Non-flat elliptic four-folds, three-form cohomology and strongly coupled theories in four dimensions

TL;DR

This paper studies non-flat elliptic Calabi-Yau four-folds, revealing their role in adding three-form cohomology and chiral singlets, and explores geometric transitions that relate to strongly coupled theories in four dimensions.

Contribution

It provides explicit constructions of non-flat fibers in elliptic four-folds and analyzes their impact on three-form cohomology and associated superconformal matter sectors.

Findings

Non-flat fibers contribute to three-form cohomology and chiral singlets.

Conifold transitions can remove non-flat fibers and alter Hodge numbers.

Birational base changes can avoid non-flat fibers, relating to tensor branch transitions.

Abstract

In this note we consider smooth elliptic Calabi-Yau four-folds whose fiber ceases to be flat over compact Riemann surfaces of genus $g$ in the base. These non-flat fibers contribute Kaehler moduli to the four-fold but also add to the three-form cohomology for $g>0$. In F-/M-theory these sectors are to be interpreted as compactifications of six/five dimensional $\mathcal{N}=(1,0)$ superconformal matter theories. The three-form cohomology leads to additional chiral singlets proportional to the dimension of five dimensional Coulomb branch of those sectors. We construct explicit examples for E-string theories as well as higher rank cases. For the E-string theories we further investigate conifold transitions that remove those non-flat fibers. First, we show how non-flat fibers can be deformed from curves down to isolated points in the base. This removes the chiral singlet of the three-forms and leads to non-perturbative four-point couplings among matter fields which can be understood as remnants of the former E-string. Alternatively, the non-flat fibers can be avoided by performing birational base changes, analogous to 6D tensor branches. For compact bases these transitions alternate all Hodge numbers but leave the Euler number invariant.