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 Kähler 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 Kähler moduli of $X$ do not receive quantum corrections if and only if $X$ has infinite fundamental group. In all other cases the Kähler moduli have instanton corrections in (essentially) all possible degrees.

94 pages, 2 figures

References in corpus (10)

• On the Geometry of the String Landscape and the Swampland
• Symmetries and Strings in Field Theory and Gravity
• The Swampland: Introduction and Review
• The String Landscape and the Swampland
• Cobordism Classes and the Swampland
• Supersymmetric Protection and the Swampland
• Maximal automorphisms of Calabi-Yau manifolds versus maximally unipotent monodromy
• Variations of Hodge Structure Considered as an Exterior Differential System: Old and New Results
• On the Hodge Metric of the Universal Deformation Space of Calabi-Yau Threefolds
• Weil-Petersson geometry on moduli space of polarized Calabi-Yau manifolds

Cited by in corpus (21)

• Merging the Weak Gravity and Distance Conjectures Using BPS Extremal Black Holes
• Swampland Conjectures for Strings and Membranes
• Cobordism Conjecture, Anomalies, and the String Lamppost Principle
• The Convex Hull Swampland Distance Conjecture and Bounds on Non-geodesics
• Supersymmetric Protection and the Swampland
• Moduli Space Holography and the Finiteness of Flux Vacua
• Weak Gravity Bounds in Asymptotic String Compactifications
• On Higher-Spin Points and Infinite Distances in Conformal Manifolds
• On TCS $G_2$ manifolds and 4D Emergent Strings
• Bulk Reconstruction in Moduli Space Holography
• Revisiting the Scalar Weak Gravity Conjecture
• 4d strings at strong coupling
• Moduli spaces of Calabi-Yau $d$-folds as gravitational-chiral instantons
• Black Holes as Probes of Moduli Space Geometry
• Swampland geometry and the gauge couplings
• A CFT Distance Conjecture
• Deformed WZW Models and Hodge Theory -- Part I
• Hitchhiker's Guide to the Swampland: The Cosmologist's Handbook to the string-theoretical Swampland Programme
• Stability of BPS States and Weak Coupling Limits
• S-duality and the universal isometries of instanton corrected q-map spaces
• A Perfectoid Duality Between M-Theory and F-Theory