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xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="quantum_systems">Quantum systems</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/quantum+logic">quantum logic</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/linear+logic">linear logic</a>, <a class="existingWikiWord" href="/nlab/show/dependent+linear+type+theory">dependent</a> <a class="existingWikiWord" href="/nlab/show/linear+type+theory">linear type theory</a> <a class="existingWikiWord" href="/nlab/show/string+diagrams">string diagrams</a> in <a class="existingWikiWord" href="/nlab/show/quantum+information+theory+via+dagger-compact+categories">†-compact categories</a> <a class="existingWikiWord" href="/nlab/show/tensor+networks">tensor networks</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a> <a class="existingWikiWord" href="/nlab/show/order-theoretic+structure+in+quantum+mechanics">order-theoretic structure</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+probability">quantum probability</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/quantum+physics">quantum physics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantum+systems">quantum systems</a> (<a class="existingWikiWord" href="/nlab/show/parameterized+quantum+systems">parameterized</a>, <a class="existingWikiWord" href="/nlab/show/open+quantum+system">open</a>) <a class="existingWikiWord" href="/nlab/show/quantum+measurement">quantum measurement</a> <a class="existingWikiWord" href="/nlab/show/quantum+state+collapse">quantum state collapse</a> <a class="existingWikiWord" href="/nlab/show/quantum+decoherence">quantum decoherence</a> <a class="existingWikiWord" href="/nlab/show/quantum+adiabatic+theorem">quantum adiabatic theorem</a> <a class="existingWikiWord" href="/nlab/show/Berry+phases">Berry phases</a> <a class="existingWikiWord" href="/nlab/show/Dyson+formula">Dyson formula</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+many-body+physics">quantum many-body physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> <a class="existingWikiWord" href="/nlab/show/functorial+quantum+field+theory">functorial quantum field theory</a> <a class="existingWikiWord" href="/nlab/show/algebraic+quantum+field+theory">algebraic quantum field theory</a> (<a class="existingWikiWord" href="/nlab/show/non-perturbative+quantum+field+theory">non-</a>)<a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/solid+state+physics">solid state physics</a> <a class="existingWikiWord" href="/nlab/show/quantum+material">quantum material</a> (<a class="existingWikiWord" href="/nlab/show/topological+phases+of+matter">topological</a>) <a class="existingWikiWord" href="/nlab/show/phases+of+matter">phases of matter</a> </li> </ul> <div> <a class="existingWikiWord" href="/nlab/show/quantum+probability+theory">quantum probability theory</a> – <a class="existingWikiWord" href="/nlab/show/observables">observables</a> and <a class="existingWikiWord" href="/nlab/show/states">states</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/states">states</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/classical+state">classical state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+state">quantum state</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/space+of+states+%28in+geometric+quantization%29">space of states (in geometric quantization)</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/state+on+a+star-algebra">state on a star-algebra</a>, <a class="existingWikiWord" href="/nlab/show/quasi-state">quasi-state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/qbit">qbit</a>, <a class="existingWikiWord" href="/nlab/show/Bell+state">Bell state</a> <a class="existingWikiWord" href="/nlab/show/dimer">dimer</a>, <a class="existingWikiWord" href="/nlab/show/tensor+network+state">tensor network state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+state+preparation">quantum state preparation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/probability+amplitude">probability amplitude</a>, <a class="existingWikiWord" href="/nlab/show/quantum+fluctuation">quantum fluctuation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/pure+state">pure state</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/wave+function">wave function</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/bra-ket">bra-ket</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bell+state">Bell state</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+superposition">quantum superposition</a>, <a class="existingWikiWord" href="/nlab/show/quantum+interference">quantum interference</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+entanglement">quantum entanglement</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+measurement">quantum measurement</a> <a class="existingWikiWord" href="/nlab/show/wave+function+collapse">wave function collapse</a> <a class="existingWikiWord" href="/nlab/show/Born+rule">Born rule</a> <a class="existingWikiWord" href="/nlab/show/deferred+measurement+principle">deferred measurement principle</a> <a class="existingWikiWord" href="/nlab/show/quantum+reader+monad">quantum reader monad</a> <a class="existingWikiWord" href="/nlab/show/measurement+problem">measurement problem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/superselection+sector">superselection sector</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/mixed+state">mixed state</a>, <a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a> <a class="existingWikiWord" href="/nlab/show/entanglement+entropy">entanglement entropy</a> <a class="existingWikiWord" href="/nlab/show/holographic+entanglement+entropy">holographic entanglement entropy</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/coherent+quantum+state">coherent quantum state</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/ground+state">ground state</a>, <a class="existingWikiWord" href="/nlab/show/excited+state">excited state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quasi-free+state">quasi-free state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Fock+space">Fock space</a>, <a class="existingWikiWord" href="/nlab/show/second+quantization">second quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a>, <a class="existingWikiWord" href="/nlab/show/vacuum+state">vacuum state</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Hadamard+state">Hadamard state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+diagram">vacuum diagram</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+expectation+value">vacuum expectation value</a>, <a class="existingWikiWord" href="/nlab/show/vacuum+amplitude">vacuum amplitude</a>, <a class="existingWikiWord" href="/nlab/show/vacuum+fluctuation">vacuum fluctuation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+energy">vacuum energy</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+polarization">vacuum polarization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/interacting+vacuum">interacting vacuum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/thermal+vacuum">thermal vacuum</a>, <a class="existingWikiWord" href="/nlab/show/KMS+state">KMS state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+stability">vacuum stability</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/false+vacuum">false vacuum</a>, <a class="existingWikiWord" href="/nlab/show/tachyon">tachyon</a>, <a class="existingWikiWord" href="/nlab/show/Coleman-De+Luccia+instanton">Coleman-De Luccia instanton</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/theta+vacuum">theta vacuum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/perturbative+string+theory+vacuum">perturbative string theory vacuum</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/non-geometric+string+theory+vacuum">non-geometric string theory vacuum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/landscape+of+string+theory+vacua">landscape of string theory vacua</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/entangled+state">entangled state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/tensor+network+state">tensor network state</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/matrix+product+state">matrix product state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/tree+tensor+network+state">tree tensor network state</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/observables">observables</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantum+observable">quantum observable</a>, <a class="existingWikiWord" href="/nlab/show/beable">beable</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/algebra+of+observables">algebra of observables</a>, <a class="existingWikiWord" href="/nlab/show/star-algebra">star-algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+operator+%28in+geometric+quantization%29">quantum operator (in geometric quantization)</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+operation">quantum operation</a>, <a class="existingWikiWord" href="/nlab/show/quantum+effect">quantum effect</a>, <a class="existingWikiWord" href="/nlab/show/effect+algebra">effect algebra</a> </li> <li> in <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/local+observable">local observable</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/polynomial+observable">polynomial observable</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/linear+observable">linear observable</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/field+observable">field observable</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/regular+observable">regular observable</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/microcausal+observable">microcausal observable</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/normal-ordered+product">normal-ordered product</a>, <a class="existingWikiWord" href="/nlab/show/time-ordered+products">time-ordered products</a>, <a class="existingWikiWord" href="/nlab/show/retarded+product">retarded product</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/interacting+field+algebra+of+observables">interacting field algebra of observables</a>, <a class="existingWikiWord" href="/nlab/show/Bogoliubov%27s+formula">Bogoliubov's formula</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/theorems">theorems</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/order-theoretic+structure+in+quantum+mechanics">order-theoretic structure in quantum mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Alfsen-Shultz+theorem">Alfsen-Shultz theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Harding-D%C3%B6ring-Hamhalter+theorem">Harding-Döring-Hamhalter theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Nuiten%27s+lemma">Nuiten's lemma</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wigner%27s+theorem">Wigner's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/no-cloning+theorem">no-cloning theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a> </li> </ul> </li> </ul> </div> <a class="existingWikiWord" href="/nlab/show/quantum+information">quantum information</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantum+information+via+dagger-compact+categories">quantum information via dagger-compact categories</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+operation">quantum operation</a>, <a class="existingWikiWord" href="/nlab/show/quantum+channel">quantum channel</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+teleportation">quantum teleportation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+entanglement">quantum entanglement</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/entanglement+entropy">entanglement entropy</a> <a class="existingWikiWord" href="/nlab/show/holographic+entanglement+entropy">holographic entanglement entropy</a> <a class="existingWikiWord" href="/nlab/show/topological+entanglement+entropy">topological entanglement entropy</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/quantum+technology">quantum technology</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantum+sensing">quantum sensing</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+communication">quantum communication</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+cryptography">quantum cryptography</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+machine+learning">quantum machine learning</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/quantum+computing">quantum computing</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/qbit">qbit</a>, <a class="existingWikiWord" href="/nlab/show/qdit">qdit</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+gate">quantum gate</a>, <a class="existingWikiWord" href="/nlab/show/quantum+circuit">quantum circuit</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/adiabatic+quantum+computation">adiabatic quantum computation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/measurement-based+quantum+computation">measurement-based quantum computation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/topological+quantum+computation">topological quantum computation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+programming+language">quantum programming language</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+error+correction">quantum error correction</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/HaPPY+code">HaPPY code</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Majorana+dimer+code">Majorana dimer code</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/spin+resonance+qbit">spin resonance qbit</a> </li> <li> quantum algorithms: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Grover%27s+algorithm">Grover's algorithm</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Shor%27s+algorithm">Shor's algorithm</a> </li> </ul> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <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='#ReferencesWeightSystemsOnChordDiagramsInPhysics'>Chord diagrams and weight systems in Physics</a></li> <ul> <li><a href='#ReferencesWeightSystemsInChernSimonsTheory'>In Chern-Simons theory</a></li> <li><a href='#ForSingleTraceOperatorsInAdSCFTDuality'>For single trace operators in AdS/CFT duality</a></li> <li><a href='#ReferencesWeightSystemsInSYKModel'>In <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>AdS</mi> <mn>2</mn></msub><mo stretchy="false">/</mo><msub><mi>CFT</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">AdS_2/CFT_1</annotation></semantics></math>, JT-gravity/SYK-model</a></li> <li><a href='#ReferencesWeightSystemsInDpDp2BraneIntersections'>In D<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>/D<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(p+2)</annotation></semantics></math>-brane intersections</a></li> <li><a href='#ForHolographicEntanglementEntropy'>As codes for holographic entanglement entropy</a></li> <li><a href='#ForDysonSchwingerEquations'>For Dyson-Schwinger equations</a></li> <li><a href='#other'>Other</a></li> </ul> </ul> </ul> </div> <h2 id="idea">Idea</h2> A class of <a class="existingWikiWord" href="/nlab/show/quantum+error+correcting+codes">quantum error correcting codes</a> whose <a class="existingWikiWord" href="/nlab/show/code+subspaces">code subspaces</a> are the <a class="existingWikiWord" href="/nlab/show/images">images</a> of <a class="existingWikiWord" href="/nlab/show/linear+maps">linear maps</a> given by <a class="existingWikiWord" href="/nlab/show/tensor+networks">tensor networks</a> of the shape of <a class="existingWikiWord" href="/nlab/show/round+chord+diagrams">round chord diagrams</a> with chords acting as <a class="existingWikiWord" href="/nlab/show/Majorana+dimer">Majorana dimers</a>. The Majorana dimer code subsumes the <a class="existingWikiWord" href="/nlab/show/HaPPY+code">HaPPY code</a>: <div style="margin: -40px 0px 20px 10px"> <figure style="margin: 0 0 0 0"> <img src="/nlab/files/HaPPYCodesAsDimerCode.jpg" width="680px" /> <figcaption style="text-align: center">From <a href="#JGPE19">JGPE 19</a></figcaption> </figure> </div> and makes the nature of <a class="existingWikiWord" href="/nlab/show/holographic+entanglement+entropy">holographic entanglement entropy</a> particularly manifest (see <a href="holographic+entanglement+entropy#ChordDiagramRepresentation">there</a> for more). <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/chord+diagram">chord diagram</a></li> </ul> <h2 id="references">References</h2> <h3 id="general">General</h3> Due to: <ul> <li id="JGPE19"><a class="existingWikiWord" href="/nlab/show/Alexander+Jahn">Alexander Jahn</a>, <a class="existingWikiWord" href="/nlab/show/Marek+Gluza">Marek Gluza</a>, <a class="existingWikiWord" href="/nlab/show/Fernando+Pastawski">Fernando Pastawski</a>, <a class="existingWikiWord" href="/nlab/show/Jens+Eisert">Jens Eisert</a>, Majorana dimers and holographic quantum error-correcting code, Phys. Rev. Research 1, 033079 (2019) (<a href="https://arxiv.org/abs/1905.03268">arXiv:1905.03268</a>)</li> </ul> based on: <ul> <li><a class="existingWikiWord" href="/nlab/show/Alexander+Jahn">Alexander Jahn</a>, <a class="existingWikiWord" href="/nlab/show/Marek+Gluza">Marek Gluza</a>, <a class="existingWikiWord" href="/nlab/show/Fernando+Pastawski">Fernando Pastawski</a>, <a class="existingWikiWord" href="/nlab/show/Jens+Eisert">Jens Eisert</a>, Holography and criticality in matchgate tensor networks, Science Advances 5, eaaw0092 (2019) (<a href="https://arxiv.org/abs/1711.03109">arXiv:1711.03109</a>, <a href="https://advances.sciencemag.org/content/5/8/eaaw0092">doi:10.1126/sciadv.aaw0092</a>)</li> </ul> Review in: <ul> <li><a class="existingWikiWord" href="/nlab/show/Alexander+Jahn">Alexander Jahn</a>, <a class="existingWikiWord" href="/nlab/show/Jens+Eisert">Jens Eisert</a>, Holographic tensor network models and quantum error correction: A topical review (<a href="https://arxiv.org/abs/2102.02619">arXiv:2102.02619</a>)</li> </ul> See also: <ul> <li id="KahnZimborasEisert19"> <a class="existingWikiWord" href="/nlab/show/Alexander+Jahn">Alexander Jahn</a>, <a class="existingWikiWord" href="/nlab/show/Zolt%C3%A1n+Zimbor%C3%A1s">Zoltán Zimborás</a>, <a class="existingWikiWord" href="/nlab/show/Jens+Eisert">Jens Eisert</a>, Central charges of aperiodic holographic tensor network models, Phys. Rev. A 102, 042407 (<a href="https://arxiv.org/abs/1911.03485">arXiv:1911.03485</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Han+Yan">Han Yan</a>, Geodesic string condensation from symmetric tensor gauge theory: a unifying framework of holographic toy models, Phys. Rev. B 102, 161119 (2020) (<a href="https://arxiv.org/abs/1911.01007">arXiv:1911.01007</a>) </li> </ul> <div> <h3 id="ReferencesWeightSystemsOnChordDiagramsInPhysics">Chord diagrams and weight systems in Physics</h3> The following is a list of references that involve (<a class="existingWikiWord" href="/nlab/show/weight+systems">weight systems</a> on) <a class="existingWikiWord" href="/nlab/show/chord+diagrams">chord diagrams</a>/<a class="existingWikiWord" href="/nlab/show/Jacobi+diagrams">Jacobi diagrams</a> in <a class="existingWikiWord" href="/nlab/show/physics">physics</a>: <ol> <li> <a href="#ReferencesWeightSystemsInChernSimonsTheory">In Chern-Simons theory</a> </li> <li> <a href="#ReferencesWeightSystemsInDpDp2BraneIntersections">In Dp-D(p+2) brane intersections</a> </li> <li> In <a class="existingWikiWord" href="/nlab/show/AdS-CFT+in+condensed+matter+physics">quantum many body models</a> for <a class="existingWikiWord" href="/nlab/show/AdS-CFT+correspondence">for holographic</a> <a class="existingWikiWord" href="/nlab/show/brane">brane</a>/<a class="existingWikiWord" href="/nlab/show/bulk">bulk</a> correspondence: <ol> <li> <a href="#ReferencesWeightSystemsInSYKModel">In AdS2/CFT1, JT-gravity/SYK-model</a> </li> <li> <a href="#ForHolographicEntanglementEntropy">As dimer/bit thread codes for holographic entanglement entropy</a> </li> </ol> </li> </ol> For a unifying perspective (via <a class="existingWikiWord" href="/nlab/show/Hypothesis+H">Hypothesis H</a>) and further pointers, see: <ul> <li id="SatiSchreiber19c"> <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/schreiber/show/Differential+Cohomotopy+implies+intersecting+brane+observables">Differential Cohomotopy implies intersecting brane observables</a>, Adv. Theor. Math. Phys. 26 4 (2022) [<a href="https://dx.doi.org/10.4310/ATMP.2022.v26.n4.a4">doi:10.4310/ATMP.2022.v26.n4.a4</a><a href="https://arxiv.org/abs/1912.10425">arXiv:1912.10425</a>] </li> <li id="CSS21"> <a class="existingWikiWord" href="/nlab/show/David+Corfield">David Corfield</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>: <a class="existingWikiWord" href="/schreiber/show/Fundamental+weight+systems+are+quantum+states">Fundamental weight systems are quantum states</a> Lett. Math. Phys. 113 112 (2023) [<a href="https://arxiv.org/abs/2105.02871">arXiv:2105.02871</a>, <a href="https://doi.org/10.1007/s11005-023-01725-4">doi:10.1007/s11005-023-01725-4</a>] </li> <li id="Collari2023"> <a class="existingWikiWord" href="/nlab/show/Carlo+Collari">Carlo Collari</a>, A note on weight systems which are quantum states, Can. Math. Bull. 66 4 (2023) [<a href="https://doi.org/10.4153/S0008439523000206">doi:10.4153/S0008439523000206</a>, <a href="https://arxiv.org/abs/2210.05399">arXiv:2210.05399</a>] </li> </ul> Review: <ul> <li><a class="existingWikiWord" href="/nlab/show/Carlo+Collari">Carlo Collari</a>, Weight systems which are quantum states, talk at <a href="https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html">QFT and Cobordism</a>, <a class="existingWikiWord" href="/nlab/show/CQTS">CQTS</a> (Mar 2023) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="Center+for+Quantum+and+Topological+Systems#CollariMar2023">web</a>, <a class="existingWikiWord" href="/nlab/files/Collari-WeightSystems.pdf" title="pdf">pdf</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> <h4 id="ReferencesWeightSystemsInChernSimonsTheory">In Chern-Simons theory</h4> Since <a class="existingWikiWord" href="/nlab/show/weight+systems+are+the+associated+graded+of+Vassiliev+invariants">weight systems are the associated graded of Vassiliev invariants</a>, and since <a class="existingWikiWord" href="/nlab/show/Vassiliev+invariants">Vassiliev invariants</a> are <a class="existingWikiWord" href="/nlab/show/knot+invariants">knot invariants</a> arising as certain <a class="existingWikiWord" href="/nlab/show/correlators">correlators</a>/<a class="existingWikiWord" href="/nlab/show/Feynman+amplitudes">Feynman amplitudes</a> of <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> in the presence of <a class="existingWikiWord" href="/nlab/show/Wilson+lines">Wilson lines</a>, there is a close relation between <a class="existingWikiWord" href="/nlab/show/weight+systems">weight systems</a> and <a class="existingWikiWord" href="/nlab/show/quantization+of+3d+Chern-Simons+theory">quantum Chern-Simons theory</a>. Historically this is the original application of <a class="existingWikiWord" href="/nlab/show/chord+diagrams">chord diagrams</a>/<a class="existingWikiWord" href="/nlab/show/Jacobi+diagrams">Jacobi diagrams</a> and their <a class="existingWikiWord" href="/nlab/show/weight+systems">weight systems</a>, see also at <a class="existingWikiWord" href="/nlab/show/graph+complex">graph complex</a> and <a class="existingWikiWord" href="/nlab/show/Kontsevich+integral">Kontsevich integral</a>. <div class="float_right_image" style="margin: -40px 0px 20px 14px"> <figure style="margin: 0 0 0 0"> <img src="/nlab/files/BarNatanChernSimonsChordDiagrams.jpg" width="330px" /> <figcaption style="text-align: center">from <a href="#BarNatan91">Bar-Natan 1991</a></figcaption> </figure> </div> <ul> <li id="BarNatan91"> <a class="existingWikiWord" href="/nlab/show/Dror+Bar-Natan">Dror Bar-Natan</a>, Perturbative aspects of the Chern-Simons topological quantum field theory, thesis 1991 (<a href="http://inspirehep.net/record/323500">spire:323500</a>, <a href="https://search.proquest.com/docview/303979053">proquest:303979053</a>, <a class="existingWikiWord" href="/nlab/files/BarNatanPerturbativeCS91.pdf" title="">BarNatanPerturbativeCS91.pdf</a>) </li> <li id="Kontsevich93"> <a class="existingWikiWord" href="/nlab/show/Maxim+Kontsevich">Maxim Kontsevich</a>, Vassiliev’s knot invariants, Advances in Soviet Mathematics, Volume 16, Part 2, 1993 (<a href="http://pagesperso.ihes.fr/~maxim/TEXTS/VassilievKnot.pdf">pdf</a>) </li> <li> Daniel Altschuler, Laurent Freidel, Vassiliev knot invariants and Chern-Simons perturbation theory to all orders, Commun. Math. Phys. 187 (1997) 261-287 (<a href="https://arxiv.org/abs/q-alg/9603010">arxiv:q-alg/9603010</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Alberto+Cattaneo">Alberto Cattaneo</a>, Paolo Cotta-Ramusino, Riccardo Longoni, Configuration spaces and Vassiliev classes in any dimension, Algebr. Geom. Topol. 2 (2002) 949-1000 (<a href="https://arxiv.org/abs/math/9910139">arXiv:math/9910139</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Alberto+Cattaneo">Alberto Cattaneo</a>, Paolo Cotta-Ramusino, Riccardo Longoni, Algebraic structures on graph cohomology, Journal of Knot Theory and Its Ramifications, Vol. 14, No. 5 (2005) 627-640 (<a href="https://arxiv.org/abs/math/0307218">arXiv:math/0307218</a>) </li> </ul> Reviewed in: <ul> <li id="Volic13"><a class="existingWikiWord" href="/nlab/show/Ismar+Voli%C4%87">Ismar Volić</a>, Section 4 of: Configuration space integrals and the topology of knot and link spaces, <a href="https://fdocuments.co/amp/document/morfismos-vol-17-no-2-2013.html">Morfismos, Vol 17, no 2, 2013</a> (<a href="https://arxiv.org/abs/1310.7224">arxiv:1310.7224</a>)</li> </ul> Applied to <a class="existingWikiWord" href="/nlab/show/Gopakumar-Vafa+duality">Gopakumar-Vafa duality</a>: <ul> <li>Dave Auckly, Sergiy Koshkin, Introduction to the Gopakumar-Vafa Large <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> Duality, Geom. Topol. Monogr. 8 (2006) 195-456 (<a href="https://arxiv.org/abs/math/0701568">arXiv:0701568</a>)</li> </ul> See also <ul> <li> <a class="existingWikiWord" href="/nlab/show/Marcos+Mari%C3%B1o">Marcos Mariño</a>, Chern-Simons theory, matrix integrals, and perturbative three-manifold invariants, Commun. Math. Phys. 253 (2004) 25-49 (<a href="https://arxiv.org/abs/hep-th/0207096">arXiv:hep-th/0207096</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Stavros+Garoufalidis">Stavros Garoufalidis</a>, <a class="existingWikiWord" href="/nlab/show/Marcos+Mari%C3%B1o">Marcos Mariño</a>, On Chern-Simons matrix models (<a href="http://people.mpim-bonn.mpg.de/stavros/publications/matrixmodels.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/GaroufalidisMarinoChernSimonsMatrixModel.pdf" title="pdf">pdf</a>) </li> </ul> <h4 id="ForSingleTraceOperatorsInAdSCFTDuality">For single trace operators in AdS/CFT duality</h4> Interpretation of <a class="existingWikiWord" href="/nlab/show/Lie+algebra+weight+systems">Lie algebra weight systems</a> on <a class="existingWikiWord" href="/nlab/show/chord+diagrams">chord diagrams</a> as certain <a class="existingWikiWord" href="/nlab/show/single+trace+operators">single trace operators</a>, in particular in application to <a class="existingWikiWord" href="/nlab/show/black+hole+thermodynamics">black hole thermodynamics</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Micha+Berkooz">Micha Berkooz</a>, <a class="existingWikiWord" href="/nlab/show/Prithvi+Narayan">Prithvi Narayan</a>, <a class="existingWikiWord" href="/nlab/show/Joan+Sim%C3%B3n">Joan Simón</a>, Section 2.1 of Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction, JHEP 08 (2018) 192 (<a href="https://arxiv.org/abs/1806.04380">arxiv:1806.04380</a>)</li> </ul> <h4 id="ReferencesWeightSystemsInSYKModel">In <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>AdS</mi> <mn>2</mn></msub><mo stretchy="false">/</mo><msub><mi>CFT</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">AdS_2/CFT_1</annotation></semantics></math>, JT-gravity/SYK-model</h4> Discussion of (<a class="existingWikiWord" href="/nlab/show/Lie+algebra+weight+system">Lie algebra</a>-)<a class="existingWikiWord" href="/nlab/show/weight+systems">weight systems</a> on <a class="existingWikiWord" href="/nlab/show/chord+diagrams">chord diagrams</a> as <a class="existingWikiWord" href="/nlab/show/SYK+model">SYK model</a> <a class="existingWikiWord" href="/nlab/show/single+trace+operators">single trace operators</a>: <ul> <li> Antonio M. García-García, Yiyang Jia, Jacobus J. M. Verbaarschot, Exact moments of the Sachdev-Ye-Kitaev model up to order <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo stretchy="false">/</mo><msup><mi>N</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">1/N^2</annotation></semantics></math>, JHEP 04 (2018) 146 (<a href="https://arxiv.org/abs/1801.02696">arXiv:1801.02696</a>) </li> <li> Yiyang Jia, Jacobus J. M. Verbaarschot, Section 4 of: Large <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> expansion of the moments and free energy of Sachdev-Ye-Kitaev model, and the enumeration of intersection graphs, JHEP 11 (2018) 031 (<a href="https://arxiv.org/abs/1806.03271">arXiv:1806.03271</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Micha+Berkooz">Micha Berkooz</a>, <a class="existingWikiWord" href="/nlab/show/Prithvi+Narayan">Prithvi Narayan</a>, <a class="existingWikiWord" href="/nlab/show/Joan+Sim%C3%B3n">Joan Simón</a>, Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction, JHEP 08 (2018) 192 (<a href="https://arxiv.org/abs/1806.04380">arxiv:1806.04380</a>) </li> </ul> following: <ul> <li>László Erdős, Dominik Schröder, Phase Transition in the Density of States of Quantum Spin Glasses, D. Math Phys Anal Geom (2014) 17: 9164 (<a href="https://arxiv.org/abs/1407.1552">arXiv:1407.1552</a>)</li> </ul> which in turn follows <ul> <li>Philippe Flajolet, Marc Noy, Analytic Combinatorics of Chord Diagrams, pages 191–201 in Daniel Krob, Alexander A. Mikhalev,and Alexander V. Mikhalev, (eds.), Formal Power Series and Algebraic Combinatorics, Springer 2000 (<a href="https://doi.org/10.1007/978-3-662-04166-6_17">doi:10.1007/978-3-662-04166-6_17</a>)</li> </ul> <div class="float_right_image" style="margin: -40px 0px 20px 14px"> <figure style="margin: 0 0 0 0"> <img src="/nlab/files/ChordDiagramHolographic.jpg" width="400px" /> <figcaption style="text-align: center">from <a href="#Narovlansky19">Narovlansky 2019</a></figcaption> </figure> </div> With emphasis on the <a class="existingWikiWord" href="/nlab/show/AdS-CFT">holographic content</a>: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Micha+Berkooz">Micha Berkooz</a>, Mikhail Isachenkov, Vladimir Narovlansky, Genis Torrents, Section 5 of: Towards a full solution of the large <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> double-scaled SYK model, JHEP 03 (2019) 079 (<a href="https://arxiv.org/abs/1811.02584">arxiv:1811.02584</a>) </li> <li id="Narovlansky19"> Vladimir Narovlansky, Slide 23 (of 28) of: Towards a Solution of Large <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> Double-Scaled SYK, 2019 (<a class="existingWikiWord" href="/nlab/files/NarovlanskySYK19.pdf" title="pdf">pdf</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Micha+Berkooz">Micha Berkooz</a>, Mikhail Isachenkov, Prithvi Narayan, Vladimir Narovlansky, Quantum groups, non-commutative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>AdS</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">AdS_2</annotation></semantics></math>, and chords in the double-scaled SYK model [<a href="https://arxiv.org/abs/2212.13668">arXiv:2212.13668</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Herman+Verlinde">Herman Verlinde</a>, Double-scaled SYK, Chords and de Sitter Gravity [<a href="https://arxiv.org/abs/2402.00635">arXiv:2402.00635</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Micha+Berkooz">Micha Berkooz</a>, Nadav Brukner, Yiyang Jia, Ohad Mamroud, A Path Integral for Chord Diagrams and Chaotic-Integrable Transitions in Double Scaled SYK [<a href="https://arxiv.org/abs/2403.05980">arXiv:2403.05980</a>] </li> </ul> and specifically in relation, under <a class="existingWikiWord" href="/nlab/show/AdS2%2FCFT1">AdS2/CFT1</a>, to <a class="existingWikiWord" href="/nlab/show/Jackiw-Teitelboim+gravity">Jackiw-Teitelboim gravity</a>: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Andreas+Blommaert">Andreas Blommaert</a>, <a class="existingWikiWord" href="/nlab/show/Thomas+Mertens">Thomas Mertens</a>, <a class="existingWikiWord" href="/nlab/show/Henri+Verschelde">Henri Verschelde</a>, The Schwarzian Theory - A Wilson Line Perspective, JHEP 1812 (2018) 022 (<a href="https://arxiv.org/abs/1806.07765">arXiv:1806.07765</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Andreas+Blommaert">Andreas Blommaert</a>, <a class="existingWikiWord" href="/nlab/show/Thomas+Mertens">Thomas Mertens</a>, <a class="existingWikiWord" href="/nlab/show/Henri+Verschelde">Henri Verschelde</a>, Fine Structure of Jackiw-Teitelboim Quantum Gravity, JHEP 1909 (2019) 066 (<a href="https://arxiv.org/abs/1812.00918">arXiv:1812.00918</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Henry+W.+Lin">Henry W. Lin</a>, The bulk Hilbert space of double scaled SYK, J. High Energ. Phys. 2022 60 (2022) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/2208.07032">arXiv:2208.07032</a>, <a href="https://doi.org/10.1007/JHEP11(2022)060">doi:10.1007/JHEP11(2022)060</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> <a class="existingWikiWord" href="/nlab/show/Henry+W.+Lin">Henry W. Lin</a>, <a class="existingWikiWord" href="/nlab/show/Douglas+Stanford">Douglas Stanford</a>, A symmetry algebra in double-scaled SYK <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/2307.15725">arXiv:2307.15725</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> <a class="existingWikiWord" href="/nlab/show/Micha+Berkooz">Micha Berkooz</a>, Ohad Mamroud: A Cordial Introduction to Double Scaled SYK [<a href="https://arxiv.org/abs/2407.09396">arXiv:2407.09396</a>] </li> <li> Adel A. Rahman, <a class="existingWikiWord" href="/nlab/show/Leonard+Susskind">Leonard Susskind</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>-Chords, Wee-Chords, and de Sitter Space [<a href="https://arxiv.org/abs/2407.12988">arXiv:2407.12988</a>] </li> </ul> <h4 id="ReferencesWeightSystemsInDpDp2BraneIntersections">In D<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>/D<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(p+2)</annotation></semantics></math>-brane intersections</h4> <div class="float_right_image" style="margin: -40px 0px 20px 14px"> <figure style="margin: 0 0 0 0"> <img src="/nlab/files/WeightSystemsAsShapeObservabesOnFuzzySphereII.jpg" width="330px" /> <figcaption style="text-align: center">from <a href="#SatiSchreiber19c">Sati and Schreiber 19c</a></figcaption> </figure> </div> Discussion of <a class="existingWikiWord" href="/nlab/show/weight+systems">weight systems</a> on <a class="existingWikiWord" href="/nlab/show/chord+diagrams">chord diagrams</a> as <a class="existingWikiWord" href="/nlab/show/single+trace+observables">single trace observables</a> for the <a class="existingWikiWord" href="/nlab/show/non-abelian+DBI+action">non-abelian DBI action</a> on the <a class="existingWikiWord" href="/nlab/show/fuzzy+funnel">fuzzy funnel</a>/<a class="existingWikiWord" href="/nlab/show/fuzzy+sphere">fuzzy sphere</a> <a class="existingWikiWord" href="/nlab/show/non-commutative+geometry">non-commutative geometry</a> of <a class="existingWikiWord" href="/nlab/show/Dp-D%28p%2B2%29-brane+intersections">Dp-D(p+2)-brane intersections</a> (<a href="Dp-Dp+2-brane+bound+states#ReferencesRelationToMonopoles">hence</a> <a class="existingWikiWord" href="/nlab/show/Yang-Mills+monopoles">Yang-Mills monopoles</a>): <ul> <li> <a class="existingWikiWord" href="/nlab/show/Sanyaje+Ramgoolam">Sanyaje Ramgoolam</a>, <a class="existingWikiWord" href="/nlab/show/Bill+Spence">Bill Spence</a>, S. Thomas, Section 3.2 of: Resolving brane collapse with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo stretchy="false">/</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">1/N</annotation></semantics></math> corrections in non-Abelian DBI, Nucl. Phys. B703 (2004) 236-276 (<a href="https://arxiv.org/abs/hep-th/0405256">arxiv:hep-th/0405256</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Simon+McNamara">Simon McNamara</a>, <a class="existingWikiWord" href="/nlab/show/Constantinos+Papageorgakis">Constantinos Papageorgakis</a>, <a class="existingWikiWord" href="/nlab/show/Sanyaje+Ramgoolam">Sanyaje Ramgoolam</a>, <a class="existingWikiWord" href="/nlab/show/Bill+Spence">Bill Spence</a>, Appendix A of: Finite <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> effects on the collapse of fuzzy spheres, JHEP 0605:060, 2006 (<a href="https://arxiv.org/abs/hep-th/0512145">arxiv:hep-th/0512145</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Simon+McNamara">Simon McNamara</a>, Section 4 of: Twistor Inspired Methods in Perturbative FieldTheory and Fuzzy Funnels, 2006 (<a href="http://inspirehep.net/record/1351861">spire:1351861</a>, <a href="https://strings.ph.qmul.ac.uk/sites/default/files/Mcnamaraphd.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/McNamara06.pdf" title="pdf">pdf</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Constantinos+Papageorgakis">Constantinos Papageorgakis</a>, p. 161-162 of: On matrix D-brane dynamics and fuzzy spheres, 2006 (<a class="existingWikiWord" href="/nlab/files/Papageorgakis06.pdf" title="pdf">pdf</a>) </li> </ul> <h4 id="ForHolographicEntanglementEntropy">As codes for holographic entanglement entropy</h4> <div class="float_right_image" style="margin: -40px 0px 20px 10px"> <figure style="margin: 0 0 0 0"> <img src="/nlab/files/HanUniversalHolographicCode.jpg" width="300px" /> <figcaption style="text-align: center">From <a href="#Yan20">Yan 20</a></figcaption> </figure> </div> <a class="existingWikiWord" href="/nlab/show/chord+diagram">Chord diagrams</a> encoding <a class="existingWikiWord" href="/nlab/show/Majorana+dimer+codes">Majorana dimer codes</a> and other <a class="existingWikiWord" href="/nlab/show/quantum+error+correcting+codes">quantum error correcting codes</a> via <a class="existingWikiWord" href="/nlab/show/tensor+networks">tensor networks</a> exhibiting <a class="existingWikiWord" href="/nlab/show/holographic+entanglement+entropy">holographic entanglement entropy</a>: <ul> <li id="JGPE19"> <a class="existingWikiWord" href="/nlab/show/Alexander+Jahn">Alexander Jahn</a>, <a class="existingWikiWord" href="/nlab/show/Marek+Gluza">Marek Gluza</a>, <a class="existingWikiWord" href="/nlab/show/Fernando+Pastawski">Fernando Pastawski</a>, <a class="existingWikiWord" href="/nlab/show/Jens+Eisert">Jens Eisert</a>, Majorana dimers and holographic quantum error-correcting code, Phys. Rev. Research 1, 033079 (2019) (<a href="https://arxiv.org/abs/1905.03268">arXiv:1905.03268</a>) </li> <li id="Yan20"> <a class="existingWikiWord" href="/nlab/show/Han+Yan">Han Yan</a>, Geodesic string condensation from symmetric tensor gauge theory: a unifying framework of holographic toy models, Phys. Rev. B 102, 161119 (2020) (<a href="https://arxiv.org/abs/1911.01007">arXiv:1911.01007</a>) </li> </ul> <div style="margin: -40px 0px 20px 10px"> <figure style="margin: 0 0 0 0"> <img src="/nlab/files/HaPPYCodesAsDimerCode.jpg" width="570px" /> <figcaption style="text-align: center">From <a href="holographic+entanglement+entropy#JahnEisert21">Jahn and Eisert 21</a></figcaption> </figure> </div> <h4 id="ForDysonSchwingerEquations">For Dyson-Schwinger equations</h4> Discussion of <a class="existingWikiWord" href="/nlab/show/round+chord+diagrams">round chord diagrams</a> organizing <a class="existingWikiWord" href="/nlab/show/Dyson-Schwinger+equations">Dyson-Schwinger equations</a>: <ul> <li> Nicolas Marie, <a class="existingWikiWord" href="/nlab/show/Karen+Yeats">Karen Yeats</a>, A chord diagram expansion coming from some Dyson-Schwinger equations, Communications in Number Theory and Physics, 7(2):251291, 2013 (<a href="https://arxiv.org/abs/1210.5457">arXiv:1210.5457</a>) </li> <li> Markus Hihn, <a class="existingWikiWord" href="/nlab/show/Karen+Yeats">Karen Yeats</a>, Generalized chord diagram expansions of Dyson-Schwinger equations, Ann. Inst. Henri Poincar Comb. Phys. Interact. 6 no 4:573-605 (<a href="https://arxiv.org/abs/1602.02550">arXiv:1602.02550</a>) </li> <li> Paul-Hermann Balduf, Amelia Cantwell, <a class="existingWikiWord" href="/nlab/show/Kurusch+Ebrahimi-Fard">Kurusch Ebrahimi-Fard</a>, Lukas Nabergall, Nicholas Olson-Harris, <a class="existingWikiWord" href="/nlab/show/Karen+Yeats">Karen Yeats</a>, Tubings, chord diagrams, and Dyson-Schwinger equations [<a href="https://arxiv.org/abs/2302.02019">arXiv:2302.02019</a>] </li> </ul> Review in: <ul> <li>Ali Assem Mahmoud, Section 3 of: On the Enumerative Structures in Quantum Field Theory (<a href="https://arxiv.org/abs/2008.11661">arXiv:2008.11661</a>)</li> </ul> <h4 id="other">Other</h4> Appearance of <a class="existingWikiWord" href="/nlab/show/horizontal+chord+diagrams">horizontal chord diagrams</a> in discussion of <a class="existingWikiWord" href="/nlab/show/neutrino">neutrino</a> interactions in <a class="existingWikiWord" href="/nlab/show/supernovae">supernovae</a>: <ul> <li>Duff Neill, Hanqing Liu, Joshua Martin, Alessandro Roggero: Scattering Neutrinos, Spin Models, and Permutations [<a href="https://arxiv.org/abs/2406.18677">arXiv:2406.18677</a>]</li> </ul> </div></body></html> </div> <div class="revisedby"> Last revised on May 14, 2021 at 14:18:16. 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