines an inner product

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle v,w\rangle =g_{ij}v^{i}w^{j}}</annotation>

</semantics>

</math><img src="./572616bd7d72999504bc637f90b8ed8cac409cbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.821ex; height:3.343ex;" alt="{\displaystyle \langle v,w\rangle =g_{ij}v^{i}w^{j}}" loading="lazy"></dd></dl>

Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} \phi }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \phi }</annotation>

</semantics>

</math><img src="./eee4bd0f85e672337902fc76e0854573d36ae34c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.678ex; height:2.509ex;" alt="{\displaystyle \mathrm {d} \phi }" loading="lazy"> is vector-valued <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} \phi =(\mathrm {d} \phi ^{1},\cdots ,\mathrm {d} \phi ^{n})}">

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \phi =(\mathrm {d} \phi ^{1},\cdots ,\mathrm {d} \phi ^{n})}</annotation>

</semantics>

</math><img src="./e5639931b2a6688f712e841b018840685490a76c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.393ex; height:3.176ex;" alt="{\displaystyle \mathrm {d} \phi =(\mathrm {d} \phi ^{1},\cdots ,\mathrm {d} \phi ^{n})}" loading="lazy"> on local charts, one also takes the inner product there as well. More verbosely,

</mrow>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}={\frac {1}{2}}\int _{M}g_{ij}(\phi )\;\mathrm {d} \phi ^{i}\wedge {\star \mathrm {d} \phi ^{j}}}</annotation>

</semantics>

</math><img src="./85f9067713b2c680c56f6d4c7e32db7ddb5a7081.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.852ex; height:5.676ex;" alt="{\displaystyle {\mathcal {S}}={\frac {1}{2}}\int _{M}g_{ij}(\phi )\;\mathrm {d} \phi ^{i}\wedge {\star \mathrm {d} \phi ^{j}}}" loading="lazy">.</dd></dl>

The tension between these two inner products can be made even more explicit by noting that

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{\mu \nu }(\phi )=g_{ij}\partial _{\mu }\phi ^{i}\partial _{\nu }\phi ^{j}}</annotation>

</semantics>

</math><img src="./b806aea106ba96c0e7e78bab4e4c97a6cfdb31a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.014ex; height:3.343ex;" alt="{\displaystyle B_{\mu \nu }(\phi )=g_{ij}\partial _{\mu }\phi ^{i}\partial _{\nu }\phi ^{j}}" loading="lazy"></dd></dl>

is a <a href="Bilinear_form" title="Bilinear form">bilinear form</a>; it is a <a href="Pullback" title="Pullback">pullback</a> of the Riemann metric <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{ij}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e7c1130c3dec178129b287a3672c72f88e773832.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.586ex; height:2.343ex;" alt="{\displaystyle g_{ij}}" loading="lazy">. The individual <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{\mu }\phi ^{i}}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e5ceb5305cb06091953b7f0b60918bfb8b2f3134.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.643ex; height:3.343ex;" alt="{\displaystyle \partial _{\mu }\phi ^{i}}" loading="lazy"> can be taken as <a href="Vielbein" class="mw-redirect" title="Vielbein">vielbeins</a>. The Lagrangian density of the sigma model is then

</mrow>

<mfrac>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}={\frac {1}{2}}g^{\mu \nu }B_{\mu \nu }}</annotation>

</semantics>

</math><img src="./da3c0c5d13887b0e98339a9dfc9b316fbcaca945.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.772ex; height:5.176ex;" alt="{\displaystyle {\mathcal {L}}={\frac {1}{2}}g^{\mu \nu }B_{\mu \nu }}" loading="lazy"></dd></dl>

for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{\mu \nu }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e5bf4140993a891f5782167dc8a0c236dc7667b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.204ex; height:2.343ex;" alt="{\displaystyle g_{\mu \nu }}" loading="lazy"> the metric on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M.}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b865c33e30eb83000cd6387517c66dbbf3c3df9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.089ex; height:2.176ex;" alt="{\displaystyle M.}" loading="lazy"> Given this gluing-together, the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} \phi }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \phi }</annotation>

</semantics>

</math><img src="./eee4bd0f85e672337902fc76e0854573d36ae34c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.678ex; height:2.509ex;" alt="{\displaystyle \mathrm {d} \phi }" loading="lazy"> can be interpreted as a <a href="Solder_form" title="Solder form">solder form</a>; this is articulated more fully below.

<div class="mw-heading mw-heading2"><h2 id="Motivations_and_basic_interpretations">Motivations and basic interpretations</h2></div>

Several interpretational and foundational remarks can be made about the classical (non-quantized) sigma model. The first of these is that the classical sigma model can be interpreted as a model of non-interacting quantum mechanics. The second concerns the interpretation of energy.

<div class="mw-heading mw-heading3"><h3 id="Interpretation_as_quantum_mechanics">Interpretation as quantum mechanics</h3></div>

This follows directly from the expression

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}={\frac {1}{2}}\langle \!\langle \phi ,\Delta \phi \rangle \!\rangle }</annotation>

</semantics>

</math><img src="./530cb341409925f396de00536a68f99c2eaf1660.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.174ex; height:5.176ex;" alt="{\displaystyle {\mathcal {S}}={\frac {1}{2}}\langle \!\langle \phi ,\Delta \phi \rangle \!\rangle }" loading="lazy"></dd></dl>

given above. Taking <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi =\mathbb {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi =\mathbb {C} }</annotation>

</semantics>

</math><img src="./2ebc2a4f47e9a2b2f6d630cad3b3eb77d4e68bb1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.455ex; height:2.176ex;" alt="{\displaystyle \Phi =\mathbb {C} }" loading="lazy">, the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi :M\to \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi :M\to \mathbb {C} }</annotation>

</semantics>

</math><img src="./ef1ed9dd36ec7dce451023e8fc4848ca17aaaa15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.057ex; height:2.509ex;" alt="{\displaystyle \phi :M\to \mathbb {C} }" loading="lazy"> can be interpreted as a <a href="Wave_function" title="Wave function">wave function</a>, and its Laplacian the kinetic energy of that wave function. The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \!\langle \cdot ,\cdot \rangle \!\rangle }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle \!\langle \cdot ,\cdot \rangle \!\rangle }</annotation>

</semantics>

</math><img src="./88f393dc6083da6f02e235158ae5230e929c931a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.172ex; height:2.843ex;" alt="{\displaystyle \langle \!\langle \cdot ,\cdot \rangle \!\rangle }" loading="lazy"> is just some geometric machinery reminding one to integrate over all space. The corresponding quantum mechanical notation is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi =|\psi \rangle .}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f09682e736c8bcddf4a63cfa947b0f6d0c994ea8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.195ex; height:2.843ex;" alt="{\displaystyle \phi =|\psi \rangle .}" loading="lazy"> In flat space, the Laplacian is conventionally written as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta =\nabla ^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta =\nabla ^{2}}</annotation>

</semantics>

</math><img src="./30822e8684827364a4f67d1125a277cd3328ba42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.024ex; height:2.676ex;" alt="{\displaystyle \Delta =\nabla ^{2}}" loading="lazy">. Assembling all these pieces together, the sigma model action is equivalent to

</mrow>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}={\frac {1}{2}}\int _{M}\langle \psi |\nabla ^{2}|\psi \rangle dx^{m}={\frac {1}{2}}\int _{M}\psi ^{\dagger }(x)\nabla ^{2}\psi (x)dx^{m}}</annotation>

</semantics>

</math><img src="./cfa9019f33299cb367504b538d8a5892f78e98e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:50.168ex; height:5.676ex;" alt="{\displaystyle {\mathcal {S}}={\frac {1}{2}}\int _{M}\langle \psi |\nabla ^{2}|\psi \rangle dx^{m}={\frac {1}{2}}\int _{M}\psi ^{\dagger }(x)\nabla ^{2}\psi (x)dx^{m}}" loading="lazy"></dd></dl>

which is just the grand-total kinetic energy of the wave-function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (x)}">

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

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</math><img src="./a596a1fb4130a47f6b88c66150497338bd6cbccc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.652ex; height:2.843ex;" alt="{\displaystyle \psi (x)}" loading="lazy">, up to a factor of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar /m}">

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

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</math><img src="./d7f91faa393342382a1706f8ccda0f23a005eff8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.509ex; height:2.843ex;" alt="{\displaystyle \hbar /m}" loading="lazy">. To conclude, the classical sigma model on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

</mrow>

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</mrow>

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

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</math><img src="./f9add4085095b9b6d28d045fd9c92c2c09f549a7.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 \mathbb {C} }" loading="lazy"> can be interpreted as the quantum mechanics of a free, non-interacting quantum particle. Obviously, adding a term of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(\phi )}">

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

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</math><img src="./604c7510110c1f3d80638df3f720999eba78615c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.982ex; height:2.843ex;" alt="{\displaystyle V(\phi )}" loading="lazy"> to the Lagrangian results in the quantum mechanics of a wave-function in a potential. Taking <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi =\mathbb {C} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \Phi =\mathbb {C} ^{n}}</annotation>

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</math><img src="./903cb8a529312bdc128705367d52169227bf7f86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.673ex; height:2.343ex;" alt="{\displaystyle \Phi =\mathbb {C} ^{n}}" loading="lazy"> is not enough to describe the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-particle system, in that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> particles require <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> distinct coordinates, which are not provided by the base manifold. This can be solved by taking <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> copies of the base manifold.

<div class="mw-heading mw-heading3"><h3 id="The_solder_form">The solder form</h3></div>

It is very well-known that the <a href="Geodesic" title="Geodesic">geodesic</a> structure of a Riemannian manifold is described by the <a href="Hamilton%E2%80%93Jacobi_equations" class="mw-redirect" title="Hamilton–Jacobi equations">Hamilton–Jacobi equations</a>.<a href="#cite_note-3">[3]</a> In thumbnail form, the construction is as follows. Both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }">

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</mrow>

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

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</math><img src="./aed80a2011a3912b028ba32a52dfa57165455f24.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 \Phi }" loading="lazy"> are Riemannian manifolds; the below is written for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }">

</mstyle>

</mrow>

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

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</math><img src="./aed80a2011a3912b028ba32a52dfa57165455f24.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 \Phi }" loading="lazy">, the same can be done for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">. The <a href="Cotangent_bundle" title="Cotangent bundle">cotangent bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{*}\Phi }">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

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</math><img src="./c4b61585a511a88b5bb7c5880e2938b18155f59d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.452ex; height:2.343ex;" alt="{\displaystyle T^{*}\Phi }" loading="lazy">, supplied with <a href="Coordinate_chart" class="mw-redirect" title="Coordinate chart">coordinate charts</a>, can always be <a href="Local_trivialization" class="mw-redirect" title="Local trivialization">locally trivialized</a>, i.e.

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<annotation encoding="application/x-tex">{\displaystyle \left.T^{*}\Phi \right|_{U}\cong U\times \mathbb {R} ^{n}}</annotation>

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</math><img src="./3337f749d9db9c15e74ee77952de26f7eb681448.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.21ex; height:3.009ex;" alt="{\displaystyle \left.T^{*}\Phi \right|_{U}\cong U\times \mathbb {R} ^{n}}" loading="lazy"></dd></dl>

The trivialization supplies <a href="Canonical_coordinates" title="Canonical coordinates">canonical coordinates</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (q^{1},\cdots ,q^{n},p_{1},\cdots ,p_{n})}">

<msup>

</mrow>

</msup>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle (q^{1},\cdots ,q^{n},p_{1},\cdots ,p_{n})}</annotation>

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</math><img src="./b0e73042c9956f38e7e0c153eae055d8eca60a72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.243ex; height:3.176ex;" alt="{\displaystyle (q^{1},\cdots ,q^{n},p_{1},\cdots ,p_{n})}" loading="lazy"> on the cotangent bundle. Given the <a href="Metric_tensor" title="Metric tensor">metric tensor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{ij}}">

<msub>

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

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</math><img src="./e7c1130c3dec178129b287a3672c72f88e773832.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.586ex; height:2.343ex;" alt="{\displaystyle g_{ij}}" loading="lazy"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }">

</mstyle>

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

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</math><img src="./aed80a2011a3912b028ba32a52dfa57165455f24.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 \Phi }" loading="lazy">, define the Hamiltonian function

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<annotation encoding="application/x-tex">{\displaystyle H(q,p)={\frac {1}{2}}g^{ij}(q)p_{i}p_{j}}</annotation>

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</math><img src="./68dce97559bacdc93bcfff5db31323e29435cd70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.765ex; height:5.176ex;" alt="{\displaystyle H(q,p)={\frac {1}{2}}g^{ij}(q)p_{i}p_{j}}" loading="lazy"></dd></dl>

where, as always, one is careful to note that the inverse of the metric is used in this definition: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g^{ij}g_{jk}=\delta _{k}^{i}.}">

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<annotation encoding="application/x-tex">{\displaystyle g^{ij}g_{jk}=\delta _{k}^{i}.}</annotation>

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</math><img src="./96db6c0a2d391701170b41ef4b83b7f538335dbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.337ex; height:3.343ex;" alt="{\displaystyle g^{ij}g_{jk}=\delta _{k}^{i}.}" loading="lazy"> Famously, the <a href="Geodesic_flow" class="mw-redirect" title="Geodesic flow">geodesic flow</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }">

</mstyle>

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

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</math><img src="./aed80a2011a3912b028ba32a52dfa57165455f24.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 \Phi }" loading="lazy"> is given by the <a href="Hamilton%E2%80%93Jacobi_equations" class="mw-redirect" title="Hamilton–Jacobi equations">Hamilton–Jacobi equations</a>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle {\dot {q}}^{i}={\frac {\partial H}{\partial p_{i}}}\quad }</annotation>

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</math><img src="./fd622a64d49380515cb441bfc034f4b62ecfc560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.816ex; height:5.843ex;" alt="{\displaystyle {\dot {q}}^{i}={\frac {\partial H}{\partial p_{i}}}\quad }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad {\dot {p}}_{i}=-{\frac {\partial H}{\partial q^{i}}}}">

<msub>

<mover>

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<mfrac>

<mrow>

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<mrow>

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<annotation encoding="application/x-tex">{\displaystyle \quad {\dot {p}}_{i}=-{\frac {\partial H}{\partial q^{i}}}}</annotation>

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</math><img src="./8e1da29c0c193c65d9960c0a7f0767357913bb25.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:13.606ex; height:6.009ex;" alt="{\displaystyle \quad {\dot {p}}_{i}=-{\frac {\partial H}{\partial q^{i}}}}" loading="lazy"></dd></dl>

The geodesic flow is the <a href="Hamiltonian_flow" class="mw-redirect" title="Hamiltonian flow">Hamiltonian flow</a>; the solutions to the above are the geodesics of the manifold. Note, incidentally, that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dH/dt=0}">

</mrow>

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

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</math><img src="./aac7ef597f98a785fbfd756c41ea0420e6fa931d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.758ex; height:2.843ex;" alt="{\displaystyle dH/dt=0}" loading="lazy"> along geodesics; the time parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

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

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> is the distance along the geodesic.

The sigma model takes the momenta in the two manifolds <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{*}\Phi }">

<msup>

</mrow>

</msup>

</mstyle>

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

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</math><img src="./c4b61585a511a88b5bb7c5880e2938b18155f59d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.452ex; height:2.343ex;" alt="{\displaystyle T^{*}\Phi }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{*}M}">

<msup>

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</msup>

</mstyle>

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

</semantics>

</math><img src="./02e9815aa22bca801ff9618269fbbb247575ae86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.216ex; height:2.343ex;" alt="{\displaystyle T^{*}M}" loading="lazy"> and solders them together, in that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} \phi }">

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \phi }</annotation>

</semantics>

</math><img src="./eee4bd0f85e672337902fc76e0854573d36ae34c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.678ex; height:2.509ex;" alt="{\displaystyle \mathrm {d} \phi }" loading="lazy"> is a <a href="Solder_form" title="Solder form">solder form</a>. In this sense, the interpretation of the sigma model as an <a href="Energy_functional" class="mw-redirect" title="Energy functional">energy functional</a> is not surprising; it is in fact the gluing together of two energy functionals. Caution: the precise definition of a solder form requires it to be an isomorphism; this can only happen if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./aed80a2011a3912b028ba32a52dfa57165455f24.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 \Phi }" loading="lazy"> have the same real dimension. Furthermore, the conventional definition of a solder form takes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./aed80a2011a3912b028ba32a52dfa57165455f24.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 \Phi }" loading="lazy"> to be a Lie group. Both conditions are satisfied in various applications.

<div class="mw-heading mw-heading2"><h2 id="Results_on_various_spaces">Results on various spaces</h2></div>

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

</mstyle>

</mrow>

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

</semantics>

</math><img src="./aed80a2011a3912b028ba32a52dfa57165455f24.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 \Phi }" loading="lazy"> is often taken to be a <a href="Lie_group" title="Lie group">Lie group</a>, usually <a href="SU(N)" class="mw-redirect" title="SU(N)">SU(N)</a>, in the conventional particle physics models, <a href="Orthogonal_group" title="Orthogonal group">O(N)</a> in condensed matter theories, or as a <a href="Symmetric_space" title="Symmetric space">symmetric space</a> in <a href="Supergravity" title="Supergravity">supergravity</a> models. Since symmetric spaces are defined in terms of their <a href="Involution_(mathematics)" title="Involution (mathematics)">involution</a>, their tangent space (i.e. the place where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} \phi }">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \phi }</annotation>

</semantics>

</math><img src="./eee4bd0f85e672337902fc76e0854573d36ae34c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.678ex; height:2.509ex;" alt="{\displaystyle \mathrm {d} \phi }" loading="lazy"> lives) naturally splits into even and odd parity subspaces. This splitting helps propel the <a href="Dimensional_reduction" title="Dimensional reduction">dimensional reduction</a> of <a href="Kaluza%E2%80%93Klein" class="mw-redirect" title="Kaluza–Klein">Kaluza–Klein</a> theories.

<div class="mw-heading mw-heading3"><h3 id="On_Lie_groups">On Lie groups</h3></div>

For the special case of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./aed80a2011a3912b028ba32a52dfa57165455f24.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 \Phi }" loading="lazy"> being a <a href="Lie_group" title="Lie group">Lie group</a>, the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{ij}}">

<msub>

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

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</math><img src="./e7c1130c3dec178129b287a3672c72f88e773832.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.586ex; height:2.343ex;" alt="{\displaystyle g_{ij}}" loading="lazy"> is the <a href="Metric_tensor" title="Metric tensor">metric tensor</a> on the Lie group, formally called the Cartan tensor or the <a href="Killing_form" title="Killing form">Killing form</a>. The Lagrangian can then be written as the pullback of the Killing form. Note that the Killing form can be written as a trace over two matrices from the corresponding <a href="Lie_algebra" title="Lie algebra">Lie algebra</a>; thus, the Lagrangian can also be written in a form involving the trace. With slight re-arrangements, it can also be written as the pullback of the <a href="Maurer%E2%80%93Cartan_form" title="Maurer–Cartan form">Maurer–Cartan form</a>.

<div class="mw-heading mw-heading3"><h3 id="On_symmetric_spaces">On symmetric spaces</h3></div>

A common variation of the sigma model is to present it on a <a href="Symmetric_space" title="Symmetric space">symmetric space</a>. The prototypical example is the <a href="Chiral_model" title="Chiral model">chiral model</a>, which takes the product

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

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</math><img src="./f30d578cbfb208256b0256bc38af5d8d89960790.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.075ex; height:2.843ex;" alt="{\displaystyle G=SU(N)\times SU(N)}" loading="lazy"></dd></dl>

of the "left" and "right" chiral fields, and then constructs the sigma model on the "diagonal"

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<annotation encoding="application/x-tex">{\displaystyle \Phi ={\frac {SU(N)\times SU(N)}{SU(N)}}}</annotation>

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</math><img src="./50897c6e45cb88fe35c5a22edd52778e1e3d1184.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.763ex; height:6.509ex;" alt="{\displaystyle \Phi ={\frac {SU(N)\times SU(N)}{SU(N)}}}" loading="lazy"></dd></dl>

Such a quotient space is a symmetric space, and so one can generically take <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi =G/H}">

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

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</math><img src="./11c5187e048563be2144e3b142974c45844ba348.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.829ex; height:2.843ex;" alt="{\displaystyle \Phi =G/H}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\subset G}">

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

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</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"> is the maximal subgroup 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"> that is invariant under the <a href="Cartan_involution" class="mw-redirect" title="Cartan involution">Cartan involution</a>. The Lagrangian is still written exactly as the above, either in terms of the pullback of the metric on <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"> to a metric on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G/H}">

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

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</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"> or as a pullback of the Maurer–Cartan form.

<div class="mw-heading mw-heading3"><h3 id="Trace_notation">Trace notation</h3></div>

In physics, the most common and conventional statement of the sigma model begins with the definition

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<annotation encoding="application/x-tex">{\displaystyle L_{\mu }=\pi _{\mathfrak {m}}\circ \left(g^{-1}\partial _{\mu }g\right)}</annotation>

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</math><img src="./b9f0a0d139702e9ecef86667c52035cb81a55fae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.072ex; height:3.343ex;" alt="{\displaystyle L_{\mu }=\pi _{\mathfrak {m}}\circ \left(g^{-1}\partial _{\mu }g\right)}" loading="lazy"></dd></dl>

Here, the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g^{-1}\partial _{\mu }g}">

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<annotation encoding="application/x-tex">{\displaystyle g^{-1}\partial _{\mu }g}</annotation>

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</math><img src="./e1a1c9253e984583e54c148353faaf00c96a273e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.025ex; height:3.343ex;" alt="{\displaystyle g^{-1}\partial _{\mu }g}" loading="lazy"> is the pullback of the <a href="Maurer%E2%80%93Cartan_form" title="Maurer–Cartan form">Maurer–Cartan form</a>, for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in G}">

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

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</math><img src="./b1be73903416a0dd94b8cbc2268ce480810c0e62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.783ex; height:2.509ex;" alt="{\displaystyle g\in G}" loading="lazy">, onto the spacetime manifold. The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{\mathfrak {m}}}">

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<annotation encoding="application/x-tex">{\displaystyle \pi _{\mathfrak {m}}}</annotation>

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</math><img src="./f8119c5b09d0f551b75c208bd03c8a98cba6c742.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.009ex;" alt="{\displaystyle \pi _{\mathfrak {m}}}" loading="lazy"> is a projection onto the odd-parity piece of the Cartan involution. That is, given the Lie algebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}">

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

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</math><img src="./40a913b1503ed9ec94361b99f7fd59ef60705c28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}" loading="lazy"> 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 involution decomposes the space into odd and even parity components <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}={\mathfrak {m}}\oplus {\mathfrak {h}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}={\mathfrak {m}}\oplus {\mathfrak {h}}}</annotation>

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</math><img src="./b26ae36b95dd94b936c15c1b8bd2b6fe630b16d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.104ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {g}}={\mathfrak {m}}\oplus {\mathfrak {h}}}" loading="lazy"> corresponding to the two eigenstates of the involution. The sigma model Lagrangian can then be written as

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}={\frac {1}{2}}\mathrm {tr} \left(L_{\mu }L^{\mu }\right)}</annotation>

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</math><img src="./f0459687e894004aae94e85f9a1542cde2b3d92a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.326ex; height:5.176ex;" alt="{\displaystyle {\mathcal {L}}={\frac {1}{2}}\mathrm {tr} \left(L_{\mu }L^{\mu }\right)}" loading="lazy"></dd></dl>

This is instantly recognizable as the first term of the <a href="Skyrme_model" class="mw-redirect" title="Skyrme model">Skyrme model</a>.

<div class="mw-heading mw-heading3"><h3 id="Metric_form">Metric form</h3></div>

The equivalent metric form of this is to write a group element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in G}">

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

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</math><img src="./b1be73903416a0dd94b8cbc2268ce480810c0e62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.783ex; height:2.509ex;" alt="{\displaystyle g\in G}" loading="lazy"> as the geodesic <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g=\exp(\theta ^{i}T_{i})}">

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<annotation encoding="application/x-tex">{\displaystyle g=\exp(\theta ^{i}T_{i})}</annotation>

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</math><img src="./f5093243dffef3aa43576357ad2cf4af631990c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.624ex; height:3.176ex;" alt="{\displaystyle g=\exp(\theta ^{i}T_{i})}" loading="lazy"> of an element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta ^{i}T_{i}\in {\mathfrak {g}}}">

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<annotation encoding="application/x-tex">{\displaystyle \theta ^{i}T_{i}\in {\mathfrak {g}}}</annotation>

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</math><img src="./546fb916f813702d25b00ddd135da978733afb0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.06ex; height:3.009ex;" alt="{\displaystyle \theta ^{i}T_{i}\in {\mathfrak {g}}}" loading="lazy"> of the Lie algebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}">

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

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</math><img src="./40a913b1503ed9ec94361b99f7fd59ef60705c28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}" loading="lazy">. The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [T_{i},T_{j}]={f_{ij}}^{k}T_{k}}">

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<annotation encoding="application/x-tex">{\displaystyle [T_{i},T_{j}]={f_{ij}}^{k}T_{k}}</annotation>

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</math><img src="./713b645b81a3a1ed02bb4c9ffa69c76ef0db307b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.002ex; height:3.343ex;" alt="{\displaystyle [T_{i},T_{j}]={f_{ij}}^{k}T_{k}}" loading="lazy"> are the basis elements for the Lie algebra; the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {f_{ij}}^{k}}">

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

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</math><img src="./6b100e043bf45a76533baa0489a830f8bba5b46c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.705ex; height:3.343ex;" alt="{\displaystyle {f_{ij}}^{k}}" loading="lazy"> are the <a href="Structure_constant" class="mw-redirect" title="Structure constant">structure constants</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}">

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

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</math><img src="./40a913b1503ed9ec94361b99f7fd59ef60705c28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}" loading="lazy">.

Plugging this directly into the above and applying the infinitesimal form of the <a href="Baker%E2%80%93Campbell%E2%80%93Hausdorff_formula" title="Baker–Campbell–Hausdorff formula">Baker–Campbell–Hausdorff formula</a> promptly leads to the equivalent expression

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}={\frac {1}{2}}g_{ij}(\phi )\;\mathrm {d} \phi _{i}\wedge \star {\mathrm {d} \phi _{j}}={\frac {1}{2}}\;{W_{i}}^{m}{W^{n}}_{j}\;\;\mathrm {d} \phi _{i}\wedge \star {\mathrm {d} \phi _{j}}\;\;\mathrm {tr} (T_{m}T_{n})}</annotation>

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</math><img src="./605ecdb917e3362de4aeeaeb5d5a0b2a2461d648.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:61.609ex; height:5.176ex;" alt="{\displaystyle {\mathcal {L}}={\frac {1}{2}}g_{ij}(\phi )\;\mathrm {d} \phi _{i}\wedge \star {\mathrm {d} \phi _{j}}={\frac {1}{2}}\;{W_{i}}^{m}{W^{n}}_{j}\;\;\mathrm {d} \phi _{i}\wedge \star {\mathrm {d} \phi _{j}}\;\;\mathrm {tr} (T_{m}T_{n})}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {tr} (T_{m}T_{n})}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {tr} (T_{m}T_{n})}</annotation>

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</math><img src="./bb768af5bd5828ecbb5c1d9217a667f577ea4bf8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.234ex; height:2.843ex;" alt="{\displaystyle \mathrm {tr} (T_{m}T_{n})}" loading="lazy"> is now obviously (proportional to) the Killing form, and the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {W_{i}}^{m}}">

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

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</math><img src="./0700e4f54ababef6adb9765a25f2c260a7a5767b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.668ex; height:2.676ex;" alt="{\displaystyle {W_{i}}^{m}}" loading="lazy"> are the <a href="Vielbein" class="mw-redirect" title="Vielbein">vielbeins</a> that express the "curved" metric <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{ij}}">

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

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</math><img src="./e7c1130c3dec178129b287a3672c72f88e773832.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.586ex; height:2.343ex;" alt="{\displaystyle g_{ij}}" loading="lazy"> in terms of the "flat" metric <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {tr} (T_{m}T_{n})}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {tr} (T_{m}T_{n})}</annotation>

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</math><img src="./bb768af5bd5828ecbb5c1d9217a667f577ea4bf8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.234ex; height:2.843ex;" alt="{\displaystyle \mathrm {tr} (T_{m}T_{n})}" loading="lazy">. The article on the <a href="Baker%E2%80%93Campbell%E2%80%93Hausdorff_formula" title="Baker–Campbell–Hausdorff formula">Baker–Campbell–Hausdorff formula</a> provides an explicit expression for the vielbeins. They can be written as

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<annotation encoding="application/x-tex">{\displaystyle W=\sum _{n=0}^{\infty }{\frac {(-1)^{n}M^{n}}{(n+1)!}}=(I-e^{-M})M^{-1}}</annotation>

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</math><img src="./3980f984f9e9384bc3dc66f7776c6b73149fe0ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:37.9ex; height:6.843ex;" alt="{\displaystyle W=\sum _{n=0}^{\infty }{\frac {(-1)^{n}M^{n}}{(n+1)!}}=(I-e^{-M})M^{-1}}" loading="lazy"></dd></dl>

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

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

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> is a matrix whose matrix elements are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {M_{j}}^{k}=\theta ^{i}{f_{ij}}^{k}}">

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<annotation encoding="application/x-tex">{\displaystyle {M_{j}}^{k}=\theta ^{i}{f_{ij}}^{k}}</annotation>

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</math><img src="./1f344c53c91997bfa6018e29d83152b7acc0f749.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.946ex; height:3.343ex;" alt="{\displaystyle {M_{j}}^{k}=\theta ^{i}{f_{ij}}^{k}}" loading="lazy">.

For the sigma model on a symmetric space, as opposed to a Lie group, the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{i}}">

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

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</math><img src="./a8dd1c50cb9436474f83624c3f679ccf3eebbfef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.157ex; height:2.509ex;" alt="{\displaystyle T_{i}}" loading="lazy"> are limited to span the subspace <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./adc0e9162e96758157a34a6e44967288b481a7cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}" loading="lazy"> instead of all of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}={\mathfrak {m}}\oplus {\mathfrak {h}}}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}={\mathfrak {m}}\oplus {\mathfrak {h}}}</annotation>

</semantics>

</math><img src="./b26ae36b95dd94b936c15c1b8bd2b6fe630b16d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.104ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {g}}={\mathfrak {m}}\oplus {\mathfrak {h}}}" loading="lazy">. The Lie commutator on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./adc0e9162e96758157a34a6e44967288b481a7cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}" loading="lazy"> will not be within <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./adc0e9162e96758157a34a6e44967288b481a7cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}" loading="lazy">; indeed, one has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [{\mathfrak {m}},{\mathfrak {m}}]\subset {\mathfrak {h}}}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [{\mathfrak {m}},{\mathfrak {m}}]\subset {\mathfrak {h}}}</annotation>

</semantics>

</math><img src="./f2d127ad607c6d87c0d4fbe298eef101571a2b6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.202ex; height:2.843ex;" alt="{\displaystyle [{\mathfrak {m}},{\mathfrak {m}}]\subset {\mathfrak {h}}}" loading="lazy"> and so a projection is still needed.

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

The model can be extended in a variety of ways. Besides the aforementioned <a href="Skyrme_model" class="mw-redirect" title="Skyrme model">Skyrme model</a>, which introduces quartic terms, the model may be augmented by a <a href="Torsion_tensor" title="Torsion tensor">torsion</a> term to yield the <a href="Wess%E2%80%93Zumino%E2%80%93Witten_model" title="Wess–Zumino–Witten model">Wess–Zumino–Witten model</a>.

Another possibility is frequently seen in <a href="Supergravity" title="Supergravity">supergravity</a> models. Here, one notes that the Maurer–Cartan form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g^{-1}dg}">

<msup>

</mrow>

</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle g^{-1}dg}</annotation>

</semantics>

</math><img src="./fa4728319049f014f5932dd94d7df6b5900c39ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.783ex; height:3.009ex;" alt="{\displaystyle g^{-1}dg}" loading="lazy"> looks like "pure gauge". In the construction above for symmetric spaces, one can also consider the other projection

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

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<annotation encoding="application/x-tex">{\displaystyle A_{\mu }=\pi _{\mathfrak {h}}\circ \left(g^{-1}\partial _{\mu }g\right)}</annotation>

</semantics>

</math><img src="./b4163f6eb0b081cc53ad359a4e11699c7bf23848.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.828ex; height:3.343ex;" alt="{\displaystyle A_{\mu }=\pi _{\mathfrak {h}}\circ \left(g^{-1}\partial _{\mu }g\right)}" loading="lazy"></dd></dl>

where, as before, the symmetric space corresponded to the split <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}={\mathfrak {m}}\oplus {\mathfrak {h}}}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}={\mathfrak {m}}\oplus {\mathfrak {h}}}</annotation>

</semantics>

</math><img src="./b26ae36b95dd94b936c15c1b8bd2b6fe630b16d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.104ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {g}}={\mathfrak {m}}\oplus {\mathfrak {h}}}" loading="lazy">. This extra term can be interpreted as a <a href="Connection_(principal_bundle)" title="Connection (principal bundle)">connection</a> on the fiber bundle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\times H}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./9c370972363dcc8e1115f63468a4bc40c06a0f4b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.346ex; height:2.176ex;" alt="{\displaystyle M\times H}" loading="lazy"> (it transforms as a gauge field). It is what is "left over" from the connection on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

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<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">. It can be endowed with its own dynamics, by writing

</mrow>

<msub>

</mrow>

</msub>

<msup>

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</msup>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}=g_{ij}F^{i}\wedge {\star }F^{j}}</annotation>

</semantics>

</math><img src="./6a336ad3a8baaf2f835935046a4d41133aafa9a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.372ex; height:3.343ex;" alt="{\displaystyle {\mathcal {L}}=g_{ij}F^{i}\wedge {\star }F^{j}}" loading="lazy"></dd></dl>

with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F^{i}=dA^{i}}">

<msup>

</mrow>

</msup>

<msup>

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

</semantics>

</math><img src="./d55f743538ecdff646a334472b068a8540f9e94c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.472ex; height:2.676ex;" alt="{\displaystyle F^{i}=dA^{i}}" loading="lazy">. Note that the differential here is just "d", and not a covariant derivative; this is not the Yang–Mills stress-energy tensor. This term is not gauge invariant by itself; it must be taken together with the part of the connection that embeds into <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{\mu }}">

<msub>

</mrow>

</msub>

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

</semantics>

</math><img src="./71b946f5c2dac157f8041edaeb58f284c0cb9f71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.806ex; height:2.843ex;" alt="{\displaystyle L_{\mu }}" loading="lazy">, so that taken together, the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{\mu }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./71b946f5c2dac157f8041edaeb58f284c0cb9f71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.806ex; height:2.843ex;" alt="{\displaystyle L_{\mu }}" loading="lazy">, now with the connection as a part of it, together with this term, forms a complete gauge invariant Lagrangian (which does have the Yang–Mills terms in it, when expanded out).

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

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<li id="cite_note-1"><a href="#cite_ref-1">^</a> page 114, <a href="David_Tong_(physicist)" title="David Tong (physicist)">David Tong</a>: Lectures on Statistical Field Theory

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> Julian S. Schwinger, "A Theory of the Fundamental Interactions", Ann. Phys. 2(407), 1957.

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> Jurgen Jost (1991) Riemannian Geometry and Geometric Analysis, Springer

</li>

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

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</style><cite id="CITEREFGell-MannLévy1960" class="citation cs2">Gell-Mann, M.; Lévy, M. (1960), "The axial vector current in beta decay", <a href="Il_Nuovo_Cimento" class="mw-redirect" title="Il Nuovo Cimento">Il Nuovo Cimento</a>, 16 (4): 705–726, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1960NCim...16..705G">1960NCim...16..705G</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02859738">10.1007/BF02859738</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122945049">122945049</a></cite></li></ul>

<ul><li><cite id="CITEREFKetov2009" class="citation journal cs1">Ketov, Sergei (2009). <a rel="nofollow" class="external text" href="https://doi.org/10.4249%2Fscholarpedia.8508">"Nonlinear Sigma model"</a>. Scholarpedia. 4 (1): 8508. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2009SchpJ...4.8508K">2009SchpJ...4.8508K</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4249%2Fscholarpedia.8508">10.4249/scholarpedia.8508</a>.</cite></li></ul></div><div><div class="zim-footer">

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