A Note on $h^{2,1}$ of Divisors of CY Fourfolds I

TL;DR

This paper derives combinatorial formulas for the Hodge number $h^{2,1}$ of prime toric divisors in Calabi-Yau fourfolds and demonstrates the existence of hypersurfaces with more non-perturbative superpotential terms than the divisor's $h^{1,1}$.

Contribution

It provides new combinatorial formulas for $h^{2,1}$ of divisors in toric hypersurface Calabi-Yau fourfolds and explores the implications for superpotential terms.

Findings

Formulas for $h^{2,1}$ of prime toric divisors.

Existence of Calabi-Yau hypersurfaces with more superpotential terms than $h^{1,1}$.

Analysis of Hodge numbers of divisors in toric CICYs in part two.

Abstract

In this note, we prove combinatorial formulas for $h^{2,1}$ of prime toric divisors in an arbitrary toric hypersurface Calabi-Yau fourfold $Y_4.$ We show that it is possible to find a toric hypersurface Calabi-Yau in which there are more than $h^{1,1}(Y_4)$ non-perturbative superpotential terms with trivial intermediate Jacobian. Hodge numbers of divisors in toric CICYs are the subjects of the part two.