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

TL;DR

This paper analyzes the middle cohomology of Calabi-Yau fourfolds in F-theory, decomposing it into components to understand flux effects on gauge symmetry and matter, with combinatorial formulas and implications for string theory vacua.

Contribution

It introduces a mirror symmetry-based combinatorial formula for the cohomology decomposition components in toric Calabi-Yau fourfolds and explores their role in F-theory flux vacua.

Findings

A flux along remaining or vertical components can break symmetries.

Purely horizontal flux does not affect unbroken gauge groups or chirality.

The horizontal component's dimension helps estimate flux vacua distributions.

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 Poincar\'e 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.