Machine Learning and Algebraic Approaches towards Complete Matter Spectra in 4d F-theory

TL;DR

This paper combines machine learning and algebraic geometry to analyze line bundle cohomologies in 4d F-theory, enabling better understanding and prediction of matter spectra and Higgs pairs in string compactifications.

Contribution

It introduces a novel integration of machine learning with algebraic geometry to analyze cohomology jumps and matter spectra in F-theory models, including a diagrammatic method for stratification.

Findings

Generated 1.8 million line bundle-curve pairs and computed their cohomologies.

Machine learning provides intuition for cohomology jumps due to curve splittings.

A diagrammatic approach reflects the stratification of the moduli space.

Abstract

Motivated by engineering vector-like (Higgs) pairs in the spectrum of 4d F-theory compactifications, we combine machine learning and algebraic geometry techniques to analyze line bundle cohomologies on families of holomorphic curves. To quantify jumps of these cohomologies, we first generate 1.8 million pairs of line bundles and curves embedded in $dP_3$, for which we compute the cohomologies. A white-box machine learning approach trained on this data provides intuition for jumps due to curve splittings, which we use to construct additional vector-like Higgs-pairs in an F-Theory toy model. We also find that, in order to explain quantitatively the full dataset, further tools from algebraic geometry, in particular Brill--Noether theory, are required. Using these ingredients, we introduce a diagrammatic way to express cohomology jumps across the parameter space of each family of matter curves, which reflects a stratification of the F-theory complex structure moduli space in terms of the vector-like spectrum. Furthermore, these insights provide an algorithmically efficient way to estimate the possible cohomology dimensions across the entire parameter space.