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href="#Gauged_supergravity_versus_Yang–Mills_supergravity"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Gauged supergravity versus Yang–Mills supergravity</span> </div> </a> <ul id="toc-Gauged_supergravity_versus_Yang–Mills_supergravity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Counting_gravitinos" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Counting_gravitinos"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Counting gravitinos</span> </div> </a> <ul id="toc-Counting_gravitinos-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-A_classification_of_spinors" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#A_classification_of_spinors"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>A classification of spinors</span> </div> </a> <ul id="toc-A_classification_of_spinors-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Choosing_chiralities" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Choosing_chiralities"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Choosing chiralities</span> </div> </a> <ul id="toc-Choosing_chiralities-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Counting_supersymmetries" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Counting_supersymmetries"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Counting supersymmetries</span> </div> </a> <ul id="toc-Counting_supersymmetries-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Examples</span> </div> </a> <button aria-controls="toc-Examples-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Examples subsection</span> </button> <ul id="toc-Examples-sublist" class="vector-toc-list"> <li id="toc-Maximal_supergravity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Maximal_supergravity"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Maximal supergravity</span> </div> </a> <ul id="toc-Maximal_supergravity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-A_12-dimensional_two-time_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#A_12-dimensional_two-time_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>A 12-dimensional two-time theory</span> </div> </a> <ul id="toc-A_12-dimensional_two-time_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-11-dimensional_maximal_SUGRA" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#11-dimensional_maximal_SUGRA"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3</span> <span>11-dimensional maximal SUGRA</span> </div> </a> <ul id="toc-11-dimensional_maximal_SUGRA-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-10d_SUGRA_theories" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#10d_SUGRA_theories"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.4</span> <span>10d SUGRA theories</span> </div> </a> <ul id="toc-10d_SUGRA_theories-sublist" class="vector-toc-list"> <li id="toc-Type_IIA_SUGRA:_N_=_(1,_1)" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Type_IIA_SUGRA:_N_=_(1,_1)"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.4.1</span> <span>Type IIA SUGRA: <i>N</i> = (1, 1)</span> </div> </a> <ul id="toc-Type_IIA_SUGRA:_N_=_(1,_1)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-IIA_SUGRA_from_11d_SUGRA" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#IIA_SUGRA_from_11d_SUGRA"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.4.2</span> <span>IIA SUGRA from 11d SUGRA</span> </div> </a> <ul id="toc-IIA_SUGRA_from_11d_SUGRA-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Type_IIB_SUGRA:_N_=_(2,_0)" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Type_IIB_SUGRA:_N_=_(2,_0)"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.4.3</span> <span>Type IIB SUGRA: <i>N</i> = (2, 0)</span> </div> </a> <ul id="toc-Type_IIB_SUGRA:_N_=_(2,_0)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Type_I_gauged_SUGRA:_N_=_(1,_0)" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Type_I_gauged_SUGRA:_N_=_(1,_0)"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.4.4</span> <span>Type I gauged SUGRA: <i>N</i> = (1, 0)</span> </div> </a> <ul id="toc-Type_I_gauged_SUGRA:_N_=_(1,_0)-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-9d_SUGRA_theories" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#9d_SUGRA_theories"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.5</span> <span>9d SUGRA theories</span> </div> </a> <ul id="toc-9d_SUGRA_theories-sublist" class="vector-toc-list"> <li id="toc-Maximal_9d_SUGRA_from_10d" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Maximal_9d_SUGRA_from_10d"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.5.1</span> <span>Maximal 9d SUGRA from 10d</span> </div> </a> <ul id="toc-Maximal_9d_SUGRA_from_10d-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-T-duality" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#T-duality"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.5.2</span> <span>T-duality</span> </div> </a> <ul id="toc-T-duality-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-N_=_1_gauged_SUGRA" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#N_=_1_gauged_SUGRA"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.5.3</span> <span><i>N</i> = 1 gauged SUGRA</span> </div> </a> <ul id="toc-N_=_1_gauged_SUGRA-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-The_mathematics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#The_mathematics"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>The mathematics</span> </div> </a> <ul id="toc-The_mathematics-sublist" class="vector-toc-list"> </ul> </li> <li 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.ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_footnotes_needed plainlinks metadata ambox ambox-style ambox-More_footnotes_needed" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article includes a list of <a href="/wiki/Wikipedia:Citing_sources#General_references" title="Wikipedia:Citing sources">general references</a>, but <b>it lacks sufficient corresponding <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a></b>.<span class="hide-when-compact"> Please help to <a href="/wiki/Wikipedia:WikiProject_Reliability" title="Wikipedia:WikiProject Reliability">improve</a> this article by <a href="/wiki/Wikipedia:When_to_cite" title="Wikipedia:When to cite">introducing</a> more precise citations.</span> <span class="date-container"><i>(<span class="date">August 2009</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p><b>Higher-dimensional <a href="/wiki/Supergravity" title="Supergravity">supergravity</a></b> is the supersymmetric generalization of <a href="/wiki/General_relativity" title="General relativity">general relativity</a> in higher dimensions. Supergravity can be formulated in any number of dimensions up to eleven. This article focuses upon supergravity (SUGRA) in greater than four dimensions. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Supermultiplets">Supermultiplets</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Higher-dimensional_supergravity&action=edit&section=1" title="Edit section: Supermultiplets"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Fields related by supersymmetry transformations form a <a href="/wiki/Supermultiplet" title="Supermultiplet">supermultiplet</a>; the one that contains a graviton is called the <a href="/w/index.php?title=Supergravity_multiplet&action=edit&redlink=1" class="new" title="Supergravity multiplet (page does not exist)">supergravity multiplet</a>. </p><p>The name of a supergravity theory generally includes the number of dimensions of <a href="/wiki/Spacetime" title="Spacetime">spacetime</a> that it inhabits, and also the number <span class="mwe-math-element"><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 {N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7551c7bed2cd2ee83e10536d157c94a5f8f72fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.062ex; width:2.337ex; height:2.509ex;" alt="{\displaystyle {\mathcal {N}}}"></span> of <a href="/wiki/Gravitino" title="Gravitino">gravitinos</a> that it has. Sometimes one also includes the choices of supermultiplets in the name of theory. For example, an <span class="mwe-math-element"><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 {N}}=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81f52d38c18106a1322d74903137ab9a0c87b4d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.062ex; width:6.597ex; height:2.509ex;" alt="{\displaystyle {\mathcal {N}}=2}"></span>, (9 + 1)-dimensional supergravity enjoys 9 spatial dimensions, one time and 2 <a href="/wiki/Gravitinos" class="mw-redirect" title="Gravitinos">gravitinos</a>. While the field content of different supergravity theories varies considerably, all supergravity theories contain at least one gravitino and they all contain a single <a href="/wiki/Graviton" title="Graviton">graviton</a>. Thus every supergravity theory contains a single supergravity supermultiplet. It is still not known whether one can construct theories with multiple gravitons that are not equivalent to multiple decoupled theories with a single graviton in each<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (March 2007)">citation needed</span></a></i>]</sup>. In <a href="/wiki/Maximal_supergravity" class="mw-redirect" title="Maximal supergravity">maximal supergravity</a> theories (see below), all fields are related by supersymmetry transformations so that there is only one supermultiplet: the supergravity multiplet. </p> <div class="mw-heading mw-heading2"><h2 id="Gauged_supergravity_versus_Yang–Mills_supergravity"><span id="Gauged_supergravity_versus_Yang.E2.80.93Mills_supergravity"></span>Gauged supergravity versus Yang–Mills supergravity</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Higher-dimensional_supergravity&action=edit&section=2" title="Edit section: Gauged supergravity versus Yang–Mills supergravity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Often an abuse of nomenclature is used when "gauge supergravity" refers to a supergravity theory in which fields in the theory are charged with respect to vector fields in the theory. However, when the distinction is important, the following is the correct nomenclature. If a global (i.e. rigid) <a href="/wiki/R-symmetry" title="R-symmetry">R-symmetry</a> is gauged, the gravitino is charged with respect to some vector fields, and the theory is called <a href="/wiki/Gauged_supergravity" title="Gauged supergravity">gauged supergravity</a>. When other global (rigid) symmetries (e.g., if the theory is a <a href="/wiki/Non-linear_sigma_model" title="Non-linear sigma model">non-linear sigma model</a>) of the theory are gauged such that some (non-gravitino) fields are charged with respect to vectors, it is known as a Yang–Mills–Einstein supergravity theory. Of course, one can imagine having a "gauged Yang–Mills–Einstein" theory using a combination of the above gaugings. </p> <div class="mw-heading mw-heading2"><h2 id="Counting_gravitinos">Counting gravitinos</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Higher-dimensional_supergravity&action=edit&section=3" title="Edit section: Counting gravitinos"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Gravitinos are fermions, which means that according to the <a href="/wiki/Spin-statistics_theorem" class="mw-redirect" title="Spin-statistics theorem">spin-statistics theorem</a> they must have an odd number of spinorial indices. In fact the gravitino field has one <a href="/wiki/Spinor" title="Spinor">spinor</a> and one <a href="/wiki/Vector_(geometric)" class="mw-redirect" title="Vector (geometric)">vector</a> index, which means that gravitinos transform as a <a href="/wiki/Tensor_product" title="Tensor product">tensor product</a> of a spinorial <a href="/wiki/Group_representation" title="Group representation">representation</a> and the vector representation of the <a href="/wiki/Lorentz_group" title="Lorentz group">Lorentz group</a>. This is a <a href="/wiki/Rarita%E2%80%93Schwinger_spinor" class="mw-redirect" title="Rarita–Schwinger spinor">Rarita–Schwinger spinor</a>. </p><p>While there is only one vector representation for each Lorentz group, in general there are several different spinorial representations. Technically these are really representations of the <a href="/wiki/Double_covering_group" class="mw-redirect" title="Double covering group">double cover</a> of the Lorentz group called a <a href="/wiki/Spin_group" title="Spin group">spin group</a>. </p><p>The canonical example of a spinorial representation is the <a href="/wiki/Dirac_spinor" title="Dirac spinor">Dirac spinor</a>, which exists in every number of space-time dimensions. However the Dirac spinor representation is not always irreducible. When calculating the number <span class="mwe-math-element"><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 {N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7551c7bed2cd2ee83e10536d157c94a5f8f72fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.062ex; width:2.337ex; height:2.509ex;" alt="{\displaystyle {\mathcal {N}}}"></span>, one always counts the number of <i>real</i> irreducible representations. The spinors with spins less than 3/2 that exist in each number of dimensions will be classified in the following subsection. </p> <div class="mw-heading mw-heading2"><h2 id="A_classification_of_spinors">A classification of spinors</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Higher-dimensional_supergravity&action=edit&section=4" title="Edit section: A classification of spinors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The available spinor representations depends on <i>k</i>; the <a href="/wiki/Maximal_compact_subgroup" title="Maximal compact subgroup">maximal compact subgroup</a> of the <a href="/wiki/Little_group" class="mw-redirect" title="Little group">little group</a> of the <a href="/wiki/Lorentz_group" title="Lorentz group">Lorentz group</a> that preserves the <a href="/wiki/Momentum" title="Momentum">momentum</a> of a massless <a href="/wiki/Elementary_particle" title="Elementary particle">particle</a> is Spin(<i>d</i> − 1) × Spin(<i>d</i> − <i>k</i> − 1), where <i>k</i> is equal to the number <i>d</i> of spatial dimensions minus the number <i>d</i> − <i>k</i> of time dimensions. (See <a href="/wiki/Helicity_(particle_physics)" title="Helicity (particle physics)">helicity (particle physics)</a>) For example, in our world, this is 3 − 1 = 2. Due to the mod 8 <a href="/wiki/Bott_periodicity" class="mw-redirect" title="Bott periodicity">Bott periodicity</a> of the <a href="/wiki/Homotopy_group" title="Homotopy group">homotopy groups</a> of the Lorentz group, really we only need to consider <i>k</i> modulo 8. </p><p>For any value of <i>k</i> there is a Dirac representation, which is always of real dimension <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{1+\lfloor {\frac {2d-k}{2}}\rfloor }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>+</mo> <mo fence="false" stretchy="false">⌊</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>d</mi> <mo>−</mo> <mi>k</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo fence="false" stretchy="false">⌋</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{1+\lfloor {\frac {2d-k}{2}}\rfloor }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d90f7fad554ecf2cf3c319303290f3b10a1cef2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.89ex; height:3.676ex;" alt="{\displaystyle 2^{1+\lfloor {\frac {2d-k}{2}}\rfloor }}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lfloor x\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⌊</mo> <mi>x</mi> <mo fence="false" stretchy="false">⌋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lfloor x\rfloor }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/738c94c88678dd08a289f90a47a609ce44eedf14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.394ex; height:2.843ex;" alt="{\displaystyle \lfloor x\rfloor }"></span> is the greatest integer less than or equal to x. When <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -2\leq k\leq 2{\pmod {8}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>2</mn> <mo>≤</mo> <mi>k</mi> <mo>≤</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>8</mn> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -2\leq k\leq 2{\pmod {8}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ad185257b0e621502a0a92235d5b0ef0e4ccfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.388ex; height:2.843ex;" alt="{\displaystyle -2\leq k\leq 2{\pmod {8}}}"></span> there is a real <a href="/wiki/Majorana_spinor" class="mw-redirect" title="Majorana spinor">Majorana spinor</a> representation, whose dimension is half that of the Dirac representation. When <i>k</i> is even there is a <a href="/wiki/Weyl_spinor" class="mw-redirect" title="Weyl spinor">Weyl spinor</a> representation, whose real dimension is again half that of the Dirac spinor. Finally when <i>k</i> is divisible by eight, that is, when <i>k</i> is zero modulo eight, there is a <a href="/wiki/Majorana%E2%80%93Weyl_spinor" class="mw-redirect" title="Majorana–Weyl spinor">Majorana–Weyl spinor</a>, whose real dimension is one quarter that of the Dirac spinor. </p><p>Occasionally one also considers <a href="/w/index.php?title=Symplectic_Majorana_spinor&action=edit&redlink=1" class="new" title="Symplectic Majorana spinor (page does not exist)">symplectic Majorana spinor</a> which exist when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\leq k\leq 5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>≤</mo> <mi>k</mi> <mo>≤</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3\leq k\leq 5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a287ba2c248790873f55ace5af8189e5c910d1d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.733ex; height:2.343ex;" alt="{\displaystyle 3\leq k\leq 5}"></span>, which have half has many components as Dirac spinors. When <i>k</i>=4 these may also be Weyl, yielding Weyl symplectic Majorana spinors which have one quarter as many components as Dirac spinors. </p> <div class="mw-heading mw-heading2"><h2 id="Choosing_chiralities">Choosing chiralities</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Higher-dimensional_supergravity&action=edit&section=5" title="Edit section: Choosing chiralities"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Spinors in <i>n</i>-dimensions are <a href="/wiki/Representations" title="Representations">representations</a> (really <a href="/wiki/Module_(mathematics)" title="Module (mathematics)">modules</a>) not only of the <i>n</i>-dimensional Lorentz group, but also of a Lie algebra called the <i>n</i>-dimensional <a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebra</a>. The most commonly used basis of the complex <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{\lfloor n\rfloor }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⌊</mo> <mi>n</mi> <mo fence="false" stretchy="false">⌋</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{\lfloor n\rfloor }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69c6fb25b63a31ea3d9da4307ef5c31d54bffa39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.841ex; height:2.843ex;" alt="{\displaystyle 2^{\lfloor n\rfloor }}"></span>-dimensional representation of the Clifford algebra, the representation that acts on the Dirac spinors, consists of the <a href="/wiki/Gamma_matrices" title="Gamma matrices">gamma matrices</a>. </p><p>When <i>n</i> is even the product of all of the gamma matrices, which is often referred to as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma _{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/995f52bbcfd25c6cbc757bcef1bc21d63d96ac88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.507ex; height:2.509ex;" alt="{\displaystyle \Gamma _{5}}"></span> as it was first considered in the case <i>n</i> = 4, is not itself a member of the Clifford algebra. However, being a product of elements of the Clifford algebra, it is in the algebra's universal cover and so has an action on the Dirac spinors. </p><p>In particular, the Dirac spinors may be decomposed into eigenspaces of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma _{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/995f52bbcfd25c6cbc757bcef1bc21d63d96ac88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.507ex; height:2.509ex;" alt="{\displaystyle \Gamma _{5}}"></span> with eigenvalues equal to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm (-1)^{-k/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm (-1)^{-k/2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc9cacb4b0c7c5cae3cfe99bcd72fe20a4b6693d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.599ex; height:3.343ex;" alt="{\displaystyle \pm (-1)^{-k/2}}"></span>, where <i>k</i> is the number of spatial minus temporal dimensions in the spacetime. The spinors in these two eigenspaces each form projective representations of the Lorentz group, known as <a href="/wiki/Weyl_spinor" class="mw-redirect" title="Weyl spinor">Weyl spinors</a>. The eigenvalue under <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma _{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/995f52bbcfd25c6cbc757bcef1bc21d63d96ac88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.507ex; height:2.509ex;" alt="{\displaystyle \Gamma _{5}}"></span> is known as the <a href="/wiki/Chirality_(mathematics)" title="Chirality (mathematics)">chirality</a> of the spinor, which can be left or right-handed. </p><p>A particle that transforms as a single Weyl spinor is said to be chiral. The <a href="/wiki/CPT_theorem" class="mw-redirect" title="CPT theorem">CPT theorem</a>, which is required by Lorentz invariance in <a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a>, implies that when there is a single time direction such particles have antiparticles of the opposite chirality. </p><p>Recall that the eigenvalues of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma _{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/995f52bbcfd25c6cbc757bcef1bc21d63d96ac88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.507ex; height:2.509ex;" alt="{\displaystyle \Gamma _{5}}"></span>, whose eigenspaces are the two chiralities, are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm (-1)^{-k/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm (-1)^{-k/2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc9cacb4b0c7c5cae3cfe99bcd72fe20a4b6693d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.599ex; height:3.343ex;" alt="{\displaystyle \pm (-1)^{-k/2}}"></span>. In particular, when <i>k</i> is equal to <a href="/wiki/Singly_even" class="mw-redirect" title="Singly even">two modulo four</a> the two eigenvalues are complex conjugate and so the two chiralities of Weyl representations are complex conjugate representations. </p><p>Complex conjugation in quantum theories corresponds to time inversion. Therefore, the CPT theorem implies that when the number of Minkowski dimensions is <a href="/wiki/Doubly_even" class="mw-redirect" title="Doubly even">divisible by four</a> (so that <i>k</i> is equal to 2 modulo 4) there be an equal number of left-handed and right-handed supercharges. On the other hand, if the dimension is equal to 2 modulo 4, there can be different numbers of left and right-handed supercharges, and so often one labels the theory by a doublet <span class="mwe-math-element"><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 {N}}=({\mathcal {N}}_{L},{\mathcal {N}}_{R})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}=({\mathcal {N}}_{L},{\mathcal {N}}_{R})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b999ea32543a62e7b3bc92791fe160e3cbdf9ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.062ex; width:14.921ex; height:3.009ex;" alt="{\displaystyle {\mathcal {N}}=({\mathcal {N}}_{L},{\mathcal {N}}_{R})}"></span> where <span class="mwe-math-element"><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 {N}}_{L}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}_{L}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a60ac5041c6d2be281de8d457c422deec869f55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.062ex; width:3.319ex; height:2.843ex;" alt="{\displaystyle {\mathcal {N}}_{L}}"></span> and <span class="mwe-math-element"><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 {N}}_{R}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}_{R}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/533f4bfe3a3bbf30f8323385e8075bb8d466a494" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.062ex; width:3.447ex; height:2.843ex;" alt="{\displaystyle {\mathcal {N}}_{R}}"></span> are the number of left-handed and right-handed supercharges respectively. </p> <div class="mw-heading mw-heading2"><h2 id="Counting_supersymmetries">Counting supersymmetries</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Higher-dimensional_supergravity&action=edit&section=6" title="Edit section: Counting supersymmetries"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>All supergravity theories are invariant under transformations in the <a href="/wiki/Super-Poincar%C3%A9_algebra" title="Super-Poincaré algebra">super-Poincaré algebra</a>, although individual configurations are not in general invariant under every transformation in this group. The super-Poincaré group is generated by the <a href="/wiki/Super-Poincar%C3%A9_algebra" title="Super-Poincaré algebra">Super-Poincaré algebra</a>, which is a <a href="/wiki/Lie_superalgebra" title="Lie superalgebra">Lie superalgebra</a>. A Lie superalgebra is a <span class="mwe-math-element"><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 {Z} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Z} _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f3b518a93ad59911a077e8f316a4ea4aebc870c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.688ex; height:2.509ex;" alt="{\displaystyle \mathbf {Z} _{2}}"></span> graded algebra in which the elements of degree zero are called bosonic and those of degree one are called fermionic. A commutator, that is an antisymmetric bracket satisfying the <a href="/wiki/Jacobi_identity" title="Jacobi identity">Jacobi identity</a> is defined between each pair of generators of fixed degree except for pairs of fermionic generators, for which instead one defines a symmetric bracket called an anticommutator. </p><p>The fermionic generators are also called <a href="/wiki/Supercharges" class="mw-redirect" title="Supercharges">supercharges</a>. Any configuration which is invariant under any of the supercharges is said to be <a href="/wiki/Bogomol%27nyi%E2%80%93Prasad%E2%80%93Sommerfield_bound" title="Bogomol'nyi–Prasad–Sommerfield bound">BPS</a>, and often <a href="/wiki/Nonrenormalization_theorems" class="mw-redirect" title="Nonrenormalization theorems">nonrenormalization theorems</a> demonstrate that such states are particularly easily treated because they are unaffected by many quantum corrections. </p><p>The supercharges transform as spinors, and the number of irreducible spinors of these fermionic generators is equal to the number of gravitinos <span class="mwe-math-element"><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 {N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7551c7bed2cd2ee83e10536d157c94a5f8f72fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.062ex; width:2.337ex; height:2.509ex;" alt="{\displaystyle {\mathcal {N}}}"></span> defined above. Often <span class="mwe-math-element"><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 {N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7551c7bed2cd2ee83e10536d157c94a5f8f72fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.062ex; width:2.337ex; height:2.509ex;" alt="{\displaystyle {\mathcal {N}}}"></span> is defined to be the number of fermionic generators, instead of the number of gravitinos, because this definition extends to supersymmetric theories without gravity. </p><p>Sometimes it is convenient to characterize theories not by the number <span class="mwe-math-element"><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 {N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7551c7bed2cd2ee83e10536d157c94a5f8f72fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.062ex; width:2.337ex; height:2.509ex;" alt="{\displaystyle {\mathcal {N}}}"></span> of irreducible representations of gravitinos or supercharges, but instead by the total <i>Q</i> of their dimensions. This is because some features of the theory have the same <i>Q</i>-dependence in any number of dimensions. For example, one is often only interested in theories in which all particles have <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin</a> less than or equal to two. This requires that <i>Q</i> not exceed 32, except possibly in special cases in which the supersymmetry is realized in an unconventional, nonlinear fashion with products of bosonic generators in the anticommutators of the fermionic generators. </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Higher-dimensional_supergravity&action=edit&section=7" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Maximal_supergravity">Maximal supergravity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Higher-dimensional_supergravity&action=edit&section=8" title="Edit section: Maximal supergravity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The supergravity theories that have attracted the most interest contain no spins higher than two. This means, in particular, that they do not contain any fields that transform as symmetric tensors of rank higher than two under Lorentz transformations. The consistency of interacting <a href="/wiki/Higher-spin_theory" title="Higher-spin theory">higher spin field theories</a> is, however, presently a field of very active interest. </p><p>The supercharges in every super-Poincaré algebra are generated by a multiplicative basis of <i>m</i> fundamental supercharges, and an additive basis of the supercharges (this definition of supercharges is a bit more broad than that given above) is given by a product of any subset of these <i>m</i> fundamental supercharges. The number of subsets of <i>m</i> elements is 2<sup><i>m</i></sup>, thus the space of supercharges is 2<sup><i>m</i></sup>-dimensional. </p><p>The fields in a supersymmetric theory form representations of the super-Poincaré algebra. It can be shown that when <i>m</i> is greater than 5 there are no representations that contain only fields of spin less than or equal to two. Thus we are interested in the case in which <i>m</i> is less than or equal to 5, which means that the maximal number of supercharges is 32. A supergravity theory with precisely 32 supersymmetries is known as a <b>maximal supergravity</b>. </p><p>Above we saw that the number of supercharges in a spinor depends on the dimension and the signature of spacetime. The supercharges occur in spinors. Thus the above limit on the number of supercharges cannot be satisfied in a spacetime of arbitrary dimension. Below we will describe some of the cases in which it is satisfied. </p> <div class="mw-heading mw-heading3"><h3 id="A_12-dimensional_two-time_theory">A 12-dimensional two-time theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Higher-dimensional_supergravity&action=edit&section=9" title="Edit section: A 12-dimensional two-time theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The highest dimension in which spinors exist with only 32 supercharges is 12. If there are 11 spatial directions and 1 time direction then there will be Weyl and Majorana spinors which both are of dimension 64, and so are too large. However, some authors have considered nonlinear actions of the supersymmetry in which higher spin fields may not appear. </p><p>If instead one considers 10 spatial direction and a <a href="/wiki/Multiple_time_dimensions" title="Multiple time dimensions">second temporal dimension</a> then there is a Majorana–Weyl spinor, which as desired has only 32 components. For an overview of two-time theories by one of their main proponents, <a href="/wiki/Itzhak_Bars" title="Itzhak Bars">Itzhak Bars</a>, see his paper <a rel="nofollow" class="external text" href="http://www.arxiv.org/abs/hep-th/9809034">Two-Time Physics</a> and <a rel="nofollow" class="external text" href="http://xstructure.inr.ac.ru/x-bin/theme2.py?arxiv=hep-th&level=1&index1=4029232">Two-Time Physics on arxiv.org</a>. He considered 12-dimensional supergravity in <a rel="nofollow" class="external text" href="http://www.arxiv.org/abs/hep-th/9604139">Supergravity, p-brane duality and hidden space and time dimensions</a>. </p><p>It was widely, but not universally, thought that two-time theories may have problems. For example, there could be causality problems (disconnect between cause and effect) and unitarity problems (negative probability, ghosts). Also, the <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a>-based approach to quantum mechanics may have to be modified in the presence of a second Hamiltonian for the other time. However, in Two-Time Physics it was demonstrated that such potential problems are solved with an appropriate gauge symmetry. </p><p>Some other two time theories describe low-energy behavior, such as <a href="/wiki/Cumrun_Vafa" title="Cumrun Vafa">Cumrun Vafa</a>'s <a href="/wiki/F-theory" title="F-theory">F-theory</a> that is also formulated with the help of 12 dimensions. F-theory itself however is not a two-time theory. One can understand 2 of the 12-dimensions of F-theory as a bookkeeping device; they should not be confused with the other 10 spacetime coordinates. These two dimensions are somehow dual to each other and should not be treated independently. </p> <div class="mw-heading mw-heading3"><h3 id="11-dimensional_maximal_SUGRA">11-dimensional maximal SUGRA</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Higher-dimensional_supergravity&action=edit&section=10" title="Edit section: 11-dimensional maximal SUGRA"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Eleven-dimensional_supergravity" title="Eleven-dimensional supergravity">Eleven-dimensional supergravity</a></div> <p>This maximal supergravity is the classical limit of <a href="/wiki/M-theory" title="M-theory">M-theory</a>. Classically, we have only one 11-dimensional supergravity theory: 7D hyperspace + 4 common dimensions. Like all maximal supergravities, it contains a single supermultiplet, the supergravity supermultiplet containing the graviton, a Majorana gravitino, and a 3-form gauge field often called the C-field. </p><p>It contains two <a href="/wiki/P-brane" class="mw-redirect" title="P-brane">p-brane</a> solutions, a 2-brane and a 5-brane, which are electrically and magnetically charged, respectively, with respect to the C-field. This means that 2-brane and 5-brane charge are the violations of the Bianchi identities for the dual C-field and original C-field respectively. The supergravity 2-brane and 5-brane are the <a href="/w/index.php?title=Long-wavelength_limits&action=edit&redlink=1" class="new" title="Long-wavelength limits (page does not exist)">long-wavelength limits</a> (see also the historical survey above) of the <a href="/wiki/M2-brane" title="M2-brane">M2-brane</a> and <a href="/wiki/M5-brane" title="M5-brane">M5-brane</a> in M-theory. </p> <div class="mw-heading mw-heading3"><h3 id="10d_SUGRA_theories">10d SUGRA theories</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Higher-dimensional_supergravity&action=edit&section=11" title="Edit section: 10d SUGRA theories"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Type_IIA_SUGRA:_N_=_(1,_1)"><span id="Type_IIA_SUGRA:_N_.3D_.281.2C_1.29"></span>Type IIA SUGRA: <i>N</i> = (1, 1)</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Higher-dimensional_supergravity&action=edit&section=12" title="Edit section: Type IIA SUGRA: N = (1, 1)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Type_IIA_supergravity" title="Type IIA supergravity">Type IIA supergravity</a></div> <p>This maximal supergravity is the classical limit of <a href="/wiki/Type_IIA_string_theory" class="mw-redirect" title="Type IIA string theory">type IIA string theory</a>. The field content of the supergravity supermultiplet consists of a graviton, a Majorana gravitino, a <a href="/wiki/Kalb%E2%80%93Ramond_field" title="Kalb–Ramond field">Kalb–Ramond field</a>, odd-dimensional <a href="/wiki/Ramond%E2%80%93Ramond_field" title="Ramond–Ramond field">Ramond–Ramond</a> gauge potentials, a <a href="/wiki/Dilaton" title="Dilaton">dilaton</a> and a <a href="/wiki/Dilatino" class="mw-redirect" title="Dilatino">dilatino</a>. </p><p>The Bianchi identities of the Ramond–Ramond gauge potentials <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{2k-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{2k-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a454c0738318f0f2b66e282f661e87495b8b6f1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.673ex; height:2.509ex;" alt="{\displaystyle C_{2k-1}}"></span> can be violated by adding sources <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span>, which are called D(8 − 2<i>k</i>)-branes </p> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ddC_{2k-1}=\rho .\,\,\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>d</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>ρ</mi> <mo>.</mo> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ddC_{2k-1}=\rho .\,\,\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7da937168a1b1d4ca1b64b27866c1001dcd5a8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.213ex; height:2.676ex;" alt="{\displaystyle ddC_{2k-1}=\rho .\,\,\,}"></span></dd></dl></dd></dl> <p>In the <a href="/w/index.php?title=Democratic_formulation&action=edit&redlink=1" class="new" title="Democratic formulation (page does not exist)">democratic formulation</a> of type IIA supergravity there exist Ramond–Ramond gauge potentials for 0 < <i>k</i> < 6, which leads to D0-branes (also called D-particles), D2-branes, D4-branes, D6-branes and, if one includes the case <i>k</i> = 0, D8-branes. In addition there are fundamental strings and their electromagnetic duals, which are called <a href="/wiki/NS5-brane" title="NS5-brane">NS5-branes</a>. </p><p>Although obviously there are no −1-form gauge connections, the corresponding 0-form field strength, <i>G</i><sub>0</sub> may exist. This field strength is called the <i>Romans mass</i> and when it is not equal to zero the supergravity theory is called <a href="/w/index.php?title=Massive_IIA_supergravity&action=edit&redlink=1" class="new" title="Massive IIA supergravity (page does not exist)">massive IIA supergravity</a> or <a href="/w/index.php?title=Romans_IIA_supergravity&action=edit&redlink=1" class="new" title="Romans IIA supergravity (page does not exist)">Romans IIA supergravity</a>. From the above Bianchi identity we see that a D8-brane is a domain wall between zones of differing <i>G</i><sub>0</sub>, thus in the presence of a D8-brane at least part of the spacetime will be described by the Romans theory. </p> <div class="mw-heading mw-heading4"><h4 id="IIA_SUGRA_from_11d_SUGRA">IIA SUGRA from 11d SUGRA</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Higher-dimensional_supergravity&action=edit&section=13" title="Edit section: IIA SUGRA from 11d SUGRA"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>IIA SUGRA is the <a href="/wiki/Kaluza%E2%80%93Klein_theory" title="Kaluza–Klein theory">dimensional reduction</a> of 11-dimensional supergravity on a circle. This means that 11d supergravity on the spacetime <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M^{10}\times S^{1}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msup> <mo>×</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M^{10}\times S^{1}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5b017c2279ee1c68781f16df04850d22898cae1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.178ex; height:2.676ex;" alt="{\displaystyle M^{10}\times S^{1}\,}"></span> is equivalent to IIA supergravity on the 10-manifold <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M^{10}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M^{10}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cf1fe719ae664f796a23f374c1313eb705d76d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.762ex; height:2.676ex;" alt="{\displaystyle M^{10}\,}"></span> where one eliminates modes with masses proportional to the inverse radius of the circle <i>S</i><sup>1</sup>. </p><p>In particular the field and brane content of IIA supergravity can be derived via this dimensional reduction procedure. The field <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ccc7daa92085410e0eaef658bb5b07173816962" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.881ex; height:2.509ex;" alt="{\displaystyle G_{0}}"></span> however does not arise from the dimensional reduction, massive IIA is not known to be the dimensional reduction of any higher-dimensional theory. The 1-form Ramond–Ramond potential <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{1}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{1}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3d4f3c02d474547c70098d2529c8636829776a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.103ex; height:2.509ex;" alt="{\displaystyle C_{1}\,}"></span> is the usual 1-form connection that arises from the Kaluza–Klein procedure, it arises from the components of the 11-d metric that contain one index along the compactified circle. The IIA 3-form gauge potential <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{3}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{3}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c069c037a7bc6ec56f89c5903ef582ca7969c120" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.103ex; height:2.509ex;" alt="{\displaystyle C_{3}\,}"></span> is the reduction of the 11d 3-form gauge potential components with indices that do not lie along the circle, while the IIA Kalb–Ramond 2-form B-field consists of those components of the 11-dimensional 3-form with one index along the circle. The higher forms in IIA are not independent degrees of freedom, but are obtained from the lower forms using Hodge duality. </p><p>Similarly the IIA branes descend from the 11-dimension branes and geometry. The IIA D0-brane is a Kaluza–Klein momentum mode along the compactified circle. The IIA fundamental string is an 11-dimensional membrane which wraps the compactified circle. The IIA D2-brane is an 11-dimensional membrane that does not wrap the compactified circle. The IIA D4-brane is an 11-dimensional 5-brane that wraps the compactified circle. The IIA NS5-brane is an 11-dimensional 5-brane that does not wrap the compactified circle. The IIA D6-brane is a Kaluza–Klein monopole, that is, a topological defect in the compact circle fibration. The lift of the IIA D8-brane to 11-dimensions is not known, as one side of the IIA geometry as a nontrivial Romans mass, and an 11-dimensional original of the Romans mass is unknown. </p> <div class="mw-heading mw-heading4"><h4 id="Type_IIB_SUGRA:_N_=_(2,_0)"><span id="Type_IIB_SUGRA:_N_.3D_.282.2C_0.29"></span>Type IIB SUGRA: <i>N</i> = (2, 0)</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Higher-dimensional_supergravity&action=edit&section=14" title="Edit section: Type IIB SUGRA: N = (2, 0)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Type_IIB_supergravity" title="Type IIB supergravity">Type IIB supergravity</a></div> <p>This maximal supergravity is the classical limit of <a href="/wiki/Type_IIB_string_theory" class="mw-redirect" title="Type IIB string theory">type IIB string theory</a>. The field content of the supergravity supermultiplet consists of a graviton, a Weyl gravitino, a <a href="/wiki/Kalb%E2%80%93Ramond_field" title="Kalb–Ramond field">Kalb–Ramond field</a>, even-dimensional Ramond–Ramond gauge potentials, a <a href="/wiki/Dilaton" title="Dilaton">dilaton</a> and a <a href="/wiki/Dilatino" class="mw-redirect" title="Dilatino">dilatino</a>. </p><p>The Ramond–Ramond fields are sourced by odd-dimensional D(2<i>k</i> + 1)-branes, which host supersymmetric <i>U</i>(1) gauge theories. As in IIA supergravity, the fundamental string is an electric source for the Kalb–Ramond B-field and the <a href="/wiki/NS5-brane" title="NS5-brane">NS5-brane</a> is a magnetic source. Unlike that of the IIA theory, the NS5-brane hosts a worldvolume <i>U</i>(1) supersymmetric gauge theory with <span class="mwe-math-element"><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 {N}}=(1,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}=(1,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3da3d2b138cfe832a2e23c3442bc53ae9754070e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.062ex; width:10.603ex; height:3.009ex;" alt="{\displaystyle {\mathcal {N}}=(1,1)}"></span> supersymmetry, although some of this supersymmetry may be broken depending on the geometry of the spacetime and the other branes that are present. </p><p>This theory enjoys an SL(2, <b>R</b>) symmetry known as <a href="/wiki/S-duality" title="S-duality">S-duality</a> that interchanges the Kalb–Ramond field and the RR 2-form and also mixes the dilaton and the RR 0-form <a href="/wiki/Axion" title="Axion">axion</a>. </p> <div class="mw-heading mw-heading4"><h4 id="Type_I_gauged_SUGRA:_N_=_(1,_0)"><span id="Type_I_gauged_SUGRA:_N_.3D_.281.2C_0.29"></span>Type I gauged SUGRA: <i>N</i> = (1, 0)</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Higher-dimensional_supergravity&action=edit&section=15" title="Edit section: Type I gauged SUGRA: N = (1, 0)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Type_I_supergravity" title="Type I supergravity">Type I supergravity</a></div> <p>These are the classical limits of <a href="/wiki/Type_I_string_theory" title="Type I string theory">type I string theory</a> and the two <a href="/wiki/Heterotic_string_theory" title="Heterotic string theory">heterotic string theories</a>. There is a single <a href="/wiki/Majorana%E2%80%93Weyl_spinor" class="mw-redirect" title="Majorana–Weyl spinor">Majorana–Weyl spinor</a> of supercharges, which in 10 dimensions contains 16 supercharges. As 16 is less than 32, the maximal number of supercharges, type I is not a maximal supergravity theory. </p><p>In particular this implies that there is more than one variety of supermultiplet. In fact, there are two. As usual, there is a supergravity supermultiplet. This is smaller than the supergravity supermultiplet in type II, it contains only the <a href="/wiki/Graviton" title="Graviton">graviton</a>, a Majorana–Weyl <a href="/wiki/Gravitino" title="Gravitino">gravitino</a>, a 2-form gauge potential, the dilaton and a dilatino. Whether this 2-form is considered to be a Kalb–Ramond field or <a href="/wiki/Ramond%E2%80%93Ramond_field" title="Ramond–Ramond field">Ramond–Ramond field</a> depends on whether one considers the supergravity theory to be a classical limit of a <a href="/wiki/Heterotic_string_theory" title="Heterotic string theory">heterotic string theory</a> or <a href="/wiki/Type_I_string_theory" title="Type I string theory">type I string theory</a>. There is also a <a href="/wiki/Vector_supermultiplet" class="mw-redirect" title="Vector supermultiplet">vector supermultiplet</a>, which contains a one-form gauge potential called a <a href="/wiki/Gluon" title="Gluon">gluon</a> and also a Majorana–Weyl <a href="/wiki/Gluino" title="Gluino">gluino</a>. </p><p>Unlike type IIA and IIB supergravities, for which the classical theory is unique, as a classical theory <span class="mwe-math-element"><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 {N}}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e733b2655ea8edffe0763ec9314deb788293e3ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.062ex; width:6.597ex; height:2.509ex;" alt="{\displaystyle {\mathcal {N}}=1}"></span> supergravity is consistent with a single supergravity supermultiplet and any number of vector multiplets. It is also consistent without the supergravity supermultiplet, but then it would contain no graviton and so would not be a supergravity theory. While one may add multiple supergravity supermultiplets, it is not known if they may consistently interact. One is free not only to determine the number, if any, of vector supermultiplets, but also there is some freedom in determining their couplings. They must describe a classical <a href="/wiki/N_%3D_1_supersymmetric_Yang%E2%80%93Mills_theory" title="N = 1 supersymmetric Yang–Mills theory">super Yang–Mills</a> <a href="/wiki/Gauge_theory" title="Gauge theory">gauge theory</a>, but the choice of gauge group is arbitrary. In addition one is free to make some choices of gravitational couplings in the classical theory. </p><p>While there are many varieties of classical <span class="mwe-math-element"><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 {N}}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e733b2655ea8edffe0763ec9314deb788293e3ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.062ex; width:6.597ex; height:2.509ex;" alt="{\displaystyle {\mathcal {N}}=1}"></span> supergravities, not all of these varieties are the classical limits of quantum theories. Generically the quantum versions of these theories suffer from various anomalies, as can be seen already at 1-loop in the <a href="/wiki/Hexagon" title="Hexagon">hexagon</a> <a href="/wiki/Feynman_diagrams" class="mw-redirect" title="Feynman diagrams">Feynman diagrams</a>. In 1984 and 1985 <a href="/wiki/Michael_Green_(physicist)" title="Michael Green (physicist)">Michael Green</a> and <a href="/wiki/John_H._Schwarz" class="mw-redirect" title="John H. Schwarz">John H. Schwarz</a> have shown that if one includes precisely 496 vector supermultiplets and chooses certain couplings of the 2-form and the metric then the <a href="/wiki/Gravitational_anomaly" title="Gravitational anomaly">gravitational anomalies</a> cancel. This is called the <a href="/wiki/Green%E2%80%93Schwarz_mechanism" title="Green–Schwarz mechanism">Green–Schwarz anomaly cancellation mechanism</a>. </p><p>In addition, anomaly cancellation requires one to cancel the <a href="/wiki/Gauge_anomaly" title="Gauge anomaly">gauge anomalies</a>. This fixes the gauge symmetry algebra to be either <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {so}}(32)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">o</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>32</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {so}}(32)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d5c5091f1015752efabf1efdab413cf71c311fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:6.342ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {so}}(32)}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {e}}_{8}\oplus {\mathfrak {e}}_{8}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> <mo>⊕</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {e}}_{8}\oplus {\mathfrak {e}}_{8}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eaf4d0edc77f354e5c3a9d250761dd5530c98c59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.814ex; height:2.343ex;" alt="{\displaystyle {\mathfrak {e}}_{8}\oplus {\mathfrak {e}}_{8}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {e}}_{8}\oplus 248{\mathfrak {u}}(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> <mo>⊕</mo> <mn>248</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">u</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {e}}_{8}\oplus 248{\mathfrak {u}}(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89b029a6304ca65600d7a50d1dcc43bae3bc9dae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.488ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {e}}_{8}\oplus 248{\mathfrak {u}}(1)}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 496{\mathfrak {u}}(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>496</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">u</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 496{\mathfrak {u}}(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91784a9fa48b6bed13268dc5542eb662409a8045" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.661ex; height:2.843ex;" alt="{\displaystyle 496{\mathfrak {u}}(1)}"></span>. However, only the first two Lie algebras can be gotten from superstring theory<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (March 2007)">citation needed</span></a></i>]</sup>. Quantum theories with at least 8 supercharges tend to have continuous <a href="/wiki/Moduli_space" title="Moduli space">moduli spaces</a> of vacua. In <a href="/wiki/Compactification_(physics)" title="Compactification (physics)">compactifications</a> of these theories, which have 16 supercharges, there exist degenerate vacua with different values of various Wilson loops. Such Wilson loops may be used to break the gauge symmetries to various subgroups. In particular the above gauge symmetries may be broken to obtain not only the standard model gauge symmetry but also symmetry groups such as SO(10) and SU(5) that are popular in <a href="/wiki/GUT_theories" class="mw-redirect" title="GUT theories">GUT theories</a>. </p> <div class="mw-heading mw-heading3"><h3 id="9d_SUGRA_theories">9d SUGRA theories</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Higher-dimensional_supergravity&action=edit&section=16" title="Edit section: 9d SUGRA theories"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 9-dimensional Minkowski space the only irreducible spinor representation is the <a href="/wiki/Majorana_spinor" class="mw-redirect" title="Majorana spinor">Majorana spinor</a>, which has 16 components. Thus supercharges inhabit Majorana spinors of which there are at most two. </p> <div class="mw-heading mw-heading4"><h4 id="Maximal_9d_SUGRA_from_10d">Maximal 9d SUGRA from 10d</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Higher-dimensional_supergravity&action=edit&section=17" title="Edit section: Maximal 9d SUGRA from 10d"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In particular, if there are two Majorana spinors then one obtains the 9-dimensional maximal supergravity theory. Recall that in 10 dimensions there were two inequivalent maximal supergravity theories, IIA and IIB. The <a href="/wiki/Dimensional_reduction" title="Dimensional reduction">dimensional reduction</a> of either IIA or IIB on a circle is the unique 9-dimensional supergravity. In other words, IIA or IIB on the product of a 9-dimensional space <i>M</i><sup>9</sup> and a circle is equivalent to the 9-dimension theory on <i>M</i><sup>9</sup>, with Kaluza–Klein modes if one does not take the limit in which the circle shrinks to zero. </p> <div class="mw-heading mw-heading4"><h4 id="T-duality">T-duality</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Higher-dimensional_supergravity&action=edit&section=18" title="Edit section: T-duality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>More generally one could consider the 10-dimensional theory on a nontrivial <a href="/wiki/Circle_bundle" title="Circle bundle">circle bundle</a> over <i>M</i><sup>9</sup>. Dimensional reduction still leads to a 9-dimensional theory on <i>M</i><sup>9</sup>, but with a 1-form <a href="/wiki/Gauge_potential" class="mw-redirect" title="Gauge potential">gauge potential</a> equal to the <a href="/wiki/Connection_(mathematics)" title="Connection (mathematics)">connection</a> of the circle bundle and a 2-form <a href="/wiki/Field_strength" title="Field strength">field strength</a> which is equal to the <a href="/wiki/Chern_class" title="Chern class">Chern class</a> of the old circle bundle. One may then lift this theory to the other 10-dimensional theory, in which case one finds that the 1-form gauge potential lifts to the Kalb–Ramond field. Similarly, the connection of the fibration of the circle in the second 10-dimensional theory is the integral of the Kalb–Ramond field of the original theory over the compactified circle. </p><p>This transformation between the two 10-dimensional theories is known as <a href="/wiki/T-duality" title="T-duality">T-duality</a>. While T-duality in supergravity involves dimensional reduction and so loses information, in the full quantum <a href="/wiki/String_theory" title="String theory">string theory</a> the extra information is stored in string winding modes and so T-duality is a <a href="/wiki/String_duality" title="String duality">duality</a> between the two 10-dimensional theories. The above construction can be used to obtain the relation between the circle bundle's connection and dual Kalb–Ramond field even in the full quantum theory. </p> <div class="mw-heading mw-heading4"><h4 id="N_=_1_gauged_SUGRA"><span id="N_.3D_1_gauged_SUGRA"></span><i>N</i> = 1 gauged SUGRA</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Higher-dimensional_supergravity&action=edit&section=19" title="Edit section: N = 1 gauged SUGRA"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Gauged_supergravity" title="Gauged supergravity">Gauged supergravity</a></div> <p>As was the case in the parent 10-dimensional theory, 9-dimensional N=1 supergravity contains a single supergravity multiplet and an arbitrary number of vector multiplets. These vector multiplets may be coupled so as to admit arbitrary gauge theories, although not all possibilities have quantum completions. Unlike the 10-dimensional theory, as was described in the previous subsection, the supergravity multiplet itself contains a vector and so there will always be at least a U(1) gauge symmetry, even in the N=2 case. </p> <div class="mw-heading mw-heading2"><h2 id="The_mathematics">The mathematics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Higher-dimensional_supergravity&action=edit&section=20" title="Edit section: The mathematics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Lagrangian_(field_theory)" title="Lagrangian (field theory)">Lagrangian</a> for <a href="/wiki/Eleven-dimensional_supergravity" title="Eleven-dimensional supergravity">11D supergravity</a> found by brute force by Cremmer, Julia and Scherk<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> is: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{rcl}L&=&+{\frac {1}{2\kappa ^{2}}}eR-{\frac {1}{2}}e{\overline {\psi }}_{M}\Gamma ^{MNP}D_{N}[{\frac {1}{2}}(\omega -{\overline {\omega }})]\psi _{P}\\&&+{\frac {1}{48}}eF_{MNPQ}^{2}+{\frac {{\sqrt {2}}\kappa }{384}}e({\overline {\psi }}_{M}\Gamma ^{MNPQRS}\psi _{S}\\&&+12{\overline {\psi }}^{N}\Gamma ^{PQ}\psi ^{R})(F+{\overline {F}})_{NPQR}+{\frac {{\sqrt {2}}\kappa }{3456}}\varepsilon ^{M_{1}\dots M_{11}}F_{M_{1}\dots M_{4}}F_{M_{5}\dots M_{8}}A_{M_{9}M_{10}M_{11}}\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right center left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>L</mi> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <msup> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>e</mi> <mi>R</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>e</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <msup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> <mi>N</mi> <mi>P</mi> </mrow> </msup> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd /> <mtd> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>48</mn> </mfrac> </mrow> <mi>e</mi> <msubsup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> <mi>N</mi> <mi>P</mi> <mi>Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mi>κ</mi> </mrow> <mn>384</mn> </mfrac> </mrow> <mi>e</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <msup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> <mi>N</mi> <mi>P</mi> <mi>Q</mi> <mi>R</mi> <mi>S</mi> </mrow> </msup> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd /> <mtd> <mo>+</mo> <mn>12</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msup> <msup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>Q</mi> </mrow> </msup> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>F</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo accent="false">¯</mo> </mover> </mrow> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mi>P</mi> <mi>Q</mi> <mi>R</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mi>κ</mi> </mrow> <mn>3456</mn> </mfrac> </mrow> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mrow> </msup> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msub> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rcl}L&=&+{\frac {1}{2\kappa ^{2}}}eR-{\frac {1}{2}}e{\overline {\psi }}_{M}\Gamma ^{MNP}D_{N}[{\frac {1}{2}}(\omega -{\overline {\omega }})]\psi _{P}\\&&+{\frac {1}{48}}eF_{MNPQ}^{2}+{\frac {{\sqrt {2}}\kappa }{384}}e({\overline {\psi }}_{M}\Gamma ^{MNPQRS}\psi _{S}\\&&+12{\overline {\psi }}^{N}\Gamma ^{PQ}\psi ^{R})(F+{\overline {F}})_{NPQR}+{\frac {{\sqrt {2}}\kappa }{3456}}\varepsilon ^{M_{1}\dots M_{11}}F_{M_{1}\dots M_{4}}F_{M_{5}\dots M_{8}}A_{M_{9}M_{10}M_{11}}\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d11f95d3ddc85b3fc909e2987761f99766a527b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.058ex; margin-bottom: -0.28ex; width:81.553ex; height:13.843ex;" alt="{\displaystyle {\begin{array}{rcl}L&=&+{\frac {1}{2\kappa ^{2}}}eR-{\frac {1}{2}}e{\overline {\psi }}_{M}\Gamma ^{MNP}D_{N}[{\frac {1}{2}}(\omega -{\overline {\omega }})]\psi _{P}\\&&+{\frac {1}{48}}eF_{MNPQ}^{2}+{\frac {{\sqrt {2}}\kappa }{384}}e({\overline {\psi }}_{M}\Gamma ^{MNPQRS}\psi _{S}\\&&+12{\overline {\psi }}^{N}\Gamma ^{PQ}\psi ^{R})(F+{\overline {F}})_{NPQR}+{\frac {{\sqrt {2}}\kappa }{3456}}\varepsilon ^{M_{1}\dots M_{11}}F_{M_{1}\dots M_{4}}F_{M_{5}\dots M_{8}}A_{M_{9}M_{10}M_{11}}\end{array}}}"></span></dd></dl> <p>which contains the three types of field: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{M}^{A},\psi _{M},A_{MNP}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msubsup> <mo>,</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> <mi>N</mi> <mi>P</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{M}^{A},\psi _{M},A_{MNP}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eaa9a7005a7383a52683ed034b7a03a498e4823a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.979ex; height:3.176ex;" alt="{\displaystyle e_{M}^{A},\psi _{M},A_{MNP}}"></span></dd></dl> <p>The symmetry of this supergravity theory is given by the supergroup OSp(1|32) which gives the subgroups O(1) for the bosonic symmetry and Sp(32) for the fermion symmetry. This is because <a href="/wiki/Spinors" class="mw-redirect" title="Spinors">spinors</a> need 32 components in 11 dimensions. 11D supergravity can be compactified down to 4 dimensions which then has OSp(8|4) symmetry. (We still have 8 × 4 = 32 so there are still the same number of components.) Spinors need 4 components in 4 dimensions. This gives O(8) for the gauge group which is too small to contain the <a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a> gauge group U(1) × SU(2) × SU(3) which would need at least O(10). </p> <div class="mw-heading mw-heading2"><h2 id="Notes_and_references">Notes and references</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Higher-dimensional_supergravity&action=edit&section=21" title="Edit section: Notes and references"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFCremmerJuliaScherk1978" class="citation journal cs1">Cremmer, E.; Julia, B.; Scherk, J. (1978). "Supergravity in theory in 11 dimensions". <i>Physics Letters B</i>. <b>76</b> (4). Elsevier BV: 409–412. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1978PhLB...76..409C">1978PhLB...76..409C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0370-2693%2878%2990894-8">10.1016/0370-2693(78)90894-8</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0370-2693">0370-2693</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Letters+B&rft.atitle=Supergravity+in+theory+in+11+dimensions&rft.volume=76&rft.issue=4&rft.pages=409-412&rft.date=1978&rft.issn=0370-2693&rft_id=info%3Adoi%2F10.1016%2F0370-2693%2878%2990894-8&rft_id=info%3Abibcode%2F1978PhLB...76..409C&rft.aulast=Cremmer&rft.aufirst=E.&rft.au=Julia%2C+B.&rft.au=Scherk%2C+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHigher-dimensional+supergravity" class="Z3988"></span></span> </li> </ol></div></div> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output 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theory">Axiomatic QFT</a></li> <li><a href="/wiki/Conformal_field_theory" title="Conformal field theory">Conformal field theory</a></li> <li><a href="/wiki/Lattice_field_theory" title="Lattice field theory">Lattice field theory</a></li> <li><a href="/wiki/Noncommutative_quantum_field_theory" title="Noncommutative quantum field theory">Noncommutative QFT</a></li> <li><a href="/wiki/Gauge_theory" title="Gauge theory">Gauge theory</a></li> <li><a href="/wiki/Quantum_field_theory_in_curved_spacetime" title="Quantum field theory in curved spacetime">QFT in curved spacetime</a></li> <li><a href="/wiki/String_theory" title="String theory">String theory</a></li> <li><a href="/wiki/Supergravity" title="Supergravity">Supergravity</a></li> <li><a href="/wiki/Thermal_quantum_field_theory" title="Thermal quantum field theory">Thermal QFT</a></li> <li><a href="/wiki/Topological_quantum_field_theory" title="Topological quantum field theory">Topological QFT</a></li> <li><a href="/wiki/Two-dimensional_conformal_field_theory" title="Two-dimensional conformal field theory">Two-dimensional conformal field theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Models</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Regular</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Born%E2%80%93Infeld_model" title="Born–Infeld model">Born–Infeld</a></li> <li><a href="/wiki/Euler%E2%80%93Heisenberg_Lagrangian" title="Euler–Heisenberg Lagrangian">Euler–Heisenberg</a></li> <li><a href="/wiki/Ginzburg%E2%80%93Landau_theory" title="Ginzburg–Landau theory">Ginzburg–Landau</a></li> <li><a href="/wiki/Non-linear_sigma_model" title="Non-linear sigma model">Non-linear sigma</a></li> <li><a href="/wiki/Proca_action" title="Proca action">Proca</a></li> <li><a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">Quantum electrodynamics</a></li> <li><a href="/wiki/Quantum_chromodynamics" title="Quantum chromodynamics">Quantum chromodynamics</a></li> <li><a href="/wiki/Quartic_interaction" title="Quartic interaction">Quartic interaction</a></li> <li><a href="/wiki/Scalar_electrodynamics" title="Scalar electrodynamics">Scalar electrodynamics</a></li> <li><a href="/wiki/Scalar_chromodynamics" title="Scalar chromodynamics">Scalar chromodynamics</a></li> <li><a href="/wiki/Soler_model" title="Soler model">Soler</a></li> <li><a href="/wiki/Yang%E2%80%93Mills_theory" title="Yang–Mills theory">Yang–Mills</a></li> <li><a href="/wiki/Yang%E2%80%93Mills%E2%80%93Higgs_equations" title="Yang–Mills–Higgs equations">Yang–Mills–Higgs</a></li> <li><a href="/wiki/Yukawa_interaction" title="Yukawa interaction">Yukawa</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Low dimensional</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Two-dimensional_Yang%E2%80%93Mills_theory" title="Two-dimensional Yang–Mills theory">2D Yang–Mills</a></li> <li><a href="/wiki/Bullough%E2%80%93Dodd_model" title="Bullough–Dodd model">Bullough–Dodd</a></li> <li><a href="/wiki/Gross%E2%80%93Neveu_model" title="Gross–Neveu model">Gross–Neveu</a></li> <li><a href="/wiki/Schwinger_model" title="Schwinger model">Schwinger</a></li> <li><a href="/wiki/Sine-Gordon_equation" title="Sine-Gordon equation">Sine-Gordon</a></li> <li><a href="/wiki/Thirring_model" title="Thirring model">Thirring</a></li> <li><a href="/wiki/Thirring%E2%80%93Wess_model" title="Thirring–Wess model">Thirring–Wess</a></li> <li><a href="/wiki/Toda_field_theory" title="Toda field theory">Toda</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Conformal</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Massless_free_scalar_bosons_in_two_dimensions" title="Massless free scalar bosons in two dimensions">2D free massless scalar</a></li> <li><a href="/wiki/Liouville_field_theory" title="Liouville field theory">Liouville</a></li> <li><a href="/wiki/Minimal_model_(physics)" title="Minimal model (physics)">Minimal</a></li> <li><a href="/wiki/Polyakov_action" title="Polyakov action">Polyakov</a></li> <li><a href="/wiki/Wess%E2%80%93Zumino%E2%80%93Witten_model" title="Wess–Zumino–Witten model">Wess–Zumino–Witten</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Supersymmetric</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/4D_N_%3D_1_global_supersymmetry" title="4D N = 1 global supersymmetry">4D N = 1</a></li> <li><a href="/wiki/N_%3D_1_supersymmetric_Yang%E2%80%93Mills_theory" title="N = 1 supersymmetric Yang–Mills theory">N = 1 super Yang–Mills</a></li> <li><a href="/wiki/Seiberg%E2%80%93Witten_theory" title="Seiberg–Witten theory">Seiberg–Witten</a></li> <li><a href="/wiki/Super_QCD" title="Super QCD">Super QCD</a></li> <li><a href="/wiki/Wess%E2%80%93Zumino_model" title="Wess–Zumino model">Wess–Zumino</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Superconformal</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/6D_(2,0)_superconformal_field_theory" title="6D (2,0) superconformal field theory">6D (2,0)</a></li> <li><a href="/wiki/ABJM_superconformal_field_theory" title="ABJM superconformal field theory">ABJM</a></li> <li><a href="/wiki/N_%3D_4_supersymmetric_Yang%E2%80%93Mills_theory" title="N = 4 supersymmetric Yang–Mills theory">N = 4 super Yang–Mills</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Supergravity</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pure_4D_N_%3D_1_supergravity" title="Pure 4D N = 1 supergravity">Pure 4D N = 1</a></li> <li><a href="/wiki/4D_N_%3D_1_supergravity" title="4D N = 1 supergravity">4D N = 1</a></li> <li><a href="/wiki/N_%3D_8_supergravity" title="N = 8 supergravity">4D N = 8</a></li> <li><a class="mw-selflink selflink">Higher dimensional</a></li> <li><a href="/wiki/Type_I_supergravity" title="Type I supergravity">Type I</a></li> <li><a href="/wiki/Type_IIA_supergravity" title="Type IIA supergravity">Type IIA</a></li> <li><a href="/wiki/Type_IIB_supergravity" title="Type IIB supergravity">Type IIB</a></li> <li><a href="/wiki/Eleven-dimensional_supergravity" title="Eleven-dimensional supergravity">11D</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Topological</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/BF_model" title="BF model">BF</a></li> <li><a href="/wiki/Chern%E2%80%93Simons_theory" title="Chern–Simons theory">Chern–Simons</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Particle theory</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chiral_model" title="Chiral model">Chiral</a></li> <li><a href="/wiki/Fermi%27s_interaction" title="Fermi's interaction">Fermi</a></li> <li><a href="/wiki/Minimal_Supersymmetric_Standard_Model" title="Minimal Supersymmetric Standard Model">MSSM</a></li> <li><a href="/wiki/Nambu%E2%80%93Jona-Lasinio_model" title="Nambu–Jona-Lasinio model">Nambu–Jona-Lasinio</a></li> <li><a href="/wiki/Next-to-Minimal_Supersymmetric_Standard_Model" title="Next-to-Minimal Supersymmetric Standard Model">NMSSM</a></li> <li><a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a></li> <li><a href="/wiki/Stueckelberg_action" title="Stueckelberg action">Stueckelberg</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Casimir_effect" title="Casimir effect">Casimir effect</a></li> <li><a href="/wiki/Cosmic_string" title="Cosmic string">Cosmic string</a></li> <li><a href="/wiki/History_of_quantum_field_theory" title="History of quantum field theory">History</a></li> <li><a href="/wiki/Loop_quantum_gravity" title="Loop quantum gravity">Loop quantum gravity</a></li> <li><a href="/wiki/Loop_quantum_cosmology" title="Loop quantum cosmology">Loop quantum cosmology</a></li> <li><a href="/wiki/On_shell_and_off_shell" title="On shell and off shell">On shell and off shell</a></li> <li><a href="/wiki/Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li> <li><a href="/wiki/Quantum_dynamics" title="Quantum dynamics">Quantum dynamics</a></li> <li><a href="/wiki/Quantum_foam" title="Quantum foam">Quantum foam</a></li> <li><a href="/wiki/Quantum_fluctuation" title="Quantum fluctuation">Quantum fluctuations</a> <ul><li><a href="/wiki/Template:Quantum_electrodynamics" title="Template:Quantum electrodynamics">links</a></li></ul></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a> <ul><li><a href="/wiki/Template:Quantum_gravity" title="Template:Quantum gravity">links</a></li></ul></li> <li><a href="/wiki/Quantum_hadrodynamics" title="Quantum hadrodynamics">Quantum hadrodynamics</a></li> <li><a href="/wiki/Quantum_hydrodynamics" title="Quantum hydrodynamics">Quantum hydrodynamics</a></li> <li><a href="/wiki/Quantum_information" title="Quantum information">Quantum information</a></li> <li><a href="/wiki/Quantum_information_science" title="Quantum information science">Quantum information science</a> <ul><li><a href="/wiki/Template:Quantum_information" title="Template:Quantum information">links</a></li></ul></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Quantum_thermodynamics" title="Quantum thermodynamics">Quantum thermodynamics</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><i>See also:</i> <span class="noviewer" typeof="mw:File"><span title="Template"><img alt="" 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