{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/statistics-of-limit-root-bundles-relevant-for","title":"Statistics of Limit Root Bundles Relevant for Exact Matter Spectra of F-Theory MSSMs","arxiv_id":"2104.08297","date":"2021-04-16","proceeding":null,"authors":["Martin Bies","Mirjam Cvetič","Muyang Liu"],"abstract":"In the largest, currently known, class of one Quadrillion globally consistent F-theory Standard Models with gauge coupling unification and no chiral exotics, the vector-like spectra are counted by cohomologies of root bundles. In this work, we apply a previously proposed method to identify toric base 3-folds, which are promising to establish F-theory Standard Models with exactly three quark-doublets and no vector-like exotics in this representation. The base spaces in question are obtained from triangulations of 708 polytopes. By studying root bundles on the quark doublet curve $C_{(\\mathbf{3},\\mathbf{2})_{1/6}}$ and employing well-known results about desingularizations of toric K3-surfaces, we derive a \\emph{triangulation independent lower bound} $\\check{N}_P^{(3)}$ for the number $N_P^{(3)}$ of root bundles on $C_{(\\mathbf{3},\\mathbf{2})_{1/6}}$ with exactly three sections. The ratio $\\check{N}_P^{(3)} / N_P$, where $N_P$ is the total number of roots on $C_{(\\mathbf{3},\\mathbf{2})_{1/6}}$, is largest for base spaces associated with triangulations of the 8-th 3-dimensional polytope $\\Delta^\\circ_8$ in the Kreuzer-Skarke list. For each of these $\\mathcal{O}( 10^{15} )$ 3-folds, we expect that many root bundles on $C_{(\\mathbf{3},\\mathbf{2})_{1/6}}$ are induced from F-theory gauge potentials and that at least every 3000th root on $C_{(\\mathbf{3},\\mathbf{2})_{1/6}}$ has exactly three global sections and thus no exotic vector-like quark-doublet modes.","url_abs":"https://arxiv.org/abs/2104.08297v1","url_pdf":"https://arxiv.org/pdf/2104.08297v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"statistics-of-limit-root-bundles-relevant-for","repo_url":"https://github.com/homalg-project/ToricVarieties_project","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"statistics-of-limit-root-bundles-relevant-for","repo_url":"https://github.com/HereAround/SheafCohomologyOnToricVarieties","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}