# Gauge Backgrounds and Zero-Mode Counting in F-Theory

## TL;DR

This paper develops a mathematical framework using algebraic geometry to precisely compute the charged massless matter spectrum in F-theory compactifications, demonstrating its application with concrete examples.

## Contribution

It introduces a novel algebraic geometric approach to determine zero-mode spectra in F-theory, utilizing Chow rings and cohomology computations implemented in software.

## Key findings

- Observed spectrum jumps with moduli variation
- Applied framework to models with Abelian and non-Abelian groups
- Validated the method with explicit examples

## Abstract

Computing the exact spectrum of charged massless matter is a crucial step towards understanding the effective field theory describing F-theory vacua in four dimensions. In this work we further develop a coherent framework to determine the charged massless matter in F-theory compactified on elliptic fourfolds, and demonstrate its application in a concrete example. The gauge background is represented, via duality with M-theory, by algebraic cycles modulo rational equivalence. Intersection theory within the Chow ring allows us to extract coherent sheaves on the base of the elliptic fibration whose cohomology groups encode the charged zero-mode spectrum. The dimensions of these cohomology groups are computed with the help of modern techniques from algebraic geometry, which we implement in the software gap. We exemplify this approach in models with an Abelian and non-Abelian gauge group and observe jumps in the exact massless spectrum as the complex structure moduli are varied. An extended mathematical appendix gives a self-contained introduction to the algebro-geometric concepts underlying our framework.