CINXE.COM

<!DOCTYPE html> <html lang="en" class="no-js"> <head> <meta charset="UTF-8"> <meta http-equiv="X-UA-Compatible" content="IE=edge"> <meta name="applicable-device" content="pc,mobile"> <meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="robots" content="max-image-preview:large"> <meta name="access" content="Yes"> <meta name="360-site-verification" content="1268d79b5e96aecf3ff2a7dac04ad990" /> <title>The Poincare Lemma, Antiexact Forms, and Fermionic Quantum Harmonic Oscillator | Results in Mathematics </title> <meta name="twitter:site" content="@SpringerLink"/> <meta name="twitter:card" content="summary_large_image"/> <meta name="twitter:image:alt" content="Content cover image"/> <meta name="twitter:title" content="The Poincare Lemma, Antiexact Forms, and Fermionic Quantum Harmonic Oscillator"/> <meta name="twitter:description" content="Results in Mathematics - The paper focuses on various properties and applications of the homotopy operator, which occurs in the Poincaré lemma. In the first part, an abstract operator calculus..."/> <meta name="twitter:image" content="https://static-content.springer.com/image/art%3A10.1007%2Fs00025-020-01247-8/MediaObjects/25_2020_1247_Fig1_HTML.png"/> <meta name="journal_id" content="25"/> <meta name="dc.title" content="The Poincare Lemma, Antiexact Forms, and Fermionic Quantum Harmonic Oscillator"/> <meta name="dc.source" content="Results in Mathematics 2020 75:3"/> <meta name="dc.format" content="text/html"/> <meta name="dc.publisher" content="Springer"/> <meta name="dc.date" content="2020-07-11"/> <meta name="dc.type" content="OriginalPaper"/> <meta name="dc.language" content="En"/> <meta name="dc.copyright" content="2020 The Author(s)"/> <meta name="dc.rights" content="2020 The Author(s)"/> <meta name="dc.rightsAgent" content="journalpermissions@springernature.com"/> <meta name="dc.description" content="The paper focuses on various properties and applications of the homotopy operator, which occurs in the Poincaré lemma. In the first part, an abstract operator calculus is constructed, where the exterior derivative is an abstract derivative and the homotopy operator plays the role of an abstract integral. This operator calculus can be used to formulate abstract differential equations. An example of the eigenvalue problem that resembles the fermionic quantum harmonic oscillator is presented. The second part presents the dual complex to the Dolbeault bicomplex generated by the homotopy operator on complex manifolds."/> <meta name="prism.issn" content="1420-9012"/> <meta name="prism.publicationName" content="Results in Mathematics"/> <meta name="prism.publicationDate" content="2020-07-11"/> <meta name="prism.volume" content="75"/> <meta name="prism.number" content="3"/> <meta name="prism.section" content="OriginalPaper"/> <meta name="prism.startingPage" content="1"/> <meta name="prism.endingPage" content="14"/> <meta name="prism.copyright" content="2020 The Author(s)"/> <meta name="prism.rightsAgent" content="journalpermissions@springernature.com"/> <meta name="prism.url" content="https://link.springer.com/article/10.1007/s00025-020-01247-8"/> <meta name="prism.doi" content="doi:10.1007/s00025-020-01247-8"/> <meta name="citation_pdf_url" content="https://link.springer.com/content/pdf/10.1007/s00025-020-01247-8.pdf"/> <meta name="citation_fulltext_html_url" content="https://link.springer.com/article/10.1007/s00025-020-01247-8"/> <meta name="citation_journal_title" content="Results in Mathematics"/> <meta name="citation_journal_abbrev" content="Results Math"/> <meta name="citation_publisher" content="Springer International Publishing"/> <meta name="citation_issn" content="1420-9012"/> <meta name="citation_title" content="The Poincare Lemma, Antiexact Forms, and Fermionic Quantum Harmonic Oscillator"/> <meta name="citation_volume" content="75"/> <meta name="citation_issue" content="3"/> <meta name="citation_publication_date" content="2020/09"/> <meta name="citation_online_date" content="2020/07/11"/> <meta name="citation_firstpage" content="1"/> <meta name="citation_lastpage" content="14"/> <meta name="citation_article_type" content="Article"/> <meta name="citation_fulltext_world_readable" content=""/> <meta name="citation_language" content="en"/> <meta name="dc.identifier" content="doi:10.1007/s00025-020-01247-8"/> <meta name="DOI" content="10.1007/s00025-020-01247-8"/> <meta name="size" content="231302"/> <meta name="citation_doi" content="10.1007/s00025-020-01247-8"/> <meta name="citation_springer_api_url" content="http://api.springer.com/xmldata/jats?q=doi:10.1007/s00025-020-01247-8&api_key="/> <meta name="description" content="The paper focuses on various properties and applications of the homotopy operator, which occurs in the Poincaré lemma. In the first part, an abstract "/> <meta name="dc.creator" content="Kycia, Radosław Antoni"/> <meta name="dc.subject" content="Mathematics, general"/> <meta name="citation_reference" content="citation_journal_title=J. Am. Math. Soc.; citation_title=p-adic periods and derived de Rham cohomology; citation_author=A Beilinson; citation_volume=25; citation_publication_date=2012; citation_pages=715-738; citation_doi=10.1090/S0894-0347-2012-00729-2; citation_id=CR1"/> <meta name="citation_reference" content="citation_journal_title=Stud. Math.; citation_title=Operational calculus in linear spaces; citation_author=R Bittner; citation_volume=20; citation_publication_date=1961; citation_pages=1-18; citation_doi=10.4064/sm-20-1-1-18; citation_id=CR2"/> <meta name="citation_reference" content="citation_title=Differential Forms in Algebraic Topology; citation_publication_date=1995; citation_id=CR3; citation_author=R Bott; citation_author=LW Tu; citation_publisher=Springer"/> <meta name="citation_reference" content="citation_title=Field Theory, A Path Integral Approach; citation_publication_date=1993; citation_id=CR4; citation_author=A Das; citation_publisher=World Scientific"/> <meta name="citation_reference" content="citation_journal_title=Appl. Numer. Math.; citation_title=Discrete Poincaré lemma; citation_author=M Desbrun, M Leok, JE Marsden; citation_volume=53; citation_issue=2–4; citation_publication_date=2005; citation_pages=231-248; citation_doi=10.1016/j.apnum.2004.09.035; citation_id=CR5"/> <meta name="citation_reference" content="Desbrun, M., Hirani, A.N., Leok, M., Marsden, J.E.: Discrete Exterior Calculus, arXiv:math/0508341 [math.DG]"/> <meta name="citation_reference" content="citation_journal_title=Ill. J. Math.; citation_title=Weighted Poincaré inequalities for solutions to A-harmonic equations; citation_author=S Ding, CA Nolder; citation_volume=46; citation_issue=1; citation_publication_date=2002; citation_pages=199-205; citation_doi=10.1215/ijm/1258136150; citation_id=CR7"/> <meta name="citation_reference" content="citation_title=Applied Exterior Calculus; citation_publication_date=2011; citation_id=CR8; citation_author=DGB Edelen; citation_publisher=Dover Publications"/> <meta name="citation_reference" content="citation_title=Isovector Methods for Equations of Balance; citation_publication_date=1980; citation_id=CR9; citation_author=DGB Edelen; citation_publisher=Springer"/> <meta name="citation_reference" content="citation_journal_title=J. Geom. Anal.; citation_title=Operator calculus of differential chains and differential forms; citation_author=J Harrison; citation_volume=25; citation_issue=1; citation_publication_date=2015; citation_pages=357-420; citation_doi=10.1007/s12220-013-9433-6; citation_id=CR10"/> <meta name="citation_reference" content="Harrison, J.: Geometric Poincaré Lemma, arXiv:1101.0313 [math.AT]"/> <meta name="citation_reference" content="citation_journal_title=Arch. Ration. Mech. Anal.; citation_title=Integral estimates for null Lagrangians; citation_author=T Iwaniec, A Lutoborski; citation_volume=125; citation_publication_date=1993; citation_pages=25-79; citation_doi=10.1007/BF00411477; citation_id=CR12"/> <meta name="citation_reference" content="citation_journal_title=Math. Z.; citation_title=A Poincaré lemma for real-valued differential forms on Berkovich spaces; citation_author=P Jell; citation_volume=282; citation_publication_date=2016; citation_pages=1149-1167; citation_doi=10.1007/s00209-015-1583-8; citation_id=CR13"/> <meta name="citation_reference" content="citation_title=Introduction to Smooth Manifolds; citation_publication_date=2012; citation_id=CR14; citation_author=J Lee; citation_publisher=Springer"/> <meta name="citation_reference" content="Lesfari, A.: On Poincaré lemma or Volterra theorem about differential forms and cohomology groups, arXiv:1905.13347 [math.GM]"/> <meta name="citation_reference" content="citation_title=Geometry, Topology and Physics; citation_publication_date=2003; citation_id=CR16; citation_author=M Nakahara; citation_publisher=CRC Press"/> <meta name="citation_reference" content="citation_journal_title=Proc. Amer. Math. Soc.; citation_title=On a non-Abelian Poincaré lemma; citation_author=T Voronov; citation_volume=140; citation_publication_date=2012; citation_pages=2855-2872; citation_doi=10.1090/S0002-9939-2011-11116-X; citation_id=CR17"/> <meta name="citation_reference" content="citation_title=An Introduction to Manifolds; citation_publication_date=2010; citation_id=CR18; citation_author=LW Tu; citation_publisher=Springer"/> <meta name="citation_author" content="Kycia, Radosław Antoni"/> <meta name="citation_author_email" content="kycia.radoslaw@gmail.com"/> <meta name="citation_author_institution" content="Department of Mathematics and Statistics, Masaryk University, Brno, Czech Republic"/> <meta name="citation_author_institution" content="Faculty of Materials Engineering and Physics, Cracow University of Technology, Kraków, Poland"/> <meta name="format-detection" content="telephone=no"/> <meta name="citation_cover_date" content="2020/09/01"/> <meta property="og:url" content="https://link.springer.com/article/10.1007/s00025-020-01247-8"/> <meta property="og:type" content="article"/> <meta property="og:site_name" content="SpringerLink"/> <meta property="og:title" content="The Poincare Lemma, Antiexact Forms, and Fermionic Quantum Harmonic Oscillator - Results in Mathematics"/> <meta property="og:description" content="The paper focuses on various properties and applications of the homotopy operator, which occurs in the Poincaré lemma. In the first part, an abstract operator calculus is constructed, where the exterior derivative is an abstract derivative and the homotopy operator plays the role of an abstract integral. This operator calculus can be used to formulate abstract differential equations. An example of the eigenvalue problem that resembles the fermionic quantum harmonic oscillator is presented. The second part presents the dual complex to the Dolbeault bicomplex generated by the homotopy operator on complex manifolds."/> <meta property="og:image" content="https://static-content.springer.com/image/art%3A10.1007%2Fs00025-020-01247-8/MediaObjects/25_2020_1247_Fig1_HTML.png"/> <meta name="format-detection" content="telephone=no"> <link rel="apple-touch-icon" sizes="180x180" href=/oscar-static/img/favicons/darwin/apple-touch-icon-92e819bf8a.png> <link rel="icon" type="image/png" sizes="192x192" href=/oscar-static/img/favicons/darwin/android-chrome-192x192-6f081ca7e5.png> <link rel="icon" type="image/png" sizes="32x32" href=/oscar-static/img/favicons/darwin/favicon-32x32-1435da3e82.png> <link rel="icon" type="image/png" sizes="16x16" href=/oscar-static/img/favicons/darwin/favicon-16x16-ed57f42bd2.png> <link rel="shortcut icon" data-test="shortcut-icon" href=/oscar-static/img/favicons/darwin/favicon-c6d59aafac.ico> <meta name="theme-color" content="#e6e6e6">   <link rel="stylesheet" media="print" href=/oscar-static/app-springerlink/css/print-b8af42253b.css> <style> html{text-size-adjust:100%;line-height:1.15}body{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;line-height:1.8;margin:0}details,main{display:block}h1{font-size:2em;margin:.67em 0}a{background-color:transparent;color:#025e8d}sub{bottom:-.25em;font-size:75%;line-height:0;position:relative;vertical-align:baseline}img{border:0;height:auto;max-width:100%;vertical-align:middle}button,input{font-family:inherit;font-size:100%;line-height:1.15;margin:0;overflow:visible}button{text-transform:none}[type=button],[type=submit],button{-webkit-appearance:button}[type=search]{-webkit-appearance:textfield;outline-offset:-2px}summary{display:list-item}[hidden]{display:none}button{cursor:pointer}svg{height:1rem;width:1rem} </style> <style>@media only print, only all and (prefers-color-scheme: no-preference), only all and (prefers-color-scheme: light), only all and (prefers-color-scheme: dark) { body{background:#fff;color:#222;font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;line-height:1.8;min-height:100%}a{color:#025e8d;text-decoration:underline;text-decoration-skip-ink:auto}button{cursor:pointer}img{border:0;height:auto;max-width:100%;vertical-align:middle}html{box-sizing:border-box;font-size:100%;height:100%;overflow-y:scroll}h1{font-size:2.25rem}h2{font-size:1.75rem}h1,h2,h4{font-weight:700;line-height:1.2}h4{font-size:1.25rem}body{font-size:1.125rem}*{box-sizing:inherit}p{margin-bottom:2rem;margin-top:0}p:last-of-type{margin-bottom:0}.c-ad{text-align:center}@media only screen and (min-width:480px){.c-ad{padding:8px}}.c-ad--728x90{display:none}.c-ad--728x90 .c-ad__inner{min-height:calc(1.5em + 94px)}@media only screen and (min-width:876px){.js .c-ad--728x90{display:none}}.c-ad__label{color:#333;font-size:.875rem;font-weight:400;line-height:1.5;margin-bottom:4px}.c-ad__label,.c-status-message{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif}.c-status-message{align-items:center;box-sizing:border-box;display:flex;position:relative;width:100%}.c-status-message :last-child{margin-bottom:0}.c-status-message--boxed{background-color:#fff;border:1px solid #ccc;line-height:1.4;padding:16px}.c-status-message__heading{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;font-size:.875rem;font-weight:700}.c-status-message__icon{fill:currentcolor;display:inline-block;flex:0 0 auto;height:1.5em;margin-right:8px;transform:translate(0);vertical-align:text-top;width:1.5em}.c-status-message__icon--top{align-self:flex-start}.c-status-message--info .c-status-message__icon{color:#003f8d}.c-status-message--boxed.c-status-message--info{border-bottom:4px solid #003f8d}.c-status-message--error .c-status-message__icon{color:#c40606}.c-status-message--boxed.c-status-message--error{border-bottom:4px solid #c40606}.c-status-message--success .c-status-message__icon{color:#00b8b0}.c-status-message--boxed.c-status-message--success{border-bottom:4px solid #00b8b0}.c-status-message--warning .c-status-message__icon{color:#edbc53}.c-status-message--boxed.c-status-message--warning{border-bottom:4px solid #edbc53}.eds-c-header{background-color:#fff;border-bottom:2px solid #01324b;font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;font-size:1rem;line-height:1.5;padding:8px 0 0}.eds-c-header__container{align-items:center;display:flex;flex-wrap:nowrap;gap:8px 16px;justify-content:space-between;margin:0 auto 8px;max-width:1280px;padding:0 8px;position:relative}.eds-c-header__nav{border-top:2px solid #c5e0f4;padding-top:4px;position:relative}.eds-c-header__nav-container{align-items:center;display:flex;flex-wrap:wrap;margin:0 auto 4px;max-width:1280px;padding:0 8px;position:relative}.eds-c-header__nav-container>:not(:last-child){margin-right:32px}.eds-c-header__link-container{align-items:center;display:flex;flex:1 0 auto;gap:8px 16px;justify-content:space-between}.eds-c-header__list{list-style:none;margin:0;padding:0}.eds-c-header__list-item{font-weight:700;margin:0 auto;max-width:1280px;padding:8px}.eds-c-header__list-item:not(:last-child){border-bottom:2px solid #c5e0f4}.eds-c-header__item{color:inherit}@media only screen and (min-width:768px){.eds-c-header__item--menu{display:none;visibility:hidden}.eds-c-header__item--menu:first-child+*{margin-block-start:0}}.eds-c-header__item--inline-links{display:none;visibility:hidden}@media only screen and (min-width:768px){.eds-c-header__item--inline-links{display:flex;gap:16px 16px;visibility:visible}}.eds-c-header__item--divider:before{border-left:2px solid #c5e0f4;content:"";height:calc(100% - 16px);margin-left:-15px;position:absolute;top:8px}.eds-c-header__brand{padding:16px 8px}.eds-c-header__brand a{display:block;line-height:1;text-decoration:none}.eds-c-header__brand img{height:1.5rem;width:auto}.eds-c-header__link{color:inherit;display:inline-block;font-weight:700;padding:16px 8px;position:relative;text-decoration-color:transparent;white-space:nowrap;word-break:normal}.eds-c-header__icon{fill:currentcolor;display:inline-block;font-size:1.5rem;height:1em;transform:translate(0);vertical-align:bottom;width:1em}.eds-c-header__icon+*{margin-left:8px}.eds-c-header__expander{background-color:#f0f7fc}.eds-c-header__search{display:block;padding:24px 0}@media only screen and (min-width:768px){.eds-c-header__search{max-width:70%}}.eds-c-header__search-container{position:relative}.eds-c-header__search-label{color:inherit;display:inline-block;font-weight:700;margin-bottom:8px}.eds-c-header__search-input{background-color:#fff;border:1px solid #000;padding:8px 48px 8px 8px;width:100%}.eds-c-header__search-button{background-color:transparent;border:0;color:inherit;height:100%;padding:0 8px;position:absolute;right:0}.has-tethered.eds-c-header__expander{border-bottom:2px solid #01324b;left:0;margin-top:-2px;top:100%;width:100%;z-index:10}@media only screen and (min-width:768px){.has-tethered.eds-c-header__expander--menu{display:none;visibility:hidden}}.has-tethered .eds-c-header__heading{display:none;visibility:hidden}.has-tethered .eds-c-header__heading:first-child+*{margin-block-start:0}.has-tethered .eds-c-header__search{margin:auto}.eds-c-header__heading{margin:0 auto;max-width:1280px;padding:16px 16px 0}.eds-c-pagination{align-items:center;display:flex;flex-wrap:wrap;font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;font-size:.875rem;gap:16px 0;justify-content:center;line-height:1.4;list-style:none;margin:0;padding:32px 0}@media only screen and (min-width:480px){.eds-c-pagination{padding:32px 16px}}.eds-c-pagination__item{margin-right:8px}.eds-c-pagination__item--prev{margin-right:16px}.eds-c-pagination__item--next .eds-c-pagination__link,.eds-c-pagination__item--prev .eds-c-pagination__link{padding:16px 8px}.eds-c-pagination__item--next{margin-left:8px}.eds-c-pagination__item:last-child{margin-right:0}.eds-c-pagination__link{align-items:center;color:#222;cursor:pointer;display:inline-block;font-size:1rem;margin:0;padding:16px 24px;position:relative;text-align:center;transition:all .2s ease 0s}.eds-c-pagination__link:visited{color:#222}.eds-c-pagination__link--disabled{border-color:#555;color:#555;cursor:default}.eds-c-pagination__link--active{background-color:#01324b;background-image:none;border-radius:8px;color:#fff}.eds-c-pagination__link--active:focus,.eds-c-pagination__link--active:hover,.eds-c-pagination__link--active:visited{color:#fff}.eds-c-pagination__link-container{align-items:center;display:flex}.eds-c-pagination__icon{fill:#222;height:1.5rem;width:1.5rem}.eds-c-pagination__icon--disabled{fill:#555}.eds-c-pagination__visually-hidden{clip:rect(0,0,0,0);border:0;clip-path:inset(50%);height:1px;overflow:hidden;padding:0;position:absolute!important;white-space:nowrap;width:1px}.c-breadcrumbs{color:#333;font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;font-size:1rem;list-style:none;margin:0;padding:0}.c-breadcrumbs>li{display:inline}svg.c-breadcrumbs__chevron{fill:#333;height:10px;margin:0 .25rem;width:10px}.c-breadcrumbs--contrast,.c-breadcrumbs--contrast .c-breadcrumbs__link{color:#fff}.c-breadcrumbs--contrast svg.c-breadcrumbs__chevron{fill:#fff}@media only screen and (max-width:479px){.c-breadcrumbs .c-breadcrumbs__item{display:none}.c-breadcrumbs .c-breadcrumbs__item:last-child,.c-breadcrumbs .c-breadcrumbs__item:nth-last-child(2){display:inline}}.c-skip-link{background:#01324b;bottom:auto;color:#fff;font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;font-size:1rem;padding:8px;position:absolute;text-align:center;transform:translateY(-100%);width:100%;z-index:9999}@media (prefers-reduced-motion:reduce){.c-skip-link{transition:top .3s ease-in-out 0s}}@media print{.c-skip-link{display:none}}.c-skip-link:active,.c-skip-link:hover,.c-skip-link:link,.c-skip-link:visited{color:#fff}.c-skip-link:focus{transform:translateY(0)}.l-with-sidebar{display:flex;flex-wrap:wrap}.l-with-sidebar>*{margin:0}.l-with-sidebar__sidebar{flex-basis:var(--with-sidebar--basis,400px);flex-grow:1}.l-with-sidebar>:not(.l-with-sidebar__sidebar){flex-basis:0px;flex-grow:999;min-width:var(--with-sidebar--min,53%)}.l-with-sidebar>:first-child{padding-right:4rem}@supports (gap:1em){.l-with-sidebar>:first-child{padding-right:0}.l-with-sidebar{gap:var(--with-sidebar--gap,4rem)}}.c-header__link{color:inherit;display:inline-block;font-weight:700;padding:16px 8px;position:relative;text-decoration-color:transparent;white-space:nowrap;word-break:normal}.app-masthead__colour-4{--background-color:#ff9500;--gradient-light:rgba(0,0,0,.5);--gradient-dark:rgba(0,0,0,.8)}.app-masthead{background:var(--background-color,#0070a8);position:relative}.app-masthead:after{background:radial-gradient(circle at top right,var(--gradient-light,rgba(0,0,0,.4)),var(--gradient-dark,rgba(0,0,0,.7)));bottom:0;content:"";left:0;position:absolute;right:0;top:0}@media only screen and (max-width:479px){.app-masthead:after{background:linear-gradient(225deg,var(--gradient-light,rgba(0,0,0,.4)),var(--gradient-dark,rgba(0,0,0,.7)))}}.app-masthead__container{color:var(--masthead-color,#fff);margin:0 auto;max-width:1280px;padding:0 16px;position:relative;z-index:1}.u-button{align-items:center;background-color:#01324b;background-image:none;border:4px solid transparent;border-radius:32px;cursor:pointer;display:inline-flex;font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;font-size:.875rem;font-weight:700;justify-content:center;line-height:1.3;margin:0;padding:16px 32px;position:relative;transition:all .2s ease 0s;width:auto}.u-button svg,.u-button--contrast svg,.u-button--primary svg,.u-button--secondary svg,.u-button--tertiary svg{fill:currentcolor}.u-button,.u-button:visited{color:#fff}.u-button,.u-button:hover{box-shadow:0 0 0 1px #01324b;text-decoration:none}.u-button:hover{border:4px solid #fff}.u-button:focus{border:4px solid #fc0;box-shadow:none;outline:0;text-decoration:none}.u-button:focus,.u-button:hover{background-color:#fff;background-image:none;color:#01324b}.app-masthead--pastel .c-pdf-download .u-button--primary:focus svg path,.app-masthead--pastel .c-pdf-download .u-button--primary:hover svg path,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--primary:focus svg path,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--primary:hover svg path,.u-button--primary:focus svg path,.u-button--primary:hover svg path,.u-button:focus svg path,.u-button:hover svg path{fill:#01324b}.u-button--primary{background-color:#01324b;background-image:none;border:4px solid transparent;box-shadow:0 0 0 1px #01324b;color:#fff;font-weight:700}.u-button--primary:visited{color:#fff}.u-button--primary:hover{border:4px solid #fff;box-shadow:0 0 0 1px #01324b;text-decoration:none}.u-button--primary:focus{border:4px solid #fc0;box-shadow:none;outline:0;text-decoration:none}.u-button--primary:focus,.u-button--primary:hover{background-color:#fff;background-image:none;color:#01324b}.u-button--secondary{background-color:#fff;border:4px solid #fff;color:#01324b;font-weight:700}.u-button--secondary:visited{color:#01324b}.u-button--secondary:hover{border:4px solid #01324b;box-shadow:none}.u-button--secondary:focus,.u-button--secondary:hover{background-color:#01324b;color:#fff}.app-masthead--pastel .c-pdf-download .u-button--secondary:focus svg path,.app-masthead--pastel .c-pdf-download .u-button--secondary:hover svg path,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--secondary:focus svg path,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--secondary:hover svg path,.u-button--secondary:focus svg path,.u-button--secondary:hover svg path,.u-button--tertiary:focus svg path,.u-button--tertiary:hover svg path{fill:#fff}.u-button--tertiary{background-color:#ebf1f5;border:4px solid transparent;box-shadow:none;color:#666;font-weight:700}.u-button--tertiary:visited{color:#666}.u-button--tertiary:hover{border:4px solid #01324b;box-shadow:none}.u-button--tertiary:focus,.u-button--tertiary:hover{background-color:#01324b;color:#fff}.u-button--contrast{background-color:transparent;background-image:none;color:#fff;font-weight:400}.u-button--contrast:visited{color:#fff}.u-button--contrast,.u-button--contrast:focus,.u-button--contrast:hover{border:4px solid #fff}.u-button--contrast:focus,.u-button--contrast:hover{background-color:#fff;background-image:none;color:#000}.u-button--contrast:focus svg path,.u-button--contrast:hover svg path{fill:#000}.u-button--disabled,.u-button:disabled{background-color:transparent;background-image:none;border:4px solid #ccc;color:#000;cursor:default;font-weight:400;opacity:.7}.u-button--disabled svg,.u-button:disabled svg{fill:currentcolor}.u-button--disabled:visited,.u-button:disabled:visited{color:#000}.u-button--disabled:focus,.u-button--disabled:hover,.u-button:disabled:focus,.u-button:disabled:hover{border:4px solid #ccc;text-decoration:none}.u-button--disabled:focus,.u-button--disabled:hover,.u-button:disabled:focus,.u-button:disabled:hover{background-color:transparent;background-image:none;color:#000}.u-button--disabled:focus svg path,.u-button--disabled:hover svg path,.u-button:disabled:focus svg path,.u-button:disabled:hover svg path{fill:#000}.u-button--small,.u-button--xsmall{font-size:.875rem;padding:2px 8px}.u-button--small{padding:8px 16px}.u-button--large{font-size:1.125rem;padding:10px 35px}.u-button--full-width{display:flex;width:100%}.u-button--icon-left svg{margin-right:8px}.u-button--icon-right svg{margin-left:8px}.u-clear-both{clear:both}.u-container{margin:0 auto;max-width:1280px;padding:0 16px}.u-justify-content-space-between{justify-content:space-between}.u-display-none{display:none}.js .u-js-hide,.u-hide{display:none;visibility:hidden}.u-visually-hidden{clip:rect(0,0,0,0);border:0;clip-path:inset(50%);height:1px;overflow:hidden;padding:0;position:absolute!important;white-space:nowrap;width:1px}.u-icon{fill:currentcolor;display:inline-block;height:1em;transform:translate(0);vertical-align:text-top;width:1em}.u-list-reset{list-style:none;margin:0;padding:0}.u-ma-16{margin:16px}.u-mt-0{margin-top:0}.u-mt-24{margin-top:24px}.u-mt-32{margin-top:32px}.u-mb-8{margin-bottom:8px}.u-mb-32{margin-bottom:32px}.u-button-reset{background-color:transparent;border:0;padding:0}.u-sans-serif{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif}.u-serif{font-family:Merriweather,serif}h1,h2,h4{-webkit-font-smoothing:antialiased}p{overflow-wrap:break-word;word-break:break-word}.u-h4{font-size:1.25rem;font-weight:700;line-height:1.2}.u-mbs-0{margin-block-start:0!important}.c-article-header{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif}.c-article-identifiers{color:#6f6f6f;display:flex;flex-wrap:wrap;font-size:1rem;line-height:1.3;list-style:none;margin:0 0 8px;padding:0}.c-article-identifiers__item{border-right:1px solid #6f6f6f;list-style:none;margin-right:8px;padding-right:8px}.c-article-identifiers__item:last-child{border-right:0;margin-right:0;padding-right:0}@media only screen and (min-width:876px){.c-article-title{font-size:1.875rem;line-height:1.2}}.c-article-author-list{display:inline;font-size:1rem;list-style:none;margin:0 8px 0 0;padding:0;width:100%}.c-article-author-list__item{display:inline;padding-right:0}.c-article-author-list__show-more{display:none;margin-right:4px}.c-article-author-list__button,.js .c-article-author-list__item--hide,.js .c-article-author-list__show-more{display:none}.js .c-article-author-list--long .c-article-author-list__show-more,.js .c-article-author-list--long+.c-article-author-list__button{display:inline}@media only screen and (max-width:767px){.js .c-article-author-list__item--hide-small-screen{display:none}.js .c-article-author-list--short .c-article-author-list__show-more,.js .c-article-author-list--short+.c-article-author-list__button{display:inline}}#uptodate-client,.js .c-article-author-list--expanded .c-article-author-list__show-more{display:none!important}.js .c-article-author-list--expanded .c-article-author-list__item--hide-small-screen{display:inline!important}.c-article-author-list__button,.c-button-author-list{background:#ebf1f5;border:4px solid #ebf1f5;border-radius:20px;color:#666;font-size:.875rem;line-height:1.4;padding:2px 11px 2px 8px;text-decoration:none}.c-article-author-list__button svg,.c-button-author-list svg{margin:1px 4px 0 0}.c-article-author-list__button:hover,.c-button-author-list:hover{background:#025e8d;border-color:transparent;color:#fff}.c-article-body .c-article-access-provider{padding:8px 16px}.c-article-body .c-article-access-provider,.c-notes{border:1px solid #d5d5d5;border-image:initial;border-left:none;border-right:none;margin:24px 0}.c-article-body .c-article-access-provider__text{color:#555}.c-article-body .c-article-access-provider__text,.c-notes__text{font-size:1rem;margin-bottom:0;padding-bottom:2px;padding-top:2px;text-align:center}.c-article-body .c-article-author-affiliation__address{color:inherit;font-weight:700;margin:0}.c-article-body .c-article-author-affiliation__authors-list{list-style:none;margin:0;padding:0}.c-article-body .c-article-author-affiliation__authors-item{display:inline;margin-left:0}.c-article-authors-search{margin-bottom:24px;margin-top:0}.c-article-authors-search__item,.c-article-authors-search__title{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif}.c-article-authors-search__title{color:#626262;font-size:1.05rem;font-weight:700;margin:0;padding:0}.c-article-authors-search__item{font-size:1rem}.c-article-authors-search__text{margin:0}.c-code-block{border:1px solid #fff;font-family:monospace;margin:0 0 24px;padding:20px}.c-code-block__heading{font-weight:400;margin-bottom:16px}.c-code-block__line{display:block;overflow-wrap:break-word;white-space:pre-wrap}.c-article-share-box{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;margin-bottom:24px}.c-article-share-box__description{font-size:1rem;margin-bottom:8px}.c-article-share-box__no-sharelink-info{font-size:.813rem;font-weight:700;margin-bottom:24px;padding-top:4px}.c-article-share-box__only-read-input{border:1px solid #d5d5d5;box-sizing:content-box;display:inline-block;font-size:.875rem;font-weight:700;height:24px;margin-bottom:8px;padding:8px 10px}.c-article-share-box__additional-info{color:#626262;font-size:.813rem}.c-article-share-box__button{background:#fff;box-sizing:content-box;text-align:center}.c-article-share-box__button--link-like{background-color:transparent;border:0;color:#025e8d;cursor:pointer;font-size:.875rem;margin-bottom:8px;margin-left:10px}.c-article-associated-content__container .c-article-associated-content__collection-label{font-size:.875rem;line-height:1.4}.c-article-associated-content__container .c-article-associated-content__collection-title{line-height:1.3}.c-reading-companion{clear:both;min-height:389px}.c-reading-companion__figures-list,.c-reading-companion__references-list{list-style:none;min-height:389px;padding:0}.c-reading-companion__references-list--numeric{list-style:decimal inside}.c-reading-companion__figure-item{border-top:1px solid #d5d5d5;font-size:1rem;padding:16px 8px 16px 0}.c-reading-companion__figure-item:first-child{border-top:none;padding-top:8px}.c-reading-companion__reference-item{font-size:1rem}.c-reading-companion__reference-item:first-child{border-top:none}.c-reading-companion__reference-item a{word-break:break-word}.c-reading-companion__reference-citation{display:inline}.c-reading-companion__reference-links{font-size:.813rem;font-weight:700;list-style:none;margin:8px 0 0;padding:0;text-align:right}.c-reading-companion__reference-links>a{display:inline-block;padding-left:8px}.c-reading-companion__reference-links>a:first-child{display:inline-block;padding-left:0}.c-reading-companion__figure-title{display:block;font-size:1.25rem;font-weight:700;line-height:1.2;margin:0 0 8px}.c-reading-companion__figure-links{display:flex;justify-content:space-between;margin:8px 0 0}.c-reading-companion__figure-links>a{align-items:center;display:flex}.c-article-section__figure-caption{display:block;margin-bottom:8px;word-break:break-word}.c-article-section__figure .video,p.app-article-masthead__access--above-download{margin:0 0 16px}.c-article-section__figure-description{font-size:1rem}.c-article-section__figure-description>*{margin-bottom:0}.c-cod{display:block;font-size:1rem;width:100%}.c-cod__form{background:#ebf0f3}.c-cod__prompt{font-size:1.125rem;line-height:1.3;margin:0 0 24px}.c-cod__label{display:block;margin:0 0 4px}.c-cod__row{display:flex;margin:0 0 16px}.c-cod__row:last-child{margin:0}.c-cod__input{border:1px solid #d5d5d5;border-radius:2px;flex-shrink:0;margin:0;padding:13px}.c-cod__input--submit{background-color:#025e8d;border:1px solid #025e8d;color:#fff;flex-shrink:1;margin-left:8px;transition:background-color .2s ease-out 0s,color .2s ease-out 0s}.c-cod__input--submit-single{flex-basis:100%;flex-shrink:0;margin:0}.c-cod__input--submit:focus,.c-cod__input--submit:hover{background-color:#fff;color:#025e8d}.save-data .c-article-author-institutional-author__sub-division,.save-data .c-article-equation__number,.save-data .c-article-figure-description,.save-data .c-article-fullwidth-content,.save-data .c-article-main-column,.save-data .c-article-satellite-article-link,.save-data .c-article-satellite-subtitle,.save-data .c-article-table-container,.save-data .c-blockquote__body,.save-data .c-code-block__heading,.save-data .c-reading-companion__figure-title,.save-data .c-reading-companion__reference-citation,.save-data .c-site-messages--nature-briefing-email-variant .serif,.save-data .c-site-messages--nature-briefing-email-variant.serif,.save-data .serif,.save-data .u-serif,.save-data h1,.save-data h2,.save-data h3{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif}.c-pdf-download__link{display:flex;flex:1 1 0%;padding:13px 24px}.c-pdf-download__link:hover{text-decoration:none}@media only screen and (min-width:768px){.c-context-bar--sticky .c-pdf-download__link{align-items:center;flex:1 1 183px}}@media only screen and (max-width:320px){.c-context-bar--sticky .c-pdf-download__link{padding:16px}}.c-article-body .c-article-recommendations-list,.c-book-body .c-article-recommendations-list{display:flex;flex-direction:row;gap:16px 16px;margin:0;max-width:100%;padding:16px 0 0}.c-article-body .c-article-recommendations-list__item,.c-book-body .c-article-recommendations-list__item{flex:1 1 0%}@media only screen and (max-width:767px){.c-article-body .c-article-recommendations-list,.c-book-body .c-article-recommendations-list{flex-direction:column}}.c-article-body .c-article-recommendations-card__authors{display:none;font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;font-size:.875rem;line-height:1.5;margin:0 0 8px}@media only screen and (max-width:767px){.c-article-body .c-article-recommendations-card__authors{display:block;margin:0}}.c-article-body .c-article-history{margin-top:24px}.app-article-metrics-bar p{margin:0}.app-article-masthead{display:flex;flex-direction:column;gap:16px 16px;padding:16px 0 24px}.app-article-masthead__info{display:flex;flex-direction:column;flex-grow:1}.app-article-masthead__brand{border-top:1px solid hsla(0,0%,100%,.8);display:flex;flex-direction:column;flex-shrink:0;gap:8px 8px;min-height:96px;padding:16px 0 0}.app-article-masthead__brand img{border:1px solid #fff;border-radius:8px;box-shadow:0 4px 15px 0 hsla(0,0%,50%,.25);height:auto;left:0;position:absolute;width:72px}.app-article-masthead__journal-link{display:block;font-size:1.125rem;font-weight:700;margin:0 0 8px;max-width:400px;padding:0 0 0 88px;position:relative}.app-article-masthead__journal-title{-webkit-box-orient:vertical;-webkit-line-clamp:3;display:-webkit-box;overflow:hidden}.app-article-masthead__submission-link{align-items:center;display:flex;font-size:1rem;gap:4px 4px;margin:0 0 0 88px}.app-article-masthead__access{align-items:center;display:flex;flex-wrap:wrap;font-size:.875rem;font-weight:300;gap:4px 4px;margin:0}.app-article-masthead__buttons{display:flex;flex-flow:column wrap;gap:16px 16px}.app-article-masthead__access svg,.app-masthead--pastel .c-pdf-download .u-button--primary svg,.app-masthead--pastel .c-pdf-download .u-button--secondary svg,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--primary svg,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--secondary svg{fill:currentcolor}.app-article-masthead a{color:#fff}.app-masthead--pastel .c-pdf-download .u-button--primary,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--primary{background-color:#025e8d;background-image:none;border:2px solid transparent;box-shadow:none;color:#fff;font-weight:700}.app-masthead--pastel .c-pdf-download .u-button--primary:visited,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--primary:visited{color:#fff}.app-masthead--pastel .c-pdf-download .u-button--primary:hover,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--primary:hover{text-decoration:none}.app-masthead--pastel .c-pdf-download .u-button--primary:focus,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--primary:focus{border:4px solid #fc0;box-shadow:none;outline:0;text-decoration:none}.app-masthead--pastel .c-pdf-download .u-button--primary:focus,.app-masthead--pastel .c-pdf-download .u-button--primary:hover,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--primary:focus,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--primary:hover{background-color:#fff;background-image:none;color:#01324b}.app-masthead--pastel .c-pdf-download .u-button--primary:hover,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--primary:hover{background:0 0;border:2px solid #025e8d;box-shadow:none;color:#025e8d}.app-masthead--pastel .c-pdf-download .u-button--secondary,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--secondary{background:0 0;border:2px solid #025e8d;color:#025e8d;font-weight:700}.app-masthead--pastel .c-pdf-download .u-button--secondary:visited,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--secondary:visited{color:#01324b}.app-masthead--pastel .c-pdf-download .u-button--secondary:hover,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--secondary:hover{background-color:#01324b;background-color:#025e8d;border:2px solid transparent;box-shadow:none;color:#fff}.app-masthead--pastel .c-pdf-download .u-button--secondary:focus,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--secondary:focus{background-color:#fff;background-image:none;border:4px solid #fc0;color:#01324b}@media only screen and (min-width:768px){.app-article-masthead{flex-direction:row;gap:64px 64px;padding:24px 0}.app-article-masthead__brand{border:0;padding:0}.app-article-masthead__brand img{height:auto;position:static;width:auto}.app-article-masthead__buttons{align-items:center;flex-direction:row;margin-top:auto}.app-article-masthead__journal-link{display:flex;flex-direction:column;gap:24px 24px;margin:0 0 8px;padding:0}.app-article-masthead__submission-link{margin:0}}@media only screen and (min-width:1024px){.app-article-masthead__brand{flex-basis:400px}}.app-article-masthead .c-article-identifiers{font-size:.875rem;font-weight:300;line-height:1;margin:0 0 8px;overflow:hidden;padding:0}.app-article-masthead .c-article-identifiers--cite-list{margin:0 0 16px}.app-article-masthead .c-article-identifiers *{color:#fff}.app-article-masthead .c-cod{display:none}.app-article-masthead .c-article-identifiers__item{border-left:1px solid #fff;border-right:0;margin:0 17px 8px -9px;padding:0 0 0 8px}.app-article-masthead .c-article-identifiers__item--cite{border-left:0}.app-article-metrics-bar{display:flex;flex-wrap:wrap;font-size:1rem;padding:16px 0 0;row-gap:24px}.app-article-metrics-bar__item{padding:0 16px 0 0}.app-article-metrics-bar__count{font-weight:700}.app-article-metrics-bar__label{font-weight:400;padding-left:4px}.app-article-metrics-bar__icon{height:auto;margin-right:4px;margin-top:-4px;width:auto}.app-article-metrics-bar__arrow-icon{margin:4px 0 0 4px}.app-article-metrics-bar a{color:#000}.app-article-metrics-bar .app-article-metrics-bar__item--metrics{padding-right:0}.app-overview-section .c-article-author-list,.app-overview-section__authors{line-height:2}.app-article-metrics-bar{margin-top:8px}.c-book-toc-pagination+.c-book-section__back-to-top{margin-top:0}.c-article-body .c-article-access-provider__text--chapter{color:#222;font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;padding:20px 0}.c-article-body .c-article-access-provider__text--chapter svg.c-status-message__icon{fill:#003f8d;vertical-align:middle}.c-article-body-section__content--separator{padding-top:40px}.c-pdf-download__link{max-height:44px}.app-article-access .u-button--primary,.app-article-access .u-button--primary:visited{color:#fff}.c-article-sidebar{display:none}@media only screen and (min-width:1024px){.c-article-sidebar{display:block}}.c-cod__form{border-radius:12px}.c-cod__label{font-size:.875rem}.c-cod .c-status-message{align-items:center;justify-content:center;margin-bottom:16px;padding-bottom:16px}@media only screen and (min-width:1024px){.c-cod .c-status-message{align-items:inherit}}.c-cod .c-status-message__icon{margin-top:4px}.c-cod .c-cod__prompt{font-size:1rem;margin-bottom:16px}.c-article-body .app-article-access,.c-book-body .app-article-access{display:block}@media only screen and (min-width:1024px){.c-article-body .app-article-access,.c-book-body .app-article-access{display:none}}.c-article-body .app-card-service{margin-bottom:32px}@media only screen and (min-width:1024px){.c-article-body .app-card-service{display:none}}.app-article-access .buybox__buy .u-button--secondary,.app-article-access .u-button--primary,.c-cod__row .u-button--primary{background-color:#025e8d;border:2px solid #025e8d;box-shadow:none;font-size:1rem;font-weight:700;gap:8px 8px;justify-content:center;line-height:1.5;padding:8px 24px}.app-article-access .buybox__buy .u-button--secondary,.app-article-access .u-button--primary:hover,.c-cod__row .u-button--primary:hover{background-color:#fff;color:#025e8d}.app-article-access .buybox__buy .u-button--secondary:hover{background-color:#025e8d;color:#fff}.buybox__buy .c-notes__text{color:#666;font-size:.875rem;padding:0 16px 8px}.c-cod__input{flex-basis:auto;width:100%}.c-article-title{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;font-size:2.25rem;font-weight:700;line-height:1.2;margin:12px 0}.c-reading-companion__figure-item figure{margin:0}@media only screen and (min-width:768px){.c-article-title{margin:16px 0}}.app-article-access{border:1px solid #c5e0f4;border-radius:12px}.app-article-access__heading{border-bottom:1px solid #c5e0f4;font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;font-size:1.125rem;font-weight:700;margin:0;padding:16px;text-align:center}.app-article-access .buybox__info svg{vertical-align:middle}.c-article-body .app-article-access p{margin-bottom:0}.app-article-access .buybox__info{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;font-size:1rem;margin:0}.app-article-access{margin:0 0 32px}@media only screen and (min-width:1024px){.app-article-access{margin:0 0 24px}}.c-status-message{font-size:1rem}.c-article-body{font-size:1.125rem}.c-article-body dl,.c-article-body ol,.c-article-body p,.c-article-body ul{margin-bottom:32px;margin-top:0}.c-article-access-provider__text:last-of-type,.c-article-body .c-notes__text:last-of-type{margin-bottom:0}.c-article-body ol p,.c-article-body ul p{margin-bottom:16px}.c-article-section__figure-caption{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif}.c-reading-companion__figure-item{border-top-color:#c5e0f4}.c-reading-companion__sticky{max-width:400px}.c-article-section .c-article-section__figure-description>*{font-size:1rem;margin-bottom:16px}.c-reading-companion__reference-item{border-top:1px solid #d5d5d5;padding:16px 0}.c-reading-companion__reference-item:first-child{padding-top:0}.c-article-share-box__button,.js .c-article-authors-search__item .c-article-button{background:0 0;border:2px solid #025e8d;border-radius:32px;box-shadow:none;color:#025e8d;font-size:1rem;font-weight:700;line-height:1.5;margin:0;padding:8px 24px;transition:all .2s ease 0s}.c-article-authors-search__item .c-article-button{width:100%}.c-pdf-download .u-button{background-color:#fff;border:2px solid #fff;color:#01324b;justify-content:center}.c-context-bar__container .c-pdf-download .u-button svg,.c-pdf-download .u-button svg{fill:currentcolor}.c-pdf-download .u-button:visited{color:#01324b}.c-pdf-download .u-button:hover{border:4px solid #01324b;box-shadow:none}.c-pdf-download .u-button:focus,.c-pdf-download .u-button:hover{background-color:#01324b}.c-pdf-download .u-button:focus svg path,.c-pdf-download .u-button:hover svg path{fill:#fff}.c-context-bar__container .c-pdf-download .u-button{background-image:none;border:2px solid;color:#fff}.c-context-bar__container .c-pdf-download .u-button:visited{color:#fff}.c-context-bar__container .c-pdf-download .u-button:hover{text-decoration:none}.c-context-bar__container .c-pdf-download .u-button:focus{box-shadow:none;outline:0;text-decoration:none}.c-context-bar__container .c-pdf-download .u-button:focus,.c-context-bar__container .c-pdf-download .u-button:hover{background-color:#fff;background-image:none;color:#01324b}.c-context-bar__container .c-pdf-download .u-button:focus svg path,.c-context-bar__container .c-pdf-download .u-button:hover svg path{fill:#01324b}.c-context-bar__container .c-pdf-download .u-button,.c-pdf-download .u-button{box-shadow:none;font-size:1rem;font-weight:700;line-height:1.5;padding:8px 24px}.c-context-bar__container .c-pdf-download .u-button{background-color:#025e8d}.c-pdf-download .u-button:hover{border:2px solid #fff}.c-pdf-download .u-button:focus,.c-pdf-download .u-button:hover{background:0 0;box-shadow:none;color:#fff}.c-context-bar__container .c-pdf-download .u-button:hover{border:2px solid #025e8d;box-shadow:none;color:#025e8d}.c-context-bar__container .c-pdf-download .u-button:focus,.c-pdf-download .u-button:focus{border:2px solid #025e8d}.c-article-share-box__button:focus:focus,.c-article__pill-button:focus:focus,.c-context-bar__container .c-pdf-download .u-button:focus:focus,.c-pdf-download .u-button:focus:focus{outline:3px solid #08c;will-change:transform}.c-pdf-download__link .u-icon{padding-top:0}.c-bibliographic-information__column button{margin-bottom:16px}.c-article-body .c-article-author-affiliation__list p,.c-article-body .c-article-author-information__list p,figure{margin:0}.c-article-share-box__button{margin-right:16px}.c-status-message--boxed{border-radius:12px}.c-article-associated-content__collection-title{font-size:1rem}.app-card-service__description,.c-article-body .app-card-service__description{color:#222;margin-bottom:0;margin-top:8px}.app-article-access__subscriptions a,.app-article-access__subscriptions a:visited,.app-book-series-listing__item a,.app-book-series-listing__item a:hover,.app-book-series-listing__item a:visited,.c-article-author-list a,.c-article-author-list a:visited,.c-article-buy-box a,.c-article-buy-box a:visited,.c-article-peer-review a,.c-article-peer-review a:visited,.c-article-satellite-subtitle a,.c-article-satellite-subtitle a:visited,.c-breadcrumbs__link,.c-breadcrumbs__link:hover,.c-breadcrumbs__link:visited{color:#000}.c-article-author-list svg{height:24px;margin:0 0 0 6px;width:24px}.c-article-header{margin-bottom:32px}@media only screen and (min-width:876px){.js .c-ad--conditional{display:block}}.u-lazy-ad-wrapper{background-color:#fff;display:none;min-height:149px}@media only screen and (min-width:876px){.u-lazy-ad-wrapper{display:block}}p.c-ad__label{margin-bottom:4px}.c-ad--728x90{background-color:#fff;border-bottom:2px solid #cedbe0} } </style> <style>@media only print, only all and (prefers-color-scheme: no-preference), only all and (prefers-color-scheme: light), only all and (prefers-color-scheme: dark) { .eds-c-header__brand img{height:24px;width:203px}.app-article-masthead__journal-link img{height:93px;width:72px}@media only screen and (min-width:769px){.app-article-masthead__journal-link img{height:161px;width:122px}} } </style> <link rel="stylesheet" data-test="critical-css-handler" data-inline-css-source="critical-css" href=/oscar-static/app-springerlink/css/core-darwin-9fe647df8f.css media="print" onload="this.media='all';this.onload=null"> <link rel="stylesheet" data-test="critical-css-handler" data-inline-css-source="critical-css" href="/oscar-static/app-springerlink/css/enhanced-darwin-article-7d5c36806c.css" media="print" onload="this.media='only print, only all and (prefers-color-scheme: no-preference), only all and (prefers-color-scheme: light), only all and (prefers-color-scheme: dark)';this.onload=null"> <script type="text/javascript"> config = { env: 'live', site: '25.springer.com', siteWithPath: '25.springer.com' + window.location.pathname, twitterHashtag: '25', cmsPrefix: 'https://studio-cms.springernature.com/studio/', publisherBrand: 'Springer', mustardcut: false }; </script> <script> window.dataLayer = [{"GA Key":"UA-26408784-1","DOI":"10.1007/s00025-020-01247-8","Page":"article","springerJournal":true,"Publishing Model":"Hybrid Access","Country":"SG","japan":false,"doi":"10.1007-s00025-020-01247-8","Journal Id":25,"Journal Title":"Results in Mathematics","imprint":"Birkhäuser","Keywords":"Poincare lemma, antiexact differential forms, homotopy operator, fermionic harmonic oscillator, complex manifold, 58A12, 58Z05","kwrd":["Poincare_lemma","antiexact_differential_forms","homotopy_operator","fermionic_harmonic_oscillator","complex_manifold","58A12","58Z05"],"Labs":"Y","ksg":"Krux.segments","kuid":"Krux.uid","Has Body":"Y","Features":[],"Open Access":"Y","hasAccess":"Y","bypassPaywall":"N","user":{"license":{"businessPartnerID":[],"businessPartnerIDString":""}},"Access Type":"open","Bpids":"","Bpnames":"","BPID":["1"],"VG Wort Identifier":"vgzm.415900-10.1007-s00025-020-01247-8","Full HTML":"Y","Subject Codes":["SCM","SCM00009"],"pmc":["M","M00009"],"session":{"authentication":{"loginStatus":"N"},"attributes":{"edition":"academic"}},"content":{"serial":{"eissn":"1420-9012","pissn":"1422-6383"},"type":"Article","category":{"pmc":{"primarySubject":"Mathematics","primarySubjectCode":"M","secondarySubjects":{"1":"Mathematics, general"},"secondarySubjectCodes":{"1":"M00009"}},"sucode":"SC10","articleType":"Article"},"attributes":{"deliveryPlatform":"oscar"}},"page":{"attributes":{"environment":"live"},"category":{"pageType":"article"}},"Event Category":"Article"}]; </script> <script data-test="springer-link-article-datalayer"> window.dataLayer = window.dataLayer || []; window.dataLayer.push({ ga4MeasurementId: 'G-B3E4QL2TPR', ga360TrackingId: 'UA-26408784-1', twitterId: 'o47a7', baiduId: 'aef3043f025ccf2305af8a194652d70b', ga4ServerUrl: 'https://collect.springer.com', imprint: 'springerlink', page: { attributes:{ featureFlags: [{ name: 'darwin-orion', active: true }, { name: 'chapter-books-recs', active: true } ], darwinAvailable: true } } }); </script> <script> (function(w, d) { w.config = w.config || {}; w.config.mustardcut = false; if (w.matchMedia && w.matchMedia('only print, only all and (prefers-color-scheme: no-preference), only all and (prefers-color-scheme: light), only all and (prefers-color-scheme: dark)').matches) { w.config.mustardcut = true; d.classList.add('js'); d.classList.remove('grade-c'); d.classList.remove('no-js'); } })(window, document.documentElement); </script> <script class="js-entry"> if (window.config.mustardcut) { (function(w, d) { window.Component = {}; window.suppressShareButton = false; window.onArticlePage = true; var currentScript = d.currentScript || d.head.querySelector('script.js-entry'); function catchNoModuleSupport() { var scriptEl = d.createElement('script'); return (!('noModule' in scriptEl) && 'onbeforeload' in scriptEl) } var headScripts = [ {'src': '/oscar-static/js/polyfill-es5-bundle-572d4fec60.js', 'async': false} ]; var bodyScripts = [ {'src': '/oscar-static/js/global-article-es5-bundle-dad1690b0d.js', 'async': false, 'module': false}, {'src': '/oscar-static/js/global-article-es6-bundle-e7d03c4cb3.js', 'async': false, 'module': true} ]; function createScript(script) { var scriptEl = d.createElement('script'); scriptEl.src = script.src; scriptEl.async = script.async; if (script.module === true) { scriptEl.type = "module"; if (catchNoModuleSupport()) { scriptEl.src = ''; } } else if (script.module === false) { scriptEl.setAttribute('nomodule', true) } if (script.charset) { scriptEl.setAttribute('charset', script.charset); } return scriptEl; } for (var i = 0; i < headScripts.length; ++i) { var scriptEl = createScript(headScripts[i]); currentScript.parentNode.insertBefore(scriptEl, currentScript.nextSibling); } d.addEventListener('DOMContentLoaded', function() { for (var i = 0; i < bodyScripts.length; ++i) { var scriptEl = createScript(bodyScripts[i]); d.body.appendChild(scriptEl); } }); // Webfont repeat view var config = w.config; if (config && config.publisherBrand && sessionStorage.fontsLoaded === 'true') { d.documentElement.className += ' webfonts-loaded'; } })(window, document); } </script> <script data-src="https://cdn.optimizely.com/js/27195530232.js" data-cc-script="C03"></script> <script data-test="gtm-head"> window.initGTM = function() { if (window.config.mustardcut) { (function (w, d, s, l, i) { w[l] = w[l] || []; w[l].push({'gtm.start': new Date().getTime(), event: 'gtm.js'}); var f = d.getElementsByTagName(s)[0], j = d.createElement(s), dl = l != 'dataLayer' ? '&l=' + l : ''; j.async = true; j.src = 'https://www.googletagmanager.com/gtm.js?id=' + i + dl; f.parentNode.insertBefore(j, f); })(window, document, 'script', 'dataLayer', 'GTM-MRVXSHQ'); } } </script> <script> (function (w, d, t) { function cc() { var h = w.location.hostname; var e = d.createElement(t), s = d.getElementsByTagName(t)[0]; if (h.indexOf('springer.com') > -1 && h.indexOf('biomedcentral.com') === -1 && h.indexOf('springeropen.com') === -1) { if (h.indexOf('link-qa.springer.com') > -1 || h.indexOf('test-www.springer.com') > -1) { e.src = 'https://cmp.springer.com/production_live/en/consent-bundle-17-52.js'; e.setAttribute('onload', "initGTM(window,document,'script','dataLayer','GTM-MRVXSHQ')"); } else { e.src = 'https://cmp.springer.com/production_live/en/consent-bundle-17-52.js'; e.setAttribute('onload', "initGTM(window,document,'script','dataLayer','GTM-MRVXSHQ')"); } } else if (h.indexOf('biomedcentral.com') > -1) { if (h.indexOf('biomedcentral.com.qa') > -1) { e.src = 'https://cmp.biomedcentral.com/production_live/en/consent-bundle-15-38.js'; e.setAttribute('onload', "initGTM(window,document,'script','dataLayer','GTM-MRVXSHQ')"); } else { e.src = 'https://cmp.biomedcentral.com/production_live/en/consent-bundle-15-38.js'; e.setAttribute('onload', "initGTM(window,document,'script','dataLayer','GTM-MRVXSHQ')"); } } else if (h.indexOf('springeropen.com') > -1) { if (h.indexOf('springeropen.com.qa') > -1) { e.src = 'https://cmp.springernature.com/production_live/en/consent-bundle-16-35.js'; e.setAttribute('onload', "initGTM(window,document,'script','dataLayer','GTM-MRVXSHQ')"); } else { e.src = 'https://cmp.springernature.com/production_live/en/consent-bundle-16-35.js'; e.setAttribute('onload', "initGTM(window,document,'script','dataLayer','GTM-MRVXSHQ')"); } } else if (h.indexOf('springernature.com') > -1) { if (h.indexOf('beta-qa.springernature.com') > -1) { e.src = 'https://cmp.springernature.com/production_live/en/consent-bundle-49-43.js'; e.setAttribute('onload', "initGTM(window,document,'script','dataLayer','GTM-NK22KLS')"); } else { e.src = 'https://cmp.springernature.com/production_live/en/consent-bundle-49-43.js'; e.setAttribute('onload', "initGTM(window,document,'script','dataLayer','GTM-NK22KLS')"); } } else { e.src = '/oscar-static/js/cookie-consent-es5-bundle-cb57c2c98a.js'; e.setAttribute('data-consent', h); } s.insertAdjacentElement('afterend', e); } cc(); })(window, document, 'script'); </script> <link rel="canonical" href="https://link.springer.com/article/10.1007/s00025-020-01247-8"/> <script type="application/ld+json">{"mainEntity":{"headline":"The Poincare Lemma, Antiexact Forms, and Fermionic Quantum Harmonic Oscillator","description":"The paper focuses on various properties and applications of the homotopy operator, which occurs in the Poincaré lemma. In the first part, an abstract operator calculus is constructed, where the exterior derivative is an abstract derivative and the homotopy operator plays the role of an abstract integral. This operator calculus can be used to formulate abstract differential equations. An example of the eigenvalue problem that resembles the fermionic quantum harmonic oscillator is presented. The second part presents the dual complex to the Dolbeault bicomplex generated by the homotopy operator on complex manifolds.\n","datePublished":"2020-07-11T00:00:00Z","dateModified":"2020-07-11T00:00:00Z","pageStart":"1","pageEnd":"14","sameAs":"https://doi.org/10.1007/s00025-020-01247-8","keywords":["Poincare lemma","antiexact differential forms","homotopy operator","fermionic harmonic oscillator","complex manifold","58A12","58Z05","Mathematics","general"],"image":["https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs00025-020-01247-8/MediaObjects/25_2020_1247_Fig1_HTML.png","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs00025-020-01247-8/MediaObjects/25_2020_1247_Fig2_HTML.png","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs00025-020-01247-8/MediaObjects/25_2020_1247_Fig3_HTML.png","https://media.springernature.com/lw1200/springer-static/image/art%3A10.1007%2Fs00025-020-01247-8/MediaObjects/25_2020_1247_Fig4_HTML.png"],"isPartOf":{"name":"Results in Mathematics","issn":["1420-9012","1422-6383"],"volumeNumber":"75","@type":["Periodical","PublicationVolume"]},"publisher":{"name":"Springer International Publishing","logo":{"url":"https://www.springernature.com/app-sn/public/images/logo-springernature.png","@type":"ImageObject"},"@type":"Organization"},"author":[{"name":"Radosław Antoni Kycia","url":"http://orcid.org/0000-0002-6390-4627","affiliation":[{"name":"Masaryk University","address":{"name":"Department of Mathematics and Statistics, Masaryk University, Brno, Czech Republic","@type":"PostalAddress"},"@type":"Organization"},{"name":"Cracow University of Technology","address":{"name":"Faculty of Materials Engineering and Physics, Cracow University of Technology, Kraków, Poland","@type":"PostalAddress"},"@type":"Organization"}],"email":"kycia.radoslaw@gmail.com","@type":"Person"}],"isAccessibleForFree":true,"@type":"ScholarlyArticle"},"@context":"https://schema.org","@type":"WebPage"}</script> </head> <body class="" >  <noscript> <iframe src="https://www.googletagmanager.com/ns.html?id=GTM-MRVXSHQ" height="0" width="0" style="display:none;visibility:hidden"></iframe> </noscript>   <noscript data-test="gtm-body"> <iframe src="https://www.googletagmanager.com/ns.html?id=GTM-MRVXSHQ" height="0" width="0" style="display:none;visibility:hidden"></iframe> </noscript>  <div class="u-visually-hidden" aria-hidden="true" data-test="darwin-icons"> <?xml version="1.0" encoding="UTF-8"?><!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN" "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd"><svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><symbol id="icon-eds-i-accesses-medium" viewBox="0 0 24 24"><path d="M15.59 1a1 1 0 0 1 .706.291l5.41 5.385a1 1 0 0 1 .294.709v13.077c0 .674-.269 1.32-.747 1.796a2.549 2.549 0 0 1-1.798.742H15a1 1 0 0 1 0-2h4.455a.549.549 0 0 0 .387-.16.535.535 0 0 0 .158-.378V7.8L15.178 3H5.545a.543.543 0 0 0-.538.451L5 3.538v8.607a1 1 0 0 1-2 0V3.538A2.542 2.542 0 0 1 5.545 1h10.046ZM8 13c2.052 0 4.66 1.61 6.36 3.4l.124.141c.333.41.516.925.516 1.459 0 .6-.232 1.178-.64 1.599C12.666 21.388 10.054 23 8 23c-2.052 0-4.66-1.61-6.353-3.393A2.31 2.31 0 0 1 1 18c0-.6.232-1.178.64-1.6C3.34 14.61 5.948 13 8 13Zm0 2c-1.369 0-3.552 1.348-4.917 2.785A.31.31 0 0 0 3 18c0 .083.031.161.09.222C4.447 19.652 6.631 21 8 21c1.37 0 3.556-1.35 4.917-2.785A.31.31 0 0 0 13 18a.32.32 0 0 0-.048-.17l-.042-.052C11.553 16.348 9.369 15 8 15Zm0 1a2 2 0 1 1 0 4 2 2 0 0 1 0-4Z"/></symbol><symbol id="icon-eds-i-altmetric-medium" viewBox="0 0 24 24"><path d="M12 1c5.978 0 10.843 4.77 10.996 10.712l.004.306-.002.022-.002.248C22.843 18.23 17.978 23 12 23 5.925 23 1 18.075 1 12S5.925 1 12 1Zm-1.726 9.246L8.848 12.53a1 1 0 0 1-.718.461L8.003 13l-4.947.014a9.001 9.001 0 0 0 17.887-.001L16.553 13l-2.205 3.53a1 1 0 0 1-1.735-.068l-.05-.11-2.289-6.106ZM12 3a9.001 9.001 0 0 0-8.947 8.013l4.391-.012L9.652 7.47a1 1 0 0 1 1.784.179l2.288 6.104 1.428-2.283a1 1 0 0 1 .722-.462l.129-.008 4.943.012A9.001 9.001 0 0 0 12 3Z"/></symbol><symbol id="icon-eds-i-arrow-bend-down-medium" viewBox="0 0 24 24"><path d="m11.852 20.989.058.007L12 21l.075-.003.126-.017.111-.03.111-.044.098-.052.104-.074.082-.073 6-6a1 1 0 0 0-1.414-1.414L13 17.585v-12.2C13 4.075 11.964 3 10.667 3H4a1 1 0 1 0 0 2h6.667c.175 0 .333.164.333.385v12.2l-4.293-4.292a1 1 0 0 0-1.32-.083l-.094.083a1 1 0 0 0 0 1.414l6 6c.035.036.073.068.112.097l.11.071.114.054.105.035.118.025Z"/></symbol><symbol id="icon-eds-i-arrow-bend-down-small" viewBox="0 0 16 16"><path d="M1 2a1 1 0 0 0 1 1h5v8.585L3.707 8.293a1 1 0 0 0-1.32-.083l-.094.083a1 1 0 0 0 0 1.414l5 5 .063.059.093.069.081.048.105.048.104.035.105.022.096.01h.136l.122-.018.113-.03.103-.04.1-.053.102-.07.052-.043 5.04-5.037a1 1 0 1 0-1.415-1.414L9 11.583V3a2 2 0 0 0-2-2H2a1 1 0 0 0-1 1Z"/></symbol><symbol id="icon-eds-i-arrow-bend-up-medium" viewBox="0 0 24 24"><path d="m11.852 3.011.058-.007L12 3l.075.003.126.017.111.03.111.044.098.052.104.074.082.073 6 6a1 1 0 1 1-1.414 1.414L13 6.415v12.2C13 19.925 11.964 21 10.667 21H4a1 1 0 0 1 0-2h6.667c.175 0 .333-.164.333-.385v-12.2l-4.293 4.292a1 1 0 0 1-1.32.083l-.094-.083a1 1 0 0 1 0-1.414l6-6c.035-.036.073-.068.112-.097l.11-.071.114-.054.105-.035.118-.025Z"/></symbol><symbol id="icon-eds-i-arrow-bend-up-small" viewBox="0 0 16 16"><path d="M1 13.998a1 1 0 0 1 1-1h5V4.413L3.707 7.705a1 1 0 0 1-1.32.084l-.094-.084a1 1 0 0 1 0-1.414l5-5 .063-.059.093-.068.081-.05.105-.047.104-.035.105-.022L7.94 1l.136.001.122.017.113.03.103.04.1.053.102.07.052.043 5.04 5.037a1 1 0 1 1-1.415 1.414L9 4.415v8.583a2 2 0 0 1-2 2H2a1 1 0 0 1-1-1Z"/></symbol><symbol id="icon-eds-i-arrow-diagonal-medium" viewBox="0 0 24 24"><path d="M14 3h6l.075.003.126.017.111.03.111.044.098.052.096.067.09.08c.036.035.068.073.097.112l.071.11.054.114.035.105.03.148L21 4v6a1 1 0 0 1-2 0V6.414l-4.293 4.293a1 1 0 0 1-1.414-1.414L17.584 5H14a1 1 0 0 1-.993-.883L13 4a1 1 0 0 1 1-1ZM4 13a1 1 0 0 1 1 1v3.584l4.293-4.291a1 1 0 1 1 1.414 1.414L6.414 19H10a1 1 0 0 1 .993.883L11 20a1 1 0 0 1-1 1l-6.075-.003-.126-.017-.111-.03-.111-.044-.098-.052-.096-.067-.09-.08a1.01 1.01 0 0 1-.097-.112l-.071-.11-.054-.114-.035-.105-.025-.118-.007-.058L3 20v-6a1 1 0 0 1 1-1Z"/></symbol><symbol id="icon-eds-i-arrow-diagonal-small" viewBox="0 0 16 16"><path d="m2 15-.082-.004-.119-.016-.111-.03-.111-.044-.098-.052-.096-.067-.09-.08a1.008 1.008 0 0 1-.097-.112l-.071-.11-.031-.062-.034-.081-.024-.076-.025-.118-.007-.058L1 14.02V9a1 1 0 1 1 2 0v2.584l2.793-2.791a1 1 0 1 1 1.414 1.414L4.414 13H7a1 1 0 0 1 .993.883L8 14a1 1 0 0 1-1 1H2ZM14 1l.081.003.12.017.111.03.111.044.098.052.096.067.09.08c.036.035.068.073.097.112l.071.11.031.062.034.081.024.076.03.148L15 2v5a1 1 0 0 1-2 0V4.414l-2.96 2.96A1 1 0 1 1 8.626 5.96L11.584 3H9a1 1 0 0 1-.993-.883L8 2a1 1 0 0 1 1-1h5Z"/></symbol><symbol id="icon-eds-i-arrow-down-medium" viewBox="0 0 24 24"><path d="m20.707 12.728-7.99 7.98a.996.996 0 0 1-.561.281l-.157.011a.998.998 0 0 1-.788-.384l-7.918-7.908a1 1 0 0 1 1.414-1.416L11 17.576V4a1 1 0 0 1 2 0v13.598l6.293-6.285a1 1 0 0 1 1.32-.082l.095.083a1 1 0 0 1-.001 1.414Z"/></symbol><symbol id="icon-eds-i-arrow-down-small" viewBox="0 0 16 16"><path d="m1.293 8.707 6 6 .063.059.093.069.081.048.105.049.104.034.056.013.118.017L8 15l.076-.003.122-.017.113-.03.085-.032.063-.03.098-.058.06-.043.05-.043 6.04-6.037a1 1 0 0 0-1.414-1.414L9 11.583V2a1 1 0 1 0-2 0v9.585L2.707 7.293a1 1 0 0 0-1.32-.083l-.094.083a1 1 0 0 0 0 1.414Z"/></symbol><symbol id="icon-eds-i-arrow-left-medium" viewBox="0 0 24 24"><path d="m11.272 3.293-7.98 7.99a.996.996 0 0 0-.281.561L3 12.001c0 .32.15.605.384.788l7.908 7.918a1 1 0 0 0 1.416-1.414L6.424 13H20a1 1 0 0 0 0-2H6.402l6.285-6.293a1 1 0 0 0 .082-1.32l-.083-.095a1 1 0 0 0-1.414.001Z"/></symbol><symbol id="icon-eds-i-arrow-left-small" viewBox="0 0 16 16"><path d="m7.293 1.293-6 6-.059.063-.069.093-.048.081-.049.105-.034.104-.013.056-.017.118L1 8l.003.076.017.122.03.113.032.085.03.063.058.098.043.06.043.05 6.037 6.04a1 1 0 0 0 1.414-1.414L4.417 9H14a1 1 0 0 0 0-2H4.415l4.292-4.293a1 1 0 0 0 .083-1.32l-.083-.094a1 1 0 0 0-1.414 0Z"/></symbol><symbol id="icon-eds-i-arrow-right-medium" viewBox="0 0 24 24"><path d="m12.728 3.293 7.98 7.99a.996.996 0 0 1 .281.561l.011.157c0 .32-.15.605-.384.788l-7.908 7.918a1 1 0 0 1-1.416-1.414L17.576 13H4a1 1 0 0 1 0-2h13.598l-6.285-6.293a1 1 0 0 1-.082-1.32l.083-.095a1 1 0 0 1 1.414.001Z"/></symbol><symbol id="icon-eds-i-arrow-right-small" viewBox="0 0 16 16"><path d="m8.707 1.293 6 6 .059.063.069.093.048.081.049.105.034.104.013.056.017.118L15 8l-.003.076-.017.122-.03.113-.032.085-.03.063-.058.098-.043.06-.043.05-6.037 6.04a1 1 0 0 1-1.414-1.414L11.583 9H2a1 1 0 1 1 0-2h9.585L7.293 2.707a1 1 0 0 1-.083-1.32l.083-.094a1 1 0 0 1 1.414 0Z"/></symbol><symbol id="icon-eds-i-arrow-up-medium" viewBox="0 0 24 24"><path d="m3.293 11.272 7.99-7.98a.996.996 0 0 1 .561-.281L12.001 3c.32 0 .605.15.788.384l7.918 7.908a1 1 0 0 1-1.414 1.416L13 6.424V20a1 1 0 0 1-2 0V6.402l-6.293 6.285a1 1 0 0 1-1.32.082l-.095-.083a1 1 0 0 1 .001-1.414Z"/></symbol><symbol id="icon-eds-i-arrow-up-small" viewBox="0 0 16 16"><path d="m1.293 7.293 6-6 .063-.059.093-.069.081-.048.105-.049.104-.034.056-.013.118-.017L8 1l.076.003.122.017.113.03.085.032.063.03.098.058.06.043.05.043 6.04 6.037a1 1 0 0 1-1.414 1.414L9 4.417V14a1 1 0 0 1-2 0V4.415L2.707 8.707a1 1 0 0 1-1.32.083l-.094-.083a1 1 0 0 1 0-1.414Z"/></symbol><symbol id="icon-eds-i-article-medium" viewBox="0 0 24 24"><path d="M8 7a1 1 0 0 0 0 2h4a1 1 0 1 0 0-2H8ZM8 11a1 1 0 1 0 0 2h8a1 1 0 1 0 0-2H8ZM7 16a1 1 0 0 1 1-1h8a1 1 0 1 1 0 2H8a1 1 0 0 1-1-1Z"/><path d="M5.545 1A2.542 2.542 0 0 0 3 3.538v16.924A2.542 2.542 0 0 0 5.545 23h12.91A2.542 2.542 0 0 0 21 20.462V3.5A2.5 2.5 0 0 0 18.5 1H5.545ZM5 3.538C5 3.245 5.24 3 5.545 3H18.5a.5.5 0 0 1 .5.5v16.962c0 .293-.24.538-.546.538H5.545A.542.542 0 0 1 5 20.462V3.538Z" clip-rule="evenodd"/></symbol><symbol id="icon-eds-i-book-medium" viewBox="0 0 24 24"><path d="M18.5 1A2.5 2.5 0 0 1 21 3.5v12c0 1.16-.79 2.135-1.86 2.418l-.14.031V21h1a1 1 0 0 1 .993.883L21 22a1 1 0 0 1-1 1H6.5A3.5 3.5 0 0 1 3 19.5v-15A3.5 3.5 0 0 1 6.5 1h12ZM17 18H6.5a1.5 1.5 0 0 0-1.493 1.356L5 19.5A1.5 1.5 0 0 0 6.5 21H17v-3Zm1.5-15h-12A1.5 1.5 0 0 0 5 4.5v11.837l.054-.025a3.481 3.481 0 0 1 1.254-.307L6.5 16h12a.5.5 0 0 0 .492-.41L19 15.5v-12a.5.5 0 0 0-.5-.5ZM15 6a1 1 0 0 1 0 2H9a1 1 0 1 1 0-2h6Z"/></symbol><symbol id="icon-eds-i-book-series-medium" viewBox="0 0 24 24"><path fill-rule="evenodd" d="M1 3.786C1 2.759 1.857 2 2.82 2H6.18c.964 0 1.82.759 1.82 1.786V4h3.168c.668 0 1.298.364 1.616.938.158-.109.333-.195.523-.252l3.216-.965c.923-.277 1.962.204 2.257 1.187l4.146 13.82c.296.984-.307 1.957-1.23 2.234l-3.217.965c-.923.277-1.962-.203-2.257-1.187L13 10.005v10.21c0 1.04-.878 1.785-1.834 1.785H7.833c-.291 0-.575-.07-.83-.195A1.849 1.849 0 0 1 6.18 22H2.821C1.857 22 1 21.241 1 20.214V3.786ZM3 4v11h3V4H3Zm0 16v-3h3v3H3Zm15.075-.04-.814-2.712 2.874-.862.813 2.712-2.873.862Zm1.485-5.49-2.874.862-2.634-8.782 2.873-.862 2.635 8.782ZM8 20V6h3v14H8Z" clip-rule="evenodd"/></symbol><symbol id="icon-eds-i-calendar-acceptance-medium" viewBox="0 0 24 24"><path d="M17 2a1 1 0 0 1 1 1v1h1.5C20.817 4 22 5.183 22 6.5v13c0 1.317-1.183 2.5-2.5 2.5h-15C3.183 22 2 20.817 2 19.5v-13C2 5.183 3.183 4 4.5 4a1 1 0 1 1 0 2c-.212 0-.5.288-.5.5v13c0 .212.288.5.5.5h15c.212 0 .5-.288.5-.5v-13c0-.212-.288-.5-.5-.5H18v1a1 1 0 0 1-2 0V3a1 1 0 0 1 1-1Zm-.534 7.747a1 1 0 0 1 .094 1.412l-4.846 5.538a1 1 0 0 1-1.352.141l-2.77-2.076a1 1 0 0 1 1.2-1.6l2.027 1.519 4.236-4.84a1 1 0 0 1 1.411-.094ZM7.5 2a1 1 0 0 1 1 1v1H14a1 1 0 0 1 0 2H8.5v1a1 1 0 1 1-2 0V3a1 1 0 0 1 1-1Z"/></symbol><symbol id="icon-eds-i-calendar-date-medium" viewBox="0 0 24 24"><path d="M17 2a1 1 0 0 1 1 1v1h1.5C20.817 4 22 5.183 22 6.5v13c0 1.317-1.183 2.5-2.5 2.5h-15C3.183 22 2 20.817 2 19.5v-13C2 5.183 3.183 4 4.5 4a1 1 0 1 1 0 2c-.212 0-.5.288-.5.5v13c0 .212.288.5.5.5h15c.212 0 .5-.288.5-.5v-13c0-.212-.288-.5-.5-.5H18v1a1 1 0 0 1-2 0V3a1 1 0 0 1 1-1ZM8 15a1 1 0 1 1 0 2 1 1 0 0 1 0-2Zm4 0a1 1 0 1 1 0 2 1 1 0 0 1 0-2Zm-4-4a1 1 0 1 1 0 2 1 1 0 0 1 0-2Zm4 0a1 1 0 1 1 0 2 1 1 0 0 1 0-2Zm4 0a1 1 0 1 1 0 2 1 1 0 0 1 0-2ZM7.5 2a1 1 0 0 1 1 1v1H14a1 1 0 0 1 0 2H8.5v1a1 1 0 1 1-2 0V3a1 1 0 0 1 1-1Z"/></symbol><symbol id="icon-eds-i-calendar-decision-medium" viewBox="0 0 24 24"><path d="M17 2a1 1 0 0 1 1 1v1h1.5C20.817 4 22 5.183 22 6.5v13c0 1.317-1.183 2.5-2.5 2.5h-15C3.183 22 2 20.817 2 19.5v-13C2 5.183 3.183 4 4.5 4a1 1 0 1 1 0 2c-.212 0-.5.288-.5.5v13c0 .212.288.5.5.5h15c.212 0 .5-.288.5-.5v-13c0-.212-.288-.5-.5-.5H18v1a1 1 0 0 1-2 0V3a1 1 0 0 1 1-1Zm-2.935 8.246 2.686 2.645c.34.335.34.883 0 1.218l-2.686 2.645a.858.858 0 0 1-1.213-.009.854.854 0 0 1 .009-1.21l1.05-1.035H7.984a.992.992 0 0 1-.984-1c0-.552.44-1 .984-1h5.928l-1.051-1.036a.854.854 0 0 1-.085-1.121l.076-.088a.858.858 0 0 1 1.213-.009ZM7.5 2a1 1 0 0 1 1 1v1H14a1 1 0 0 1 0 2H8.5v1a1 1 0 1 1-2 0V3a1 1 0 0 1 1-1Z"/></symbol><symbol id="icon-eds-i-calendar-impact-factor-medium" viewBox="0 0 24 24"><path d="M17 2a1 1 0 0 1 1 1v1h1.5C20.817 4 22 5.183 22 6.5v13c0 1.317-1.183 2.5-2.5 2.5h-15C3.183 22 2 20.817 2 19.5v-13C2 5.183 3.183 4 4.5 4a1 1 0 1 1 0 2c-.212 0-.5.288-.5.5v13c0 .212.288.5.5.5h15c.212 0 .5-.288.5-.5v-13c0-.212-.288-.5-.5-.5H18v1a1 1 0 0 1-2 0V3a1 1 0 0 1 1-1Zm-3.2 6.924a.48.48 0 0 1 .125.544l-1.52 3.283h2.304c.27 0 .491.215.491.483a.477.477 0 0 1-.13.327l-4.18 4.484a.498.498 0 0 1-.69.031.48.48 0 0 1-.125-.544l1.52-3.284H9.291a.487.487 0 0 1-.491-.482c0-.121.047-.238.13-.327l4.18-4.484a.498.498 0 0 1 .69-.031ZM7.5 2a1 1 0 0 1 1 1v1H14a1 1 0 0 1 0 2H8.5v1a1 1 0 1 1-2 0V3a1 1 0 0 1 1-1Z"/></symbol><symbol id="icon-eds-i-call-papers-medium" viewBox="0 0 24 24"><g><path d="m20.707 2.883-1.414 1.414a1 1 0 0 0 1.414 1.414l1.414-1.414a1 1 0 0 0-1.414-1.414Z"/><path d="M6 16.054c0 2.026 1.052 2.943 3 2.943a1 1 0 1 1 0 2c-2.996 0-5-1.746-5-4.943v-1.227a4.068 4.068 0 0 1-1.83-1.189 4.553 4.553 0 0 1-.87-1.455 4.868 4.868 0 0 1-.3-1.686c0-1.17.417-2.298 1.17-3.14.38-.426.834-.767 1.338-1 .51-.237 1.06-.36 1.617-.36L6.632 6H7l7.932-2.895A2.363 2.363 0 0 1 18 5.36v9.28a2.36 2.36 0 0 1-3.069 2.25l.084.03L7 14.997H6v1.057Zm9.637-11.057a.415.415 0 0 0-.083.008L8 7.638v5.536l7.424 1.786.104.02c.035.01.072.02.109.02.2 0 .363-.16.363-.36V5.36c0-.2-.163-.363-.363-.363Zm-9.638 3h-.874a1.82 1.82 0 0 0-.625.111l-.15.063a2.128 2.128 0 0 0-.689.517c-.42.47-.661 1.123-.661 1.81 0 .34.06.678.176.992.114.308.28.585.485.816.4.447.925.691 1.464.691h.874v-5Z" clip-rule="evenodd"/><path d="M20 8.997h2a1 1 0 1 1 0 2h-2a1 1 0 1 1 0-2ZM20.707 14.293l1.414 1.414a1 1 0 0 1-1.414 1.414l-1.414-1.414a1 1 0 0 1 1.414-1.414Z"/></g></symbol><symbol id="icon-eds-i-card-medium" viewBox="0 0 24 24"><path d="M19.615 2c.315 0 .716.067 1.14.279.76.38 1.245 1.107 1.245 2.106v15.23c0 .315-.067.716-.279 1.14-.38.76-1.107 1.245-2.106 1.245H4.385a2.56 2.56 0 0 1-1.14-.279C2.485 21.341 2 20.614 2 19.615V4.385c0-.315.067-.716.279-1.14C2.659 2.485 3.386 2 4.385 2h15.23Zm0 2H4.385c-.213 0-.265.034-.317.14A.71.71 0 0 0 4 4.385v15.23c0 .213.034.265.14.317a.71.71 0 0 0 .245.068h15.23c.213 0 .265-.034.317-.14a.71.71 0 0 0 .068-.245V4.385c0-.213-.034-.265-.14-.317A.71.71 0 0 0 19.615 4ZM17 16a1 1 0 0 1 0 2H7a1 1 0 0 1 0-2h10Zm0-3a1 1 0 0 1 0 2H7a1 1 0 0 1 0-2h10Zm-.5-7A1.5 1.5 0 0 1 18 7.5v3a1.5 1.5 0 0 1-1.5 1.5h-9A1.5 1.5 0 0 1 6 10.5v-3A1.5 1.5 0 0 1 7.5 6h9ZM16 8H8v2h8V8Z"/></symbol><symbol id="icon-eds-i-cart-medium" viewBox="0 0 24 24"><path d="M5.76 1a1 1 0 0 1 .994.902L7.155 6h13.34c.18 0 .358.02.532.057l.174.045a2.5 2.5 0 0 1 1.693 3.103l-2.069 7.03c-.36 1.099-1.398 1.823-2.49 1.763H8.65c-1.272.015-2.352-.927-2.546-2.244L4.852 3H2a1 1 0 0 1-.993-.883L1 2a1 1 0 0 1 1-1h3.76Zm2.328 14.51a.555.555 0 0 0 .55.488l9.751.001a.533.533 0 0 0 .527-.357l2.059-7a.5.5 0 0 0-.48-.642H7.351l.737 7.51ZM18 19a2 2 0 1 1 0 4 2 2 0 0 1 0-4ZM8 19a2 2 0 1 1 0 4 2 2 0 0 1 0-4Z"/></symbol><symbol id="icon-eds-i-check-circle-medium" viewBox="0 0 24 24"><path d="M12 1c6.075 0 11 4.925 11 11s-4.925 11-11 11S1 18.075 1 12 5.925 1 12 1Zm0 2a9 9 0 1 0 0 18 9 9 0 0 0 0-18Zm5.125 4.72a1 1 0 0 1 .156 1.405l-6 7.5a1 1 0 0 1-1.421.143l-3-2.5a1 1 0 0 1 1.28-1.536l2.217 1.846 5.362-6.703a1 1 0 0 1 1.406-.156Z"/></symbol><symbol id="icon-eds-i-check-filled-medium" viewBox="0 0 24 24"><path d="M12 1c6.075 0 11 4.925 11 11s-4.925 11-11 11S1 18.075 1 12 5.925 1 12 1Zm5.125 6.72a1 1 0 0 0-1.406.155l-5.362 6.703-2.217-1.846a1 1 0 1 0-1.28 1.536l3 2.5a1 1 0 0 0 1.42-.143l6-7.5a1 1 0 0 0-.155-1.406Z"/></symbol><symbol id="icon-eds-i-chevron-down-medium" viewBox="0 0 24 24"><path d="M3.305 8.28a1 1 0 0 0-.024 1.415l7.495 7.762c.314.345.757.543 1.224.543.467 0 .91-.198 1.204-.522l7.515-7.783a1 1 0 1 0-1.438-1.39L12 15.845l-7.28-7.54A1 1 0 0 0 3.4 8.2l-.096.082Z"/></symbol><symbol id="icon-eds-i-chevron-down-small" viewBox="0 0 16 16"><path d="M13.692 5.278a1 1 0 0 1 .03 1.414L9.103 11.51a1.491 1.491 0 0 1-2.188.019L2.278 6.692a1 1 0 0 1 1.444-1.384L8 9.771l4.278-4.463a1 1 0 0 1 1.318-.111l.096.081Z"/></symbol><symbol id="icon-eds-i-chevron-left-medium" viewBox="0 0 24 24"><path d="M15.72 3.305a1 1 0 0 0-1.415-.024l-7.762 7.495A1.655 1.655 0 0 0 6 12c0 .467.198.91.522 1.204l7.783 7.515a1 1 0 1 0 1.39-1.438L8.155 12l7.54-7.28A1 1 0 0 0 15.8 3.4l-.082-.096Z"/></symbol><symbol id="icon-eds-i-chevron-left-small" viewBox="0 0 16 16"><path d="M10.722 2.308a1 1 0 0 0-1.414-.03L4.49 6.897a1.491 1.491 0 0 0-.019 2.188l4.838 4.637a1 1 0 1 0 1.384-1.444L6.229 8l4.463-4.278a1 1 0 0 0 .111-1.318l-.081-.096Z"/></symbol><symbol id="icon-eds-i-chevron-right-medium" viewBox="0 0 24 24"><path d="M8.28 3.305a1 1 0 0 1 1.415-.024l7.762 7.495c.345.314.543.757.543 1.224 0 .467-.198.91-.522 1.204l-7.783 7.515a1 1 0 1 1-1.39-1.438L15.845 12l-7.54-7.28A1 1 0 0 1 8.2 3.4l.082-.096Z"/></symbol><symbol id="icon-eds-i-chevron-right-small" viewBox="0 0 16 16"><path d="M5.278 2.308a1 1 0 0 1 1.414-.03l4.819 4.619a1.491 1.491 0 0 1 .019 2.188l-4.838 4.637a1 1 0 1 1-1.384-1.444L9.771 8 5.308 3.722a1 1 0 0 1-.111-1.318l.081-.096Z"/></symbol><symbol id="icon-eds-i-chevron-up-medium" viewBox="0 0 24 24"><path d="M20.695 15.72a1 1 0 0 0 .024-1.415l-7.495-7.762A1.655 1.655 0 0 0 12 6c-.467 0-.91.198-1.204.522l-7.515 7.783a1 1 0 1 0 1.438 1.39L12 8.155l7.28 7.54a1 1 0 0 0 1.319.106l.096-.082Z"/></symbol><symbol id="icon-eds-i-chevron-up-small" viewBox="0 0 16 16"><path d="M13.692 10.722a1 1 0 0 0 .03-1.414L9.103 4.49a1.491 1.491 0 0 0-2.188-.019L2.278 9.308a1 1 0 0 0 1.444 1.384L8 6.229l4.278 4.463a1 1 0 0 0 1.318.111l.096-.081Z"/></symbol><symbol id="icon-eds-i-citations-medium" viewBox="0 0 24 24"><path d="M15.59 1a1 1 0 0 1 .706.291l5.41 5.385a1 1 0 0 1 .294.709v13.077c0 .674-.269 1.32-.747 1.796a2.549 2.549 0 0 1-1.798.742h-5.843a1 1 0 1 1 0-2h5.843a.549.549 0 0 0 .387-.16.535.535 0 0 0 .158-.378V7.8L15.178 3H5.545a.543.543 0 0 0-.538.451L5 3.538v8.607a1 1 0 0 1-2 0V3.538A2.542 2.542 0 0 1 5.545 1h10.046ZM5.483 14.35c.197.26.17.62-.049.848l-.095.083-.016.011c-.36.24-.628.45-.804.634-.393.409-.59.93-.59 1.562.077-.019.192-.028.345-.028.442 0 .84.158 1.195.474.355.316.532.716.532 1.2 0 .501-.173.9-.518 1.198-.345.298-.767.446-1.266.446-.672 0-1.209-.195-1.612-.585-.403-.39-.604-.976-.604-1.757 0-.744.11-1.39.33-1.938.222-.549.49-1.009.807-1.38a4.28 4.28 0 0 1 .992-.88c.07-.043.148-.087.232-.133a.881.881 0 0 1 1.121.245Zm5 0c.197.26.17.62-.049.848l-.095.083-.016.011c-.36.24-.628.45-.804.634-.393.409-.59.93-.59 1.562.077-.019.192-.028.345-.028.442 0 .84.158 1.195.474.355.316.532.716.532 1.2 0 .501-.173.9-.518 1.198-.345.298-.767.446-1.266.446-.672 0-1.209-.195-1.612-.585-.403-.39-.604-.976-.604-1.757 0-.744.11-1.39.33-1.938.222-.549.49-1.009.807-1.38a4.28 4.28 0 0 1 .992-.88c.07-.043.148-.087.232-.133a.881.881 0 0 1 1.121.245Z"/></symbol><symbol id="icon-eds-i-clipboard-check-medium" viewBox="0 0 24 24"><path d="M14.4 1c1.238 0 2.274.865 2.536 2.024L18.5 3C19.886 3 21 4.14 21 5.535v14.93C21 21.86 19.886 23 18.5 23h-13C4.114 23 3 21.86 3 20.465V5.535C3 4.14 4.114 3 5.5 3h1.57c.27-1.147 1.3-2 2.53-2h4.8Zm4.115 4-1.59.024A2.601 2.601 0 0 1 14.4 7H9.6c-1.23 0-2.26-.853-2.53-2H5.5c-.27 0-.5.234-.5.535v14.93c0 .3.23.535.5.535h13c.27 0 .5-.234.5-.535V5.535c0-.3-.23-.535-.485-.535Zm-1.909 4.205a1 1 0 0 1 .19 1.401l-5.334 7a1 1 0 0 1-1.344.23l-2.667-1.75a1 1 0 1 1 1.098-1.672l1.887 1.238 4.769-6.258a1 1 0 0 1 1.401-.19ZM14.4 3H9.6a.6.6 0 0 0-.6.6v.8a.6.6 0 0 0 .6.6h4.8a.6.6 0 0 0 .6-.6v-.8a.6.6 0 0 0-.6-.6Z"/></symbol><symbol id="icon-eds-i-clipboard-report-medium" viewBox="0 0 24 24"><path d="M14.4 1c1.238 0 2.274.865 2.536 2.024L18.5 3C19.886 3 21 4.14 21 5.535v14.93C21 21.86 19.886 23 18.5 23h-13C4.114 23 3 21.86 3 20.465V5.535C3 4.14 4.114 3 5.5 3h1.57c.27-1.147 1.3-2 2.53-2h4.8Zm4.115 4-1.59.024A2.601 2.601 0 0 1 14.4 7H9.6c-1.23 0-2.26-.853-2.53-2H5.5c-.27 0-.5.234-.5.535v14.93c0 .3.23.535.5.535h13c.27 0 .5-.234.5-.535V5.535c0-.3-.23-.535-.485-.535Zm-2.658 10.929a1 1 0 0 1 0 2H8a1 1 0 0 1 0-2h7.857Zm0-3.929a1 1 0 0 1 0 2H8a1 1 0 0 1 0-2h7.857ZM14.4 3H9.6a.6.6 0 0 0-.6.6v.8a.6.6 0 0 0 .6.6h4.8a.6.6 0 0 0 .6-.6v-.8a.6.6 0 0 0-.6-.6Z"/></symbol><symbol id="icon-eds-i-close-medium" viewBox="0 0 24 24"><path d="M12 1c6.075 0 11 4.925 11 11s-4.925 11-11 11S1 18.075 1 12 5.925 1 12 1Zm0 2a9 9 0 1 0 0 18 9 9 0 0 0 0-18ZM8.707 7.293 12 10.585l3.293-3.292a1 1 0 0 1 1.414 1.414L13.415 12l3.292 3.293a1 1 0 0 1-1.414 1.414L12 13.415l-3.293 3.292a1 1 0 1 1-1.414-1.414L10.585 12 7.293 8.707a1 1 0 0 1 1.414-1.414Z"/></symbol><symbol id="icon-eds-i-cloud-upload-medium" viewBox="0 0 24 24"><path d="m12.852 10.011.028-.004L13 10l.075.003.126.017.086.022.136.052.098.052.104.074.082.073 3 3a1 1 0 0 1 0 1.414l-.094.083a1 1 0 0 1-1.32-.083L14 13.416V20a1 1 0 0 1-2 0v-6.586l-1.293 1.293a1 1 0 0 1-1.32.083l-.094-.083a1 1 0 0 1 0-1.414l3-3 .112-.097.11-.071.114-.054.105-.035.118-.025Zm.587-7.962c3.065.362 5.497 2.662 5.992 5.562l.013.085.207.073c2.117.782 3.496 2.845 3.337 5.097l-.022.226c-.297 2.561-2.503 4.491-5.124 4.502a1 1 0 1 1-.009-2c1.619-.007 2.967-1.186 3.147-2.733.179-1.542-.86-2.979-2.487-3.353-.512-.149-.894-.579-.981-1.165-.21-2.237-2-4.035-4.308-4.308-2.31-.273-4.497 1.06-5.25 3.19l-.049.113c-.234.468-.718.756-1.176.743-1.418.057-2.689.857-3.32 2.084a3.668 3.668 0 0 0 .262 3.798c.796 1.136 2.169 1.764 3.583 1.635a1 1 0 1 1 .182 1.992c-2.125.194-4.193-.753-5.403-2.48a5.668 5.668 0 0 1-.403-5.86c.85-1.652 2.449-2.79 4.323-3.092l.287-.039.013-.028c1.207-2.741 4.125-4.404 7.186-4.042Z"/></symbol><symbol id="icon-eds-i-collection-medium" viewBox="0 0 24 24"><path d="M21 7a1 1 0 0 1 1 1v12.5a2.5 2.5 0 0 1-2.5 2.5H8a1 1 0 0 1 0-2h11.5a.5.5 0 0 0 .5-.5V8a1 1 0 0 1 1-1Zm-5.5-5A2.5 2.5 0 0 1 18 4.5v12a2.5 2.5 0 0 1-2.5 2.5h-11A2.5 2.5 0 0 1 2 16.5v-12A2.5 2.5 0 0 1 4.5 2h11Zm0 2h-11a.5.5 0 0 0-.5.5v12a.5.5 0 0 0 .5.5h11a.5.5 0 0 0 .5-.5v-12a.5.5 0 0 0-.5-.5ZM13 13a1 1 0 0 1 0 2H7a1 1 0 0 1 0-2h6Zm0-3.5a1 1 0 0 1 0 2H7a1 1 0 0 1 0-2h6ZM13 6a1 1 0 0 1 0 2H7a1 1 0 1 1 0-2h6Z"/></symbol><symbol id="icon-eds-i-conference-series-medium" viewBox="0 0 24 24"><path fill-rule="evenodd" d="M4.5 2A2.5 2.5 0 0 0 2 4.5v11A2.5 2.5 0 0 0 4.5 18h2.37l-2.534 2.253a1 1 0 0 0 1.328 1.494L9.88 18H11v3a1 1 0 1 0 2 0v-3h1.12l4.216 3.747a1 1 0 0 0 1.328-1.494L17.13 18h2.37a2.5 2.5 0 0 0 2.5-2.5v-11A2.5 2.5 0 0 0 19.5 2h-15ZM20 6V4.5a.5.5 0 0 0-.5-.5h-15a.5.5 0 0 0-.5.5V6h16ZM4 8v7.5a.5.5 0 0 0 .5.5h15a.5.5 0 0 0 .5-.5V8H4Z" clip-rule="evenodd"/></symbol><symbol id="icon-eds-i-delivery-medium" viewBox="0 0 24 24"><path d="M8.51 20.598a3.037 3.037 0 0 1-3.02 0A2.968 2.968 0 0 1 4.161 19L3.5 19A2.5 2.5 0 0 1 1 16.5v-11A2.5 2.5 0 0 1 3.5 3h10a2.5 2.5 0 0 1 2.45 2.004L16 5h2.527c.976 0 1.855.585 2.27 1.49l2.112 4.62a1 1 0 0 1 .091.416v4.856C23 17.814 21.889 19 20.484 19h-.523a1.01 1.01 0 0 1-.121-.007 2.96 2.96 0 0 1-1.33 1.605 3.037 3.037 0 0 1-3.02 0A2.968 2.968 0 0 1 14.161 19H9.838a2.968 2.968 0 0 1-1.327 1.597Zm-2.024-3.462a.955.955 0 0 0-.481.73L5.999 18l.001.022a.944.944 0 0 0 .388.777l.098.065c.316.181.712.181 1.028 0A.97.97 0 0 0 8 17.978a.95.95 0 0 0-.486-.842 1.037 1.037 0 0 0-1.028 0Zm10 0a.955.955 0 0 0-.481.73l-.005.156a.944.944 0 0 0 .388.777l.098.065c.316.181.712.181 1.028 0a.97.97 0 0 0 .486-.886.95.95 0 0 0-.486-.842 1.037 1.037 0 0 0-1.028 0ZM21 12h-5v3.17a3.038 3.038 0 0 1 2.51.232 2.993 2.993 0 0 1 1.277 1.45l.058.155.058-.005.581-.002c.27 0 .516-.263.516-.618V12Zm-7.5-7h-10a.5.5 0 0 0-.5.5v11a.5.5 0 0 0 .5.5h.662a2.964 2.964 0 0 1 1.155-1.491l.172-.107a3.037 3.037 0 0 1 3.022 0A2.987 2.987 0 0 1 9.843 17H13.5a.5.5 0 0 0 .5-.5v-11a.5.5 0 0 0-.5-.5Zm5.027 2H16v3h4.203l-1.224-2.677a.532.532 0 0 0-.375-.316L18.527 7Z"/></symbol><symbol id="icon-eds-i-download-medium" viewBox="0 0 24 24"><path d="M22 18.5a3.5 3.5 0 0 1-3.5 3.5h-13A3.5 3.5 0 0 1 2 18.5V18a1 1 0 0 1 2 0v.5A1.5 1.5 0 0 0 5.5 20h13a1.5 1.5 0 0 0 1.5-1.5V18a1 1 0 0 1 2 0v.5Zm-3.293-7.793-6 6-.063.059-.093.069-.081.048-.105.049-.104.034-.056.013-.118.017L12 17l-.076-.003-.122-.017-.113-.03-.085-.032-.063-.03-.098-.058-.06-.043-.05-.043-6.04-6.037a1 1 0 0 1 1.414-1.414l4.294 4.29L11 3a1 1 0 0 1 2 0l.001 10.585 4.292-4.292a1 1 0 0 1 1.32-.083l.094.083a1 1 0 0 1 0 1.414Z"/></symbol><symbol id="icon-eds-i-edit-medium" viewBox="0 0 24 24"><path d="M17.149 2a2.38 2.38 0 0 1 1.699.711l2.446 2.46a2.384 2.384 0 0 1 .005 3.38L10.01 19.906a1 1 0 0 1-.434.257l-6.3 1.8a1 1 0 0 1-1.237-1.237l1.8-6.3a1 1 0 0 1 .257-.434L15.443 2.718A2.385 2.385 0 0 1 17.15 2Zm-3.874 5.689-7.586 7.536-1.234 4.319 4.318-1.234 7.54-7.582-3.038-3.039ZM17.149 4a.395.395 0 0 0-.286.126L14.695 6.28l3.029 3.029 2.162-2.173a.384.384 0 0 0 .106-.197L20 6.864c0-.103-.04-.2-.119-.278l-2.457-2.47A.385.385 0 0 0 17.149 4Z"/></symbol><symbol id="icon-eds-i-education-medium" viewBox="0 0 24 24"><path fill-rule="evenodd" d="M12.41 2.088a1 1 0 0 0-.82 0l-10 4.5a1 1 0 0 0 0 1.824L3 9.047v7.124A3.001 3.001 0 0 0 4 22a3 3 0 0 0 1-5.83V9.948l1 .45V14.5a1 1 0 0 0 .087.408L7 14.5c-.913.408-.912.41-.912.41l.001.003.003.006.007.015a1.988 1.988 0 0 0 .083.16c.054.097.131.225.236.373.21.297.53.68.993 1.057C8.351 17.292 9.824 18 12 18c2.176 0 3.65-.707 4.589-1.476.463-.378.783-.76.993-1.057a4.162 4.162 0 0 0 .319-.533l.007-.015.003-.006v-.003h.002s0-.002-.913-.41l.913.408A1 1 0 0 0 18 14.5v-4.103l4.41-1.985a1 1 0 0 0 0-1.824l-10-4.5ZM16 11.297l-3.59 1.615a1 1 0 0 1-.82 0L8 11.297v2.94a3.388 3.388 0 0 0 .677.739C9.267 15.457 10.294 16 12 16s2.734-.543 3.323-1.024a3.388 3.388 0 0 0 .677-.739v-2.94ZM4.437 7.5 12 4.097 19.563 7.5 12 10.903 4.437 7.5ZM3 19a1 1 0 1 1 2 0 1 1 0 0 1-2 0Z" clip-rule="evenodd"/></symbol><symbol id="icon-eds-i-error-diamond-medium" viewBox="0 0 24 24"><path d="M12.002 1c.702 0 1.375.279 1.871.775l8.35 8.353a2.646 2.646 0 0 1 .001 3.744l-8.353 8.353a2.646 2.646 0 0 1-3.742 0l-8.353-8.353a2.646 2.646 0 0 1 0-3.744l8.353-8.353.156-.142c.424-.362.952-.58 1.507-.625l.21-.008Zm0 2a.646.646 0 0 0-.38.123l-.093.08-8.34 8.34a.646.646 0 0 0-.18.355L3 12c0 .171.068.336.19.457l8.353 8.354a.646.646 0 0 0 .914 0l8.354-8.354a.646.646 0 0 0-.001-.914l-8.351-8.354A.646.646 0 0 0 12.002 3ZM12 14.5a1.5 1.5 0 0 1 .144 2.993L12 17.5a1.5 1.5 0 0 1 0-3ZM12 6a1 1 0 0 1 1 1v5a1 1 0 0 1-2 0V7a1 1 0 0 1 1-1Z"/></symbol><symbol id="icon-eds-i-error-filled-medium" viewBox="0 0 24 24"><path d="M12.002 1c.702 0 1.375.279 1.871.775l8.35 8.353a2.646 2.646 0 0 1 .001 3.744l-8.353 8.353a2.646 2.646 0 0 1-3.742 0l-8.353-8.353a2.646 2.646 0 0 1 0-3.744l8.353-8.353.156-.142c.424-.362.952-.58 1.507-.625l.21-.008ZM12 14.5a1.5 1.5 0 0 0 0 3l.144-.007A1.5 1.5 0 0 0 12 14.5ZM12 6a1 1 0 0 0-1 1v5a1 1 0 0 0 2 0V7a1 1 0 0 0-1-1Z"/></symbol><symbol id="icon-eds-i-external-link-medium" viewBox="0 0 24 24"><path d="M9 2a1 1 0 1 1 0 2H4.6c-.371 0-.6.209-.6.5v15c0 .291.229.5.6.5h14.8c.371 0 .6-.209.6-.5V15a1 1 0 0 1 2 0v4.5c0 1.438-1.162 2.5-2.6 2.5H4.6C3.162 22 2 20.938 2 19.5v-15C2 3.062 3.162 2 4.6 2H9Zm6 0h6l.075.003.126.017.111.03.111.044.098.052.096.067.09.08c.036.035.068.073.097.112l.071.11.054.114.035.105.03.148L22 3v6a1 1 0 0 1-2 0V5.414l-6.693 6.693a1 1 0 0 1-1.414-1.414L18.584 4H15a1 1 0 0 1-.993-.883L14 3a1 1 0 0 1 1-1Z"/></symbol><symbol id="icon-eds-i-external-link-small" viewBox="0 0 16 16"><path d="M5 1a1 1 0 1 1 0 2l-2-.001V13L13 13v-2a1 1 0 0 1 2 0v2c0 1.15-.93 2-2.067 2H3.067C1.93 15 1 14.15 1 13V3c0-1.15.93-2 2.067-2H5Zm4 0h5l.075.003.126.017.111.03.111.044.098.052.096.067.09.08.044.047.073.093.051.083.054.113.035.105.03.148L15 2v5a1 1 0 0 1-2 0V4.414L9.107 8.307a1 1 0 0 1-1.414-1.414L11.584 3H9a1 1 0 0 1-.993-.883L8 2a1 1 0 0 1 1-1Z"/></symbol><symbol id="icon-eds-i-file-download-medium" viewBox="0 0 24 24"><path d="M14.5 1a1 1 0 0 1 .707.293l5.5 5.5A1 1 0 0 1 21 7.5v12.962A2.542 2.542 0 0 1 18.455 23H5.545A2.542 2.542 0 0 1 3 20.462V3.538A2.542 2.542 0 0 1 5.545 1H14.5Zm-.415 2h-8.54A.542.542 0 0 0 5 3.538v16.924c0 .296.243.538.545.538h12.91a.542.542 0 0 0 .545-.538V7.915L14.085 3ZM12 7a1 1 0 0 1 1 1v6.585l2.293-2.292a1 1 0 0 1 1.32-.083l.094.083a1 1 0 0 1 0 1.414l-4 4a1.008 1.008 0 0 1-.112.097l-.11.071-.114.054-.105.035-.149.03L12 18l-.075-.003-.126-.017-.111-.03-.111-.044-.098-.052-.096-.067-.09-.08-4-4a1 1 0 0 1 1.414-1.414L11 14.585V8a1 1 0 0 1 1-1Z"/></symbol><symbol id="icon-eds-i-file-report-medium" viewBox="0 0 24 24"><path d="M14.5 1a1 1 0 0 1 .707.293l5.5 5.5A1 1 0 0 1 21 7.5v12.962c0 .674-.269 1.32-.747 1.796a2.549 2.549 0 0 1-1.798.742H5.545c-.674 0-1.32-.267-1.798-.742A2.535 2.535 0 0 1 3 20.462V3.538A2.542 2.542 0 0 1 5.545 1H14.5Zm-.415 2h-8.54A.542.542 0 0 0 5 3.538v16.924c0 .142.057.278.158.379.102.102.242.159.387.159h12.91a.549.549 0 0 0 .387-.16.535.535 0 0 0 .158-.378V7.915L14.085 3ZM16 17a1 1 0 0 1 0 2H8a1 1 0 0 1 0-2h8Zm0-3a1 1 0 0 1 0 2H8a1 1 0 0 1 0-2h8Zm-4.793-6.207L13 9.585l1.793-1.792a1 1 0 0 1 1.32-.083l.094.083a1 1 0 0 1 0 1.414l-2.5 2.5a1 1 0 0 1-1.414 0L10.5 9.915l-1.793 1.792a1 1 0 0 1-1.32.083l-.094-.083a1 1 0 0 1 0-1.414l2.5-2.5a1 1 0 0 1 1.414 0Z"/></symbol><symbol id="icon-eds-i-file-text-medium" viewBox="0 0 24 24"><path d="M14.5 1a1 1 0 0 1 .707.293l5.5 5.5A1 1 0 0 1 21 7.5v12.962A2.542 2.542 0 0 1 18.455 23H5.545A2.542 2.542 0 0 1 3 20.462V3.538A2.542 2.542 0 0 1 5.545 1H14.5Zm-.415 2h-8.54A.542.542 0 0 0 5 3.538v16.924c0 .296.243.538.545.538h12.91a.542.542 0 0 0 .545-.538V7.915L14.085 3ZM16 15a1 1 0 0 1 0 2H8a1 1 0 0 1 0-2h8Zm0-4a1 1 0 0 1 0 2H8a1 1 0 0 1 0-2h8Zm-5-4a1 1 0 0 1 0 2H8a1 1 0 1 1 0-2h3Z"/></symbol><symbol id="icon-eds-i-file-upload-medium" viewBox="0 0 24 24"><path d="M14.5 1a1 1 0 0 1 .707.293l5.5 5.5A1 1 0 0 1 21 7.5v12.962A2.542 2.542 0 0 1 18.455 23H5.545A2.542 2.542 0 0 1 3 20.462V3.538A2.542 2.542 0 0 1 5.545 1H14.5Zm-.415 2h-8.54A.542.542 0 0 0 5 3.538v16.924c0 .296.243.538.545.538h12.91a.542.542 0 0 0 .545-.538V7.915L14.085 3Zm-2.233 4.011.058-.007L12 7l.075.003.126.017.111.03.111.044.098.052.104.074.082.073 4 4a1 1 0 0 1 0 1.414l-.094.083a1 1 0 0 1-1.32-.083L13 10.415V17a1 1 0 0 1-2 0v-6.585l-2.293 2.292a1 1 0 0 1-1.32.083l-.094-.083a1 1 0 0 1 0-1.414l4-4 .112-.097.11-.071.114-.054.105-.035.118-.025Z"/></symbol><symbol id="icon-eds-i-filter-medium" viewBox="0 0 24 24"><path d="M21 2a1 1 0 0 1 .82 1.573L15 13.314V18a1 1 0 0 1-.31.724l-.09.076-4 3A1 1 0 0 1 9 21v-7.684L2.18 3.573a1 1 0 0 1 .707-1.567L3 2h18Zm-1.921 2H4.92l5.9 8.427a1 1 0 0 1 .172.45L11 13v6l2-1.5V13a1 1 0 0 1 .117-.469l.064-.104L19.079 4Z"/></symbol><symbol id="icon-eds-i-funding-medium" viewBox="0 0 24 24"><path fill-rule="evenodd" d="M23 8A7 7 0 1 0 9 8a7 7 0 0 0 14 0ZM9.006 12.225A4.07 4.07 0 0 0 6.12 11.02H2a.979.979 0 1 0 0 1.958h4.12c.558 0 1.094.222 1.489.617l2.207 2.288c.27.27.27.687.012.944a.656.656 0 0 1-.928 0L7.744 15.67a.98.98 0 0 0-1.386 1.384l1.157 1.158c.535.536 1.244.791 1.946.765l.041.002h6.922c.874 0 1.597.748 1.597 1.688 0 .203-.146.354-.309.354H7.755c-.487 0-.96-.178-1.339-.504L2.64 17.259a.979.979 0 0 0-1.28 1.482L5.137 22c.733.631 1.66.979 2.618.979h9.957c1.26 0 2.267-1.043 2.267-2.312 0-2.006-1.584-3.646-3.555-3.646h-4.529a2.617 2.617 0 0 0-.681-2.509l-2.208-2.287ZM16 3a5 5 0 1 0 0 10 5 5 0 0 0 0-10Zm.979 3.5a.979.979 0 1 0-1.958 0v3a.979.979 0 1 0 1.958 0v-3Z" clip-rule="evenodd"/></symbol><symbol id="icon-eds-i-hashtag-medium" viewBox="0 0 24 24"><path d="M12 1c6.075 0 11 4.925 11 11s-4.925 11-11 11S1 18.075 1 12 5.925 1 12 1Zm0 2a9 9 0 1 0 0 18 9 9 0 0 0 0-18ZM9.52 18.189a1 1 0 1 1-1.964-.378l.437-2.274H6a1 1 0 1 1 0-2h2.378l.592-3.076H6a1 1 0 0 1 0-2h3.354l.51-2.65a1 1 0 1 1 1.964.378l-.437 2.272h3.04l.51-2.65a1 1 0 1 1 1.964.378l-.438 2.272H18a1 1 0 0 1 0 2h-1.917l-.592 3.076H18a1 1 0 0 1 0 2h-2.893l-.51 2.652a1 1 0 1 1-1.964-.378l.437-2.274h-3.04l-.51 2.652Zm.895-4.652h3.04l.591-3.076h-3.04l-.591 3.076Z"/></symbol><symbol id="icon-eds-i-home-medium" viewBox="0 0 24 24"><path d="M5 22a1 1 0 0 1-1-1v-8.586l-1.293 1.293a1 1 0 0 1-1.32.083l-.094-.083a1 1 0 0 1 0-1.414l10-10a1 1 0 0 1 1.414 0l10 10a1 1 0 0 1-1.414 1.414L20 12.415V21a1 1 0 0 1-1 1H5Zm7-17.585-6 5.999V20h5v-4a1 1 0 0 1 2 0v4h5v-9.585l-6-6Z"/></symbol><symbol id="icon-eds-i-image-medium" viewBox="0 0 24 24"><path d="M19.615 2A2.385 2.385 0 0 1 22 4.385v15.23A2.385 2.385 0 0 1 19.615 22H4.385A2.385 2.385 0 0 1 2 19.615V4.385A2.385 2.385 0 0 1 4.385 2h15.23Zm0 2H4.385A.385.385 0 0 0 4 4.385v15.23c0 .213.172.385.385.385h1.244l10.228-8.76a1 1 0 0 1 1.254-.037L20 13.392V4.385A.385.385 0 0 0 19.615 4Zm-3.07 9.283L8.703 20h10.912a.385.385 0 0 0 .385-.385v-3.713l-3.455-2.619ZM9.5 6a3.5 3.5 0 1 1 0 7 3.5 3.5 0 0 1 0-7Zm0 2a1.5 1.5 0 1 0 0 3 1.5 1.5 0 0 0 0-3Z"/></symbol><symbol id="icon-eds-i-impact-factor-medium" viewBox="0 0 24 24"><path d="M16.49 2.672c.74.694.986 1.765.632 2.712l-.04.1-1.549 3.54h1.477a2.496 2.496 0 0 1 2.485 2.34l.005.163c0 .618-.23 1.21-.642 1.675l-7.147 7.961a2.48 2.48 0 0 1-3.554.165 2.512 2.512 0 0 1-.633-2.712l.042-.103L9.108 15H7.46c-1.393 0-2.379-1.11-2.455-2.369L5 12.473c0-.593.142-1.145.628-1.692l7.307-7.944a2.48 2.48 0 0 1 3.555-.165ZM14.43 4.164l-7.33 7.97c-.083.093-.101.214-.101.34 0 .277.19.526.46.526h4.163l.097-.009c.015 0 .03.003.046.009.181.078.264.32.186.5l-2.554 5.817a.512.512 0 0 0 .127.552.48.48 0 0 0 .69-.033l7.155-7.97a.513.513 0 0 0 .13-.34.497.497 0 0 0-.49-.502h-3.988a.355.355 0 0 1-.328-.497l2.555-5.844a.512.512 0 0 0-.127-.552.48.48 0 0 0-.69.033Z"/></symbol><symbol id="icon-eds-i-info-circle-medium" viewBox="0 0 24 24"><path d="M12 1c6.075 0 11 4.925 11 11s-4.925 11-11 11S1 18.075 1 12 5.925 1 12 1Zm0 2a9 9 0 1 0 0 18 9 9 0 0 0 0-18Zm0 7a1 1 0 0 1 1 1v5h1.5a1 1 0 0 1 0 2h-5a1 1 0 0 1 0-2H11v-4h-.5a1 1 0 0 1-.993-.883L9.5 11a1 1 0 0 1 1-1H12Zm0-4.5a1.5 1.5 0 0 1 .144 2.993L12 8.5a1.5 1.5 0 0 1 0-3Z"/></symbol><symbol id="icon-eds-i-info-filled-medium" viewBox="0 0 24 24"><path d="M12 1c6.075 0 11 4.925 11 11s-4.925 11-11 11S1 18.075 1 12 5.925 1 12 1Zm0 9h-1.5a1 1 0 0 0-1 1l.007.117A1 1 0 0 0 10.5 12h.5v4H9.5a1 1 0 0 0 0 2h5a1 1 0 0 0 0-2H13v-5a1 1 0 0 0-1-1Zm0-4.5a1.5 1.5 0 0 0 0 3l.144-.007A1.5 1.5 0 0 0 12 5.5Z"/></symbol><symbol id="icon-eds-i-journal-medium" viewBox="0 0 24 24"><path d="M18.5 1A2.5 2.5 0 0 1 21 3.5v14a2.5 2.5 0 0 1-2.5 2.5h-13a.5.5 0 1 0 0 1H20a1 1 0 0 1 0 2H5.5A2.5 2.5 0 0 1 3 20.5v-17A2.5 2.5 0 0 1 5.5 1h13ZM7 3H5.5a.5.5 0 0 0-.5.5v14.549l.016-.002c.104-.02.211-.035.32-.042L5.5 18H7V3Zm11.5 0H9v15h9.5a.5.5 0 0 0 .5-.5v-14a.5.5 0 0 0-.5-.5ZM16 5a1 1 0 0 1 1 1v4a1 1 0 0 1-1 1h-5a1 1 0 0 1-1-1V6a1 1 0 0 1 1-1h5Zm-1 2h-3v2h3V7Z"/></symbol><symbol id="icon-eds-i-mail-medium" viewBox="0 0 24 24"><path d="M20.462 3C21.875 3 23 4.184 23 5.619v12.762C23 19.816 21.875 21 20.462 21H3.538C2.125 21 1 19.816 1 18.381V5.619C1 4.184 2.125 3 3.538 3h16.924ZM21 8.158l-7.378 6.258a2.549 2.549 0 0 1-3.253-.008L3 8.16v10.222c0 .353.253.619.538.619h16.924c.285 0 .538-.266.538-.619V8.158ZM20.462 5H3.538c-.264 0-.5.228-.534.542l8.65 7.334c.2.165.492.165.684.007l8.656-7.342-.001-.025c-.044-.3-.274-.516-.531-.516Z"/></symbol><symbol id="icon-eds-i-mail-send-medium" viewBox="0 0 24 24"><path d="M20.444 5a2.562 2.562 0 0 1 2.548 2.37l.007.078.001.123v7.858A2.564 2.564 0 0 1 20.444 18H9.556A2.564 2.564 0 0 1 7 15.429l.001-7.977.007-.082A2.561 2.561 0 0 1 9.556 5h10.888ZM21 9.331l-5.46 3.51a1 1 0 0 1-1.08 0L9 9.332v6.097c0 .317.251.571.556.571h10.888a.564.564 0 0 0 .556-.571V9.33ZM20.444 7H9.556a.543.543 0 0 0-.32.105l5.763 3.706 5.766-3.706a.543.543 0 0 0-.32-.105ZM4.308 5a1 1 0 1 1 0 2H2a1 1 0 1 1 0-2h2.308Zm0 5.5a1 1 0 0 1 0 2H2a1 1 0 0 1 0-2h2.308Zm0 5.5a1 1 0 0 1 0 2H2a1 1 0 0 1 0-2h2.308Z"/></symbol><symbol id="icon-eds-i-mentions-medium" viewBox="0 0 24 24"><path d="m9.452 1.293 5.92 5.92 2.92-2.92a1 1 0 0 1 1.415 1.414l-2.92 2.92 5.92 5.92a1 1 0 0 1 0 1.415 10.371 10.371 0 0 1-10.378 2.584l.652 3.258A1 1 0 0 1 12 23H2a1 1 0 0 1-.874-1.486l4.789-8.62C4.194 9.074 4.9 4.43 8.038 1.292a1 1 0 0 1 1.414 0Zm-2.355 13.59L3.699 21h7.081l-.689-3.442a10.392 10.392 0 0 1-2.775-2.396l-.22-.28Zm1.69-11.427-.07.09a8.374 8.374 0 0 0 11.737 11.737l.089-.071L8.787 3.456Z"/></symbol><symbol id="icon-eds-i-menu-medium" viewBox="0 0 24 24"><path d="M21 4a1 1 0 0 1 0 2H3a1 1 0 1 1 0-2h18Zm-4 7a1 1 0 0 1 0 2H3a1 1 0 0 1 0-2h14Zm4 7a1 1 0 0 1 0 2H3a1 1 0 0 1 0-2h18Z"/></symbol><symbol id="icon-eds-i-metrics-medium" viewBox="0 0 24 24"><path d="M3 22a1 1 0 0 1-1-1V3a1 1 0 0 1 1-1h6a1 1 0 0 1 1 1v7h4V8a1 1 0 0 1 1-1h6a1 1 0 0 1 1 1v13a1 1 0 0 1-.883.993L21 22H3Zm17-2V9h-4v11h4Zm-6-8h-4v8h4v-8ZM8 4H4v16h4V4Z"/></symbol><symbol id="icon-eds-i-news-medium" viewBox="0 0 24 24"><path d="M17.384 3c.975 0 1.77.787 1.77 1.762v13.333c0 .462.354.846.815.899l.107.006.109-.006a.915.915 0 0 0 .809-.794l.006-.105V8.19a1 1 0 0 1 2 0v9.905A2.914 2.914 0 0 1 20.077 21H3.538a2.547 2.547 0 0 1-1.644-.601l-.147-.135A2.516 2.516 0 0 1 1 18.476V4.762C1 3.787 1.794 3 2.77 3h14.614Zm-.231 2H3v13.476c0 .11.035.216.1.304l.054.063c.101.1.24.157.384.157l13.761-.001-.026-.078a2.88 2.88 0 0 1-.115-.655l-.004-.17L17.153 5ZM14 15.021a.979.979 0 1 1 0 1.958H6a.979.979 0 1 1 0-1.958h8Zm0-8c.54 0 .979.438.979.979v4c0 .54-.438.979-.979.979H6A.979.979 0 0 1 5.021 12V8c0-.54.438-.979.979-.979h8Zm-.98 1.958H6.979v2.041h6.041V8.979Z"/></symbol><symbol id="icon-eds-i-newsletter-medium" viewBox="0 0 24 24"><path d="M21 10a1 1 0 0 1 1 1v9.5a2.5 2.5 0 0 1-2.5 2.5h-15A2.5 2.5 0 0 1 2 20.5V11a1 1 0 0 1 2 0v.439l8 4.888 8-4.889V11a1 1 0 0 1 1-1Zm-1 3.783-7.479 4.57a1 1 0 0 1-1.042 0l-7.48-4.57V20.5a.5.5 0 0 0 .501.5h15a.5.5 0 0 0 .5-.5v-6.717ZM15 9a1 1 0 0 1 0 2H9a1 1 0 0 1 0-2h6Zm2.5-8A2.5 2.5 0 0 1 20 3.5V9a1 1 0 0 1-2 0V3.5a.5.5 0 0 0-.5-.5h-11a.5.5 0 0 0-.5.5V9a1 1 0 1 1-2 0V3.5A2.5 2.5 0 0 1 6.5 1h11ZM15 5a1 1 0 0 1 0 2H9a1 1 0 1 1 0-2h6Z"/></symbol><symbol id="icon-eds-i-notifcation-medium" viewBox="0 0 24 24"><path d="M14 20a1 1 0 0 1 0 2h-4a1 1 0 0 1 0-2h4ZM3 18l-.133-.007c-1.156-.124-1.156-1.862 0-1.986l.3-.012C4.32 15.923 5 15.107 5 14V9.5C5 5.368 8.014 2 12 2s7 3.368 7 7.5V14c0 1.107.68 1.923 1.832 1.995l.301.012c1.156.124 1.156 1.862 0 1.986L21 18H3Zm9-14C9.17 4 7 6.426 7 9.5V14c0 .671-.146 1.303-.416 1.858L6.51 16h10.979l-.073-.142a4.192 4.192 0 0 1-.412-1.658L17 14V9.5C17 6.426 14.83 4 12 4Z"/></symbol><symbol id="icon-eds-i-publish-medium" viewBox="0 0 24 24"><g><path d="M16.296 1.291A1 1 0 0 0 15.591 1H5.545A2.542 2.542 0 0 0 3 3.538V13a1 1 0 1 0 2 0V3.538l.007-.087A.543.543 0 0 1 5.545 3h9.633L20 7.8v12.662a.534.534 0 0 1-.158.379.548.548 0 0 1-.387.159H11a1 1 0 1 0 0 2h8.455c.674 0 1.32-.267 1.798-.742A2.534 2.534 0 0 0 22 20.462V7.385a1 1 0 0 0-.294-.709l-5.41-5.385Z"/><path d="M10.762 16.647a1 1 0 0 0-1.525-1.294l-4.472 5.271-2.153-1.665a1 1 0 1 0-1.224 1.582l2.91 2.25a1 1 0 0 0 1.374-.144l5.09-6ZM16 10a1 1 0 1 1 0 2H8a1 1 0 1 1 0-2h8ZM12 7a1 1 0 0 0-1-1H8a1 1 0 1 0 0 2h3a1 1 0 0 0 1-1Z"/></g></symbol><symbol id="icon-eds-i-refresh-medium" viewBox="0 0 24 24"><g><path d="M7.831 5.636H6.032A8.76 8.76 0 0 1 9 3.631 8.549 8.549 0 0 1 12.232 3c.603 0 1.192.063 1.76.182C17.979 4.017 21 7.632 21 12a1 1 0 1 0 2 0c0-5.296-3.674-9.746-8.591-10.776A10.61 10.61 0 0 0 5 3.851V2.805a1 1 0 0 0-.987-1H4a1 1 0 0 0-1 1v3.831a1 1 0 0 0 1 1h3.831a1 1 0 0 0 .013-2h-.013ZM17.968 18.364c-1.59 1.632-3.784 2.636-6.2 2.636C6.948 21 3 16.993 3 12a1 1 0 1 0-2 0c0 6.053 4.799 11 10.768 11 2.788 0 5.324-1.082 7.232-2.85v1.045a1 1 0 1 0 2 0v-3.831a1 1 0 0 0-1-1h-3.831a1 1 0 0 0 0 2h1.799Z"/></g></symbol><symbol id="icon-eds-i-search-medium" viewBox="0 0 24 24"><path d="M11 1c5.523 0 10 4.477 10 10 0 2.4-.846 4.604-2.256 6.328l3.963 3.965a1 1 0 0 1-1.414 1.414l-3.965-3.963A9.959 9.959 0 0 1 11 21C5.477 21 1 16.523 1 11S5.477 1 11 1Zm0 2a8 8 0 1 0 0 16 8 8 0 0 0 0-16Z"/></symbol><symbol id="icon-eds-i-settings-medium" viewBox="0 0 24 24"><path d="M11.382 1h1.24a2.508 2.508 0 0 1 2.334 1.63l.523 1.378 1.59.933 1.444-.224c.954-.132 1.89.3 2.422 1.101l.095.155.598 1.066a2.56 2.56 0 0 1-.195 2.848l-.894 1.161v1.896l.92 1.163c.6.768.707 1.812.295 2.674l-.09.17-.606 1.08a2.504 2.504 0 0 1-2.531 1.25l-1.428-.223-1.589.932-.523 1.378a2.512 2.512 0 0 1-2.155 1.625L12.65 23h-1.27a2.508 2.508 0 0 1-2.334-1.63l-.524-1.379-1.59-.933-1.443.225c-.954.132-1.89-.3-2.422-1.101l-.095-.155-.598-1.066a2.56 2.56 0 0 1 .195-2.847l.891-1.161v-1.898l-.919-1.162a2.562 2.562 0 0 1-.295-2.674l.09-.17.606-1.08a2.504 2.504 0 0 1 2.531-1.25l1.43.223 1.618-.938.524-1.375.07-.167A2.507 2.507 0 0 1 11.382 1Zm.003 2a.509.509 0 0 0-.47.338l-.65 1.71a1 1 0 0 1-.434.51L7.6 6.85a1 1 0 0 1-.655.123l-1.762-.275a.497.497 0 0 0-.498.252l-.61 1.088a.562.562 0 0 0 .04.619l1.13 1.43a1 1 0 0 1 .216.62v2.585a1 1 0 0 1-.207.61L4.15 15.339a.568.568 0 0 0-.036.634l.601 1.072a.494.494 0 0 0 .484.26l1.78-.278a1 1 0 0 1 .66.126l2.2 1.292a1 1 0 0 1 .43.507l.648 1.71a.508.508 0 0 0 .467.338h1.263a.51.51 0 0 0 .47-.34l.65-1.708a1 1 0 0 1 .428-.507l2.201-1.292a1 1 0 0 1 .66-.126l1.763.275a.497.497 0 0 0 .498-.252l.61-1.088a.562.562 0 0 0-.04-.619l-1.13-1.43a1 1 0 0 1-.216-.62v-2.585a1 1 0 0 1 .207-.61l1.105-1.437a.568.568 0 0 0 .037-.634l-.601-1.072a.494.494 0 0 0-.484-.26l-1.78.278a1 1 0 0 1-.66-.126l-2.2-1.292a1 1 0 0 1-.43-.507l-.649-1.71A.508.508 0 0 0 12.62 3h-1.234ZM12 8a4 4 0 1 1 0 8 4 4 0 0 1 0-8Zm0 2a2 2 0 1 0 0 4 2 2 0 0 0 0-4Z"/></symbol><symbol id="icon-eds-i-shipping-medium" viewBox="0 0 24 24"><path d="M16.515 2c1.406 0 2.706.728 3.352 1.902l2.02 3.635.02.042.036.089.031.105.012.058.01.073.004.075v11.577c0 .64-.244 1.255-.683 1.713a2.356 2.356 0 0 1-1.701.731H4.386a2.356 2.356 0 0 1-1.702-.731 2.476 2.476 0 0 1-.683-1.713V7.948c.01-.217.083-.43.22-.6L4.2 3.905C4.833 2.755 6.089 2.032 7.486 2h9.029ZM20 9H4v10.556a.49.49 0 0 0 .075.26l.053.07a.356.356 0 0 0 .257.114h15.23c.094 0 .186-.04.258-.115a.477.477 0 0 0 .127-.33V9Zm-2 7.5a1 1 0 0 1 0 2h-4a1 1 0 0 1 0-2h4ZM16.514 4H13v3h6.3l-1.183-2.13c-.288-.522-.908-.87-1.603-.87ZM11 3.999H7.51c-.679.017-1.277.36-1.566.887L4.728 7H11V3.999Z"/></symbol><symbol id="icon-eds-i-step-guide-medium" viewBox="0 0 24 24"><path d="M11.394 9.447a1 1 0 1 0-1.788-.894l-.88 1.759-.019-.02a1 1 0 1 0-1.414 1.415l1 1a1 1 0 0 0 1.601-.26l1.5-3ZM12 11a1 1 0 0 1 1-1h3a1 1 0 1 1 0 2h-3a1 1 0 0 1-1-1ZM12 17a1 1 0 0 1 1-1h3a1 1 0 1 1 0 2h-3a1 1 0 0 1-1-1ZM10.947 14.105a1 1 0 0 1 .447 1.342l-1.5 3a1 1 0 0 1-1.601.26l-1-1a1 1 0 1 1 1.414-1.414l.02.019.879-1.76a1 1 0 0 1 1.341-.447Z"/><path d="M5.545 1A2.542 2.542 0 0 0 3 3.538v16.924A2.542 2.542 0 0 0 5.545 23h12.91A2.542 2.542 0 0 0 21 20.462V7.5a1 1 0 0 0-.293-.707l-5.5-5.5A1 1 0 0 0 14.5 1H5.545ZM5 3.538C5 3.245 5.24 3 5.545 3h8.54L19 7.914v12.547c0 .294-.24.539-.546.539H5.545A.542.542 0 0 1 5 20.462V3.538Z" clip-rule="evenodd"/></symbol><symbol id="icon-eds-i-submission-medium" viewBox="0 0 24 24"><g><path d="M5 3.538C5 3.245 5.24 3 5.545 3h9.633L20 7.8v12.662a.535.535 0 0 1-.158.379.549.549 0 0 1-.387.159H6a1 1 0 0 1-1-1v-2.5a1 1 0 1 0-2 0V20a3 3 0 0 0 3 3h13.455c.673 0 1.32-.266 1.798-.742A2.535 2.535 0 0 0 22 20.462V7.385a1 1 0 0 0-.294-.709l-5.41-5.385A1 1 0 0 0 15.591 1H5.545A2.542 2.542 0 0 0 3 3.538V7a1 1 0 0 0 2 0V3.538Z"/><path d="m13.707 13.707-4 4a1 1 0 0 1-1.414 0l-.083-.094a1 1 0 0 1 .083-1.32L10.585 14 2 14a1 1 0 1 1 0-2l8.583.001-2.29-2.294a1 1 0 0 1 1.414-1.414l4.037 4.04.043.05.043.06.059.098.03.063.031.085.03.113.017.122L14 13l-.004.087-.017.118-.013.056-.034.104-.049.105-.048.081-.07.093-.058.063Z"/></g></symbol><symbol id="icon-eds-i-table-1-medium" viewBox="0 0 24 24"><path d="M4.385 22a2.56 2.56 0 0 1-1.14-.279C2.485 21.341 2 20.614 2 19.615V4.385c0-.315.067-.716.279-1.14C2.659 2.485 3.386 2 4.385 2h15.23c.315 0 .716.067 1.14.279.76.38 1.245 1.107 1.245 2.106v15.23c0 .315-.067.716-.279 1.14-.38.76-1.107 1.245-2.106 1.245H4.385ZM4 19.615c0 .213.034.265.14.317a.71.71 0 0 0 .245.068H8v-4H4v3.615ZM20 16H10v4h9.615c.213 0 .265-.034.317-.14a.71.71 0 0 0 .068-.245V16Zm0-2v-4H10v4h10ZM4 14h4v-4H4v4ZM19.615 4H10v4h10V4.385c0-.213-.034-.265-.14-.317A.71.71 0 0 0 19.615 4ZM8 4H4.385l-.082.002c-.146.01-.19.047-.235.138A.71.71 0 0 0 4 4.385V8h4V4Z"/></symbol><symbol id="icon-eds-i-table-2-medium" viewBox="0 0 24 24"><path d="M4.384 22A2.384 2.384 0 0 1 2 19.616V4.384A2.384 2.384 0 0 1 4.384 2h15.232A2.384 2.384 0 0 1 22 4.384v15.232A2.384 2.384 0 0 1 19.616 22H4.384ZM10 15H4v4.616c0 .212.172.384.384.384H10v-5Zm5 0h-3v5h3v-5Zm5 0h-3v5h2.616a.384.384 0 0 0 .384-.384V15ZM10 9H4v4h6V9Zm5 0h-3v4h3V9Zm5 0h-3v4h3V9Zm-.384-5H4.384A.384.384 0 0 0 4 4.384V7h16V4.384A.384.384 0 0 0 19.616 4Z"/></symbol><symbol id="icon-eds-i-tag-medium" viewBox="0 0 24 24"><path d="m12.621 1.998.127.004L20.496 2a1.5 1.5 0 0 1 1.497 1.355L22 3.5l-.005 7.669c.038.456-.133.905-.447 1.206l-9.02 9.018a2.075 2.075 0 0 1-2.932 0l-6.99-6.99a2.075 2.075 0 0 1 .001-2.933L11.61 2.47c.246-.258.573-.418.881-.46l.131-.011Zm.286 2-8.885 8.886a.075.075 0 0 0 0 .106l6.987 6.988c.03.03.077.03.106 0l8.883-8.883L19.999 4l-7.092-.002ZM16 6.5a1.5 1.5 0 0 1 .144 2.993L16 9.5a1.5 1.5 0 0 1 0-3Z"/></symbol><symbol id="icon-eds-i-trash-medium" viewBox="0 0 24 24"><path d="M12 1c2.717 0 4.913 2.232 4.997 5H21a1 1 0 0 1 0 2h-1v12.5c0 1.389-1.152 2.5-2.556 2.5H6.556C5.152 23 4 21.889 4 20.5V8H3a1 1 0 1 1 0-2h4.003l.001-.051C7.114 3.205 9.3 1 12 1Zm6 7H6v12.5c0 .238.19.448.454.492l.102.008h10.888c.315 0 .556-.232.556-.5V8Zm-4 3a1 1 0 0 1 1 1v6.005a1 1 0 0 1-2 0V12a1 1 0 0 1 1-1Zm-4 0a1 1 0 0 1 1 1v6a1 1 0 0 1-2 0v-6a1 1 0 0 1 1-1Zm2-8c-1.595 0-2.914 1.32-2.996 3h5.991v-.02C14.903 4.31 13.589 3 12 3Z"/></symbol><symbol id="icon-eds-i-user-account-medium" viewBox="0 0 24 24"><path d="M12 1c6.075 0 11 4.925 11 11s-4.925 11-11 11S1 18.075 1 12 5.925 1 12 1Zm0 16c-1.806 0-3.52.994-4.664 2.698A8.947 8.947 0 0 0 12 21a8.958 8.958 0 0 0 4.664-1.301C15.52 17.994 13.806 17 12 17Zm0-14a9 9 0 0 0-6.25 15.476C7.253 16.304 9.54 15 12 15s4.747 1.304 6.25 3.475A9 9 0 0 0 12 3Zm0 3a4 4 0 1 1 0 8 4 4 0 0 1 0-8Zm0 2a2 2 0 1 0 0 4 2 2 0 0 0 0-4Z"/></symbol><symbol id="icon-eds-i-user-add-medium" viewBox="0 0 24 24"><path d="M9 1a5 5 0 1 1 0 10A5 5 0 0 1 9 1Zm0 2a3 3 0 1 0 0 6 3 3 0 0 0 0-6Zm9 10a1 1 0 0 1 1 1v3h3a1 1 0 0 1 0 2h-3v3a1 1 0 0 1-2 0v-3h-3a1 1 0 0 1 0-2h3v-3a1 1 0 0 1 1-1Zm-5.545-.15a1 1 0 1 1-.91 1.78 5.713 5.713 0 0 0-5.705.282c-1.67 1.068-2.728 2.927-2.832 4.956L3.004 20 11.5 20a1 1 0 0 1 .993.883L12.5 21a1 1 0 0 1-1 1H2a1 1 0 0 1-1-1v-.876c.028-2.812 1.446-5.416 3.763-6.897a7.713 7.713 0 0 1 7.692-.378Z"/></symbol><symbol id="icon-eds-i-user-assign-medium" viewBox="0 0 24 24"><path d="M16.226 13.298a1 1 0 0 1 1.414-.01l.084.093a1 1 0 0 1-.073 1.32L15.39 17H22a1 1 0 0 1 0 2h-6.611l2.262 2.298a1 1 0 0 1-1.425 1.404l-3.939-4a1 1 0 0 1 0-1.404l3.94-4Zm-3.771-.449a1 1 0 1 1-.91 1.781 5.713 5.713 0 0 0-5.705.282c-1.67 1.068-2.728 2.927-2.832 4.956L3.004 20 10.5 20a1 1 0 0 1 .993.883L11.5 21a1 1 0 0 1-1 1H2a1 1 0 0 1-1-1v-.876c.028-2.812 1.446-5.416 3.763-6.897a7.713 7.713 0 0 1 7.692-.378ZM9 1a5 5 0 1 1 0 10A5 5 0 0 1 9 1Zm0 2a3 3 0 1 0 0 6 3 3 0 0 0 0-6Z"/></symbol><symbol id="icon-eds-i-user-block-medium" viewBox="0 0 24 24"><path d="M9 1a5 5 0 1 1 0 10A5 5 0 0 1 9 1Zm0 2a3 3 0 1 0 0 6 3 3 0 0 0 0-6Zm9 10a5 5 0 1 1 0 10 5 5 0 0 1 0-10Zm-5.545-.15a1 1 0 1 1-.91 1.78 5.713 5.713 0 0 0-5.705.282c-1.67 1.068-2.728 2.927-2.832 4.956L3.004 20 11.5 20a1 1 0 0 1 .993.883L12.5 21a1 1 0 0 1-1 1H2a1 1 0 0 1-1-1v-.876c.028-2.812 1.446-5.416 3.763-6.897a7.713 7.713 0 0 1 7.692-.378ZM15 18a3 3 0 0 0 4.294 2.707l-4.001-4c-.188.391-.293.83-.293 1.293Zm3-3c-.463 0-.902.105-1.294.293l4.001 4A3 3 0 0 0 18 15Z"/></symbol><symbol id="icon-eds-i-user-check-medium" viewBox="0 0 24 24"><path d="M9 1a5 5 0 1 1 0 10A5 5 0 0 1 9 1Zm0 2a3 3 0 1 0 0 6 3 3 0 0 0 0-6Zm13.647 12.237a1 1 0 0 1 .116 1.41l-5.091 6a1 1 0 0 1-1.375.144l-2.909-2.25a1 1 0 1 1 1.224-1.582l2.153 1.665 4.472-5.271a1 1 0 0 1 1.41-.116Zm-8.139-.977c.22.214.428.44.622.678a1 1 0 1 1-1.548 1.266 6.025 6.025 0 0 0-1.795-1.49.86.86 0 0 1-.163-.048l-.079-.036a5.721 5.721 0 0 0-2.62-.63l-.194.006c-2.76.134-5.022 2.177-5.592 4.864l-.035.175-.035.213c-.03.201-.05.405-.06.61L3.003 20 10 20a1 1 0 0 1 .993.883L11 21a1 1 0 0 1-1 1H2a1 1 0 0 1-1-1v-.876l.005-.223.02-.356.02-.222.03-.248.022-.15c.02-.133.044-.265.071-.397.44-2.178 1.725-4.105 3.595-5.301a7.75 7.75 0 0 1 3.755-1.215l.12-.004a7.908 7.908 0 0 1 5.87 2.252Z"/></symbol><symbol id="icon-eds-i-user-delete-medium" viewBox="0 0 24 24"><path d="M9 1a5 5 0 1 1 0 10A5 5 0 0 1 9 1Zm0 2a3 3 0 1 0 0 6 3 3 0 0 0 0-6ZM4.763 13.227a7.713 7.713 0 0 1 7.692-.378 1 1 0 1 1-.91 1.781 5.713 5.713 0 0 0-5.705.282c-1.67 1.068-2.728 2.927-2.832 4.956L3.004 20H11.5a1 1 0 0 1 .993.883L12.5 21a1 1 0 0 1-1 1H2a1 1 0 0 1-1-1v-.876c.028-2.812 1.446-5.416 3.763-6.897Zm11.421 1.543 2.554 2.553 2.555-2.553a1 1 0 0 1 1.414 1.414l-2.554 2.554 2.554 2.555a1 1 0 0 1-1.414 1.414l-2.555-2.554-2.554 2.554a1 1 0 0 1-1.414-1.414l2.553-2.555-2.553-2.554a1 1 0 0 1 1.414-1.414Z"/></symbol><symbol id="icon-eds-i-user-edit-medium" viewBox="0 0 24 24"><path d="m19.876 10.77 2.831 2.83a1 1 0 0 1 0 1.415l-7.246 7.246a1 1 0 0 1-.572.284l-3.277.446a1 1 0 0 1-1.125-1.13l.461-3.277a1 1 0 0 1 .283-.567l7.23-7.246a1 1 0 0 1 1.415-.001Zm-7.421 2.08a1 1 0 1 1-.91 1.78 5.713 5.713 0 0 0-5.705.282c-1.67 1.068-2.728 2.927-2.832 4.956L3.004 20 7.5 20a1 1 0 0 1 .993.883L8.5 21a1 1 0 0 1-1 1H2a1 1 0 0 1-1-1v-.876c.028-2.812 1.446-5.416 3.763-6.897a7.713 7.713 0 0 1 7.692-.378Zm6.715.042-6.29 6.3-.23 1.639 1.633-.222 6.302-6.302-1.415-1.415ZM9 1a5 5 0 1 1 0 10A5 5 0 0 1 9 1Zm0 2a3 3 0 1 0 0 6 3 3 0 0 0 0-6Z"/></symbol><symbol id="icon-eds-i-user-linked-medium" viewBox="0 0 24 24"><path d="M15.65 6c.31 0 .706.066 1.122.274C17.522 6.65 18 7.366 18 8.35v12.3c0 .31-.066.706-.274 1.122-.375.75-1.092 1.228-2.076 1.228H3.35a2.52 2.52 0 0 1-1.122-.274C1.478 22.35 1 21.634 1 20.65V8.35c0-.31.066-.706.274-1.122C1.65 6.478 2.366 6 3.35 6h12.3Zm0 2-12.376.002c-.134.007-.17.04-.21.12A.672.672 0 0 0 3 8.35v12.3c0 .198.028.24.122.287.09.044.2.063.228.063h.887c.788-2.269 2.814-3.5 5.263-3.5 2.45 0 4.475 1.231 5.263 3.5h.887c.198 0 .24-.028.287-.122.044-.09.063-.2.063-.228V8.35c0-.198-.028-.24-.122-.287A.672.672 0 0 0 15.65 8ZM9.5 19.5c-1.36 0-2.447.51-3.06 1.5h6.12c-.613-.99-1.7-1.5-3.06-1.5ZM20.65 1A2.35 2.35 0 0 1 23 3.348V15.65A2.35 2.35 0 0 1 20.65 18H20a1 1 0 0 1 0-2h.65a.35.35 0 0 0 .35-.35V3.348A.35.35 0 0 0 20.65 3H8.35a.35.35 0 0 0-.35.348V4a1 1 0 1 1-2 0v-.652A2.35 2.35 0 0 1 8.35 1h12.3ZM9.5 10a3.5 3.5 0 1 1 0 7 3.5 3.5 0 0 1 0-7Zm0 2a1.5 1.5 0 1 0 0 3 1.5 1.5 0 0 0 0-3Z"/></symbol><symbol id="icon-eds-i-user-multiple-medium" viewBox="0 0 24 24"><path d="M9 1a5 5 0 1 1 0 10A5 5 0 0 1 9 1Zm6 0a5 5 0 0 1 0 10 1 1 0 0 1-.117-1.993L15 9a3 3 0 0 0 0-6 1 1 0 0 1 0-2ZM9 3a3 3 0 1 0 0 6 3 3 0 0 0 0-6Zm8.857 9.545a7.99 7.99 0 0 1 2.651 1.715A8.31 8.31 0 0 1 23 20.134V21a1 1 0 0 1-1 1h-3a1 1 0 0 1 0-2h1.995l-.005-.153a6.307 6.307 0 0 0-1.673-3.945l-.204-.209a5.99 5.99 0 0 0-1.988-1.287 1 1 0 1 1 .732-1.861Zm-3.349 1.715A8.31 8.31 0 0 1 17 20.134V21a1 1 0 0 1-1 1H2a1 1 0 0 1-1-1v-.877c.044-4.343 3.387-7.908 7.638-8.115a7.908 7.908 0 0 1 5.87 2.252ZM9.016 14l-.285.006c-3.104.15-5.58 2.718-5.725 5.9L3.004 20h11.991l-.005-.153a6.307 6.307 0 0 0-1.673-3.945l-.204-.209A5.924 5.924 0 0 0 9.3 14.008L9.016 14Z"/></symbol><symbol id="icon-eds-i-user-notify-medium" viewBox="0 0 24 24"><path d="M9 1a5 5 0 1 1 0 10A5 5 0 0 1 9 1Zm0 2a3 3 0 1 0 0 6 3 3 0 0 0 0-6Zm10 18v1a1 1 0 0 1-2 0v-1h-3a1 1 0 0 1 0-2v-2.818C14 13.885 15.777 12 18 12s4 1.885 4 4.182V19a1 1 0 0 1 0 2h-3Zm-6.545-8.15a1 1 0 1 1-.91 1.78 5.713 5.713 0 0 0-5.705.282c-1.67 1.068-2.728 2.927-2.832 4.956L3.004 20 11.5 20a1 1 0 0 1 .993.883L12.5 21a1 1 0 0 1-1 1H2a1 1 0 0 1-1-1v-.876c.028-2.812 1.446-5.416 3.763-6.897a7.713 7.713 0 0 1 7.692-.378ZM18 14c-1.091 0-2 .964-2 2.182V19h4v-2.818c0-1.165-.832-2.098-1.859-2.177L18 14Z"/></symbol><symbol id="icon-eds-i-user-remove-medium" viewBox="0 0 24 24"><path d="M9 1a5 5 0 1 1 0 10A5 5 0 0 1 9 1Zm0 2a3 3 0 1 0 0 6 3 3 0 0 0 0-6Zm3.455 9.85a1 1 0 1 1-.91 1.78 5.713 5.713 0 0 0-5.705.282c-1.67 1.068-2.728 2.927-2.832 4.956L3.004 20 11.5 20a1 1 0 0 1 .993.883L12.5 21a1 1 0 0 1-1 1H2a1 1 0 0 1-1-1v-.876c.028-2.812 1.446-5.416 3.763-6.897a7.713 7.713 0 0 1 7.692-.378ZM22 17a1 1 0 0 1 0 2h-8a1 1 0 0 1 0-2h8Z"/></symbol><symbol id="icon-eds-i-user-single-medium" viewBox="0 0 24 24"><path d="M12 1a5 5 0 1 1 0 10 5 5 0 0 1 0-10Zm0 2a3 3 0 1 0 0 6 3 3 0 0 0 0-6Zm-.406 9.008a8.965 8.965 0 0 1 6.596 2.494A9.161 9.161 0 0 1 21 21.025V22a1 1 0 0 1-1 1H4a1 1 0 0 1-1-1v-.985c.05-4.825 3.815-8.777 8.594-9.007Zm.39 1.992-.299.006c-3.63.175-6.518 3.127-6.678 6.775L5 21h13.998l-.009-.268a7.157 7.157 0 0 0-1.97-4.573l-.214-.213A6.967 6.967 0 0 0 11.984 14Z"/></symbol><symbol id="icon-eds-i-warning-circle-medium" viewBox="0 0 24 24"><path d="M12 1c6.075 0 11 4.925 11 11s-4.925 11-11 11S1 18.075 1 12 5.925 1 12 1Zm0 2a9 9 0 1 0 0 18 9 9 0 0 0 0-18Zm0 11.5a1.5 1.5 0 0 1 .144 2.993L12 17.5a1.5 1.5 0 0 1 0-3ZM12 6a1 1 0 0 1 1 1v5a1 1 0 0 1-2 0V7a1 1 0 0 1 1-1Z"/></symbol><symbol id="icon-eds-i-warning-filled-medium" viewBox="0 0 24 24"><path d="M12 1c6.075 0 11 4.925 11 11s-4.925 11-11 11S1 18.075 1 12 5.925 1 12 1Zm0 13.5a1.5 1.5 0 0 0 0 3l.144-.007A1.5 1.5 0 0 0 12 14.5ZM12 6a1 1 0 0 0-1 1v5a1 1 0 0 0 2 0V7a1 1 0 0 0-1-1Z"/></symbol><symbol id="icon-chevron-left-medium" viewBox="0 0 24 24"><path d="M15.7194 3.3054C15.3358 2.90809 14.7027 2.89699 14.3054 3.28061L6.54342 10.7757C6.19804 11.09 6 11.5335 6 12C6 12.4665 6.19804 12.91 6.5218 13.204L14.3054 20.7194C14.7027 21.103 15.3358 21.0919 15.7194 20.6946C16.103 20.2973 16.0919 19.6642 15.6946 19.2806L8.155 12L15.6946 4.71939C16.0614 4.36528 16.099 3.79863 15.8009 3.40105L15.7194 3.3054Z"/></symbol><symbol id="icon-chevron-right-medium" viewBox="0 0 24 24"><path d="M8.28061 3.3054C8.66423 2.90809 9.29729 2.89699 9.6946 3.28061L17.4566 10.7757C17.802 11.09 18 11.5335 18 12C18 12.4665 17.802 12.91 17.4782 13.204L9.6946 20.7194C9.29729 21.103 8.66423 21.0919 8.28061 20.6946C7.89699 20.2973 7.90809 19.6642 8.3054 19.2806L15.845 12L8.3054 4.71939C7.93865 4.36528 7.90098 3.79863 8.19908 3.40105L8.28061 3.3054Z"/></symbol><symbol id="icon-eds-alerts" viewBox="0 0 32 32"><path d="M28 12.667c.736 0 1.333.597 1.333 1.333v13.333A3.333 3.333 0 0 1 26 30.667H6a3.333 3.333 0 0 1-3.333-3.334V14a1.333 1.333 0 1 1 2.666 0v1.252L16 21.769l10.667-6.518V14c0-.736.597-1.333 1.333-1.333Zm-1.333 5.71-9.972 6.094c-.427.26-.963.26-1.39 0l-9.972-6.094v8.956c0 .368.299.667.667.667h20a.667.667 0 0 0 .667-.667v-8.956ZM19.333 12a1.333 1.333 0 1 1 0 2.667h-6.666a1.333 1.333 0 1 1 0-2.667h6.666Zm4-10.667a3.333 3.333 0 0 1 3.334 3.334v6.666a1.333 1.333 0 1 1-2.667 0V4.667A.667.667 0 0 0 23.333 4H8.667A.667.667 0 0 0 8 4.667v6.666a1.333 1.333 0 1 1-2.667 0V4.667a3.333 3.333 0 0 1 3.334-3.334h14.666Zm-4 5.334a1.333 1.333 0 0 1 0 2.666h-6.666a1.333 1.333 0 1 1 0-2.666h6.666Z"/></symbol><symbol id="icon-eds-arrow-up" viewBox="0 0 24 24"><path fill-rule="evenodd" d="m13.002 7.408 4.88 4.88a.99.99 0 0 0 1.32.08l.09-.08c.39-.39.39-1.03 0-1.42l-6.58-6.58a1.01 1.01 0 0 0-1.42 0l-6.58 6.58a1 1 0 0 0-.09 1.32l.08.1a1 1 0 0 0 1.42-.01l4.88-4.87v11.59a.99.99 0 0 0 .88.99l.12.01c.55 0 1-.45 1-1V7.408z" class="layer"/></symbol><symbol id="icon-eds-checklist" viewBox="0 0 32 32"><path d="M19.2 1.333a3.468 3.468 0 0 1 3.381 2.699L24.667 4C26.515 4 28 5.52 28 7.38v19.906c0 1.86-1.485 3.38-3.333 3.38H7.333c-1.848 0-3.333-1.52-3.333-3.38V7.38C4 5.52 5.485 4 7.333 4h2.093A3.468 3.468 0 0 1 12.8 1.333h6.4ZM9.426 6.667H7.333c-.36 0-.666.312-.666.713v19.906c0 .401.305.714.666.714h17.334c.36 0 .666-.313.666-.714V7.38c0-.4-.305-.713-.646-.714l-2.121.033A3.468 3.468 0 0 1 19.2 9.333h-6.4a3.468 3.468 0 0 1-3.374-2.666Zm12.715 5.606c.586.446.7 1.283.253 1.868l-7.111 9.334a1.333 1.333 0 0 1-1.792.306l-3.556-2.333a1.333 1.333 0 1 1 1.463-2.23l2.517 1.651 6.358-8.344a1.333 1.333 0 0 1 1.868-.252ZM19.2 4h-6.4a.8.8 0 0 0-.8.8v1.067a.8.8 0 0 0 .8.8h6.4a.8.8 0 0 0 .8-.8V4.8a.8.8 0 0 0-.8-.8Z"/></symbol><symbol id="icon-eds-citation" viewBox="0 0 36 36"><path d="M23.25 1.5a1.5 1.5 0 0 1 1.06.44l8.25 8.25a1.5 1.5 0 0 1 .44 1.06v19.5c0 2.105-1.645 3.75-3.75 3.75H18a1.5 1.5 0 0 1 0-3h11.25c.448 0 .75-.302.75-.75V11.873L22.628 4.5H8.31a.811.811 0 0 0-.8.68l-.011.13V16.5a1.5 1.5 0 0 1-3 0V5.31A3.81 3.81 0 0 1 8.31 1.5h14.94ZM8.223 20.358a.984.984 0 0 1-.192 1.378l-.048.034c-.54.36-.942.676-1.206.951-.59.614-.885 1.395-.885 2.343.115-.028.288-.042.518-.042.662 0 1.26.237 1.791.711.533.474.799 1.074.799 1.799 0 .753-.259 1.352-.777 1.799-.518.446-1.151.669-1.9.669-1.006 0-1.812-.293-2.417-.878C3.302 28.536 3 27.657 3 26.486c0-1.115.165-2.085.496-2.907.331-.823.734-1.513 1.209-2.071.475-.558.971-.997 1.49-1.318a6.01 6.01 0 0 1 .347-.2 1.321 1.321 0 0 1 1.681.368Zm7.5 0a.984.984 0 0 1-.192 1.378l-.048.034c-.54.36-.942.676-1.206.951-.59.614-.885 1.395-.885 2.343.115-.028.288-.042.518-.042.662 0 1.26.237 1.791.711.533.474.799 1.074.799 1.799 0 .753-.259 1.352-.777 1.799-.518.446-1.151.669-1.9.669-1.006 0-1.812-.293-2.417-.878-.604-.586-.906-1.465-.906-2.636 0-1.115.165-2.085.496-2.907.331-.823.734-1.513 1.209-2.071.475-.558.971-.997 1.49-1.318a6.01 6.01 0 0 1 .347-.2 1.321 1.321 0 0 1 1.681.368Z"/></symbol><symbol id="icon-eds-i-access-indicator" viewBox="0 0 16 16"><circle cx="4.5" cy="11.5" r="3.5" style="fill:currentColor"/><path fill-rule="evenodd" d="M4 3v3a1 1 0 0 1-2 0V2.923C2 1.875 2.84 1 3.909 1h5.909a1 1 0 0 1 .713.298l3.181 3.231a1 1 0 0 1 .288.702v7.846c0 .505-.197.993-.554 1.354a1.902 1.902 0 0 1-1.355.569H10a1 1 0 1 1 0-2h2V5.64L9.4 3H4Z" clip-rule="evenodd" style="fill:#222"/></symbol><symbol id="icon-eds-i-copy-link" viewBox="0 0 24 24"><path fill-rule="evenodd" clip-rule="evenodd" d="M19.4594 8.57015C19.0689 8.17963 19.0689 7.54646 19.4594 7.15594L20.2927 6.32261C20.2927 6.32261 20.2927 6.32261 20.2927 6.32261C21.0528 5.56252 21.0528 4.33019 20.2928 3.57014C19.5327 2.81007 18.3004 2.81007 17.5404 3.57014L16.7071 4.40347C16.3165 4.794 15.6834 4.794 15.2928 4.40348C14.9023 4.01296 14.9023 3.3798 15.2928 2.98927L16.1262 2.15594C17.6673 0.614803 20.1659 0.614803 21.707 2.15593C23.2481 3.69705 23.248 6.19569 21.707 7.7368L20.8737 8.57014C20.4831 8.96067 19.85 8.96067 19.4594 8.57015Z"/><path fill-rule="evenodd" clip-rule="evenodd" d="M18.0944 5.90592C18.4849 6.29643 18.4849 6.9296 18.0944 7.32013L16.4278 8.9868C16.0373 9.37733 15.4041 9.37734 15.0136 8.98682C14.6231 8.59631 14.6231 7.96314 15.0136 7.57261L16.6802 5.90594C17.0707 5.51541 17.7039 5.5154 18.0944 5.90592Z"/><path fill-rule="evenodd" clip-rule="evenodd" d="M13.5113 6.32243C13.9018 6.71295 13.9018 7.34611 13.5113 7.73664L12.678 8.56997C12.678 8.56997 12.678 8.56997 12.678 8.56997C11.9179 9.33006 11.9179 10.5624 12.6779 11.3224C13.438 12.0825 14.6703 12.0825 15.4303 11.3224L16.2636 10.4891C16.6542 10.0986 17.2873 10.0986 17.6779 10.4891C18.0684 10.8796 18.0684 11.5128 17.6779 11.9033L16.8445 12.7366C15.3034 14.2778 12.8048 14.2778 11.2637 12.7366C9.72262 11.1955 9.72266 8.69689 11.2637 7.15578L12.097 6.32244C12.4876 5.93191 13.1207 5.93191 13.5113 6.32243Z"/><path d="M8 20V22H19.4619C20.136 22 20.7822 21.7311 21.2582 21.2529C21.7333 20.7757 22 20.1289 22 19.4549V15C22 14.4477 21.5523 14 21 14C20.4477 14 20 14.4477 20 15V19.4549C20 19.6004 19.9426 19.7397 19.8408 19.842C19.7399 19.9433 19.6037 20 19.4619 20H8Z"/><path d="M4 13H2V19.4619C2 20.136 2.26889 20.7822 2.74705 21.2582C3.22434 21.7333 3.87105 22 4.5451 22H9C9.55228 22 10 21.5523 10 21C10 20.4477 9.55228 20 9 20H4.5451C4.39957 20 4.26028 19.9426 4.15804 19.8408C4.05668 19.7399 4 19.6037 4 19.4619V13Z"/><path d="M4 13H2V4.53808C2 3.86398 2.26889 3.21777 2.74705 2.74178C3.22434 2.26666 3.87105 2 4.5451 2H9C9.55228 2 10 2.44772 10 3C10 3.55228 9.55228 4 9 4H4.5451C4.39957 4 4.26028 4.05743 4.15804 4.15921C4.05668 4.26011 4 4.39633 4 4.53808V13Z"/></symbol><symbol id="icon-eds-i-github-medium" viewBox="0 0 24 24"><path d="M 11.964844 0 C 5.347656 0 0 5.269531 0 11.792969 C 0 17.003906 3.425781 21.417969 8.179688 22.976562 C 8.773438 23.09375 8.992188 22.722656 8.992188 22.410156 C 8.992188 22.136719 8.972656 21.203125 8.972656 20.226562 C 5.644531 20.929688 4.953125 18.820312 4.953125 18.820312 C 4.417969 17.453125 3.625 17.101562 3.625 17.101562 C 2.535156 16.378906 3.703125 16.378906 3.703125 16.378906 C 4.914062 16.457031 5.546875 17.589844 5.546875 17.589844 C 6.617188 19.386719 8.339844 18.878906 9.03125 18.566406 C 9.132812 17.804688 9.449219 17.277344 9.785156 16.984375 C 7.132812 16.710938 4.339844 15.695312 4.339844 11.167969 C 4.339844 9.878906 4.8125 8.824219 5.566406 8.003906 C 5.445312 7.710938 5.03125 6.5 5.683594 4.878906 C 5.683594 4.878906 6.695312 4.566406 8.972656 6.089844 C 9.949219 5.832031 10.953125 5.703125 11.964844 5.699219 C 12.972656 5.699219 14.003906 5.835938 14.957031 6.089844 C 17.234375 4.566406 18.242188 4.878906 18.242188 4.878906 C 18.898438 6.5 18.480469 7.710938 18.363281 8.003906 C 19.136719 8.824219 19.589844 9.878906 19.589844 11.167969 C 19.589844 15.695312 16.796875 16.691406 14.125 16.984375 C 14.558594 17.355469 14.933594 18.058594 14.933594 19.171875 C 14.933594 20.753906 14.914062 22.019531 14.914062 22.410156 C 14.914062 22.722656 15.132812 23.09375 15.726562 22.976562 C 20.480469 21.414062 23.910156 17.003906 23.910156 11.792969 C 23.929688 5.269531 18.558594 0 11.964844 0 Z M 11.964844 0 "/></symbol><symbol id="icon-eds-i-institution-medium" viewBox="0 0 24 24"><g><path fill-rule="evenodd" clip-rule="evenodd" d="M11.9967 1C11.6364 1 11.279 1.0898 10.961 1.2646C10.9318 1.28061 10.9035 1.29806 10.8761 1.31689L2.79765 6.87C2.46776 7.08001 2.20618 7.38466 2.07836 7.76668C1.94823 8.15561 1.98027 8.55648 2.12665 8.90067C2.42086 9.59246 3.12798 10 3.90107 10H4.99994V16H4.49994C3.11923 16 1.99994 17.1193 1.99994 18.5V19.5C1.99994 20.8807 3.11923 22 4.49994 22H19.4999C20.8807 22 21.9999 20.8807 21.9999 19.5V18.5C21.9999 17.1193 20.8807 16 19.4999 16H18.9999V10H20.0922C20.8653 10 21.5725 9.59252 21.8667 8.90065C22.0131 8.55642 22.0451 8.15553 21.9149 7.7666C21.7871 7.38459 21.5255 7.07997 21.1956 6.86998L13.1172 1.31689C13.0898 1.29806 13.0615 1.28061 13.0324 1.2646C12.7143 1.0898 12.357 1 11.9967 1ZM4.6844 8L11.9472 3.00755C11.9616 3.00295 11.9783 3 11.9967 3C12.015 3 12.0318 3.00295 12.0461 3.00755L19.3089 8H4.6844ZM16.9999 16V10H14.9999V16H16.9999ZM12.9999 16V10H10.9999V16H12.9999ZM8.99994 16V10H6.99994V16H8.99994ZM3.99994 18.5C3.99994 18.2239 4.2238 18 4.49994 18H19.4999C19.7761 18 19.9999 18.2239 19.9999 18.5V19.5C19.9999 19.7761 19.7761 20 19.4999 20H4.49994C4.2238 20 3.99994 19.7761 3.99994 19.5V18.5Z"/></g></symbol><symbol id="icon-eds-i-limited-access" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 3v3a1 1 0 0 1-2 0V2.923C2 1.875 2.84 1 3.909 1h5.909a1 1 0 0 1 .713.298l3.181 3.231a1 1 0 0 1 .288.702V6a1 1 0 1 1-2 0v-.36L9.4 3H4ZM3 8a1 1 0 0 1 1 1v1a1 1 0 1 1-2 0V9a1 1 0 0 1 1-1Zm10 0a1 1 0 0 1 1 1v1a1 1 0 1 1-2 0V9a1 1 0 0 1 1-1Zm-3.5 6a1 1 0 0 1-1 1h-1a1 1 0 1 1 0-2h1a1 1 0 0 1 1 1Zm2.441-1a1 1 0 0 1 2 0c0 .73-.246 1.306-.706 1.664a1.61 1.61 0 0 1-.876.334l-.032.002H11.5a1 1 0 1 1 0-2h.441ZM4 13a1 1 0 0 0-2 0c0 .73.247 1.306.706 1.664a1.609 1.609 0 0 0 .876.334l.032.002H4.5a1 1 0 1 0 0-2H4Z" clip-rule="evenodd"/></symbol><symbol id="icon-eds-i-search-category-medium" viewBox="0 0 32 32"><path fill-rule="evenodd" d="M2 5.306A3.306 3.306 0 0 1 5.306 2h5.833a3.306 3.306 0 0 1 3.306 3.306v5.833a3.306 3.306 0 0 1-3.306 3.305H5.306A3.306 3.306 0 0 1 2 11.14V5.306Zm3.306-.584a.583.583 0 0 0-.584.584v5.833c0 .322.261.583.584.583h5.833a.583.583 0 0 0 .583-.583V5.306a.583.583 0 0 0-.583-.584H5.306Zm15.555 8.945a7.194 7.194 0 1 0 4.034 13.153l2.781 2.781a1.361 1.361 0 1 0 1.925-1.925l-2.781-2.781a7.194 7.194 0 0 0-5.958-11.228Zm3.173 10.346a4.472 4.472 0 1 0-.021.021l.01-.01.011-.011Zm-5.117-19.29a.583.583 0 0 0-.584.583v5.833a1.361 1.361 0 0 1-2.722 0V5.306A3.306 3.306 0 0 1 18.917 2h5.833a3.306 3.306 0 0 1 3.306 3.306v5.833c0 .6-.161 1.166-.443 1.654a1.361 1.361 0 1 1-2.357-1.363.575.575 0 0 0 .078-.291V5.306a.583.583 0 0 0-.584-.584h-5.833ZM2 18.916a3.306 3.306 0 0 1 3.306-3.306h5.833a1.361 1.361 0 1 1 0 2.722H5.306a.583.583 0 0 0-.584.584v5.833c0 .322.261.583.584.583h5.833a.574.574 0 0 0 .29-.077 1.361 1.361 0 1 1 1.364 2.356 3.296 3.296 0 0 1-1.654.444H5.306A3.306 3.306 0 0 1 2 24.75v-5.833Z" clip-rule="evenodd"/></symbol><symbol id="icon-eds-i-subjects-medium" viewBox="0 0 24 24"><g id="icon-subjects-copy" stroke="none" stroke-width="1" fill-rule="evenodd"><path d="M13.3846154,2 C14.7015971,2 15.7692308,3.06762994 15.7692308,4.38461538 L15.7692308,7.15384615 C15.7692308,8.47082629 14.7015955,9.53846154 13.3846154,9.53846154 L13.1038388,9.53925278 C13.2061091,9.85347965 13.3815528,10.1423885 13.6195822,10.3804178 C13.9722182,10.7330539 14.436524,10.9483278 14.9293854,10.9918129 L15.1153846,11 C16.2068332,11 17.2535347,11.433562 18.0254647,12.2054189 C18.6411944,12.8212361 19.0416785,13.6120766 19.1784166,14.4609738 L19.6153846,14.4615385 C20.932386,14.4615385 22,15.5291672 22,16.8461538 L22,19.6153846 C22,20.9323924 20.9323924,22 19.6153846,22 L16.8461538,22 C15.5291672,22 14.4615385,20.932386 14.4615385,19.6153846 L14.4615385,16.8461538 C14.4615385,15.5291737 15.5291737,14.4615385 16.8461538,14.4615385 L17.126925,14.460779 C17.0246537,14.1465537 16.8492179,13.857633 16.6112344,13.6196157 C16.2144418,13.2228606 15.6764136,13 15.1153846,13 C14.0239122,13 12.9771569,12.5664197 12.2053686,11.7946314 C12.1335167,11.7227795 12.0645962,11.6485444 11.9986839,11.5721119 C11.9354038,11.6485444 11.8664833,11.7227795 11.7946314,11.7946314 C11.0228431,12.5664197 9.97608778,13 8.88461538,13 C8.323576,13 7.78552852,13.2228666 7.38881294,13.6195822 C7.15078359,13.8576115 6.97533988,14.1465203 6.8730696,14.4607472 L7.15384615,14.4615385 C8.47082629,14.4615385 9.53846154,15.5291737 9.53846154,16.8461538 L9.53846154,19.6153846 C9.53846154,20.932386 8.47083276,22 7.15384615,22 L4.38461538,22 C3.06762347,22 2,20.9323876 2,19.6153846 L2,16.8461538 C2,15.5291721 3.06762994,14.4615385 4.38461538,14.4615385 L4.8215823,14.4609378 C4.95831893,13.6120029 5.3588057,12.8211623 5.97459937,12.2053686 C6.69125996,11.488708 7.64500941,11.0636656 8.6514968,11.0066017 L8.88461538,11 C9.44565477,11 9.98370225,10.7771334 10.3804178,10.3804178 C10.6184472,10.1423885 10.7938909,9.85347965 10.8961612,9.53925278 L10.6153846,9.53846154 C9.29840448,9.53846154 8.23076923,8.47082629 8.23076923,7.15384615 L8.23076923,4.38461538 C8.23076923,3.06762994 9.29840286,2 10.6153846,2 L13.3846154,2 Z M7.15384615,16.4615385 L4.38461538,16.4615385 C4.17220099,16.4615385 4,16.63374 4,16.8461538 L4,19.6153846 C4,19.8278134 4.17218833,20 4.38461538,20 L7.15384615,20 C7.36626945,20 7.53846154,19.8278103 7.53846154,19.6153846 L7.53846154,16.8461538 C7.53846154,16.6337432 7.36625679,16.4615385 7.15384615,16.4615385 Z M19.6153846,16.4615385 L16.8461538,16.4615385 C16.6337432,16.4615385 16.4615385,16.6337432 16.4615385,16.8461538 L16.4615385,19.6153846 C16.4615385,19.8278103 16.6337306,20 16.8461538,20 L19.6153846,20 C19.8278229,20 20,19.8278229 20,19.6153846 L20,16.8461538 C20,16.6337306 19.8278103,16.4615385 19.6153846,16.4615385 Z M13.3846154,4 L10.6153846,4 C10.4029708,4 10.2307692,4.17220099 10.2307692,4.38461538 L10.2307692,7.15384615 C10.2307692,7.36625679 10.402974,7.53846154 10.6153846,7.53846154 L13.3846154,7.53846154 C13.597026,7.53846154 13.7692308,7.36625679 13.7692308,7.15384615 L13.7692308,4.38461538 C13.7692308,4.17220099 13.5970292,4 13.3846154,4 Z" id="Shape" fill-rule="nonzero"/></g></symbol><symbol id="icon-eds-small-arrow-left" viewBox="0 0 16 17"><path stroke="currentColor" stroke-linecap="round" stroke-linejoin="round" stroke-width="2" d="M14 8.092H2m0 0L8 2M2 8.092l6 6.035"/></symbol><symbol id="icon-eds-small-arrow-right" viewBox="0 0 16 16"><g fill-rule="evenodd" stroke="currentColor" stroke-linecap="round" stroke-linejoin="round" stroke-width="2"><path d="M2 8.092h12M8 2l6 6.092M8 14.127l6-6.035"/></g></symbol><symbol id="icon-orcid-logo" viewBox="0 0 40 40"><path fill-rule="evenodd" d="M12.281 10.453c.875 0 1.578-.719 1.578-1.578 0-.86-.703-1.578-1.578-1.578-.875 0-1.578.703-1.578 1.578 0 .86.703 1.578 1.578 1.578Zm-1.203 18.641h2.406V12.359h-2.406v16.735Z"/><path fill-rule="evenodd" d="M17.016 12.36h6.5c6.187 0 8.906 4.421 8.906 8.374 0 4.297-3.36 8.375-8.875 8.375h-6.531V12.36Zm6.234 14.578h-3.828V14.53h3.703c4.688 0 6.828 2.844 6.828 6.203 0 2.063-1.25 6.203-6.703 6.203Z" clip-rule="evenodd"/></symbol></svg> </div> <a class="c-skip-link" href="#main">Skip to main content</a> <header class="eds-c-header" data-eds-c-header> <div class="eds-c-header__container" data-eds-c-header-expander-anchor> <div class="eds-c-header__brand"> <a href="https://link.springer.com" data-test=springerlink-logo data-track="click_imprint_logo" data-track-context="unified header" data-track-action="click logo link" data-track-category="unified header" data-track-label="link" > <img src="/oscar-static/images/darwin/header/img/logo-springer-nature-link-3149409f62.svg" alt="Springer Nature Link"> </a> </div> <a class="c-header__link eds-c-header__link" id="identity-account-widget" data-track="click_login" data-track-context="header" href='https://idp.springer.com/auth/personal/springernature?redirect_uri=https://link.springer.com/article/10.1007/s00025-020-01247-8?'><span class="eds-c-header__widget-fragment-title">Log in</span></a> </div> <nav class="eds-c-header__nav" aria-label="header navigation"> <div class="eds-c-header__nav-container"> <div class="eds-c-header__item eds-c-header__item--menu"> <a href="#eds-c-header-nav" class="eds-c-header__link" data-eds-c-header-expander> <svg class="eds-c-header__icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-menu-medium"></use> </svg><span>Menu</span> </a> </div> <div class="eds-c-header__item eds-c-header__item--inline-links"> <a class="eds-c-header__link" href="https://link.springer.com/journals/" data-track="nav_find_a_journal" data-track-context="unified header" data-track-action="click find a journal" data-track-category="unified header" data-track-label="link" > Find a journal </a> <a class="eds-c-header__link" href="https://www.springernature.com/gp/authors" data-track="nav_how_to_publish" data-track-context="unified header" data-track-action="click publish with us link" data-track-category="unified header" data-track-label="link" > Publish with us </a> <a class="eds-c-header__link" href="https://link.springernature.com/home/" data-track="nav_track_your_research" data-track-context="unified header" data-track-action="click track your research" data-track-category="unified header" data-track-label="link" > Track your research </a> </div> <div class="eds-c-header__link-container"> <div class="eds-c-header__item eds-c-header__item--divider"> <a href="#eds-c-header-popup-search" class="eds-c-header__link" data-eds-c-header-expander data-eds-c-header-test-search-btn> <svg class="eds-c-header__icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-search-medium"></use> </svg><span>Search</span> </a> </div> <div id="ecommerce-header-cart-icon-link" class="eds-c-header__item ecommerce-cart" style="display:inline-block"> <a class="eds-c-header__link" href="https://order.springer.com/public/cart" style="appearance:none;border:none;background:none;color:inherit;position:relative"> <svg id="eds-i-cart" class="eds-c-header__icon" xmlns="http://www.w3.org/2000/svg" height="24" width="24" viewBox="0 0 24 24" aria-hidden="true" focusable="false"> <path fill="currentColor" fill-rule="nonzero" d="M2 1a1 1 0 0 0 0 2l1.659.001 2.257 12.808a2.599 2.599 0 0 0 2.435 2.185l.167.004 9.976-.001a2.613 2.613 0 0 0 2.61-1.748l.03-.106 1.755-7.82.032-.107a2.546 2.546 0 0 0-.311-1.986l-.108-.157a2.604 2.604 0 0 0-2.197-1.076L6.042 5l-.56-3.17a1 1 0 0 0-.864-.82l-.12-.007L2.001 1ZM20.35 6.996a.63.63 0 0 1 .54.26.55.55 0 0 1 .082.505l-.028.1L19.2 15.63l-.022.05c-.094.177-.282.299-.526.317l-10.145.002a.61.61 0 0 1-.618-.515L6.394 6.999l13.955-.003ZM18 19a2 2 0 1 0 0 4 2 2 0 0 0 0-4ZM8 19a2 2 0 1 0 0 4 2 2 0 0 0 0-4Z"></path> </svg><span>Cart</span><span class="cart-info" style="display:none;position:absolute;top:10px;right:45px;background-color:#C65301;color:#fff;width:18px;height:18px;font-size:11px;border-radius:50%;line-height:17.5px;text-align:center"></span></a> <script>(function () { var exports = {}; if (window.fetch) { "use strict"; Object.defineProperty(exports, "__esModule", { value: true }); exports.headerWidgetClientInit = void 0; var headerWidgetClientInit = function (getCartInfo) { document.body.addEventListener("updatedCart", function () { updateCartIcon(); }, false); return updateCartIcon(); function updateCartIcon() { return getCartInfo() .then(function (res) { return res.json(); }) .then(refreshCartState) .catch(function (_) { }); } function refreshCartState(json) { var indicator = document.querySelector("#ecommerce-header-cart-icon-link .cart-info"); /* istanbul ignore else */ if (indicator && json.itemCount) { indicator.style.display = 'block'; indicator.textContent = json.itemCount > 9 ? '9+' : json.itemCount.toString(); var moreThanOneItem = json.itemCount > 1; indicator.setAttribute('title', "there ".concat(moreThanOneItem ? "are" : "is", " ").concat(json.itemCount, " item").concat(moreThanOneItem ? "s" : "", " in your cart")); } return json; } }; exports.headerWidgetClientInit = headerWidgetClientInit; headerWidgetClientInit( function () { return window.fetch("https://cart.springer.com/cart-info", { credentials: "include", headers: { Accept: "application/json" } }) } ) }})()</script> </div> </div> </div> </nav> </header> <article lang="en" id="main" class="app-masthead__colour-16"> <section class="app-masthead " aria-label="article masthead"> <div class="app-masthead__container"> <div class="app-article-masthead u-sans-serif js-context-bar-sticky-point-masthead" data-track-component="article" data-test="masthead-component"> <div class="app-article-masthead__info"> <nav aria-label="breadcrumbs" data-test="breadcrumbs"> <ol class="c-breadcrumbs c-breadcrumbs--contrast" itemscope itemtype="https://schema.org/BreadcrumbList"> <li class="c-breadcrumbs__item" id="breadcrumb0" itemprop="itemListElement" itemscope="" itemtype="https://schema.org/ListItem"> <a href="/" class="c-breadcrumbs__link" itemprop="item" data-track="click_breadcrumb" data-track-context="article page" data-track-category="article" data-track-action="breadcrumbs" data-track-label="breadcrumb1"><span itemprop="name">Home</span></a><meta itemprop="position" content="1"> <svg class="c-breadcrumbs__chevron" role="img" aria-hidden="true" focusable="false" width="10" height="10" viewBox="0 0 10 10"> <path d="m5.96738168 4.70639573 2.39518594-2.41447274c.37913917-.38219212.98637524-.38972225 1.35419292-.01894278.37750606.38054586.37784436.99719163-.00013556 1.37821513l-4.03074001 4.06319683c-.37758093.38062133-.98937525.38100976-1.367372-.00003075l-4.03091981-4.06337806c-.37759778-.38063832-.38381821-.99150444-.01600053-1.3622839.37750607-.38054587.98772445-.38240057 1.37006824.00302197l2.39538588 2.4146743.96295325.98624457z" fill-rule="evenodd" transform="matrix(0 -1 1 0 0 10)"/> </svg> </li> <li class="c-breadcrumbs__item" id="breadcrumb1" itemprop="itemListElement" itemscope="" itemtype="https://schema.org/ListItem"> <a href="/journal/25" class="c-breadcrumbs__link" itemprop="item" data-track="click_breadcrumb" data-track-context="article page" data-track-category="article" data-track-action="breadcrumbs" data-track-label="breadcrumb2"><span itemprop="name">Results in Mathematics</span></a><meta itemprop="position" content="2"> <svg class="c-breadcrumbs__chevron" role="img" aria-hidden="true" focusable="false" width="10" height="10" viewBox="0 0 10 10"> <path d="m5.96738168 4.70639573 2.39518594-2.41447274c.37913917-.38219212.98637524-.38972225 1.35419292-.01894278.37750606.38054586.37784436.99719163-.00013556 1.37821513l-4.03074001 4.06319683c-.37758093.38062133-.98937525.38100976-1.367372-.00003075l-4.03091981-4.06337806c-.37759778-.38063832-.38381821-.99150444-.01600053-1.3622839.37750607-.38054587.98772445-.38240057 1.37006824.00302197l2.39538588 2.4146743.96295325.98624457z" fill-rule="evenodd" transform="matrix(0 -1 1 0 0 10)"/> </svg> </li> <li class="c-breadcrumbs__item" id="breadcrumb2" itemprop="itemListElement" itemscope="" itemtype="https://schema.org/ListItem"> <span itemprop="name">Article</span><meta itemprop="position" content="3"> </li> </ol> </nav> <h1 class="c-article-title" data-test="article-title" data-article-title="">The Poincare Lemma, Antiexact Forms, and Fermionic Quantum Harmonic Oscillator</h1> <ul class="c-article-identifiers"> <li class="c-article-identifiers__item"> <a href="https://www.springernature.com/gp/open-research/about/the-fundamentals-of-open-access-and-open-research" data-track="click" data-track-action="open access" data-track-label="link" class="u-color-open-access" data-test="open-access">Open access</a> </li> <li class="c-article-identifiers__item"> Published: <time datetime="2020-07-11">11 July 2020</time> </li> </ul> <ul class="c-article-identifiers c-article-identifiers--cite-list"> <li class="c-article-identifiers__item"> <span data-test="journal-volume">Volume 75</span>, article number <span data-test="article-number">122</span>, (<span data-test="article-publication-year">2020</span>) </li> <li class="c-article-identifiers__item c-article-identifiers__item--cite"> <a href="#citeas" data-track="click" data-track-action="cite this article" data-track-category="article body" data-track-label="link">Cite this article</a> </li> </ul> <div class="app-article-masthead__buttons" data-test="download-article-link-wrapper" data-track-context="masthead"> <div class="c-pdf-container"> <div class="c-pdf-download u-clear-both u-mb-16"> <a href="/content/pdf/10.1007/s00025-020-01247-8.pdf" class="u-button u-button--full-width u-button--primary u-justify-content-space-between c-pdf-download__link" data-article-pdf="true" data-readcube-pdf-url="true" data-test="pdf-link" data-draft-ignore="true" data-track="content_download" data-track-type="article pdf download" data-track-action="download pdf" data-track-label="button" data-track-external download> <span class="c-pdf-download__text">Download PDF</span> <svg aria-hidden="true" focusable="false" width="16" height="16" class="u-icon"><use xlink:href="#icon-eds-i-download-medium"/></svg> </a> </div> </div> <p class="app-article-masthead__access"> <svg width="16" height="16" focusable="false" role="img" aria-hidden="true"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-check-filled-medium"></use></svg> You have full access to this <a href="https://www.springernature.com/gp/open-research/about/the-fundamentals-of-open-access-and-open-research" data-track="click" data-track-action="open access" data-track-label="link">open access</a> article</p> </div> </div> <div class="app-article-masthead__brand"> <a href="/journal/25" class="app-article-masthead__journal-link" data-track="click_journal_home" data-track-action="journal homepage" data-track-context="article page" data-track-label="link"> <picture> <source type="image/webp" media="(min-width: 768px)" width="120" height="159" srcset="https://media.springernature.com/w120/springer-static/cover-hires/journal/25?as=webp, https://media.springernature.com/w316/springer-static/cover-hires/journal/25?as=webp 2x"> <img width="72" height="95" src="https://media.springernature.com/w72/springer-static/cover-hires/journal/25?as=webp" srcset="https://media.springernature.com/w144/springer-static/cover-hires/journal/25?as=webp 2x" alt=""> </picture> <span class="app-article-masthead__journal-title">Results in Mathematics</span> </a> <a href="https://link.springer.com/journal/25/aims-and-scope" class="app-article-masthead__submission-link" data-track="click_aims_and_scope" data-track-action="aims and scope" data-track-context="article page" data-track-label="link"> Aims and scope <svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-arrow-right-medium"></use></svg> </a> <a href="https://www.editorialmanager.com/rima/" class="app-article-masthead__submission-link" data-track="click_submit_manuscript" data-track-context="article masthead on springerlink article page" data-track-action="submit manuscript" data-track-label="link"> Submit manuscript <svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-arrow-right-medium"></use></svg> </a> </div> </div> </div> </section> <div class="c-article-main u-container u-mt-24 u-mb-32 l-with-sidebar" id="main-content" data-component="article-container"> <main class="u-serif js-main-column" data-track-component="article body"> <div class="c-context-bar u-hide" data-test="context-bar" data-context-bar aria-hidden="true"> <div class="c-context-bar__container u-container"> <div class="c-context-bar__title"> The Poincare Lemma, Antiexact Forms, and Fermionic Quantum Harmonic Oscillator </div> <div data-test="inCoD" data-track-context="sticky banner"> <div class="c-pdf-container"> <div class="c-pdf-download u-clear-both u-mb-16"> <a href="/content/pdf/10.1007/s00025-020-01247-8.pdf" class="u-button u-button--full-width u-button--primary u-justify-content-space-between c-pdf-download__link" data-article-pdf="true" data-readcube-pdf-url="true" data-test="pdf-link" data-draft-ignore="true" data-track="content_download" data-track-type="article pdf download" data-track-action="download pdf" data-track-label="button" data-track-external download> <span class="c-pdf-download__text">Download PDF</span> <svg aria-hidden="true" focusable="false" width="16" height="16" class="u-icon"><use xlink:href="#icon-eds-i-download-medium"/></svg> </a> </div> </div> </div> </div> </div> <div class="c-article-header"> <header> <ul class="c-article-author-list c-article-author-list--short" data-test="authors-list" data-component-authors-activator="authors-list"><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Rados_aw_Antoni-Kycia-Aff1-Aff2" data-author-popup="auth-Rados_aw_Antoni-Kycia-Aff1-Aff2" data-author-search="Kycia, Radosław Antoni" data-corresp-id="c1">Radosław Antoni Kycia<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-mail-medium"></use></svg></a><span class="u-js-hide"> <a class="js-orcid" href="http://orcid.org/0000-0002-6390-4627"><span class="u-visually-hidden">ORCID: </span>orcid.org/0000-0002-6390-4627</a></span><sup class="u-js-hide"><a href="#Aff1">1</a>,<a href="#Aff2">2</a></sup> </li></ul> <div data-test="article-metrics"> <ul class="app-article-metrics-bar u-list-reset"> <li class="app-article-metrics-bar__item"> <p class="app-article-metrics-bar__count"><svg class="u-icon app-article-metrics-bar__icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-accesses-medium"></use> </svg>2326 <span class="app-article-metrics-bar__label">Accesses</span></p> </li> <li class="app-article-metrics-bar__item"> <p class="app-article-metrics-bar__count"><svg class="u-icon app-article-metrics-bar__icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-citations-medium"></use> </svg>1 <span class="app-article-metrics-bar__label">Citation</span></p> </li> <li class="app-article-metrics-bar__item"> <p class="app-article-metrics-bar__count"><svg class="u-icon app-article-metrics-bar__icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-altmetric-medium"></use> </svg>5 <span class="app-article-metrics-bar__label">Altmetric</span></p> </li> <li class="app-article-metrics-bar__item app-article-metrics-bar__item--metrics"> <p class="app-article-metrics-bar__details"><a href="/article/10.1007/s00025-020-01247-8/metrics" data-track="click" data-track-action="view metrics" data-track-label="link" rel="nofollow">Explore all metrics <svg class="u-icon app-article-metrics-bar__arrow-icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-arrow-right-medium"></use> </svg></a></p> </li> </ul> </div> <div class="u-mt-32"> </div> </header> </div> <div data-article-body="true" data-track-component="article body" class="c-article-body"> <section aria-labelledby="Abs1" data-title="Abstract" lang="en"><div class="c-article-section" id="Abs1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Abs1">Abstract</h2><div class="c-article-section__content" id="Abs1-content"><p>The paper focuses on various properties and applications of the homotopy operator, which occurs in the Poincaré lemma. In the first part, an abstract operator calculus is constructed, where the exterior derivative is an abstract derivative and the homotopy operator plays the role of an abstract integral. This operator calculus can be used to formulate abstract differential equations. An example of the eigenvalue problem that resembles the fermionic quantum harmonic oscillator is presented. The second part presents the dual complex to the Dolbeault bicomplex generated by the homotopy operator on complex manifolds. </p></div></div></section> <div data-test="cobranding-download"> </div> <section aria-labelledby="inline-recommendations" data-title="Inline Recommendations" class="c-article-recommendations" data-track-component="inline-recommendations"> <h3 class="c-article-recommendations-title" id="inline-recommendations">Similar content being viewed by others</h3> <div class="c-article-recommendations-list"> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3Aplaceholder%2Fimages/placeholder-figure-springernature.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://link.springer.com/10.1007/s00006-020-01057-9?fromPaywallRec=false" data-track="select_recommendations_1" data-track-context="inline recommendations" data-track-action="click recommendations inline - 1" data-track-label="10.1007/s00006-020-01057-9">Conjugate Harmonic Functions of Fueter Type </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__date">16 April 2020</span> </div> </div> </article> </div> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3Aplaceholder%2Fimages/placeholder-figure-springernature.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://link.springer.com/10.1007/JHEP02(2024)137?fromPaywallRec=false" data-track="select_recommendations_2" data-track-context="inline recommendations" data-track-action="click recommendations inline - 2" data-track-label="10.1007/JHEP02(2024)137">Homological quantum mechanics </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__access-type">Open access</span> <span class="c-article-meta-recommendations__date">20 February 2024</span> </div> </div> </article> </div> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3Aplaceholder%2Fimages/placeholder-figure-springernature.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://link.springer.com/10.1007/s11425-019-1668-4?fromPaywallRec=false" data-track="select_recommendations_3" data-track-context="inline recommendations" data-track-action="click recommendations inline - 3" data-track-label="10.1007/s11425-019-1668-4">Extension formulae on almost complex manifolds </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__date">30 April 2020</span> </div> </div> </article> </div> </div> </section> <script> window.dataLayer = window.dataLayer || []; window.dataLayer.push({ recommendations: { recommender: 'semantic', model: 'specter', policy_id: 'NA', timestamp: 1740969109, embedded_user: 'null' } }); </script> <div class="app-card-service" data-test="article-checklist-banner"> <div> <a class="app-card-service__link" data-track="click_presubmission_checklist" data-track-context="article page top of reading companion" data-track-category="pre-submission-checklist" data-track-action="clicked article page checklist banner test 2 old version" data-track-label="link" href="https://beta.springernature.com/pre-submission?journalId=25" data-test="article-checklist-banner-link"> <span class="app-card-service__link-text">Use our pre-submission checklist</span> <svg class="app-card-service__link-icon" aria-hidden="true" focusable="false"><use xlink:href="#icon-eds-i-arrow-right-small"></use></svg> </a> <p class="app-card-service__description">Avoid common mistakes on your manuscript.</p> </div> <div class="app-card-service__icon-container"> <svg class="app-card-service__icon" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-clipboard-check-medium"></use> </svg> </div> </div> <div class="main-content"> <section data-title="Introduction"><div class="c-article-section" id="Sec1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec1"><span class="c-article-section__title-number">1 </span>Introduction</h2><div class="c-article-section__content" id="Sec1-content"><p>The Poincaré lemma is one of the most important tools of exterior calculus. Although it is a very old result it is continuously generalized in various ways [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title="Beilinson, A.: p-adic periods and derived de Rham cohomology. J. Am. Math. Soc. 25, 715–738 (2012). 
https://doi.org/10.1090/S0894-0347-2012-00729-2

" href="/article/10.1007/s00025-020-01247-8#ref-CR1" id="ref-link-section-d158109130e383">1</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 5" title="Desbrun, M., Leok, M., Marsden, J.E.: Discrete Poincaré lemma. Appl. Numer. Math. 53(2–4), 231–248 (2005). 
https://doi.org/10.1016/j.apnum.2004.09.035

" href="/article/10.1007/s00025-020-01247-8#ref-CR5" id="ref-link-section-d158109130e386">5</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 6" title="Desbrun, M., Hirani, A.N., Leok, M., Marsden, J.E.: Discrete Exterior Calculus, 
arXiv:math/0508341

 [math.DG]" href="/article/10.1007/s00025-020-01247-8#ref-CR6" id="ref-link-section-d158109130e389">6</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 13" title="Jell, P.: A Poincaré lemma for real-valued differential forms on Berkovich spaces. Math. Z. 282, 1149–1167 (2016). 
https://doi.org/10.1007/s00209-015-1583-8

" href="/article/10.1007/s00025-020-01247-8#ref-CR13" id="ref-link-section-d158109130e392">13</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 15" title="Lesfari, A.: On Poincaré lemma or Volterra theorem about differential forms and cohomology groups, 
arXiv:1905.13347

 [math.GM]" href="/article/10.1007/s00025-020-01247-8#ref-CR15" id="ref-link-section-d158109130e395">15</a>], including non-Abelian cases [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 17" title="Voronov, T.: On a non-Abelian Poincaré lemma. Proc. Amer. Math. Soc. 140, 2855–2872 (2012). 
https://doi.org/10.1090/S0002-9939-2011-11116-X

" href="/article/10.1007/s00025-020-01247-8#ref-CR17" id="ref-link-section-d158109130e399">17</a>] or general approach to (dis)continuous cases [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 10" title="Harrison, J.: Operator calculus of differential chains and differential forms. J. Geom. Anal. 25(1), 357–420 (2015). 
https://doi.org/10.1007/s12220-013-9433-6

" href="/article/10.1007/s00025-020-01247-8#ref-CR10" id="ref-link-section-d158109130e402">10</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 11" title="Harrison, J.: Geometric Poincaré Lemma, 
arXiv:1101.0313

 [math.AT]" href="/article/10.1007/s00025-020-01247-8#ref-CR11" id="ref-link-section-d158109130e405">11</a>]. In this paper, we will focus on the ’operator’ approach to this lemma as well as on extension to complex manifolds.</p><p>As an introduction, we review some basic facts about the Poincaré lemma in order to fix notation. There are various formulations of the lemma and the most general one is the following well-known form</p> <h3 class="c-article__sub-heading" id="FPar1">Theorem 1</h3> <p>(Corollary 4.1.1 of [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 3" title="Bott, R., Tu, L.W.: Differential Forms in Algebraic Topology. Springer, Berlin (1995)" href="/article/10.1007/s00025-020-01247-8#ref-CR3" id="ref-link-section-d158109130e418">3</a>]) (The Poincaré lemma) </p><div id="Equ35" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} H^{*}({\mathbb {R}}^n) = H^{*}(point)=\left\{ \begin{array}{l} {\mathbb {R}}, \quad (n=0) \\ 0 \quad (n>0) \end{array}\right. \end{aligned}$$</span></div></div> <p>It can be formulated in another way by introducing (open) star-shaped region <i>U</i> of <span class="mathjax-tex">${\mathbb {R}}^{n}$</span> with respect to <span class="mathjax-tex">$x_{0}\in U$</span>. It is an open region (<span class="mathjax-tex">$dim(U)=n$</span>) where any other point <span class="mathjax-tex">$x\in U$</span> can be connected with <span class="mathjax-tex">$x_{0}$</span> using the line segment that lies entirely inside <i>U</i>. For a smooth manifold <i>M</i> without boundary and of dimension <span class="mathjax-tex">$n=dim(M)$</span>, we define a star-shaped region <i>U</i> as (see, e.g., [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 14" title="Lee, J.: Introduction to Smooth Manifolds, 2nd edn. Springer, Berlin (2012)" href="/article/10.1007/s00025-020-01247-8#ref-CR14" id="ref-link-section-d158109130e756">14</a>]) the region diffeomorphic to an open ball in <span class="mathjax-tex">${\mathbb {R}}^{n}$</span>. For <i>M</i> having boundary the star-shaped region can be also diffeomorphic to an open half-ball. Moreover, locally for a smooth manifold each point has a neighbourhood that is star-shaped.</p><p>Then we have</p> <h3 class="c-article__sub-heading" id="FPar2">Theorem 2</h3> <p>(e.g. Theorem 11.49 of [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 14" title="Lee, J.: Introduction to Smooth Manifolds, 2nd edn. Springer, Berlin (2012)" href="/article/10.1007/s00025-020-01247-8#ref-CR14" id="ref-link-section-d158109130e797">14</a>])(The Poincaré lemma) If <i>U</i> is a star-shaped open subset of <span class="mathjax-tex">${\mathbb {R}}^n$</span>, then every closed covector field on <i>U</i> is exact.</p> <p>As an existential statement, it is not useful in computations. The other approach relies on homotopy operator which is a local notion. To begin with, introduce the operator</p><div id="Equ1" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} G\omega := \int _{0}^{1} (\partial _{t} \lrcorner \omega ) \mathrm{d}t, \end{aligned}$$</span></div><div class="c-article-equation__number"> (1) </div></div><p>for <span class="mathjax-tex">$\omega \in \Omega (M \times {\mathbb {R}})$</span>, where <span class="mathjax-tex">$\Omega (M \times {\mathbb {R}})$</span> is the module of forms, and where <i>M</i> is a smooth manifold with or without boundary. Next, choose a homotopy <span class="mathjax-tex">$F:[0,1]\times M \rightarrow M$</span> between <i>f</i> and <i>g</i>, that is, <span class="mathjax-tex">$F(0,.)=f(.)$</span> and <span class="mathjax-tex">$F(1,.)=g(.)$</span>. Using the homotopy we can define the operator</p><div id="Equ2" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {\tilde{H}}\omega = G\circ F^{*} (\omega ), \end{aligned}$$</span></div><div class="c-article-equation__number"> (2) </div></div><p>for <span class="mathjax-tex">$\omega \in \Omega (M)$</span>. This operator has important property, which can be introduced using the Homotopy Invariance Formula, namely,</p> <h3 class="c-article__sub-heading" id="FPar3">Theorem 3</h3> <p>(Paragraph 29 of [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 18" title="Tu, L.W.: An Introduction to Manifolds, 2nd edn. Springer, Berlin (2010)" href="/article/10.1007/s00025-020-01247-8#ref-CR18" id="ref-link-section-d158109130e1222">18</a>])(Homotopy Invariance Formula for the de Rham complex) </p><div id="Equ3" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} dG + Gd = i_{1}^{*}-i_{0}^{*}, \end{aligned}$$</span></div><div class="c-article-equation__number"> (3) </div></div><p>where <span class="mathjax-tex">$i_{t}(x)=(t,x)$</span> for <span class="mathjax-tex">$t \in {\mathbb {R}}$</span> and <span class="mathjax-tex">$x \in M$</span>.</p> <p>Using this formula we have the well known identity</p><div id="Equ4" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} ({\tilde{H}}d+d{\tilde{H}})\omega = GdF^{*}\omega +dGF^{*}\omega =i_{1}^{*}F^{*}\omega -i_{0}^{*}F^{*}\omega =g^{*}\omega -f^{*}\omega . \end{aligned}$$</span></div><div class="c-article-equation__number"> (4) </div></div><p>The crucial observation was made by D.G.B. Edelen (see [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="Edelen, D.G.B.: Applied Exterior Calculus, Revised edn. Dover Publications, New York (2011)" href="/article/10.1007/s00025-020-01247-8#ref-CR8" id="ref-link-section-d158109130e1609">8</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 9" title="Edelen, D.G.B.: Isovector Methods for Equations of Balance. Springer, Berlin (1980)" href="/article/10.1007/s00025-020-01247-8#ref-CR9" id="ref-link-section-d158109130e1612">9</a>]), that after a special choice of the homotopy, the definition (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00025-020-01247-8#Equ2">2</a>) has a particularly simple form with deep implications. In order to derive Edelen’s version of homotopy operator <i>H</i> one have to choose the homotopy between the identity (<span class="mathjax-tex">$g(x)=x$</span>) and the constant map (<span class="mathjax-tex">$f(x)=x_{0}$</span>) for some fixed point <span class="mathjax-tex">$x_{0}\in U \subset M$</span>. To provide the correct definition it is assumed that <i>U</i> is a star-shaped region with respect to <span class="mathjax-tex">$x_{0}$</span>. For such homotopy Edelen rewrote (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00025-020-01247-8#Equ4">4</a>) [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="Edelen, D.G.B.: Applied Exterior Calculus, Revised edn. Dover Publications, New York (2011)" href="/article/10.1007/s00025-020-01247-8#ref-CR8" id="ref-link-section-d158109130e1745">8</a>] as</p> <h3 class="c-article__sub-heading" id="FPar4">Definition 1</h3> <p><i>(Edelen’s homotopy operator)</i> </p><div id="Equ5" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} H\omega := \int _{0}^{1} {\mathcal {K}} \lrcorner \omega _{F(t,x)} t^{k-1} \mathrm{d}t, \end{aligned}$$</span></div><div class="c-article-equation__number"> (5) </div></div><p>for a <i>k</i>-form <span class="mathjax-tex">$\omega \in \Omega ^{k}(U)$</span>, <span class="mathjax-tex">${\mathcal {K}}:=(x-x_{0})^{i}\partial _{i}$</span>, <span class="mathjax-tex">$k=deg(\omega )$</span>, and <span class="mathjax-tex">$F(t,x)=x_{0}+t(x-x_{0})$</span> is a homotopy between the constant map <span class="mathjax-tex">$x\rightarrow x_{0}$</span> and the identity map <span class="mathjax-tex">$I:x\rightarrow x$</span>. The form <span class="mathjax-tex">$\omega $</span> under the integral is evaluated at the point <i>F</i>(<i>t</i>, <i>x</i>).</p> <p>The form of the operator is a special case of <span class="mathjax-tex">${\tilde{H}}$</span> for the homotopy <span class="mathjax-tex">$F(t,x)=x_{0}+t(x-x_{0})$</span> and its explicit derivation is simple application of the pullback along <i>F</i>. <i>H</i> has various properties described by Theorem 5–3.1 of [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="Edelen, D.G.B.: Applied Exterior Calculus, Revised edn. Dover Publications, New York (2011)" href="/article/10.1007/s00025-020-01247-8#ref-CR8" id="ref-link-section-d158109130e2227">8</a>], from which the most important in later use is its nilpotency, <span class="mathjax-tex">$H^{2}=0$</span>, which results from the double application of the insertion of <span class="mathjax-tex">${\mathcal {K}}$</span> under the integral of (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00025-020-01247-8#Equ5">5</a>).</p><p>The operator <i>H</i> has its own Homotopy Invariance Formula, which can be written in a more compact form than in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="Edelen, D.G.B.: Applied Exterior Calculus, Revised edn. Dover Publications, New York (2011)" href="/article/10.1007/s00025-020-01247-8#ref-CR8" id="ref-link-section-d158109130e2281">8</a>], namely,</p> <h3 class="c-article__sub-heading" id="FPar5">Theorem 4</h3> <p>(Homotopy Invariance Formula for <i>H</i> operator) </p><div id="Equ6" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} dH+Hd=I^{*} - s_{x_{0}}^{*}, \end{aligned}$$</span></div><div class="c-article-equation__number"> (6) </div></div><p>where <span class="mathjax-tex">$s_{x_{0}}(x)=x_{0}$</span> is a constant map and <i>I</i> is the identity.</p> <p>This formula was provided in Theorem 5–3.1 of [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="Edelen, D.G.B.: Applied Exterior Calculus, Revised edn. Dover Publications, New York (2011)" href="/article/10.1007/s00025-020-01247-8#ref-CR8" id="ref-link-section-d158109130e2428">8</a>] as a piecewise definition</p><div id="Equ7" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \left\{ \begin{array}{c} Hd+dH=I, \quad \mathrm {on} \quad \Omega ^k, k>0, \\ (Hdf)(x) = f(x)-f(x_{0}) \quad \mathrm {for} \quad f \in \Omega ^{0}. \end{array} \right. \end{aligned}$$</span></div><div class="c-article-equation__number"> (7) </div></div><p>It results from the fact that the pullback along the constant function <span class="mathjax-tex">$s_{x_{0}}^{*}\omega =0$</span> for <span class="mathjax-tex">$deg(\omega )>0$</span>, and from the fact that <span class="mathjax-tex">${\mathcal {K}} \lrcorner f =0$</span>.</p><p>One can also note that (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00025-020-01247-8#Equ6">6</a>) is correct for any, not necessarily linear, homotopy <i>F</i> between the identity and the constant map, however, in such a case, the explicit formula (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00025-020-01247-8#Equ5">5</a>) is not valid.</p><p>If a form <span class="mathjax-tex">$\omega $</span> fulfils <span class="mathjax-tex">$d\omega = 0$</span> then it is called <i>closed</i>. It is a well-known fact that in the star-shaped region <i>U</i> (which we will assume hereafter), by the Poincaré lemma, it is also <i>exact</i>, which means that there is a form <span class="mathjax-tex">$\alpha $</span> of degree <span class="mathjax-tex">$deg(\alpha )=deg(\omega )-1$</span> such that <span class="mathjax-tex">$\omega =d\alpha $</span>. The exact (and hence closed) forms form a subspace <span class="mathjax-tex">${\mathcal {E}}(U)$</span> of <span class="mathjax-tex">$\Omega (U)$</span>.</p><p>The following Lemmas will be useful in formulating operator calculus for <i>d</i> and <i>H</i>:</p> <h3 class="c-article__sub-heading" id="FPar6">Lemma 1</h3> <p>(Lemma 5–4.1 of [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="Edelen, D.G.B.: Applied Exterior Calculus, Revised edn. Dover Publications, New York (2011)" href="/article/10.1007/s00025-020-01247-8#ref-CR8" id="ref-link-section-d158109130e2935">8</a>]) The operator <i>dH</i> maps <span class="mathjax-tex">${\mathcal {E}}^{k}$</span> onto <span class="mathjax-tex">${\mathcal {E}}^{k}$</span> and <span class="mathjax-tex">$\Omega $</span> onto <span class="mathjax-tex">${\mathcal {E}}$</span>.</p> <h3 class="c-article__sub-heading" id="FPar7">Lemma 2</h3> <p>(Lemma 5–4.2 of [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="Edelen, D.G.B.: Applied Exterior Calculus, Revised edn. Dover Publications, New York (2011)" href="/article/10.1007/s00025-020-01247-8#ref-CR8" id="ref-link-section-d158109130e3026">8</a>]) The operator <i>d</i> is the inverse of the operator <i>H</i> when the domain of <i>H</i> is restricted to <span class="mathjax-tex">${\mathcal {E}}^{k}$</span>.</p> <p>In addition, <span class="mathjax-tex">${\mathcal {E}}^{0}(U)$</span>—the set of exact functions over <i>U</i> is empty.</p><p>The less-known fact [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="Edelen, D.G.B.: Applied Exterior Calculus, Revised edn. Dover Publications, New York (2011)" href="/article/10.1007/s00025-020-01247-8#ref-CR8" id="ref-link-section-d158109130e3108">8</a>] is that, the <i>H</i> singles out the so-called, <i>antiexact</i> forms, that are the image of the complementary projection operator <span class="mathjax-tex">$Hd = I^{*}-dH-s_{x_{0}}^{*}$</span>. This means that for an antiexact form <span class="mathjax-tex">$\omega $</span> there is an exact form <span class="mathjax-tex">$\alpha =d\beta $</span> such that <span class="mathjax-tex">$\omega = H\alpha $</span>. The antiexact forms compose into the submodule <span class="mathjax-tex">${\mathcal {A}}(U)$</span> of <span class="mathjax-tex">$\Omega (U)$</span> which is characterized by</p> <h3 class="c-article__sub-heading" id="FPar8">Lemma 3</h3> <p>(Lemma 5–5.1 of [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="Edelen, D.G.B.: Applied Exterior Calculus, Revised edn. Dover Publications, New York (2011)" href="/article/10.1007/s00025-020-01247-8#ref-CR8" id="ref-link-section-d158109130e3301">8</a>])</p><div id="Equ8" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {\mathcal {A}}^{k}=\{ \alpha \in \Omega ^{k}, {\mathcal {K}}\lrcorner \alpha =0, \quad \alpha |_{x_{0}}=0, k>0\}. \end{aligned}$$</span></div><div class="c-article-equation__number"> (8) </div></div> <p>In addition, <span class="mathjax-tex">${\mathcal {A}}^{n}(U)$</span> for <span class="mathjax-tex">$n=dim(U)$</span> is the empty set. Antiexactness is a local notion on star-shaped regions.</p><p>In this paper, we start from these simple properties and build on them additional abstract structures. The first aim is to formulate operator calculus in terms of <i>d</i> and <i>H</i>. It is suitable to use the work of R. Bittner [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Bittner, R.: Operational calculus in linear spaces. Stud. Math. 20, 1–18 (1961). 
https://doi.org/10.4064/sm-20-1-1-18

" href="/article/10.1007/s00025-020-01247-8#ref-CR2" id="ref-link-section-d158109130e3500">2</a>] who generalized differentiation and integration operations. This will allow us to formulate abstract differential equations and eigenvalue problems. We will show an example of such equations, which behaves similarly to the fermionic quantum harmonic oscillator (see, e.g., Chapter 5 of [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 4" title="Das, A.: Field Theory, A Path Integral Approach. World Scientific, Singapore (1993)" href="/article/10.1007/s00025-020-01247-8#ref-CR4" id="ref-link-section-d158109130e3503">4</a>]) used in quantum mechanics.</p><p>The second aim is to show how the Dolbeault bicomplex on complex manifolds interplay with the complex-valued version of <i>H</i>, which will be defined.</p><p>In summary, our aims are as follows:</p><ul class="u-list-style-bullet"> <li> <p>Make a more detailed characterization of exact and antiexact complexes.</p> </li> <li> <p>Construct operator calculus, where the exterior derivative plays a role of a ’derivative’ and the homotopy operator is an ’integral’. This allows us to construct and solve abstract differential equations in these terms.</p> </li> <li> <p>Construct the homotopy operator for complex manifolds and describe its action on the Dolbeault complex.</p> </li> </ul><p>The paper is organized as follows. In the next section, a detailed description of <span class="mathjax-tex">$\Omega (U)$</span> decomposition into the exact vector space and the antiexact module will be given. Then in the following section, we will present the connection of these formulas with operator calculus and the fermionic quantum harmonic oscillator. Next, we develop the theory of the homotopy operator for complex manifolds.</p></div></div></section><section data-title="Homotopy Operator and (Anti)Exact Forms"><div class="c-article-section" id="Sec2-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec2"><span class="c-article-section__title-number">2 </span>Homotopy Operator and (Anti)Exact Forms</h2><div class="c-article-section__content" id="Sec2-content"><p>In [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="Edelen, D.G.B.: Applied Exterior Calculus, Revised edn. Dover Publications, New York (2011)" href="/article/10.1007/s00025-020-01247-8#ref-CR8" id="ref-link-section-d158109130e3570">8</a>] the decomposition <span class="mathjax-tex">$\Omega ={\mathcal {E}} \oplus {\mathcal {A}}$</span> on some star-shaped region <i>U</i> of a smooth manifold <i>M</i> is stated, however, using Lemmas <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s00025-020-01247-8#FPar6">1</a>, <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s00025-020-01247-8#FPar7">2</a>, <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s00025-020-01247-8#FPar8">3</a> and above properties, we have a finer characterization of this decomposition</p> <h3 class="c-article__sub-heading" id="FPar9">Corollary 1</h3> <p><span class="mathjax-tex">$\Omega (U)$</span> for <span class="mathjax-tex">$n=dim(U)>0$</span> is decomposed into the direct sum of exact and antiexact parts with respect to grading in the way presented in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s00025-020-01247-8#Fig1">1</a>. At each degree <span class="mathjax-tex">$k \ge 0$</span> there is <span class="mathjax-tex">$\Omega ^{k} = {\mathcal {E}}^k \oplus {\mathcal {A}}^k$</span>. The relations <span class="mathjax-tex">$d^2=0 = H^2$</span> when moving along the arrows are also visible.</p> <div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-1" data-title="Fig. 1"><figure><figcaption><b id="Fig1" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 1</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1007/s00025-020-01247-8/figures/1" rel="nofollow"><picture><img aria-describedby="Fig1" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs00025-020-01247-8/MediaObjects/25_2020_1247_Fig1_HTML.png" alt="figure 1" loading="lazy" width="685" height="1005"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-1-desc"><p>Decomposition of <span class="mathjax-tex">$\Omega $</span> into exact and antiexact subspaces with respect to the degree. Here <span class="mathjax-tex">${\mathbb {R}}$</span> is treated as a space of constant functions</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1007/s00025-020-01247-8/figures/1" data-track-dest="link:Figure1 Full size image" aria-label="Full size image figure 1" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <p>It is easily visible from the figure that for fixed <span class="mathjax-tex">$0<k<n=dim(U)$</span> there is separate ’subdiagram’ depicted in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s00025-020-01247-8#Fig2">2</a>, which will be the starting point for construction of operator calculus in the next section.</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-2" data-title="Fig. 2"><figure><figcaption><b id="Fig2" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 2</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1007/s00025-020-01247-8/figures/2" rel="nofollow"><picture><img aria-describedby="Fig2" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs00025-020-01247-8/MediaObjects/25_2020_1247_Fig2_HTML.png" alt="figure 2" loading="lazy" width="577" height="120"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-2-desc"><p>Part of the decomposition from Fig.<a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s00025-020-01247-8#Fig1">1</a> for <span class="mathjax-tex">$0<k<n$</span></p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1007/s00025-020-01247-8/figures/2" data-track-dest="link:Figure2 Full size image" aria-label="Full size image figure 2" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><p>For <span class="mathjax-tex">$k=0$</span>, the kernel <i>Ker</i>(<i>d</i>) consists of constant functions with values in the field over which <span class="mathjax-tex">$\Omega $</span> is the vector space, i.e., <span class="mathjax-tex">${\mathbb {R}}$</span>. This field can also be treated as constant 0-forms. They are closed although not exact, which is a peculiarity in this decomposition.</p><p>The general formula (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00025-020-01247-8#Equ6">6</a>) is the starting point for considering the operator algebra of <i>H</i>, <i>d</i>, <i>I</i> and <span class="mathjax-tex">$s_{x_{0}}$</span> in terms of operator calculus of Bittner [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Bittner, R.: Operational calculus in linear spaces. Stud. Math. 20, 1–18 (1961). 
https://doi.org/10.4064/sm-20-1-1-18

" href="/article/10.1007/s00025-020-01247-8#ref-CR2" id="ref-link-section-d158109130e4043">2</a>] which will be the subject of the next section.</p></div></div></section><section data-title="Bittner’s Operator Calculus"><div class="c-article-section" id="Sec3-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec3"><span class="c-article-section__title-number">3 </span>Bittner’s Operator Calculus</h2><div class="c-article-section__content" id="Sec3-content"><h3 class="c-article__sub-heading" id="Sec4"><span class="c-article-section__title-number">3.1 </span>General Setup</h3><p>The Bittner’s operator calculus [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Bittner, R.: Operational calculus in linear spaces. Stud. Math. 20, 1–18 (1961). 
https://doi.org/10.4064/sm-20-1-1-18

" href="/article/10.1007/s00025-020-01247-8#ref-CR2" id="ref-link-section-d158109130e4058">2</a>] is a way to redefine derivative and integral in abstract terms. It mimics the well-known formulas</p><div id="Equ9" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \frac{d}{dx} \int _{q}^{x} f(x')\mathrm{d}x' = f(x), \quad \int _{q}^{x} \frac{df(x')}{dx'} \mathrm{d}x' = f(x)-f(q), \end{aligned}$$</span></div><div class="c-article-equation__number"> (9) </div></div><p>for <i>f</i> being e.g. <span class="mathjax-tex">$C^{1}$</span> function. It is defined as follows</p> <h3 class="c-article__sub-heading" id="FPar10">Definition 2</h3> <p> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Bittner, R.: Operational calculus in linear spaces. Stud. Math. 20, 1–18 (1961). 
https://doi.org/10.4064/sm-20-1-1-18

" href="/article/10.1007/s00025-020-01247-8#ref-CR2" id="ref-link-section-d158109130e4281">2</a>] Consider two linear spaces <span class="mathjax-tex">$L^{0}$</span> and <span class="mathjax-tex">$L^{1}$</span> and define an abstract derivative as surjective mapping <span class="mathjax-tex">$S \in Hom(L^{1},L^{0})$</span>. Elements of <i>Ker</i>(<i>S</i>) are called constants of the derivative <i>S</i>. Define also <span class="mathjax-tex">$T_{q} \in Hom(L^{0},L^{1})$</span> for some constant <span class="mathjax-tex">$q \in Ker(S)$</span> such that</p><div id="Equ10" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} ST_{q}=I, \quad T_{q}S = I - s_{q}, \end{aligned}$$</span></div><div class="c-article-equation__number"> (10) </div></div><p>where <span class="mathjax-tex">$s_{q}$</span> is the projection operator on <i>Ker</i>(<i>d</i>) associated with <i>q</i>. <span class="mathjax-tex">$T_{q}$</span> is called an abstract integral.</p> <p>For instance, in (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00025-020-01247-8#Equ9">9</a>) <span class="mathjax-tex">$s_{q}f = f(q)\in ker\left( \frac{d}{dx}\right) $</span> is understood as a constant function.</p><p>We want to underline that this is not the derivation occurring in Differential Geometry [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 14" title="Lee, J.: Introduction to Smooth Manifolds, 2nd edn. Springer, Berlin (2012)" href="/article/10.1007/s00025-020-01247-8#ref-CR14" id="ref-link-section-d158109130e4662">14</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Nakahara, M.: Geometry, Topology and Physics, 2nd edn. CRC Press, Boca Raton (2003)" href="/article/10.1007/s00025-020-01247-8#ref-CR16" id="ref-link-section-d158109130e4665">16</a>] since, e.g., no Leibnitz rule is implemented in this definition. Instead, it is a set of three operators that fulfill the requirements (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00025-020-01247-8#Equ10">10</a>) that generalize (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00025-020-01247-8#Equ9">9</a>) from standard Calculus. It would be better to call them Bittner’s derivative, integral, and projection on boundary conditions to distinguish them, however, we will not adopt this convention.</p><p>Let us consider the diagram from Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s00025-020-01247-8#Fig2">2</a> for <span class="mathjax-tex">$k>1$</span>. In this case <span class="mathjax-tex">$Hd=I$</span> on <span class="mathjax-tex">${\mathcal {A}}^{k-1}$</span> and <span class="mathjax-tex">$dH=I$</span> on <span class="mathjax-tex">${\mathcal {E}}^{k}$</span> and therefore there is no projection on boundary data. In this case, restricted <i>H</i> and <i>d</i> are inverses to each other.</p><p>For <span class="mathjax-tex">$k>0$</span> this can be also seen as a mapping between the spaces on Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s00025-020-01247-8#Fig3">3</a>. In this case at the level <span class="mathjax-tex">$k-1$</span> we have <span class="mathjax-tex">$\Omega ^{k-1}={\mathcal {A}}^{k-1}\oplus {\mathcal {E}}^{k-1}$</span>. This is decomposition according to the action of <i>d</i> since <span class="mathjax-tex">$Ker(d)={\mathcal {E}}^{k-1}$</span> and <span class="mathjax-tex">$Im(d)={\mathcal {E}}^{k}$</span>. We therefore have <span class="mathjax-tex">$Hd=I^{*}$</span> and <span class="mathjax-tex">$dH=I^{*}$</span>, which is the special case of the second formula of (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00025-020-01247-8#Equ10">10</a>) where <span class="mathjax-tex">$s_{x_{0}}^{*}=0$</span>.</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-3" data-title="Fig. 3"><figure><figcaption><b id="Fig3" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 3</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1007/s00025-020-01247-8/figures/3" rel="nofollow"><picture><img aria-describedby="Fig3" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs00025-020-01247-8/MediaObjects/25_2020_1247_Fig3_HTML.png" alt="figure 3" loading="lazy" width="685" height="192"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-3-desc"><p>Part of decomposition of <span class="mathjax-tex">$\Omega ^{k-1}={\mathcal {A}}^{k-1}\oplus {\mathcal {E}}^{k-1}$</span> for fixed <span class="mathjax-tex">$0<k<n$</span>. Note that <span class="mathjax-tex">$ker(d)={\mathcal {E}}^{k-1}$</span> and <span class="mathjax-tex">$im(d) = {\mathcal {E}}^{k}$</span></p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1007/s00025-020-01247-8/figures/3" data-track-dest="link:Figure3 Full size image" aria-label="Full size image figure 3" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div><p>For <span class="mathjax-tex">$k=1$</span> for the case in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s00025-020-01247-8#Fig3">3</a> the pullback along the constant function <span class="mathjax-tex">$s_{x_{0}}^{*}$</span> is indispensable, and therefore, the formula (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00025-020-01247-8#Equ6">6</a>) leads to</p><div id="Equ11" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} dH = I, \quad Hd = I^{*} - s_{x_{0}}^{*}. \end{aligned}$$</span></div><div class="c-article-equation__number"> (11) </div></div><p>In this case the resemblance to (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00025-020-01247-8#Equ10">10</a>) is even closer with <span class="mathjax-tex">$s_{q} = s_{x_{0}}^{*}$</span>. In case of <span class="mathjax-tex">$\Omega ^{0}$</span> the ’constants’ of <i>d</i> are the constant functions with values in <span class="mathjax-tex">${\mathbb {R}}$</span>, and pullback projects on them any function from <span class="mathjax-tex">$\Omega ^{0}$</span> since it is evaluation of a function at <span class="mathjax-tex">$x=x_{0}$</span> and <span class="mathjax-tex">${\mathcal {A}}^{0}\in ker(s_{x_{0}}^{*})$</span>.</p><p>Therefore we have,</p> <h3 class="c-article__sub-heading" id="FPar11">Lemma 4</h3> <p>For <span class="mathjax-tex">$k>0$</span> the operator calculus of Definition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s00025-020-01247-8#FPar10">2</a> with the abstract derivative <i>d</i> and the abstract integral <i>H</i> is realized on the spaces of Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s00025-020-01247-8#Fig3">3</a> as</p><div id="Equ12" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} Hd=I \quad \mathrm {and} \quad dH=I^{*}. \end{aligned}$$</span></div><div class="c-article-equation__number"> (12) </div></div><p>For <span class="mathjax-tex">$k=0$</span> the Bittner’s operator calculus is realized by (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00025-020-01247-8#Equ11">11</a>).</p> <p>This observation allows us to formulate abstract differential equations on <span class="mathjax-tex">$\Omega $</span> using <i>d</i> and <i>H</i> as a ’derivative’ and an ’integral’ respectively. However, these operations are nilpotent, which put additional constraints on this ’operator calculus’ and suggest that they resemble the situation appearing in the fermionic harmonic oscillator. This idea will be followed in the next subsection.</p><p>The idea of applying a combination of the exterior derivative and the homotopy operator to variational differential equations was discussed in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 12" title="Iwaniec, T., Lutoborski, A.: Integral estimates for null Lagrangians. Arch. Ration. Mech. Anal. 125, 25–79 (1993). 
https://doi.org/10.1007/BF00411477

" href="/article/10.1007/s00025-020-01247-8#ref-CR12" id="ref-link-section-d158109130e5860">12</a>] and [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 7" title="Ding, S., Nolder, C.A.: Weighted Poincaré inequalities for solutions to A-harmonic equations. Ill. J. Math. 46(1), 199–205 (2002). 
https://doi.org/10.1215/ijm/1258136150

" href="/article/10.1007/s00025-020-01247-8#ref-CR7" id="ref-link-section-d158109130e5863">7</a>], and this presentation uses methods of functional analysis. However, we follow a different direction focusing on connecting Edelen’s idea and abstract Bittner’s calculus.</p><h3 class="c-article__sub-heading" id="Sec5"><span class="c-article-section__title-number">3.2 </span>Homotopical Harmonic Oscillator</h3><p>We first recall a basic structure of the fermionic quantum harmonic oscillator. Consider a two-dimensional Hilbert space over <span class="mathjax-tex">${\mathbb {C}}$</span>. Then the fermionic quantum harmonic oscillator is defined by the Hamilton operator [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 4" title="Das, A.: Field Theory, A Path Integral Approach. World Scientific, Singapore (1993)" href="/article/10.1007/s00025-020-01247-8#ref-CR4" id="ref-link-section-d158109130e5889">4</a>]</p><div id="Equ13" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {\hat{H}}=a^{\dagger }a-aa^{\dagger }, \end{aligned}$$</span></div><div class="c-article-equation__number"> (13) </div></div><p>where creation <span class="mathjax-tex">$a^{\dagger }$</span> and annihilation <i>a</i> operator fulfills the anticommutation rules (<span class="mathjax-tex">$\{A,B\}:=AB+BA$</span>)</p><div id="Equ14" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \{a,a\}=0, \quad \{a^{\dagger }, a^{\dagger }\}=0, \quad \{a, a^{\dagger } \}= I. \end{aligned}$$</span></div><div class="c-article-equation__number"> (14) </div></div><p>The standard representation is</p><div id="Equ15" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} a = \left( \begin{array}{cc} 0 &{} 1 \\ 0 &{} 0 \end{array} \right) , \quad a^{\dagger }=\left( \begin{array}{cc} 0 &{} 0 \\ 1 &{} 0 \end{array} \right) . \end{aligned}$$</span></div><div class="c-article-equation__number"> (15) </div></div><p>In this representation <span class="mathjax-tex">${\hat{H}}$</span> is diagonal with eigenvalues <span class="mathjax-tex">$\pm 1$</span>.</p><p>The algebra of <i>d</i> and <i>H</i> is the same<sup><a href="#Fn1"><span class="u-visually-hidden">Footnote </span>1</a></sup> as for <i>a</i> and <span class="mathjax-tex">$a^{\dagger }$</span>, namely,</p><div id="Equ16" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} dd=0, \quad HH=0, \quad Hd+dH = I^{*}-s_{x_{0}}^{*}, \end{aligned}$$</span></div><div class="c-article-equation__number"> (16) </div></div><p>where the term <span class="mathjax-tex">$s_{x_{0}}^{*}$</span> is zero when <span class="mathjax-tex">$deg(\omega )=k>0$</span>. It is therefore natural, by analogy, to define the Hamiltonian operator for the ’homotopical’ harmonic oscillator</p><div id="Equ17" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {\bar{H}}:=Hd-dH. \end{aligned}$$</span></div><div class="c-article-equation__number"> (17) </div></div><p>We can now solve the eigenvalue problem for (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00025-020-01247-8#Equ17">17</a>), namely,</p><div id="Equ18" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {\bar{H}}\omega = \lambda \omega , \end{aligned}$$</span></div><div class="c-article-equation__number"> (18) </div></div><p>where <span class="mathjax-tex">$\lambda \in {\mathbb {R}}$</span> and <span class="mathjax-tex">$\omega \in \Omega ^{k}$</span>. We have to consider three cases:</p><ul class="u-list-style-bullet"> <li> <p><span class="mathjax-tex">$0<k<n$</span>: The equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00025-020-01247-8#Equ17">17</a>) is of the form </p><div id="Equ19" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} 2Hd\omega = (\lambda +1)\omega , \end{aligned}$$</span></div><div class="c-article-equation__number"> (19) </div></div><p> and we are left with two cases:</p><ul class="u-list-style-dash"> <li> <p><span class="mathjax-tex">$\lambda = -1$</span>: for which <span class="mathjax-tex">$Hd\omega = 0$</span> that is <span class="mathjax-tex">$\omega \in Ker(Hd)$</span>, which gives that <span class="mathjax-tex">$\omega \in {\mathcal {E}}^{k}$</span>.</p> </li> <li> <p><span class="mathjax-tex">$\lambda \ne -1$</span>: since <span class="mathjax-tex">$Hd\omega \in {\mathcal {A}}^{k}$</span> so <span class="mathjax-tex">$\omega \in {\mathcal {A}}^{k}$</span>. Therefore <span class="mathjax-tex">$Hd\omega =\omega $</span> and the equation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00025-020-01247-8#Equ17">17</a>) is <span class="mathjax-tex">$2\omega =(\lambda +1)\omega $</span>, which gives <span class="mathjax-tex">$\lambda =1$</span>.</p> </li> </ul> </li> <li> <p><span class="mathjax-tex">$k=0$</span>: take <span class="mathjax-tex">$f\in \Omega ^{0}$</span>, then <span class="mathjax-tex">$Hf=0$</span>. If <span class="mathjax-tex">$f \in {\mathcal {E}}^{0}=ker(d)$</span> is a constant function then the eigenvalue problem for (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00025-020-01247-8#Equ17">17</a>) has the trivial solution <span class="mathjax-tex">$f=0$</span>. Therefore we assume that <span class="mathjax-tex">$f\in {\mathcal {A}}^{0}$</span>. Then <span class="mathjax-tex">$Hdf = f-f_{x_{0}}$</span> and the eigenvalue problem is </p><div id="Equ20" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} f-f_{x_{0}}=\lambda f \quad \Leftrightarrow \quad (1-\lambda )f=f_{x_{0}}. \end{aligned}$$</span></div><div class="c-article-equation__number"> (20) </div></div><p> For <span class="mathjax-tex">$f \in {\mathcal {A}}^{0}$</span> we have <span class="mathjax-tex">$f_{x_{0}}=0$</span> and there are two cases</p><ul class="u-list-style-dash"> <li> <p><span class="mathjax-tex">$\lambda =1$</span>: then <i>f</i> is an arbitrary element of <span class="mathjax-tex">${\mathcal {A}}^{0}$</span>.</p> </li> <li> <p><span class="mathjax-tex">$\lambda \ne 1$</span>: then <span class="mathjax-tex">$f=0$</span>.</p> </li> </ul> </li> <li> <p><span class="mathjax-tex">$k=n$</span>: let <span class="mathjax-tex">$\mu \in {\mathcal {E}}^{n}$</span>, then <span class="mathjax-tex">$d\mu =0$</span> and <span class="mathjax-tex">$dH\mu =\mu $</span>. The eigenvalue problem for (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00025-020-01247-8#Equ17">17</a>) has the form </p><div id="Equ21" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} -dH\mu =\lambda \mu \quad \Leftrightarrow \quad (\lambda +1)\mu =0, \end{aligned}$$</span></div><div class="c-article-equation__number"> (21) </div></div><p> which gives two cases:</p><ul class="u-list-style-dash"> <li> <p><span class="mathjax-tex">$\lambda =-1$</span>: then <span class="mathjax-tex">$\mu \in {\mathcal {E}}^{n}$</span> is arbitrary.</p> </li> <li> <p><span class="mathjax-tex">$\lambda \ne -1$</span>: then <span class="mathjax-tex">$\mu =0$</span>.</p> </li> </ul> </li> </ul><p>The above computation shows that the homotopical harmonic oscillator for <span class="mathjax-tex">$0<k<n$</span> only picks exact (<span class="mathjax-tex">$\lambda = -1$</span>) or antiexact (<span class="mathjax-tex">$\lambda =1$</span>) form and does not impose additional conditions on their functional form. For <span class="mathjax-tex">$k=0$</span> and <span class="mathjax-tex">$k=n$</span> there is only antiexact or exact solution respectively. This is a result of the fact that the tower from Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s00025-020-01247-8#Fig1">1</a> has a deficiency at the top and the bottom. It is an additional obstacle in making the analogy to the quantum mechanical system (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00025-020-01247-8#Equ13">13</a>) and (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00025-020-01247-8#Equ17">17</a>).</p><p>As in quantum mechanics [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 4" title="Das, A.: Field Theory, A Path Integral Approach. World Scientific, Singapore (1993)" href="/article/10.1007/s00025-020-01247-8#ref-CR4" id="ref-link-section-d158109130e8014">4</a>] there is also a top-down method for base generation where <i>H</i> rises eigenstate and <i>d</i> lowers eigenstate in the following sense:</p><ul class="u-list-style-bullet"> <li> <p>Let <span class="mathjax-tex">$\omega \in {\mathcal {E}}^{k}$</span>, <span class="mathjax-tex">$k>0$</span>. Then (since <span class="mathjax-tex">$d\omega =0$</span>) locally <span class="mathjax-tex">$\omega =d\mu $</span> for <span class="mathjax-tex">$\mu \in {\mathcal {A}}^{k-1}$</span>. Then <span class="mathjax-tex">${\hat{H}} \omega = -dHd\omega = -d\mu = -\omega $</span>, where the property <span class="mathjax-tex">$dHd=d$</span> from [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="Edelen, D.G.B.: Applied Exterior Calculus, Revised edn. Dover Publications, New York (2011)" href="/article/10.1007/s00025-020-01247-8#ref-CR8" id="ref-link-section-d158109130e8242">8</a>] was used. Therefore such <span class="mathjax-tex">$\omega $</span> is <span class="mathjax-tex">$\lambda =-1$</span> eigenvector. It originates from <span class="mathjax-tex">$\mu $</span> which is <span class="mathjax-tex">$\lambda =1$</span> eigenvector.</p> </li> <li> <p>Likewise, let <span class="mathjax-tex">$\omega \in {\mathcal {A}}^{k}$</span>, <span class="mathjax-tex">$k<n$</span>. Then <span class="mathjax-tex">$H\omega =0$</span> and therefore <span class="mathjax-tex">$\omega = H\mu $</span>. Finally, <span class="mathjax-tex">${\hat{H}}\omega = HdH\mu = H\mu =\omega $</span>, where the property <span class="mathjax-tex">$HdH=H$</span> of was used. Therefore <span class="mathjax-tex">$\omega $</span> is an eigenvector to the eigenvalue <span class="mathjax-tex">$\lambda = 1$</span>. It originates from <span class="mathjax-tex">$\mu $</span> for eigenvalue <span class="mathjax-tex">$-1$</span>.</p> </li> </ul><p>These two cases completely describe the diagram from Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s00025-020-01247-8#Fig2">2</a> and show how starting from one eigenvalue obtain the remaining one.</p><p>The homotopical fermionic quantum harmonic oscillator is, in some sense, similar to the Laplace–Beltrami operator known from Riemannian geometryrm [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Nakahara, M.: Geometry, Topology and Physics, 2nd edn. CRC Press, Boca Raton (2003)" href="/article/10.1007/s00025-020-01247-8#ref-CR16" id="ref-link-section-d158109130e8571">16</a>]. However, in our case, there is no metric structure and, as a result, no Hodge star, and therefore no codifferential can be constructed in a natural way. The homotopy operator is treated as a (local) alternative to the adjoint operator to the exterior derivative—the role that is played by the codifferential on a Riemannian manifold.</p></div></div></section><section data-title="Homotopy Operator for Complex Manifolds"><div class="c-article-section" id="Sec6-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec6"><span class="c-article-section__title-number">4 </span>Homotopy Operator for Complex Manifolds</h2><div class="c-article-section__content" id="Sec6-content"><p>This section contains an extension of the above theory for complex manifolds. We use the fact that Edelen’s homotopy operator (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00025-020-01247-8#Equ5">5</a>) does not ’feel’ the underlying field of numbers.</p><p>First, we summarize the facts about complex manifolds in order to fix notation. Complex manifold [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Nakahara, M.: Geometry, Topology and Physics, 2nd edn. CRC Press, Boca Raton (2003)" href="/article/10.1007/s00025-020-01247-8#ref-CR16" id="ref-link-section-d158109130e8589">16</a>] is a smooth even dimensional manifold <i>M</i> with holomorphic structure (of transition maps between coordinate patches). Such manifold has a complex structure <i>J</i> which eigenspaces define the split of tangent space <span class="mathjax-tex">$T_{p}M = T_{p}M^{+} \oplus T_{p}M^{-}$</span>, where the <span class="mathjax-tex">$+$</span> denotes the space spanned by holomorphic vector fields with the base <span class="mathjax-tex">$\{\partial _{z^{\mu }}\}_{\mu =1}^{n}$</span> and the space—is spanned by anti-holomorphic vector fields with the base <span class="mathjax-tex">$\{\partial _{{\bar{z}}^{\mu }}\}_{\mu =1}^{n}$</span>, where <span class="mathjax-tex">$2n = dim(M)$</span>. We have</p><div id="Equ22" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \partial _{z^{\mu }}:=\frac{1}{2}\left( \partial _{x^{\mu }}-i \partial _{y^{\nu }}\right) , \quad \partial _{{\bar{z}}^{\mu }}:=\frac{1}{2}\left( \partial _{x^{\mu }}+i \partial _{y^{\nu }}\right) , \end{aligned}$$</span></div><div class="c-article-equation__number"> (22) </div></div><p>where <span class="mathjax-tex">$\{z^{1},\ldots ,z^{n},{\bar{z}}^{1},\ldots ,{\bar{z}}^{n}\}$</span> and <span class="mathjax-tex">$\{x^{1},\ldots ,x^{n},y^{1},\ldots ,y^{n}\}$</span> are local complex and real coordinates related by the standard formula <span class="mathjax-tex">$z^{\mu }=x^{\mu }+iy^{\mu }$</span>.</p><p>This induces similar structure on the cotangent space, where the dual base has the <i>n</i>-dimensional covector base <span class="mathjax-tex">$dz^{\mu }$</span> of bidegree (1, 0) and the covector base <span class="mathjax-tex">$d{\bar{z}}^{\mu }$</span> of bidegree (0, 1). This constitutes the base of 1-forms <span class="mathjax-tex">$\Omega ^{1}(M)=\Omega ^{1,0}(M)\oplus \Omega ^{0,1}(M)$</span>. Using exterior product, higher bidegree spaces can be constructed.</p><p>The exterior derivative <i>d</i> can be decomposed as <span class="mathjax-tex">$d=\partial + {\bar{\partial }}$</span>, where the Dolbeault operators are defined as</p><div id="Equ23" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \begin{array}{cc} \partial :\Omega ^{p,q}(M)\rightarrow \Omega ^{p+1,q}(M), &{} \partial := dz^{\mu } \wedge \frac{\partial }{\partial z^{\mu }}, \\ {\bar{\partial }}:\Omega ^{p,q}(M)\rightarrow \Omega ^{p,q+1}(M), &{} {\bar{\partial }} := d{\bar{z}}^{\mu } \wedge \frac{\partial }{\partial {\bar{z}}^{\mu }}. \end{array} \end{aligned}$$</span></div><div class="c-article-equation__number"> (23) </div></div><p>Since from <span class="mathjax-tex">$d^{2}=0$</span> it results that <span class="mathjax-tex">$\partial ^{2}=0$</span>, <span class="mathjax-tex">${\bar{\partial }}^{2}=0$</span> and <span class="mathjax-tex">$\partial {\bar{\partial }}+ {\bar{\partial }}\partial =0$</span> therefore they define a double complex on <span class="mathjax-tex">$\Omega ^{p,q}(M)$</span>.</p><p>Selecting a star-shaped region <span class="mathjax-tex">$U \subset M$</span> we can inside define, by analogy to (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00025-020-01247-8#Equ5">5</a>), the homotopy operator where now <span class="mathjax-tex">${\mathcal {K}} := (x-x_{0})^{\mu }\partial _{x^{\mu }}+(y-y_{0})^{\mu }\partial _{y^{\mu }}$</span>, and the homotopy is <span class="mathjax-tex">$F(t,x,y)^{\mu }:=(x_{0}^{\mu }+t(x-x_{0})^{\mu }, y_{0}^{\mu }+t(y-y_{0})^{\mu })$</span>. It is however more instructive to reformulate <i>H</i> in terms of <span class="mathjax-tex">$z^{\mu }$</span> and <span class="mathjax-tex">${\bar{z}}^{\mu }$</span> variables. In this case</p><div id="Equ24" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {\mathcal {K}}={\mathcal {K}}^{+}+{\mathcal {K}}^{-}, \end{aligned}$$</span></div><div class="c-article-equation__number"> (24) </div></div><p>where</p><div id="Equ25" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {\mathcal {K}}^{+}=(z-z_{0})^{\mu }\partial _{z^{\mu }}, \quad {\mathcal {K}}^{-}=\overline{{\mathcal {K}}^{+}}. \end{aligned}$$</span></div><div class="c-article-equation__number"> (25) </div></div><p>Then the homotopy is <span class="mathjax-tex">$F(t,z)=z_{0}+t(z-z_{0})$</span> and similar for its complex conjugate. In this setup we have</p> <h3 class="c-article__sub-heading" id="FPar12">Proposition 1</h3> <p><i>H</i> splits into</p><div id="Equ26" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} H=H^{+}+H^{-}, \end{aligned}$$</span></div><div class="c-article-equation__number"> (26) </div></div><p>where</p><div id="Equ27" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} H^{\pm }\omega = \int _{0}^{1}{\mathcal {K}}^{\pm } \lrcorner \omega _{F(t,z)} t^{k-1} \mathrm{d}t. \end{aligned}$$</span></div><div class="c-article-equation__number"> (27) </div></div><p>These operators act as follows</p><div id="Equ28" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} H^{+}:\Omega ^{p,q}(U) \rightarrow \Omega ^{p-1,q}(U), \quad H^{-}:\Omega ^{p,q}(U) \rightarrow \Omega ^{p,q-1}(U), \end{aligned}$$</span></div><div class="c-article-equation__number"> (28) </div></div><p>which vanish when <span class="mathjax-tex">$p-1<0$</span> or <span class="mathjax-tex">$q-1<0$</span>, respectively.</p> <p>Then similarly to <i>H</i> we have obvious properties</p><div id="Equ29" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} H^{+}H^{+}=0 = H^{-}H^{-}, \end{aligned}$$</span></div><div class="c-article-equation__number"> (29) </div></div><p>and</p><div id="Equ30" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} H^{+}H^{-}+H^{-}H^{+}=0, \end{aligned}$$</span></div><div class="c-article-equation__number"> (30) </div></div><p>which result from <span class="mathjax-tex">$HH=0$</span>.</p> <div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-4" data-title="Fig. 4"><figure><figcaption><b id="Fig4" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 4</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/article/10.1007/s00025-020-01247-8/figures/4" rel="nofollow"><picture><img aria-describedby="Fig4" src="//media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs00025-020-01247-8/MediaObjects/25_2020_1247_Fig4_HTML.png" alt="figure 4" loading="lazy" width="685" height="727"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-4-desc"><p>Dolbeault complex and its homotopy dual</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="article-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/article/10.1007/s00025-020-01247-8/figures/4" data-track-dest="link:Figure4 Full size image" aria-label="Full size image figure 4" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <p>As a conclusion from the above Theorem and Corollary <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s00025-020-01247-8#FPar9">1</a> we have</p> <h3 class="c-article__sub-heading" id="FPar13">Corollary 2</h3> <p><span class="mathjax-tex">$H^{\pm }$</span> define a double complex dual to the Dolbeault complex on a start-shaped region of a complex manifold. The complex is visualized in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s00025-020-01247-8#Fig4">4</a>.</p> <p>In the complex case the formula (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00025-020-01247-8#Equ6">6</a>) becomes more elaborate</p><div id="Equ31" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \begin{array}{c} Id-s_{(z_{0},\bar{z_{0}})}^{*} = (H^{+}+H^{-})(\partial +{\bar{\partial }})+(\partial +{\bar{\partial }})(H^{+}+H^{-}) \\ =(H^{+}\partial + \partial H^{+}) + (H^{-}{\bar{\partial }} + {\bar{\partial }} H^{-}) + (H^{-}\partial + \partial H^{-}) + (H^{+}{\bar{\partial }} + {\bar{\partial }} H^{+}). \end{array} \end{aligned}$$</span></div><div class="c-article-equation__number"> (31) </div></div><p>The formula (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00025-020-01247-8#Equ31">31</a>) in general cannot be simplified to corresponding formulas for the pairs <span class="mathjax-tex">$(\partial , H^{+})$</span> and <span class="mathjax-tex">$({\bar{\partial }}, H^{-})$</span> as it is presented in the following example. Consider a differential (1, 0) form <span class="mathjax-tex">$\omega = {\bar{z}}dz$</span>. Nonzero elements of (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00025-020-01247-8#Equ31">31</a>) are</p><div id="Equ32" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \begin{array}{c} \partial H^{+} \omega = (\bar{z_{0}}+\frac{1}{2}({\bar{z}}-\bar{z_{0}}))\mathrm{d}z, \\ {\bar{\partial }} H^{+} \omega = \frac{1}{2}(z-z_{0})\mathrm{d}{\bar{z}}, \\ H^{+}{\bar{\partial }} \omega = -\frac{1}{2}(z-z_{0})\mathrm{d}{\bar{z}}, \\ H^{-}{\bar{\partial }} \omega = \frac{1}{2}({\bar{z}}-\bar{z_{0}})\mathrm{d}z. \end{array} \end{aligned}$$</span></div><div class="c-article-equation__number"> (32) </div></div><p>Summing these terms up we get <span class="mathjax-tex">$(Hd+dH)\omega = {\bar{z}} dz = I({\bar{z}}dz) - s_{z_{0},\bar{z_{0}}}^{*}({\bar{z}} dz)$</span> as required. Therefore all ingredients of (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00025-020-01247-8#Equ31">31</a>) must be taken into account in the general case.</p><p>There are however two important cases when there is a split into <span class="mathjax-tex">$H^\pm $</span> subcomplexes.</p> <h3 class="c-article__sub-heading" id="FPar14">Corollary 3</h3> <p>There are two subcomplexes for <span class="mathjax-tex">$H^{+}$</span> and <span class="mathjax-tex">$H^{-}$</span>, namely,</p><ul class="u-list-style-bullet"> <li> <p><span class="mathjax-tex">${\bar{\partial }}\omega = 0$</span> (holomorphic), <span class="mathjax-tex">$\omega \in \Omega ^{p,0}, p\in {\mathbb {N}}$</span>—with no <span class="mathjax-tex">$d{\bar{z}}$</span> terms in the local representation, that is, <span class="mathjax-tex">$\omega = \omega (z)_{\mu _{1},\ldots ,\mu _{p}}dz^{\mu _{1}}\wedge \ldots \wedge dz^{\mu _{p}}$</span>. In this case <span class="mathjax-tex">$H^{-}\omega =0$</span> (anti-<span class="mathjax-tex">${\bar{\partial }}$</span>-exact), and <span class="mathjax-tex">${\bar{\partial }}H^{+}\omega =0$</span>. Then (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00025-020-01247-8#Equ31">31</a>) has the simple form </p><div id="Equ33" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} H^{+}\partial + \partial H^{+}=I - s_{z_{0}}^{*}. \end{aligned}$$</span></div><div class="c-article-equation__number"> (33) </div></div><p> This defines the subcomplex <span class="mathjax-tex">$(\Omega ^{p,0}, \partial , H^{+})$</span>.</p> </li> <li> <p><span class="mathjax-tex">$\partial \omega = 0$</span> (antiholomorphic), <span class="mathjax-tex">$\omega \in \Omega ^{0,p}, p\in {\mathbb {N}}$</span>—with no <i>dz</i> terms in the local representation, that is, <span class="mathjax-tex">$\omega = \omega ({\bar{z}})_{\mu _{1},\ldots ,\mu _{p}}d{\bar{z}}^{\mu _{1}}\wedge \ldots \wedge d{\bar{z}}^{\mu _{p}}$</span>. In this case <span class="mathjax-tex">$H^{+}\omega =0$</span> (anti-<span class="mathjax-tex">$\partial $</span>-exact), and <span class="mathjax-tex">$\partial H^{-}\omega =0$</span>. Then (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s00025-020-01247-8#Equ31">31</a>) has the simple form </p><div id="Equ34" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} H^{-}{\bar{\partial }} + {\bar{\partial }} H^{-}=I - s_{\bar{z_{0}}}^{*}. \end{aligned}$$</span></div><div class="c-article-equation__number"> (34) </div></div><p> Likewise, this defines the subcomplex <span class="mathjax-tex">$(\Omega ^{0,p}, {\bar{\partial }}, H^{-})$</span>.</p> </li> </ul><p>Both of these subcomplexes lie on the boundary (left and bottom part) of Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="/article/10.1007/s00025-020-01247-8#Fig4">4</a>.</p> </div></div></section><section data-title="Conclusions"><div class="c-article-section" id="Sec7-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec7"><span class="c-article-section__title-number">5 </span>Conclusions</h2><div class="c-article-section__content" id="Sec7-content"><p>In this paper, the local results related to the homotopy operator from the Poincaré lemma was used to derive a special case of operator calculus that resembles structures occurring in quantum mechanics. Moreover, the analysis of dual Dolbeault bicomplex induced on a complex manifold by complex homotopy operator was provided. Two special subcomplexes were identified. These results organize and generalize the Poincaré lemma by building additional abstract structure on the top of this classical result.</p></div></div></section> </div> <section data-title="Notes"><div class="c-article-section" id="notes-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="notes">Notes</h2><div class="c-article-section__content" id="notes-content"><ol class="c-article-footnote c-article-footnote--listed"><li class="c-article-footnote--listed__item" id="Fn1" data-counter="1."><div class="c-article-footnote--listed__content"><p>We neglect the fact that <span class="mathjax-tex">$a^{\dagger }$</span> is an operator which is not the adjoint of <i>a</i>. In the space where <i>d</i> and <i>H</i> acts, there is no inner product which can be used to form such adjoint.</p></div></li></ol></div></div></section><div id="MagazineFulltextArticleBodySuffix"><section aria-labelledby="Bib1" data-title="References"><div class="c-article-section" id="Bib1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Bib1">References</h2><div class="c-article-section__content" id="Bib1-content"><div data-container-section="references"><ol class="c-article-references" data-track-component="outbound reference" data-track-context="references section"><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="1."><p class="c-article-references__text" id="ref-CR1">Beilinson, A.: p-adic periods and derived de Rham cohomology. J. Am. Math. Soc. <b>25</b>, 715–738 (2012). <a href="https://doi.org/10.1090/S0894-0347-2012-00729-2" data-track="click_references" data-track-action="external reference" data-track-value="external reference" data-track-label="10.1090/S0894-0347-2012-00729-2">https://doi.org/10.1090/S0894-0347-2012-00729-2</a></p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1090/S0894-0347-2012-00729-2" data-track-item_id="10.1090/S0894-0347-2012-00729-2" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1090%2FS0894-0347-2012-00729-2" aria-label="Article reference 1" data-doi="10.1090/S0894-0347-2012-00729-2">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=2904571" aria-label="MathSciNet reference 1">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1247.14018" aria-label="MATH reference 1">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 1" href="http://scholar.google.com/scholar_lookup?&title=p-adic%20periods%20and%20derived%20de%20Rham%20cohomology&journal=J.%20Am.%20Math.%20Soc.&doi=10.1090%2FS0894-0347-2012-00729-2&volume=25&pages=715-738&publication_year=2012&author=Beilinson%2CA"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="2."><p class="c-article-references__text" id="ref-CR2">Bittner, R.: Operational calculus in linear spaces. Stud. Math. <b>20</b>, 1–18 (1961). <a href="https://doi.org/10.4064/sm-20-1-1-18" data-track="click_references" data-track-action="external reference" data-track-value="external reference" data-track-label="10.4064/sm-20-1-1-18">https://doi.org/10.4064/sm-20-1-1-18</a></p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.4064/sm-20-1-1-18" data-track-item_id="10.4064/sm-20-1-1-18" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.4064%2Fsm-20-1-1-18" aria-label="Article reference 2" data-doi="10.4064/sm-20-1-1-18">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=140902" aria-label="MathSciNet reference 2">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?0097.10502" aria-label="MATH reference 2">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 2" href="http://scholar.google.com/scholar_lookup?&title=Operational%20calculus%20in%20linear%20spaces&journal=Stud.%20Math.&doi=10.4064%2Fsm-20-1-1-18&volume=20&pages=1-18&publication_year=1961&author=Bittner%2CR"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="3."><p class="c-article-references__text" id="ref-CR3">Bott, R., Tu, L.W.: Differential Forms in Algebraic Topology. Springer, Berlin (1995)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?0496.55001" aria-label="MATH reference 3">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 3" href="http://scholar.google.com/scholar_lookup?&title=Differential%20Forms%20in%20Algebraic%20Topology&publication_year=1995&author=Bott%2CR&author=Tu%2CLW"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="4."><p class="c-article-references__text" id="ref-CR4">Das, A.: Field Theory, A Path Integral Approach. World Scientific, Singapore (1993)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1142/2025" data-track-item_id="10.1142/2025" data-track-value="book reference" data-track-action="book reference" href="https://doi.org/10.1142%2F2025" aria-label="Book reference 4" data-doi="10.1142/2025">Book</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 4" href="http://scholar.google.com/scholar_lookup?&title=Field%20Theory%2C%20A%20Path%20Integral%20Approach&doi=10.1142%2F2025&publication_year=1993&author=Das%2CA"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="5."><p class="c-article-references__text" id="ref-CR5">Desbrun, M., Leok, M., Marsden, J.E.: Discrete Poincaré lemma. Appl. Numer. Math. <b>53</b>(2–4), 231–248 (2005). <a href="https://doi.org/10.1016/j.apnum.2004.09.035" data-track="click_references" data-track-action="external reference" data-track-value="external reference" data-track-label="10.1016/j.apnum.2004.09.035">https://doi.org/10.1016/j.apnum.2004.09.035</a></p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1016/j.apnum.2004.09.035" data-track-item_id="10.1016/j.apnum.2004.09.035" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1016%2Fj.apnum.2004.09.035" aria-label="Article reference 5" data-doi="10.1016/j.apnum.2004.09.035">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=2128524" aria-label="MathSciNet reference 5">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1080.39021" aria-label="MATH reference 5">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 5" href="http://scholar.google.com/scholar_lookup?&title=Discrete%20Poincar%C3%A9%20lemma&journal=Appl.%20Numer.%20Math.&doi=10.1016%2Fj.apnum.2004.09.035&volume=53&issue=2%E2%80%934&pages=231-248&publication_year=2005&author=Desbrun%2CM&author=Leok%2CM&author=Marsden%2CJE"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="6."><p class="c-article-references__text" id="ref-CR6">Desbrun, M., Hirani, A.N., Leok, M., Marsden, J.E.: Discrete Exterior Calculus, <a href="http://arxiv.org/abs/math/0508341" data-track="click_references" data-track-action="external reference" data-track-value="external reference" data-track-label="http://arxiv.org/abs/math/0508341">arXiv:math/0508341</a> [math.DG]</p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="7."><p class="c-article-references__text" id="ref-CR7">Ding, S., Nolder, C.A.: Weighted Poincaré inequalities for solutions to A-harmonic equations. Ill. J. Math. <b>46</b>(1), 199–205 (2002). <a href="https://doi.org/10.1215/ijm/1258136150" data-track="click_references" data-track-action="external reference" data-track-value="external reference" data-track-label="10.1215/ijm/1258136150">https://doi.org/10.1215/ijm/1258136150</a></p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1215/ijm/1258136150" data-track-item_id="10.1215/ijm/1258136150" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1215%2Fijm%2F1258136150" aria-label="Article reference 7" data-doi="10.1215/ijm/1258136150">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1071.35520" aria-label="MATH reference 7">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 7" href="http://scholar.google.com/scholar_lookup?&title=Weighted%20Poincar%C3%A9%20inequalities%20for%20solutions%20to%20A-harmonic%20equations&journal=Ill.%20J.%20Math.&doi=10.1215%2Fijm%2F1258136150&volume=46&issue=1&pages=199-205&publication_year=2002&author=Ding%2CS&author=Nolder%2CCA"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="8."><p class="c-article-references__text" id="ref-CR8">Edelen, D.G.B.: Applied Exterior Calculus, Revised edn. Dover Publications, New York (2011)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1101.58301" aria-label="MATH reference 8">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 8" href="http://scholar.google.com/scholar_lookup?&title=Applied%20Exterior%20Calculus&publication_year=2011&author=Edelen%2CDGB"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="9."><p class="c-article-references__text" id="ref-CR9">Edelen, D.G.B.: Isovector Methods for Equations of Balance. Springer, Berlin (1980)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?0452.58001" aria-label="MATH reference 9">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 9" href="http://scholar.google.com/scholar_lookup?&title=Isovector%20Methods%20for%20Equations%20of%20Balance&publication_year=1980&author=Edelen%2CDGB"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="10."><p class="c-article-references__text" id="ref-CR10">Harrison, J.: Operator calculus of differential chains and differential forms. J. Geom. Anal. <b>25</b>(1), 357–420 (2015). <a href="https://doi.org/10.1007/s12220-013-9433-6" data-track="click_references" data-track-action="external reference" data-track-value="external reference" data-track-label="10.1007/s12220-013-9433-6">https://doi.org/10.1007/s12220-013-9433-6</a></p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="noopener" data-track-label="10.1007/s12220-013-9433-6" data-track-item_id="10.1007/s12220-013-9433-6" data-track-value="article reference" data-track-action="article reference" href="https://link.springer.com/doi/10.1007/s12220-013-9433-6" aria-label="Article reference 10" data-doi="10.1007/s12220-013-9433-6">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=3299287" aria-label="MathSciNet reference 10">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1325.58003" aria-label="MATH reference 10">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 10" href="http://scholar.google.com/scholar_lookup?&title=Operator%20calculus%20of%20differential%20chains%20and%20differential%20forms&journal=J.%20Geom.%20Anal.&doi=10.1007%2Fs12220-013-9433-6&volume=25&issue=1&pages=357-420&publication_year=2015&author=Harrison%2CJ"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="11."><p class="c-article-references__text" id="ref-CR11">Harrison, J.: Geometric Poincaré Lemma, <a href="http://arxiv.org/abs/1101.0313" data-track="click_references" data-track-action="external reference" data-track-value="external reference" data-track-label="http://arxiv.org/abs/1101.0313">arXiv:1101.0313</a> [math.AT]</p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="12."><p class="c-article-references__text" id="ref-CR12">Iwaniec, T., Lutoborski, A.: Integral estimates for null Lagrangians. Arch. Ration. Mech. Anal. <b>125</b>, 25–79 (1993). <a href="https://doi.org/10.1007/BF00411477" data-track="click_references" data-track-action="external reference" data-track-value="external reference" data-track-label="10.1007/BF00411477">https://doi.org/10.1007/BF00411477</a></p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="noopener" data-track-label="10.1007/BF00411477" data-track-item_id="10.1007/BF00411477" data-track-value="article reference" data-track-action="article reference" href="https://link.springer.com/doi/10.1007/BF00411477" aria-label="Article reference 12" data-doi="10.1007/BF00411477">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=1241286" aria-label="MathSciNet reference 12">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?0793.58002" aria-label="MATH reference 12">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 12" href="http://scholar.google.com/scholar_lookup?&title=Integral%20estimates%20for%20null%20Lagrangians&journal=Arch.%20Ration.%20Mech.%20Anal.&doi=10.1007%2FBF00411477&volume=125&pages=25-79&publication_year=1993&author=Iwaniec%2CT&author=Lutoborski%2CA"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="13."><p class="c-article-references__text" id="ref-CR13">Jell, P.: A Poincaré lemma for real-valued differential forms on Berkovich spaces. Math. Z. <b>282</b>, 1149–1167 (2016). <a href="https://doi.org/10.1007/s00209-015-1583-8" data-track="click_references" data-track-action="external reference" data-track-value="external reference" data-track-label="10.1007/s00209-015-1583-8">https://doi.org/10.1007/s00209-015-1583-8</a></p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="noopener" data-track-label="10.1007/s00209-015-1583-8" data-track-item_id="10.1007/s00209-015-1583-8" data-track-value="article reference" data-track-action="article reference" href="https://link.springer.com/doi/10.1007/s00209-015-1583-8" aria-label="Article reference 13" data-doi="10.1007/s00209-015-1583-8">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=3473662" aria-label="MathSciNet reference 13">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1388.58001" aria-label="MATH reference 13">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 13" href="http://scholar.google.com/scholar_lookup?&title=A%20Poincar%C3%A9%20lemma%20for%20real-valued%20differential%20forms%20on%20Berkovich%20spaces&journal=Math.%20Z.&doi=10.1007%2Fs00209-015-1583-8&volume=282&pages=1149-1167&publication_year=2016&author=Jell%2CP"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="14."><p class="c-article-references__text" id="ref-CR14">Lee, J.: Introduction to Smooth Manifolds, 2nd edn. Springer, Berlin (2012)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="noopener" data-track-label="10.1007/978-1-4419-9982-5" data-track-item_id="10.1007/978-1-4419-9982-5" data-track-value="book reference" data-track-action="book reference" href="https://link.springer.com/doi/10.1007/978-1-4419-9982-5" aria-label="Book reference 14" data-doi="10.1007/978-1-4419-9982-5">Book</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 14" href="http://scholar.google.com/scholar_lookup?&title=Introduction%20to%20Smooth%20Manifolds&doi=10.1007%2F978-1-4419-9982-5&publication_year=2012&author=Lee%2CJ"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="15."><p class="c-article-references__text" id="ref-CR15">Lesfari, A.: On Poincaré lemma or Volterra theorem about differential forms and cohomology groups, <a href="http://arxiv.org/abs/1905.13347" data-track="click_references" data-track-action="external reference" data-track-value="external reference" data-track-label="http://arxiv.org/abs/1905.13347">arXiv:1905.13347</a> [math.GM]</p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="16."><p class="c-article-references__text" id="ref-CR16">Nakahara, M.: Geometry, Topology and Physics, 2nd edn. CRC Press, Boca Raton (2003)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1090.53001" aria-label="MATH reference 16">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 16" href="http://scholar.google.com/scholar_lookup?&title=Geometry%2C%20Topology%20and%20Physics&publication_year=2003&author=Nakahara%2CM"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="17."><p class="c-article-references__text" id="ref-CR17">Voronov, T.: On a non-Abelian Poincaré lemma. Proc. Amer. Math. Soc. <b>140</b>, 2855–2872 (2012). <a href="https://doi.org/10.1090/S0002-9939-2011-11116-X" data-track="click_references" data-track-action="external reference" data-track-value="external reference" data-track-label="10.1090/S0002-9939-2011-11116-X">https://doi.org/10.1090/S0002-9939-2011-11116-X</a></p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="10.1090/S0002-9939-2011-11116-X" data-track-item_id="10.1090/S0002-9939-2011-11116-X" data-track-value="article reference" data-track-action="article reference" href="https://doi.org/10.1090%2FS0002-9939-2011-11116-X" aria-label="Article reference 17" data-doi="10.1090/S0002-9939-2011-11116-X">Article</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=2910772" aria-label="MathSciNet reference 17">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1282.58007" aria-label="MATH reference 17">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 17" href="http://scholar.google.com/scholar_lookup?&title=On%20a%20non-Abelian%20Poincar%C3%A9%20lemma&journal=Proc.%20Amer.%20Math.%20Soc.&doi=10.1090%2FS0002-9939-2011-11116-X&volume=140&pages=2855-2872&publication_year=2012&author=Voronov%2CT"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="18."><p class="c-article-references__text" id="ref-CR18">Tu, L.W.: An Introduction to Manifolds, 2nd edn. Springer, Berlin (2010)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 18" href="http://scholar.google.com/scholar_lookup?&title=An%20Introduction%20to%20Manifolds&publication_year=2010&author=Tu%2CLW"> Google Scholar</a> </p></li></ol><p class="c-article-references__download u-hide-print"><a data-track="click" data-track-action="download citation references" data-track-label="link" rel="nofollow" href="https://citation-needed.springer.com/v2/references/10.1007/s00025-020-01247-8?format=refman&flavour=references">Download references<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-download-medium"></use></svg></a></p></div></div></div></section></div><section data-title="Acknowledgements"><div class="c-article-section" id="Ack1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Ack1">Acknowledgements</h2><div class="c-article-section__content" id="Ack1-content"><p>I would like to thank Josef Šilhan for long stimulating discussions during the preparation of the paper, and Lukáš Vokřínek for discussion about Homological algebra. I would also like to thank Jan Slovák and Henrik Winther for useful suggestions. Last but not least, I thank anonymous Referee, whose vital and precise comments and suggestions improve this paper.</p></div></div></section><section data-title="Funding"><div class="c-article-section" id="Fun-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Fun">Funding</h2><div class="c-article-section__content" id="Fun-content"><p>This research was supported by the GACR grant GA19-06357S and Masaryk University grant MUNI/A/0885/2019. I also thank the PHAROS COST Action (CA16214), and SyMat COST Action (CA18223) for partial support.</p></div></div></section><section aria-labelledby="author-information" data-title="Author information"><div class="c-article-section" id="author-information-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="author-information">Author information</h2><div class="c-article-section__content" id="author-information-content"><h3 class="c-article__sub-heading" id="affiliations">Authors and Affiliations</h3><ol class="c-article-author-affiliation__list"><li id="Aff1"><p class="c-article-author-affiliation__address">Department of Mathematics and Statistics, Masaryk University, Kotlářská 267/2, 611 37, Brno, Czech Republic</p><p class="c-article-author-affiliation__authors-list">Radosław Antoni Kycia</p></li><li id="Aff2"><p class="c-article-author-affiliation__address">Faculty of Materials Engineering and Physics, Cracow University of Technology, Warszawska 24, 31-155, Kraków, Poland</p><p class="c-article-author-affiliation__authors-list">Radosław Antoni Kycia</p></li></ol><div class="u-js-hide u-hide-print" data-test="author-info"><span class="c-article__sub-heading">Authors</span><ol class="c-article-authors-search u-list-reset"><li id="auth-Rados_aw_Antoni-Kycia-Aff1-Aff2"><span class="c-article-authors-search__title u-h3 js-search-name">Radosław Antoni Kycia</span><div class="c-article-authors-search__list"><div class="c-article-authors-search__item c-article-authors-search__list-item--left"><a href="/search?dc.creator=Rados%C5%82aw%20Antoni%20Kycia" class="c-article-button" data-track="click" data-track-action="author link - publication" data-track-label="link" rel="nofollow">View author publications</a></div><div class="c-article-authors-search__item c-article-authors-search__list-item--right"><p class="search-in-title-js c-article-authors-search__text">You can also search for this author in <span class="c-article-identifiers"><a class="c-article-identifiers__item" href="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=search&term=Rados%C5%82aw%20Antoni%20Kycia" data-track="click" data-track-action="author link - pubmed" data-track-label="link" rel="nofollow">PubMed</a><span class="u-hide"> </span><a class="c-article-identifiers__item" href="http://scholar.google.co.uk/scholar?as_q=&num=10&btnG=Search+Scholar&as_epq=&as_oq=&as_eq=&as_occt=any&as_sauthors=%22Rados%C5%82aw%20Antoni%20Kycia%22&as_publication=&as_ylo=&as_yhi=&as_allsubj=all&hl=en" data-track="click" data-track-action="author link - scholar" data-track-label="link" rel="nofollow">Google Scholar</a></span></p></div></div></li></ol></div><h3 class="c-article__sub-heading" id="corresponding-author">Corresponding author</h3><p id="corresponding-author-list">Correspondence to <a id="corresp-c1" href="mailto:kycia.radoslaw@gmail.com">Radosław Antoni Kycia</a>.</p></div></div></section><section data-title="Additional information"><div class="c-article-section" id="additional-information-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="additional-information">Additional information</h2><div class="c-article-section__content" id="additional-information-content"><h3 class="c-article__sub-heading">Publisher's Note</h3><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p></div></div></section><section data-title="Rights and permissions"><div class="c-article-section" id="rightslink-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="rightslink">Rights and permissions</h2><div class="c-article-section__content" id="rightslink-content"> <p><b>Open Access</b> This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.</p> <p class="c-article-rights"><a data-track="click" data-track-action="view rights and permissions" data-track-label="link" href="https://s100.copyright.com/AppDispatchServlet?title=The%20Poincare%20Lemma%2C%20Antiexact%20Forms%2C%20and%20Fermionic%20Quantum%20Harmonic%20Oscillator&author=Rados%C5%82aw%20Antoni%20Kycia&contentID=10.1007%2Fs00025-020-01247-8&copyright=The%20Author%28s%29&publication=1422-6383&publicationDate=2020-07-11&publisherName=SpringerNature&orderBeanReset=true&oa=CC%20BY">Reprints and permissions</a></p></div></div></section><section aria-labelledby="article-info" data-title="About this article"><div class="c-article-section" id="article-info-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="article-info">About this article</h2><div class="c-article-section__content" id="article-info-content"><div class="c-bibliographic-information"><div class="u-hide-print c-bibliographic-information__column c-bibliographic-information__column--border"><a data-crossmark="10.1007/s00025-020-01247-8" target="_blank" rel="noopener" href="https://crossmark.crossref.org/dialog/?doi=10.1007/s00025-020-01247-8" data-track="click" data-track-action="Click Crossmark" data-track-label="link" data-test="crossmark"><img loading="lazy" width="57" height="81" alt="Check for updates. Verify currency and authenticity via CrossMark" src="data:image/svg+xml;base64,<svg height="81" width="57" xmlns="http://www.w3.org/2000/svg"><g fill="none" fill-rule="evenodd"><path d="m17.35 35.45 21.3-14.2v-17.03h-21.3" fill="#989898"/><path d="m38.65 35.45-21.3-14.2v-17.03h21.3" fill="#747474"/><path d="m28 .5c-12.98 0-23.5 10.52-23.5 23.5s10.52 23.5 23.5 23.5 23.5-10.52 23.5-23.5c0-6.23-2.48-12.21-6.88-16.62-4.41-4.4-10.39-6.88-16.62-6.88zm0 41.25c-9.8 0-17.75-7.95-17.75-17.75s7.95-17.75 17.75-17.75 17.75 7.95 17.75 17.75c0 4.71-1.87 9.22-5.2 12.55s-7.84 5.2-12.55 5.2z" fill="#535353"/><path d="m41 36c-5.81 6.23-15.23 7.45-22.43 2.9-7.21-4.55-10.16-13.57-7.03-21.5l-4.92-3.11c-4.95 10.7-1.19 23.42 8.78 29.71 9.97 6.3 23.07 4.22 30.6-4.86z" fill="#9c9c9c"/><path d="m.2 58.45c0-.75.11-1.42.33-2.01s.52-1.09.91-1.5c.38-.41.83-.73 1.34-.94.51-.22 1.06-.32 1.65-.32.56 0 1.06.11 1.51.35.44.23.81.5 1.1.81l-.91 1.01c-.24-.24-.49-.42-.75-.56-.27-.13-.58-.2-.93-.2-.39 0-.73.08-1.05.23-.31.16-.58.37-.81.66-.23.28-.41.63-.53 1.04-.13.41-.19.88-.19 1.39 0 1.04.23 1.86.68 2.46.45.59 1.06.88 1.84.88.41 0 .77-.07 1.07-.23s.59-.39.85-.68l.91 1c-.38.43-.8.76-1.28.99-.47.22-1 .34-1.58.34-.59 0-1.13-.1-1.64-.31-.5-.2-.94-.51-1.31-.91-.38-.4-.67-.9-.88-1.48-.22-.59-.33-1.26-.33-2.02zm8.4-5.33h1.61v2.54l-.05 1.33c.29-.27.61-.51.96-.72s.76-.31 1.24-.31c.73 0 1.27.23 1.61.71.33.47.5 1.14.5 2.02v4.31h-1.61v-4.1c0-.57-.08-.97-.25-1.21-.17-.23-.45-.35-.83-.35-.3 0-.56.08-.79.22-.23.15-.49.36-.78.64v4.8h-1.61zm7.37 6.45c0-.56.09-1.06.26-1.51.18-.45.42-.83.71-1.14.29-.3.63-.54 1.01-.71.39-.17.78-.25 1.18-.25.47 0 .88.08 1.23.24.36.16.65.38.89.67s.42.63.54 1.03c.12.41.18.84.18 1.32 0 .32-.02.57-.07.76h-4.36c.07.62.29 1.1.65 1.44.36.33.82.5 1.38.5.29 0 .57-.04.83-.13s.51-.21.76-.37l.55 1.01c-.33.21-.69.39-1.09.53-.41.14-.83.21-1.26.21-.48 0-.92-.08-1.34-.25-.41-.16-.76-.4-1.07-.7-.31-.31-.55-.69-.72-1.13-.18-.44-.26-.95-.26-1.52zm4.6-.62c0-.55-.11-.98-.34-1.28-.23-.31-.58-.47-1.06-.47-.41 0-.77.15-1.07.45-.31.29-.5.73-.58 1.3zm2.5.62c0-.57.09-1.08.28-1.53.18-.44.43-.82.75-1.13s.69-.54 1.1-.71c.42-.16.85-.24 1.31-.24.45 0 .84.08 1.17.23s.61.34.85.57l-.77 1.02c-.19-.16-.38-.28-.56-.37-.19-.09-.39-.14-.61-.14-.56 0-1.01.21-1.35.63-.35.41-.52.97-.52 1.67 0 .69.17 1.24.51 1.66.34.41.78.62 1.32.62.28 0 .54-.06.78-.17.24-.12.45-.26.64-.42l.67 1.03c-.33.29-.69.51-1.08.65-.39.15-.78.23-1.18.23-.46 0-.9-.08-1.31-.24-.4-.16-.75-.39-1.05-.7s-.53-.69-.7-1.13c-.17-.45-.25-.96-.25-1.53zm6.91-6.45h1.58v6.17h.05l2.54-3.16h1.77l-2.35 2.8 2.59 4.07h-1.75l-1.77-2.98-1.08 1.23v1.75h-1.58zm13.69 1.27c-.25-.11-.5-.17-.75-.17-.58 0-.87.39-.87 1.16v.75h1.34v1.27h-1.34v5.6h-1.61v-5.6h-.92v-1.2l.92-.07v-.72c0-.35.04-.68.13-.98.08-.31.21-.57.4-.79s.42-.39.71-.51c.28-.12.63-.18 1.04-.18.24 0 .48.02.69.07.22.05.41.1.57.17zm.48 5.18c0-.57.09-1.08.27-1.53.17-.44.41-.82.72-1.13.3-.31.65-.54 1.04-.71.39-.16.8-.24 1.23-.24s.84.08 1.24.24c.4.17.74.4 1.04.71s.54.69.72 1.13c.19.45.28.96.28 1.53s-.09 1.08-.28 1.53c-.18.44-.42.82-.72 1.13s-.64.54-1.04.7-.81.24-1.24.24-.84-.08-1.23-.24-.74-.39-1.04-.7c-.31-.31-.55-.69-.72-1.13-.18-.45-.27-.96-.27-1.53zm1.65 0c0 .69.14 1.24.43 1.66.28.41.68.62 1.18.62.51 0 .9-.21 1.19-.62.29-.42.44-.97.44-1.66 0-.7-.15-1.26-.44-1.67-.29-.42-.68-.63-1.19-.63-.5 0-.9.21-1.18.63-.29.41-.43.97-.43 1.67zm6.48-3.44h1.33l.12 1.21h.05c.24-.44.54-.79.88-1.02.35-.24.7-.36 1.07-.36.32 0 .59.05.78.14l-.28 1.4-.33-.09c-.11-.01-.23-.02-.38-.02-.27 0-.56.1-.86.31s-.55.58-.77 1.1v4.2h-1.61zm-47.87 15h1.61v4.1c0 .57.08.97.25 1.2.17.24.44.35.81.35.3 0 .57-.07.8-.22.22-.15.47-.39.73-.73v-4.7h1.61v6.87h-1.32l-.12-1.01h-.04c-.3.36-.63.64-.98.86-.35.21-.76.32-1.24.32-.73 0-1.27-.24-1.61-.71-.33-.47-.5-1.14-.5-2.02zm9.46 7.43v2.16h-1.61v-9.59h1.33l.12.72h.05c.29-.24.61-.45.97-.63.35-.17.72-.26 1.1-.26.43 0 .81.08 1.15.24.33.17.61.4.84.71.24.31.41.68.53 1.11.13.42.19.91.19 1.44 0 .59-.09 1.11-.25 1.57-.16.47-.38.85-.65 1.16-.27.32-.58.56-.94.73-.35.16-.72.25-1.1.25-.3 0-.6-.07-.9-.2s-.59-.31-.87-.56zm0-2.3c.26.22.5.37.73.45.24.09.46.13.66.13.46 0 .84-.2 1.15-.6.31-.39.46-.98.46-1.77 0-.69-.12-1.22-.35-1.61-.23-.38-.61-.57-1.13-.57-.49 0-.99.26-1.52.77zm5.87-1.69c0-.56.08-1.06.25-1.51.16-.45.37-.83.65-1.14.27-.3.58-.54.93-.71s.71-.25 1.08-.25c.39 0 .73.07 1 .2.27.14.54.32.81.55l-.06-1.1v-2.49h1.61v9.88h-1.33l-.11-.74h-.06c-.25.25-.54.46-.88.64-.33.18-.69.27-1.06.27-.87 0-1.56-.32-2.07-.95s-.76-1.51-.76-2.65zm1.67-.01c0 .74.13 1.31.4 1.7.26.38.65.58 1.15.58.51 0 .99-.26 1.44-.77v-3.21c-.24-.21-.48-.36-.7-.45-.23-.08-.46-.12-.7-.12-.45 0-.82.19-1.13.59-.31.39-.46.95-.46 1.68zm6.35 1.59c0-.73.32-1.3.97-1.71.64-.4 1.67-.68 3.08-.84 0-.17-.02-.34-.07-.51-.05-.16-.12-.3-.22-.43s-.22-.22-.38-.3c-.15-.06-.34-.1-.58-.1-.34 0-.68.07-1 .2s-.63.29-.93.47l-.59-1.08c.39-.24.81-.45 1.28-.63.47-.17.99-.26 1.54-.26.86 0 1.51.25 1.93.76s.63 1.25.63 2.21v4.07h-1.32l-.12-.76h-.05c-.3.27-.63.48-.98.66s-.73.27-1.14.27c-.61 0-1.1-.19-1.48-.56-.38-.36-.57-.85-.57-1.46zm1.57-.12c0 .3.09.53.27.67.19.14.42.21.71.21.28 0 .54-.07.77-.2s.48-.31.73-.56v-1.54c-.47.06-.86.13-1.18.23-.31.09-.57.19-.76.31s-.33.25-.41.4c-.09.15-.13.31-.13.48zm6.29-3.63h-.98v-1.2l1.06-.07.2-1.88h1.34v1.88h1.75v1.27h-1.75v3.28c0 .8.32 1.2.97 1.2.12 0 .24-.01.37-.04.12-.03.24-.07.34-.11l.28 1.19c-.19.06-.4.12-.64.17-.23.05-.49.08-.76.08-.4 0-.74-.06-1.02-.18-.27-.13-.49-.3-.67-.52-.17-.21-.3-.48-.37-.78-.08-.3-.12-.64-.12-1.01zm4.36 2.17c0-.56.09-1.06.27-1.51s.41-.83.71-1.14c.29-.3.63-.54 1.01-.71.39-.17.78-.25 1.18-.25.47 0 .88.08 1.23.24.36.16.65.38.89.67s.42.63.54 1.03c.12.41.18.84.18 1.32 0 .32-.02.57-.07.76h-4.37c.08.62.29 1.1.65 1.44.36.33.82.5 1.38.5.3 0 .58-.04.84-.13.25-.09.51-.21.76-.37l.54 1.01c-.32.21-.69.39-1.09.53s-.82.21-1.26.21c-.47 0-.92-.08-1.33-.25-.41-.16-.77-.4-1.08-.7-.3-.31-.54-.69-.72-1.13-.17-.44-.26-.95-.26-1.52zm4.61-.62c0-.55-.11-.98-.34-1.28-.23-.31-.58-.47-1.06-.47-.41 0-.77.15-1.08.45-.31.29-.5.73-.57 1.3zm3.01 2.23c.31.24.61.43.92.57.3.13.63.2.98.2.38 0 .65-.08.83-.23s.27-.35.27-.6c0-.14-.05-.26-.13-.37-.08-.1-.2-.2-.34-.28-.14-.09-.29-.16-.47-.23l-.53-.22c-.23-.09-.46-.18-.69-.3-.23-.11-.44-.24-.62-.4s-.33-.35-.45-.55c-.12-.21-.18-.46-.18-.75 0-.61.23-1.1.68-1.49.44-.38 1.06-.57 1.83-.57.48 0 .91.08 1.29.25s.71.36.99.57l-.74.98c-.24-.17-.49-.32-.73-.42-.25-.11-.51-.16-.78-.16-.35 0-.6.07-.76.21-.17.15-.25.33-.25.54 0 .14.04.26.12.36s.18.18.31.26c.14.07.29.14.46.21l.54.19c.23.09.47.18.7.29s.44.24.64.4c.19.16.34.35.46.58.11.23.17.5.17.82 0 .3-.06.58-.17.83-.12.26-.29.48-.51.68-.23.19-.51.34-.84.45-.34.11-.72.17-1.15.17-.48 0-.95-.09-1.41-.27-.46-.19-.86-.41-1.2-.68z" fill="#535353"/></g></svg>"></a></div><div class="c-bibliographic-information__column"><h3 class="c-article__sub-heading" id="citeas">Cite this article</h3><p class="c-bibliographic-information__citation">Kycia, R.A. The Poincare Lemma, Antiexact Forms, and Fermionic Quantum Harmonic Oscillator. <i>Results Math</i> <b>75</b>, 122 (2020). https://doi.org/10.1007/s00025-020-01247-8</p><p class="c-bibliographic-information__download-citation u-hide-print"><a data-test="citation-link" data-track="click" data-track-action="download article citation" data-track-label="link" data-track-external="" rel="nofollow" href="https://citation-needed.springer.com/v2/references/10.1007/s00025-020-01247-8?format=refman&flavour=citation">Download citation<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-download-medium"></use></svg></a></p><ul class="c-bibliographic-information__list" data-test="publication-history"><li class="c-bibliographic-information__list-item"><p>Received<span class="u-hide">: </span><span class="c-bibliographic-information__value"><time datetime="2019-11-14">14 November 2019</time></span></p></li><li class="c-bibliographic-information__list-item"><p>Accepted<span class="u-hide">: </span><span class="c-bibliographic-information__value"><time datetime="2020-06-26">26 June 2020</time></span></p></li><li class="c-bibliographic-information__list-item"><p>Published<span class="u-hide">: </span><span class="c-bibliographic-information__value"><time datetime="2020-07-11">11 July 2020</time></span></p></li><li class="c-bibliographic-information__list-item c-bibliographic-information__list-item--full-width"><p><abbr title="Digital Object Identifier">DOI</abbr><span class="u-hide">: </span><span class="c-bibliographic-information__value">https://doi.org/10.1007/s00025-020-01247-8</span></p></li></ul><div data-component="share-box"><div class="c-article-share-box u-display-none" hidden=""><h3 class="c-article__sub-heading">Share this article</h3><p class="c-article-share-box__description">Anyone you share the following link with will be able to read this content:</p><button class="js-get-share-url c-article-share-box__button" type="button" id="get-share-url" data-track="click" data-track-label="button" data-track-external="" data-track-action="get shareable link">Get shareable link</button><div class="js-no-share-url-container u-display-none" hidden=""><p class="js-c-article-share-box__no-sharelink-info c-article-share-box__no-sharelink-info">Sorry, a shareable link is not currently available for this article.</p></div><div class="js-share-url-container u-display-none" hidden=""><p class="js-share-url c-article-share-box__only-read-input" id="share-url" data-track="click" data-track-label="button" data-track-action="select share url"></p><button class="js-copy-share-url c-article-share-box__button--link-like" type="button" id="copy-share-url" data-track="click" data-track-label="button" data-track-action="copy share url" data-track-external="">Copy to clipboard</button></div><p class="js-c-article-share-box__additional-info c-article-share-box__additional-info"> Provided by the Springer Nature SharedIt content-sharing initiative </p></div></div><h3 class="c-article__sub-heading">Keywords</h3><ul class="c-article-subject-list"><li class="c-article-subject-list__subject"><span><a href="/search?query=Poincare%20lemma&facet-discipline="Mathematics"" data-track="click" data-track-action="view keyword" data-track-label="link">Poincare lemma</a></span></li><li class="c-article-subject-list__subject"><span><a href="/search?query=antiexact%20differential%20forms&facet-discipline="Mathematics"" data-track="click" data-track-action="view keyword" data-track-label="link">antiexact differential forms</a></span></li><li class="c-article-subject-list__subject"><span><a href="/search?query=homotopy%20operator&facet-discipline="Mathematics"" data-track="click" data-track-action="view keyword" data-track-label="link">homotopy operator</a></span></li><li class="c-article-subject-list__subject"><span><a href="/search?query=fermionic%20harmonic%20oscillator&facet-discipline="Mathematics"" data-track="click" data-track-action="view keyword" data-track-label="link">fermionic harmonic oscillator</a></span></li><li class="c-article-subject-list__subject"><span><a href="/search?query=complex%20manifold&facet-discipline="Mathematics"" data-track="click" data-track-action="view keyword" data-track-label="link">complex manifold</a></span></li></ul><h3 class="c-article__sub-heading">Mathematics Subject Classification</h3><ul class="c-article-subject-list"><li class="c-article-subject-list__subject"><span><a href="/search?query=58A12&facet-discipline="Mathematics"" data-track="click" data-track-action="view keyword" data-track-label="link">58A12</a></span></li><li class="c-article-subject-list__subject"><span><a href="/search?query=58Z05&facet-discipline="Mathematics"" data-track="click" data-track-action="view keyword" data-track-label="link">58Z05</a></span></li></ul><div data-component="article-info-list"></div></div></div></div></div></section> </div> </main> <div class="c-article-sidebar u-text-sm u-hide-print l-with-sidebar__sidebar" id="sidebar" data-container-type="reading-companion" data-track-component="reading companion"> <aside> <div class="app-card-service" data-test="article-checklist-banner"> <div> <a class="app-card-service__link" data-track="click_presubmission_checklist" data-track-context="article page top of reading companion" data-track-category="pre-submission-checklist" data-track-action="clicked article page checklist banner test 2 old version" data-track-label="link" href="https://beta.springernature.com/pre-submission?journalId=25" data-test="article-checklist-banner-link"> <span class="app-card-service__link-text">Use our pre-submission checklist</span> <svg class="app-card-service__link-icon" aria-hidden="true" focusable="false"><use xlink:href="#icon-eds-i-arrow-right-small"></use></svg> </a> <p class="app-card-service__description">Avoid common mistakes on your manuscript.</p> </div> <div class="app-card-service__icon-container"> <svg class="app-card-service__icon" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-clipboard-check-medium"></use> </svg> </div> </div> <div data-test="collections"> </div> <div data-test="editorial-summary"> </div> <div class="c-reading-companion"> <div class="c-reading-companion__sticky" data-component="reading-companion-sticky" data-test="reading-companion-sticky"> <div class="c-reading-companion__panel c-reading-companion__sections c-reading-companion__panel--active" id="tabpanel-sections"> <div class="u-lazy-ad-wrapper u-mt-16 u-hide" data-component-mpu><div class="c-ad c-ad--300x250"> <div class="c-ad__inner"> <p class="c-ad__label">Advertisement</p> <div id="div-gpt-ad-MPU1" class="div-gpt-ad grade-c-hide" data-pa11y-ignore data-gpt data-gpt-unitpath="/270604982/springerlink/25/article" data-gpt-sizes="300x250" data-test="MPU1-ad" data-gpt-targeting="pos=MPU1;articleid=s00025-020-01247-8;"> </div> </div> </div> </div> </div> <div class="c-reading-companion__panel c-reading-companion__figures c-reading-companion__panel--full-width" id="tabpanel-figures"></div> <div class="c-reading-companion__panel c-reading-companion__references c-reading-companion__panel--full-width" id="tabpanel-references"></div> </div> </div> </aside> </div> </div> </article> <div class="app-elements"> <div class="eds-c-header__expander eds-c-header__expander--search" id="eds-c-header-popup-search"> <h2 class="eds-c-header__heading">Search</h2> <div class="u-container"> <search class="eds-c-header__search" role="search" aria-label="Search from the header"> <form method="GET" action="//link.springer.com/search" data-test="header-search" data-track="search" data-track-context="search from header" data-track-action="submit search form" data-track-category="unified header" data-track-label="form" > <label for="eds-c-header-search" class="eds-c-header__search-label">Search by keyword or author</label> <div class="eds-c-header__search-container"> <input id="eds-c-header-search" class="eds-c-header__search-input" autocomplete="off" name="query" type="search" value="" required> <button class="eds-c-header__search-button" type="submit"> <svg class="eds-c-header__icon" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-search-medium"></use> </svg> <span class="u-visually-hidden">Search</span> </button> </div> </form> </search> </div> </div> <div class="eds-c-header__expander eds-c-header__expander--menu" id="eds-c-header-nav"> <h2 class="eds-c-header__heading">Navigation</h2> <ul class="eds-c-header__list"> <li class="eds-c-header__list-item"> <a class="eds-c-header__link" href="https://link.springer.com/journals/" data-track="nav_find_a_journal" data-track-context="unified header" data-track-action="click find a journal" data-track-category="unified header" data-track-label="link" > Find a journal </a> </li> <li class="eds-c-header__list-item"> <a class="eds-c-header__link" href="https://www.springernature.com/gp/authors" data-track="nav_how_to_publish" data-track-context="unified header" data-track-action="click publish with us link" data-track-category="unified header" data-track-label="link" > Publish with us </a> </li> <li class="eds-c-header__list-item"> <a class="eds-c-header__link" href="https://link.springernature.com/home/" data-track="nav_track_your_research" data-track-context="unified header" data-track-action="click track your research" data-track-category="unified header" data-track-label="link" > Track your research </a> </li> </ul> </div> <footer > <div class="eds-c-footer" > <div class="eds-c-footer__container"> <div class="eds-c-footer__grid eds-c-footer__group--separator"> <div class="eds-c-footer__group"> <h3 class="eds-c-footer__heading">Discover content</h3> <ul class="eds-c-footer__list"> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://link.springer.com/journals/a/1" data-track="nav_journals_a_z" data-track-action="journals a-z" data-track-context="unified footer" data-track-label="link">Journals A-Z</a></li> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://link.springer.com/books/a/1" data-track="nav_books_a_z" data-track-action="books a-z" data-track-context="unified footer" data-track-label="link">Books A-Z</a></li> </ul> </div> <div class="eds-c-footer__group"> <h3 class="eds-c-footer__heading">Publish with us</h3> <ul class="eds-c-footer__list"> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://link.springer.com/journals" data-track="nav_journal_finder" data-track-action="journal finder" data-track-context="unified footer" data-track-label="link">Journal finder</a></li> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.springernature.com/gp/authors" data-track="nav_publish_your_research" data-track-action="publish your research" data-track-context="unified footer" data-track-label="link">Publish your research</a></li> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.springernature.com/gp/open-research/about/the-fundamentals-of-open-access-and-open-research" data-track="nav_open_access_publishing" data-track-action="open access publishing" data-track-context="unified footer" data-track-label="link">Open access publishing</a></li> </ul> </div> <div class="eds-c-footer__group"> <h3 class="eds-c-footer__heading">Products and services</h3> <ul class="eds-c-footer__list"> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.springernature.com/gp/products" data-track="nav_our_products" data-track-action="our products" data-track-context="unified footer" data-track-label="link">Our products</a></li> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.springernature.com/gp/librarians" data-track="nav_librarians" data-track-action="librarians" data-track-context="unified footer" data-track-label="link">Librarians</a></li> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.springernature.com/gp/societies" data-track="nav_societies" data-track-action="societies" data-track-context="unified footer" data-track-label="link">Societies</a></li> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.springernature.com/gp/partners" data-track="nav_partners_and_advertisers" data-track-action="partners and advertisers" data-track-context="unified footer" data-track-label="link">Partners and advertisers</a></li> </ul> </div> <div class="eds-c-footer__group"> <h3 class="eds-c-footer__heading">Our imprints</h3> <ul class="eds-c-footer__list"> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.springer.com/" data-track="nav_imprint_Springer" data-track-action="Springer" data-track-context="unified footer" data-track-label="link">Springer</a></li> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.nature.com/" data-track="nav_imprint_Nature_Portfolio" data-track-action="Nature Portfolio" data-track-context="unified footer" data-track-label="link">Nature Portfolio</a></li> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.biomedcentral.com/" data-track="nav_imprint_BMC" data-track-action="BMC" data-track-context="unified footer" data-track-label="link">BMC</a></li> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.palgrave.com/" data-track="nav_imprint_Palgrave_Macmillan" data-track-action="Palgrave Macmillan" data-track-context="unified footer" data-track-label="link">Palgrave Macmillan</a></li> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.apress.com/" data-track="nav_imprint_Apress" data-track-action="Apress" data-track-context="unified footer" data-track-label="link">Apress</a></li> </ul> </div> </div> </div> <div class="eds-c-footer__container"> <nav aria-label="footer navigation"> <ul class="eds-c-footer__links"> <li class="eds-c-footer__item"> <button class="eds-c-footer__link" data-cc-action="preferences" data-track="dialog_manage_cookies" data-track-action="Manage cookies" data-track-context="unified footer" data-track-label="link"><span class="eds-c-footer__button-text">Your privacy choices/Manage cookies</span></button> </li> <li class="eds-c-footer__item"> <a class="eds-c-footer__link" href="https://www.springernature.com/gp/legal/ccpa" data-track="nav_california_privacy_statement" data-track-action="california privacy statement" data-track-context="unified footer" data-track-label="link">Your US state privacy rights</a> </li> <li class="eds-c-footer__item"> <a class="eds-c-footer__link" href="https://www.springernature.com/gp/info/accessibility" data-track="nav_accessibility_statement" data-track-action="accessibility statement" data-track-context="unified footer" data-track-label="link">Accessibility statement</a> </li> <li class="eds-c-footer__item"> <a class="eds-c-footer__link" href="https://link.springer.com/termsandconditions" data-track="nav_terms_and_conditions" data-track-action="terms and conditions" data-track-context="unified footer" data-track-label="link">Terms and conditions</a> </li> <li class="eds-c-footer__item"> <a class="eds-c-footer__link" href="https://link.springer.com/privacystatement" data-track="nav_privacy_policy" data-track-action="privacy policy" data-track-context="unified footer" data-track-label="link">Privacy policy</a> </li> <li class="eds-c-footer__item"> <a class="eds-c-footer__link" href="https://support.springernature.com/en/support/home" data-track="nav_help_and_support" data-track-action="help and support" data-track-context="unified footer" data-track-label="link">Help and support</a> </li> <li class="eds-c-footer__item"> <a class="eds-c-footer__link" href="https://link.springer.com/legal-notice" data-track="nav_legal_notice" data-track-action="legal notice" data-track-context="unified footer" data-track-label="link">Legal notice</a> </li> <li class="eds-c-footer__item"> <a class="eds-c-footer__link" href="https://support.springernature.com/en/support/solutions/articles/6000255911-subscription-cancellations" data-track-action="cancel contracts here">Cancel contracts here</a> </li> </ul> </nav> <div class="eds-c-footer__user"> <p class="eds-c-footer__user-info"> <span data-test="footer-user-ip">8.222.208.146</span> </p> <p class="eds-c-footer__user-info" data-test="footer-business-partners">Not affiliated</p> </div> <a href="https://www.springernature.com/" class="eds-c-footer__link"> <img src="/oscar-static/images/logo-springernature-white-19dd4ba190.svg" alt="Springer Nature" loading="lazy" width="200" height="20"/> </a> <p class="eds-c-footer__legal" data-test="copyright">© 2025 Springer Nature</p> </div> </div> </footer> </div> </body> </html>