Moduli spaces of Calabi-Yau dd-folds as gravitational-chiral instantons

Motivated by the swampland program, we show that the Weil-Petersson geometry of the moduli space of a Calabi-Yau manifold of complex dimension $d\leq4$ is a gravitational instanton (i.e. a finite-action solution of the Euclidean equations of motion of gravity with matter). More precisely, the moduli geometry of Calabi-Yau $d$-folds ($d\leq4$) describes instantons of (E)AdS Einstein gravity coupled to a standard chiral model. From the point of view of the low-energy physics of string/M-theory compactified on the Calabi-Yau $X$, the various fields propagating on its moduli space are the couplings appearing in the effective Lagrangian $\mathscr{L}_\text{eff}$.Comment: 18 pages; more details on the geometry and finite actio

Fuchsian ODEs as Seiberg dualities

The classical theory of Fuchsian differential equations is largely equivalent to the theory of Seiberg dualities for quiver SUSY gauge theories. In particular: all known integral representations of solutions, and their connection formulae, are immediate consequences of (analytically continued) Seiberg duality in view of the dictionary between linear ODEs and gauge theories with 4 supersymmetries. The purpose of this divertissement is to explain "physically'' this remarkable relation in the spirit of Physical Mathematics. The connection goes through a "mirror-theoretic'' identification of irreducible logarithmic connections on $\mathbb{P}^1$ with would-be BPS dyons of 4d $\mathcal{N}=2$ $SU(2)$ SYM coupled to a certain Argyres-Douglas "matter''. When the underlying bundle is trivial, i.e. the log-connection is a Fuchs system, the world-line theory of the dyon simplifies and the action of Seiberg duality on the Fuchsian ODEs becomes quite explicit. The duality action is best described in terms of Representation Theory of Kac-Moody Lie algebras (and their affinizations).Comment: 51 pages; clarifications required by the refere

Special Geometry and the Swampland

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.Comment: 94 pages, 2 figure