Vector-Like Pairs and Brill--Noether Theory

TL;DR

This paper investigates the likelihood and constraints of vector-like pairs in low-energy spectra within F-theory and Heterotic compactifications, using Brill--Noether theory to set bounds on their number.

Contribution

It introduces a geometric approach employing Brill--Noether theory to bound vector-like pairs in F-theory compactifications, linking topology and particle spectrum.

Findings

Upper bound of approximately 4 vector-like pairs for SU(5) models

Range of vector-like pairs compatible with current F-theory understanding

Analysis of quantisation of net chiralities in geometric compactifications

Abstract

How likely is it that there are particles in a vector-like pair of representations in low-energy spectrum, when neither symmetry nor anomaly consideration motivates their presence? We address this question in the context of supersymmetric and geometric phase compactification of F-theory and Heterotic dual. Quantisation of the number of generations (or net chiralities in more general term) is also discussed along the way. Self-dual nature of the fourth cohomology of Calabi--Yau fourfolds is essential for the latter issue, while we employ Brill--Noether theory to set upper bounds on the number $\ell$ of vector-like pairs of chiral multiplets in the SU(5) 5+5bar representations. For typical topological choices of geometry for F-theory compactification for SU(5) unification, the range of $0 \leq \ell \lesssim 4$ for perturbative unification is not in immediate conflict with what is already understood about F-theory compactification at this moment.