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

Understand

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.