[2105.11565] Chern-Simons perturbative series revisited

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A special basis in the center of the universal enveloping algebra $ZU(\mathfrak{sl}_N)$ is introduced. This basis allows one to present group factors in an arbitrary irreducible finite-dimensional representation. Developed methods have wide applications, the most straightforward and evident ones are mentioned. Namely, Vassiliev invariants of higher orders are computed, a conjecture about existence of new symmetries of the colored HOMFLY polynomials is stated, and the recently discovered tug-the-hook symmetry of the colored HOMFLY polynomial is proved."/> <meta name="twitter:site" content="@arxiv"/> <meta name="twitter:card" content="summary"/> <meta name="twitter:title" content="Chern-Simons perturbative series revisited"/> <meta name="twitter:description" content="A group-theoretical structure in a perturbative expansion of the Wilson loops in the 3d Chern-Simons theory with $SU(N)$ gauge group is studied in symmetric approach. A special basis in the center..."/> <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="Chern-Simons perturbative series revisited" /><meta name="citation_author" content="Lanina, E." /><meta name="citation_author" content="Sleptsov, A." /><meta name="citation_author" content="Tselousov, N." /><meta name="citation_doi" content="10.1016/j.physletb.2021.136727" /><meta name="citation_date" content="2021/05/24" /><meta name="citation_online_date" content="2021/10/26" /><meta name="citation_pdf_url" content="http://arxiv.org/pdf/2105.11565" /><meta name="citation_arxiv_id" content="2105.11565" /><meta name="citation_abstract" content="A group-theoretical structure in a perturbative expansion of the Wilson loops in the 3d Chern-Simons theory with $SU(N)$ gauge group is studied in symmetric approach. A special basis in the center of the universal enveloping algebra $ZU(\mathfrak{sl}_N)$ is introduced. This basis allows one to present group factors in an arbitrary irreducible finite-dimensional representation. Developed methods have wide applications, the most straightforward and evident ones are mentioned. Namely, Vassiliev invariants of higher orders are computed, a conjecture about existence of new symmetries of the colored HOMFLY polynomials is stated, and the recently discovered tug-the-hook symmetry of the colored HOMFLY polynomial is proved." /> </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 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revisited</h1> <div class="authors"><span class="descriptor">Authors:</span><a href="https://arxiv.org/search/hep-th?searchtype=author&query=Lanina,+E" rel="nofollow">E. Lanina</a>, <a href="https://arxiv.org/search/hep-th?searchtype=author&query=Sleptsov,+A" rel="nofollow">A. Sleptsov</a>, <a href="https://arxiv.org/search/hep-th?searchtype=author&query=Tselousov,+N" rel="nofollow">N. Tselousov</a></div> <div id="download-button-info" hidden>View a PDF of the paper titled Chern-Simons perturbative series revisited, by E. Lanina and 1 other authors</div> <a class="mobile-submission-download" href="/pdf/2105.11565">View PDF</a> <blockquote class="abstract mathjax"> <span class="descriptor">Abstract:</span>A group-theoretical structure in a perturbative expansion of the Wilson loops in the 3d Chern-Simons theory with $SU(N)$ gauge group is studied in symmetric approach. A special basis in the center of the universal enveloping algebra $ZU(\mathfrak{sl}_N)$ is introduced. This basis allows one to present group factors in an arbitrary irreducible finite-dimensional representation. Developed methods have wide applications, the most straightforward and evident ones are mentioned. Namely, Vassiliev invariants of higher orders are computed, a conjecture about existence of new symmetries of the colored HOMFLY polynomials is stated, and the recently discovered tug-the-hook symmetry of the colored HOMFLY polynomial is proved. </blockquote>  <div class="metatable"> <table summary="Additional metadata"> <tr> <td class="tablecell label">Comments:</td> <td class="tablecell comments mathjax">10 pages</td> </tr> <tr> <td class="tablecell label">Subjects:</td> <td class="tablecell subjects"> <span class="primary-subject">High Energy Physics - Theory (hep-th)</span>; Mathematical Physics (math-ph); Geometric Topology (math.GT); Representation Theory (math.RT)</td> </tr><tr> <td class="tablecell label">Cite as:</td> <td class="tablecell arxivid"><span class="arxivid"><a href="https://arxiv.org/abs/2105.11565">arXiv:2105.11565</a> [hep-th]</span></td> </tr> <tr> <td class="tablecell label"> </td> <td class="tablecell arxividv">(or <span class="arxivid"> <a href="https://arxiv.org/abs/2105.11565v2">arXiv:2105.11565v2</a> [hep-th]</span> for this version) </td> </tr> <tr> <td class="tablecell label"> </td> <td class="tablecell arxivdoi"> <a href="https://doi.org/10.48550/arXiv.2105.11565" id="arxiv-doi-link">https://doi.org/10.48550/arXiv.2105.11565</a><div class="button-and-tooltip"> <button class="more-info" aria-describedby="more-info-desc-1"> <svg height="15" role="presentation" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><path fill="currentColor" d="M256 8C119.043 8 8 119.083 8 256c0 136.997 111.043 248 248 248s248-111.003 248-248C504 119.083 392.957 8 256 8zm0 110c23.196 0 42 18.804 42 42s-18.804 42-42 42-42-18.804-42-42 18.804-42 42-42zm56 254c0 6.627-5.373 12-12 12h-88c-6.627 0-12-5.373-12-12v-24c0-6.627 5.373-12 12-12h12v-64h-12c-6.627 0-12-5.373-12-12v-24c0-6.627 5.373-12 12-12h64c6.627 0 12 5.373 12 12v100h12c6.627 0 12 5.373 12 12v24z" class=""></path></svg> <span class="visually-hidden">Focus to learn more</span> </button>  <div role="tooltip" id="more-info-desc-1"> <span class="left-corner"></span> arXiv-issued DOI via DataCite</div> </div> </td> </tr> <tr> <td class="tablecell label">Journal reference:</td> <td class="tablecell jref">Physics Letters B 823 (2021) 136727</td> </tr> <tr> <td class="tablecell label"> <abbr title="Digital Object Identifier">Related DOI</abbr>: </td> <td class="tablecell doi"><a href="https://doi.org/10.1016/j.physletb.2021.136727" data-doi="10.1016/j.physletb.2021.136727" class="link-https link-external" rel="external noopener nofollow">https://doi.org/10.1016/j.physletb.2021.136727</a>  <div class="button-and-tooltip"> <button class="more-info" aria-describedby="more-info-desc-1"> <svg height="15" role="presentation" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><path fill="currentColor" d="M256 8C119.043 8 8 119.083 8 256c0 136.997 111.043 248 248 248s248-111.003 248-248C504 119.083 392.957 8 256 8zm0 110c23.196 0 42 18.804 42 42s-18.804 42-42 42-42-18.804-42-42 18.804-42 42-42zm56 254c0 6.627-5.373 12-12 12h-88c-6.627 0-12-5.373-12-12v-24c0-6.627 5.373-12 12-12h12v-64h-12c-6.627 0-12-5.373-12-12v-24c0-6.627 5.373-12 12-12h64c6.627 0 12 5.373 12 12v100h12c6.627 0 12 5.373 12 12v24z" class=""></path></svg> <span class="visually-hidden">Focus to learn more</span> </button>  <div role="tooltip" id="more-info-desc-1"> <span class="left-corner"></span> DOI(s) linking to related resources </div> </div> </td> </tr> </table> </div> </div> </div> <div class="submission-history"> <h2>Submission history</h2> From: Elena Lanina [<a href="/show-email/c5e32c0c/2105.11565" rel="nofollow">view email</a>] <br/> <strong><a href="/abs/2105.11565v1" rel="nofollow">[v1]</a></strong> Mon, 24 May 2021 22:57:12 UTC (50 KB)<br/> <strong>[v2]</strong> Tue, 26 Oct 2021 11:41:51 UTC (28 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 Chern-Simons perturbative series revisited, by E. 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[2105.11565] Chern-Simons perturbative series revisited