The Type IIA Flux Potential, 4-forms and Freed-Witten anomalies

TL;DR

This paper derives the full classical scalar potential for type IIA Calabi-Yau orientifolds with fluxes and D6-branes, revealing a bilinear structure linked to Freed-Witten anomalies and providing a universal polynomial framework.

Contribution

It introduces a bilinear form for the scalar potential expressed through flux-dependent invariants derived from Freed-Witten anomalies, unifying the superpotential with a master polynomial.

Findings

Scalar potential expressed as a bilinear form $V = Z^{AB} ho_A ho_B$.

$ ho_A$ invariants depend on fluxes, axions, and topological data.

The superpotential is uniquely determined by a universal polynomial.

Abstract

We compute the full classical 4d scalar potential of type IIA Calabi-Yau orientifolds in the presence of fluxes and D6-branes. We show that it can be written as a bilinear form $V = Z^{AB} \rho_A\rho_B$, where the $\rho_A$ are in one-to-one correspondence with the 4-form fluxes of the 4d effective theory. The $\rho_A$ only depend on the internal fluxes, the axions and the topological data of the compactification, and are fully determined by the Freed-Witten anomalies of branes that appear as 4d string defects. The quadratic form $Z^{AB}$ only depends on the saxionic partners of these axions. In general, the $\rho_A$ can be seen as the basic invariants under the discrete shift symmetries of the 4d effective theory, and therefore the building blocks of any flux-dependent quantity. All these polynomials may be obtained by derivation from one of them, associated to a universal 4-form. The standard N=1 supergravity flux superpotential is uniquely determined from this {\it master polynomial}, and vice versa.