[1910.07444] Symmetry Breaking and Link Homologies II

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Evaluating this filtered spectrum on suitable U(r)-equivariant cohomology theories gives rise to a spectral sequence of link invariants that converges to the cohomology of the limiting spectrum. In this paper, we evaluate our construction on Borel equivariant singular cohomology HU(r). We show that the E_2-term of the spectral sequence is isomorphic to an unreduced verision of triply-graded link homology. More precisely, we show that the E_1-term of the spectral sequence is isomorphic to the Hochschild-homology of Soergel bimodules that was shown by M. Khovanov in to compute triply-graded link homology. We also set up the theory that allows for evaluating our construction on equivariant cohomologies that are twisted by adjoint-equivariant local systems on U(r). This allows us to apply Borel equivariant cohomology twisted by a universal power series p(x) with coefficients given by formal variables, and no constant term. The specialization of p(x) to x^n gives rise to a link homology that has recently been shown by T. Mej谋a Gomez to be isomorphic to sl(n)-link homology. In other words, the power series p(x) can be interpreted as the (differential of the) universal potential function with no linear term, in the language of matrix factorizations."/> <meta name="twitter:site" content="@arxiv"/> <meta name="twitter:card" content="summary"/> <meta name="twitter:title" content="Symmetry Breaking and Link Homologies II"/> <meta name="twitter:description" content="In the first part of this paper, we constructed a filtered U(r)-equivariant stable homotopy type called the spectrum of strict broken symmetries sB(L) of links L given by closing a braid with r..."/> <meta name="twitter:image" content="https://static.arxiv.org/icons/twitter/arxiv-logo-twitter-square.png"/> <meta name="twitter:image:alt" content="arXiv logo"/> <link rel="stylesheet" media="screen" type="text/css" href="/static/browse/0.3.4/css/tooltip.css"/><link rel="stylesheet" media="screen" type="text/css" href="https://static.arxiv.org/js/bibex-dev/bibex.css?20200709"/> <script src="/static/browse/0.3.4/js/mathjaxToggle.min.js" type="text/javascript"></script> <script src="//code.jquery.com/jquery-latest.min.js" type="text/javascript"></script> <script src="//cdn.jsdelivr.net/npm/js-cookie@2/src/js.cookie.min.js" type="text/javascript"></script> <script src="//cdn.jsdelivr.net/npm/dompurify@2.3.5/dist/purify.min.js"></script> <script src="/static/browse/0.3.4/js/toggle-labs.js?20241022" type="text/javascript"></script> <script src="/static/browse/0.3.4/js/cite.js" type="text/javascript"></script><meta name="citation_title" content="Symmetry Breaking and Link Homologies II" /><meta name="citation_author" content="Kitchloo, Nitu" /><meta name="citation_date" content="2019/10/14" /><meta name="citation_online_date" content="2023/09/07" /><meta name="citation_pdf_url" content="http://arxiv.org/pdf/1910.07444" /><meta name="citation_arxiv_id" content="1910.07444" /><meta name="citation_abstract" content="In the first part of this paper, we constructed a filtered U(r)-equivariant stable homotopy type called the spectrum of strict broken symmetries sB(L) of links L given by closing a braid with r strands. Evaluating this filtered spectrum on suitable U(r)-equivariant cohomology theories gives rise to a spectral sequence of link invariants that converges to the cohomology of the limiting spectrum. In this paper, we evaluate our construction on Borel equivariant singular cohomology HU(r). We show that the E_2-term of the spectral sequence is isomorphic to an unreduced verision of triply-graded link homology. More precisely, we show that the E_1-term of the spectral sequence is isomorphic to the Hochschild-homology of Soergel bimodules that was shown by M. Khovanov in to compute triply-graded link homology. We also set up the theory that allows for evaluating our construction on equivariant cohomologies that are twisted by adjoint-equivariant local systems on U(r). This allows us to apply Borel equivariant cohomology twisted by a universal power series p(x) with coefficients given by formal variables, and no constant term. The specialization of p(x) to x^n gives rise to a link homology that has recently been shown by T. Mej{\i}a Gomez to be isomorphic to sl(n)-link homology. In other words, the power series p(x) can be interpreted as the (differential of the) universal potential function with no linear term, in the language of matrix factorizations." /> </head> <body class="with-cu-identity"> <div class="flex-wrap-footer"> <header> <a href="#content" class="is-sr-only">Skip to main content</a>  <div class="columns is-vcentered is-hidden-mobile" id="cu-identity"> <div class="column" id="cu-logo"> <a href="https://www.cornell.edu/"><img src="/static/browse/0.3.4/images/icons/cu/cornell-reduced-white-SMALL.svg" alt="Cornell University" /></a> </div><div class="column" id="support-ack"> <span id="support-ack-url">We gratefully acknowledge support from the Simons Foundation, <a href="https://info.arxiv.org/about/ourmembers.html">member institutions</a>, and all contributors.</span> <a href="https://info.arxiv.org/about/donate.html" class="btn-header-donate">Donate</a> </div> </div> <div id="header" class="is-hidden-mobile"> <a aria-hidden="true" tabindex="-1" 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class="descriptor">Authors:</span><a href="https://arxiv.org/search/math?searchtype=author&query=Kitchloo,+N" rel="nofollow">Nitu Kitchloo</a></div> <div id="download-button-info" hidden>View a PDF of the paper titled Symmetry Breaking and Link Homologies II, by Nitu Kitchloo</div> <a class="mobile-submission-download" href="/pdf/1910.07444">View PDF</a> <blockquote class="abstract mathjax"> <span class="descriptor">Abstract:</span>In the first part of this paper, we constructed a filtered U(r)-equivariant stable homotopy type called the spectrum of strict broken symmetries sB(L) of links L given by closing a braid with r strands. Evaluating this filtered spectrum on suitable U(r)-equivariant cohomology theories gives rise to a spectral sequence of link invariants that converges to the cohomology of the limiting spectrum. In this paper, we evaluate our construction on Borel equivariant singular cohomology HU(r). We show that the E_2-term of the spectral sequence is isomorphic to an unreduced verision of triply-graded link homology. More precisely, we show that the E_1-term of the spectral sequence is isomorphic to the Hochschild-homology of Soergel bimodules that was shown by M. Khovanov in to compute triply-graded link homology. We also set up the theory that allows for evaluating our construction on equivariant cohomologies that are twisted by adjoint-equivariant local systems on U(r). This allows us to apply Borel equivariant cohomology twisted by a universal power series p(x) with coefficients given by formal variables, and no constant term. The specialization of p(x) to x^n gives rise to a link homology that has recently been shown by T. Mej谋a Gomez to be isomorphic to sl(n)-link homology. In other words, the power series p(x) can be interpreted as the (differential of the) universal potential function with no linear term, in the language of matrix factorizations. </blockquote>  <div class="metatable"> <table summary="Additional metadata"> <tr> <td class="tablecell label">Comments:</td> <td class="tablecell comments mathjax">This represents a substantial revision of Parts II and III of the original posting. We have removed an erroneous computation given in the Appendix of Part II and III. An alternate argument makes our results independent of this computation. We also reference forthcoming work by Gomez that proves our conjecture that the link homologies we construct specialize to sl(n)-link homology for any n</td> </tr> <tr> <td class="tablecell label">Subjects:</td> <td class="tablecell subjects"> <span class="primary-subject">Algebraic Topology (math.AT)</span>; Quantum Algebra (math.QA)</td> </tr><tr> <td class="tablecell label">Cite as:</td> <td class="tablecell arxivid"><span class="arxivid"><a href="https://arxiv.org/abs/1910.07444">arXiv:1910.07444</a> [math.AT]</span></td> </tr> <tr> <td class="tablecell label"> </td> <td class="tablecell arxividv">(or <span class="arxivid"> <a href="https://arxiv.org/abs/1910.07444v4">arXiv:1910.07444v4</a> [math.AT]</span> for this version) </td> </tr> <tr> <td class="tablecell label"> </td> <td class="tablecell arxivdoi"> <a href="https://doi.org/10.48550/arXiv.1910.07444" id="arxiv-doi-link">https://doi.org/10.48550/arXiv.1910.07444</a><div class="button-and-tooltip"> <button class="more-info" 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href="/abs/1910.07444v1" rel="nofollow">[v1]</a></strong> Mon, 14 Oct 2019 18:01:37 UTC (21 KB)<br/> <strong><a href="/abs/1910.07444v2" rel="nofollow">[v2]</a></strong> Tue, 19 Nov 2019 18:09:31 UTC (32 KB)<br/> <strong><a href="/abs/1910.07444v3" rel="nofollow">[v3]</a></strong> Tue, 28 Jul 2020 01:37:01 UTC (33 KB)<br/> <strong>[v4]</strong> Thu, 7 Sep 2023 16:18:27 UTC (41 KB)<br/> </div> </div>  <div class="extra-services"> <div class="full-text"> <a name="other"></a> <span class="descriptor">Full-text links:</span> <h2>Access Paper:</h2> <ul> <div id="download-button-info" hidden> View a PDF of the paper titled Symmetry Breaking and Link Homologies II, by Nitu Kitchloo</div><li><a href="/pdf/1910.07444" aria-describedby="download-button-info" accesskey="f" class="abs-button download-pdf">View PDF</a></li><li><a href="/src/1910.07444" class="abs-button download-eprint">TeX Source</a></li><li><a href="/format/1910.07444" class="abs-button download-format">Other 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[1910.07444] Symmetry Breaking and Link Homologies II