The N=1 effective action of F-theory compactifications

TL;DR

This work develops a framework to obtain the four-dimensional $\mathcal{N}=1$ effective action for F-theory on elliptically fibered Calabi–Yau fourfolds by lifting a three-dimensional M‑theory compactification and taking the F‑theory limit via a Legendre transform. It identifies two source sectors for gauge dynamics: non‑Abelian seven‑branes from resolved elliptic singularities and abelian RR‑sector $U(1)^r$ fields tied to harmonic three-forms, with holomorphic gauge couplings that depend on complex-structure moduli including open-string data. The leading actions are encoded in a three-dimensional kinetic potential $\mathbf{K}^{\rm M}$, from which the four-dimensional Kähler potential $K^{\rm F}$ and gauge-kinetic functions $f$ follow in the F‑theory limit; crucial results include $f_{G}(T)=T_{S'}$ for the seven-brane sector and $f^{\rm RR}_{\kappa\lambda}=f_{\, obrace{ extkappa extlambda}}(z)$ for RR U(1)s, with $K^{\rm F}(z,T)= -2\log \mathcal V_{\rm b} - \log \int_{\hat X_4} \Omega\wedge \bar\Omega$. The paper also treats adjoint matter and Wilson lines on seven-branes, deriving their impact on D‑terms and the Kähler potential, and discusses electromagnetic duality constraints arising from the three-dimensional theory. Overall, it provides a concrete, duality‑guided path to connect M‑theory data to the phenomenologically relevant 4D $\mathcal{N}=1$ F‑theory effective action and sets the stage for moduli stabilization and GUT model building within this framework.

Abstract

The four-dimensional N=1 effective action of F-theory compactified on a Calabi-Yau fourfold is studied by lifting a three-dimensional M-theory compactification. The lift is performed by using T-duality realized via a Legendre transform on the level of the effective action, and the application of vector-scalar duality in three dimensions. The leading order Kahler potential and gauge-kinetic coupling functions are determined. In these compactifications two sources of gauge theories are present. Space-time filling non-Abelian seven-branes arise at the singularities of the elliptic fibration of the fourfold. Their couplings are included by resolving the singular fourfold. Generically a U(1)^r gauge theory arises from the R-R bulk sector if the base of the elliptically fibered Calabi-Yau fourfold supports 2r harmonic three-forms. The gauge coupling functions depend holomorphically on the complex structure moduli of the fourfold, comprising closed and open string degrees of freedom. The four-dimensional electro-magnetic duality is studied in the three-dimensional effective theory obtained after M-theory compactification. A discussion of matter couplings transforming in the adjoint of the seven-brane gauge group is included.