From M-theory higher curvature terms to α' corrections in F-theory

TL;DR

This work computes higher-derivative corrections to M-theory compactifications on Calabi–Yau fourfolds, focusing on $R^4$- and $G_4^2R^3$ terms, and derives their impact on the 3d $\mathcal{N}=2$ effective action. A key result is that the Kähler potential’s functional form remains unchanged while the Kähler coordinates receive non-trivial corrections tied to the third Chern class via $\chi_\Sigma$, and that a quantum-corrected volume $\mathcal{V}$ governs these effects. Through the F-theory limit, these corrections uplift to 4d $\mathcal{N}=1$ theories, yielding $\alpha'^2$-order corrections expressed in terms of two-cycle volumes and a curve $\mathcal{C}$ in the base geometry; in the weak coupling limit these corrections have an open-string interpretation as arising from D7-brane self-intersections. A systematic algorithm and broad survey of Abelian and non-Abelian seven-brane configurations support explicit formulas for $\mathcal{C}$, strengthening the link between higher-derivative M-theory corrections and stringy open-sector effects in F-theory compactifications.

Abstract

We perform a Kaluza-Klein reduction of eleven-dimensional supergravity on a Calabi-Yau fourfold including terms quartic and cubic in the Riemann curvature and determine the induced corrections to the three-dimensional N=2 effective action. We focus on the effective Einstein-Hilbert term and the kinetic terms for vectors. Dualizing the vectors into scalars, we derive the resulting Kahler potential and complex coordinates. The classical expressions for the Kahler coordinates are non-trivially modified, while the functional form of the Kahler potential is shown to be uncorrected. For elliptically fibered Calabi-Yau fourfolds the corrections can be uplifted to a four-dimensional F-theory compactification. We argue that also the four-dimensional N=1 Kahler coordinates receive non-trivial corrections. We find a simple expression for the induced corrections for different Abelian and non-Abelian seven-brane configurations by scanning over many Calabi-Yau fourfolds with resolved singularities. The interpretation of this expression leads us to conjecture that the higher-curvature corrections correspond to α'^2 corrections that arise from open strings at the self-intersection of seven-branes.

Figures (1)

• Figure 1: Outline of the algorithm for determining $c_3$ for a wide range of examples. Note that the three diamond-shaped nodes corresponds to three nested loops and that the input to the algorithm is symbolized by the three green boxes.