WHAT IS...a Thin Group?

Kontorovich, Alex; Long, D. D.; Lubotzky, Alexander; Reid, Alan W.

doi:10.1090/noti1900

Cited by 8 publications

(1 citation statement)

References 31 publications

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“…• Γ i is of infinite index in the arithmetic subgroup M i (Z) of its Q-Zariski closure M i . In this case the monodromy group is said to be thin [97].…”

Section: Jhep09(2020)147mentioning

confidence: 99%

Special geometry and the swampland

Cecotti

2020

J. High Energ. Phys.

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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 $$ \mathcal{N} $$ 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 quantum-consistent $$ \mathcal{N} $$ N = 2 supergravity with a cubic pre-potential is necessarily a truncation of a higher-$$ \mathcal{N} $$ N 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 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 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.

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“…• Γ i is of infinite index in the arithmetic subgroup M i (Z) of its Q-Zariski closure M i . In this case the monodromy group is said to be thin [97].…”

Section: Jhep09(2020)147mentioning

confidence: 99%