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<span>Examples</span> </div> </a> <ul id="toc-Examples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Analytical_character" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Analytical_character"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Analytical character</span> </div> </a> <ul id="toc-Analytical_character-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Topology_of_Ricci-flat_Riemannian_manifolds" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Topology_of_Ricci-flat_Riemannian_manifolds"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Topology of Ricci-flat Riemannian manifolds</span> </div> </a> <ul id="toc-Topology_of_Ricci-flat_Riemannian_manifolds-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ricci-flatness_and_holonomy" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ricci-flatness_and_holonomy"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Ricci-flatness and holonomy</span> </div> </a> <ul id="toc-Ricci-flatness_and_holonomy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input 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class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In the <a href="/wiki/Mathematics" title="Mathematics">mathematical</a> field of <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a>, <b>Ricci-flatness</b> is a condition on the <a href="/wiki/Curvature" title="Curvature">curvature</a> of a <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a>. <b>Ricci-flat manifolds</b> are a special kind of <a href="/wiki/Einstein_manifold" title="Einstein manifold">Einstein manifold</a>. In <a href="/wiki/Theoretical_physics" title="Theoretical physics">theoretical physics</a>, Ricci-flat <a href="/wiki/Lorentzian_manifold" class="mw-redirect" title="Lorentzian manifold">Lorentzian manifolds</a> are of fundamental interest, as they are the solutions of <a href="/wiki/Einstein%27s_field_equation" class="mw-redirect" title="Einstein's field equation">Einstein's field equations</a> in a <a href="/wiki/Vacuum" title="Vacuum">vacuum</a> with vanishing <a href="/wiki/Cosmological_constant" title="Cosmological constant">cosmological constant</a>. </p><p>In Lorentzian geometry, a number of Ricci-flat metrics are known from works of <a href="/wiki/Karl_Schwarzschild" title="Karl Schwarzschild">Karl Schwarzschild</a>, <a href="/wiki/Roy_Kerr" title="Roy Kerr">Roy Kerr</a>, and <a href="/wiki/Yvonne_Choquet-Bruhat" title="Yvonne Choquet-Bruhat">Yvonne Choquet-Bruhat</a>. In <a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a>, <a href="/wiki/Shing-Tung_Yau" title="Shing-Tung Yau">Shing-Tung Yau</a>'s resolution of the <a href="/wiki/Calabi_conjecture" title="Calabi conjecture">Calabi conjecture</a> produced a number of Ricci-flat metrics on <a href="/wiki/K%C3%A4hler_manifold" title="Kähler manifold">Kähler manifolds</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ricci-flat_manifold&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">pseudo-Riemannian manifold</a> is said to be <b>Ricci-flat</b> if its <a href="/wiki/Ricci_curvature" title="Ricci curvature">Ricci curvature</a> is zero.<sup id="cite_ref-FOOTNOTEO'Neill198387_1-0" class="reference"><a href="#cite_note-FOOTNOTEO'Neill198387-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> It is direct to verify that, except in dimension two, a metric is Ricci-flat if and only if its <a href="/wiki/Einstein_tensor" title="Einstein tensor">Einstein tensor</a> is zero.<sup id="cite_ref-FOOTNOTEO'Neill1983336_2-0" class="reference"><a href="#cite_note-FOOTNOTEO'Neill1983336-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Ricci-flat manifolds are one of three special types of <a href="/wiki/Einstein_manifold" title="Einstein manifold">Einstein manifold</a>, arising as the special case of <a href="/wiki/Scalar_curvature" title="Scalar curvature">scalar curvature</a> equaling zero. </p><p>From the definition of the <a href="/wiki/Weyl_curvature_tensor" class="mw-redirect" title="Weyl curvature tensor">Weyl curvature tensor</a>, it is direct to see that any Ricci-flat metric has Weyl curvature equal to <a href="/wiki/Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a>. By taking <a href="/wiki/Trace_(linear_algebra)" title="Trace (linear algebra)">traces</a>, it is straightforward to see that the converse also holds. This may also be phrased as saying that Ricci-flatness is characterized by the vanishing of the two non-Weyl parts of the <a href="/wiki/Ricci_decomposition" title="Ricci decomposition">Ricci decomposition</a>. </p><p>Since the Weyl curvature vanishes in two or three dimensions, every Ricci-flat metric in these dimensions is <a href="/wiki/Flat_manifold" title="Flat manifold">flat</a>. Conversely, it is automatic from the definitions that any flat metric is Ricci-flat. The study of flat metrics is usually considered as a topic unto itself. As such, the study of Ricci-flat metrics is only a distinct topic in dimension four and above. </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=Ricci-flat_manifold&action=edit&section=2" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As noted above, any flat metric is Ricci-flat. However it is nontrivial to identify Ricci-flat manifolds whose full curvature is nonzero. </p><p>In 1916, <a href="/wiki/Karl_Schwarzschild" title="Karl Schwarzschild">Karl Schwarzschild</a> found the <a href="/wiki/Schwarzschild_metric" title="Schwarzschild metric">Schwarzschild metrics</a>, which are Ricci-flat <a href="/wiki/Lorentzian_manifold" class="mw-redirect" title="Lorentzian manifold">Lorentzian manifolds</a> of nonzero curvature.<sup id="cite_ref-FOOTNOTEBesse1987Section_3FMisnerThorneWheeler1973Chapter_31O'Neill1983Chapter_13Schwarzschild1916_3-0" class="reference"><a href="#cite_note-FOOTNOTEBesse1987Section_3FMisnerThorneWheeler1973Chapter_31O'Neill1983Chapter_13Schwarzschild1916-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Roy_Kerr" title="Roy Kerr">Roy Kerr</a> later found the <a href="/wiki/Kerr_metric" title="Kerr metric">Kerr metrics</a>, a two-parameter family containing the Schwarzschild metrics as a special case.<sup id="cite_ref-FOOTNOTEKerr1963MisnerThorneWheeler1973Chapter_33_4-0" class="reference"><a href="#cite_note-FOOTNOTEKerr1963MisnerThorneWheeler1973Chapter_33-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> These metrics are fully explicit and are of fundamental interest in the mathematics and physics of <a href="/wiki/Black_hole" title="Black hole">black holes</a>. More generally, in <a href="/wiki/General_relativity" title="General relativity">general relativity</a>, Ricci-flat Lorentzian manifolds represent the <a href="/wiki/Vacuum_solution" class="mw-redirect" title="Vacuum solution">vacuum solutions</a> of <a href="/wiki/Einstein%27s_field_equations" class="mw-redirect" title="Einstein's field equations">Einstein's field equations</a> with vanishing <a href="/wiki/Cosmological_constant" title="Cosmological constant">cosmological constant</a>.<sup id="cite_ref-FOOTNOTEBesse1987Section_3C_5-0" class="reference"><a href="#cite_note-FOOTNOTEBesse1987Section_3C-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>Many pseudo-Riemannian manifolds are constructed as <a href="/wiki/Homogeneous_space" title="Homogeneous space">homogeneous spaces</a>. However, these constructions are not directly helpful for Ricci-flat Riemannian metrics, in the sense that any homogeneous Riemannian manifold which is Ricci-flat must be flat.<sup id="cite_ref-FOOTNOTEBesse1987Theorem_7.61_6-0" class="reference"><a href="#cite_note-FOOTNOTEBesse1987Theorem_7.61-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> However, there are homogeneous (and even <a href="/wiki/Symmetric_space" title="Symmetric space">symmetric</a>) Lorentzian manifolds which are Ricci-flat but not flat, as follows from an explicit construction and computation of <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebras</a>.<sup id="cite_ref-FOOTNOTEBesse1987Theorem_7.118_7-0" class="reference"><a href="#cite_note-FOOTNOTEBesse1987Theorem_7.118-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>Until <a href="/wiki/Shing-Tung_Yau" title="Shing-Tung Yau">Shing-Tung Yau</a>'s resolution of the <a href="/wiki/Calabi_conjecture" title="Calabi conjecture">Calabi conjecture</a> in the 1970s, it was not known whether every Ricci-flat Riemannian metric on a <a href="/wiki/Closed_manifold" title="Closed manifold">closed manifold</a> is flat.<sup id="cite_ref-FOOTNOTEBesse1987Paragraph_0.30_8-0" class="reference"><a href="#cite_note-FOOTNOTEBesse1987Paragraph_0.30-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> His work, using techniques of <a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equations</a>, established a comprehensive existence theory for Ricci-flat metrics in the special case of <a href="/wiki/K%C3%A4hler_metric" class="mw-redirect" title="Kähler metric">Kähler metrics</a> on closed <a href="/wiki/Complex_manifold" title="Complex manifold">complex manifolds</a>. Due to his analytical techniques, the metrics are non-explicit even in the simplest cases. Such Riemannian manifolds are often called <a href="/wiki/Calabi%E2%80%93Yau_manifold" title="Calabi–Yau manifold">Calabi–Yau manifolds</a>, although various authors use this name in slightly different ways.<sup id="cite_ref-FOOTNOTEBesse1987Sections_11B–CYau1978_9-0" class="reference"><a href="#cite_note-FOOTNOTEBesse1987Sections_11B–CYau1978-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Analytical_character">Analytical character</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ricci-flat_manifold&action=edit&section=3" title="Edit section: Analytical character"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Relative to <a href="/wiki/Harmonic_coordinate" class="mw-redirect" title="Harmonic coordinate">harmonic coordinates</a>, the condition of Ricci-flatness for a Riemannian metric can be interpreted as a system of <a href="/wiki/Elliptic_partial_differential_equation" title="Elliptic partial differential equation">elliptic partial differential equations</a>. It is a straightforward consequence of standard <i>elliptic regularity</i> results that any Ricci-flat Riemannian metric on a smooth manifold is analytic, in the sense that <a href="/wiki/Harmonic_coordinate" class="mw-redirect" title="Harmonic coordinate">harmonic coordinates</a> define a compatible <a href="/wiki/Analytic_manifold" title="Analytic manifold">analytic structure</a>, and the local representation of the metric is <a href="/wiki/Analytic_function" title="Analytic function">real-analytic</a>. This also holds in the broader setting of Einstein Riemannian metrics.<sup id="cite_ref-FOOTNOTEBesse1987Section_5F_10-0" class="reference"><a href="#cite_note-FOOTNOTEBesse1987Section_5F-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>Analogously, relative to harmonic coordinates, Ricci-flatness of a Lorentzian metric can be interpreted as a system of <a href="/wiki/Hyperbolic_partial_differential_equation" title="Hyperbolic partial differential equation">hyperbolic partial differential equations</a>. Based on this perspective, <a href="/wiki/Yvonne_Choquet-Bruhat" title="Yvonne Choquet-Bruhat">Yvonne Choquet-Bruhat</a> developed the <a href="/wiki/Well-posed_problem" title="Well-posed problem">well-posedness</a> of the Ricci-flatness condition. She reached a definitive result in collaboration with <a href="/wiki/Robert_Geroch" title="Robert Geroch">Robert Geroch</a> in the 1960s, establishing how a certain class of <i>maximally extended</i> Ricci-flat Lorentzian metrics are prescribed and constructed by certain Riemannian data. These are known as <i>maximal <a href="/wiki/Globally_hyperbolic_manifold" title="Globally hyperbolic manifold">globally hyperbolic</a> developments</i>. In general relativity, this is typically interpreted as an <a href="/wiki/Initial_value_formulation_(general_relativity)" title="Initial value formulation (general relativity)">initial value formulation</a> of Einstein's field equations for gravitation.<sup id="cite_ref-FOOTNOTEHawkingEllis1973Sections_7.5–7.6_11-0" class="reference"><a href="#cite_note-FOOTNOTEHawkingEllis1973Sections_7.5–7.6-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>The study of Ricci-flatness in the Riemannian and Lorentzian cases are quite distinct. This is already indicated by the fundamental distinction between the geodesically complete metrics which are typical of Riemannian geometry and the maximal globally hyperbolic developments which arise from Choquet-Bruhat and Geroch's work. Moreover, the analyticity and corresponding unique continuation of a Ricci-flat Riemannian metric has a fundamentally different character than Ricci-flat Lorentzian metrics, which have finite speeds of propagation and fully localizable phenomena. This can be viewed as a nonlinear geometric analogue of the difference between the <a href="/wiki/Laplace_equation" class="mw-redirect" title="Laplace equation">Laplace equation</a> and the <a href="/wiki/Wave_equation" title="Wave equation">wave equation</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Topology_of_Ricci-flat_Riemannian_manifolds">Topology of Ricci-flat Riemannian manifolds</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ricci-flat_manifold&action=edit&section=4" title="Edit section: Topology of Ricci-flat Riemannian manifolds"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Yau's existence theorem for Ricci-flat Kähler metrics established the precise topological condition under which such a metric exists on a given closed <a href="/wiki/Complex_manifold" title="Complex manifold">complex manifold</a>: the <a href="/wiki/First_Chern_class" class="mw-redirect" title="First Chern class">first Chern class</a> of the <a href="/wiki/Holomorphic_tangent_bundle" title="Holomorphic tangent bundle">holomorphic tangent bundle</a> must be zero. The necessity of this condition was previously known by <a href="/wiki/Chern%E2%80%93Weil_theory" class="mw-redirect" title="Chern–Weil theory">Chern–Weil theory</a>. </p><p>Beyond Kähler geometry, the situation is not as well understood. A four-dimensional closed and oriented manifold supporting any Einstein Riemannian metric must satisfy the <a href="/wiki/Hitchin%E2%80%93Thorpe_inequality" title="Hitchin–Thorpe inequality">Hitchin–Thorpe inequality</a> on its topological data. As particular cases of well-known theorems on Riemannian manifolds of nonnegative Ricci curvature, any manifold with a complete Ricci-flat Riemannian metric must:<sup id="cite_ref-FOOTNOTEBesse1987Sections_6D–E_12-0" class="reference"><a href="#cite_note-FOOTNOTEBesse1987Sections_6D–E-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <ul><li>have first <a href="/wiki/Betti_number" title="Betti number">Betti number</a> less than or equal to the dimension, whenever the manifold is closed</li> <li>have <a href="/wiki/Fundamental_group" title="Fundamental group">fundamental group</a> of polynomial growth.</li></ul> <p><a href="/wiki/Mikhael_Gromov_(mathematician)" title="Mikhael Gromov (mathematician)">Mikhael Gromov</a> and <a href="/wiki/Blaine_Lawson" class="mw-redirect" title="Blaine Lawson">Blaine Lawson</a> introduced the notion of <i>enlargeability</i> of a closed manifold. The class of enlargeable manifolds is closed under <a href="/wiki/Homotopy_equivalence" class="mw-redirect" title="Homotopy equivalence">homotopy equivalence</a>, the taking of products, and under the <a href="/wiki/Connected_sum" title="Connected sum">connected sum</a> with an arbitrary closed manifold. Every Ricci-flat Riemannian manifold in this class is flat, which is a corollary of Cheeger and Gromoll's <a href="/wiki/Splitting_theorem" title="Splitting theorem">splitting theorem</a>.<sup id="cite_ref-FOOTNOTELawsonMichelsohn1989Section_IV.5_13-0" class="reference"><a href="#cite_note-FOOTNOTELawsonMichelsohn1989Section_IV.5-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Ricci-flatness_and_holonomy">Ricci-flatness and holonomy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ricci-flat_manifold&action=edit&section=5" title="Edit section: Ricci-flatness and holonomy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>On a <a href="/wiki/Simply-connected" class="mw-redirect" title="Simply-connected">simply-connected</a> Kähler manifold, a Kähler metric is Ricci-flat if and only if the <a href="/wiki/Holonomy_group" class="mw-redirect" title="Holonomy group">holonomy group</a> is contained in the <a href="/wiki/Special_unitary_group" title="Special unitary group">special unitary group</a>. On a general Kähler manifold, the <i>if</i> direction still holds, but only the <i>restricted</i> holonomy group of a Ricci-flat Kähler metric is necessarily contained in the special unitary group.<sup id="cite_ref-FOOTNOTEBesse1987Proposition_10.29_14-0" class="reference"><a href="#cite_note-FOOTNOTEBesse1987Proposition_10.29-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>A <a href="/wiki/Hyperk%C3%A4hler_manifold" title="Hyperkähler manifold">hyperkähler manifold</a> is a Riemannian manifold whose holonomy group is contained in the <a href="/wiki/Symplectic_group" title="Symplectic group">symplectic group</a>. This condition on a Riemannian manifold may also be characterized (roughly speaking) by the existence of a 2-sphere of <a href="/wiki/Complex_manifold" title="Complex manifold">complex structures</a> which are all <a href="/wiki/Parallel_transport" title="Parallel transport">parallel</a>. This says in particular that every hyperkähler metric is Kähler; furthermore, via the <a href="/wiki/Holonomy_group" class="mw-redirect" title="Holonomy group">Ambrose–Singer theorem</a>, every such metric is Ricci-flat. The Calabi–Yau theorem specializes to this context, giving a general existence and uniqueness theorem for hyperkähler metrics on compact Kähler manifolds admitting holomorphically symplectic structures. Examples of hyperkähler metrics on noncompact spaces had earlier been obtained by <a href="/wiki/Eugenio_Calabi" title="Eugenio Calabi">Eugenio Calabi</a>. The <a href="/wiki/Eguchi%E2%80%93Hanson_space" title="Eguchi–Hanson space">Eguchi–Hanson space</a>, discovered at the same time, is a special case of his construction.<sup id="cite_ref-FOOTNOTEBesse1987Sections_14A–C_15-0" class="reference"><a href="#cite_note-FOOTNOTEBesse1987Sections_14A–C-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p>A <a href="/wiki/Quaternion-K%C3%A4hler_manifold" title="Quaternion-Kähler manifold">quaternion-Kähler manifold</a> is a Riemannian manifold whose holonomy group is contained in the <a href="/wiki/Lie_group" title="Lie group">Lie group</a> <span class="texhtml">Sp(n)·Sp(1)</span>. <a href="/wiki/Marcel_Berger" title="Marcel Berger">Marcel Berger</a> showed that any such metric must be Einstein. Furthermore, any Ricci-flat quaternion-Kähler manifold must be <i>locally</i> hyperkähler, meaning that the <i>restricted</i> holonomy group is contained in the symplectic group.<sup id="cite_ref-FOOTNOTEBesse1987Section_14D_16-0" class="reference"><a href="#cite_note-FOOTNOTEBesse1987Section_14D-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p>A <a href="/wiki/G2_manifold" title="G2 manifold"><span class="texhtml">G<sub>2</sub></span> manifold</a> or <a href="/wiki/Spin(7)-manifold" title="Spin(7)-manifold"><span class="texhtml">Spin(7)</span> manifold</a> is a Riemannian manifold whose holonomy group is contained in the Lie groups <a href="/wiki/Spin_group" title="Spin group"><span class="texhtml">Spin(7)</span></a> or <a href="/wiki/G2_(mathematics)" title="G2 (mathematics)"><span class="texhtml">G<sub>2</sub></span></a>. The <a href="/wiki/Holonomy_group" class="mw-redirect" title="Holonomy group">Ambrose–Singer theorem</a> implies that any such manifold is Ricci-flat.<sup id="cite_ref-FOOTNOTEBesse1987Section_10F_17-0" class="reference"><a href="#cite_note-FOOTNOTEBesse1987Section_10F-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> The existence of closed manifolds of this type was established by <a href="/wiki/Dominic_Joyce" title="Dominic Joyce">Dominic Joyce</a> in the 1990s.<sup id="cite_ref-FOOTNOTEBerger2003Section_13.5.1Joyce2000_18-0" class="reference"><a href="#cite_note-FOOTNOTEBerger2003Section_13.5.1Joyce2000-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Marcel_Berger" title="Marcel Berger">Marcel Berger</a> commented that all known examples of irreducible Ricci-flat Riemannian metrics on simply-connected closed manifolds have special holonomy groups, according to the above possibilities. It is not known whether this suggests an unknown general theorem or simply a limitation of known techniques. For this reason, Berger considered Ricci-flat manifolds to be "extremely mysterious."<sup id="cite_ref-FOOTNOTEBerger2003Section_11.4.6_19-0" class="reference"><a href="#cite_note-FOOTNOTEBerger2003Section_11.4.6-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ricci-flat_manifold&action=edit&section=6" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Notes.</b> </p> <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 mw-references-columns"><ol class="references"> <li id="cite_note-FOOTNOTEO'Neill198387-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEO'Neill198387_1-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFO'Neill1983">O'Neill 1983</a>, p. 87.</span> </li> <li id="cite_note-FOOTNOTEO'Neill1983336-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEO'Neill1983336_2-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFO'Neill1983">O'Neill 1983</a>, p. 336.</span> </li> <li id="cite_note-FOOTNOTEBesse1987Section_3FMisnerThorneWheeler1973Chapter_31O'Neill1983Chapter_13Schwarzschild1916-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBesse1987Section_3FMisnerThorneWheeler1973Chapter_31O'Neill1983Chapter_13Schwarzschild1916_3-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBesse1987">Besse 1987</a>, Section 3F; <a href="#CITEREFMisnerThorneWheeler1973">Misner, Thorne & Wheeler 1973</a>, Chapter 31; <a href="#CITEREFO'Neill1983">O'Neill 1983</a>, Chapter 13; <a href="#CITEREFSchwarzschild1916">Schwarzschild 1916</a>.</span> </li> <li id="cite_note-FOOTNOTEKerr1963MisnerThorneWheeler1973Chapter_33-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEKerr1963MisnerThorneWheeler1973Chapter_33_4-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFKerr1963">Kerr 1963</a>; <a href="#CITEREFMisnerThorneWheeler1973">Misner, Thorne & Wheeler 1973</a>, Chapter 33.</span> </li> <li id="cite_note-FOOTNOTEBesse1987Section_3C-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBesse1987Section_3C_5-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBesse1987">Besse 1987</a>, Section 3C.</span> </li> <li id="cite_note-FOOTNOTEBesse1987Theorem_7.61-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBesse1987Theorem_7.61_6-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBesse1987">Besse 1987</a>, Theorem 7.61.</span> </li> <li id="cite_note-FOOTNOTEBesse1987Theorem_7.118-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBesse1987Theorem_7.118_7-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBesse1987">Besse 1987</a>, Theorem 7.118.</span> </li> <li id="cite_note-FOOTNOTEBesse1987Paragraph_0.30-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBesse1987Paragraph_0.30_8-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBesse1987">Besse 1987</a>, Paragraph 0.30.</span> </li> <li id="cite_note-FOOTNOTEBesse1987Sections_11B–CYau1978-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBesse1987Sections_11B–CYau1978_9-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBesse1987">Besse 1987</a>, Sections 11B–C; <a href="#CITEREFYau1978">Yau 1978</a>.</span> </li> <li id="cite_note-FOOTNOTEBesse1987Section_5F-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBesse1987Section_5F_10-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBesse1987">Besse 1987</a>, Section 5F.</span> </li> <li id="cite_note-FOOTNOTEHawkingEllis1973Sections_7.5–7.6-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHawkingEllis1973Sections_7.5–7.6_11-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHawkingEllis1973">Hawking & Ellis 1973</a>, Sections 7.5–7.6.</span> </li> <li id="cite_note-FOOTNOTEBesse1987Sections_6D–E-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBesse1987Sections_6D–E_12-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBesse1987">Besse 1987</a>, Sections 6D–E.</span> </li> <li id="cite_note-FOOTNOTELawsonMichelsohn1989Section_IV.5-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTELawsonMichelsohn1989Section_IV.5_13-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFLawsonMichelsohn1989">Lawson & Michelsohn 1989</a>, Section IV.5.</span> </li> <li id="cite_note-FOOTNOTEBesse1987Proposition_10.29-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBesse1987Proposition_10.29_14-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBesse1987">Besse 1987</a>, Proposition 10.29.</span> </li> <li id="cite_note-FOOTNOTEBesse1987Sections_14A–C-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBesse1987Sections_14A–C_15-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBesse1987">Besse 1987</a>, Sections 14A–C.</span> </li> <li id="cite_note-FOOTNOTEBesse1987Section_14D-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBesse1987Section_14D_16-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBesse1987">Besse 1987</a>, Section 14D.</span> </li> <li id="cite_note-FOOTNOTEBesse1987Section_10F-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBesse1987Section_10F_17-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBesse1987">Besse 1987</a>, Section 10F.</span> </li> <li id="cite_note-FOOTNOTEBerger2003Section_13.5.1Joyce2000-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBerger2003Section_13.5.1Joyce2000_18-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBerger2003">Berger 2003</a>, Section 13.5.1; <a href="#CITEREFJoyce2000">Joyce 2000</a>.</span> </li> <li id="cite_note-FOOTNOTEBerger2003Section_11.4.6-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBerger2003Section_11.4.6_19-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBerger2003">Berger 2003</a>, Section 11.4.6.</span> </li> </ol></div></div> <p><b>Sources.</b> </p> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><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="CITEREFBerger2003" class="citation book cs1"><a href="/wiki/Marcel_Berger" title="Marcel Berger">Berger, Marcel</a> (2003). <i>A panoramic view of Riemannian geometry</i>. Berlin: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-18245-7">10.1007/978-3-642-18245-7</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-65317-1" title="Special:BookSources/3-540-65317-1"><bdi>3-540-65317-1</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2002701">2002701</a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1038.53002">1038.53002</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+panoramic+view+of+Riemannian+geometry&rft.place=Berlin&rft.pub=Springer-Verlag&rft.date=2003&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1038.53002%23id-name%3DZbl&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2002701%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-3-642-18245-7&rft.isbn=3-540-65317-1&rft.aulast=Berger&rft.aufirst=Marcel&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARicci-flat+manifold" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBesse1987" class="citation book cs1"><a href="/wiki/Arthur_Besse" title="Arthur Besse">Besse, Arthur L.</a> (1987). <i>Einstein manifolds</i>. Ergebnisse der Mathematik und ihrer Grenzgebiete (3). Vol. 10. Reprinted in 2008. 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"On the Ricci curvature of a compact Kähler manifold and the complex Monge−Ampère equation. I". <i><a href="/wiki/Communications_on_Pure_and_Applied_Mathematics" title="Communications on Pure and Applied Mathematics">Communications on Pure and Applied Mathematics</a></i>. <b>31</b> (3): 339–411. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fcpa.3160310304">10.1002/cpa.3160310304</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0480350">0480350</a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0369.53059">0369.53059</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Communications+on+Pure+and+Applied+Mathematics&rft.atitle=On+the+Ricci+curvature+of+a+compact+K%C3%A4hler+manifold+and+the+complex+Monge%E2%88%92Amp%C3%A8re+equation.+I&rft.volume=31&rft.issue=3&rft.pages=339-411&rft.date=1978&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0369.53059%23id-name%3DZbl&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0480350%23id-name%3DMR&rft_id=info%3Adoi%2F10.1002%2Fcpa.3160310304&rft.aulast=Yau&rft.aufirst=Shing+Tung&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARicci-flat+manifold" class="Z3988"></span></li></ul> </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 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href="/wiki/Template:String_theory_topics" title="Template:String theory topics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:String_theory_topics" title="Template talk:String theory topics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:String_theory_topics" title="Special:EditPage/Template:String theory topics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="String_theory" style="font-size:114%;margin:0 4em"><a href="/wiki/String_theory" title="String theory">String theory</a></div></th></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Background</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/String_(physics)" title="String (physics)">Strings</a></li> <li><a href="/wiki/Cosmic_string" title="Cosmic string">Cosmic strings</a></li> <li><a href="/wiki/History_of_string_theory" title="History of string theory">History of string theory</a> <ul><li><a href="/wiki/First_superstring_revolution" class="mw-redirect" title="First superstring revolution">First superstring revolution</a></li> <li><a href="/wiki/Second_superstring_revolution" class="mw-redirect" title="Second superstring revolution">Second superstring revolution</a></li></ul></li> <li><a href="/wiki/String_theory_landscape" title="String theory landscape">String theory landscape</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Theory</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nambu%E2%80%93Goto_action" title="Nambu–Goto action">Nambu–Goto action</a></li> <li><a href="/wiki/Polyakov_action" title="Polyakov action">Polyakov action</a></li> <li><a href="/wiki/Bosonic_string_theory" title="Bosonic string theory">Bosonic string theory</a></li> <li><a href="/wiki/Superstring_theory" title="Superstring theory">Superstring theory</a> <ul><li><a href="/wiki/Type_I_string_theory" title="Type I string theory">Type I string</a></li> <li><a href="/wiki/Type_II_string_theory" title="Type II string theory">Type II string</a> <ul><li><a href="/wiki/Type_II_string_theory" title="Type II string theory">Type IIA string</a></li> <li><a href="/wiki/Type_II_string_theory" title="Type II string theory">Type IIB string</a></li></ul></li> <li><a href="/wiki/Heterotic_string_theory" title="Heterotic string theory">Heterotic string</a></li></ul></li> <li><a href="/wiki/N%3D2_superstring" class="mw-redirect" title="N=2 superstring">N=2 superstring</a></li> <li><a href="/wiki/F-theory" title="F-theory">F-theory</a></li> <li><a href="/wiki/String_field_theory" title="String field theory">String field theory</a></li> <li><a href="/wiki/Matrix_string_theory" title="Matrix string theory">Matrix string theory</a></li> <li><a href="/wiki/Non-critical_string_theory" title="Non-critical string theory">Non-critical string theory</a></li> <li><a href="/wiki/Non-linear_sigma_model" title="Non-linear sigma model">Non-linear sigma model</a></li> <li><a href="/wiki/Tachyon_condensation" title="Tachyon condensation">Tachyon condensation</a></li> <li><a href="/wiki/RNS_formalism" title="RNS formalism">RNS formalism</a></li> <li><a href="/wiki/GS_formalism" title="GS formalism">GS formalism</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/String_duality" title="String duality">String duality</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/T-duality" title="T-duality">T-duality</a></li> <li><a href="/wiki/S-duality" title="S-duality">S-duality</a></li> <li><a href="/wiki/U-duality" title="U-duality">U-duality</a></li> <li><a href="/wiki/Montonen%E2%80%93Olive_duality" title="Montonen–Olive duality">Montonen–Olive duality</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Particles and fields</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Graviton" title="Graviton">Graviton</a></li> <li><a href="/wiki/Dilaton" title="Dilaton">Dilaton</a></li> <li><a href="/wiki/Tachyon" title="Tachyon">Tachyon</a></li> <li><a href="/wiki/Ramond%E2%80%93Ramond_field" title="Ramond–Ramond field">Ramond–Ramond field</a></li> <li><a href="/wiki/Kalb%E2%80%93Ramond_field" title="Kalb–Ramond field">Kalb–Ramond field</a></li> <li><a href="/wiki/Magnetic_monopole" title="Magnetic monopole">Magnetic monopole</a></li> <li><a href="/wiki/Dual_graviton" title="Dual graviton">Dual graviton</a></li> <li><a href="/wiki/Dual_photon" title="Dual photon">Dual photon</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/Brane" title="Brane">Branes</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/D-brane" title="D-brane">D-brane</a></li> <li><a href="/wiki/NS5-brane" title="NS5-brane">NS5-brane</a></li> <li><a href="/wiki/M2-brane" title="M2-brane">M2-brane</a></li> <li><a href="/wiki/M5-brane" title="M5-brane">M5-brane</a></li> <li><a href="/wiki/S-brane" title="S-brane">S-brane</a></li> <li><a href="/wiki/Black_brane" title="Black brane">Black brane</a></li> <li><a href="/wiki/Black_hole" title="Black hole">Black holes</a></li> <li><a href="/wiki/Black_string" class="mw-redirect" title="Black string">Black string</a></li> <li><a href="/wiki/Brane_cosmology" title="Brane cosmology">Brane cosmology</a></li> <li><a href="/wiki/Quiver_diagram" title="Quiver diagram">Quiver diagram</a></li> <li><a href="/wiki/Hanany%E2%80%93Witten_transition" title="Hanany–Witten transition">Hanany–Witten transition</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/Conformal_field_theory" title="Conformal field theory">Conformal field theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Virasoro_algebra" title="Virasoro algebra">Virasoro algebra</a></li> <li><a href="/wiki/Mirror_symmetry_(string_theory)" title="Mirror symmetry (string theory)">Mirror symmetry</a></li> <li><a href="/wiki/Conformal_anomaly" title="Conformal anomaly">Conformal anomaly</a></li> <li><a href="/wiki/Conformal_symmetry" title="Conformal symmetry">Conformal algebra</a></li> <li><a href="/wiki/Superconformal_algebra" title="Superconformal algebra">Superconformal algebra</a></li> <li><a href="/wiki/Vertex_operator_algebra" title="Vertex operator algebra">Vertex operator algebra</a></li> <li><a href="/wiki/Loop_algebra" title="Loop algebra">Loop algebra</a></li> <li><a href="/wiki/Kac%E2%80%93Moody_algebra" title="Kac–Moody algebra">Kac–Moody algebra</a></li> <li><a href="/wiki/Wess%E2%80%93Zumino%E2%80%93Witten_model" title="Wess–Zumino–Witten model">Wess–Zumino–Witten model</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/Gauge_theory" title="Gauge theory">Gauge theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Anomaly_(physics)" title="Anomaly (physics)">Anomalies</a></li> <li><a href="/wiki/Instanton" title="Instanton">Instantons</a></li> <li><a href="/wiki/Chern%E2%80%93Simons_form" title="Chern–Simons form">Chern–Simons form</a></li> <li><a href="/wiki/Bogomol%27nyi%E2%80%93Prasad%E2%80%93Sommerfield_bound" title="Bogomol'nyi–Prasad–Sommerfield bound">Bogomol'nyi–Prasad–Sommerfield bound</a></li> <li><a href="/wiki/Exceptional_Lie_group" class="mw-redirect" title="Exceptional Lie group">Exceptional Lie groups</a> (<a href="/wiki/G2_(mathematics)" title="G2 (mathematics)">G<sub>2</sub></a>, <a href="/wiki/F4_(mathematics)" title="F4 (mathematics)">F<sub>4</sub></a>, <a href="/wiki/E6_(mathematics)" title="E6 (mathematics)">E<sub>6</sub></a>, <a href="/wiki/E7_(mathematics)" title="E7 (mathematics)">E<sub>7</sub></a>, <a href="/wiki/E8_(mathematics)" title="E8 (mathematics)">E<sub>8</sub></a>)</li> <li><a href="/wiki/ADE_classification" title="ADE classification">ADE classification</a></li> <li><a href="/wiki/Dirac_string" title="Dirac string">Dirac string</a></li> <li><a href="/wiki/P-form_electrodynamics" title="P-form electrodynamics"><i>p</i>-form electrodynamics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Geometry</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Worldsheet" title="Worldsheet">Worldsheet</a></li> <li><a href="/wiki/Kaluza%E2%80%93Klein_theory" title="Kaluza–Klein theory">Kaluza–Klein theory</a></li> <li><a href="/wiki/Compactification_(physics)" title="Compactification (physics)">Compactification</a></li> <li><a href="/wiki/Why_10_dimensions" class="mw-redirect" title="Why 10 dimensions">Why 10 dimensions</a>?</li> <li><a href="/wiki/K%C3%A4hler_manifold" title="Kähler manifold">Kähler manifold</a></li> <li><a class="mw-selflink selflink">Ricci-flat manifold</a> <ul><li><a href="/wiki/Calabi%E2%80%93Yau_manifold" title="Calabi–Yau manifold">Calabi–Yau manifold</a></li> <li><a href="/wiki/Hyperk%C3%A4hler_manifold" title="Hyperkähler manifold">Hyperkähler manifold</a> <ul><li><a href="/wiki/K3_surface" title="K3 surface">K3 surface</a></li></ul></li> <li><a href="/wiki/G2_manifold" title="G2 manifold">G<sub>2</sub> manifold</a></li> <li><a href="/wiki/Spin(7)-manifold" title="Spin(7)-manifold">Spin(7)-manifold</a></li></ul></li> <li><a href="/wiki/Generalized_complex_structure" title="Generalized complex structure">Generalized complex manifold</a></li> <li><a href="/wiki/Orbifold" title="Orbifold">Orbifold</a></li> <li><a href="/wiki/Conifold" title="Conifold">Conifold</a></li> <li><a href="/wiki/Orientifold" title="Orientifold">Orientifold</a></li> <li><a href="/wiki/Moduli_space" title="Moduli space">Moduli space</a></li> <li><a href="/wiki/Ho%C5%99ava%E2%80%93Witten_theory" title="Hořava–Witten theory">Hořava–Witten theory</a></li> <li><a href="/wiki/K-theory_(physics)" title="K-theory (physics)">K-theory (physics)</a></li> <li><a href="/wiki/Twisted_K-theory" title="Twisted K-theory">Twisted K-theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/Supersymmetry" title="Supersymmetry">Supersymmetry</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Supergravity" title="Supergravity">Supergravity</a></li> <li><a href="/wiki/Eleven-dimensional_supergravity" title="Eleven-dimensional supergravity">Eleven-dimensional supergravity</a></li> <li><a href="/wiki/Type_I_supergravity" title="Type I supergravity">Type I supergravity</a></li> <li><a href="/wiki/Type_IIA_supergravity" title="Type IIA supergravity">Type IIA supergravity</a></li> <li><a href="/wiki/Type_IIB_supergravity" title="Type IIB supergravity">Type IIB supergravity</a></li> <li><a href="/wiki/Superspace" title="Superspace">Superspace</a></li> <li><a href="/wiki/Lie_superalgebra" title="Lie superalgebra">Lie superalgebra</a></li> <li><a href="/wiki/Lie_supergroup" class="mw-redirect" title="Lie supergroup">Lie supergroup</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/Holography" title="Holography">Holography</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Holographic_principle" title="Holographic principle">Holographic principle</a></li> <li><a href="/wiki/AdS/CFT_correspondence" title="AdS/CFT correspondence">AdS/CFT correspondence</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/M-theory" title="M-theory">M-theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Matrix_theory_(physics)" title="Matrix theory (physics)">Matrix theory</a></li> <li><a href="/wiki/Introduction_to_M-theory" title="Introduction to M-theory">Introduction to M-theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">String theorists</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mina_Aganagi%C4%87" title="Mina Aganagić">Aganagić</a></li> <li><a href="/wiki/Nima_Arkani-Hamed" title="Nima Arkani-Hamed">Arkani-Hamed</a></li> <li><a href="/wiki/Michael_Atiyah" title="Michael Atiyah">Atiyah</a></li> <li><a href="/wiki/Tom_Banks_(physicist)" title="Tom Banks (physicist)">Banks</a></li> <li><a href="/wiki/David_Berenstein" title="David Berenstein">Berenstein</a></li> <li><a href="/wiki/Raphael_Bousso" title="Raphael Bousso">Bousso</a></li> <li><a href="/wiki/Thomas_Curtright" title="Thomas Curtright">Curtright</a></li> <li><a href="/wiki/Robbert_Dijkgraaf" title="Robbert Dijkgraaf">Dijkgraaf</a></li> <li><a href="/wiki/Jacques_Distler" title="Jacques Distler">Distler</a></li> <li><a href="/wiki/Michael_R._Douglas" title="Michael R. Douglas">Douglas</a></li> <li><a href="/wiki/Michael_Duff_(physicist)" title="Michael Duff (physicist)">Duff</a></li> <li><a href="/wiki/Gia_Dvali" class="mw-redirect" title="Gia Dvali">Dvali</a></li> <li><a href="/wiki/Sergio_Ferrara" title="Sergio Ferrara">Ferrara</a></li> <li><a href="/wiki/Willy_Fischler" title="Willy Fischler">Fischler</a></li> <li><a href="/wiki/Daniel_Friedan" title="Daniel Friedan">Friedan</a></li> <li><a href="/wiki/Sylvester_James_Gates" title="Sylvester James Gates">Gates</a></li> <li><a href="/wiki/Ferdinando_Gliozzi" title="Ferdinando Gliozzi">Gliozzi</a></li> <li><a href="/wiki/Rajesh_Gopakumar" title="Rajesh Gopakumar">Gopakumar</a></li> <li><a href="/wiki/Michael_Green_(physicist)" title="Michael Green (physicist)">Green</a></li> <li><a href="/wiki/Brian_Greene" title="Brian Greene">Greene</a></li> <li><a href="/wiki/David_Gross" title="David Gross">Gross</a></li> <li><a href="/wiki/Steven_Gubser" title="Steven Gubser">Gubser</a></li> <li><a href="/wiki/Sergei_Gukov" title="Sergei Gukov">Gukov</a></li> <li><a href="/wiki/Alan_Guth" title="Alan Guth">Guth</a></li> <li><a href="/wiki/Andrew_J._Hanson" title="Andrew J. Hanson">Hanson</a></li> <li><a href="/wiki/Jeffrey_A._Harvey" title="Jeffrey A. Harvey">Harvey</a></li> <li><a href="/wiki/Gerard_%27t_Hooft" title="Gerard 't Hooft">'t Hooft</a></li> <li><a href="/wiki/Petr_Ho%C5%99ava_(theorist)" class="mw-redirect" title="Petr Hořava (theorist)">Hořava</a></li> <li><a href="/wiki/Gary_Gibbons" title="Gary Gibbons">Gibbons</a></li> <li><a href="/wiki/Shamit_Kachru" title="Shamit Kachru">Kachru</a></li> <li><a href="/wiki/Michio_Kaku" title="Michio Kaku">Kaku</a></li> <li><a href="/wiki/Renata_Kallosh" title="Renata Kallosh">Kallosh</a></li> <li><a href="/wiki/Theodor_Kaluza" title="Theodor Kaluza">Kaluza</a></li> <li><a href="/wiki/Anton_Kapustin" title="Anton Kapustin">Kapustin</a></li> <li><a href="/wiki/Igor_Klebanov" title="Igor Klebanov">Klebanov</a></li> <li><a href="/wiki/Vadim_Knizhnik" title="Vadim Knizhnik">Knizhnik</a></li> <li><a href="/wiki/Maxim_Kontsevich" title="Maxim Kontsevich">Kontsevich</a></li> <li><a href="/wiki/Oskar_Klein" title="Oskar Klein">Klein</a></li> <li><a href="/wiki/Andrei_Linde" title="Andrei Linde">Linde</a></li> <li><a href="/wiki/Juan_Mart%C3%ADn_Maldacena" class="mw-redirect" title="Juan Martín Maldacena">Maldacena</a></li> <li><a href="/wiki/Stanley_Mandelstam" title="Stanley Mandelstam">Mandelstam</a></li> <li><a href="/wiki/Donald_Marolf" title="Donald Marolf">Marolf</a></li> <li><a href="/wiki/Emil_Martinec" title="Emil Martinec">Martinec</a></li> <li><a href="/wiki/Shiraz_Minwalla" title="Shiraz Minwalla">Minwalla</a></li> <li><a href="/wiki/Greg_Moore_(physicist)" title="Greg Moore (physicist)">Moore</a></li> <li><a href="/wiki/Lubo%C5%A1_Motl" title="Luboš Motl">Motl</a></li> <li><a href="/wiki/Sunil_Mukhi" title="Sunil Mukhi">Mukhi</a></li> <li><a href="/wiki/Robert_Myers_(physicist)" title="Robert Myers (physicist)">Myers</a></li> <li><a href="/wiki/Dimitri_Nanopoulos" title="Dimitri Nanopoulos">Nanopoulos</a></li> <li><a href="/wiki/Hora%C8%9Biu_N%C4%83stase" title="Horațiu Năstase">Năstase</a></li> <li><a href="/wiki/Nikita_Nekrasov" title="Nikita Nekrasov">Nekrasov</a></li> <li><a href="/wiki/Andr%C3%A9_Neveu" title="André Neveu">Neveu</a></li> <li><a href="/wiki/Holger_Bech_Nielsen" title="Holger Bech Nielsen">Nielsen</a></li> <li><a href="/wiki/Peter_van_Nieuwenhuizen" title="Peter van Nieuwenhuizen">van Nieuwenhuizen</a></li> <li><a href="/wiki/Sergei_Novikov_(mathematician)" title="Sergei Novikov (mathematician)">Novikov</a></li> <li><a href="/wiki/David_Olive" title="David Olive">Olive</a></li> <li><a href="/wiki/Hirosi_Ooguri" title="Hirosi Ooguri">Ooguri</a></li> <li><a href="/wiki/Burt_Ovrut" title="Burt Ovrut">Ovrut</a></li> <li><a href="/wiki/Joseph_Polchinski" title="Joseph Polchinski">Polchinski</a></li> <li><a href="/wiki/Alexander_Markovich_Polyakov" title="Alexander Markovich Polyakov">Polyakov</a></li> <li><a href="/wiki/Arvind_Rajaraman" title="Arvind Rajaraman">Rajaraman</a></li> <li><a href="/wiki/Pierre_Ramond" title="Pierre Ramond">Ramond</a></li> <li><a href="/wiki/Lisa_Randall" title="Lisa Randall">Randall</a></li> <li><a href="/wiki/Seifallah_Randjbar-Daemi" title="Seifallah Randjbar-Daemi">Randjbar-Daemi</a></li> <li><a href="/wiki/Martin_Ro%C4%8Dek" title="Martin Roček">Roček</a></li> <li><a href="/wiki/Ryan_Rohm" title="Ryan Rohm">Rohm</a></li> <li><a href="/wiki/Augusto_Sagnotti" title="Augusto Sagnotti">Sagnotti</a></li> <li><a href="/wiki/Jo%C3%ABl_Scherk" title="Joël Scherk">Scherk</a></li> <li><a href="/wiki/John_Henry_Schwarz" title="John Henry Schwarz">Schwarz</a></li> <li><a href="/wiki/Nathan_Seiberg" title="Nathan Seiberg">Seiberg</a></li> <li><a href="/wiki/Ashoke_Sen" title="Ashoke Sen">Sen</a></li> <li><a href="/wiki/Stephen_Shenker" title="Stephen Shenker">Shenker</a></li> <li><a href="/wiki/Warren_Siegel" title="Warren Siegel">Siegel</a></li> <li><a href="/wiki/Eva_Silverstein" title="Eva Silverstein">Silverstein</a></li> <li><a href="/wiki/%C4%90%C3%A0m_Thanh_S%C6%A1n" title="Đàm Thanh Sơn">Sơn</a></li> <li><a href="/wiki/Matthias_Staudacher" title="Matthias Staudacher">Staudacher</a></li> <li><a href="/wiki/Paul_Steinhardt" title="Paul Steinhardt">Steinhardt</a></li> <li><a href="/wiki/Andrew_Strominger" title="Andrew Strominger">Strominger</a></li> <li><a href="/wiki/Raman_Sundrum" title="Raman Sundrum">Sundrum</a></li> <li><a href="/wiki/Leonard_Susskind" title="Leonard Susskind">Susskind</a></li> <li><a href="/wiki/Paul_Townsend" title="Paul Townsend">Townsend</a></li> <li><a href="/wiki/Sandip_Trivedi" title="Sandip Trivedi">Trivedi</a></li> <li><a href="/wiki/Neil_Turok" title="Neil Turok">Turok</a></li> <li><a href="/wiki/Cumrun_Vafa" title="Cumrun Vafa">Vafa</a></li> <li><a href="/wiki/Gabriele_Veneziano" title="Gabriele Veneziano">Veneziano</a></li> <li><a href="/wiki/Erik_Verlinde" title="Erik Verlinde">Verlinde</a></li> <li><a href="/wiki/Herman_Verlinde" title="Herman Verlinde">Verlinde</a></li> <li><a href="/wiki/Julius_Wess" title="Julius Wess">Wess</a></li> <li><a href="/wiki/Edward_Witten" title="Edward Witten">Witten</a></li> <li><a href="/wiki/Shing-Tung_Yau" title="Shing-Tung Yau">Yau</a></li> <li><a href="/wiki/Tamiaki_Yoneya" title="Tamiaki Yoneya">Yoneya</a></li> <li><a href="/wiki/Alexander_Zamolodchikov" title="Alexander Zamolodchikov">Zamolodchikov</a></li> <li><a href="/wiki/Alexei_Zamolodchikov" title="Alexei Zamolodchikov">Zamolodchikov</a></li> <li><a href="/wiki/Eric_Zaslow" title="Eric Zaslow">Zaslow</a></li> <li><a href="/wiki/Bruno_Zumino" title="Bruno Zumino">Zumino</a></li> <li><a href="/wiki/Barton_Zwiebach" title="Barton Zwiebach">Zwiebach</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" 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