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<title>Generalized complex structure</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Generalized complex structure</span></span>

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<p>In the field of <a href="Mathematics" title="Mathematics">mathematics</a> known as <a href="Differential_geometry" title="Differential geometry">differential geometry</a>, a <b>generalized complex structure</b> is a property of a <a href="Differential_manifold" class="mw-redirect" title="Differential manifold">differential manifold</a> that includes as special cases a <a href="Linear_complex_structure" title="Linear complex structure">complex structure</a> and a <a href="Symplectic_structure" class="mw-redirect" title="Symplectic structure">symplectic structure</a>. Generalized complex structures were introduced by <a href="Nigel_Hitchin" title="Nigel Hitchin">Nigel Hitchin</a> in 2002 and further developed by his students Marco Gualtieri and Gil Cavalcanti.

</p><p>These structures first arose in Hitchin's program of characterizing geometrical structures via <a href="Functional_(mathematics)" title="Functional (mathematics)">functionals</a> of <a href="Differential_forms" class="mw-redirect" title="Differential forms">differential forms</a>, a connection which formed the basis of <a href="Robbert_Dijkgraaf" title="Robbert Dijkgraaf">Robbert Dijkgraaf</a>, <a href="Sergei_Gukov" title="Sergei Gukov">Sergei Gukov</a>, <a href="Andrew_Neitzke" title="Andrew Neitzke">Andrew Neitzke</a> and <a href="Cumrun_Vafa" title="Cumrun Vafa">Cumrun Vafa</a>'s 2004 proposal that <a href="Topological_string_theory" title="Topological string theory">topological string theories</a> are special cases of a <a href="Topological_M-theory" class="mw-redirect" title="Topological M-theory">topological M-theory</a>. Today generalized complex structures also play a leading role in physical <a href="String_theory" title="String theory">string theory</a>, as <a href="Supersymmetry" title="Supersymmetry">supersymmetric</a> <a href="Compactification_(physics)#Flux_compactification" title="Compactification (physics)">flux compactifications</a>, which relate 10-dimensional physics to 4-dimensional worlds like ours, require (possibly twisted) generalized complex structures.

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

<div class="mw-heading mw-heading3"><h3 id="The_generalized_tangent_bundle">The generalized tangent bundle</h3></div>

<p>Consider an <a href="Manifold" title="Manifold"><i>N</i>-manifold</a> <i>M</i>. The <a href="Tangent_bundle" title="Tangent bundle">tangent bundle</a> of <i>M</i>, which will be denoted <b>T</b>, is the <a href="Vector_bundle" title="Vector bundle">vector bundle</a> over <i>M</i> whose fibers consist of all <a href="Tangent_vector" title="Tangent vector">tangent vectors</a> to <i>M</i>. A <a href="Fiber_bundle#Sections" title="Fiber bundle">section</a> of <b>T</b> is a <a href="Vector_field" title="Vector field">vector field</a> on <i>M</i>. The <a href="Cotangent_bundle" title="Cotangent bundle">cotangent bundle</a> of <i>M</i>, denoted <b>T</b><sup>*</sup>, is the vector bundle over <i>M</i> whose sections are <a href="Differential_form" title="Differential form">one-forms</a> on <i>M</i>.

</p><p>In <a href="Complex_geometry" title="Complex geometry">complex geometry</a> one considers structures on the tangent bundles of manifolds. In <a href="Symplectic_geometry" title="Symplectic geometry">symplectic geometry</a> one is instead interested in <a href="Exterior_algebra#Exterior_power" title="Exterior algebra">exterior powers</a> of the cotangent bundle. Generalized geometry unites these two fields by treating sections of the <b>generalized tangent bundle</b>, which is the <a href="Direct_sum_of_vector_bundles" class="mw-redirect" title="Direct sum of vector bundles">direct sum</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {T} \oplus \mathbf {T} ^{*}}">

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

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</math></span><img src="./708862574aa8cca21669fb8470938d29397ade31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.613ex; height:2.509ex;" alt="{\displaystyle \mathbf {T} \oplus \mathbf {T} ^{*}}" loading="lazy"></span> of the tangent and cotangent bundles, which are formal sums of a vector field and a one-form.

</p><p>The fibers are endowed with a natural <a href="Inner_product" class="mw-redirect" title="Inner product">inner product</a> with <a href="Metric_signature" title="Metric signature">signature</a> (<i>N</i>, <i>N</i>). If <i>X</i> and <i>Y</i> are vector fields and <i>ξ</i> and <i>η</i> are one-forms then the inner product of <i>X+ξ</i> and <i>Y+η</i> is defined as

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<annotation encoding="application/x-tex">{\displaystyle \langle X+\xi ,Y+\eta \rangle ={\frac {1}{2}}(\xi (Y)+\eta (X)).}</annotation>

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</math></span><img src="./cbed90d80bd34847d11e92ac963384931653f17d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.442ex; height:5.176ex;" alt="{\displaystyle \langle X+\xi ,Y+\eta \rangle ={\frac {1}{2}}(\xi (Y)+\eta (X)).}" loading="lazy"></span></dd></dl>

<p>A <b>generalized almost complex structure</b> is just an <a href="Almost_complex_structure" class="mw-redirect" title="Almost complex structure">almost complex structure</a> of the generalized tangent bundle which preserves the natural inner product:

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {J}}:\mathbf {T} \oplus \mathbf {T} ^{*}\to \mathbf {T} \oplus \mathbf {T} ^{*}}</annotation>

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</math></span><img src="./26c478aa2b78565bed7cb956a6672be4765c1455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:22.727ex; height:2.509ex;" alt="{\displaystyle {\mathcal {J}}:\mathbf {T} \oplus \mathbf {T} ^{*}\to \mathbf {T} \oplus \mathbf {T} ^{*}}" loading="lazy"></span></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {J}}^{2}=-{\rm {Id}},}</annotation>

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</math></span><img src="./41e4f3cb5ff01e0fa0630810c4dad4ff385f6007.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.802ex; height:3.009ex;" alt="{\displaystyle {\mathcal {J}}^{2}=-{\rm {Id}},}" loading="lazy"></span> and

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<annotation encoding="application/x-tex">{\displaystyle \langle {\mathcal {J}}(X+\xi ),{\mathcal {J}}(Y+\eta )\rangle =\langle X+\xi ,Y+\eta \rangle .}</annotation>

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</math></span><img src="./f91d50850f0bb2a9c3cd1debec214aba456a0dd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.217ex; height:2.843ex;" alt="{\displaystyle \langle {\mathcal {J}}(X+\xi ),{\mathcal {J}}(Y+\eta )\rangle =\langle X+\xi ,Y+\eta \rangle .}" loading="lazy"></span></dd></dl>

<p>Like in the case of an ordinary <a href="Almost_complex_structure" class="mw-redirect" title="Almost complex structure">almost complex structure</a>, a generalized almost complex structure is uniquely determined by its <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {-1}}}">

<msqrt>

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

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</math></span><img src="./4ea1ea9ac61e6e1e84ac39130f78143c18865719.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.906ex; height:3.009ex;" alt="{\displaystyle {\sqrt {-1}}}" loading="lazy"></span>-<a href="Vector_bundle#Operations_on_vector_bundles" title="Vector bundle">eigenbundle</a>, i.e. a subbundle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

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

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</math></span><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"></span> of the complexified generalized tangent bundle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} }">

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<annotation encoding="application/x-tex">{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} }</annotation>

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</math></span><img src="./98b9e2cbee826ad17c2106649c3d7565f14e5d79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.941ex; height:2.843ex;" alt="{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} }" loading="lazy"></span>

given by

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<annotation encoding="application/x-tex">{\displaystyle L=\{X+\xi \in (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} \ :\ {\mathcal {J}}(X+\xi )={\sqrt {-1}}(X+\xi )\}}</annotation>

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</math></span><img src="./819c39c4a8e6b80b8d6277da08318de44741bdc2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:58.011ex; height:3.009ex;" alt="{\displaystyle L=\{X+\xi \in (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} \ :\ {\mathcal {J}}(X+\xi )={\sqrt {-1}}(X+\xi )\}}" loading="lazy"></span></dd></dl>

<p>Such subbundle <i>L</i> satisfies the following properties:

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<div><ol style="list-style-type:lower-roman"><li>the intersection with its <a href="Complex_conjugate" title="Complex conjugate">complex conjugate</a> is the zero section: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L\cap {\overline {L}}=0}">

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<annotation encoding="application/x-tex">{\displaystyle L\cap {\overline {L}}=0}</annotation>

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</math></span><img src="./26b5b97ee9e4b62c78de2da1b0497df90c987eb3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.124ex; height:3.009ex;" alt="{\displaystyle L\cap {\overline {L}}=0}" loading="lazy"></span>;</li><li><i>L</i> is <b>maximal isotropic</b>, i.e. its complex <a href="Rank_(linear_algebra)" title="Rank (linear algebra)">rank</a> equals <i>N</i> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \ell ,\ell '\rangle =0}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \langle \ell ,\ell '\rangle =0}</annotation>

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</math></span><img src="./e44a4969def0ed092b8300e60bef733a3666677e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.728ex; height:3.009ex;" alt="{\displaystyle \langle \ell ,\ell '\rangle =0}" loading="lazy"></span> for all <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell ,\ell '\in L.}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \ell ,\ell '\in L.}</annotation>

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</math></span><img src="./dfb2f4ff181f4fb052a0ff0fc0139686975c007d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.728ex; height:2.843ex;" alt="{\displaystyle \ell ,\ell '\in L.}" loading="lazy"></span></li></ol></div>

<p>Vice versa, any subbundle <i>L</i> satisfying (i), (ii) is the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {-1}}}">

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

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</math></span><img src="./4ea1ea9ac61e6e1e84ac39130f78143c18865719.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.906ex; height:3.009ex;" alt="{\displaystyle {\sqrt {-1}}}" loading="lazy"></span>-eigenbundle of a unique generalized almost complex structure, so that the properties (i), (ii) can be considered as an alternative definition of generalized almost complex structure.

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<div class="mw-heading mw-heading3"><h3 id="Courant_bracket">Courant bracket</h3></div>

<p>In ordinary complex geometry, an <a href="Almost_complex_structure" class="mw-redirect" title="Almost complex structure">almost complex structure</a> is <a href="Integrable_system" title="Integrable system">integrable</a> to a <a href="Linear_complex_structure" title="Linear complex structure">complex structure</a> if and only if the <a href="Lie_derivative" title="Lie derivative">Lie bracket</a> of two sections of the <a href="Holomorphic_function" title="Holomorphic function">holomorphic</a> subbundle is another section of the holomorphic subbundle.

</p><p>In generalized complex geometry one is not interested in vector fields, but rather in the formal sums of vector fields and one-forms. A kind of Lie bracket for such formal sums was introduced in 1990 and is called the <a href="Courant_bracket" title="Courant bracket">Courant bracket</a> which is defined by

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<annotation encoding="application/x-tex">{\displaystyle [X+\xi ,Y+\eta ]=[X,Y]+{\mathcal {L}}_{X}\eta -{\mathcal {L}}_{Y}\xi -{\frac {1}{2}}d(i(X)\eta -i(Y)\xi )}</annotation>

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</math></span><img src="./2caa5b2c87a8ea267002ab118123164a18b3c015.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:59.228ex; height:5.176ex;" alt="{\displaystyle [X+\xi ,Y+\eta ]=[X,Y]+{\mathcal {L}}_{X}\eta -{\mathcal {L}}_{Y}\xi -{\frac {1}{2}}d(i(X)\eta -i(Y)\xi )}" loading="lazy"></span></dd></dl>

<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}_{X}}">

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

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</math></span><img src="./ce4db2d17b365a7321dbfdb8f8bc512dd911ea54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.236ex; height:2.509ex;" alt="{\displaystyle {\mathcal {L}}_{X}}" loading="lazy"></span> is the <a href="Lie_derivative" title="Lie derivative">Lie derivative</a> along the vector field <i>X</i>, <i>d</i> is the <a href="Exterior_derivative" title="Exterior derivative">exterior derivative</a> and <i>i</i> is the <a href="Exterior_algebra#The_interior_product_or_insertion_operator" title="Exterior algebra">interior product</a>.

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<div class="mw-heading mw-heading3"><h3 id="Definition_2">Definition</h3></div>

<p>A <b>generalized complex structure</b> is a generalized almost complex structure such that the space of smooth sections of <i>L</i> is closed under the Courant bracket.

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<div class="mw-heading mw-heading2"><h2 id="Maximal_isotropic_subbundles">Maximal isotropic subbundles</h2></div>

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

<p>There is a one-to-one correspondence between maximal isotropic <a href="Subbundle" title="Subbundle">subbundle</a> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {T} \oplus \mathbf {T} ^{*}}">

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

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</math></span><img src="./708862574aa8cca21669fb8470938d29397ade31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.613ex; height:2.509ex;" alt="{\displaystyle \mathbf {T} \oplus \mathbf {T} ^{*}}" loading="lazy"></span> and pairs <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {E} ,\varepsilon )}">

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

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</math></span><img src="./fabb83d7642ec172983a2c1ca8127f932bd0d50b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.684ex; height:2.843ex;" alt="{\displaystyle (\mathbf {E} ,\varepsilon )}" loading="lazy"></span> where <b>E</b> is a subbundle of <b>T</b> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }">

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

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</math></span><img src="./a30c89172e5b88edbd45d3e2772c7f5e562e5173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }" loading="lazy"></span> is a 2-form. This correspondence extends straightforwardly to the complex case.

</p><p>Given a pair <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {E} ,\varepsilon )}">

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

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</math></span><img src="./fabb83d7642ec172983a2c1ca8127f932bd0d50b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.684ex; height:2.843ex;" alt="{\displaystyle (\mathbf {E} ,\varepsilon )}" loading="lazy"></span> one can construct a maximally isotropic subbundle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(\mathbf {E} ,\varepsilon )}">

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

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</math></span><img src="./1373cd57f1004f8536ca0523f8a7e34734690f74.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.267ex; height:2.843ex;" alt="{\displaystyle L(\mathbf {E} ,\varepsilon )}" loading="lazy"></span> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {T} \oplus \mathbf {T} ^{*}}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./708862574aa8cca21669fb8470938d29397ade31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.613ex; height:2.509ex;" alt="{\displaystyle \mathbf {T} \oplus \mathbf {T} ^{*}}" loading="lazy"></span> as follows. The elements of the subbundle are the <a href="Formal_sum" title="Formal sum">formal sums</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X+\xi }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./23d213ece3557dbe425f7949fe7901f6f507d2f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.85ex; height:2.509ex;" alt="{\displaystyle X+\xi }" loading="lazy"></span> where the <a href="Vector_field" title="Vector field">vector field</a> <i>X</i> is a section of <b>E</b> and the one-form <i>ξ</i> restricted to the <a href="Dual_space" title="Dual space">dual space</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} ^{*}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./08870ad967c7c0da943c966d88cdee0b2b3131d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.811ex; height:2.343ex;" alt="{\displaystyle \mathbf {E} ^{*}}" loading="lazy"></span> is equal to the one-form <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon (X).}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./dbc5749e8f66c589bce991c3ae1649f429326d85.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.52ex; height:2.843ex;" alt="{\displaystyle \varepsilon (X).}" loading="lazy"></span>

</p><p>To see that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(\mathbf {E} ,\varepsilon )}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./1373cd57f1004f8536ca0523f8a7e34734690f74.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.267ex; height:2.843ex;" alt="{\displaystyle L(\mathbf {E} ,\varepsilon )}" loading="lazy"></span> is isotropic, notice that if <i>Y</i> is a section of <b>E</b> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./e0b461aaf61091abd5d2c808931c48b8ff9647db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.03ex; height:2.509ex;" alt="{\displaystyle \xi }" loading="lazy"></span> restricted to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} ^{*}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./08870ad967c7c0da943c966d88cdee0b2b3131d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.811ex; height:2.343ex;" alt="{\displaystyle \mathbf {E} ^{*}}" loading="lazy"></span> is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon (X)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./62fcef1dcef9014105a3fb2c6905838e08709ccb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.873ex; height:2.843ex;" alt="{\displaystyle \varepsilon (X)}" loading="lazy"></span> then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi (Y)=\varepsilon (X,Y),}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \xi (Y)=\varepsilon (X,Y),}</annotation>

</semantics>

</math></span><img src="./c2ff741a609b500cb820df9cef0d54afa7c5d3d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.038ex; height:2.843ex;" alt="{\displaystyle \xi (Y)=\varepsilon (X,Y),}" loading="lazy"></span> as the part of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./e0b461aaf61091abd5d2c808931c48b8ff9647db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.03ex; height:2.509ex;" alt="{\displaystyle \xi }" loading="lazy"></span> orthogonal to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} ^{*}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./08870ad967c7c0da943c966d88cdee0b2b3131d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.811ex; height:2.343ex;" alt="{\displaystyle \mathbf {E} ^{*}}" loading="lazy"></span> annihilates <i>Y</i>. Therefore if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X+\xi }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./23d213ece3557dbe425f7949fe7901f6f507d2f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.85ex; height:2.509ex;" alt="{\displaystyle X+\xi }" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y+\eta }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./97a14893b249b6efc9ddeb27ead69b1faeb03496.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.783ex; height:2.676ex;" alt="{\displaystyle Y+\eta }" loading="lazy"></span> are sections of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {T} \oplus \mathbf {T} ^{*}}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./708862574aa8cca21669fb8470938d29397ade31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.613ex; height:2.509ex;" alt="{\displaystyle \mathbf {T} \oplus \mathbf {T} ^{*}}" loading="lazy"></span> then

</p>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle X+\xi ,Y+\eta \rangle ={\frac {1}{2}}(\xi (Y)+\eta (X))={\frac {1}{2}}(\varepsilon (Y,X)+\varepsilon (X,Y))=0}</annotation>

</semantics>

</math></span><img src="./a337886679a365934a27c53f4572add7bc7e754c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:63.163ex; height:5.176ex;" alt="{\displaystyle \langle X+\xi ,Y+\eta \rangle ={\frac {1}{2}}(\xi (Y)+\eta (X))={\frac {1}{2}}(\varepsilon (Y,X)+\varepsilon (X,Y))=0}" loading="lazy"></span></dd></dl>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./1373cd57f1004f8536ca0523f8a7e34734690f74.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.267ex; height:2.843ex;" alt="{\displaystyle L(\mathbf {E} ,\varepsilon )}" loading="lazy"></span> is isotropic. Furthermore, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(\mathbf {E} ,\varepsilon )}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./1373cd57f1004f8536ca0523f8a7e34734690f74.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.267ex; height:2.843ex;" alt="{\displaystyle L(\mathbf {E} ,\varepsilon )}" loading="lazy"></span> is maximal because there are <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dim(\mathbf {E} )}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./47984cf12fef1b9271b758abfd4d5e47cdc13fb6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.442ex; height:2.843ex;" alt="{\displaystyle \dim(\mathbf {E} )}" loading="lazy"></span> (complex) dimensions of choices for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} ,}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./a7391d7d5c650e0e177e4e0e9cbcc1c872bd1be3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.404ex; height:2.509ex;" alt="{\displaystyle \mathbf {E} ,}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./a30c89172e5b88edbd45d3e2772c7f5e562e5173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }" loading="lazy"></span> is unrestricted on the <a href="Complement_(complexity)" title="Complement (complexity)">complement</a> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} ^{*},}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./80e26b105c5e55478a88aee5aeb2f6c9df910388.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.458ex; height:2.676ex;" alt="{\displaystyle \mathbf {E} ^{*},}" loading="lazy"></span> which is of (complex) dimension <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-\dim(\mathbf {E} ).}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n-\dim(\mathbf {E} ).}</annotation>

</semantics>

</math></span><img src="./cc34520c8f34183d49db889457a93e448b27483f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.324ex; height:2.843ex;" alt="{\displaystyle n-\dim(\mathbf {E} ).}" loading="lazy"></span> Thus the total (complex) dimension is <i>n</i>. Gualtieri has proven that all maximal isotropic subbundles are of the form <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(\mathbf {E} ,\varepsilon )}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./1373cd57f1004f8536ca0523f8a7e34734690f74.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.267ex; height:2.843ex;" alt="{\displaystyle L(\mathbf {E} ,\varepsilon )}" loading="lazy"></span> for some <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./0d7f22b39d51f780fc02859059c1757c606b9de2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.757ex; height:2.176ex;" alt="{\displaystyle \mathbf {E} }" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon .}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./5807913813d5188ce49b63a9b26d43f7a7763c19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.73ex; height:1.676ex;" alt="{\displaystyle \varepsilon .}" loading="lazy"></span>

</p>

<p>The <b>type</b> of a maximal isotropic subbundle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(\mathbf {E} ,\varepsilon )}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./1373cd57f1004f8536ca0523f8a7e34734690f74.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.267ex; height:2.843ex;" alt="{\displaystyle L(\mathbf {E} ,\varepsilon )}" loading="lazy"></span> is the real dimension of the subbundle that annihilates <b>E</b>. Equivalently it is 2<i>N</i> minus the real dimension of the <a href="Projection_(mathematics)" title="Projection (mathematics)">projection</a> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(\mathbf {E} ,\varepsilon )}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./1373cd57f1004f8536ca0523f8a7e34734690f74.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.267ex; height:2.843ex;" alt="{\displaystyle L(\mathbf {E} ,\varepsilon )}" loading="lazy"></span> onto the tangent bundle <b>T</b>. In other words, the type of a maximal isotropic subbundle is the codimension of its projection onto the tangent bundle. In the complex case one uses the complex dimension and the type is sometimes referred to as the <b>complex type</b>. While the type of a subbundle can in principle be any integer between 0 and 2<i>N</i>, generalized almost complex structures cannot have a type greater than <i>N</i> because the sum of the subbundle and its complex conjugate must be all of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} .}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} .}</annotation>

</semantics>

</math></span><img src="./aa9c0bdf1d452d0f396570a8ed42166f53b20d5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.588ex; height:2.843ex;" alt="{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} .}" loading="lazy"></span>

</p><p>The type of a maximal isotropic subbundle is <a href="Invariant_(mathematics)" title="Invariant (mathematics)">invariant</a> under <a href="Diffeomorphisms" class="mw-redirect" title="Diffeomorphisms">diffeomorphisms</a> and also under shifts of the <a href="Kalb%E2%80%93Ramond_field" title="Kalb–Ramond field">B-field</a>, which are <a href="Isometry" title="Isometry">isometries</a> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {T} \oplus \mathbf {T} ^{*}}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./708862574aa8cca21669fb8470938d29397ade31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.613ex; height:2.509ex;" alt="{\displaystyle \mathbf {T} \oplus \mathbf {T} ^{*}}" loading="lazy"></span> of the form

</p>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X+\xi \longrightarrow X+\xi +i_{X}B}</annotation>

</semantics>

</math></span><img src="./8152805830c4f5e412dd15f70183c50a6d41c2c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.836ex; height:2.509ex;" alt="{\displaystyle X+\xi \longrightarrow X+\xi +i_{X}B}" loading="lazy"></span></dd></dl>

<p>where <i>B</i> is an arbitrary closed 2-form called the B-field in the <a href="String_theory" title="String theory">string theory</a> literature.

</p><p>The type of a generalized almost complex structure is in general not constant, it can jump by any even <a href="Integer" title="Integer">integer</a>. However it is upper <a href="Semi-continuous" class="mw-redirect" title="Semi-continuous">semi-continuous</a>, which means that each point has an open neighborhood in which the type does not increase. In practice this means that subsets of greater type than the ambient type occur on submanifolds with positive <a href="Codimension" title="Codimension">codimension</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Real_index">Real index</h3></div>

<p>The real index <i>r</i> of a maximal isotropic subspace <i>L</i> is the complex dimension of the <a href="Intersection_(set_theory)" title="Intersection (set theory)">intersection</a> of <i>L</i> with its complex conjugate. A maximal isotropic subspace of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} }">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} }</annotation>

</semantics>

</math></span><img src="./98b9e2cbee826ad17c2106649c3d7565f14e5d79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.941ex; height:2.843ex;" alt="{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} }" loading="lazy"></span> is a generalized almost complex structure if and only if <i>r</i> = 0.

</p>

<div class="mw-heading mw-heading2"><h2 id="Canonical_bundle">Canonical bundle</h2></div>

<p>As in the case of ordinary complex geometry, there is a correspondence between generalized almost complex structures and <a href="Complex_line_bundle" class="mw-redirect" title="Complex line bundle">complex line bundles</a>. The complex line bundle corresponding to a particular generalized almost complex structure is often referred to as the <b>canonical bundle</b>, as it generalizes the <a href="Canonical_bundle" title="Canonical bundle">canonical bundle</a> in the ordinary case. It is sometimes also called the pure spinor bundle, as its sections are <a href="Pure_spinor" title="Pure spinor">pure spinors</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Generalized_almost_complex_structures">Generalized almost complex structures</h3></div>

<p>The canonical bundle is a one complex dimensional subbundle of the bundle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\Lambda } ^{*}\mathbf {T} \otimes \mathbb {C} }">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {\Lambda } ^{*}\mathbf {T} \otimes \mathbb {C} }</annotation>

</semantics>

</math></span><img src="./41171fc870354f4d1662ea2e6d2d4b25a2ad9de6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.305ex; height:2.509ex;" alt="{\displaystyle \mathbf {\Lambda } ^{*}\mathbf {T} \otimes \mathbb {C} }" loading="lazy"></span> of complex differential forms on <i>M</i>. Recall that the <a href="Gamma_matrices" title="Gamma matrices">gamma matrices</a> define an <a href="Isomorphism" title="Isomorphism">isomorphism</a> between differential forms and spinors. In particular even and odd forms map to the two chiralities of <a href="Spinor#Weyl_spinors" title="Spinor">Weyl spinors</a>. Vectors have an action on differential forms given by the interior product. One-forms have an action on forms given by the wedge product. Thus sections of the bundle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} }">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} }</annotation>

</semantics>

</math></span><img src="./98b9e2cbee826ad17c2106649c3d7565f14e5d79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.941ex; height:2.843ex;" alt="{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} }" loading="lazy"></span> act on differential forms. This action is a <a href="Group_representation" title="Group representation">representation</a> of the action of the <a href="Clifford_algebra" title="Clifford algebra">Clifford algebra</a> on spinors.

</p><p>A spinor is said to be a <b>pure spinor</b> if it is annihilated by half of a set of generators of the Clifford algebra. Spinors are sections of our bundle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\Lambda } ^{*}\mathbf {T} ,}">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./9802901afa95d2cef735867377f15eb938b47855.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.433ex; height:2.676ex;" alt="{\displaystyle \mathbf {\Lambda } ^{*}\mathbf {T} ,}" loading="lazy"></span> and generators of the Clifford algebra are the fibers of our other bundle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} .}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} .}</annotation>

</semantics>

</math></span><img src="./aa9c0bdf1d452d0f396570a8ed42166f53b20d5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.588ex; height:2.843ex;" alt="{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} .}" loading="lazy"></span> Therefore, a given pure spinor is annihilated by a half-dimensional subbundle <b>E</b> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} .}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} .}</annotation>

</semantics>

</math></span><img src="./aa9c0bdf1d452d0f396570a8ed42166f53b20d5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.588ex; height:2.843ex;" alt="{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} .}" loading="lazy"></span> Such subbundles are always isotropic, so to define an almost complex structure one must only impose that the sum of <b>E</b> and its complex conjugate is all of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} .}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} .}</annotation>

</semantics>

</math></span><img src="./aa9c0bdf1d452d0f396570a8ed42166f53b20d5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.588ex; height:2.843ex;" alt="{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} .}" loading="lazy"></span> This is true whenever the <a href="Wedge_product" class="mw-redirect" title="Wedge product">wedge product</a> of the pure spinor and its complex conjugate contains a top-dimensional component. Such pure spinors determine generalized almost complex structures.

</p><p>Given a generalized almost complex structure, one can also determine a pure spinor up to multiplication by an arbitrary <a href="Complex_function" class="mw-redirect" title="Complex function">complex function</a>. These choices of pure spinors are defined to be the sections of the canonical bundle.

</p>

<div class="mw-heading mw-heading3"><h3 id="Integrability_and_other_structures">Integrability and other structures</h3></div>

<p>If a pure spinor that determines a particular complex structure is <a href="Closed_and_exact_differential_forms" title="Closed and exact differential forms">closed</a>, or more generally if its exterior derivative is equal to the action of a gamma matrix on itself, then the almost complex structure is integrable and so such pure spinors correspond to generalized complex structures.

</p><p>If one further imposes that the canonical bundle is holomorphically trivial, meaning that it is global sections which are closed forms, then it defines a generalized Calabi-Yau structure and <i>M</i> is said to be a <b>generalized Calabi-Yau manifold</b>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Local_classification">Local classification</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Canonical_bundle_2">Canonical bundle</h3></div>

<p>Locally all pure spinors can be written in the same form, depending on an integer <i>k</i>, the B-field 2-form <i>B</i>, a nondegenerate symplectic form ω and a <i>k</i>-form Ω. In a local neighborhood of any point a <a href="Pure_spinor" title="Pure spinor">pure spinor</a> Φ which generates the canonical bundle may always be put in the form

</p>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi =e^{B+i\omega }\Omega }</annotation>

</semantics>

</math></span><img src="./c8592a5d875a98a1c5279b80d40da4fd15b2e8f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.886ex; height:2.676ex;" alt="{\displaystyle \Phi =e^{B+i\omega }\Omega }" loading="lazy"></span></dd></dl>

<p>where Ω is decomposable as the <a href="Wedge_product" class="mw-redirect" title="Wedge product">wedge product</a> of one-forms.

</p>

<div class="mw-heading mw-heading3"><h3 id="Regular_point">Regular point</h3></div>

<p>Define the subbundle <b>E</b> of the complexified tangent bundle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {T} \otimes \mathbb {C} }">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./9caa76717b1903b08fa790e2233c931b1a06e34d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.378ex; height:2.343ex;" alt="{\displaystyle \mathbf {T} \otimes \mathbb {C} }" loading="lazy"></span> to be the projection of the holomorphic subbundle <b>L</b> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} }">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} }</annotation>

</semantics>

</math></span><img src="./98b9e2cbee826ad17c2106649c3d7565f14e5d79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.941ex; height:2.843ex;" alt="{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} }" loading="lazy"></span> to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {T} \otimes \mathbb {C} .}">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./7eef7369cbc00e95e811c22ef319322f51d08ae5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.025ex; height:2.343ex;" alt="{\displaystyle \mathbf {T} \otimes \mathbb {C} .}" loading="lazy"></span> In the definition of a generalized almost complex structure we have imposed that the intersection of <b>L</b> and its conjugate contains only the origin, otherwise they would be unable to span the entirety of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} .}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} .}</annotation>

</semantics>

</math></span><img src="./aa9c0bdf1d452d0f396570a8ed42166f53b20d5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.588ex; height:2.843ex;" alt="{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} .}" loading="lazy"></span> However the intersection of their projections need not be trivial. In general this intersection is of the form

</p>

<mover>

</mover>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E\cap {\overline {E}}=\Delta \otimes \mathbb {C} }</annotation>

</semantics>

</math></span><img src="./fa4ce3d3e80a482702cbc6f8882d0be8dba10fdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.841ex; height:3.176ex;" alt="{\displaystyle E\cap {\overline {E}}=\Delta \otimes \mathbb {C} }" loading="lazy"></span></dd></dl>

<p>for some subbundle Δ. A point which has an <a href="Open_set" title="Open set">open</a> <a href="Neighborhood_(mathematics)" class="mw-redirect" title="Neighborhood (mathematics)">neighborhood</a> in which the dimension of the fibers of Δ is constant is said to be a <b>regular point</b>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Darboux's_theorem">Darboux's theorem</h3></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Darboux's_theorem" title="Darboux's theorem">Darboux's theorem</a></div>

<p>Every regular point in a generalized complex manifold has an open neighborhood which, after a diffeomorphism and shift of the B-field, has the same generalized complex structure as the <a href="Cartesian_product" title="Cartesian product">Cartesian product</a> of the <a href="Linear_complex_structure" title="Linear complex structure">complex vector space</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{k}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./81536cae4ef687e2b0228fb5867e4541fccb768c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.767ex; height:2.676ex;" alt="{\displaystyle \mathbb {C} ^{k}}" loading="lazy"></span> and the standard symplectic space <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2n-2k}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./70926fbc4f9de9fc9a16e51f05c1d1740e0c93f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.676ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2n-2k}}" loading="lazy"></span> with the standard symplectic form, which is the <a href="Direct_sum_of_matrices" title="Direct sum of matrices">direct sum</a> of the two by two off-diagonal matrices with entries 1 and −1.

</p>

<div class="mw-heading mw-heading3"><h3 id="Local_holomorphicity">Local holomorphicity</h3></div>

<p>Near non-regular points, the above <a href="Classification_theorem" title="Classification theorem">classification theorem</a> does not apply. However, about any point, a generalized complex manifold is, up to diffeomorphism and B-field, a product of a symplectic manifold with a generalized complex manifold which is of complex type at the point, much like Weinstein's theorem for the local structure of <a href="Poisson_manifold" title="Poisson manifold">Poisson manifolds</a>. The remaining question of the local structure is: what does a generalized complex structure look like near a point of complex type? In fact, it will be induced by a holomorphic <a href="Poisson_manifold" title="Poisson manifold">Poisson structure</a>.

</p>

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

<div class="mw-heading mw-heading3"><h3 id="Complex_manifolds">Complex manifolds</h3></div>

<p>The space of complex differential forms <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\Lambda } ^{*}\mathbf {T} \otimes \mathbb {C} }">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {\Lambda } ^{*}\mathbf {T} \otimes \mathbb {C} }</annotation>

</semantics>

</math></span><img src="./41171fc870354f4d1662ea2e6d2d4b25a2ad9de6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.305ex; height:2.509ex;" alt="{\displaystyle \mathbf {\Lambda } ^{*}\mathbf {T} \otimes \mathbb {C} }" loading="lazy"></span> has a complex conjugation operation given by complex conjugation in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} .}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./8f4d5d3ec97eee8b915d3b14d3fb38579ee639d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} .}" loading="lazy"></span> This allows one to define <a href="Holomorphic_function" title="Holomorphic function">holomorphic</a> and <a href="Antiholomorphic" class="mw-redirect" title="Antiholomorphic">antiholomorphic</a> one-forms and (<i>m</i>, <i>n</i>)-forms, which are homogeneous polynomials in these one-forms with <i>m</i> holomorphic factors and <i>n</i> antiholomorphic factors. In particular, all (<i>n</i>, 0)-forms are related locally by multiplication by a complex function and so they form a complex line bundle.

</p><p>(<i>n</i>, 0)-forms are pure spinors, as they are annihilated by antiholomorphic tangent vectors and by holomorphic one-forms. Thus this line bundle can be used as a canonical bundle to define a generalized complex structure. Restricting the annihilator from <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} }">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} }</annotation>

</semantics>

</math></span><img src="./98b9e2cbee826ad17c2106649c3d7565f14e5d79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.941ex; height:2.843ex;" alt="{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} }" loading="lazy"></span> to the complexified tangent bundle one gets the subspace of antiholomorphic vector fields. Therefore, this generalized complex structure on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} }">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} }</annotation>

</semantics>

</math></span><img src="./98b9e2cbee826ad17c2106649c3d7565f14e5d79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.941ex; height:2.843ex;" alt="{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} }" loading="lazy"></span> defines an ordinary <a href="Linear_complex_structure" title="Linear complex structure">complex structure</a> on the tangent bundle.

</p><p>As only half of a basis of vector fields are holomorphic, these complex structures are of type <i>N</i>. In fact complex manifolds, and the manifolds obtained by multiplying the pure spinor bundle defining a complex manifold by a complex, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./62b4e7c1cedb9564609aefd2aa2309972f455c24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.318ex; height:2.176ex;" alt="{\displaystyle \partial }" loading="lazy"></span>-closed (2,0)-form, are the only type <i>N</i> generalized complex manifolds.

</p>

<div class="mw-heading mw-heading3"><h3 id="Symplectic_manifolds">Symplectic manifolds</h3></div>

<p>The pure spinor bundle generated by

</p>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./7efccc3a3a8cd27006039198580e558b02e27c49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.389ex; height:3.009ex;" alt="{\displaystyle \phi =e^{i\omega }}" loading="lazy"></span></dd></dl>

<p>for a nondegenerate two-form <i>ω</i> defines a symplectic structure on the tangent space. Thus symplectic manifolds are also generalized complex manifolds.

</p><p>The above pure spinor is globally defined, and so the canonical bundle is trivial. This means that symplectic manifolds are not only generalized complex manifolds but in fact are generalized Calabi-Yau manifolds.

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

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> is related to a pure spinor which is just a number by an imaginary shift of the B-field, which is a shift of the <a href="K%C3%A4hler_form" class="mw-redirect" title="Kähler form">Kähler form</a>. Therefore, these generalized complex structures are of the same type as those corresponding to a <a href="Scalar_(mathematics)" title="Scalar (mathematics)">scalar</a> pure spinor. A scalar is annihilated by the entire tangent space, and so these structures are of type <i>0</i>.

</p><p>Up to a shift of the B-field, which corresponds to multiplying the pure spinor by the exponential of a closed, real 2-form, symplectic manifolds are the only type 0 generalized complex manifolds. Manifolds which are symplectic up to a shift of the B-field are sometimes called <b>B-symplectic</b>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Relation_to_G-structures">Relation to G-structures</h2></div>

<p>Some of the almost structures in generalized complex geometry may be rephrased in the language of <a href="G-structure" class="mw-redirect" title="G-structure">G-structures</a>. The word "almost" is removed if the structure is integrable.

</p><p>The bundle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} }">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} }</annotation>

</semantics>

</math></span><img src="./98b9e2cbee826ad17c2106649c3d7565f14e5d79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.941ex; height:2.843ex;" alt="{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} }" loading="lazy"></span> with the above inner product is an <span class="texhtml">O(2<i>n</i>, 2<i>n</i>)</span> structure. A generalized almost complex structure is a reduction of this structure to a <span class="texhtml">U(<i>n</i>, <i>n</i>)</span> structure. Therefore, the space of generalized complex structures is the coset

</p>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {O(2n,2n)}{U(n,n)}}.}</annotation>

</semantics>

</math></span><img src="./652e6f58f9d992bd77d075389f5118f5802b86aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:11.214ex; height:6.509ex;" alt="{\displaystyle {\frac {O(2n,2n)}{U(n,n)}}.}" loading="lazy"></span></dd></dl>

<p>A generalized almost Kähler structure is a pair of <a href="Commutative_operation" class="mw-redirect" title="Commutative operation">commuting</a> generalized complex structures such that minus the product of the corresponding tensors is a positive definite metric on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} .}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} .}</annotation>

</semantics>

</math></span><img src="./aa9c0bdf1d452d0f396570a8ed42166f53b20d5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.588ex; height:2.843ex;" alt="{\displaystyle (\mathbf {T} \oplus \mathbf {T} ^{*})\otimes \mathbb {C} .}" loading="lazy"></span> Generalized Kähler structures are reductions of the structure group to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(n)\times U(n).}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./b565538385b1d420e97a4f8d9cd2d70c6637730a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.46ex; height:2.843ex;" alt="{\displaystyle U(n)\times U(n).}" loading="lazy"></span> Generalized Kähler manifolds, and their twisted counterparts, are equivalent to the bihermitian manifolds discovered by <a href="Sylvester_James_Gates" title="Sylvester James Gates">Sylvester James Gates</a>, <a href="Chris_Hull_(physicist)" class="mw-redirect" title="Chris Hull (physicist)">Chris Hull</a> and <a href="Martin_Rocek" class="mw-redirect" title="Martin Rocek">Martin Roček</a> in the context of 2-dimensional <a href="Supersymmetry" title="Supersymmetry">supersymmetric</a> <a href="Quantum_field_theory" title="Quantum field theory">quantum field theories</a> in 1984.

</p><p>Finally, a generalized almost Calabi-Yau metric structure is a further reduction of the structure group to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle SU(n)\times SU(n).}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./b6a1216d4613393316150f8d4b376488671dece2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.459ex; height:2.843ex;" alt="{\displaystyle SU(n)\times SU(n).}" loading="lazy"></span>

</p>

<div class="mw-heading mw-heading3"><h3 id="Calabi_versus_Calabi–Yau_metric">Calabi versus Calabi–Yau metric</h3></div>

<p>Notice that a generalized Calabi metric structure, which was introduced by Marco Gualtieri, is a stronger condition than a generalized Calabi–Yau structure, which was introduced by <a href="Nigel_Hitchin" title="Nigel Hitchin">Nigel Hitchin</a>. In particular a generalized Calabi–Yau metric structure implies the existence of two commuting generalized almost complex structures.

</p>

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

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</style><cite id="CITEREFHitchin2003" class="citation journal cs1"><a href="Nigel_Hitchin" title="Nigel Hitchin">Hitchin, Nigel</a> (2003). "Generalized Calabi-Yau manifolds". <i><a href="Quarterly_Journal_of_Mathematics" title="Quarterly Journal of Mathematics">Quarterly Journal of Mathematics</a></i>. <b>54</b> (3): <span class="nowrap">281–</span>308. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fqmath%2Fhag025">10.1093/qmath/hag025</a>.</cite></li>

<li><cite id="CITEREFGualtieri2004" class="citation thesis cs1">Gualtieri, Marco (2004). <i>Generalized complex geometry</i> (PhD Thesis). <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math.DG/0401221">math.DG/0401221</a></span>.</cite></li>

<li><cite id="CITEREFGualtieri2011" class="citation journal cs1">Gualtieri, Marco (2011). <a rel="nofollow" class="external text" href="https://doi.org/10.4007%2Fannals.2011.174.1.3">"Generalized complex geometry"</a>. <i><a href="Annals_of_Mathematics" title="Annals of Mathematics">Annals of Mathematics</a></i>. (2). <b>174</b> (1): <span class="nowrap">75–</span>123. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0911.0993">0911.0993</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.4007%2Fannals.2011.174.1.3">10.4007/annals.2011.174.1.3</a></span>.</cite></li>

<li><cite id="CITEREFGraña2006" class="citation journal cs1">Graña, Mariana (2006). "Flux compactifications in string theory: a comprehensive review". <i>Phys. Rep</i>. <b>423</b> (3): <span class="nowrap">91–</span>158. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/hep-th/0509003">hep-th/0509003</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.physrep.2005.10.008">10.1016/j.physrep.2005.10.008</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:119508517">119508517</a>.</cite></li>

<li><cite id="CITEREFDijkgraafGukovNeitzkeVafa2005" class="citation journal cs1"><a href="Robbert_Dijkgraaf" title="Robbert Dijkgraaf">Dijkgraaf, Robbert</a>; <a href="Sergei_Gukov" title="Sergei Gukov">Gukov, Sergei</a>; Neitzke, Andrew; <a href="Cumrun_Vafa" title="Cumrun Vafa">Vafa, Cumrun</a> (2005). <a rel="nofollow" class="external text" href="https://doi.org/10.4310%2FATMP.2005.v9.n4.a5">"Topological M-theory as unification of form theories of gravity"</a>. <i><a href="Advances_in_Theoretical_and_Mathematical_Physics" title="Advances in Theoretical and Mathematical Physics">Advances in Theoretical and Mathematical Physics</a></i>. <b>9</b> (4): <span class="nowrap">603–</span>665. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/hep-th/0411073">hep-th/0411073</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.4310%2FATMP.2005.v9.n4.a5">10.4310/ATMP.2005.v9.n4.a5</a></span>.</cite></li></ul>

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</style><div id="String_theory190" style="font-size:114%;margin:0 4em"><a href="String_theory" title="String theory">String theory</a></div></th></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Background</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

<li><a href="Exceptional_Lie_group" class="mw-redirect" title="Exceptional Lie group">Exceptional Lie groups</a> (<a href="G2_(mathematics)" title="G2 (mathematics)">G<sub>2</sub></a>, <a href="F4_(mathematics)" title="F4 (mathematics)">F<sub>4</sub></a>, <a href="E6_(mathematics)" title="E6 (mathematics)">E<sub>6</sub></a>, <a href="E7_(mathematics)" title="E7 (mathematics)">E<sub>7</sub></a>, <a href="E8_(mathematics)" title="E8 (mathematics)">E<sub>8</sub></a>)</li>

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

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

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

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

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

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

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

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

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

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

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

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

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

<li><a href="G2_manifold" title="G2 manifold">G<sub>2</sub> manifold</a></li>

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

This article is issued from <a class="external text" title="Last edited on 2025-04-29" href="https://en.wikipedia.org/wiki/?title=Generalized_complex_structure&oldid=1288019306">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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