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T-duality</a>, <a class="existingWikiWord" href="/nlab/show/non-abelian+T-duality">non-abelian T-duality</a>)</p> <p><a class="existingWikiWord" href="/nlab/show/mirror+symmetry">mirror symmetry</a></p> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/S-duality">S-duality</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/electric-magnetic+duality">electric-magnetic duality</a>, <a class="existingWikiWord" href="/nlab/show/Montonen-Olive+duality">Montonen-Olive duality</a>, <a class="existingWikiWord" href="/nlab/show/geometric+Langlands+duality">geometric Langlands duality</a></p> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/U-duality">U-duality</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/exceptional+generalized+geometry">exceptional generalized geometry</a></p> </li> </ul> <p><strong>string-fivebrane duality</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/dual+heterotic+string+theory">dual heterotic string theory</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/string-string+dualities">string-string dualities</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/HET+-+I">HET - I</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/HET+-+II">HET - II</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/HET+-+M">HET - M</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/duality+between+M-theory+and+type+IIA+superstring+theory">M - IIA</a>.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/duality+between+M-theory+on+Z2-orbifolds+and+type+IIB+string+theory+on+K3-fibrations">M- IIB</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/F-theory">F-theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/duality+between+M-theory+and+F-theory">M - F</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/duality+between+F-theory+and+heterotic+string+theory">F - HET</a></p> </li> </ul> <p><strong>string-QFT duality</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/AdS-CFT+duality">AdS-CFT duality</a></p> <p>(an <a class="existingWikiWord" href="/nlab/show/open%2Fclosed+string+duality">open/closed string duality</a>)</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/AdS2%2FCFT1+duality">AdS2/CFT1 duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/AdS3%2FCFT2+duality">AdS3/CFT2 duality</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/p-adic+AdS%2FCFT+duality">p-adic AdS/CFT duality</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/AdS-QCD+correspondence">AdS-QCD correspondence</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory+as+topological+string+theory">Chern-Simons theory as topological string theory</a></p> </li> </ul> <p><strong>QFT-QFT duality</strong>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+QFT">effective QFT</a> incarnations of <a class="existingWikiWord" href="/nlab/show/open%2Fclosed+string+duality">open/closed string duality</a>,</p> <p>relating (<a class="existingWikiWord" href="/nlab/show/supergravity">super</a>-)<a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> to (<a class="existingWikiWord" href="/nlab/show/super+Yang-Mills+theory">super</a>-)<a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BCJ+relations">BCJ relations</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+double+copy">classical double copy</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kramers-Wannier+duality">Kramers-Wannier duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Seiberg+duality">Seiberg duality</a> (swapping <a class="existingWikiWord" href="/nlab/show/NS5-branes">NS5-branes</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/AGT+correspondence">AGT correspondence</a> (<a class="existingWikiWord" href="/nlab/show/wrapped+brane">wrapped</a> <a class="existingWikiWord" href="/nlab/show/M5-branes">M5-branes</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/3d-3d+correspondence">3d-3d correspondence</a> (<a class="existingWikiWord" href="/nlab/show/wrapped+brane">wrapped</a> <a class="existingWikiWord" href="/nlab/show/M5-branes">M5-branes</a>)</p> </li> </ul> </div></div> <h4 id="string_theory">String theory</h4> <div class="hide"><div> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a></strong></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/books+about+string+theory">books about string theory</a></p> </li> </ul> <h3 id="ingredients">Ingredients</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a>,</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/CFT">CFT</a>, <a class="existingWikiWord" href="/nlab/show/perturbation+theory">perturbation theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+QFT">effective background QFT</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/gravity">gravity</a>, <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a>, <a class="existingWikiWord" href="/nlab/show/quantum+gravity">quantum gravity</a></li> </ul> </li> </ul> <h3 id="critical_string_models">Critical string models</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a>, <a class="existingWikiWord" href="/nlab/show/differential+string+structure">differential string structure</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dual+heterotic+string+theory">dual heterotic string theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+fivebrane+structure">differential fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/type+IIA+string+theory">type IIA string theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/type+IIB+string+theory">type IIB string theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/F-theory">F-theory</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+field+theory">string field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/duality+in+string+theory">duality in string theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/T-duality">T-duality</a>, <a class="existingWikiWord" href="/nlab/show/mirror+symmetry">mirror symmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/S-duality">S-duality</a>, <a class="existingWikiWord" href="/nlab/show/electric-magnetic+duality">electric-magnetic duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/U-duality">U-duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open%2Fclosed+string+duality">open/closed string duality</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/AdS%2FCFT+correspondence">AdS/CFT correspondence</a>, <a class="existingWikiWord" href="/nlab/show/holographic+principle">holographic principle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a>, <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%C5%99ava-Witten+theory">Hořava-Witten theory</a></li> </ul> </li> </ul> <h3 id="extended_objects">Extended objects</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/brane">brane</a></p> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/D0-brane">D0-brane</a>, <a class="existingWikiWord" href="/nlab/show/D2-brane">D2-brane</a>, <a class="existingWikiWord" href="/nlab/show/D4-brane">D4-brane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D1-brane">D1-brane</a>, <a class="existingWikiWord" href="/nlab/show/D3-brane">D3-brane</a>, <a class="existingWikiWord" href="/nlab/show/D5-brane">D5-brane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/RR-field">RR-field</a>, <a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/NS-brane">NS-brane</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/string">string</a>, <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spinning+string">spinning string</a>, <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/B2-field">B2-field</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/NS5-brane">NS5-brane</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/B6-field">B6-field</a></li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/M-brane">M-brane</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/C3-field">C3-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ABJM+theory">ABJM theory</a>, <a class="existingWikiWord" href="/nlab/show/BLG+model">BLG model</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/C6-field">C6-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a></p> </li> </ul> </li> </ul> </li> </ul> <h3 id="topological_strings">Topological strings</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a>, <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+M-theory">topological M-theory</a></p> </li> </ul> <h2 id="backgrounds">Backgrounds</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/target+space">target space</a>, <a class="existingWikiWord" href="/nlab/show/background+gauge+field">background gauge field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+smooth+cohomology+in+string+theory">twisted smooth cohomology in string theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/landscape+of+string+theory+vacua">landscape of string theory vacua</a></p> </li> </ul> <h2 id="phenomenology">Phenomenology</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/string+phenomenology">string phenomenology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/moduli+stabilization">moduli stabilization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/G%E2%82%82-MSSM">G₂-MSSM</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/string+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#relation_to_tropical_geometry'>Relation to tropical geometry.</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='#CompleteProofs'>Complete proofs</a></li> <li><a href='#computation_via_topological_recursion'>Computation via topological recursion</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>To every complex 3-dimensional <a class="existingWikiWord" href="/nlab/show/Calabi-Yau+variety">Calabi-Yau variety</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> are associated two similar but differing types of <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> <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>-supersymmetric <a class="existingWikiWord" href="/nlab/show/2d+CFTs">2d CFTs</a>. There is at least for some CY <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> a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>↦</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">X \mapsto \hat X</annotation></semantics></math> which exchanges the <a class="existingWikiWord" href="/nlab/show/Hodge+numbers">Hodge numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>h</mi> <mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">h^{1,1}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>h</mi> <mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msup></mrow><annotation encoding="application/x-tex">h^{1,2}</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>SCFT</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SCFT_A(X)</annotation></semantics></math> is expected to be equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>SCFT</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SCFT_B(\hat X)</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>SCFT</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>SCFT</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> SCFT_A(X) \simeq SCFT_B(\hat X) \,. </annotation></semantics></math></div> <p>This is called <em>mirror symmetry</em>. At least in some cases this can be understood as a special case of <a class="existingWikiWord" href="/nlab/show/T-duality">T-duality</a> (<a href="#StromingerYauZaslow96">Strominger-Yau-Zaslow 96</a>).</p> <p>In this form mirror symmetry remains a conjecture, not the least because for the moment there is no complete construction of these SCFTs. But to every such <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SCFT</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SCFT(X)</annotation></semantics></math> one can associate two <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a>s, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A(X)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B(X)</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/A-model">A-model</a> and the <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a>. These <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> supersymmetric field theories were obtained by <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a> using a “topological twist”. The topological <a class="existingWikiWord" href="/nlab/show/A-model">A-model</a> can be expressed in terms of <a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a> of a variety and the topological B-model can be expressed in terms of the <a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a> of a variety.</p> <p>These topological theories are easier to understand and do retain a little bit of the information encoded in the full SCFTs. In terms of these the statement of mirror symmetry says that passing to mirror CYs <em>exchanges</em> the <a class="existingWikiWord" href="/nlab/show/A-model">A-model</a> with the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-model and conversely:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>B</mi><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mi>B</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>A</mi><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> A(X) \simeq B(\hat X),\,\,\,\,\,\,\,B(X)\simeq A(\hat X) \,. </annotation></semantics></math></div> <p>By a version of the <a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem, these <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a>s (see there) are encoded by <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+categories">A-∞ categories</a> that are <a class="existingWikiWord" href="/nlab/show/Calabi-Yau+categories">Calabi-Yau categories</a>: the <a class="existingWikiWord" href="/nlab/show/A-model">A-model</a> by the <a class="existingWikiWord" href="/nlab/show/Fukaya+category">Fukaya category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Fuk</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Fuk(X)</annotation></semantics></math> of <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> which can be understood as a <a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a> representing the Lagrangian intersection theory on the underlying <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a>; and the <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a> by an <a class="existingWikiWord" href="/nlab/show/enhanced+triangulated+category">enhancement</a> of the <a class="existingWikiWord" href="/nlab/show/derived+category">derived category</a> of <a class="existingWikiWord" href="/nlab/show/coherent+sheaves">coherent sheaves</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>D</mi> <mn>∞</mn> <mi>b</mi></msubsup><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D^b_\infty(\hat X)</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>X</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat X</annotation></semantics></math>.</p> <p>In terms of this data, mirror symmetry is the assertion that these <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+categories">A-∞ categories</a> are equivalent and simultaneously the same under exchange <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>↔</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">X\leftrightarrow \hat{X}</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Fuk</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><msubsup><mi>D</mi> <mn>∞</mn> <mi>b</mi></msubsup><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mi>and</mi><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mi>Fuk</mi><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo><mo>≃</mo><msubsup><mi>D</mi> <mn>∞</mn> <mi>b</mi></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> Fuk(X) \simeq D^b_\infty(\hat{X}), \,\,\,\, and \,\,\,\, Fuk(\hat{X}) \simeq D^b_\infty(X). </annotation></semantics></math></div> <p>This categorical formulation was introduced by <a class="existingWikiWord" href="/nlab/show/Maxim+Kontsevich">Maxim Kontsevich</a> in 1994 under the name <strong>homological mirror symmetry</strong>. The equivalence of the categorical expression of mirror symmetry to the SCFT formulation has been proven by <a class="existingWikiWord" href="/nlab/show/Maxim+Kontsevich">Maxim Kontsevich</a> and independently by <a class="existingWikiWord" href="/nlab/show/Kevin+Costello">Kevin Costello</a>, who showed how the datum of a topological conformal field theory is equivalent to the datum of a <a class="existingWikiWord" href="/nlab/show/Calabi-Yau+category">Calabi-Yau A-∞-category</a>(see <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a>).</p> <p>The mirror symmetry conjecture roughly claims that every Calabi-Yau 3-fold has a mirror. In fact one considers (mirror symmetry for) degenerating families for Calabi-Yau 3-folds in large volume limit (which may be expressed precisely via the Gromov-Hausdorff metric). The appropriate definition of (an appropriate version of) the <a class="existingWikiWord" href="/nlab/show/Fukaya+category">Fukaya category</a> of a symplectic manifold is difficult to achieve in desired generality. Invariants/tools of Fukaya category include symplectic <a class="existingWikiWord" href="/nlab/show/Floer+homology">Floer homology</a> and Gromov-Witten invariants (building up the <a class="existingWikiWord" href="/nlab/show/quantum+cohomology">quantum cohomology</a>).</p> <p>Mirror symmetry is related to the <a class="existingWikiWord" href="/nlab/show/T-duality">T-duality</a> on each fiber of an associated Lagrangian fibration <a href="#StromingerYauZaslow96">Strominger-Yau-Zaslow 96</a>.</p> <p>Although the non-Calabi-Yau case may be of lesser interest to physics, one can still formulate some mirror symmetry statements for, for instance, Fano manifolds. The mirror to a Fano manifold is a <a class="existingWikiWord" href="/nlab/show/Landau-Ginzburg+model">Landau-Ginzburg model</a> (see <a href="#HoriVafa00">Hori-Vafa 00</a>; see also work of Auroux for an explanation via the Strominger-Yau-Zaslow T-duality philosophy). Then the statements are: the <a class="existingWikiWord" href="/nlab/show/A-model">A-model</a> of the Fano (given by the Fukaya category) is equivalent to the B-model of the <a class="existingWikiWord" href="/nlab/show/Landau-Ginzburg+model">Landau-Ginzburg model</a> (given by the category of matrix factorizations); and the B-model of the Fano (given by the derived category of sheaves) is equivalent to the <a class="existingWikiWord" href="/nlab/show/A-model">A-model</a> of the <a class="existingWikiWord" href="/nlab/show/Landau-Ginzburg+model">Landau-Ginzburg model</a> (given by the Fukaya-Seidel category). A few of the relevant names: Kontsevich, Hori-Vafa, Auroux, Katzarkov, Orlov, Seidel, …</p> <h2 id="properties">Properties</h2> <h3 id="relation_to_tropical_geometry">Relation to tropical geometry.</h3> <p>Close relation to <a class="existingWikiWord" href="/nlab/show/tropical+geometry">tropical geometry</a>, see e.g. <a href="#Gross11">Gross 11</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/mirror+map">mirror map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/moduli+space+of+Calabi-Yau+spaces">moduli space of Calabi-Yau spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/3d+mirror+symmetry">3d mirror symmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/duality+in+physics">duality in physics</a>, <a class="existingWikiWord" href="/nlab/show/duality+in+string+theory">duality in string theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+Langlands+duality">geometric Langlands duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Landau-Ginzburg+model">Landau-Ginzburg model</a></p> </li> </ul> <h2 id="references">References</h2> <h3 id="general">General</h3> <p>The original statement of the homological mirror symmetry conjecture is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Maxim+Kontsevich">Maxim Kontsevich</a>, <em>Homological algebra of mirror symmetry</em>, Proc. ICM Zürich 1994, <a href="http://arxiv.org/abs/alg-geom/9411018">alg-geom/9411018</a></li> </ul> <p>See also:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Paul+Aspinwall">Paul Aspinwall</a>, <a class="existingWikiWord" href="/nlab/show/Brian+Greene">Brian Greene</a>, <a class="existingWikiWord" href="/nlab/show/David+Morrison">David Morrison</a>, <em>Calabi-Yau moduli space, mirror manifolds and spacetime topology change in string theory</em>, Nucl.Phys. B416 (1994) 414-480 (<a href="https://arxiv.org/abs/hep-th/9309097">arXiv:hep-th/9309097</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Philip+Candelas">Philip Candelas</a>, Xenia C. de la Ossa, Paul S. Green, Linda Parkes, <em>A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory</em>, Nuclear Phys. B 359 (1):21–74, 1991</p> </li> </ul> <p>Review:</p> <ul> <li id="HoriVafa00"> <p><a class="existingWikiWord" href="/nlab/show/Kentaro+Hori">Kentaro Hori</a>, <a class="existingWikiWord" href="/nlab/show/Cumrun+Vafa">Cumrun Vafa</a>, <em>Mirror Symmetry</em> (<a href="https://arxiv.org/abs/hep-th/0002222">arXiv:hep-th/0002222</a>)</p> </li> <li> <p>M. Ballard, <em>Meet homological mirror symmetry</em> (<a href="http://arxiv.org/abs/0801.2014">arxiv:0801.2014</a>)</p> </li> <li> <p>A. Port, <em>An introduction to homological mirror symmetry and the case of elliptic curves</em> (<a href="http://arxiv.org/abs/1501.00730">arXiv:1501.00730</a>)</p> </li> <li id="IbanezUranga12"> <p><a class="existingWikiWord" href="/nlab/show/Luis+Ib%C3%A1%C3%B1ez">Luis Ibáñez</a>, <a class="existingWikiWord" href="/nlab/show/Angel+Uranga">Angel Uranga</a>, section 10.1.2 of <em><a class="existingWikiWord" href="/nlab/show/String+Theory+and+Particle+Physics+--+An+Introduction+to+String+Phenomenology">String Theory and Particle Physics – An Introduction to String Phenomenology</a></em>, Cambridge University Press 2012</p> </li> </ul> <p>History:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Maxim+Kontsevich">Maxim Kontsevich</a>, <em>On the History of quantum cohomology and homological mirror symmetry</em> (2021) [video:<a href="https://youtu.be/Ml2x5NnEQ1I">YT</a>]</li> </ul> <p>Discussion amplifying the role of <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a>, and <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a>:</p> <ul> <li id="Sharpe19"><a class="existingWikiWord" href="/nlab/show/Eric+Sharpe">Eric Sharpe</a>, <em>Categorical equivalence and the renormalization group</em>, Proceedings of LMS/EPSRC Symposium <em><a href="http://www.maths.dur.ac.uk/lms/109/index.html">Higher Structures in M-Theory</a></em>, Fortschritte der Physik 2019 (<a href="https://arxiv.org/abs/1903.02880">arXiv:1903.02880</a>)</li> </ul> <p>Further review:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Paul+Aspinwall">Paul Aspinwall</a>, <a class="existingWikiWord" href="/nlab/show/Tom+Bridgeland">Tom Bridgeland</a>, <a class="existingWikiWord" href="/nlab/show/Alastair+Craw">Alastair Craw</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Douglas">Michael Douglas</a>, Mark Gross, <a class="existingWikiWord" href="/nlab/show/Anton+Kapustin">Anton Kapustin</a>, <a class="existingWikiWord" href="/nlab/show/Gregory+Moore">Gregory Moore</a>, <a class="existingWikiWord" href="/nlab/show/Graeme+Segal">Graeme Segal</a>, <a class="existingWikiWord" href="/nlab/show/Bal%C3%A1zs+Szendr%C5%91i">Balázs Szendrői</a>, P. Wilson,</p> <p><em>Dirichlet branes and mirror symmetry</em>,</p> <p>Clay Mathematics Monograph Volume 4, Amer. Math. Soc. Clay Math. Institute 2009</p> <p>(<a href="http://www.claymath.org/library/monographs/cmim04c.pdf">pdf</a>, <a href="http://www2.maths.ox.ac.uk/cmi/library/monographs/cmim04.pdf">pdf</a>) (very readable!)</p> </li> <li> <p>Felipe Espreafico G. Ramos, <em>Mirror Symmetry and Fukaya Categories</em> (2020) [<a href="https://mathi.uni-heidelberg.de/~framos/Fukaya.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Ramos-MirrorSymmetry.pdf" title="pdf">pdf</a>]</p> </li> </ul> <p>The relation to <a class="existingWikiWord" href="/nlab/show/T-duality">T-duality</a> was established in</p> <ul> <li id="StromingerYauZaslow96"><a class="existingWikiWord" href="/nlab/show/Andrew+Strominger">Andrew Strominger</a>, <a class="existingWikiWord" href="/nlab/show/Shing-Tung+Yau">Shing-Tung Yau</a>, <a class="existingWikiWord" href="/nlab/show/Eric+Zaslow">Eric Zaslow</a>, <em>Mirror Symmetry is T-Duality</em>, Nucl.Phys.B479:243-259,1996 (DOI 10.1016/0550-3213(96)00434-8) <a href="http://arxiv.org/abs/hep-th/9606040">hep-th/9606040</a></li> </ul> <p>Further references:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Cumrun+Vafa">Cumrun Vafa</a>, <a class="existingWikiWord" href="/nlab/show/Shing-Tung+Yau">Shing-Tung Yau</a> (eds.), <em>Winter school on mirror symmetry, vector bundles, and Lagrangian submanifolds</em>, Harvard 1999, AMS, Intern. Press (includes A. Strominger, S-T. Yau, E. Zaslow, <em>Mirror symmetry is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-duality</em> as pages 333–347; ).</p> </li> <li> <p>K. Hori, S. Katz, A. Klemm et al. <em>Mirror symmetry I</em>, AMS, Clay Math. Institute 2003.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Paul+Seidel">Paul Seidel</a>, <em>Fukaya categories and Picard-Lefschetz theory</em>, Zurich Lectures in Advanced Mathematics. European Mathematical Society, Zürich, 2008. viii+326 pp</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Mark+Gross">Mark Gross</a>, <a class="existingWikiWord" href="/nlab/show/Bernd+Siebert">Bernd Siebert</a>, <em>Mirror symmetry via logarithmic degeneration data I</em>, <a href="http://arxiv.org/abs/math/0309070">math.AG/0309070</a>, <em>From real affine geometry to complex geometry</em>, <a href="http://arxiv.org/abs/math/0703822">math.AG/0703822</a>, <em>Mirror symmetry via logarithmic degeneration data II</em>, <a href="http://arxiv.org/abs/0709.2290">arxiv/0709.2290</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Anton+Kapustin">Anton Kapustin</a>, <a class="existingWikiWord" href="/nlab/show/Dmitri+Orlov">Dmitri Orlov</a>, <em>Lectures on mirror symmetry, derived categories, and D-branes</em>, Uspehi Mat. Nauk <strong>59</strong> (2004), no. 5(359), 101–134; translation in Russian Math. Surveys <strong>59</strong> (2004), no. 5, 907–940, <a href="http://arxiv.org/abs/math/0308173">math.AG/0308173</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Maxim+Kontsevich">Maxim Kontsevich</a>, <a class="existingWikiWord" href="/nlab/show/Yan+Soibelman">Yan Soibelman</a>, <em>Homological mirror symmetry and torus fibrations</em>, <a href="http://arxiv.org/abs/math/0011041">math.SG/0011041</a></p> </li> <li> <p>Yong-Geun Oh, Kenji Fukaya, <em>Floer homology in symplectic geometry and mirror symmetry</em>, Proc. ICM 2006, <a href="http://www.math.wisc.edu/~oh/Oh-icm2006.pdf">pdf</a></p> </li> <li> <p>wikipedia: <a href="http://en.wikipedia.org/wiki/Mirror_symmetry_%28string_theory%29">mirror symmetry (string theory)</a>, <a href="http://en.wikipedia.org/wiki/Homological_mirror_symmetry">homological mirror symmetry</a></p> </li> <li> <p>partial notes from Miami 08 workshop: <a href="http://www-math.mit.edu/~auroux/frg/miami08-notes">miami08-notes</a> and abstracts from <a href="http://www-math.mit.edu/~auroux/frg/miami09-abstracts.html">miami09</a>, <a href="http://www-math.mit.edu/~auroux/frg/miami10-abstracts.html">miami10</a></p> </li> <li id="Gross11"> <p><a class="existingWikiWord" href="/nlab/show/Mark+Gross">Mark Gross</a>, <em>Tropical geometry and mirror symmetry</em>, CBMS regional conf. ser. 114 (2011), based on the CBMS course in Kansas, <a href="http://www.ams.org/bookstore-getitem/item=CBMS-114">AMS book page</a>, <a href="http://www.math.ucsd.edu/~mgross/kansas.pdf">pdf</a></p> </li> </ul> <p>Discussion in the context of <a class="existingWikiWord" href="/nlab/show/derived+Morita+equivalence">derived Morita equivalence</a> includes</p> <ul> <li id="Okada09">So Okada, <em>Homological mirror symmetry of Fermat polynomials</em> (<a href="http://arxiv.org/abs/0910.2014">arXiv:0910.2014</a>)</li> </ul> <h3 id="CompleteProofs">Complete proofs</h3> <p>Here is a list with references that give complete proofs of <em>homological</em> mirror symmetry on certain (types of) spaces.</p> <ul> <li> <p>M. Abouzaid, I. Smith, <em>Homological mirror symmetry for the four-torus</em>, Duke Math. J. 152 (2010), 373–440, <a href="http://arxiv.org/abs/0903.3065">arXiv:0903.3065</a></p> </li> <li> <p>A. Polishchuk and E. Zaslow, <em>Categorical mirror symmetry: the elliptic curve</em>, Adv. Theor. Math. Phys. 2:443470, 1998.</p> </li> <li> <p>V. Golyshev, V. Lunts, D. Orlov, <em>Mirror symmetry for abelian varieties</em>, J. Alg. Geom. <strong>10</strong> (2001), no. 3, 433–496, <a href="http://arxiv.org/abs/math/9812003">math.AG/9812003</a></p> </li> <li> <p>P. Seidel, <em>Homological mirror symmetry for the quartic surface</em>, <a href="http://arxiv.org/abs/math/0310414">arXiv:0310414</a></p> </li> <li> <p>Alexander I. Efimov, <em>Homological mirror symmetry for curves of higher genus</em>, Inventiones Math. <strong>166</strong> (2006), 537–582, <a href="http://arxiv.org/abs/0907.3903">arXiv:0907.3903</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D.+Auroux">D. Auroux</a>, <a class="existingWikiWord" href="/nlab/show/L.+Katzarkov">L. Katzarkov</a>, <a class="existingWikiWord" href="/nlab/show/D.+Orlov">D. Orlov</a>, <em>Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves</em>, <a href=""></a>; <em>Mirror symmetry for weighted projective planes and their noncommutative deformations</em>, Ann. Math. <strong>167</strong> (2008), 867–943, <a href="http://arxiv.org/abs/math/0404281">math.AG/0404281</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Mohammed+Abouzaid">Mohammed Abouzaid</a>, Denis Auroux, Alexander I. Efimov, Ludmil Katzarkov, Dmitri Orlov, <em>Homological mirror symmetry for punctured spheres</em>, <a href="http://arxiv.org/abs/1103.4322">arxiv/1103.4322</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Paul+Seidel">Paul Seidel</a>, <em>Homological mirror symmetry for the genus two curve</em>, J. Algebraic Geometry, to appear, <a href="http://arxiv.org/abs/0812.1171">arXiv:0812.1171</a></p> </li> </ul> <h3 id="computation_via_topological_recursion">Computation via topological recursion</h3> <p>Computation via <a class="existingWikiWord" href="/nlab/show/topological+recursion">topological recursion</a> in <a class="existingWikiWord" href="/nlab/show/matrix+models">matrix models</a> and all-<a class="existingWikiWord" href="/nlab/show/genus+of+a+surface">genus</a> proofs of mirror symmetry is due to</p> <ul> <li id="BouchardKlemmMarinoPasquetti09"> <p><a class="existingWikiWord" href="/nlab/show/Vincent+Bouchard">Vincent Bouchard</a>, <a class="existingWikiWord" href="/nlab/show/Albrecht+Klemm">Albrecht Klemm</a>, <a class="existingWikiWord" href="/nlab/show/Marcos+Marino">Marcos Marino</a>, <a class="existingWikiWord" href="/nlab/show/Sara+Pasquetti">Sara Pasquetti</a>, <em>Remodeling the B-model</em>, Commun.Math.Phys.287:117-178, 2009 (<a href="https://arxiv.org/abs/0709.1453">arXiv:0709.1453</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bertrand+Eynard">Bertrand Eynard</a>, <a class="existingWikiWord" href="/nlab/show/Amir-Kian+Kashani-Poor">Amir-Kian Kashani-Poor</a>, Olivier Marchal, <em>A matrix model for the topological string I: Deriving the matrix model</em> (<a href="https://arxiv.org/abs/1003.1737">arXiv:1003.1737</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bertrand+Eynard">Bertrand Eynard</a>, <a class="existingWikiWord" href="/nlab/show/Amir-Kian+Kashani-Poor">Amir-Kian Kashani-Poor</a>, Olivier Marchal, <em>A matrix model for the topological string II: The spectral curve and mirror geometry</em> (<a href="https://arxiv.org/abs/1007.2194">arXiv:1007.2194</a>)</p> </li> <li id="EynardOrantin12"> <p><a class="existingWikiWord" href="/nlab/show/Bertrand+Eynard">Bertrand Eynard</a>, <a class="existingWikiWord" href="/nlab/show/Nicolas+Orantin">Nicolas Orantin</a>, <em>Computation of open Gromov-Witten invariants for toric Calabi-Yau 3-folds by topological recursion, a proof of the BKMP conjecture</em> (<a href="https://arxiv.org/abs/1205.1103">arXiv:1205.1103</a>)</p> </li> <li id="FangLiuZong13"> <p>Bohan Fang, Chiu-Chu Melissa Liu, Zhengyu Zong, <em>All Genus Open-Closed Mirror Symmetry for Affine Toric Calabi-Yau 3-Orbifolds</em> (<a href="https://arxiv.org/abs/1310.4818">arXiv:1310.4818</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on April 1, 2024 at 09:01:48. 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