Special Geometry and the Swampland

Abstract: In the context of 4d effective gravity theories with 8 supersymmetries, we propose to unify, strenghten, and refine the several swampland conjectures into a single statement: the structural criterion, modelled on the structure theorem in Hodge theory. In its most abstract form the new swampland criterion applies to all 4d $\mathcal{N}=2$ effective theories (having a quantum-consistent UV completion) whether supersymmetry is \emph{local} or rigid: indeed it may be regarded as the more general version of Seiberg-Witten geometry which holds both in the rigid and local cases. As a first application of the new swampland criterion we show that a quantum-consistent $\mathcal{N}=2$ supergravity with a cubic pre-potential is necessarily a truncation of a higher-$\mathcal{N}$ \textsc{sugra}. More precisely: its moduli space is a Shimura variety of `magic' type. In all other cases a quantum-consistent special K\"ahler geometry is either an arithmetic quotient of the complex hyperbolic space $SU(1,m)/U(m)$ or has no \emph{local} Killing vector. Applied to Calabi-Yau 3-folds this result implies (assuming mirror symmetry) the validity of the Oguiso-Sakurai conjecture in Algebraic Geometry: all Calabi-Yau 3-folds $X$ without rational curves have Picard number $\rho=2,3$; in facts they are finite quotients of Abelian varieties. More generally: the K\"ahler moduli of $X$ do not receive quantum corrections if and only if $X$ has infinite fundamental group. In all other cases the K\"ahler moduli have instanton corrections in (essentially) all possible degrees.