Chow groups, Deligne cohomology and massless matter in F-theory

TL;DR

This paper introduces a novel method using Deligne cohomology and Chow groups to precisely compute the number of massless matter states in F-theory compactifications, linking algebraic geometry with string theory spectra.

Contribution

It develops a new approach combining Deligne cohomology, Chow groups, and intersection theory to determine charged matter spectra in F-theory models.

Findings

Successfully computed massless spectrum in an SU(5) x U(1) model

Established a conjectural link between cohomology groups and massless states

Demonstrated the method with explicit cohomology calculations using cohomCalg

Abstract

We propose a method to compute the exact number of charged localized massless matter states in an F-theory compactification on a Calabi-Yau 4-fold with non-trivial 3-form data. Our starting point is the description of the 3-form data via Deligne cohomology. A refined cycle map allows us to specify concrete elements therein in terms of the second Chow group of the 4-fold, i.e. rational equivalence classes of algebraic 2-cycles. We use intersection theory within the Chow ring to extract from this data a line bundle class on the curves in the base of the fibration on which charged matter is localized. The associated cohomology groups are conjectured to count the exact massless spectrum, in agreement with general patterns in Type IIB compactifications with 7-branes. We exemplify our approach by calculating the massless spectrum in an SU(5) x U(1) toy model based on an elliptic 4-fold with an extra section. The explicit evaluation of the cohomology classes is performed with the help of the cohomCalg-algorithm by Blumenhagen et al.