The Vertical, the Horizontal and the Rest: anatomy of the middle cohomology of Calabi-Yau fourfolds and F-theory applications

TL;DR

This work analyzes the middle cohomology $H^4$ of Calabi–Yau fourfolds in F-theory, decomposing it into vertical, horizontal, and remaining subspaces $(H^4_V, H^4_H, H^4_{RM})$ to understand flux vacua, symmetry breaking, and chirality. Using mirror symmetry and toric stratification, the authors derive combinatorial formulas for the dimensions of these components in toric CY4 hypersurfaces and provide a geometric characterization of the remaining component. They demonstrate triangulation-independence of the final cohomology decomposition and illustrate the framework with concrete examples (the sextic, elliptic fourfolds with SU(5) and SO(10) along $P^2$), as well as extensions to non-toric CY4s. The paper further connects these geometric structures to the statistics of flux vacua, showing how the horizontal component governs the distribution of generations and the unification gauge group, with implications for the relative abundance of vacua supporting different GUT scenarios. Collectively, the results offer a tractable, combinatorial handle on the landscape of F-theory flux vacua via precise control of the horizontal and remaining cohomology contributions.

Abstract

The four-form field strength in F-theory compactifications on Calabi-Yau fourfolds takes its value in the middle cohomology group $H^4$. The middle cohomology is decomposed into a vertical, a horizontal and a remaining component, all three of which are present in general. We argue that a flux along the remaining or vertical component may break some symmetry, while a purely horizontal flux does not influence the unbroken part of the gauge group or the net chirality of charged matter fields. This makes the decomposition crucial to the counting of flux vacua in the context of F-theory GUTs. We use mirror symmetry to derive a combinatorial formula for the dimensions of these components applicable to any toric Calabi--Yau hypersurface, and also make a partial attempt at providing a geometric characterization of the four-cycles Poincaré dual to the remaining component of $H^4$. It is also found in general elliptic Calabi-Yau fourfolds supporting SU(5) gauge symmetry that a remaining component can be present, for example, in a form crucial to the symmetry breaking ${\rm SU}(5) \longrightarrow {\rm SU}(3)_C \times {\rm SU}(2)_L \times {\rm U}(1)_Y$. The dimension of the horizontal component is used to derive an estimate of the statistical distribution of the number of generations and the rank of 7-brane gauge groups in the landscape of F-theory flux vacua.

Figures (10)

• Figure 1: Two fine regular triangulations of a three-dimensional cube. The 'central' 3-simplex of the triangulation on the left has lattice volume 2. As there are no extra lattice points, this triangulation cannot be further refined to become unimodular. Of course the cube shown above does have a unimodular triangulation, as shown on the right. It may happen, however, that the central simplex of the triangulation shown on the left arises as a face of a polytope, in which case no unimodular triangulation can exist.
• Figure 2: $\hat{Y}_i$ as a fibration of $E^{\geq \nu_i}_{\widetilde{\Theta}_i^{[2]}}$ over the curve $Z_{\Theta_i^{[2]}}$ with $k_{\Theta_i^{[2]}}$ punctures. Divisors of a generic fibre correspond to divisors of $\hat{Y}_i$ by sweeping them over the whole base.
• Figure 3: Over specific points, the fibre degenerates and becomes reducible. All fibres are algebraically (and hence homologically) equivalent, however.
• Figure 4: (colour online) Fibre components corresponding to points $\nu_i$ contributing to $\Theta^{[3]}_v$ (left hand side) and $\Theta^{[3]}_{nv}$ (right hand side). The fibre components drawn in red, blue and green correspond to fibre components which are obtained by intersecting appropriate divisors. These components arise from one-simplices connecting $\nu_i$ to an interior point of the face $\widetilde{\Theta}^{[3]}_a$. When there is a one-simplex connecting $\nu_i$ to a point on the boundary of $\widetilde{\Theta}^{[3]}_a$, only the sum of several components of different fibres, drawn in grey, arises from an intersection of divisors (so that it should be considered vertical). Again, the two degenerate fibres on the left hand side are algebraically equivalent, as are the two degenerate fibres on the right hand side.
• Figure 5: (colour online) The lattice points at the distance-2 layer are in one-to-one correspondence with the lattice points and 1-simplices at the distance-1 layer.