Brill-Noether-general Limit Root Bundles: Absence of vector-like Exotics in F-theory Standard Models

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)$_{1/6}$, ($ \overline{\textbf{3}} $,1)$_{−2/3}$ and (1,1)$_{1}$. For the family B$_{3}$($ {\Delta }_4^{{}^{\circ}} $) consisting of $ \mathcal{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 $ \mathcal{O} $(10$^{11}$) QSM geometries B$_{3}$($ {\Delta }_4^{{}^{\circ}} $).The QSM geometries come in families of toric 3-folds B$_{3}$(∆$^{°}$) obtained from triangulations of certain 3-dimensional polytopes ∆$^{°}$. The matter curves in X$_{Σ}$ ∈ B$_{3}$(∆$^{°}$) can be deformed to nodal curves which are the same for all spaces in B$_{3}$(∆$^{°}$). Therefore, one can probe the vector-like spectra on the entire family B$_{3}$(∆$^{°}$) 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^{{}^{\circ}} $) admits a jumping circuit, but the root bundle constraints pick the canonical bundle and no jump happens.