{"id":211,"date":"2020-04-17T12:54:11","date_gmt":"2020-04-17T11:54:11","guid":{"rendered":"http:\/\/e595.lan\/?page_id=211"},"modified":"2025-03-11T15:32:08","modified_gmt":"2025-03-11T13:32:08","slug":"mathematical-apects-of-qft","status":"publish","type":"page","link":"https:\/\/www2.hu-berlin.de\/rtg2575\/research\/mathematical-apects-of-qft\/","title":{"rendered":"Mathematical aspects of QFT"},"content":{"rendered":"<h3>Principal investigators: Gaetan Borot, Olaf Hohm, Matthias Staudacher<\/h3>\n<h3>Post-doctoral researcher: Rob Klabbers<\/h3>\n<h3>Doctoral researchers: Giuseppe Casale, Maria Kallimani, Davide Scazzuso, Tobias Scherdin<\/h3>\n<h2>Topological recursion, enumerative geometry and QFT<\/h2>\n<p>In this research area we focus on the interrelations between various aspects of algebra, geometry and quantum field theory investigated from the unifying perspective of topological recursion, which can be thought as a recursion on the topology of the worldsheet.<\/p>\n<h3>Main topics<\/h3>\n<ul>\n<li>Intersection theory on the moduli space of curves<\/li>\n<li>Topological string theory and mirror symmetry<\/li>\n<li>Topological quantum field theories<\/li>\n<li>2d conformal field theory and W-algebras<\/li>\n<li>N = 2 supersymmetric gauge theories of class S<\/li>\n<li>Geometry of spectral curves<\/li>\n<li>Integrable systems<\/li>\n<li>2d statistical physics on random surfaces<\/li>\n<li>Random matrix models, combinatorics and geometry of random surfaces<\/li>\n<\/ul>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1519 lazyload\" data-src=\"https:\/\/www2.hu-berlin.de\/rtg2575\/wp-content\/uploads\/2023\/01\/Summary_research_borot-1024x768.jpg\" alt=\"\" width=\"900\" height=\"675\" data-srcset=\"https:\/\/www2.hu-berlin.de\/rtg2575\/wp-content\/uploads\/2023\/01\/Summary_research_borot-1024x768.jpg 1024w, https:\/\/www2.hu-berlin.de\/rtg2575\/wp-content\/uploads\/2023\/01\/Summary_research_borot-300x225.jpg 300w, https:\/\/www2.hu-berlin.de\/rtg2575\/wp-content\/uploads\/2023\/01\/Summary_research_borot-768x576.jpg 768w, https:\/\/www2.hu-berlin.de\/rtg2575\/wp-content\/uploads\/2023\/01\/Summary_research_borot-1536x1152.jpg 1536w, https:\/\/www2.hu-berlin.de\/rtg2575\/wp-content\/uploads\/2023\/01\/Summary_research_borot-2048x1536.jpg 2048w\" data-sizes=\"(max-width: 900px) 100vw, 900px\" src=\"data:image\/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==\" style=\"--smush-placeholder-width: 900px; --smush-placeholder-aspect-ratio: 900\/675;\" \/><noscript><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1519\" src=\"https:\/\/www2.hu-berlin.de\/rtg2575\/wp-content\/uploads\/2023\/01\/Summary_research_borot-1024x768.jpg\" alt=\"\" width=\"900\" height=\"675\" srcset=\"https:\/\/www2.hu-berlin.de\/rtg2575\/wp-content\/uploads\/2023\/01\/Summary_research_borot-1024x768.jpg 1024w, https:\/\/www2.hu-berlin.de\/rtg2575\/wp-content\/uploads\/2023\/01\/Summary_research_borot-300x225.jpg 300w, https:\/\/www2.hu-berlin.de\/rtg2575\/wp-content\/uploads\/2023\/01\/Summary_research_borot-768x576.jpg 768w, https:\/\/www2.hu-berlin.de\/rtg2575\/wp-content\/uploads\/2023\/01\/Summary_research_borot-1536x1152.jpg 1536w, https:\/\/www2.hu-berlin.de\/rtg2575\/wp-content\/uploads\/2023\/01\/Summary_research_borot-2048x1536.jpg 2048w\" sizes=\"(max-width: 900px) 100vw, 900px\" \/><\/noscript><\/p>\n<h2><i>L<\/i><sub>\u221e<\/sub>-algebras in Quantum Field Theory<\/h2>\n<p>This aspect of the research program is concerned with higher algebraic structures in QFT and, more generally, in theoretical and mathematical physics.<span class=\"Apple-converted-space\">\u00a0<\/span><\/p>\n<p>Lie groups and Lie algebras are the structures underlying most of contemporary fundamental physics, from the standard model of particle physics to Einstein\u2019s theory of general relativity. Recent developments have shown that higher algebraic structures in form of<br \/>\n<b><i>L<\/i><\/b><b><sub>\u221e<\/sub><\/b><b>-algebras<\/b> could play an important role for the physics of the 21st century.<span class=\"Apple-converted-space\">\u00a0<\/span><\/p>\n<ul>\n<li><i>L<\/i><sub>\u221e<\/sub>-algebras are a generalisation of Lie algebras satisfying higher Jacobi identities involving potentially infinitely many higher brackets. It has become clear recently that <i>L<\/i><sub>\u221e<\/sub>-algebras can function naturally as the organisational principle for arbitrary field theories<\/li>\n<li>A powerful machinery for <i>L<\/i><sub>\u221e<\/sub>-algebras exists \u2014<b>homotopy transfer<\/b>\u2014that has recently been argued to govern dualities of QFT<span class=\"Apple-converted-space\">\u00a0<\/span><\/li>\n<li>There are rich sub-theories of full string theory that include <b>massive string modes<\/b> in a duality covariant formulation that are not captured by the usual spacetime theories such as supergravity. The explicit construction of such theories and the determination of their underlying <i>L<\/i><sub>\u221e<\/sub>-algebras is the focus of current research<\/li>\n<li><b>Higher-derivative <\/b><b><i>\u03b1<\/i><\/b><b><i><sup>\u2019<\/sup><\/i><\/b><b>-corrections<\/b> in string inspired gravitational theories could play a role in resolving the big-bang singularity or in providing a non-perturbative mechanism to generate de Sitter vacua.<br \/>\n<i>L<\/i><sub>\u221e<\/sub>-algebras within the framework of <b>double field theory<\/b> can help in understanding the symmetry principles of these corrections<\/li>\n<\/ul>\n<figure id=\"attachment_682\" aria-describedby=\"caption-attachment-682\" style=\"width: 440px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-682 lazyload\" data-src=\"https:\/\/www2.hu-berlin.de\/rtg2575\/wp-content\/uploads\/2020\/09\/jacobi_identity-1024x276.jpeg\" alt=\"\" width=\"450\" height=\"121\" data-srcset=\"https:\/\/www2.hu-berlin.de\/rtg2575\/wp-content\/uploads\/2020\/09\/jacobi_identity-1024x276.jpeg 1024w, https:\/\/www2.hu-berlin.de\/rtg2575\/wp-content\/uploads\/2020\/09\/jacobi_identity-300x81.jpeg 300w, https:\/\/www2.hu-berlin.de\/rtg2575\/wp-content\/uploads\/2020\/09\/jacobi_identity-768x207.jpeg 768w, https:\/\/www2.hu-berlin.de\/rtg2575\/wp-content\/uploads\/2020\/09\/jacobi_identity.jpeg 1315w\" data-sizes=\"(max-width: 450px) 100vw, 450px\" src=\"data:image\/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==\" style=\"--smush-placeholder-width: 450px; --smush-placeholder-aspect-ratio: 450\/121;\" \/><noscript><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-682\" src=\"https:\/\/www2.hu-berlin.de\/rtg2575\/wp-content\/uploads\/2020\/09\/jacobi_identity-1024x276.jpeg\" alt=\"\" width=\"450\" height=\"121\" srcset=\"https:\/\/www2.hu-berlin.de\/rtg2575\/wp-content\/uploads\/2020\/09\/jacobi_identity-1024x276.jpeg 1024w, https:\/\/www2.hu-berlin.de\/rtg2575\/wp-content\/uploads\/2020\/09\/jacobi_identity-300x81.jpeg 300w, https:\/\/www2.hu-berlin.de\/rtg2575\/wp-content\/uploads\/2020\/09\/jacobi_identity-768x207.jpeg 768w, https:\/\/www2.hu-berlin.de\/rtg2575\/wp-content\/uploads\/2020\/09\/jacobi_identity.jpeg 1315w\" sizes=\"(max-width: 450px) 100vw, 450px\" \/><\/noscript><figcaption id=\"caption-attachment-682\" class=\"wp-caption-text\">Graphical presentation of higher Jacobi identity<\/figcaption><\/figure>\n<h3><\/h3>\n<h3><strong>Integration into the RTG<\/strong><\/h3>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-678 lazyload\" data-src=\"https:\/\/www2.hu-berlin.de\/rtg2575\/wp-content\/uploads\/2020\/09\/ellips_math.jpeg\" alt=\"\" width=\"550\" height=\"290\" data-srcset=\"https:\/\/www2.hu-berlin.de\/rtg2575\/wp-content\/uploads\/2020\/09\/ellips_math.jpeg 750w, https:\/\/www2.hu-berlin.de\/rtg2575\/wp-content\/uploads\/2020\/09\/ellips_math-300x158.jpeg 300w\" data-sizes=\"(max-width: 550px) 100vw, 550px\" src=\"data:image\/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==\" style=\"--smush-placeholder-width: 550px; --smush-placeholder-aspect-ratio: 550\/290;\" \/><noscript><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-678\" src=\"https:\/\/www2.hu-berlin.de\/rtg2575\/wp-content\/uploads\/2020\/09\/ellips_math.jpeg\" alt=\"\" width=\"550\" height=\"290\" srcset=\"https:\/\/www2.hu-berlin.de\/rtg2575\/wp-content\/uploads\/2020\/09\/ellips_math.jpeg 750w, https:\/\/www2.hu-berlin.de\/rtg2575\/wp-content\/uploads\/2020\/09\/ellips_math-300x158.jpeg 300w\" sizes=\"(max-width: 550px) 100vw, 550px\" \/><\/noscript><\/p>\n<p>This research topic integrates well into the RTG as it provides mathematical foundations for phenomenology, scattering amplitudes, the AdS\/CFT correspondence and for perturbative expansions in terms of Feynman diagrams.<br \/>\nThe spacetime formulation provided by double field theory is a promising framework in order to understand the principles underlying the \u201cdouble copy\u201d construction for the computation of amplitudes.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Principal investigators: Gaetan Borot, Olaf Hohm, Matthias Staudacher Post-doctoral researcher: Rob Klabbers Doctoral researchers: Giuseppe Casale, Maria Kallimani, Davide Scazzuso, Tobias Scherdin Topological recursion, enumerative geometry and QFT In this research area we focus on the interrelations between various aspects of algebra, geometry and quantum field theory investigated from the unifying perspective of topological recursion, &#8230; <a title=\"Mathematical aspects of QFT\" class=\"read-more\" href=\"https:\/\/www2.hu-berlin.de\/rtg2575\/research\/mathematical-apects-of-qft\/\">Read more <span class=\"screen-reader-text\">Mathematical aspects of QFT<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"parent":291,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"_links":{"self":[{"href":"https:\/\/www2.hu-berlin.de\/rtg2575\/wp-json\/wp\/v2\/pages\/211"}],"collection":[{"href":"https:\/\/www2.hu-berlin.de\/rtg2575\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www2.hu-berlin.de\/rtg2575\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www2.hu-berlin.de\/rtg2575\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www2.hu-berlin.de\/rtg2575\/wp-json\/wp\/v2\/comments?post=211"}],"version-history":[{"count":21,"href":"https:\/\/www2.hu-berlin.de\/rtg2575\/wp-json\/wp\/v2\/pages\/211\/revisions"}],"predecessor-version":[{"id":2222,"href":"https:\/\/www2.hu-berlin.de\/rtg2575\/wp-json\/wp\/v2\/pages\/211\/revisions\/2222"}],"up":[{"embeddable":true,"href":"https:\/\/www2.hu-berlin.de\/rtg2575\/wp-json\/wp\/v2\/pages\/291"}],"wp:attachment":[{"href":"https:\/\/www2.hu-berlin.de\/rtg2575\/wp-json\/wp\/v2\/media?parent=211"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}