On the intermediate Jacobian of M5-branes

TL;DR

This paper develops a combinatorial method to analyze the Hodge structure and intermediate Jacobian of vertical divisors in elliptic Calabi-Yau fourfolds relevant for M/F-theory compactifications, using data from type IIB orientifolds.

Contribution

It introduces a new formula for the Euler characteristic of vertical divisors, including a conjectured correction for singularities, validated through explicit examples.

Findings

The method accurately computes the intermediate Jacobian dimensions.

The Euler characteristic formula matches direct computations in tested cases.

The correction term accounts for $ ext{Z}_2$ singularities at O3-planes.

Abstract

We study Euclidean M5-branes wrapping vertical divisors in elliptic Calabi-Yau fourfold compactifications of M/F-theory that admit a Sen limit. We construct these Calabi-Yau fourfolds as elliptic fibrations over coordinate flip O3/O7 orientifolds of toric hypersurface Calabi-Yau threefolds. We devise a method to analyze the Hodge structure (and hence the dimension of the intermediate Jacobian) of vertical divisors in these fourfolds, using only the data available from a type IIB compactification on the O3/O7 Calabi-Yau orientifold. Our method utilizes simple combinatorial formulae (that we prove) for the equivariant Hodge numbers of the Calabi-Yau orientifolds and their prime toric divisors, along with a formula for the Euler characteristic of vertical divisors in the corresponding elliptic Calabi-Yau fourfold. Our formula for the Euler characteristic includes a conjectured correction term that accounts for the contributions of pointlike terminal $\mathbb{Z}_2$ singularities corresponding to perturbative O3-planes. We check our conjecture in a number of explicit examples and find perfect agreement with the results of direct computations.