$6$D Anomaly-Free Matter Spectrum in F-theory on Singular Spaces

TL;DR

This paper develops a method to determine the charged matter spectrum in 6D F-theory models directly on singular spaces using string junctions, matching anomaly cancellation predictions without smoothing the geometry.

Contribution

It introduces a general prescription for computing localized matter spectra on singular elliptic Calabi-Yau 3-folds using string junction technology, bypassing resolution.

Findings

Agreement with 6D anomaly cancellation in all tested cases

Applicable to generic Weierstrass models with various Kodaira fibers

Effective for intersecting non-abelian seven-branes

Abstract

In this paper we study the 6d localized charged matter spectrum of F-theory directly on a singular elliptic Calabi-Yau 3-fold, i.e. without smoothing via resolution or deformation of the entire fibration. Given only the base surface, discriminant locus, and the $SL(2,\mathbb{Z})$ local system, we propose a general prescription for determining the charged matter spectrum localized at intersections of seven-branes, using the technology of string junctions. More precisely, at each codimension-$2$ collision of seven-branes, we determine the local seven-brane content and compute the number of massless string junctions modulo the action of the $SL(2,\mathbb{Z})$ monodromy. We find agreement with the predicted results from $6$d anomaly cancellation in all cases considered. Examples include a generic Weierstrass model with arbitrary Kodaira fiber intersecting an $I_1$, as well as cases with jointly charged matter localized at intersections of non-abelian seven-branes.