BrillNoethergeneral Limit Root Bundles: Absence of vectorlike Exotics in Ftheory Standard Models
Abstract
Root bundles appear prominently in studies of vectorlike spectra of 4d Ftheory compactifications. Of particular importance to phenomenology are the Quadrillion Ftheory Standard Models (Ftheory QSMs). In this work, we analyze a superset of the physical root bundles whose cohomologies encode the vectorlike spectra for the matter representations $(\mathbf{3}, \mathbf{2})_{1/6}$, $(\mathbf{\overline{3}}, \mathbf{1})_{2/3}$ and $(\mathbf{1}, \mathbf{1})_{1}$. For the family $B_3( \Delta_4^\circ )$ consisting of $\mathcal{O}(10^{11})$ Ftheory QSM geometries, we argue that more than $99.995\%$ of the roots in this superset have no vectorlike exotics. This indicates that absence of vectorlike exotics in those representations is a very likely scenario. The QSM geometries come in families of toric 3folds $B_3( \Delta^\circ )$ obtained from triangulations of certain 3dimensional polytopes $\Delta^\circ$. The matter curves in $X_\Sigma \in B_3( \Delta^\circ )$ can be deformed to nodal curves which are the same for all spaces in $B_3( \Delta^\circ )$. Therefore, one can probe the vectorlike spectra on the entire family $B_3( \Delta^\circ )$ 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 treelike curves. For this, we present a computer algorithm. The remaining limit roots, corresponding to circuitlike graphs, are handled by hand. The cohomologies are independent of the relative position of the nodes, except for a few circuits. On these \emph{jumping circuits}, line bundle cohomologies can jump if nodes are specially aligned. This mirrors classical BrillNoether jumps. $B_3( \Delta_4^\circ )$ admits a jumping circuit, but the root bundle constraints pick the canonical bundle and no jump happens.
 Publication:

arXiv eprints
 Pub Date:
 April 2022
 arXiv:
 arXiv:2205.00008
 Bibcode:
 2022arXiv220500008B
 Keywords:

 High Energy Physics  Theory;
 Mathematics  Algebraic Geometry
 EPrint:
 37 pages + appendix