Machine learning and algebraic approaches towards complete matter spectra in 4d F-theory

Ling Lin,Fabian Ruehle,Muyang Liu,Mirjam Cvetič,Martin Bies,Ron Donagi

doi:10.1007/jhep01(2021)196

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 dP3, 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.

Highlights

The spectrum of light chiral particles is a defining feature of any four dimensional quantum field theory
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
We find a unified perspective on jumps due to curve splittings and non-generic line bundles described by Brill-Noether theory, and introduce a diagrammatic way to illustrate the natural stratification of the complex structure parameter space in terms of the vector-like spectrum

Summary

Introduction

The spectrum of light chiral particles is a defining feature of any four dimensional quantum field theory. To fully understand the results of the machine learning, we further employ “formal” techniques from algebraic geometry, in the form of Brill-Noether theory This allows to identify “microscopically” the sources for jumps in cohomology, either from the curve CR or the line bundle LR becoming non-generic. With these insights, we provide an algorithmic way to estimate the admissible numbers of vector-like pairs over the entire parameter space of a matter curve in a global F-theory model with given gauge background. We find a unified perspective on jumps due to curve splittings and non-generic line bundles described by Brill-Noether theory, and introduce a diagrammatic way to illustrate the natural stratification of the complex structure parameter space in terms of the vector-like spectrum. In contrast to currently existing exact methods, such as [42], our implementation [44] has a much lower demand of computational resources and run times

Introduction to decision trees

Divisors and line bundles on dP3

Decision trees to learn cohomology jumps

Jumps from curve splittings

Unpredicted jumps

Application

Engineering jumps in cohomology

Single vector-like pair from Brill-Noether theory

Cohomology jumps throughout the moduli space

Example: one additional section

An h0-gap

Jump from non-rigid curve splitting