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

TL;DR

The paper develops a concrete framework to determine the exact number of charged localized massless matter in F-theory by refining 3-form gauge data through Deligne cohomology and its description via the second Chow group ${\rm CH}^2(\hat{Y}_4)$. A refined cycle map from ${\rm CH}^2$ to $H^{4}_D$ yields gauge-data classes whose intersections with matter surfaces produce line bundles $L_R$ on matter curves $C_R$, and the massless spectrum is conjectured to be counted by $H^i(C_R, L_R \otimes \sqrt{K_{C_R}})$; this general construction recovers known Type IIB expectations in abelian cases. The authors illustrate the method in a 3-generation SU(5) × U(1)_X toy model, computing the relevant cohomologies with the cohomCalg algorithm and Koszul spectral sequences to obtain the full spectrum. The work provides a practical, globally defined tool to extract localized massless spectra from non-trivial 3-form data in F-theory, with clear paths to generalizations beyond pullback fluxes and to more intricate geometries.

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.

Figures (12)

• Figure 1: Rational equivalence between the union of two lines in $\mathbb{CP}^2$ and a hyperbola. This picture is inspired by EisenbudInt.
• Figure 2: If $a + b + c \sim d +e+f \sim 2g+h$, then the pushforwards are equivalent cycles, i.e. $a^\prime + b^\prime + c^\prime \sim 3 d^\prime \sim 2g^\prime + h^\prime$. This picture is based on EisenbudInt.
• Figure 3: The $E_0$-sheet of the Koszul spectral sequence for codimension $3$.
• Figure 4: The $E_1$-sheet of the Koszul spectral sequence for codimension $3$.
• Figure 5: The $E_2$-sheet of the Koszul spectral sequence for codimension $3$.