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Root bundles and towards exact matter spectra of F-theory MSSMs
Martin Bies, Mirjam Cvetič, Ron Donagi, Mingqiang Liu, Marielle Ong, 2021, izvirni znanstveni članek

Opis: Motivated by the appearance of fractional powers of line bundles in studies of vector-like spectra in 4d F-theory compactifications, we analyze the structure and origin of these bundles. Fractional powers of line bundles are also known as root bundles and can be thought of as generalizations of spin bundles. We explain how these root bundles are linked to inequivalent F-theory gauge potentials of a G(4)-flux. While this observation is interesting in its own right, it is particularly valuable for F-theory Standard Model constructions. In aiming for MSSMs, it is desired to argue for the absence of vector-like exotics. We work out the root bundle constraints on all matter curves in the largest class of currently-known F-theory Standard Model constructions without chiral exotics and gauge coupling unification. On each matter curve, we conduct a systematic "bottom"-analysis of all solutions to the root bundle constraints and all spin bundles. Thereby, we derive a lower bound for the number of combinations of root bundles and spin bundles whose cohomologies satisfy the physical demand of absence of vector-like pairs. On a technical level, this systematic study is achieved by a well-known diagrammatic description of root bundles on nodal curves. We extend this description by a counting procedure, which determines the cohomologies of so-called limit root bundles on full blow-ups of nodal curves. By use of deformation theory, these results constrain the vector-like spectra on the smooth matter curves in the actual F-theory geometry.
Ključne besede: F-theory, differential geometry, algebraic geometry
Objavljeno v DKUM: 16.10.2023; Ogledov: 236; Prenosov: 15
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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, izvirni znanstveni članek

Opis: 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.
Ključne besede: astrophysics, compactification, string theory models, geometry, higher-dimensional field theories, mathematical physics, quantum fields in curved spacetime, string phenomenology, supersymmetric models, topology
Objavljeno v DKUM: 16.10.2023; Ogledov: 217; Prenosov: 11
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Machine learning and algebraic approaches towards complete matter spectra in 4d F-theory
Martin Bies, Mirjam Cvetič, Ron Donagi, Ling Lin, Mingqiang Liu, Fabian Ruehle, 2021, izvirni znanstveni članek

Opis: Motivated by engineering vector-like (Higgs) pairs in the spectrum of 4d Ftheory compactifications, we combine machine learning and algebraic geometry techniques to analyze line bundle cohomologies on families of holomorphic curves. To quantify jumps of these cohomologies, we first generate 1.8 million pairs of line bundles and curves embedded in dP3, for which we compute the cohomologies. A white-box machine learning approach trained on this data provides intuition for jumps due to curve splittings, which we use to construct additional vector-like Higgs-pairs in an F-Theory toy model. We also find that, in order to explain quantitatively the full dataset, further tools from algebraic geometry, in particular Brill-Noether theory, are required. Using these ingredients, we introduce a diagrammatic way to express cohomology jumps across the parameter space of each family of matter curves, which reflects a stratification of the F-theory complex structure moduli space in terms of the vector-like spectrum. Furthermore, these insights provide an algorithmically efficient way to estimate the possible cohomology dimensions across the entire parameter space.
Ključne besede: Differential Geometry, Algebraic Geometry, F-Theory, Flux Compactifications, Field Theories, Higher Dimensions
Objavljeno v DKUM: 13.10.2023; Ogledov: 221; Prenosov: 13
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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, izvirni znanstveni članek

Opis: 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.
Ključne besede: differential geometry, algebraic geometry, F-theory, string phenomenology, brane phenomenology
Objavljeno v DKUM: 17.08.2023; Ogledov: 263; Prenosov: 18
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