On Flux Quantization in F-Theory

TL;DR

This paper proves that four-form flux in smooth elliptically fibered Calabi-Yau fourfolds is always integrally quantized, with half-integral effects arising only from singularities associated with 7-branes, connecting F-theory and IIB string theory.

Contribution

It establishes the integral quantization of flux in smooth F-theory compactifications and analyzes half-integral effects due to 7-branes with explicit examples and formulas.

Findings

Flux is always integrally quantized in smooth cases.

Half-integral quantization occurs at non-spin 7-brane surfaces.

Derived formulas for D3 tadpoles in singular fourfolds.

Abstract

We study the problem of four-form flux quantization in F-theory compactifications. We prove that for smooth, elliptically fibered Calabi-Yau fourfolds with a Weierstrass representation, the flux is always integrally quantized. This implies that any possible half-integral quantization effects must come from 7-branes, i.e. from singularities of the fourfold. We subsequently analyze the quantization rule on explicit fourfolds with Sp(N) singularities, and connect our findings via Sen's limit to IIB string theory. Via direct computations we find that the four-form is half-integrally quantized whenever the corresponding 7-brane stacks wrap non-spin complex surfaces, in accordance with the perturbative Freed-Witten anomaly. Our calculations on the fourfolds are done via toric techniques, whereas in IIB we rely on Sen's tachyon condensation picture to treat bound states of branes. Finally, we give general formulae for the curvature- and flux-induced D3 tadpoles for general fourfolds with Sp(N) singularities.