The making of Calabi‐Yau spaces: Beyond toric hypersurfaces

Kreuzer, Maximilian

doi:10.1002/prop.200900053

Cited by 4 publications

(3 citation statements)

References 33 publications

(49 reference statements)

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“…We make the assumption that the element s 9 ∈ H 0 (X Σ(T ) , K X Σ(T ) ) is generic. In [20], it was argued that the set V (x i , s 9 ) is either empty, consists of a single irreducible component, or is a finite collection of smooth P 1 s (further background on this topic can be found in [11,40,76,87,91]). The number of intersection points among the irreducible components of C • (3,2) 1/6 is equal to their topological intersection number, which in turn depends only on ∆ • [20].…”

Section: Frst-invariant Graphsmentioning

confidence: 99%

Root bundles: Applications to F-theory Standard Models

Bies¹

2023

Preprint

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The study of vector-like spectra in 4-dimensional F-theory compactifications involves root bundles, which are important for understanding the Quadrillion F-theory Standard Models (F-theory QSMs) and their potential implications in physics. Recent studies focused on a superset of physical root bundles whose cohomologies encode the vector-like spectra for certain matter representations. It was found that more than 99.995% of the roots in this superset for the family B 3 (∆ • 4 ) of O(10 11 ) different F-theory QSM geometries had no vector-like exotics, indicating that this scenario is highly likely.To study the vector-like spectra, the matter curves in the F-theory QSMs were analyzed. It was found that each of them can be deformed to nodal curve that is identical across all spaces in B 3 (∆ • ). Therefore, from studying a few nodal curves, one can probe the vector-like spectra of a large fraction of F-theory QSMs. To this end, the cohomologies of all limit roots were determined, with line bundle cohomology on rational nodal curves playing a major role. A computer algorithm was used to enumerate all limit roots and analyze the global sections of all tree-like limit roots. For the remaining circuit-like limit roots, the global sections were manually determined. These results were organized into tables, which represent -to the best knowledge of the authorthe first arithmetic steps towards Brill-Noether theory of limit roots.

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Section: Frst-invariant Graphsmentioning

confidence: 99%

Root bundles: Applications to F-theory Standard Models

Bies¹

2023

Preprint

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“…poly.x has been extended with several features that include information about the facets of the polytope, data of Fano varieties and conifold Calabi-Yaus. In [51,52] this extension of PALP has been used to find new Calabi-Yau manifolds with small h 1,1 which are obtained from known Calabi-Yau threefolds via conifold transitions. The full set of options in PALP can be obtained with poly.x -h and poly.x -x for extended options.…”

Section: Toric Calculations Using Palp and Other Softwarementioning

confidence: 99%

Toric Methods in F-Theory Model Building

Knapp

Kreuzer

2011

Advances in High Energy Physics

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In this review article we discuss recent constructions of global F-theory GUT models and explain how to make use of toric geometry to do calculations within this framework. After introducing the basic properties of global F-theory GUTs we give a self-contained review of toric geometry and introduce all the tools that are necessary to construct and analyze global F-theory models. We will explain how to systematically obtain a large class of compact Calabi-Yau fourfolds which can support F-theory GUTs by using the software package PALP. 1

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“…On the subject of F-theory there were papers with Knapp and co-workers [26][27][28]. Work on extending the constructions of Calabi-Yau manifolds beyond hypersurfaces in toric varieties included a consideration of complete intersections in toric varieties [29,30] and work with Batyrev [31,32] on a construction of Calabi-Yau threefolds via conifold transition from varieties described by reflexive polyhedra.…”

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confidence: 99%