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[2005.01841] Representation variety for the rank one affine group

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We perform this calculation by three different approaches: the..."/> <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="Representation variety for the rank one affine group" /><meta name="citation_author" content="Gonz谩lez-Prieto, Angel" /><meta name="citation_author" content="Logares, Marina" /><meta name="citation_author" content="Mu帽oz, Vicente" /><meta name="citation_doi" content="10.1007/978-3-030-84721-0_18" /><meta name="citation_date" content="2020/05/04" /><meta name="citation_online_date" content="2020/09/11" /><meta name="citation_pdf_url" content="http://arxiv.org/pdf/2005.01841" /><meta name="citation_arxiv_id" content="2005.01841" /><meta name="citation_abstract" content="The aim of this paper is to study the virtual classes of representation varieties of surface groups onto the rank one affine group. We perform this calculation by three different approaches: the geometric method, based on stratifying the representation variety into simpler pieces; the arithmetic method, focused on counting their number of points over finite fields; and the quantum method, which performs the computation by means of a Topological Quantum Field Theory. We also discuss the corresponding moduli spaces of representations and character varieties, which turn out to be non-equivalent due to the non-reductiveness of the underlying group." /> </head> <body class="with-cu-identity"> <div class="flex-wrap-footer"> <header> <a href="#content" class="is-sr-only">Skip to main content</a> <!-- start desktop header --> <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 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<strong>arXiv:2005.01841</strong> (math) </div> <link rel="stylesheet" type="text/css" href="/static/base/1.0.1/css/abs.css"> <div id="content-inner"> <div id="abs"> <div class="dateline"> [Submitted on 4 May 2020 (<a href="https://arxiv.org/abs/2005.01841v1">v1</a>), last revised 11 Sep 2020 (this version, v3)]</div> <h1 class="title mathjax"><span class="descriptor">Title:</span>Representation variety for the rank one affine group</h1> <div class="authors"><span class="descriptor">Authors:</span><a href="https://arxiv.org/search/math?searchtype=author&amp;query=Gonz%C3%A1lez-Prieto,+A" rel="nofollow">Angel Gonz谩lez-Prieto</a>, <a href="https://arxiv.org/search/math?searchtype=author&amp;query=Logares,+M" rel="nofollow">Marina Logares</a>, <a href="https://arxiv.org/search/math?searchtype=author&amp;query=Mu%C3%B1oz,+V" rel="nofollow">Vicente Mu帽oz</a></div> <div id="download-button-info" hidden>View a PDF of the paper titled Representation variety for the rank one affine group, by Angel Gonz\&#39;alez-Prieto and 1 other authors</div> <a class="mobile-submission-download" href="/pdf/2005.01841">View PDF</a> <blockquote class="abstract mathjax"> <span class="descriptor">Abstract:</span>The aim of this paper is to study the virtual classes of representation varieties of surface groups onto the rank one affine group. We perform this calculation by three different approaches: the geometric method, based on stratifying the representation variety into simpler pieces; the arithmetic method, focused on counting their number of points over finite fields; and the quantum method, which performs the computation by means of a Topological Quantum Field Theory. We also discuss the corresponding moduli spaces of representations and character varieties, which turn out to be non-equivalent due to the non-reductiveness of the underlying group. </blockquote> <!--CONTEXT--> <div class="metatable"> <table summary="Additional metadata"> <tr> <td class="tablecell label">Comments:</td> <td class="tablecell comments mathjax">28 pages, 3 figures. References added</td> </tr> <tr> <td class="tablecell label">Subjects:</td> <td class="tablecell subjects"> <span class="primary-subject">Algebraic Geometry (math.AG)</span></td> </tr> <tr> <td class="tablecell label"><abbr title="Mathematical Subject Classification">MSC</abbr> classes:</td> <td class="tablecell msc-classes">57R56, 14C30, 14D07, 14D21</td> </tr> <tr> <td class="tablecell label">Cite as:</td> <td class="tablecell arxivid"><span class="arxivid"><a href="https://arxiv.org/abs/2005.01841">arXiv:2005.01841</a> [math.AG]</span></td> </tr> <tr> <td class="tablecell label">&nbsp;</td> <td class="tablecell arxividv">(or <span class="arxivid"> <a href="https://arxiv.org/abs/2005.01841v3">arXiv:2005.01841v3</a> [math.AG]</span> for this version) </td> </tr> <tr> <td class="tablecell label">&nbsp;</td> <td class="tablecell arxivdoi"> <a href="https://doi.org/10.48550/arXiv.2005.01841" id="arxiv-doi-link">https://doi.org/10.48550/arXiv.2005.01841</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> <!-- tooltip description --> <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&nbsp;reference:</td> <td class="tablecell jref">Mathematical Analysis in Interdisciplinary Research. Springer Optimization and Its Applications, vol 179, 2021</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/10.1007/978-3-030-84721-0_18" data-doi="10.1007/978-3-030-84721-0_18" class="link-https link-external" rel="external noopener nofollow">https://doi.org/10.1007/978-3-030-84721-0_18</a> <!-- accessible tooltip example --> <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> <!-- tooltip description --> <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: 脕ngel Gonz谩lez-Prieto [<a href="/show-email/8d771b13/2005.01841" rel="nofollow">view email</a>] <br/> <strong><a href="/abs/2005.01841v1" rel="nofollow">[v1]</a></strong> Mon, 4 May 2020 20:47:36 UTC (138 KB)<br/> <strong><a href="/abs/2005.01841v2" rel="nofollow">[v2]</a></strong> Tue, 2 Jun 2020 11:29:35 UTC (138 KB)<br/> <strong>[v3]</strong> Fri, 11 Sep 2020 15:11:13 UTC (138 KB)<br/> </div> </div> <!--end leftcolumn--> <div class="extra-services"> <div class="full-text"> <a name="other"></a> <span class="descriptor">Full-text links:</span> <h2>Access Paper:</h2> <ul> 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