Special geometry and the swampland
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Springer
Received: April 30, 2020 Accepted: August 25, 2020 Published: September 22, 2020
Special geometry and the swampland
SISSA, via Bonomea 265, I-34100 Trieste, Italy
E-mail: [email protected] 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 N = 2 effective theories (having a quantum-consistent UV completion) whether supersymmetry is 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 quantumconsistent N = 2 supergravity with a cubic pre-potential is necessarily a truncation of a higher-N 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 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 ρ = 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. Keywords: Differential and Algebraic Geometry, Effective Field Theories, Supergravity Models ArXiv ePrint: 2004.06929
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP09(2020)147
JHEP09(2020)147
Sergio Cecotti
Contents
2 Review of special K¨ ahler geometry 2.1 Basic geometric structures 2.2 Special K¨ ahler symmetries
14 15 22
3 Warm-up: two weak results in the DG paradigm 3.1 A restricted class of special K¨ahler geometries 3.2 The quantum corrections cannot be trivial 3.3 Symmetric rank-3 tube domains
24 25 28 30
4 The 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8
swampland structural criterion The rough physical idea Intuitive view: tt∗ solitons and brane rigidity The structure theorem of algebro-geometric VHS The case of “motivic” special K¨ahler geometries The structural swampland criterion A direct physical proof? Recovering the Ooguri-Vafa swampland statements More on “no free parameter”
31 31 32 36 39 40 40 41 47
5 Functional equations for quantum-consistent F ’s
48
6 Dicothomy 6.1 Proof of dicothomy 6.2 Existence of quantum-consistent symmetric geometries
51 51 53
7 Swampland criterion mirror symmetry 7.1 Necessity of non-perturbative corrections 7.2 Non-empty instanton-charge sectors 7.3 Applications to Type IIA compactifications
54 59 60 61
8 Answering the Oguiso-Sakurai question 8.1 Th
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