{"title":"Combinatorial 2d higher topological quantum field theory from a local cyclic \\(A_\\infty \\) algebra","authors":"Justin Beck, Andrey Losev, Pavel Mnev","doi":"10.1007/s11005-024-01874-0","DOIUrl":null,"url":null,"abstract":"<div><p>We construct combinatorial analogs of 2d higher topological quantum field theories. We consider triangulations as vertices of a certain CW complex <span>\\(\\Xi \\)</span>. In the “flip theory,” cells of <span>\\(\\Xi _\\textrm{flip}\\)</span> correspond to polygonal decompositions obtained by erasing the edges in a triangulation. These theories assign to a cobordism <span>\\(\\Sigma \\)</span> a cochain <i>Z</i> on <span>\\(\\Xi _\\textrm{flip}\\)</span> constructed as a contraction of structure tensors of a cyclic <span>\\(A_\\infty \\)</span> algebra <i>V</i> assigned to polygons. The cyclic <span>\\(A_\\infty \\)</span> equations imply the closedness equation <span>\\((\\delta +Q)Z=0\\)</span>. In this context, we define combinatorial BV operators and give examples with coefficients in <span>\\(\\mathbb {Z}_2\\)</span>. In the “secondary polytope theory,” <span>\\(\\Xi _\\textrm{sp}\\)</span> is the secondary polytope (due to Gelfand–Kapranov–Zelevinsky) and the cyclic <span>\\(A_\\infty \\)</span> algebra has to be replaced by an appropriate refinement that we call an <span>\\(\\widehat{A}_\\infty \\)</span> algebra. We conjecture the existence of a good Pachner CW complex <span>\\(\\Xi \\)</span> for any cobordism, whose local combinatorics is described by secondary polytopes and the homotopy type is that of Zwiebach’s moduli space of complex structures. Depending on this conjecture, one has an “ideal model” of combinatorial 2d HTQFT determined by a local <span>\\(\\widehat{A}_\\infty \\)</span> algebra.</p></div>","PeriodicalId":685,"journal":{"name":"Letters in Mathematical Physics","volume":"114 6","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2024-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11005-024-01874-0.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Letters in Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1007/s11005-024-01874-0","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0
Abstract
We construct combinatorial analogs of 2d higher topological quantum field theories. We consider triangulations as vertices of a certain CW complex \(\Xi \). In the “flip theory,” cells of \(\Xi _\textrm{flip}\) correspond to polygonal decompositions obtained by erasing the edges in a triangulation. These theories assign to a cobordism \(\Sigma \) a cochain Z on \(\Xi _\textrm{flip}\) constructed as a contraction of structure tensors of a cyclic \(A_\infty \) algebra V assigned to polygons. The cyclic \(A_\infty \) equations imply the closedness equation \((\delta +Q)Z=0\). In this context, we define combinatorial BV operators and give examples with coefficients in \(\mathbb {Z}_2\). In the “secondary polytope theory,” \(\Xi _\textrm{sp}\) is the secondary polytope (due to Gelfand–Kapranov–Zelevinsky) and the cyclic \(A_\infty \) algebra has to be replaced by an appropriate refinement that we call an \(\widehat{A}_\infty \) algebra. We conjecture the existence of a good Pachner CW complex \(\Xi \) for any cobordism, whose local combinatorics is described by secondary polytopes and the homotopy type is that of Zwiebach’s moduli space of complex structures. Depending on this conjecture, one has an “ideal model” of combinatorial 2d HTQFT determined by a local \(\widehat{A}_\infty \) algebra.
期刊介绍:
The aim of Letters in Mathematical Physics is to attract the community''s attention on important and original developments in the area of mathematical physics and contemporary theoretical physics. The journal publishes letters and longer research articles, occasionally also articles containing topical reviews. We are committed to both fast publication and careful refereeing. In addition, the journal offers important contributions to modern mathematics in fields which have a potential physical application, and important developments in theoretical physics which have potential mathematical impact.