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Statistics of limit root bundles relevant for exact matter spectra of F-theory MSSMs
Martin Bies, Mirjam Cvetič, Mingqiang Liu, 2021, original scientific article

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 vectorlike spectra are counted by cohomologies of root bundles. In this work, we apply a previously proposed method to identify toric base threefolds, which are promising to establish F-theory Standard Models with exactly three quark doublets and no vectorlike 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 Cð3;2Þ1=6 and employing well-known results about desingularizations of toric K3 surfaces, we derive a triangulation independent lower bound Nˇ ð3Þ P for the number Nð3Þ P of root bundles on Cð3;2Þ1=6 with exactly three sections. The ratio Nˇ ð3Þ P =NP, where NP is the total number of roots on Cð3;2Þ1=6 , is largest for base spaces associated with triangulations of the eighth three-dimensional polytope Δ∘ 8 in the Kreuzer-Skarke list. For each of these Oð1015Þ threefolds, we expect that many root bundles on Cð3;2Þ1=6 are induced from F-theory gauge potentials and that at least every 3000th root on Cð3;2Þ1=6 has exactly three global sections and thus no exotic vectorlike quark-doublet modes.
Keywords: astrophysics, compactification, string theory models, geometry, higher-dimensional field theories, mathematical physics, quantum fields in curved spacetime, string phenomenology, supersymmetric models, topology
Published in DKUM: 16.10.2023; Views: 239; Downloads: 11
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Brill-Noether-general limit root bundles: absence of vector-like exotics in F-theory standard models
Martin Bies, Mirjam Cvetič, Ron Donagi, Marielle Ong, 2022, original scientific article

Abstract: Root bundles appear prominently in studies of vector-like spectra of 4d F-theory compactifications. Of particular importance to phenomenology are the Quadrillion F-theory Standard Models (F-theory QSMs). In this work, we analyze a superset of the physical root bundles whose cohomologies encode the vector-like spectra for the matter representations (3, 2)(1/6), ((3) over bar ,1)(-2/3) and (1,1)(1) For the family B-3(Delta(4)degrees) consisting of O(10(11)) F-theory QSM geometries, we argue that more than 99.995% of the roots in this superset have no vector-like exotics. This indicates that absence of vector-like exotics in those representations is a very likely scenario in the O(10(11)) QSM geometries B-3(Delta(4)degrees). The QSM geometries come in families of toric 3-folds B-3(Delta(4)degrees) obtained from triangulations of certain 3-dimensional polytopes Delta degrees. The matter curves in X-Sigma is an element of B-3(Delta(4)degrees) can be deformed to nodal curves which are the same for all spaces in B-3(Delta(4)degrees). Therefore, one can probe the vector-like spectra on the entire family B-3(Delta(4)degrees) from studies of a few nodal curves. We compute the cohomologies of all limit roots on these nodal curves. In our applications, for the majority of limit roots the cohomologies are determined by line bundle cohomology on rational tree-like curves. For this, we present a computer algorithm. The remaining limit roots, corresponding to circuit-like graphs, are handled by hand. The cohomologies are independent of the relative position of the nodes, except for a few circuits. On these jumping circuits, line bundle cohomologies can jump if nodes are specially aligned. This mirrors classical Brill-Noether jumps. B-3(Delta(4)degrees) admits a jumping circuit, but the root bundle constraints pick the canonical bundle and no jump happens.
Keywords: differential geometry, algebraic geometry, F-theory, string phenomenology, brane phenomenology
Published in DKUM: 17.08.2023; Views: 276; Downloads: 20
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