2016

DOI: 10.48550/arxiv.1607.07280

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Looijenga's weighted projective space, Tate's algorithm and Mordell-Weil Lattice in F-theory and heterotic string theory

Shun’ya Mizoguchi¹,

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Discussion1

"Codimension-one" Singularities Of the I N Models1

Unbroken E 8 E 7 Gauge Algebra1

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Cited by 9 publications

(16 citation statements)

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“…F-theory/heterotic duality is strictly formulated when the stable degeneration limit [71,79] of a K3 fibration is considered for the F-theory. There have been recent studies on stable degenerations of F-theory/heterotic duality, such as [80,81,76,22,82].…”

Section: Discussionmentioning

confidence: 99%

$\frac{1}{2}$Calabi-Yau 4-folds and four-dimensional F-theory on Calabi-Yau 4-folds with U(1) factors

Kimura

2019

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In this study, four-dimensional N = 1 F-theory models with multiple U(1) gauge group factors are constructed. A class of rational elliptic 4-folds, which we call as " 1 2 Calabi-Yau 4-folds," is introduced, and we construct the elliptically fibered 4-folds by utilizing them. This yields a novel approach for building families of elliptically fibered Calabi-Yau 4-folds with positive Mordell-Weil ranks. The introduced 1 2 Calabi-Yau 4folds possess the characteristic property wherein the sum of the ranks of the singularity type and the Mordell-Weil group is always equal to six. This interesting property enables us to construct the elliptically fibered Calabi-Yau 4-folds of various positive Mordell-Weil ranks. From one to six U(1) factors form in four-dimensional F-theory on the resulting Calabi-Yau 4-folds. We also propose the geometric condition on the base 3-fold of the built Calabi-Yau 4-folds that allows four-dimensional F-theory models that have heterotic duals to be distinguished from those that do not.

“…F-theory/heterotic duality is strictly formulated when the stable degeneration limit [71,79] of a K3 fibration is considered for the F-theory. There have been recent studies on stable degenerations of F-theory/heterotic duality, such as [80,81,76,22,82].…”

Section: Discussionmentioning

confidence: 99%

$\frac{1}{2}$Calabi-Yau 4-folds and four-dimensional F-theory on Calabi-Yau 4-folds with U(1) factors

Kimura

2019

Preprint

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In this study, four-dimensional N = 1 F-theory models with multiple U(1) gauge group factors are constructed. A class of rational elliptic 4-folds, which we call as " 1 2 Calabi-Yau 4-folds," is introduced, and we construct the elliptically fibered 4-folds by utilizing them. This yields a novel approach for building families of elliptically fibered Calabi-Yau 4-folds with positive Mordell-Weil ranks. The introduced 1 2 Calabi-Yau 4folds possess the characteristic property wherein the sum of the ranks of the singularity type and the Mordell-Weil group is always equal to six. This interesting property enables us to construct the elliptically fibered Calabi-Yau 4-folds of various positive Mordell-Weil ranks. From one to six U(1) factors form in four-dimensional F-theory on the resulting Calabi-Yau 4-folds. We also propose the geometric condition on the base 3-fold of the built Calabi-Yau 4-folds that allows four-dimensional F-theory models that have heterotic duals to be distinguished from those that do not.

“…Thus, according to the general rule [5], the zero loci of b 8,2k are the places (on the base) where a hypermultiplet transforming in 2k of A 2k−1 arises, and that of c 1,0 are where a hypermultiplet in k(2k − 1) appears. In general, a section c i,j or b i,j or whatever with a subscript (i, j) is expressed as a polynomial of degree (i − j)n + 2i in w [45], so we have (8 − 2k)n + 16 hypermultiplets in the 2k representation, and n + 2 hypermultiplets in the k(2k − 1) representation. We will focus on the singularity enhancement to D 2k at the zero loci of c 1,0 since it is this singularity enhancement that its associated conifold singularities and their transitions are closely related to the split/non-split transitions in F-theory.…”

Section: "Codimension-one" Singularities Of the I N Modelsmentioning

confidence: 99%

Non-split singularities and conifold transitions in F-theory

2021

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In F-theory, if a fiber type of an elliptic fibration involves a condition that requires an exceptional curve to split into two irreducible components, it is called "split" or "non-split" type depending on whether it is globally possible or not. In the latter case, the gauge symmetry is reduced to a non-simply-laced Lie algebra due to monodromy. We show that this split/non-split transition is, except in certain exceptional cases, a conifold transition from the resolved to the deformed side, associated with the conifold singularities emerging where the codimension-one singularity is enhanced to D 2k+2 (k ≥ 1) or E 7 . This clarifies the origin of nonlocal matter in the non-split case, which has been a mystery for many years.

“…where [t : s] is the homogeneous coordinate of the base P 1 . In the form (21), the type II * fiber is at [t : s] = [1 : 0] and the type…”

Section: Unbroken E 8 E 7 Gauge Algebramentioning

confidence: 99%

Nongeometric heterotic strings and dual F-theory with enhanced gauge groups

¹

2019

J. High Energ. Phys.

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Eight-dimensional nongeometric heterotic strings were constructed as duals of Ftheory on Λ 1,1 ⊕ E 8 ⊕ E 7 lattice polarized K3 surfaces by Malmendier and Morrison. We study the structure of the moduli space of this construction. There are special points in this space at which the ranks of the non-Abelian gauge groups on the 7-branes in F-theory are enhanced to 18. We demonstrate that the enhanced rank-18 non-Abelian gauge groups arise as a consequence of the coincident 7-branes, which deform stable degenerations on the F-theory side. This observation suggests that the non-geometric heterotic strings include nonperturbative effects of the coincident 7-branes on the Ftheory side. The gauge groups that arise at these special points in the moduli space do not allow for perturbative descriptions on the heterotic side.We also construct a family of elliptically fibered Calabi-Yau 3-folds by fibering K3 surfaces with enhanced singularities over P 1 . Highly enhanced gauge groups arise in F-theory compactifications on the resulting Calabi-Yau 3-folds. 5 Connections of lattice polarized K3 surfaces, O + (Λ 2,2 )-modular forms, and non-geometric heterotic strings were discussed in [31]. K3 surfaces with Λ 1,1 ⊕ E 7 ⊕ E 7 lattice polarization and the construction of nongeometric heterotic strings were studied in [32].6 See [42] for the construction of the Jacobians of elliptic curves.

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