Statistics of Limit Root Bundles Relevant for Exact Matter Spectra of F-Theory MSSMs

TL;DR

This paper investigates the statistics of root bundles on certain curves in F-theory models, providing bounds and estimates that suggest many models naturally avoid exotic matter, advancing understanding of realistic string compactifications.

Contribution

It introduces a triangulation-independent lower bound for root bundles with three sections on specific curves, aiding the construction of realistic F-theory Standard Models.

Findings

Many root bundles with exactly three sections are expected on key curves.

The ratio of such bundles is maximized for certain toric base spaces.

At least one in 3000 roots has exactly three global sections, avoiding exotics.

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.