Statistics of limit root bundles relevant for exact matter spectra of F-theory MSSMs

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 ${C}_{(\mathbf{3},\mathbf{2}{)}_{1/6}}$ and employing well-known results about desingularizations of toric K3 surfaces, we derive a triangulation independent lower bound ${\stackrel{\textasciicaron{}}{N}}_{P}^{(3)}$ for the number ${N}_{P}^{(3)}$ of root bundles on ${C}_{(\mathbf{3},\mathbf{2}{)}_{1/6}}$ with exactly three sections. The ratio ${\stackrel{\textasciicaron{}}{N}}_{P}^{(3)}/{N}_{P}$, 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 eighth three-dimensional polytope ${\mathrm{\ensuremath{\Delta}}}_{8}^{\ensuremath{\circ}}$ in the Kreuzer-Skarke list. For each of these $\mathcal{O}({10}^{15})$ threefolds, 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 vectorlike quark-doublet modes.