Super fiber bundles, connection forms, and parallel transport | Journal of Mathematical Physics

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<div class="content-inner-wrap"> <div class="widget-ArticleTopInfo widget-instance-ArticleTopInfo"> <div class="module-widget article-top-widget content-metadata_wrap"> <div class="article-groups left-flag"> <span class="article-client_type">Research Article</span><span class="pipe">|</span> <span class="article-date">June 03 2021</span> </div> <div class="widget-items"> <h1 class="wi-article-title article-title-main"> Super fiber bundles, connection forms, and parallel transport <i class='icon-availability_cart' title='Available to Purchase' ></i> </h1> <div class="wi-authors"> <div class="al-authors-list"> <div class="al-author-name"> <a class="linked-name js-linked-name stats-author-info-trigger" href="javascript:;">Konstantin Eder</a> <div class="al-orcid-info-wrap"> <a href="https://orcid.org/0000-0002-3776-6777" class="al-orcid-url"> <img class="orchid-icon" alt="ORCID logo" src="//aipp.silverchair-cdn.com/Themes/Silver/app/img/mini-icon.png" /> </a> <span class="al-orcid-id"> <a href="https://orcid.org/0000-0002-3776-6777" class="al-orcid-url"> 0000-0002-3776-6777 </a> </span> </div> <div class="al-author-info-wrap arrow-up stats-author-info-panel"> <div class="info-card-author authorInfo_ArticleTopInfo"> <div class="name-role-wrap"> <div class="info-card-name"> Konstantin Eder <span class="info-card-footnote"><a reveal-id="cor1" data-open="cor1" class="link link-ref link-reveal js-reveal-distinct xref-corresp"><sup>a)</sup></a></span> </div> </div> <div class="info-card-affilitation"> <div class="aff"><div class="institution">Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Institute for Quantum Gravity (IQG)</div>, Staudtstr. 7, D-91058 Erlangen, <div class="country">Germany</div></div> </div> <div class="info-author-correspondence"> <div content-id="cor1"><span class="label title-label"><sup>a)</sup></span>Author to whom correspondence should be addressed: <a href="mailto:konstantin.eder@gravity.fau.de" target="_blank">konstantin.eder@gravity.fau.de</a></div> </div> <div class="info-card-search-label"> Search for other works by this author on: </div> <div class="info-card-search info-card-search-internal"> <a href="/search-results?f_AllAuthors=Konstantin+Eder" rel="nofollow">This Site</a> </div> <div class="info-card-search info-card-search-pubmed"> <a href="http://www.ncbi.nlm.nih.gov/pubmed?cmd=search&term=Konstantin Eder">PubMed</a> </div> <div class="info-card-search info-card-search-google"> <a href="http://scholar.google.com/scholar?q=author:"Konstantin Eder"">Google Scholar</a> </div> </div> </div> </div> </div> </div> <div class="widget-EditorInformation widget-instance-ArticleTopInfo_EditorInformation"> </div> <div class="wi-crossmark-button"> <script src="https://crossmark-cdn.crossref.org/widget/v2.0/widget.js"></script> <a href="javascript:;" class="stats-crossmark-updates" data-target="crossmark"> <img 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Math. Phys.</em> 62, 063506 (2021)</div> </div> <div class="pub-history-row citation-wrap-row clearfix"> <div class="ww-citation-wrap-doi"> <div class="citation-doi"> <a href="https://doi.org/10.1063/5.0044343">https://doi.org/10.1063/5.0044343</a> </div> </div> <div class="ww-citation-history-wrap js-history-dropdown-wrap"> <a href="javascript:;" class="history-label js-history-dropdown-trigger"> <span>Article history</span><i class="icon-history-small"></i> </a> <div class="ww-history js-history-entries-wrap"> <div class="history-entry"> <div class="wi-state">Received:</div> <div class="wi-date">January 15 2021</div> </div> <div class="history-entry"> <div class="wi-state">Accepted:</div> <div class="wi-date">May 13 2021</div> </div> </div> </div> </div> </div> </div> </div> <script> $(document).ready(function () { $('.article-top-widget').on('click', '.ati-toggle-trigger', function () { $(this).find('.icon-general-add, .icon-minus').toggleClass('icon-minus icon-general-add'); 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class="widget-items" data-widgetname="ArticleFulltext"> <div id="79348482" class="article-section-wrapper js-article-section js-content-section " data-section-parent-id="0"> <section class="abstract"><p>The present work provides a mathematically rigorous account on super fiber bundle theory, connection forms, and their parallel transport, which ties together various approaches. We begin with a detailed introduction to super fiber bundles. We then introduce the concept of so-called relative supermanifolds as well as bundles and connections defined in these categories. Studying these objects turns out to be of utmost importance in order to, among other things, model anticommuting classical fermionic fields in mathematical physics. We then construct the parallel transport map corresponding to such connections and compare the results with those found by other means in the mathematical literature. Finally, applications of these methods to supergravity will be discussed, such as the Cartan geometric formulation of Poincaré supergravity as well as the description of Killing vector fields and Killing spinors of super Riemannian manifolds arising from metric reductive super Cartan geometries.</p></section> </div> <div class="article-metadata-panel clearfix rs_skip"> <div class="widget-SolrResourceMetadata widget-instance-ContentMetadata_ArticleFulltext_Article"> <div class="content-metadata-taxonomies"> <div class="content-metadata"> <div class="content-metadata-taxonomies-title">Topics</div> <a href="/jmp/search-results?page=1&tax=P2163P2164">General relativity</a><span>,</span> <a href="/jmp/search-results?page=1&tax=P3437C1806">Differentiable manifold</a><span>,</span> <a href="/jmp/search-results?page=1&tax=P3437C1818">Lie algebras</a><span>,</span> <a href="/jmp/search-results?page=1&tax=P3437P1330">Differential geometry</a><span>,</span> <a href="/jmp/search-results?page=1&tax=P3437P1335">Differential topology</a><span>,</span> <a href="/jmp/search-results?page=1&tax=P3437P2318">Group theory</a><span>,</span> <a href="/jmp/search-results?page=1&tax=P3437P3437">Mathematical physics</a><span>,</span> <a href="/jmp/search-results?page=1&tax=P4951C4884">Gauge theory</a><span>,</span> <a href="/jmp/search-results?page=1&tax=P4951P1903">Field theory</a><span>,</span> <a href="/jmp/search-results?page=1&tax=P4951P88367">Fermionic field</a> </div> </div> </div> </div> <h2 scrollto-destination="79348909" id="79348909" data-section-title="REFERENCES" class="backreferences-title jumplink-heading">REFERENCES</h2> <div class=""> <div id="79348909" class="article-section-wrapper js-article-section js-content-section " data-section-parent-id="0"> <div class="ref-list js-splitview-ref-list"><div data-content-id="c1" xmlns:helper="urn:XsltStringHelper"><div class="ref false"><div class="ref-content "><span class="label title-label">1.</span><div class="citation mixed-citation"><span class="name string-name"><div class="given-names">F.</div> <div class="surname">Berezin</div></span> and <span class="name string-name"><div class="given-names">D.</div> <div class="surname">Leites</div></span>, “<div class="article-title">Supermanifolds</div>,” <div class="source">Sov. 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