Lectures on the Ax–Schanuel conjecture

Bakker, Benjamin; Tsimerman, Jacob

doi:10.1007/978-3-030-49864-1_1

Cited by 5 publications

(11 citation statements)

References 41 publications

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“…For the type II string such theories can be obtained by compactifying on Calabi-Yau three-folds in the presence of fluxes, where the choice of fluxes is constrained by the tadpole cancellation condition (for a review see for instance [5]). Remarkably, using this tadpole condition it has been argued that for a given compactification manifold the number of flux vacua is finite [6,7].…”

Section: Introductionmentioning

confidence: 99%

The tadpole conjecture at large complex-structure

Plauschinn

2022

J. High Energ. Phys.

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The tadpole conjecture by Bena, Blåbäck, Graña and Lüst effectively states that for string-theory compactifications with a large number of complex-structure moduli, not all of these moduli can be stabilized by fluxes. In this note we study this conjecture in the large complex-structure regime using statistical data obtained by Demirtas, Long, McAllister and Stillman for the Kreuzer-Skarke list. We estimate a lower bound on the flux number in type IIB Calabi-Yau orientifold compactifications at large complex-structure and for large h2,1, and our results support the tadpole conjecture in this regime.

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Section: Introductionmentioning

confidence: 99%

The tadpole conjecture at large complex-structure

Plauschinn

2022

J. High Energ. Phys.

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“…These are motivated by Lawrence-Venkatesh's recent breakthrough on the non-density of integral points on the moduli space of hypersurfaces [61], and are in accordance with Lang-Vojta's conjecture for affine varieties. Our results in this section are proven using methods from Hodge theory, and are loosely related to Bakker-Tsimerman's chapter in this book [11].…”

Section: Introductionmentioning

confidence: 57%

“…Namely, in [50] it is shown that a complex algebraic variety with a quasi-finite period map is geometrically hyperbolic. For other results about period domains we refer the reader to the article of Bakker-Tsimerman in this book [11].…”

Section: Hyperbolicity Along Field Extensionsmentioning

confidence: 99%

The Lang-Vojta conjectures on projective pseudo-hyperbolic varieties

Javanpeykar¹

2020

Preprint

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“…Let us also comment on the linear scenario in the context of the general finiteness theorems for flux vacua satisfying the self-duality condition [38,63]. A first observation is that the tadpole N flux = −e L m L seems to be independent of the flux e α .…”

Section: Relations To Swampland Conjectures and Finiteness Theoremmentioning

confidence: 99%

“…This implies that if one aims to describe the structure of all flux vacua one has to leave the world of algebraic geometry. Remarkably, a delicate and powerful extension of algebraic geometry that can be use to describe flux vacua is provided by using tame topology and o-minimality [63]. The resulting tame form of geometry manifest the notion of finiteness and removes many pathologies that are allowed in geometric settings based on ordinary topology.…”

Section: Relations To Swampland Conjectures and Finiteness Theoremmentioning

confidence: 99%

Moduli Stabilization in Asymptotic Flux Compactifications

Grimm,

Plauschinn,

van de Heisteeg

2021

Preprint

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We present a novel strategy to systematically study complex-structure moduli stabilization in Type IIB and F-theory flux compactifications. In particular, we determine vacua in any asymptotic regime of the complex-structure moduli space by exploiting powerful tools of asymptotic Hodge theory. In a leading approximation the moduli dependence of the vacuum conditions are shown to be polynomial with a dependence given by sl(2)-weights of the fluxes. This simple algebraic dependence can be extracted in any asymptotic regime, even though in nearly all asymptotic regimes essential exponential corrections have to be present for consistency. We give a pedagogical introduction to the sl(2)-approximation as well as a detailed step-by-step procedure for constructing the corresponding Hodge star operator. To exemplify the construction, we present a detailed analysis of several Calabi-Yau three-and fourfold examples. For these examples we illustrate that the vacua in the sl(2)-approximation match the vacua obtained with all polynomial and essential exponential corrections rather well, and we determine the behaviour of the tadpole contribution of the fluxes. Finally, we discuss the structure of vacuum loci and their relations to several swampland conjectures.In particular, we comment on the realization of the so-called linear scenario in view of the tadpole conjecture.

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Lectures on the Ax–Schanuel conjecture

Cited by 5 publications

References 41 publications

The tadpole conjecture at large complex-structure

The tadpole conjecture at large complex-structure

The Lang-Vojta conjectures on projective pseudo-hyperbolic varieties

Moduli Stabilization in Asymptotic Flux Compactifications

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