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(PDF) Gauge Theories and Fiber Bundles: Definitions, Pictures, and Results
<!DOCTYPE html> <html > <head> <meta charset="utf-8"> <meta rel="search" type="application/opensearchdescription+xml" href="/open_search.xml" title="Academia.edu"> <meta content="width=device-width, initial-scale=1" name="viewport"> <meta name="google-site-verification" content="bKJMBZA7E43xhDOopFZkssMMkBRjvYERV-NaN4R6mrs"> <meta name="csrf-param" content="authenticity_token" /> <meta name="csrf-token" content="4oFJf8E2lHd8Fdcpp9dDuxqZwPUXUaZEVytvQN2QCQbp9ORafIHTI2EVyCYt6AFvjO0lnHON6_1vM77S7t_nVw" /> <meta name="citation_title" content="Gauge Theories and Fiber Bundles: Definitions, Pictures, and Results" /> <meta name="citation_publication_date" content="2016/01/01" /> <meta name="citation_author" content="Adam Marsh" /> <meta name="twitter:card" content="summary" /> <meta name="twitter:url" content="https://www.academia.edu/63797283/Gauge_Theories_and_Fiber_Bundles_Definitions_Pictures_and_Results" /> <meta name="twitter:title" content="Gauge Theories and Fiber Bundles: Definitions, Pictures, and Results" /> <meta name="twitter:description" content="A pedagogical but concise overview of fiber bundles and their connections is provided, in the context of gauge theories in physics. The emphasis is on defining and visualizing concepts and relationships between them, as well as listing common" /> <meta name="twitter:image" content="http://a.academia-assets.com/images/twitter-card.jpeg" /> <meta property="fb:app_id" content="2369844204" /> <meta property="og:type" content="article" /> <meta property="og:url" content="https://www.academia.edu/63797283/Gauge_Theories_and_Fiber_Bundles_Definitions_Pictures_and_Results" /> <meta property="og:title" content="Gauge Theories and Fiber Bundles: Definitions, Pictures, and Results" /> <meta property="og:image" content="http://a.academia-assets.com/images/open-graph-icons/fb-paper.gif" /> <meta property="og:description" content="A pedagogical but concise overview of fiber bundles and their connections is provided, in the context of gauge theories in physics. The emphasis is on defining and visualizing concepts and relationships between them, as well as listing common" /> <meta property="article:author" content="https://independent.academia.edu/AdamMarsh19" /> <meta name="description" content="A pedagogical but concise overview of fiber bundles and their connections is provided, in the context of gauge theories in physics. The emphasis is on defining and visualizing concepts and relationships between them, as well as listing common" /> <title>(PDF) Gauge Theories and Fiber Bundles: Definitions, Pictures, and Results</title> <link rel="canonical" href="https://www.academia.edu/63797283/Gauge_Theories_and_Fiber_Bundles_Definitions_Pictures_and_Results" /> <script async src="https://www.googletagmanager.com/gtag/js?id=G-5VKX33P2DS"></script> <script> window.dataLayer = window.dataLayer || []; function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'G-5VKX33P2DS', { cookie_domain: 'academia.edu', send_page_view: false, }); gtag('event', 'page_view', { 'controller': "single_work", 'action': "show", 'controller_action': 'single_work#show', 'logged_in': 'false', 'edge': 'unknown', // Send nil if there is no A/B test bucket, in case some records get logged // with missing data - that way we can distinguish between the two cases. // ab_test_bucket should be of the form <ab_test_name>:<bucket> 'ab_test_bucket': null, }) </script> <script> var $controller_name = 'single_work'; var $action_name = "show"; var $rails_env = 'production'; var $app_rev = '2dbf19f283ec395370665dd1f7acd1f78b8fa59d'; var $domain = 'academia.edu'; var $app_host = "academia.edu"; var $asset_host = "academia-assets.com"; var $start_time = new Date().getTime(); var $recaptcha_key = "6LdxlRMTAAAAADnu_zyLhLg0YF9uACwz78shpjJB"; var $recaptcha_invisible_key = "6Lf3KHUUAAAAACggoMpmGJdQDtiyrjVlvGJ6BbAj"; var $disableClientRecordHit = false; </script> <script> window.require = { config: function() { return function() {} } } </script> <script> window.Aedu = window.Aedu || {}; window.Aedu.hit_data = null; window.Aedu.serverRenderTime = new Date(1740164415000); window.Aedu.timeDifference = new Date().getTime() - 1740164415000; </script> <script type="application/ld+json">{"@context":"https://schema.org","@type":"ScholarlyArticle","abstract":"A pedagogical but concise overview of fiber bundles and their connections is provided, in the context of gauge theories in physics. The emphasis is on defining and visualizing concepts and relationships between them, as well as listing common confusions, alternative notations and jargon, and relevant facts and theorems. Special attention is given to detailed figures and geometric viewpoints, some of which would seem to be novel to the literature. Topics are avoided which are well covered in textbooks, such as historical motivations, proofs and derivations, and tools for practical calculations. 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{"work":{"id":63797283,"created_at":"2021-12-10T11:30:29.778-08:00","from_world_paper_id":186521719,"updated_at":"2024-11-30T19:06:58.237-08:00","_data":{"abstract":"A pedagogical but concise overview of fiber bundles and their connections is provided, in the context of gauge theories in physics. The emphasis is on defining and visualizing concepts and relationships between them, as well as listing common confusions, alternative notations and jargon, and relevant facts and theorems. Special attention is given to detailed figures and geometric viewpoints, some of which would seem to be novel to the literature. Topics are avoided which are well covered in textbooks, such as historical motivations, proofs and derivations, and tools for practical calculations. The present paper is best read in conjunction with the similar paper on Riemannian geometry cited herein.","ai_title_tag":"Overview of Fiber Bundles in Gauge Theories","publication_date":"2016,,"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"Gauge Theories and Fiber Bundles: Definitions, Pictures, and Results","broadcastable":true,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [163946664]; window.loswp.locale = "en"; window.loswp.countryCode = "SG"; window.loswp.cwvAbTestBucket = ""; window.loswp.designVariant = "ds_vanilla"; window.loswp.fullPageMobileSutdModalVariant = "control"; window.loswp.useOptimizedScribd4genScript = false; window.loginModal = {}; window.loginModal.appleClientId = 'edu.academia.applesignon'; window.userInChina = "false";</script><script defer="" src="https://accounts.google.com/gsi/client"></script><div class="ds-loswp-container"><div class="ds-work-card--grid-container"><div class="ds-work-card--container js-loswp-work-card"><div class="ds-work-card--cover"><div class="ds-work-cover--wrapper"><div class="ds-work-cover--container"><button class="ds-work-cover--clickable js-swp-download-button" data-signup-modal="{"location":"swp-splash-paper-cover","attachmentId":76100720,"attachmentType":"pdf"}"><img alt="First page of “Gauge Theories and Fiber Bundles: Definitions, Pictures, and Results”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/76100720/mini_magick20211210-11297-1bi363b.png?1639164734" /><img alt="PDF Icon" class="ds-work-cover--file-icon" src="//a.academia-assets.com/images/single_work_splash/adobe_icon.svg" /><div class="ds-work-cover--hover-container"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span><p>Download Free PDF</p></div><div class="ds-work-cover--ribbon-container">Download Free PDF</div><div class="ds-work-cover--ribbon-triangle"></div></button></div></div></div><div class="ds-work-card--work-information"><h1 class="ds-work-card--work-title">Gauge Theories and Fiber Bundles: Definitions, Pictures, and Results</h1><div class="ds-work-card--work-authors ds-work-card--detail"><a class="ds-work-card--author js-wsj-grid-card-author ds2-5-body-md ds2-5-body-link" data-author-id="163946664" href="https://independent.academia.edu/AdamMarsh19"><img alt="Profile image of Adam Marsh" class="ds-work-card--author-avatar" src="//a.academia-assets.com/images/s65_no_pic.png" />Adam Marsh</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">2016</p><div class="ds-work-card--work-metadata"><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">visibility</span><p class="ds2-5-body-sm" id="work-metadata-view-count">…</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">description</span><p class="ds2-5-body-sm">42 pages</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">link</span><p class="ds2-5-body-sm">1 file</p></div></div><script>(async () => { const workId = 63797283; 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if (!viewCountBody) { throw new Error('Failed to find work views element'); } viewCountBody.textContent = `${commaizedViewCount} views`; } catch (error) { // Remove the whole views element if there was some issue parsing. document.getElementById('work-metadata-view-count')?.parentNode?.remove(); throw new Error(`Failed to parse view count: ${viewCount}`, error); } }; // If the DOM is still loading, wait for it to be ready before updating the view count. if (document.readyState === "loading") { document.addEventListener('DOMContentLoaded', () => { updateViewCount(viewCount); }); // Otherwise, just update it immediately. } else { updateViewCount(viewCount); } })();</script></div><p class="ds-work-card--work-abstract ds-work-card--detail ds2-5-body-md">A pedagogical but concise overview of fiber bundles and their connections is provided, in the context of gauge theories in physics. The emphasis is on defining and visualizing concepts and relationships between them, as well as listing common confusions, alternative notations and jargon, and relevant facts and theorems. Special attention is given to detailed figures and geometric viewpoints, some of which would seem to be novel to the literature. Topics are avoided which are well covered in textbooks, such as historical motivations, proofs and derivations, and tools for practical calculations. The present paper is best read in conjunction with the similar paper on Riemannian geometry cited herein.</p><div class="ds-work-card--button-container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--work-card","attachmentId":76100720,"attachmentType":"pdf","workUrl":"https://www.academia.edu/63797283/Gauge_Theories_and_Fiber_Bundles_Definitions_Pictures_and_Results"}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--work-card","attachmentId":76100720,"attachmentType":"pdf","workUrl":"https://www.academia.edu/63797283/Gauge_Theories_and_Fiber_Bundles_Definitions_Pictures_and_Results"}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div><div class="ds-signup-banner-trigger-container"><div class="ds-signup-banner-trigger ds-signup-banner-trigger-control"></div></div><div class="ds-signup-banner ds-signup-banner-control"><div id="ds-signup-banner-close-button"><button class="ds2-5-button ds2-5-button--secondary ds2-5-button--inverse"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">close</span></button></div><div class="ds-signup-banner-ctas"><img src="//a.academia-assets.com/images/academia-logo-capital-white.svg" /><h4 class="ds2-5-heading-serif-sm">Sign up for access to the world's latest research</h4><button class="ds2-5-button ds2-5-button--inverse ds2-5-button--full-width js-swp-download-button" data-signup-modal="{"location":"signup-banner"}">Sign up for free<span class="material-symbols-outlined" style="font-size: 20px" translate="no">arrow_forward</span></button></div><div class="ds-signup-banner-divider"></div><div class="ds-signup-banner-reasons"><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Get notified about relevant papers</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Save papers to use in your research</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Join the discussion with peers</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Track your impact</span></div></div></div><script>(() => { // Set up signup banner show/hide behavior: // 1. 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The main problem we face is to uphold a strong and meaningful criterion of what is physical. We like to call it &quot;Field&#39;s dilemma&quot;, referring to Hartry Field&#39;s nominalist proposal which we consider to be inadaequate. The resolution to the dilemma, we believe, is implicitly</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Fiber Bundle Gauge Theories and \"Field's Dilemma","attachmentId":48216748,"attachmentType":"pdf","work_url":"https://www.academia.edu/27921637/Fiber_Bundle_Gauge_Theories_and_Fields_Dilemma","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/27921637/Fiber_Bundle_Gauge_Theories_and_Fields_Dilemma"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="3" data-entity-id="33349643" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/33349643/INSTITUTE_FOR_QUANTUM_STUDIES_Fiber_bundle_description_of_number_scaling_in_gauge_theory_and_geometry">INSTITUTE FOR QUANTUM STUDIES Fiber bundle description of number scaling in gauge theory and geometry</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="38459710" href="https://independent.academia.edu/PBenioff">Paul Benioff</a></div><p class="ds-related-work--abstract ds2-5-body-sm">This work uses fiber bundles as a framework to describe some effects of number scaling on gauge theory and some geometric quantities. A description of number scaling and fiber bundles over a flat space time manifold, M, is followed by a description of gauge theory. A fiber at point x of M contains a pair of scaled complex number and vector space structures, ¯ C c x × ¯ V c x , for each c in GL(1, C). A space time dependent scalar field, g, determines, for each x, the scaling value of the vector space structure that contains the value, ψ(x), of a vector field at x. Vertical components of connections between neighboring fibers are taken to be the gradient field, A(x) + iB(x), of g. Abelian gauge theory for these fields gives the result that B is massless, and no mass restrictions for A. Addition of an electromagnetic field does not change these results. In the Mexican hat Higgs mechanism B combines with a Goldstone boson to create massive vector bosons, the photon field, and the Higgs field. For geometric quantities the fiber bundle is a tangent bundle with pairs, ¯ R r x × ¯ T r x for each x and nonnegative real r. B is zero everywhere. The A field affects path lengths and the proper times of clocks along paths. It also appears in the geodesic equation. The lack of physical evidence for the gradient g field means that either it couples very weakly to matter fields, or that it is close to zero for all x in a local region of cosmological space and time. It says nothing about the values outside the local region.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"INSTITUTE FOR QUANTUM STUDIES Fiber bundle description of number scaling in gauge theory and geometry","attachmentId":53408264,"attachmentType":"pdf","work_url":"https://www.academia.edu/33349643/INSTITUTE_FOR_QUANTUM_STUDIES_Fiber_bundle_description_of_number_scaling_in_gauge_theory_and_geometry","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/33349643/INSTITUTE_FOR_QUANTUM_STUDIES_Fiber_bundle_description_of_number_scaling_in_gauge_theory_and_geometry"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="4" data-entity-id="18591566" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/18591566/Fiber_bundle_description_of_number_scaling_in_gauge_theory_and_geometry">Fiber bundle description of number scaling in gauge theory and geometry</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="38459710" href="https://independent.academia.edu/PBenioff">Paul Benioff</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Quantum Studies: Mathematics and Foundations, 2015</p><p class="ds-related-work--abstract ds2-5-body-sm">This work uses fiber bundles as a framework to describe some effects of number scaling on gauge theory and some geometric quantities. A description of number scaling and fiber bundles over a flat space time manifold, M, is followed by a description of gauge theory. A fiber at point x of M contains a pair of scaled complex number and vector space structures,C c x ×V c x , for each c in GL(1, C). A space time dependent scalar field, g, determines, for each x, the scaling value of the vector space structure that contains the value, ψ(x), of a vector field at x. Vertical components of connections between neighboring fibers are taken to be the gradient field, A(x) + iB(x), of g. Abelian gauge theory for these fields gives the result that B is massless, and no mass restrictions for A. Addition of an electromagnetic field does not change these results. In the Mexican hat Higgs mechanism B combines with a Goldstone boson to create massive vector bosons, the photon field, and the Higgs field. For geometric quantities the fiber bundle is a tangent bundle with pairs,R r x ×T r x for each x and nonnegative real r . B is zero everywhere. The A field affects path lengths and the proper times of clocks along paths. It also appears in the geodesic equation. The lack of physical evidence for the gradient g field means that either it couples very weakly to matter fields, or that it is close to zero for all x in a local region of cosmological space and time. It says nothing about the values outside the local region.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Fiber bundle description of number scaling in gauge theory and geometry","attachmentId":40146001,"attachmentType":"pdf","work_url":"https://www.academia.edu/18591566/Fiber_bundle_description_of_number_scaling_in_gauge_theory_and_geometry","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/18591566/Fiber_bundle_description_of_number_scaling_in_gauge_theory_and_geometry"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="5" data-entity-id="18591572" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/18591572/Principal_fiber_bundle_description_of_number_scaling_for_scalars_and_vectors_application_to_gauge_theory">Principal fiber bundle description of number scaling for scalars and vectors: application to gauge theory</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="38459710" href="https://independent.academia.edu/PBenioff">Paul Benioff</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Quantum Information and Computation XIII, 2015</p><p class="ds-related-work--abstract ds2-5-body-sm">The purpose of this paper is to put the description of number scaling and its effects on physics and geometry on a firmer foundation, and to make it more understandable. A main point is that two different concepts, number and number value are combined in the usual representations of number structures. This is valid as long as just one structure of each number type is being considered. It is not valid when different structures of each number type are being considered. Elements of base sets of number structures, considered by themselves, have no meaning. They acquire meaning or value as elements of a number structure. Fiber bundles over a space or space time manifold, M, are described. The fiber consists of a collection of many real or complex number structures and vector space structures. The structures are parameterized by a real or complex scaling factor, s. A vector space at a fiber level, s, has, as scalars, real or complex number structures at the same level. Connections are described that relate scalar and vector space structures at both neighbor M locations and at neighbor scaling levels. Scalar and vector structure valued fields are described and covariant derivatives of these fields are obtained. Two complex vector fields, each with one real and one imaginary field, appear, with one complex field associated with positions in M and the other with position dependent scaling factors. A derivation of the covariant derivative for scalar and vector valued fields gives the same vector fields. The derivation shows that the complex vector field associated with scaling fiber levels is the gradient of a complex scalar field. Use of these results in gauge theory shows that the imaginary part of the vector field associated with M positions acts like the electromagnetic field. The physical relevance of the other three fields, if any, is not known.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Principal fiber bundle description of number scaling for scalars and vectors: application to gauge theory","attachmentId":40146046,"attachmentType":"pdf","work_url":"https://www.academia.edu/18591572/Principal_fiber_bundle_description_of_number_scaling_for_scalars_and_vectors_application_to_gauge_theory","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/18591572/Principal_fiber_bundle_description_of_number_scaling_for_scalars_and_vectors_application_to_gauge_theory"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="6" data-entity-id="101652573" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/101652573/Advanced_topics_in_gauge_theory_Mathematics_and_Physics_of_Higgs_bundles">Advanced topics in gauge theory: Mathematics and Physics of Higgs bundles</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="143477408" href="https://independent.academia.edu/LauraSchaposnik">Laura Schaposnik</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Quantum Field Theory and Manifold Invariants, 2021</p><p class="ds-related-work--abstract ds2-5-body-sm">These notes have been prepared as reading material for the mini-course given by the author at the 2019 Graduate Summer School at Park City Mathematics Institute-Institute for Advanced Study. We begin by introducing Higgs bundles and their main properties (Lecture 1), and then we discuss the Hitchin fibration and its different uses (Lecture 2). The second half of the course is dedicated to studying different types of subspaces (branes) of the moduli space of complex Higgs bundles, their appearances in terms of flat connections and representations (Lecture 3), as well as correspondences between them (Lecture 4). Contents 1 Introduction 1 2 The geometry of the moduli space of Higgs bundles 3 3 The geometry of the Hitchin fibration 15 4 Branes in the moduli space of Higgs bundles 26 5 Higgs bundles and correspondences 39</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Advanced topics in gauge theory: Mathematics and Physics of Higgs bundles","attachmentId":102134477,"attachmentType":"pdf","work_url":"https://www.academia.edu/101652573/Advanced_topics_in_gauge_theory_Mathematics_and_Physics_of_Higgs_bundles","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/101652573/Advanced_topics_in_gauge_theory_Mathematics_and_Physics_of_Higgs_bundles"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="7" data-entity-id="125428687" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/125428687/Multi_Fibers_Bundles_as_a_new_model_for_high_dimensional_Spacetimes">Multi-Fibers Bundles as a new model for high-dimensional Spacetimes</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="323122585" href="https://independent.academia.edu/StephaneCollion">Stephane Collion</a></div><p class="ds-related-work--metadata ds2-5-body-xs">arXiv (Cornell University), 2017</p><p class="ds-related-work--abstract ds2-5-body-sm">Around 1920, Kaluza and Klein had the idea to add a fifth dimension to the classical 4-dimensional spacetime of general relativity to create a geometric theory of gravitation and electromagnetism. Today, theoretical evidences, like string theory, suggest the need for a spacetime with more than five dimensions. The mathematical translation of the heuristic idea of a 4-dimensional classical spacetime equipped with extra "small" dimensions, is a fiber bundle structure π : M → M on a (4 + k)-dimensional manifold M , with fiber a compact manifold F of dimension k, more shortly a F -fibration. Kaluza and Klein used a fibration with fiber the standard circle S 1 , this fiber carrying the electromagnetic potential. Inclusion of other physical interaction would therefore require a F -fibration with F of the form F = S 1 × W , W being a compact manifold. However, with such a F -fibration, they is no naturally defined fiber diffeomorphic to S 1 at each point of the manifold, and therefore one looses the possibility to define simply the electromagnetic potential. We want to present in this paper a mathematical structure generalizing the fiber bundle, that enable the possible definition of multiple naturally defined fibers at each point of the manifold, on which therefore one can define objects that depend only on one of the components, S 1 or W , of the global (S 1 × W ) fiber. Although we do not pretend here to model precisely other known physical interactions, we present this geometric structure as a possible way to model or encode deviations from standard 4-dimensional General Relativity, or "dark" effects such as dark matter or energy ; (we refer to the authors' article from which this paper is extracted, but whose purpose is different). Also this geometry was a starting point for the second author's new approach to a geometric unification of General Relativity and Quantum Physics ( see ). 1 Full details of this use of the multi-fibers structure can be found in [3] : S. Collion, M. Vaugon, A New Approach to Kaluza-Klein theory. arXiv.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Multi-Fibers Bundles as a new model for high-dimensional Spacetimes","attachmentId":119473777,"attachmentType":"pdf","work_url":"https://www.academia.edu/125428687/Multi_Fibers_Bundles_as_a_new_model_for_high_dimensional_Spacetimes","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/125428687/Multi_Fibers_Bundles_as_a_new_model_for_high_dimensional_Spacetimes"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="8" data-entity-id="101171572" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/101171572/Aspects_Of_Gauge_Field_Theory">Aspects Of Gauge Field Theory</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="267962040" href="https://independent.academia.edu/TarunPalla1">Tarun Palla</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Deccan Education Society's Willingdon College,Sangli, 2022</p><p class="ds-related-work--abstract ds2-5-body-sm">To describe successfully the properties and dynamics of the particle, we need field theory. The dynamics of gauge bosons are described by Gauge Field Theory, a type of quantum field theory. In this project, we work in four-dimensional Minkowski space. Gauge field theory is characterized by the presence of a vector field, we defined the Lagrangian for the gauge field (photon field) and showed that it is Lorentz invariant and gauge invariant. Any physical theory must be Lorentz invariant. We derived the equation of motion for the gauge field and derived all Maxwell's equations in terms of the electric and magnetic fields. We check the invariance of the action under global and local symmetry transformations using Noether's theorem. We derived the stress-energy tensor that gives us energy density, and the momentum of the field and determined the conserved charges associated with the gauge field. We then quantize the gauge field by treating each field as a harmonic oscillator. We derived the expression for vector field in terms of annihilation and creation operators. The gauge invariance implies that in four dimensions the photon has only two physical degrees of freedom</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Aspects Of Gauge Field Theory","attachmentId":101785541,"attachmentType":"pdf","work_url":"https://www.academia.edu/101171572/Aspects_Of_Gauge_Field_Theory","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/101171572/Aspects_Of_Gauge_Field_Theory"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="9" data-entity-id="16939010" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/16939010/Fiber_bundles_and_Kaluza_Klein_theory">Fiber bundles and Kaluza-Klein theory</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="1052417" href="https://ronininstitute.academia.edu/ArkadiuszJadczyk">Arkadiusz Jadczyk</a></div><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Fiber bundles and Kaluza-Klein theory","attachmentId":39267265,"attachmentType":"pdf","work_url":"https://www.academia.edu/16939010/Fiber_bundles_and_Kaluza_Klein_theory","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/16939010/Fiber_bundles_and_Kaluza_Klein_theory"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div></div></div><div class="ds-sticky-ctas--wrapper js-loswp-sticky-ctas hidden"><div class="ds-sticky-ctas--grid-container"><div class="ds-sticky-ctas--container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--sticky-ctas","attachmentId":76100720,"attachmentType":"pdf","workUrl":null}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--sticky-ctas","attachmentId":76100720,"attachmentType":"pdf","workUrl":null}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div></div></div><div class="ds-below-fold--grid-container"><div class="ds-work--container js-loswp-embedded-document"><div class="attachment_preview" data-attachment="Attachment_76100720" style="display: none"><div class="js-scribd-document-container"><div class="scribd--document-loading js-scribd-document-loader" style="display: block;"><img alt="Loading..." src="//a.academia-assets.com/images/loaders/paper-load.gif" /><p>Loading Preview</p></div></div><div style="text-align: center;"><div class="scribd--no-preview-alert js-preview-unavailable"><p>Sorry, preview is currently unavailable. 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class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="3" data-entity-id="125846009" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/125846009/Fibre_bundles_connections_general_relativity_and_Einstein_Cartan_theory">Fibre bundles, connections, general relativity, and Einstein-Cartan theory</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="329124018" href="https://independent.academia.edu/SocolovskyM">Miguel Socolovsky</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2011</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Fibre bundles, connections, general relativity, and Einstein-Cartan 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ds2-5-body-link" data-author-id="58820257" href="https://independent.academia.edu/BernsteinHerbert">Herbert Bernstein</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Scientific American, 1981</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Fiber Bundles and Quantum Theory","attachmentId":51346398,"attachmentType":"pdf","work_url":"https://www.academia.edu/30921559/Fiber_Bundles_and_Quantum_Theory","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/30921559/Fiber_Bundles_and_Quantum_Theory"><span class="ds2-5-text-link__content">View PDF</span><span 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href="https://www.academia.edu/92715225/Extended_Gauge_Principle_and_Quantization_of_Gauge_Theories">Extended Gauge Principle and Quantization of Gauge Theories</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="149267844" href="https://independent.academia.edu/JorgeAndr%C3%A9sDevoto">Jorge Andrés Devoto</a></div><p class="ds-related-work--metadata ds2-5-body-xs">AIP Conference Proceedings, 2006</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Extended Gauge Principle and Quantization of Gauge Theories","attachmentId":95651022,"attachmentType":"pdf","work_url":"https://www.academia.edu/92715225/Extended_Gauge_Principle_and_Quantization_of_Gauge_Theories","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" 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data-author-id="25024933" href="https://perimeterinstitute.academia.edu/HenriqueGomes">Henrique Gomes</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2008</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"A Gauge-theoretical Treatment of the Gravitational Field: Classical","attachmentId":86067491,"attachmentType":"pdf","work_url":"https://www.academia.edu/79309023/A_Gauge_theoretical_Treatment_of_the_Gravitational_Field_Classical","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/79309023/A_Gauge_theoretical_Treatment_of_the_Gravitational_Field_Classical"><span 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href="https://www.academia.edu/101652574/Higgs_Bundles_Recent_Applications">Higgs Bundles—Recent Applications</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="143477408" href="https://independent.academia.edu/LauraSchaposnik">Laura Schaposnik</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Notices of the American Mathematical Society</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Higgs Bundles—Recent Applications","attachmentId":102134476,"attachmentType":"pdf","work_url":"https://www.academia.edu/101652574/Higgs_Bundles_Recent_Applications","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a 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class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="57886487" href="https://sissa.academia.edu/cesarereina">cesare reina</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Geometry and Physics, 1984</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"The action of the group of bundle-automorphisms on the space of connections and the geometry of gauge theories","attachmentId":50879984,"attachmentType":"pdf","work_url":"https://www.academia.edu/30437517/The_action_of_the_group_of_bundle_automorphisms_on_the_space_of_connections_and_the_geometry_of_gauge_theories","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free 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class="ds-related-work--metadata ds2-5-body-xs">1999</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Gauges, Holes, and their `Connections","attachmentId":48216747,"attachmentType":"pdf","work_url":"https://www.academia.edu/27921636/Gauges_Holes_and_their_Connections","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/27921636/Gauges_Holes_and_their_Connections"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container 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class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="38752823" href="https://ucsc.academia.edu/ArthurFischer">Arthur Fischer</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Communications in Mathematical Physics, 1987</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"The internal symmetry group of a connection on a principal fiber bundle with applications to gauge field theories","attachmentId":105123373,"attachmentType":"pdf","work_url":"https://www.academia.edu/105055160/The_internal_symmetry_group_of_a_connection_on_a_principal_fiber_bundle_with_applications_to_gauge_field_theories","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a 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