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href="/nlab/show/string+diagram">string diagram</a>, <a class="existingWikiWord" href="/nlab/show/tensor+network">tensor network</a></p> </li> </ul> <p><strong>With braiding</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/balanced+monoidal+category">balanced monoidal category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twist">twist</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a></p> </li> </ul> <p><strong>With duals for objects</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+with+duals">category with duals</a> (list of them)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dualizable+object">dualizable object</a> (what they have)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid monoidal category</a>, a.k.a. <a class="existingWikiWord" href="/nlab/show/autonomous+category">autonomous category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pivotal+category">pivotal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spherical+category">spherical category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ribbon+category">ribbon category</a>, a.k.a. <a class="existingWikiWord" href="/nlab/show/tortile+category">tortile category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+closed+category">compact closed category</a></p> </li> </ul> <p><strong>With duals for morphisms</strong></p> <ul> <li> <p><span class="newWikiWord">monoidal dagger-category<a href="/nlab/new/monoidal+dagger-category">?</a></span></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+dagger-category">symmetric monoidal dagger-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dagger+compact+category">dagger compact category</a></p> </li> </ul> <p><strong>With traces</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/trace">trace</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/traced+monoidal+category">traced monoidal category</a></p> </li> </ul> <p><strong>Closed structure</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+closed+category">cartesian closed category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+category">closed category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/star-autonomous+category">star-autonomous category</a></p> </li> </ul> <p><strong>Special sorts of products</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+monoidal+category">cartesian monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semicartesian+monoidal+category">semicartesian monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+category+with+diagonals">monoidal category with diagonals</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multicategory">multicategory</a></p> </li> </ul> <p><strong>Semisimplicity</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/semisimple+category">semisimple category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fusion+category">fusion category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/modular+tensor+category">modular tensor category</a></p> </li> </ul> <p><strong>Morphisms</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+functor">monoidal functor</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/lax+monoidal+functor">lax</a>, <a class="existingWikiWord" href="/nlab/show/oplax+monoidal+functor">oplax</a>, <a class="existingWikiWord" href="/nlab/show/strong+monoidal+functor">strong</a> <a class="existingWikiWord" href="/nlab/show/bilax+monoidal+functor">bilax</a>, <a class="existingWikiWord" href="/nlab/show/Frobenius+monoidal+functor">Frobenius</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/braided+monoidal+functor">braided monoidal functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+functor">symmetric monoidal functor</a></p> </li> </ul> <p><strong>Internal monoids</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+in+a+monoidal+category">monoid in a monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+a+symmetric+monoidal+category">commutative monoid in a symmetric monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module+over+a+monoid">module over a monoid</a></p> </li> </ul> <p><strong id="_examples">Examples</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+monoidal+structure+on+presheaves">closed monoidal structure on presheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Day+convolution">Day convolution</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/coherence+theorem+for+monoidal+categories">coherence theorem for monoidal categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a></p> </li> </ul> <p><strong>In higher category theory</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+2-category">monoidal 2-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/braided+monoidal+2-category">braided monoidal 2-category</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+bicategory">monoidal bicategory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cartesian+bicategory">cartesian bicategory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/k-tuply+monoidal+n-category">k-tuply monoidal n-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/little+cubes+operad">little cubes operad</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2C1%29-category">monoidal (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+double+category">compact double category</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='#relation_to_weak_hopf_algebras'>Relation to weak Hopf algebras</a></li> <li><a href='#relation_to_pivotal_and_spherical_categories'>Relation to pivotal and spherical categories</a></li> <ul> <li><a href='#conjecture_etingof_nikshych_and_ostrik'>Conjecture (Etingof, Nikshych, and Ostrik)</a></li> </ul> <li><a href='#RelationtoTQFT'>Relation to extended 3d TQFT</a></li> </ul> <li><a href='#suggestions'>Suggestions</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='#AnyonicTopologicalOrderInTermsOfBraidedFusionCategoriesReferences'>Anyonic topological order in terms of braided fusion categories</a></li> <ul> <li><a href='#claim_and_status'>Claim and status</a></li> <li><a href='#AnyonicOrderInMTheoryReferences'>In string/M-theory</a></li> <li><a href='#further_discussion'>Further discussion</a></li> </ul> </ul> </ul> </div> <h2 id="definition">Definition</h2> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>A <strong>fusion category</strong> is a <a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid</a> <a class="existingWikiWord" href="/nlab/show/semisimple+category">semisimple</a> <a class="existingWikiWord" href="/nlab/show/linear+category">linear</a> (<a class="existingWikiWord" href="/nlab/show/Vect">Vect</a>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched</a>) <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> (“<a class="existingWikiWord" href="/nlab/show/tensor+category">tensor category</a>”), with only <a class="existingWikiWord" href="/nlab/show/finite+number">finitely</a> many <a class="existingWikiWord" href="/nlab/show/decategorification">isomorphism classes</a> of <a class="existingWikiWord" href="/nlab/show/simple+objects">simple objects</a>, such that the <a class="existingWikiWord" href="/nlab/show/endomorphisms">endomorphisms</a> of the unit object form just the <a class="existingWikiWord" href="/nlab/show/ground+field">ground field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>.</p> </div> <p>Often one also assumes a <a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braiding</a> and speaks of a <em>braided fusion category</em>.</p> <h2 id="Examples">Examples</h2> <p>The name “fusion category” comes from the central examples of structures whose canonical <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> is called a “fusion product”, notably <a class="existingWikiWord" href="/nlab/show/representations">representations</a> of <a class="existingWikiWord" href="/nlab/show/loop+groups">loop groups</a> and of <a class="existingWikiWord" href="/nlab/show/Hopf+algebras">Hopf algebras</a> and of <a class="existingWikiWord" href="/nlab/show/vertex+operator+algebras">vertex operator algebras</a>.</p> <p>Simple examples:</p> <p> <div class='num_remark' id='GradedVectorSpaces'> <h6>Example</h6> <p><strong>(graded vector spaces)</strong> <br /> For</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/finite+group">finite group</a>,</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕂</mi></mrow><annotation encoding="application/x-tex">\mathbb{K}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/field">field</a>,</p> </li> </ul> <p>the category of <a class="existingWikiWord" href="/nlab/show/finite-dimensional+vector+space">finite-dimensional</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/graded+vector+spaces">graded</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕂</mi></mrow><annotation encoding="application/x-tex">\mathbb{K}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/vector+spaces">vector spaces</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Vect</mi> <mi>G</mi> <mi>fdim</mi></msubsup></mrow><annotation encoding="application/x-tex"> Vect_{G}^{fdim} </annotation></semantics></math></div> <p>is a fusion category, with <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> given by the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+vector+spaces">tensor product of vector spaces</a> and the <a class="existingWikiWord" href="/nlab/show/binary+operation">binary operation</a> of the group:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>g</mi></msub><mo>⊗</mo><msub><mi>W</mi> <mi>h</mi></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mi>V</mi><mo>⊗</mo><mi>W</mi><msub><mo stretchy="false">)</mo> <mrow><mi>g</mi><mo>⋅</mo><mi>h</mi></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> V_g \otimes W_h \;\; \coloneqq \;\; (V \otimes W)_{g \cdot h} \,. </annotation></semantics></math></div> <p>More generally, for</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><msubsup><mi>H</mi> <mi>Grp</mi> <mn>3</mn></msubsup><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><msup><mi>𝕂</mi> <mo>×</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\omega \,\in\, H^3_{Grp}(G, \mathbb{K}^\times)</annotation></semantics></math> a 3-cocycle in the <a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in the <a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝕂</mi> <mo>×</mo></msup></mrow><annotation encoding="application/x-tex">\mathbb{K}^\times</annotation></semantics></math>,</li> </ul> <p>the above construction but with <a class="existingWikiWord" href="/nlab/show/associator">associator</a> multiplied by the 3-cocycle applied to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-degrees of the 3 factors is again a fusion category</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Vect</mi> <mrow><mi>G</mi><mo>,</mo><mi>ω</mi></mrow> <mi>fdim</mi></msubsup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Vect_{G, \omega}^{fdim} \,. </annotation></semantics></math></div> <p></p> </div> (<a href="#EtingofNikshychOstrik05">Etingof, Nikshych & Ostrik 2005, item 1. on p. 584</a>)</p> <p> <div class='num_remark'> <h6>Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕂</mi></mrow><annotation encoding="application/x-tex">\mathbb{K}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/field">field</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/finite+group">finite group</a> (or <a href="supergroup#finite_supergroups">finite super-group</a>), whose <a class="existingWikiWord" href="/nlab/show/order+of+a+group">order</a> is <a class="existingWikiWord" href="/nlab/show/coprime+integer">relatively prime</a> to the <a class="existingWikiWord" href="/nlab/show/characteristic+of+a+field">characteristic</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕂</mi></mrow><annotation encoding="application/x-tex">\mathbb{K}</annotation></semantics></math>, then the <a class="existingWikiWord" href="/nlab/show/category+of+representations">category of representations</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Rep</mi><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>𝕂</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Rep(G, \mathbb{K})</annotation></semantics></math> is a fusion category.</p> </div> (<a href="#EtingofNikshychOstrik05">Etingof, Nikshych & Ostrik 2005, item 2. on p. 584</a>)</p> <h2 id="properties">Properties</h2> <h3 id="relation_to_weak_hopf_algebras">Relation to weak Hopf algebras</h3> <p>Under <a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a>, every fusion category <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">C</annotation></semantics></math> arises as the <a class="existingWikiWord" href="/nlab/show/representation+category">representation category</a> of a <a class="existingWikiWord" href="/nlab/show/weak+Hopf+algebra">weak Hopf algebra</a> (<a href="#Ostrik">Ostrik</a>). However, this does not mean that every fusion category admits a <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a> to the <a class="existingWikiWord" href="/nlab/show/Vect">category of vector spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mtext>Vect</mtext><mo>=</mo><mi>k</mi><mo>−</mo><mi>Mod</mi></mrow><annotation encoding="application/x-tex">\text{Vect}= k-Mod</annotation></semantics></math>.</p> <p>Given any multi-fusion category <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">C</annotation></semantics></math>, one can always construct a fiber functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>RMod</mi></mrow><annotation encoding="application/x-tex">F:C\to RMod</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> the algebra spanned by a basis of orthogonal idempotents <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>v</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{v_i\}_{i\in I}</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> the equivalence classes of simple objects of <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">C</annotation></semantics></math>. This functor is referred to in some sources as a <em>generalized</em> fiber functor. The endomorphisms of this functor then give a weak Hopf algebra that represents <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">C</annotation></semantics></math>. In <a href="#Hayashi99">Hayashi 1999</a> (see there for the relevant definitions), this is computed as a <a class="existingWikiWord" href="/nlab/show/coend">coend</a>, where one has that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>≅</mo><mi>Rep</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C\cong Rep(A)</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mtext>coend</mtext><mo stretchy="false">(</mo><msup><mi>F</mi> <mo>*</mo></msup><mo>⊗</mo><mi>F</mi><mo>:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>×</mo><mi>C</mi><mo>→</mo><mi>Bmd</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A= \text{coend}(F^*\otimes F: C^{op} \times C \to Bmd(E))</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>=</mo><mover><mi>R</mi><mo>˙</mo></mover><mo>⊗</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">E=\dot R\otimes R</annotation></semantics></math> is equipped with a coalgebra structure</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">(</mo><mover><mi>λ</mi><mo>˙</mo></mover><mi>μ</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>ν</mi><mo>∈</mo><mi>I</mi></mrow></munder><mover><mi>λ</mi><mo>˙</mo></mover><mi>ν</mi><mo>⊗</mo><mover><mi>ν</mi><mo>˙</mo></mover><mi>μ</mi></mrow><annotation encoding="application/x-tex"> \Delta(\dot\lambda \mu) = \sum_{\nu\in I} \dot\lambda \nu\otimes \dot\nu \mu </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo stretchy="false">(</mo><mover><mi>λ</mi><mo>˙</mo></mover><mi>μ</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>δ</mi> <mrow><mi>λ</mi><mo>,</mo><mi>μ</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> \epsilon (\dot\lambda \mu) = \delta_{\lambda,\mu} </annotation></semantics></math></div> <p>It is important to note that, generally speaking, <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">C</annotation></semantics></math> may admit other fiber functor to different module categories <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>RMod</mi></mrow><annotation encoding="application/x-tex">RMod</annotation></semantics></math>, as is the case for fusion categories of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Rep</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Rep(H)</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> a Hopf algebra, which admits both the fiber functor described above, as well as a fiber functor to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mtext>Vect</mtext></mrow><annotation encoding="application/x-tex">\text{Vect}</annotation></semantics></math>.</p> <h3 id="relation_to_pivotal_and_spherical_categories">Relation to pivotal and spherical categories</h3> <p>Fusion categories were first systematically studied by <a href="#EtingofNikshychOstrik05">Etingof, Nikshych & Ostrik 2005</a>. This paper listed many examples and proved many properties of fusion categories. One of the important conjectures made in that paper was the following:</p> <div class="num_theorem"> <h6 id="conjecture_etingof_nikshych_and_ostrik">Conjecture (Etingof, Nikshych, and Ostrik)</h6> <p>Every fusion category admits a <a class="existingWikiWord" href="/nlab/show/pivotal+structure">pivotal structure</a>.</p> </div> <p>Providing a certain condition is satisfied, a pivotal structure on a fusion category can be shown to correspond to a ‘twisted’ monoidal natural endotransformation of the identity functor on the category, where the twisting is given by the <a class="existingWikiWord" href="/nlab/show/pivotal+symbols">pivotal symbols</a>.</p> <h3 id="RelationtoTQFT">Relation to extended 3d TQFT</h3> <p>Given the data of a <a class="existingWikiWord" href="/nlab/show/fusion+category">fusion category</a> one can build a 3d <a class="existingWikiWord" href="/nlab/show/extended+TQFT">extended TQFT</a> by various means. This is explained by the fact, see below, that fusion categories are (probably precisely) the <a class="existingWikiWord" href="/nlab/show/fully+dualizable+object">fully dualizable object</a>s in the <a class="existingWikiWord" href="/nlab/show/3-category">3-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>MonCat</mi></mrow><annotation encoding="application/x-tex">MonCat</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a>. By the <a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a> this explains how they induce <a class="existingWikiWord" href="/nlab/show/3d+TQFT">3d TQFT</a>s.</p> <div class="num_defn" id="MonCat"> <h6 id="definition_3">Definition</h6> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>MonCat</mi> <mi>bim</mi></msub></mrow><annotation encoding="application/x-tex">MonCat_{bim}</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/%28infinity%2Cn%29-category">(infinity,3)-category</a> which has as</p> <ul> <li> <p>objects <a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a>,</p> </li> <li> <p>morphism <a class="existingWikiWord" href="/nlab/show/bimodule">bimodule</a>s of these,</p> </li> <li> <p>and so on.</p> </li> </ul> </div> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>With its natural <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>MonCat</mi></mrow><annotation encoding="application/x-tex">MonCat</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28infinity%2Cn%29-category">symmetric monoidal (infinity,3)-category</a>.</p> </div> <div class="num_prop" id="FusionCategoriesAreFullyDualizable"> <h6 id="proposition_2">Proposition</h6> <p>A monoidal category which is fusion is <a class="existingWikiWord" href="/nlab/show/fully+dualizable+object">fully dualizable</a> in the <a class="existingWikiWord" href="/nlab/show/%28infinity%2Cn%29-category">(infinity,3)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>MonCat</mi> <mi>bim</mi></msub></mrow><annotation encoding="application/x-tex">MonCat_{bim}</annotation></semantics></math>, def. <a class="maruku-ref" href="#MonCat"></a>.</p> </div> <p>This is due to (<a href="#DSPS13">Douglas & Schommer-Pries & Snyder 13</a>).</p> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>Via the <a class="existingWikiWord" href="/nlab/show/cobordism+theorem">cobordism theorem</a> prop. <a class="maruku-ref" href="#FusionCategoriesAreFullyDualizable"></a> implies that fusion categories encode <a class="existingWikiWord" href="/nlab/show/extended+TQFTs">extended TQFTs</a> on 3-dimensional <a class="existingWikiWord" href="/nlab/show/framed+manifold">framed</a> <a class="existingWikiWord" href="/nlab/show/cobordisms">cobordisms</a>, while their <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(3)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/homotopy+fixed+points">homotopy fixed points</a> encode extended 3d TQFTs on general (not framed) cobordisms.</p> <p>These 3d TQFTs hence arise from similar algebraic data as the <a class="existingWikiWord" href="/nlab/show/Turaev-Viro+model">Turaev-Viro model</a> and the <a class="existingWikiWord" href="/nlab/show/Reshetikhin-Turaev+construction">Reshetikhin-Turaev construction</a>, however there are various slight differences. See (<a href="#DSPS13">Douglas & Schommer-Pries & Snyder 13, p. 5</a>).</p> </div> <h2 id="suggestions">Suggestions</h2> <p>Here are three things such that it’d be awesome if they were sorted out on this page:</p> <ol> <li> <p>Kuperberg’s theorem saying that <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian</a> semisimple implies <a class="existingWikiWord" href="/nlab/show/linear+category">linear</a> over some field. <a href="http://arxiv.org/abs/math/0209256">Finite, connected, semisimple, rigid tensor categories are linear</a></p> </li> <li> <p>Some correct version of the claim that abelian semisimple is the same as <a class="existingWikiWord" href="/nlab/show/idempotent+complete+category">idempotent complete</a> and nondegenerate. <a href="http://mathoverflow.net/questions/245/are-abelian-nondegenerate-tensor-categories-semisimple">Math Overflow question</a></p> </li> <li> <p>Good notation distinguishing <a class="existingWikiWord" href="/nlab/show/simple+object">simple</a> versus <span class="newWikiWord">absolutely simple<a href="/nlab/new/absolutely+simple+object">?</a></span> (is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>End</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>=</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">End(V) = k</annotation></semantics></math> or just <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> has no nontrivial proper subobjects).</p> </li> </ol> <p>Together 1 and 2 let you go between the two different obvious notions of semisimple which seem a bit muddled here when I clicked through the links.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><strong>fusion category</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+fusion+category">unitary fusion category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fusion+2-category">fusion 2-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fusion+ring">fusion ring</a>, <a class="existingWikiWord" href="/nlab/show/Frobenius-Perron+dimension">Frobenius-Perron dimension</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pivotal+category">pivotal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spherical+category">spherical category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/modular+tensor+category">modular tensor category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne%27s+theorem+on+tensor+categories">Deligne's theorem on tensor categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/graded+fusion+category">graded fusion category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tambara-Yamagami+category">Tambara-Yamagami category</a></p> </li> </ul> <h2 id="references">References</h2> <h3 id="general">General</h3> <p>Original articles:</p> <ul> <li id="EtingofNikshychOstrik05"> <p><a class="existingWikiWord" href="/nlab/show/Pavel+Etingof">Pavel Etingof</a>, <a class="existingWikiWord" href="/nlab/show/Dmitri+Nikshych">Dmitri Nikshych</a>, <a class="existingWikiWord" href="/nlab/show/Victor+Ostrik">Victor Ostrik</a>, <em>On fusion categories</em>, Annals of Mathematics Second Series <strong>162</strong> 2 (2005) 581-642 [<a href="http://arxiv.org/abs/math/0203060">arXiv:math/0203060</a>, <a href="https://www.jstor.org/stable/20159926">jstor:20159926</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pavel+Etingof">Pavel Etingof</a> and <a class="existingWikiWord" href="/nlab/show/Damien+Calaque">Damien Calaque</a>, <a href="http://arxiv.org/abs/math/0401246">Lectures on tensor categories</a>.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael+M%C3%BCger">Michael Müger</a>, <a href="http://arxiv.org/abs/0804.3587">Tensor categories: A selective guided tour</a>.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pavel+Etingof">Pavel Etingof</a>, D. Nikshych and V. Ostrik, <em>Fusion categories and homotopy theory</em> , Quantum Topology, 1(2010), 209-273. (Earlier version available as <a href="http://arxiv.org/PS_cache/arxiv/pdf/0909/0909.3140v2.pdf">ArXiv:0909.3140</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Vladimir+Drinfeld">Vladimir Drinfeld</a>, <a class="existingWikiWord" href="/nlab/show/Shlomo+Gelaki">Shlomo Gelaki</a>, <a class="existingWikiWord" href="/nlab/show/Dmitri+Nikshych">Dmitri Nikshych</a>, <a class="existingWikiWord" href="/nlab/show/Victor+Ostrik">Victor Ostrik</a>, <em>On braided fusion categories I</em>, Selecta Mathematica. New Series <strong>16</strong> 1 (2010) 1–119 [<a href="https://arxiv.org/abs/0906.0620">arXiv:0906.0620</a>, <a href="https://doi.org/10.1007/s00029-010-0017-z">doi:10.1007/s00029-010-0017-z</a>]</p> </li> </ul> <p>Further review:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Bruce+Bartlett">Bruce Bartlett</a>, chapter 6 of <p><em><a href="http://arxiv.org/abs/0901.3975">On unitary 2-representations of finite groups and topological quantum field theory</a></em>.</p> </li> </ul> <p>On the <a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a> to <a class="existingWikiWord" href="/nlab/show/weak+Hopf+algebras">weak Hopf algebras</a>:</p> <ul> <li id="Hayashi99"> <p>Takahiro Hayashi, <em>A canonical Tannaka duality for finite seimisimple tensor categories</em> (<a href="http://arxiv.org/abs/math/9904073">arXiv:math/9904073</a>)</p> </li> <li id="Ostrik"> <p><a class="existingWikiWord" href="/nlab/show/Victor+Ostrik">Victor Ostrik</a>, <em>Module categories, weak Hopf algebras and modular invariants</em> (<a href="http://arxiv.org/abs/math/0111139">arXiv:math/0111139</a>)</p> </li> </ul> <p>The relation to <a class="existingWikiWord" href="/nlab/show/3d+TQFT">3d TQFT</a> clarified via the <a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>:</p> <ul> <li id="DSPS"> <p><a class="existingWikiWord" href="/nlab/show/Chris+Schommer-Pries">Chris Schommer-Pries</a>, <em>The Structure of Fusion Categories via 3D TQFTs</em> (2001) [<a class="existingWikiWord" href="/nlab/files/DSSFusionSlides.pdf" title="">DSSFusionSlides.pdf</a>]</p> </li> <li id="DSPS13"> <p><a class="existingWikiWord" href="/nlab/show/Chris+Douglas">Chris Douglas</a>, <a class="existingWikiWord" href="/nlab/show/Chris+Schommer-Pries">Chris Schommer-Pries</a>, <a class="existingWikiWord" href="/nlab/show/Noah+Snyder">Noah Snyder</a>, <em>Dualizable tensor categories</em>, Memoirs of the AMS <strong>268</strong> 1308 (2021) [<a href="http://arxiv.org/abs/1312.7188">arXiv:1312.7188</a>, <a href="https://bookstore.ams.org/memo-268-1308">ams:memo-268-1308</a>]</p> </li> </ul> <p>and for the case of <a class="existingWikiWord" href="/nlab/show/modular+tensor+categories">modular tensor categories</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Bruce+Bartlett">Bruce Bartlett</a>, <a class="existingWikiWord" href="/nlab/show/Christopher+Douglas">Christopher Douglas</a>, <a class="existingWikiWord" href="/nlab/show/Chris+Schommer-Pries">Chris Schommer-Pries</a>, <a class="existingWikiWord" href="/nlab/show/Jamie+Vicary">Jamie Vicary</a>, <em>Modular categories as representations of the 3-dimensional bordism 2-category</em> (<a href="http://arxiv.org/abs/1509.06811">arXiv:1509.06811</a>)</li> </ul> <p>Discussion in terms of <a class="existingWikiWord" href="/nlab/show/skein+relations">skein relations</a>:</p> <ul> <li>Anup Poudel, <a class="existingWikiWord" href="/nlab/show/Sachin+J.+Valera">Sachin J. Valera</a>, <em>Skein-Theoretic Methods for Unitary Fusion Categories</em> [<a href="https://arxiv.org/abs/2008.07129">arXiv:2008.07129</a>]</li> </ul> <p>See also:</p> <ul> <li>Math Overflow, <em>Why are fusion categories interesting?</em> [<a href="https://mathoverflow.net/q/6180/381">MO:q/6180</a>]</li> </ul> <p>Further work on their classification using finite groups:</p> <ul> <li>Agustina Czenky: <em>Diagramatics for cyclic pointed fusion categories</em> [<a href="https://arxiv.org/abs/2404.08084">arXiv:2404.08084</a>]</li> </ul> <p>On a notion of <a class="existingWikiWord" href="/nlab/show/fusion+2-categories">fusion 2-categories</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Christopher+L.+Douglas">Christopher L. Douglas</a>, <a class="existingWikiWord" href="/nlab/show/David+J.+Reutter">David J. Reutter</a>, <em>Fusion 2-categories and a state-sum invariant for 4-manifolds</em> [<a href="https://arxiv.org/abs/1812.11933">arXiv:1812.11933</a>]</p> </li> <li id="DY23"> <p><a class="existingWikiWord" href="/nlab/show/Thibault+Decoppet">Thibault Decoppet</a>, <a class="existingWikiWord" href="/nlab/show/Matthew+Yu">Matthew Yu</a>. <em>Fiber 2-Functors and Tambara-Yamagami Fusion 2-Categories</em>. (2023) [<a href="https://arxiv.org/abs/2306.08117">arXiv:2306.08117</a>]</p> </li> </ul> <p>On fusion categories <a class="existingWikiWord" href="/nlab/show/invertible+object">invertible</a> with respect to the <a class="existingWikiWord" href="/nlab/show/Deligne+tensor+product+of+abelian+categories">Deligne tensor product</a>:</p> <ul> <li>Sean Sanford, <a class="existingWikiWord" href="/nlab/show/Noah+Snyder">Noah Snyder</a>: <em>Invertible Fusion Categories</em> [<a href="https://arxiv.org/abs/2407.02597">arXiv:2407.02597</a>]</li> </ul> <div> <h3 id="AnyonicTopologicalOrderInTermsOfBraidedFusionCategoriesReferences">Anyonic topological order in terms of braided fusion categories</h3> <h4 id="claim_and_status">Claim and status</h4> <p>In <a class="existingWikiWord" href="/nlab/show/condensed+matter+theory">condensed matter theory</a> it is <a class="existingWikiWord" href="/nlab/show/folklore">folklore</a> that species of <a class="existingWikiWord" href="/nlab/show/anyon">anyonic</a> <a class="existingWikiWord" href="/nlab/show/topological+order">topological order</a> correspond to <a class="existingWikiWord" href="/nlab/show/braided+monoidal+categories">braided</a> <a class="existingWikiWord" href="/nlab/show/unitary+fusion+category">unitary</a> <a class="existingWikiWord" href="/nlab/show/fusion+categories">fusion categories</a>/<a class="existingWikiWord" href="/nlab/show/modular+tensor+categories">modular tensor categories</a>.</p> <p>The origin of the claim is:</p> <ul> <li id="Kitaev06"><a class="existingWikiWord" href="/nlab/show/Alexei+Kitaev">Alexei Kitaev</a>, Section 8 and Appendix E of: <em>Anyons in an exactly solved model and beyond</em>, Annals of Physics <strong>321</strong> 1 (2006) 2-111 <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://doi.org/10.1016/j.aop.2005.10.005">doi:10.1016/j.aop.2005.10.005</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> <p>Early accounts re-stating this claim (without attribution):</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chetan+Nayak">Chetan Nayak</a>, <a class="existingWikiWord" href="/nlab/show/Steven+H.+Simon">Steven H. Simon</a>, <a class="existingWikiWord" href="/nlab/show/Ady+Stern">Ady Stern</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Freedman">Michael Freedman</a>, <a class="existingWikiWord" href="/nlab/show/Sankar+Das+Sarma">Sankar Das Sarma</a>, pp. 28 of: <em>Non-Abelian Anyons and Topological Quantum Computation</em>, Rev. Mod. Phys. <strong>80</strong> 1083 (2008) <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="http://arxiv.org/abs/0707.1889">arXiv:0707.1888</a>, <a href="https://doi.org/10.1103/RevModPhys.80.1083">doi:10.1103/RevModPhys.80.1083</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></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Zhenghan+Wang">Zhenghan Wang</a>, Section 6.3 of: <em>Topological Quantum Computation</em>, CBMS Regional Conference Series in Mathematics <strong>112</strong>, AMS (2010) <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="http://www.ams.org/publications/authors/books/postpub/cbms-112">ISBN-13: 978-0-8218-4930-9</a>, <a href="http://web.math.ucsb.edu/~zhenghwa/data/course/cbms.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></p> </li> </ul> <p>Further discussion (mostly review and mostly without attribution):</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Simon+Burton">Simon Burton</a>, <em>A Short Guide to Anyons and Modular Functors</em> <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/1610.05384">arXiv:1610.05384</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></p> <blockquote> <p>(this one stands out as still attributing the claim to <a href="#Kitaev06">Kitaev (2006), Appendix E</a>)</p> </blockquote> </li> <li id="RowellWang18"> <p><a class="existingWikiWord" href="/nlab/show/Eric+C.+Rowell">Eric C. Rowell</a>, <a class="existingWikiWord" href="/nlab/show/Zhenghan+Wang">Zhenghan Wang</a>, <em>Mathematics of Topological Quantum Computing</em>, Bull. Amer. Math. Soc. 55 (2018), 183-238 (<a href="https://arxiv.org/abs/1705.06206">arXiv:1705.06206</a>, <a href="https://doi.org/10.1090/bull/1605">doi:10.1090/bull/1605</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tian+Lan">Tian Lan</a>, <em>A Classification of (2+1)D Topological Phases with Symmetries</em> <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/1801.01210">arXiv:1801.01210</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></p> </li> <li> <p><em>From categories to anyons: a travelogue</em> [<a href="https://arxiv.org/abs/1811.06670">arXiv:1811.06670</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Colleen+Delaney">Colleen Delaney</a>, <em>A categorical perspective on symmetry, topological order, and quantum information</em> (2019) <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://crdelane.pages.iu.edu/dissertationch1-5.pdf">pdf</a>, <a href="https://escholarship.org/uc/item/5z384290">uc:5z384290</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></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Colleen+Delaney">Colleen Delaney</a>, <em>Lecture notes on modular tensor categories and braid group representations</em> (2019) <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="http://web.math.ucsb.edu/~cdelaney/MTC_Notes.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/DelaneyModularTensorCategories.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></p> </li> <li> <p>Liang Wang, <a class="existingWikiWord" href="/nlab/show/Zhenghan+Wang">Zhenghan Wang</a>, <em>In and around Abelian anyon models</em>, J. Phys. A: Math. Theor. <strong>53</strong> 505203 (2020) <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://iopscience.iop.org/article/10.1088/1751-8121/abc6c0">doi:10.1088/1751-8121/abc6c0</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></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Parsa+Bonderson">Parsa Bonderson</a>, <em>Measuring Topological Order</em>, Phys. Rev. Research <strong>3</strong>, 033110 (2021) <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/2102.05677">arXiv:2102.05677</a>, <a href="https://doi.org/10.1103/PhysRevResearch.3.033110">doi:10.1103/PhysRevResearch.3.033110</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></p> </li> <li> <p>Zhuan Li, Roger S.K. Mong, <em>Detecting topological order from modular transformations of ground states on the torus</em> <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/2203.04329">arXiv:2203.04329</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></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eric+C.+Rowell">Eric C. Rowell</a>, <em>Braids, Motions and Topological Quantum Computing</em> <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/2208.11762">arXiv:2208.11762</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></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sachin+Valera">Sachin Valera</a>, <em>A Quick Introduction to the Algebraic Theory of Anyons</em>, talk at <em><a class="existingWikiWord" href="/nlab/show/CQTS">CQTS</a> Initial Researcher Meeting</em> (Sep 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 class="existingWikiWord" href="/nlab/files/CQTS-InitialResearcherMeeting-Valera-220914.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></p> </li> <li> <p>Willie Aboumrad, <em>Quantum computing with anyons: an F-matrix and braid calculator</em> <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/2212.00831">arXiv:2212.00831</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></p> </li> </ul> <p>Emphasis that the expected description of <a class="existingWikiWord" href="/nlab/show/anyons">anyons</a> by <a class="existingWikiWord" href="/nlab/show/braided+fusion+categories">braided fusion categories</a> had remained <a class="existingWikiWord" href="/nlab/show/folklore">folklore</a>, together with a list of minimal assumptions that would need to be shown:</p> <ul> <li id="Valera21"><a class="existingWikiWord" href="/nlab/show/Sachin+J.+Valera">Sachin J. Valera</a>, <em>Fusion Structure from Exchange Symmetry in (2+1)-Dimensions</em>, Annals of Physics <strong>429</strong> (2021) [<a href="https://doi.org/10.1016/j.aop.2021.168471">doi:10.1016/j.aop.2021.168471</a>, <a href="https://arxiv.org/abs/2004.06282">arXiv:2004.06282</a>]</li> </ul> <p>An argument that the statement at least for <a class="existingWikiWord" href="/nlab/show/SU%282%29-anyons">SU(2)-anyons</a> does follow from an enhancement of the <a class="existingWikiWord" href="/nlab/show/K-theory+classification+of+topological+phases+of+matter">K-theory classification of topological phases of matter</a> to interacting <a class="existingWikiWord" href="/nlab/show/topological+order">topological order</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/Anyonic+topological+order+in+TED+K-theory">Anyonic topological order in TED K-theory</a></em>, Rev. Math. Phys. <strong>35</strong> 03 (2023) 2350001 [<a href="https://doi.org/10.1142/S0129055X23500010">doi:10.1142/S0129055X23500010</a>,<a href="https://arxiv.org/abs/2206.13563">arXiv:2206.13563</a>]</li> </ul> <h4 id="AnyonicOrderInMTheoryReferences">In string/M-theory</h4> <p>Arguments realizing such anyonic topological order in the <a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a>-<a class="existingWikiWord" href="/nlab/show/quantum+field+theory">field theory</a> on <a class="existingWikiWord" href="/nlab/show/M5-branes">M5-branes</a>:</p> <p>Via <a class="existingWikiWord" href="/nlab/show/KK-compactification">KK-compactification</a> on <a class="existingWikiWord" href="/nlab/show/closed+manifold">closed</a> <a class="existingWikiWord" href="/nlab/show/3-manifolds">3-manifolds</a> (<a class="existingWikiWord" href="/nlab/show/Seifert+manifolds">Seifert manifolds</a>) analogous to the <a class="existingWikiWord" href="/nlab/show/3d-3d+correspondence">3d-3d correspondence</a> (which instead uses <a class="existingWikiWord" href="/nlab/show/hyperbolic+3-manifolds">hyperbolic 3-manifolds</a>):</p> <ul> <li id="CGK20"> <p>Gil Young Cho, <a class="existingWikiWord" href="/nlab/show/Dongmin+Gang">Dongmin Gang</a>, Hee-Cheol Kim: <em>M-theoretic Genesis of Topological Phases</em>, J. High Energ. Phys. <strong>2020</strong> 115 (2020) [<a href="https://arxiv.org/abs/2007.01532">arXiv:2007.01532</a>, <a href=" https://doi.org/10.1007/JHEP11(2020)115">doi:10.1007/JHEP11(2020)115</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Shawn+X.+Cui">Shawn X. Cui</a>, Paul Gustafson, <a class="existingWikiWord" href="/nlab/show/Yang+Qiu">Yang Qiu</a>, Qing Zhang, <em>From Torus Bundles to Particle-Hole Equivariantization</em>, Lett Math Phys <strong>112</strong> 15 (2022) [<a href="https://doi.org/10.1007/s11005-022-01508-3">doi:10.1007/s11005-022-01508-3</a>, <a href="https://arxiv.org/abs/2106.01959">arXiv:2106.01959</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Shawn+X.+Cui">Shawn X. Cui</a>, <a class="existingWikiWord" href="/nlab/show/Yang+Qiu">Yang Qiu</a>, <a class="existingWikiWord" href="/nlab/show/Zhenghan+Wang">Zhenghan Wang</a>, <em>From Three Dimensional Manifolds to Modular Tensor Categories</em>, Commun. Math. Phys. <strong>397</strong> (2023) 1191–1235 [<a href="https://doi.org/10.1007/s00220-022-04517-4">doi:10.1007/s00220-022-04517-4</a>, <a href="https://arxiv.org/abs/2101.01674">arXiv:2101.01674</a>]</p> </li> <li> <p>Sunjin Choi, <a class="existingWikiWord" href="/nlab/show/Dongmin+Gang">Dongmin Gang</a>, Hee-Cheol Kim: <em>Infrared phases of 3D Class R theories</em>, J. High Energ. Phys. <strong>2022</strong> 151 (2022) [<a href="https://doi.org/10.1007/JHEP11(2022)151">doi:10.1007/JHEP11(2022)151</a>, <a href="https://arxiv.org/abs/2206.11982">arXiv:2206.11982</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Federico+Bonetti">Federico Bonetti</a>, <a class="existingWikiWord" href="/nlab/show/Sakura+Sch%C3%A4fer-Nameki">Sakura Schäfer-Nameki</a>, Jingxiang Wu, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>MTC</mi><mo stretchy="false">[</mo><msub><mi>M</mi> <mn>3</mn></msub><mo>,</mo><mi>G</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">MTC[M_3,G]</annotation></semantics></math>: 3d Topological Order Labeled by Seifert Manifolds</em> [<a href="https://arxiv.org/abs/2403.03973">arXiv:2403.03973</a>]</p> </li> </ul> <p>Via <a class="existingWikiWord" href="/nlab/show/3-brane+in+6d">3-brane</a> <a class="existingWikiWord" href="/nlab/show/defect+branes">defects</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>: <em><a class="existingWikiWord" href="/schreiber/show/Anyonic+defect+branes+in+TED+K-theory">Anyonic Defect Branes and Conformal Blocks in Twisted Equivariant Differential K-Theory</a></em>, Reviews in Mathematical Physics <strong>35</strong> 06 (2023) 2350009 [<a href="https://arxiv.org/abs/2203.11838">arXiv:2203.11838</a>, <a href="https://doi.org/10.1142/S0129055X23500095">doi:10.1142/S0129055X23500095</a>]</li> </ul> <h4 id="further_discussion">Further discussion</h4> <p>Relation to <a class="existingWikiWord" href="/nlab/show/ZX-calculus">ZX-calculus</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Fatimah+Rita+Ahmadi">Fatimah Rita Ahmadi</a>, <a class="existingWikiWord" href="/nlab/show/Aleks+Kissinger">Aleks Kissinger</a>, <em>Topological Quantum Computation Through the Lens of Categorical Quantum Mechanics</em> <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/2211.03855">arXiv:2211.03855</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> <p>On detection of <a class="existingWikiWord" href="/nlab/show/topological+order">topological order</a> by observing <a class="existingWikiWord" href="/nlab/show/modular+transformations">modular transformations</a> on the <a class="existingWikiWord" href="/nlab/show/ground+state">ground state</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Zhuan+Li">Zhuan Li</a>, <a class="existingWikiWord" href="/nlab/show/Roger+S.+K.+Mong">Roger S. K. Mong</a>, <em>Detecting topological order from modular transformations of ground states on the torus</em>, Phys. Rev. B <strong>106</strong> (2022) 235115 <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://doi.org/10.1103/PhysRevB.106.235115">doi:10.1103/PhysRevB.106.235115</a>, <a href="https://arxiv.org/abs/2203.04329">arXiv:2203.04329</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> <p>See also:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Liang+Kong">Liang Kong</a>, <em>Topological Wick Rotation and Holographic Dualities</em>, <a href="CQTS#KongOct2022">talk at</a> <a class="existingWikiWord" href="/nlab/show/CQTS">CQTS</a> (Oct 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 class="existingWikiWord" href="/nlab/files/Kong-TalkAtCQTS-20221019.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> </div></body></html> </div> <div class="revisedby"> <p> Last revised on July 4, 2024 at 16:52:01. 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