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value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='functorial_quantum_field_theory'>Functorial quantum field theory</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/functorial+field+theory'>functorial quantum field theory</a></strong></p> <h2 id='contents'>Contents</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/cobordism+category'>cobordism category</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/cobordism'>cobordism</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/extended+cobordism'>extended cobordism</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/%28infinity%2Cn%29-category+of+cobordisms'>(∞,n)-category of cobordisms</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/bordism+categories+following+Stolz-Teichner'>Riemannian bordism category</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/cobordism+hypothesis'>cobordism hypothesis</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/tangle+hypothesis'>generalized tangle hypothesis</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/On+the+Classification+of+Topological+Field+Theories'>classification of TQFTs</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/functorial+field+theory'>FQFT</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/extended+topological+quantum+field+theory'>extended TQFT</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/conformal+field+theory'>CFT</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/vertex+operator+algebra'>vertex operator algebra</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/topological+quantum+field+theory'>TQFT</a></p> <ul> <li> <p><span class='newWikiWord'>Reshetikhin?Turaev model<a href='/nlab/new/Reshetikhin%3FTuraev+model'>?</a></span> / <a class='existingWikiWord' href='/nlab/show/Chern-Simons+theory'>Chern-Simons theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/HQFT'>HQFT</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/TCFT'>TCFT</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/A-model'>A-model</a>, <a class='existingWikiWord' href='/nlab/show/B-model'>B-model</a>, <a class='existingWikiWord' href='/nlab/show/Gromov-Witten+invariants'>Gromov-Witten theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/mirror+symmetry'>homological mirror symmetry</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p>FQFT and <a class='existingWikiWord' href='/nlab/show/cohomology'>cohomology</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory'>(1,1)-dimensional Euclidean field theories and K-theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/%282%7C1%29-dimensional+Euclidean+field+theory'>(2,1)-dimensional Euclidean field theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/geometric+model+for+elliptic+cohomology'>geometric models for tmf</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/holographic+principle+of+higher+category+theory'>holographic principle of higher category theory</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/holographic+principle'>holographic principle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/AdS-CFT+correspondence'>AdS/CFT correspondence</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/quantization+via+the+A-model'>quantization via the A-model</a></p> </li> </ul> </li> </ul> <div> <p> <a href='/nlab/edit/functorial+quantum+field+theory+-+contents'>Edit this sidebar</a> </p> </div></div> </div> </div> <h1 id='contents_2'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#statement'>Statement</a></li><li><a href='#generalized_tangle_hypothesis'>Generalized tangle hypothesis</a></li><li><a href='#the_tangle_hypothesis_as_a_consequence_of_the_cobordism_hypothesis'>The tangle hypothesis as a consequence of the cobordism hypothesis</a></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a></li></ul></div> <h2 id='statement'>Statement</h2> <p>The <em>tangle hypothesis</em> (<a href='#BaezDolan95'>Baez and Dolan 95</a>) is as follows:</p> <div class='un_prop'> <h6 id='tangle_hypothesis'>Tangle Hypothesis</h6> <p>The <math class='maruku-mathml' display='inline' id='mathml_112171c40c8abe75b06f211bb0aff073951e6742_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-category of framed <math class='maruku-mathml' display='inline' id='mathml_112171c40c8abe75b06f211bb0aff073951e6742_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/tangle'>tangles</a> in <math class='maruku-mathml' display='inline' id='mathml_112171c40c8abe75b06f211bb0aff073951e6742_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>+</mo><mi>k</mi></mrow><annotation encoding='application/x-tex'>n+k</annotation></semantics></math> dimensions is <math class='maruku-mathml' display='inline' id='mathml_112171c40c8abe75b06f211bb0aff073951e6742_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>n</mi><mo>+</mo><mi>k</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(n+k)</annotation></semantics></math>-equivalent to the free <a class='existingWikiWord' href='/nlab/show/k-tuply+monoidal+n-category'>k-tuply monoidal n-category</a> with duals on one object.</p> </div> <p>The tangle hypothesis may be seen as a refinement of the <a class='existingWikiWord' href='/nlab/show/cobordism+hypothesis'>cobordism hypothesis</a> in the sense that the latter arises from the former in the limit <math class='maruku-mathml' display='inline' id='mathml_112171c40c8abe75b06f211bb0aff073951e6742_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mo>→</mo><mn>∞</mn></mrow><annotation encoding='application/x-tex'>k \to \infty</annotation></semantics></math>:</p> <div class='un_prop'> <h6 id='<a class='><a class='existingWikiWord' href='/nlab/show/cobordism+hypothesis'>Cobordism Hypothesis</a></h6> <p>The <math class='maruku-mathml' display='inline' id='mathml_112171c40c8abe75b06f211bb0aff073951e6742_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-category <math class='maruku-mathml' display='inline' id='mathml_112171c40c8abe75b06f211bb0aff073951e6742_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mi>Cob</mi></mrow><annotation encoding='application/x-tex'>n Cob</annotation></semantics></math> of cobordisms is the free stable <math class='maruku-mathml' display='inline' id='mathml_112171c40c8abe75b06f211bb0aff073951e6742_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-category with duals on one object (the point).</p> </div> <h2 id='generalized_tangle_hypothesis'>Generalized tangle hypothesis</h2> <p>The tangle hypothesis has been generalized to allow certain structures on the tangles.</p> <p>The <math class='maruku-mathml' display='inline' id='mathml_112171c40c8abe75b06f211bb0aff073951e6742_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math>-tuply monoidal <math class='maruku-mathml' display='inline' id='mathml_112171c40c8abe75b06f211bb0aff073951e6742_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-category of <math class='maruku-mathml' display='inline' id='mathml_112171c40c8abe75b06f211bb0aff073951e6742_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>-structured <math class='maruku-mathml' display='inline' id='mathml_112171c40c8abe75b06f211bb0aff073951e6742_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-tangles in the <math class='maruku-mathml' display='inline' id='mathml_112171c40c8abe75b06f211bb0aff073951e6742_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>n</mi><mo>+</mo><mi>k</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(n + k)</annotation></semantics></math>-cube is the <a class='existingWikiWord' href='/nlab/show/fundamental+category'>fundamental</a> <math class='maruku-mathml' display='inline' id='mathml_112171c40c8abe75b06f211bb0aff073951e6742_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>n</mi><mo>+</mo><mi>k</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(n + k)</annotation></semantics></math>-category with duals of <math class='maruku-mathml' display='inline' id='mathml_112171c40c8abe75b06f211bb0aff073951e6742_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>M</mi><mi>G</mi><mo>,</mo><mi>Z</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(M G,Z)</annotation></semantics></math>.</p> <ul> <li><math class='maruku-mathml' display='inline' id='mathml_112171c40c8abe75b06f211bb0aff073951e6742_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>M</mi><mi>G</mi></mrow><annotation encoding='application/x-tex'>M G</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/Thom+space'>Thom space</a> of group <math class='maruku-mathml' display='inline' id='mathml_112171c40c8abe75b06f211bb0aff073951e6742_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>.</li> <li><math class='maruku-mathml' display='inline' id='mathml_112171c40c8abe75b06f211bb0aff073951e6742_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> can be any group equipped with a homomorphism to <math class='maruku-mathml' display='inline' id='mathml_112171c40c8abe75b06f211bb0aff073951e6742_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>O</mi><mo stretchy='false'>(</mo><mi>k</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>O(k)</annotation></semantics></math>. (<a href='http://golem.ph.utexas.edu/category/2006/11/this_weeks_finds_in_mathematic_2.html#c006471'>comment</a>)</li> </ul> <h2 id='the_tangle_hypothesis_as_a_consequence_of_the_cobordism_hypothesis'>The tangle hypothesis as a consequence of the cobordism hypothesis</h2> <p>While the tangle hypothesis and its generalizations may be seen as refinements of the cobordism hypothesis and its generalizations, Lurie shows (<a href='#LurieCTFT'>Lurie 09, Sec. 4.4</a>) that the former may be deduced from the latter when expressed in a sufficiently general form.</p> <h2 id='related_concepts'>Related concepts</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/cobordism+hypothesis'>cobordism hypothesis</a></li> </ul> <h2 id='references'>References</h2> <ul> <li id='BaezDolan95'> <p><a class='existingWikiWord' href='/nlab/show/John+Baez'>John Baez</a> and <a class='existingWikiWord' href='/nlab/show/James+Dolan'>James Dolan</a>, <em>Higher-dimensional Algebra and Topological Quantum Field Theory</em> 1995 (<a href='http://arxiv.org/abs/q-alg/9503002'>arXiv</a>)</p> </li> <li id='LurieCTFT'> <p><a class='existingWikiWord' href='/nlab/show/Jacob+Lurie'>Jacob Lurie</a>, <a class='existingWikiWord' href='/nlab/show/On+the+Classification+of+Topological+Field+Theories'>On the Classification of Topological Field Theories</a> (<a href='http://arxiv.org/abs/0905.0465'>arXiv:0905.0465</a>)</p> </li> <li id='LurieTQFT'> <p><a class='existingWikiWord' href='/nlab/show/Jacob+Lurie'>Jacob Lurie</a>, <em>TQFT and the cobordism hypothesis</em>, videos of 4 lectures at the <a href='http://www.ma.utexas.edu/users/plowrey/dev/rtg/perspectives.html'>Geometry Research Group</a>, Mathematics Department, University of Texas Austin. Lecture notes for Lurie’s talks are available at the <a href='http://www.ma.utexas.edu/users/plowrey/dev/rtg/perspectives.html'>Geometry Research Group website</a>.</p> </li> </ul> <p>For a discussion of the generalized tangle hypothesis see</p> <ul> <li><a href='http://golem.ph.utexas.edu/category/2006/11/this_weeks_finds_in_mathematic_2.html#c006381'>n-Category Café</a></li> </ul> <p> </p> <p> </p> </div>  <div class="revisedby"> <p> Revision on March 18, 2023 at 17:17:39 by <a href="/nlab/author/David+Corfield" style="color: #005c19">David Corfield</a> See the <a href="/nlab/history/tangle+hypothesis" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="https://nforum.ncatlab.org/discussion/13677/#Item_4">Discuss</a><span class="backintime"><a href="/nlab/show/tangle+hypothesis" accesskey="F" class="navlinkbackintime" id="to_next_revision" rel="nofollow">Next revision</a> (to current)</span><span class="backintime"><a href="/nlab/revision/tangle+hypothesis/25" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a> (25 more)</span><a href="/nlab/show/tangle+hypothesis" class="navlink" id="to_current_revision">Current version of page</a><a href="/nlab/revision/diff/tangle+hypothesis/26" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/tangle+hypothesis" accesskey="S" class="navlink" id="history" rel="nofollow">History (26 revisions)</a><a href="/nlab/rollback/tangle+hypothesis?rev=26" class="navlink" id="rollback" rel="nofollow">Rollback</a> <a href="/nlab/revision/tangle+hypothesis/26/cite" style="color: black">Cite</a> <a href="/nlab/source/tangle+hypothesis/26" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>