Papers › Statistics of Limit Root Bundles Relevant for Exact Matter Spectra of F-Theory MSSMs
Statistics of Limit Root Bundles Relevant for Exact Matter Spectra of F-Theory MSSMs
Martin Bies, Mirjam Cvetič, Muyang Liu
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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_((3,2)_(1/6)) and employing well-known results about desingularizations of toric K3-surfaces, we derive a \emph{triangulation independent lower bound} Ň_P⁽³⁾ for the number N_P⁽³⁾ of root bundles on C_((3,2)_(1/6)) with exactly three sections. The ratio Ň_P⁽³⁾ / N_P, where N_P is the total number of roots on C_((3,2)_(1/6)), is largest for base spaces associated with triangulations of the 8-th 3-dimensional polytope Δ°₈ in the Kreuzer-Skarke list. For each of these 𝒪( 10¹⁵ ) 3-folds, we expect that many root bundles on C_((3,2)_(1/6)) are induced from F-theory gauge potentials and that at least every 3000th root on C_((3,2)_(1/6)) has exactly three global sections and thus no exotic vector-like quark-doublet modes.
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