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class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a>, <a class="existingWikiWord" href="/nlab/show/complex+structure">complex structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+analytic+space">complex analytic space</a></p> </li> </ul> <h3 id="variants">Variants</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+complex+geometry">generalized complex geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+supermanifold">complex supermanifold</a></p> </li> </ul> <h3 id="structures">Structures</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dolbeault+complex">Dolbeault complex</a>, <a class="existingWikiWord" href="/nlab/show/holomorphic+de+Rham+complex">holomorphic de Rham complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+structure">Hodge structure</a>, <a class="existingWikiWord" href="/nlab/show/Hodge+filtration">Hodge filtration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge-filtered+differential+cohomology">Hodge-filtered differential cohomology</a></p> </li> </ul> <h3 id="examples">Examples</h3> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">dim = 1</annotation></semantics></math>: <a class="existingWikiWord" href="/nlab/show/Riemann+surface">Riemann surface</a>, <a class="existingWikiWord" href="/nlab/show/super+Riemann+surface">super Riemann surface</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+manifold">Calabi-Yau manifold</a></p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">dim = 2</annotation></semantics></math>: <a class="existingWikiWord" href="/nlab/show/K3+surface">K3 surface</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+Calabi-Yau+manifold">generalized Calabi-Yau manifold</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#definition'>Definition</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#compact_hyperkhler_structure'>Compact hyperkähler structure</a></li> <li><a href='#homotopy'>Homotopy</a></li> <li><a href='#cohomology'>Cohomology</a></li> <li><a href='#SUBordism'>SU-Bordism</a></li> <li><a href='#CharacteristicClasses'>Characteristic classes</a></li> <ul> <li><a href='#of_'>Of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mn>3</mn></mrow><annotation encoding="application/x-tex">K3</annotation></semantics></math></a></li> <li><a href='#of__2'>Of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mn>3</mn><mo>×</mo><mi>K</mi><mn>3</mn></mrow><annotation encoding="application/x-tex">K3 \times K3</annotation></semantics></math></a></li> <li><a href='#of__3'>Of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mn>3</mn><mo>×</mo><msup><mi>X</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">K3 \times X^4</annotation></semantics></math></a></li> </ul> <li><a href='#ModuliOfHigherLineBundles'>Moduli of higher line bundles and deformation theory</a></li> <li><a href='#relation_to_third_stably_framed_bordism_group'>Relation to third stably framed bordism group</a></li> <li><a href='#elliptic_fibration'>Elliptic fibration</a></li> <li><a href='#as_a_fiber_space_in_string_compactifications'>As a fiber space in string compactifications</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#elliptic_fibrations'>Elliptic fibrations</a></li> <li><a href='#InStringTheory'>In string theory</a></li> </ul> </ul> </div> <h2 id="definition">Definition</h2> <p>A <strong>K3 surface</strong> is a <a class="existingWikiWord" href="/nlab/show/Calabi-Yau+variety">Calabi-Yau variety</a> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math> whose <a class="existingWikiWord" href="/nlab/show/Picard+variety">Picard variety</a> is zero-dimensional. In other words, it is a complex <a class="existingWikiWord" href="/nlab/show/algebraic+surface">algebraic surface</a> with trivial <a class="existingWikiWord" href="/nlab/show/canonical+bundle">canonical bundle</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mi>X</mi></msub><mo>=</mo><msup><mo>∧</mo> <mn>2</mn></msup><msub><mi>Ω</mi> <mi>X</mi></msub><mo>≃</mo><msub><mi>𝒪</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\omega_X=\wedge^2\Omega_X\simeq \mathcal{O}_X</annotation></semantics></math>) and with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>𝒪</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">H^1(X, \mathcal{O}_X)=0</annotation></semantics></math>.</p> <p>The term “K3” is</p> <blockquote> <p>in honor of Kummer, Kähler, Kodaira, and the beautiful K2 mountain in Kashmir</p> </blockquote> <p>(<a href="#Weil79">Weil 79, p. 546</a>)</p> <h2 id="examples">Examples</h2> <ul> <li> <p>A cyclic cover of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℙ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{P}^2</annotation></semantics></math> branched over a curve of degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>6</mn></mrow><annotation encoding="application/x-tex">6</annotation></semantics></math>.</p> </li> <li> <p>A nonsingular degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>4</mn></mrow><annotation encoding="application/x-tex">4</annotation></semantics></math> hypersurface in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℙ</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{P}^3</annotation></semantics></math>, such as the <span class="newWikiWord">Fermat quartic<a href="/nlab/new/Fermat+hypersurface">?</a></span> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo stretchy="false">[</mo><mi>w</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">]</mo><mo>∈</mo><msup><mi>ℙ</mi> <mn>3</mn></msup><mo stretchy="false">|</mo><msup><mi>w</mi> <mn>4</mn></msup><mo>+</mo><msup><mi>x</mi> <mn>4</mn></msup><mo>+</mo><msup><mi>y</mi> <mn>4</mn></msup><mo>+</mo><msup><mi>z</mi> <mn>4</mn></msup><mo>=</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{[w,x,y,z] \in \mathbb{P}^3 | w^4 + x^4 + y^4 + z^4 = 0\}</annotation></semantics></math> (in fact every K3 surface over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphic</a> to this example).</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/flat+orbifold">flat orbifold</a> quotient of the <a class="existingWikiWord" href="/nlab/show/4-torus">4-torus</a> (equipped with some <a class="existingWikiWord" href="/nlab/show/complex+structure">complex structure</a>) by the sign <a class="existingWikiWord" href="/nlab/show/involution">involution</a> on all four canonical <a class="existingWikiWord" href="/nlab/show/coordinates">coordinates</a> is the flat compact 4-dimensional orbifold known as a <em>Kummer surface</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>T</mi> <mn>4</mn></msup><mo>⫽</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">T^4 \sslash \mathbb{Z}_2</annotation></semantics></math>, a singular <a class="existingWikiWord" href="/nlab/show/K3-surface">K3-surface</a> (e.g. <a href="Riemannian+orbifold#BettiolDerdzinskiPiccione18">Bettiol, Derdzinski & Piccione 2018, 5.5</a>; <a href="#TaorminaWendland15">Taormina & Wendland 2015, §1</a>)</p> </li> </ul> <h2 id="properties">Properties</h2> <h3 id="compact_hyperkhler_structure">Compact hyperkähler structure</h3> <p>Over the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a> <a class="existingWikiWord" href="/nlab/show/K3+surfaces">K3 surfaces</a> are all <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+manifold">Kähler</a>, and even <a class="existingWikiWord" href="/nlab/show/hyperk%C3%A4hler+manifold">hyperkähler</a>.</p> <p>The only known examples of <a class="existingWikiWord" href="/nlab/show/compact+hyperk%C3%A4hler+manifolds">compact hyperkähler manifolds</a> are <a class="existingWikiWord" href="/nlab/show/Hilbert+schemes+of+points">Hilbert schemes of points</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mrow><mo stretchy="false">[</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup></mrow><annotation encoding="application/x-tex">X^{[n+1]}</annotation></semantics></math> (for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>) for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> either</p> <ol> <li> <p>a <a class="existingWikiWord" href="/nlab/show/K3-surface">K3-surface</a></p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/4-torus">4-torus</a> (in which case the <a class="existingWikiWord" href="/nlab/show/compact+hyperk%C3%A4hler+manifolds">compact hyperkähler manifolds</a> is really the <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>𝕋</mi> <mn>4</mn></msup><msup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow></msup><mo>→</mo><msup><mi>𝕋</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">(\mathbb{T}^4)^{[n]} \to \mathbb{T}^4</annotation></semantics></math>)</p> </li> </ol> <p>(<a href="compact+hyperkähler+manifold#Beauville83">Beauville 83</a>) and two exceptional examples (<a href="compact+hyperkähler+manifold#OGrady99">O’Grady 99</a>, <a href="compact+hyperkähler+manifold#OGrady03">O’Grady 03</a> ), see <a href="compact+hyperkähler+manifold#Sawon04">Sawon 04, Sec. 5.3</a>.</p> <h3 id="homotopy">Homotopy</h3> <ul> <li>All K3 surfaces are <a class="existingWikiWord" href="/nlab/show/simply+connected">simply connected</a>.</li> </ul> <h3 id="cohomology">Cohomology</h3> <div class="num_prop" id="IntegralCohomology"> <h6 id="proposition">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a> of <a class="existingWikiWord" href="/nlab/show/K3-surface">K3-surface</a>)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a> of a K3-surface <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mi>ℤ</mi></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>n</mi><mo>=</mo><mn>0</mn></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>n</mi><mo>=</mo><mn>1</mn></mtd></mtr> <mtr><mtd><msup><mi>ℤ</mi> <mn>22</mn></msup></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>n</mi><mo>=</mo><mn>2</mn></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>n</mi><mo>=</mo><mn>3</mn></mtd></mtr> <mtr><mtd><mi>ℤ</mi></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>n</mi><mo>=</mo><mn>4</mn></mtd></mtr></mtable></mrow></mrow></mrow><annotation encoding="application/x-tex"> H^n(X,\mathbb{Z}) \;\simeq\; \left\{ \array{ \mathbb{Z} &\vert& n = 0 \\ 0 &\vert& n = 1 \\ \mathbb{Z}^{22} &\vert& n = 2 \\ 0 &\vert& n = 3 \\ \mathbb{Z} &\vert& n = 4 } \right. </annotation></semantics></math></div></div> <p>(e.g. <a href="#BarthPetersVandenVen84">Barth-Peters-Van den Ven 84, VIII Prop. 3.2</a>)</p> <div class="num_prop" id="BettiNumbers"> <h6 id="proposition_2">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Betti+numbers">Betti numbers</a> of a <a class="existingWikiWord" href="/nlab/show/K3-surface">K3-surface</a>)</strong></p> <p>The <span class="newWikiWord">Hodge diamond<a href="/nlab/new/Hodge+diamond">?</a></span> is completely determined (even in positive <a class="existingWikiWord" href="/nlab/show/characteristic">characteristic</a>) and hence the <a class="existingWikiWord" href="/nlab/show/Hodge-de+Rham+spectral+sequence">Hodge-de Rham spectral sequence</a> degenerates at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">E_1</annotation></semantics></math>. This also implies that the <a class="existingWikiWord" href="/nlab/show/Betti+numbers">Betti numbers</a> are completely determined as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>22</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">1, 0, 22, 0, 1</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mi>h</mi> <mrow><mn>0</mn><mo>,</mo><mn>0</mn></mrow></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mi>h</mi> <mrow><mn>1</mn><mo>,</mo><mn>0</mn></mrow></msup></mtd> <mtd></mtd> <mtd><msup><mi>h</mi> <mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow></msup></mtd></mtr> <mtr><mtd><msup><mi>h</mi> <mrow><mn>2</mn><mo>,</mo><mn>0</mn></mrow></msup></mtd> <mtd></mtd> <mtd><msup><mi>h</mi> <mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup></mtd> <mtd></mtd> <mtd><msup><mi>h</mi> <mrow><mn>0</mn><mo>,</mo><mn>2</mn></mrow></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mi>h</mi> <mrow><mn>2</mn><mo>,</mo><mn>1</mn></mrow></msup></mtd> <mtd></mtd> <mtd><msup><mi>h</mi> <mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mi>h</mi> <mrow><mn>2</mn><mo>,</mo><mn>2</mn></mrow></msup></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mn>1</mn></mtd></mtr> <mtr><mtd></mtd> <mtd><mn>0</mn></mtd> <mtd></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd><mn>1</mn></mtd> <mtd></mtd> <mtd><mn>20</mn></mtd> <mtd></mtd> <mtd><mn>1</mn></mtd></mtr> <mtr><mtd></mtd> <mtd><mn>0</mn></mtd> <mtd></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mn>1</mn></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && h^{0,0} \\ & h^{1,0} && h^{0,1} \\ h^{2,0} & & h^{1,1} & & h^{0,2} \\ & h^{2,1} & & h^{1,2} \\ && h^{2,2} } \;\;\;=\;\;\; \array{ && 1 \\ & 0 && 0 \\ 1 & & 20 & & 1 \\ & 0 & & 0 \\ && 1 } </annotation></semantics></math></div></div> <p>(e.g. <a href="#BarthPetersVandenVen84">Barth-Peters-Van den Ven 84, VIII Prop. 3.3</a>)</p> <p><br /></p> <h3 id="SUBordism">SU-Bordism</h3> <div class="num_prop" id="K3SurfaceSpansSUBordismRingInDegree4"> <h6 id="proposition_3">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/K3-surface">K3-surface</a> spans <a class="existingWikiWord" href="/nlab/show/SU-bordism+ring">SU-bordism ring</a> in degree 4)</strong></p> <p>The canonical degree-4 generator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>y</mi> <mn>4</mn></msub><mo>∈</mo><msubsup><mi>Ω</mi> <mn>4</mn> <mi>SU</mi></msubsup></mrow><annotation encoding="application/x-tex">y_4 \in \Omega^{SU}_4</annotation></semantics></math> in the <a class="existingWikiWord" href="/nlab/show/SU-bordism+ring">SU-bordism ring</a> (<a href="MSU#SUBordismRingAwayFromTwo">this Prop.</a>) is represented by minus the class of any (non-<a class="existingWikiWord" href="/nlab/show/torus">torus</a>) <a class="existingWikiWord" href="/nlab/show/K3-surface">K3-surface</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mn>4</mn> <mi>SU</mi></msubsup><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>ℤ</mi><mo maxsize="1.2em" minsize="1.2em">[</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo maxsize="1.2em" minsize="1.2em">]</mo><mo maxsize="1.2em" minsize="1.2em">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">[</mo><mi>K</mi><mn>3</mn><mo stretchy="false">]</mo><mo maxsize="1.2em" minsize="1.2em">⟩</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Omega^{SU}_4 \;\simeq\; \mathbb{Z}\big[ \tfrac{1}{2}\big]\big\langle -[K3] \big\rangle \,. </annotation></semantics></math></div></div> <p>(<a href="MSU#LLP17">LLP 17, Lemma 1.5, Example 3.1</a>, <a href="MSU#CLP19">CLP 19, Theorem 13.5a</a>, see at <em><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+manifolds+in+SU-bordism+theory">Calabi-Yau manifolds in SU-bordism theory</a></em>)</p> <p><br /></p> <h3 id="CharacteristicClasses">Characteristic classes</h3> <p>We discuss some <a class="existingWikiWord" href="/nlab/show/characteristic+classes">characteristic classes</a> of (the <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a> of) <a class="existingWikiWord" href="/nlab/show/K3">K3</a>, and their evaluation on (the <a class="existingWikiWord" href="/nlab/show/fundamental+class">fundamental class</a> of) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mn>3</mn></mrow><annotation encoding="application/x-tex">K3</annotation></semantics></math> (i.e. their <a class="existingWikiWord" href="/nlab/show/integration">integration</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mn>3</mn></mrow><annotation encoding="application/x-tex">K3</annotation></semantics></math>).</p> <h4 id="of_">Of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mn>3</mn></mrow><annotation encoding="application/x-tex">K3</annotation></semantics></math></h4> <div class="num_prop" id="EulerCharacteristicOfK3"> <h6 id="proposition_4">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Euler+characteristic">Euler characteristic</a> of <a class="existingWikiWord" href="/nlab/show/K3">K3</a>)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/Euler+characteristic">Euler characteristic</a> of <a class="existingWikiWord" href="/nlab/show/K3">K3</a> is 24:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>χ</mi> <mn>4</mn></msub><mo stretchy="false">[</mo><mi>K</mi><mn>3</mn><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mn>24</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \chi_4[K3] \;=\; 24 \,. </annotation></semantics></math></div></div> <div class="num_prop" id="FirstChernClassOfK3"> <h6 id="proposition_5">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/first+Chern+class">first Chern class</a> of <a class="existingWikiWord" href="/nlab/show/K3">K3</a>)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/first+Chern+class">first Chern class</a> of <a class="existingWikiWord" href="/nlab/show/K3">K3</a> vanishes:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>K</mi><mn>3</mn><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> c_1(K3) \;=\; 0 \,. </annotation></semantics></math></div></div> <div class="num_prop" id="SecondChernClassOfK3"> <h6 id="proposition_6">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/second+Chern+class">second Chern class</a> of <a class="existingWikiWord" href="/nlab/show/K3">K3</a>)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/second+Chern+class">second Chern class</a> of <a class="existingWikiWord" href="/nlab/show/K3">K3</a> evaluates to 24:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">[</mo><mi>K</mi><mn>3</mn><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mn>24</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> c_2[K3] \;=\; 24 \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof">Proof</h6> <p>For every <a class="existingWikiWord" href="/nlab/show/closed+manifold">closed</a> <a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a> the evaluation of the top degree <a class="existingWikiWord" href="/nlab/show/Chern+class">Chern class</a> equals the <a class="existingWikiWord" href="/nlab/show/Euler+characteristic">Euler characteristic</a>. Hence the statement follows from Prop. <a class="maruku-ref" href="#EulerCharacteristicOfK3"></a>.</p> </div> <div class="num_prop" id="FirstPontryaginClassOfK3"> <h6 id="proposition_7">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/first+Chern+class">first Chern class</a> of <a class="existingWikiWord" href="/nlab/show/K3">K3</a>)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/first+Pontryagin+class">first Pontryagin class</a> of <a class="existingWikiWord" href="/nlab/show/K3">K3</a> evaluates to 48:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">[</mo><mi>K</mi><mn>3</mn><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo>−</mo><mn>48</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> p_1[K3] \;=\; -48 \,. </annotation></semantics></math></div></div> <p>(see also e.g. <a href="I8#DuffLiuMinasian95">Duff-Liu-Minasian 95 (5.10)</a>)</p> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>For a <a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a> the <a class="existingWikiWord" href="/nlab/show/first+Pontryagin+class">first Pontryagin class</a> is the following <a class="existingWikiWord" href="/nlab/show/polynomial">polynomial</a> in the <a class="existingWikiWord" href="/nlab/show/first+Chern+class">first Chern class</a> and the <a class="existingWikiWord" href="/nlab/show/second+Chern+class">second Chern class</a>. By Prop. <a class="maruku-ref" href="#FirstChernClassOfK3"></a> the first Chern class vanishes, and by Prop. <a class="maruku-ref" href="#SecondChernClassOfK3"></a> the second Chern class evaluates to 24:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">[</mo><mi>K</mi><mn>3</mn><mo stretchy="false">]</mo></mtd> <mtd><mo>=</mo><mspace width="thickmathspace"></mspace><munder><munder><mrow><msub><mi>c</mi> <mn>1</mn></msub><mo>∪</mo><msub><mi>c</mi> <mn>1</mn></msub></mrow><mo>⏟</mo></munder><mrow><mo>=</mo><mn>0</mn></mrow></munder><mo stretchy="false">[</mo><mi>K</mi><mn>3</mn><mo stretchy="false">]</mo><mo>−</mo><mn>2</mn><munder><munder><mrow><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">[</mo><mi>K</mi><mn>3</mn><mo stretchy="false">]</mo></mrow><mo>⏟</mo></munder><mn>24</mn></munder></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>48</mn></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} p_1[K3] & =\; \underset{= 0}{\underbrace{c_1 \cup c_1}}[K3] - 2 \underset{24}{\underbrace{c_2[K3]}} \\ & = - 48 \end{aligned} \,. </annotation></semantics></math></div></div> <h4 id="of__2">Of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mn>3</mn><mo>×</mo><mi>K</mi><mn>3</mn></mrow><annotation encoding="application/x-tex">K3 \times K3</annotation></semantics></math></h4> <p>Now consider the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian</a> <a class="existingWikiWord" href="/nlab/show/product+space">product space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mn>3</mn><mo>×</mo><mi>K</mi><mn>3</mn></mrow><annotation encoding="application/x-tex">K3 \times K3</annotation></semantics></math>.</p> <p>See also at <em><a class="existingWikiWord" href="/nlab/show/C-field+tadpole+cancellation">C-field tadpole cancellation</a></em> the section <em><a href="C-field+tadpole+cancellation#IntegralityOnK3TimesK3">Integrality on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mn>3</mn><mo>×</mo><mi>K</mi><mn>3</mn></mrow><annotation encoding="application/x-tex">K3 \times K3</annotation></semantics></math></a></em>.</p> <div class="num_prop" id="EulerCharacteristicOfK3TimesK3"> <h6 id="proposition_8">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Euler+characteristic">Euler characteristic</a> of <a class="existingWikiWord" href="/nlab/show/K3">K3</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>×</mo></mrow><annotation encoding="application/x-tex">\times</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/K3">K3</a>)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/Euler+characteristic">Euler characteristic</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mn>3</mn><mo>×</mo><mi>K</mi><mn>3</mn></mrow><annotation encoding="application/x-tex">K3 \times K3</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mn>24</mn> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">24^2</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>χ</mi> <mn>8</mn></msub><mo stretchy="false">[</mo><mi>K</mi><mn>3</mn><mo>×</mo><mi>K</mi><mn>3</mn><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msup><mn>24</mn> <mn>2</mn></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \chi_8[K3\times K3] \;=\; 24^2 \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>By the <a href="Euler%20class#EulerClassOfWhitneySumIsCupProductOfEulerClasses">Whitney sum formula for the Euler class</a> we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>χ</mi> <mn>8</mn></msub><mo stretchy="false">[</mo><mi>K</mi><mn>3</mn><mo>×</mo><mi>K</mi><mn>3</mn><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>χ</mi> <mn>4</mn></msub><mo stretchy="false">[</mo><mi>K</mi><mn>3</mn><mo stretchy="false">]</mo><msup><mo stretchy="false">)</mo> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\chi_8[K3 \times K3] = (\chi_4[K3])^2</annotation></semantics></math>. Hence the statement follows by Prop. <a class="maruku-ref" href="#EulerCharacteristicOfK3"></a>.</p> </div> <div class="num_prop" id="FirstPontryaginClassOfK3TimesK3"> <h6 id="proposition_9">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/first+Pontryagin+class">first Pontryagin class</a> of <a class="existingWikiWord" href="/nlab/show/K3">K3</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>×</mo></mrow><annotation encoding="application/x-tex">\times</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/K3">K3</a>)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/first+Pontryagin+class">first Pontryagin class</a> evaluated on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mn>3</mn><mo>×</mo><mi>K</mi><mn>3</mn></mrow><annotation encoding="application/x-tex">K3 \times K3</annotation></semantics></math> is:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">[</mo><mi>K</mi><mn>3</mn><mo>×</mo><mi>K</mi><mn>3</mn><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo>−</mo><mn>2</mn><mo>×</mo><mn>48</mn></mrow><annotation encoding="application/x-tex"> p_1[K3 \times K3] \;=\; - 2 \times 48 </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>By the general formula for <a class="existingWikiWord" href="/nlab/show/Pontryagin+classes">Pontryagin classes</a> of <a class="existingWikiWord" href="/nlab/show/product+spaces">product spaces</a> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>K</mi><mn>3</mn><mo>×</mo><mi>K</mi><mn>3</mn><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mspace width="thickmathspace"></mspace><msub><mi>p</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>K</mi><mn>3</mn><mo stretchy="false">)</mo><mo>⌣</mo><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>K</mi><mn>3</mn><mo stretchy="false">)</mo><mo>+</mo><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>K</mi><mn>3</mn><mo stretchy="false">)</mo><mo>⌣</mo><msub><mi>p</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>K</mi><mn>3</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mn>2</mn><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>K</mi><mn>3</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} p_1(K3 \times K3) & =\; p_0(K3) \smile p_1(K3) + p_1(K3) \smile p_0(K3) \\ & = 2 p_1(K3) \end{aligned} </annotation></semantics></math></div> <p>With this, the statement follows by Prop. <a class="maruku-ref" href="#FirstPontryaginClassOfK3"></a>.</p> </div> <h4 id="of__3">Of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mn>3</mn><mo>×</mo><msup><mi>X</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">K3 \times X^4</annotation></semantics></math></h4> <p>Now consider the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian</a> <a class="existingWikiWord" href="/nlab/show/product+space">product space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mn>3</mn><mo>×</mo><msup><mi>X</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">K3 \times X^4</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/K3">K3</a> with some <a class="existingWikiWord" href="/nlab/show/4-manifold">4-manifold</a>.</p> <div class="num_prop" id="EulerCharacteristicOfK3TimesK3"> <h6 id="proposition_10">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Euler+characteristic">Euler characteristic</a> of <a class="existingWikiWord" href="/nlab/show/K3">K3</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>×</mo><msup><mi>X</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">\times X^4</annotation></semantics></math>)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/Euler+characteristic">Euler characteristic</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mn>3</mn><mo>×</mo><mi>K</mi><mn>3</mn></mrow><annotation encoding="application/x-tex">K3 \times K3</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>24</mn></mrow><annotation encoding="application/x-tex">24</annotation></semantics></math> times the Euler characteristic of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">X^4</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>χ</mi> <mn>8</mn></msub><mo stretchy="false">[</mo><mi>K</mi><mn>3</mn><mo>×</mo><msup><mi>X</mi> <mn>4</mn></msup><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mn>24</mn><mo>⋅</mo><msub><mi>χ</mi> <mn>4</mn></msub><mo stretchy="false">[</mo><msup><mi>X</mi> <mn>4</mn></msup><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \chi_8[K3\times X^4] \;=\; 24 \cdot \chi_4[X^4] \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_5">Proof</h6> <p>By the <a href="Euler%20class#EulerClassOfWhitneySumIsCupProductOfEulerClasses">Whitney sum formula for the Euler class</a> we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>χ</mi> <mn>8</mn></msub><mo stretchy="false">[</mo><mi>K</mi><mn>3</mn><mo>×</mo><msup><mi>X</mi> <mn>4</mn></msup><mo stretchy="false">]</mo><mo>=</mo><msub><mi>χ</mi> <mn>4</mn></msub><mo stretchy="false">[</mo><mi>K</mi><mn>3</mn><mo stretchy="false">]</mo><mo>⋅</mo><msub><mi>χ</mi> <mn>4</mn></msub><mo stretchy="false">[</mo><msup><mi>X</mi> <mn>4</mn></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\chi_8[K3 \times X^4] = \chi_4[K3] \cdot \chi_4[X^4]</annotation></semantics></math>. Hence the statement follows by Prop. <a class="maruku-ref" href="#EulerCharacteristicOfK3"></a>.</p> </div> <div class="num_prop" id="FirstPontryaginClassOfK3TimesK3"> <h6 id="proposition_11">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/first+Pontryagin+class">first Pontryagin class</a> of <a class="existingWikiWord" href="/nlab/show/K3">K3</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>×</mo><msup><mi>X</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">\times X^4</annotation></semantics></math>)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/first+Pontryagin+class">first Pontryagin class</a> evaluated on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mn>3</mn><mo>×</mo><msup><mi>X</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">K3 \times X^4</annotation></semantics></math> is:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">[</mo><mi>K</mi><mn>3</mn><mo>×</mo><msup><mi>X</mi> <mn>4</mn></msup><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">[</mo><msup><mi>X</mi> <mn>4</mn></msup><mo stretchy="false">]</mo><mo>−</mo><mn>48</mn></mrow><annotation encoding="application/x-tex"> p_1[K3 \times X^4] \;=\; p_1[X^4] - 48 </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_6">Proof</h6> <p>By the general formula for <a class="existingWikiWord" href="/nlab/show/Pontryagin+classes">Pontryagin classes</a> of <a class="existingWikiWord" href="/nlab/show/product+spaces">product spaces</a> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">[</mo><mi>K</mi><mn>3</mn><mo>×</mo><msup><mi>X</mi> <mn>4</mn></msup><mo stretchy="false">]</mo></mtd> <mtd><mo>=</mo><mspace width="thickmathspace"></mspace><msub><mi>p</mi> <mn>0</mn></msub><mo stretchy="false">[</mo><mi>K</mi><mn>3</mn><mo stretchy="false">]</mo><mo>⌣</mo><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">[</mo><msup><mi>X</mi> <mn>4</mn></msup><mo stretchy="false">]</mo><mo>+</mo><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">[</mo><mi>K</mi><mn>3</mn><mo stretchy="false">]</mo><mo>⌣</mo><msub><mi>p</mi> <mn>0</mn></msub><mo stretchy="false">[</mo><msup><mi>X</mi> <mn>4</mn></msup><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">[</mo><msup><mi>X</mi> <mn>4</mn></msup><mo stretchy="false">]</mo><mo>+</mo><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">[</mo><mi>K</mi><mn>3</mn><mo stretchy="false">]</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} p_1[K3 \times X^4] & =\; p_0[K3] \smile p_1[X^4] + p_1[K3] \smile p_0[X^4] \\ & = p_1[X^4] + p_1[K3] \end{aligned} </annotation></semantics></math></div> <p>With this, the statement follows by Prop. <a class="maruku-ref" href="#FirstPontryaginClassOfK3"></a>.</p> </div> <h3 id="ModuliOfHigherLineBundles">Moduli of higher line bundles and deformation theory</h3> <p>In <a class="existingWikiWord" href="/nlab/show/positive+number">positive</a> <a class="existingWikiWord" href="/nlab/show/characteristic">characteristic</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>:</p> <p>The <a class="existingWikiWord" href="/nlab/show/N%C3%A9ron-Severi+group">Néron-Severi group</a> of a K3 is a <a class="existingWikiWord" href="/nlab/show/free+abelian+group">free abelian group</a></p> <p>The <a class="existingWikiWord" href="/nlab/show/formal+Brauer+group">formal Brauer group</a> is</p> <ul> <li> <p>either the formal <a class="existingWikiWord" href="/nlab/show/additive+group">additive group</a>, in which case it has <a class="existingWikiWord" href="/nlab/show/height+of+a+formal+group">height</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>=</mo><mn>∞</mn></mrow><annotation encoding="application/x-tex">h = \infty</annotation></semantics></math>, by definition;</p> </li> <li> <p>or its <a class="existingWikiWord" href="/nlab/show/height+of+a+formal+group">height</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>h</mi><mo>≤</mo><mn>10</mn></mrow><annotation encoding="application/x-tex">1 \leq h \leq 10</annotation></semantics></math>, and every value may occur</p> </li> </ul> <p>(<a href="#Artin74">Artin 74</a>), see also (<a href="#ArtinMazur77">Artin-Mazur 77, p. 5 (of 46)</a>)</p> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/moduli+spaces">moduli spaces</a> of <a class="existingWikiWord" href="/nlab/show/line+n-bundles+with+connection">line n-bundles with connection</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/dimension">dimensional</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></strong></p> <table><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></th><th><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+object">Calabi-Yau n-fold</a></th><th><a class="existingWikiWord" href="/nlab/show/line+n-bundle">line n-bundle</a></th><th>moduli of <a class="existingWikiWord" href="/nlab/show/line+n-bundles">line n-bundles</a></th><th>moduli of <a class="existingWikiWord" href="/nlab/show/flat+infinity-connection">flat</a>/degree-0 n-bundles</th><th><a class="existingWikiWord" href="/nlab/show/Artin-Mazur+formal+group">Artin-Mazur formal group</a> of <a class="existingWikiWord" href="/nlab/show/deformation+theory">deformation moduli</a> of <a class="existingWikiWord" href="/nlab/show/line+n-bundles">line n-bundles</a></th><th><a class="existingWikiWord" href="/nlab/show/complex+oriented+cohomology+theory">complex oriented cohomology theory</a></th><th><a class="existingWikiWord" href="/nlab/show/modular+functor">modular functor</a>/<a class="existingWikiWord" href="/nlab/show/self-dual+higher+gauge+theory">self-dual higher gauge theory</a> of <a class="existingWikiWord" href="/nlab/show/higher+dimensional+Chern-Simons+theory">higher dimensional Chern-Simons theory</a></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n = 0</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/unit">unit</a> in <a class="existingWikiWord" href="/nlab/show/structure+sheaf">structure sheaf</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/multiplicative+group">multiplicative group</a>/<a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/formal+multiplicative+group">formal multiplicative group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/complex+K-theory">complex K-theory</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n = 1</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/elliptic+curve">elliptic curve</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/line+bundle">line bundle</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Picard+group">Picard group</a>/<a class="existingWikiWord" href="/nlab/show/Picard+scheme">Picard scheme</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Jacobian">Jacobian</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/formal+Picard+group">formal Picard group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/3d+Chern-Simons+theory">3d Chern-Simons theory</a>/<a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n = 2</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/K3+surface">K3 surface</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/line+2-bundle">line 2-bundle</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Brauer+group">Brauer group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/intermediate+Jacobian">intermediate Jacobian</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/formal+Brauer+group">formal Brauer group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/K3+cohomology">K3 cohomology</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">n = 3</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+3-fold">Calabi-Yau 3-fold</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/line+3-bundle">line 3-bundle</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/intermediate+Jacobian">intermediate Jacobian</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+cohomology">CY3 cohomology</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/7d+Chern-Simons+theory">7d Chern-Simons theory</a>/<a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/intermediate+Jacobian">intermediate Jacobian</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> </tbody></table> </div> <h3 id="relation_to_third_stably_framed_bordism_group">Relation to third stably framed bordism group</h3> <p>The <a class="existingWikiWord" href="/nlab/show/third+stable+homotopy+group+of+spheres">third stable homotopy group of spheres</a> (the third <a class="existingWikiWord" href="/nlab/show/stable+stem">stable stem</a>) is the <a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a> of <a class="existingWikiWord" href="/nlab/show/order+of+a+group">order</a> 24:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msubsup><mi>π</mi> <mn>3</mn> <mi>s</mi></msubsup></mtd> <mtd><mo>≃</mo></mtd> <mtd><mi>ℤ</mi><mo stretchy="false">/</mo><mn>24</mn></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><msub><mi>h</mi> <mi>ℍ</mi></msub><mo stretchy="false">]</mo></mtd> <mtd><mo>↔</mo></mtd> <mtd><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \pi_3^s &\simeq& \mathbb{Z}/24 \\ [h_{\mathbb{H}}] &\leftrightarrow& [1] } </annotation></semantics></math></div> <p>where the generator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>∈</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>24</mn></mrow><annotation encoding="application/x-tex">[1] \in \mathbb{Z}/24</annotation></semantics></math> is represented by the <a class="existingWikiWord" href="/nlab/show/quaternionic+Hopf+fibration">quaternionic Hopf fibration</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>7</mn></msup><mover><mo>⟶</mo><mrow><msub><mi>h</mi> <mi>ℍ</mi></msub></mrow></mover><msup><mi>S</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">S^7 \overset{h_{\mathbb{H}}}{\longrightarrow} S^4</annotation></semantics></math>.</p> <p>Under the <a class="existingWikiWord" href="/nlab/show/Pontrjagin-Thom+isomorphism">Pontrjagin-Thom isomorphism</a>, identifying the <a class="existingWikiWord" href="/nlab/show/stable+homotopy+groups+of+spheres">stable homotopy groups of spheres</a> with the <a class="existingWikiWord" href="/nlab/show/bordism+ring">bordism ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mo>•</mo> <mi>fr</mi></msubsup></mrow><annotation encoding="application/x-tex">\Omega^{fr}_\bullet</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/stable+framing">stably framed</a> manifolds (see at <em><a class="existingWikiWord" href="/nlab/show/MFr">MFr</a></em>), this generator is represented by the <a class="existingWikiWord" href="/nlab/show/3-sphere">3-sphere</a> (with its left-invariant framing induced from the identification with the <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> <a class="existingWikiWord" href="/nlab/show/SU%282%29">SU(2)</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo></mrow><annotation encoding="application/x-tex">\simeq</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Sp%281%29">Sp(1)</a> )</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msubsup><mi>π</mi> <mn>3</mn> <mi>s</mi></msubsup></mtd> <mtd><mo>≃</mo></mtd> <mtd><msubsup><mi>Ω</mi> <mn>3</mn> <mi>fr</mi></msubsup></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><msub><mi>h</mi> <mi>ℍ</mi></msub><mo stretchy="false">]</mo></mtd> <mtd><mo>↔</mo></mtd> <mtd><mo stretchy="false">[</mo><msup><mi>S</mi> <mn>3</mn></msup><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \pi_3^s & \simeq & \Omega_3^{fr} \\ [h_{\mathbb{H}}] & \leftrightarrow & [S^3] \,. } </annotation></semantics></math></div> <p>Moreover, the relation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mn>4</mn><mo stretchy="false">[</mo><msup><mi>S</mi> <mn>3</mn></msup><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>≃</mo><mspace width="thinmathspace"></mspace><mn>0</mn></mrow><annotation encoding="application/x-tex">2 4 [S^3] \,\simeq\, 0</annotation></semantics></math> is represented by the <a class="existingWikiWord" href="/nlab/show/bordism">bordism</a> which is the <a class="existingWikiWord" href="/nlab/show/complement">complement</a> of 24 <a class="existingWikiWord" href="/nlab/show/open+balls">open balls</a> inside <a class="existingWikiWord" href="/nlab/show/generalized+the">the</a> <a class="existingWikiWord" href="/nlab/show/K3">K3</a>-manifold (<a href="third+stable+homotopy+group+of+spheres#WangXu10">Wang-Xu 10, Sec. 2.6</a>).</p> <h3 id="elliptic_fibration">Elliptic fibration</h3> <p>See at <em><a class="existingWikiWord" href="/nlab/show/elliptically+fibered+K3-surface">elliptically fibered K3-surface</a></em>.</p> <h3 id="as_a_fiber_space_in_string_compactifications">As a fiber space in string compactifications</h3> <p>See</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/duality+between+heterotic+and+type+II+string+theory">duality between heterotic and type II string theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/F-theory+on+K3">F-theory on K3</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/M-theory+on+Sp%281%29-manifolds">M-theory on Sp(1)-manifolds</a></p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/formal+Brauer+group">formal Brauer group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+fibration">elliptic fibration</a>, <a class="existingWikiWord" href="/nlab/show/elliptically+fibered+K3-surface">elliptically fibered K3-surface</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K3-cohomology">K3-cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/F-theory">F-theory</a>, <a class="existingWikiWord" href="/nlab/show/F-theory+on+K3">F-theory on K3</a></p> </li> </ul> <h2 id="references">References</h2> <h3 id="general">General</h3> <p>Original sources:</p> <ul> <li id="Artin74"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Artin">Michael Artin</a>, <em>Supersingular K3 Surfaces</em>, Annal. Sc. d, l’Éc Norm. Sup. 4e séries, T. 7, fasc. 4, 1974, pp. 543-568</p> </li> <li id="Weil79"> <p><a class="existingWikiWord" href="/nlab/show/Andre+Weil">Andre Weil</a>, Final report on contract AF 18 (603)-57. In Scientific works. Collected papers. Vol. II (1951-1964). 1979.</p> </li> </ul> <p>Textbook accounts include</p> <ul> <li id="BarthPetersVandenVen84">W. Barth, C. Peters, A. Van den Ven, chapter VII of <em>Compact complex surfaces</em>, Springer 1984</li> </ul> <p>Lecture notes:</p> <ul> <li id="Huybrechts16"> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Huybrechts">Daniel Huybrechts</a>, <em>Lectures on K3-surfaces</em>, Cambridge University Press 2016 (<a href="http://www.math.uni-bonn.de/people/huybrech/K3Global.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/HuybrechtsLecturesOnK3.pdf" title="pdf">pdf</a>, <a href="https://doi.org/10.1017/CBO9781316594193">doi:10.1017/CBO9781316594193</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Claire+Voisin">Claire Voisin</a>, <em><a href="http://www.ams.org/mathscinet-getitem?mr=2487743">Geometry of the Moduli Space of K3 surfaces</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/David+Morrison">David Morrison</a>, <em><a href="http://www.cgtp.duke.edu/ITP99/morrison/cortona.pdf">The geometry of K3 surfaces</a></em> Lecture notes (1988)</p> </li> <li> <p>Viacheslav Nikulin, <em>Elliptic fibrations on K3 surfaces</em> (<a href="http://arxiv.org/abs/1010.3904">arXiv:1010.3904</a>)</p> </li> </ul> <p>Discussion of the <a class="existingWikiWord" href="/nlab/show/deformation+theory">deformation theory</a> of K3-surfaces (of their <a class="existingWikiWord" href="/nlab/show/Picard+schemes">Picard schemes</a>) is (see also at <em><a class="existingWikiWord" href="/nlab/show/Artin-Mazur+formal+group">Artin-Mazur formal group</a></em>) in</p> <ul> <li id="ArtinMazur77"><a class="existingWikiWord" href="/nlab/show/Michael+Artin">Michael Artin</a>, <a class="existingWikiWord" href="/nlab/show/Barry+Mazur">Barry Mazur</a>, <em>Formal Groups Arising from Algebraic Varieties</em>, Annales scientifiques de l’École Normale Supérieure, Sér. 4, 10 no. 1 (1977), p. 87-131 <a href="http://www.numdam.org/item?id=ASENS_1977_4_10_1_87_0">numdam</a>, <a href="http://www.ams.org/mathscinet-getitem?mr=56:15663">MR56:15663</a></li> </ul> <p>Systematic construction of <a class="existingWikiWord" href="/nlab/show/Ricci+tensor">Ricci flat</a> <a class="existingWikiWord" href="/nlab/show/Riemannian+metrics">Riemannian metrics</a> on <a class="existingWikiWord" href="/nlab/show/K3">K3</a> <a class="existingWikiWord" href="/nlab/show/orbifolds">orbifolds</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Shamit+Kachru">Shamit Kachru</a>, <a class="existingWikiWord" href="/nlab/show/Arnav+Tripathy">Arnav Tripathy</a>, <a class="existingWikiWord" href="/nlab/show/Max+Zimet">Max Zimet</a>, <em>K3 metrics</em> (<a href="https://arxiv.org/abs/2006.02435">arXiv:2006.02435</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Arnav+Tripathy">Arnav Tripathy</a>, <a class="existingWikiWord" href="/nlab/show/Max+Zimet">Max Zimet</a>, <em>A plethora of K3 metrics</em> (<a href="https://arxiv.org/abs/2010.12581">arXiv:2010.12581</a>)</p> </li> </ul> <h3 id="elliptic_fibrations">Elliptic fibrations</h3> <ul> <li id="Lecacheux19"> <p>O. Lecacheux, <em>Weierstrass Equations for the Elliptic Fibrations of a K3 Surface</em> In: Balakrishnan J., Folsom A., Lalín M., Manes M. (eds.) <em>Research Directions in Number Theory Association for Women in Mathematics Series, vol 19. Springer (2019) (<a href="https://doi.org/10.1007/978-3-030-19478-9_4">doi:10.1007/978-3-030-19478-9_4</a>)</em></p> </li> <li> <p>Marie Bertin, <em>Elliptic Fibrations on K3 surfaces</em>, 2013 (<a href="https://webusers.imj-prg.fr/~marie-jose.bertin/WINEurope13.pdf">pdf</a>)</p> </li> </ul> <h3 id="InStringTheory">In string theory</h3> <p>In <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a>, the <a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactification</a> of <a class="existingWikiWord" href="/nlab/show/type+IIA+string+theory">type IIA string theory</a>/<a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a>/<a class="existingWikiWord" href="/nlab/show/F-theory">F-theory</a> on K3-<a class="existingWikiWord" href="/nlab/show/fibers">fibers</a> is supposed to exhibit te <a class="existingWikiWord" href="/nlab/show/duality+between+M%2FF-theory+and+heterotic+string+theory">duality between M/F-theory and heterotic string theory</a>, originally due to</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chris+Hull">Chris Hull</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Townsend">Paul Townsend</a>, section 6 of <em>Unity of Superstring Dualities</em>, Nucl.Phys.B438:109-137,1995 (<a href="http://arxiv.org/abs/hep-th/9410167">arXiv:hep-th/9410167</a>)</p> </li> <li id="Witten95"> <p><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, section 4 of <em><a class="existingWikiWord" href="/nlab/show/String+Theory+Dynamics+In+Various+Dimensions">String Theory Dynamics In Various Dimensions</a></em>, Nucl.Phys.B443:85-126,1995 (<a href="http://arxiv.org/abs/hep-th/9503124">arXiv:hep-th/9503124</a>)</p> </li> </ul> <p>Review includes</p> <ul> <li id="Aspinwall96"><a class="existingWikiWord" href="/nlab/show/Paul+Aspinwall">Paul Aspinwall</a>, <em>K3 Surfaces and String Duality</em>, in <a class="existingWikiWord" href="/nlab/show/Shing-Tung+Yau">Shing-Tung Yau</a> (ed.): <em>Differential geometry inspired by string theory</em> 1-95 (<a href="https://arxiv.org/abs/hep-th/9611137">arXiv:9611137</a>, <a href="http://inspirehep.net/record/426102">spire:426102</a>)</li> </ul> <p>Further discussion includes</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Paul+Aspinwall">Paul Aspinwall</a>, <a class="existingWikiWord" href="/nlab/show/David+Morrison">David Morrison</a>, <em>String Theory on K3 Surfaces</em>, in <a class="existingWikiWord" href="/nlab/show/Brian+Greene">Brian Greene</a>, <a class="existingWikiWord" href="/nlab/show/Shing-Tung+Yau">Shing-Tung Yau</a> (eds.), <em>Mirror Symmetry II</em>, International Press, Cambridge, 1997, pp. 703-716 (<a href="https://arxiv.org/abs/hep-th/9404151">arXiv:hep-th/9404151</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Paul+Aspinwall">Paul Aspinwall</a>, <em>Enhanced Gauge Symmetries and K3 Surfaces</em>, Phys.Lett. B357 (1995) 329-334 (<a href="http://arxiv.org/abs/hep-th/9507012">arXiv:hep-th/9507012</a>)</p> </li> </ul> <p>Specifically in relation to <a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a> <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a>:</p> <ul> <li id="Wendland01"> <p><a class="existingWikiWord" href="/nlab/show/Katrin+Wendland">Katrin Wendland</a>, <em>Orbifold Constructions of K3: A Link between Conformal Field Theory and Geometry</em>, in <em><a class="existingWikiWord" href="/nlab/show/Orbifolds+in+Mathematics+and+Physics">Orbifolds in Mathematics and Physics</a></em> [<a href="https://arxiv.org/abs/hep-th/0112006">arXiv:hep-th/0112006</a>]</p> </li> <li id="TaorminaWendland15"> <p>Anne Taormina, <a class="existingWikiWord" href="/nlab/show/Katrin+Wendland">Katrin Wendland</a>, <em>Symmetry-surfing the moduli space of Kummer K3s</em>, Proc. Symp. Pure Math. <strong>90</strong> (2015) 129 [<a href="https://arxiv.org/abs/1303.2931">arXiv:1303.2931</a>, <a href="https://doi.org/10.1090/pspum/090/01522">doi:10.1090/pspum/090/01522</a>]</p> </li> </ul> <p>Specifically in relation to the putative <a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a>-classification of <a class="existingWikiWord" href="/nlab/show/D-brane+charge">D-brane charge</a>:</p> <ul> <li id="GarciaUranga05">Inaki Garcia-Etxebarria, <a class="existingWikiWord" href="/nlab/show/Angel+Uranga">Angel Uranga</a>, <em>From F/M-theory to K-theory and back</em>, JHEP 0602:008,2006 (<a href="https://arxiv.org/abs/hep-th/0510073">arXiv:hep-th/0510073</a>)</li> </ul> <p>Specifically in <a class="existingWikiWord" href="/nlab/show/M-theory+on+G%E2%82%82-manifolds">M-theory on G₂-manifolds</a>:</p> <ul> <li id="AtiyahWitten01"><a class="existingWikiWord" href="/nlab/show/Michael+Atiyah">Michael Atiyah</a>, <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a> section 6.4 of <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>-Theory dynamics on a manifold of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>-holonomy</em>, Adv. Theor. Math. Phys. 6 (2001) (<a href="http://arxiv.org/abs/hep-th/0107177">arXiv:hep-th/0107177</a>)</li> </ul> <p>Specifically in relation to <a class="existingWikiWord" href="/nlab/show/Moonshine">Moonshine</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Miranda+Cheng">Miranda Cheng</a>, Sarah M. Harrison, Roberto Volpato, Max Zimet, <em>K3 String Theory, Lattices and Moonshine</em> (<a href="https://arxiv.org/abs/1612.04404">arXiv:1612.04404</a>)</li> </ul> <p>Specifically in relation to <a class="existingWikiWord" href="/nlab/show/little+string+theory">little string theory</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Shamit+Kachru">Shamit Kachru</a>, Arnav Tripathy, Max Zimet, <em>K3 metrics from little string theory</em> (<a href="https://arxiv.org/abs/1810.10540">arXiv:1810.10540</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on July 18, 2024 at 11:09:33. 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