Towards the Non-Perturbative Completion of 4d N=1 Effective Theories of Gravity

TL;DR

This work argues that 4d $\mathcal{N}=1$ effective theories of gravity from string compactifications require a non-perturbative completion, particularly in small-volume regimes where locally enhanced SUSY sectors reveal light states missing in perturbative descriptions. By analyzing subsectors with enhanced SUSY in F-theory on elliptically fibered Calabi–Yau fourfolds and their heterotic duals, the authors identify blow-up modes and D3-brane string states that complete $\mathcal{N}=2$ multiplets into the full local spectrum, while clarifying how global gravity couples these sectors and imposes tadpole and SUSY-breaking constraints. The paper develops a cohesive picture linking Kähler and complex-structure moduli sectors through birational flop transitions, flux vacua, and NS5-brane dynamics, showing that non-perturbative physics is essential for consistency in 4d $\mathcal{N}=1$ theories. The results offer a non-perturbative organizing principle for moduli spaces, domain walls, and the Swampland in string theory, with implications for how gravity couples to locally enhanced SUSY sectors and for constructing globally consistent 4d EFTs.

Abstract

We show that four-dimensional $\mathcal N=1$ effective theories of gravity obtained from string compactifications require a non-perturbative completion, as additional light states of non-perturbative origin must be incorporated in the small volume regime to obtain a consistent low-energy description. This completion becomes concrete in subsectors that locally exhibit enhanced supersymmetry, where the enhancement predicts the existence of additional light degrees of freedom absent in the perturbative description. Motivated by analogous setups in six-dimensional $\mathcal N=(1,0)$ theories, we focus on the Kähler moduli space of F-theory compactifications on Calabi--Yau fourfolds to four dimensions, where shrinkable curves not intersected by 7-planes realize such supersymmetry-enhanced subsectors. Guided by the enhanced supersymmetry, we use F-theory to identify the degrees of freedom missing in the small volume regime of the perturbative Type IIB description. A consistent embedding of these local subsectors into a four-dimensional $\mathcal N=1$ theory of gravity requires an appropriate inclusion of complex structure moduli and spacetime-filling D3-branes. We also discuss supersymmetry enhancement in the complex structure sector and study how a heterotic dual description gives a unifying picture of the different F-theory sectors with enhanced supersymmetry. Finally, we comment on cases without local supersymmetry enhancement.

Figures (4)

• Figure 1: The conifold singularity of the 4d $\mathcal{N}=2$ theory separates the Higgs from the Coulomb branch. The Higgs branch is parameterized by the volume of $\tilde{C}_0$, while the Coulomb branch is parameterized by the volume of $\mathcal{A}$. In the Higgs branch, a massive photon arises by reducing $C_4$ over the chain $\Sigma_3$. In the Coulomb branch, the same amount of bosonic degrees of freedom arises from a charged hypermultiplet corresponding to D3-brane wrapping $\mathcal{A}$, plus the dimensional reduction of $C_4$ over $\mathcal{B}$, yielding a massless gauge boson.
• Figure 2: Schematic representation of the birational factorization of a flop transition. The left diagram corresponds to the 4-fold $X_4$ containing the flop curve $C_0$, whereas the right one corresponds to $\tilde{X}_4$ after a flop transition. The diagram in the middle shows the 4-fold $\hat{X}_4$ for which the curve $C_0$ has been blown up into an extra exceptional divisor $E$.
• Figure 3: A sketch of the phase structure of $\mathcal{M}_{\widehat{X}_4}$. The Phase \ref{['phaseI']} corresponds to the classical geometric description of F-theory on $\widehat{X}_4$. Phases \ref{['phaseII,III']} correspond to a regime in which the volumes of the cycles $C'$ and $C"$ differ hierarchically. The deep interiror of these phases can respectively be identified with the moduli space of the theory compactified on $\mathcal{M}_{X_4}$ and $\mathcal{M}_{\check{X}_4}$. Phase \ref{['phaseIV']} corresponds to a small-volume regime for both curves and marks the transition into the full non-perturbative regime. The orange line represents the locus along which the mass of $\chi_+$ (or $\check\chi_+$) vanishes, as discussed in Section \ref{['sec: phase iv']}.
• Figure 4: A sketch of the fibration structure of the Calabi Yau 4-fold $\widehat{X}_4$ considered in this chapter $T^2\to \mathbb{P}^1_F \to B_2 =\mathbb{F}_1$. The fiber $\mathbb{P}^1_f$ degenerates over $C'$ in the union of two rational curves intersecting over a point corresponding to a blow-up of the base curve of $\mathbb{F}_1$.