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

TL;DR

This work recasts the classical Type IIA flux potential on Calabi–Yau orientifolds with D6-branes as a bilinear in four-dimensional four-forms, $V = \frac{1}{8\kappa_4^2} Z^{AB} \varrho_A \varrho_B$, where the $\varrho_A$ are axion–flux polynomials determined by flux quanta, axions, and Freed-Witten anomalies. The saxion dependence is captured by a saxion-only matrix $M$ with an axion-induced rotation $R$ so that $Z = R^t M R$ and $\varrho_A = (R^t)^{-1} q$, revealing a triple factorisation into saxions, axions, and flux quanta. The key axion structure is generated from a master polynomial $\rho_0$, with all other $\rho_A$ obtained by derivatives with respect to axions, and the standard $\cal N}=1$ flux superpotential $W$ derived from $\rho_0$ via $W = e^{i s^\lambda \partial_{\phi^\lambda}} \rho_0$. Freed-Witten anomalies and their associated 4d domain walls provide a microscopic origin for the axion-dependent rotation $R$, and the formalism naturally incorporates open-string moduli, showing how D6-brane data enter the same bilinear framework. The results offer a transparent handle on axion shift symmetries, vacua search, and potential generalisations to broader flux landscapes and non-geometric settings, with implications for moduli stabilization and inflation model-building in string theory.

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 $V = Z^{AB} ρ_Aρ_B$, where the $ρ_A$ are in one-to-one correspondence with the 4-form fluxes of the 4d effective theory. The $ρ_A$ 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 $ρ_A$ 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.

Figures (1)

• Figure 1: Hanany-Witten effect Hanany:1996ie: when an NS5-brane crosses $M$ D($p$+2)-branes, $M$ D$p$-branes will appear stretching between the two.