Modularity of Calabi–Yau Varieties: 2011 and Beyond

Yui, Noriko

doi:10.1007/978-1-4614-6403-7_4

Cited by 11 publications

(10 citation statements)

References 61 publications

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“…First, in Section 2.1, we will review the famous modularity theorem for elliptic curves, following [18]. Next, in Section 2.2, we will discuss recent progress in the study of modularity for threefolds, following [19,20]. Finally, in Section 2.3 we will provide an alternative perspective on these results that will be essential later.…”

Section: The Modularity Of Calabi-yau Varietiesmentioning

confidence: 99%

“…More precisely, if ρ 2 (X) is of Hodge type (3, 0) + (0, 3), then the situation is identical to the rigid case, and ρ 2 (X) is associated to a weight-four eigenform; many examples of such a split can be found in e.g. [19,20]. It was pointed out in [15] that such splits can be related to rank-two attractor points in Calabi-Yau moduli space.…”

Section: Modularity For Calabi-yau Threefoldsmentioning

confidence: 99%

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Supersymmetric Flux Compactifications and Calabi-Yau Modularity

Kachru,

Nally,

Yang

2020

Preprint

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Flux compactification of IIB string theory associates special points in Calabi-Yau moduli space to choices of (pairs of) integral three-form fluxes. In this paper, we propose that supersymmetric flux vacua are modular. That is, to a supersymmetric flux vacuum arising in a variety defined over Q, we associate a two-dimensional Galois representation that we conjecture to be modular. We provide numerical evidence for our conjecture by examining flux vacua arising on the octic hypersurface in P 4 (1, 1, 2, 2, 2).

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Section: The Modularity Of Calabi-yau Varietiesmentioning

confidence: 99%

Section: Modularity For Calabi-yau Threefoldsmentioning

confidence: 99%

Supersymmetric Flux Compactifications and Calabi-Yau Modularity

Kachru,

Nally,

Yang

2020

Preprint

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“…The case k = 4 is similarly related to rigid Calabi-Yau three-folds [GY11] and examples have been systematically pursued [Mey05]; it seems to us premature to speculate whether # 4 is infinite or finite. The cases k ≥ 6 have been studied [Yui13,PR15], but there do not seem to be any systematic non-modular sources: this lack of sources contributes to our expectation of finiteness for these # k .…”

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confidence: 90%

Newforms with rational coefficients

Roberts

2018

Ramanujan J

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We consider the set of classical newforms with rational coefficients and no complex multiplication. We study the distribution of quadratic-twist classes of these forms with respect to weight k and minimal level N . We conjecture that for each weight k ≥ 6, there are only finitely many classes. In large weights, we make this conjecture effective: in weights 18 ≤ k ≤ 24, all classes have N ≤ 30, in weights 26 ≤ k ≤ 50, all classes have N ∈ {2, 6}, and in weights k ≥ 52, there are no classes at all. We study some of the newforms appearing on our conjecturally complete list in more detail, especially in the cases N = 2, 3, 4, 6, and 8, where formulas can be kept nearly as simple as those for the classical case N = 1.

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“…One of the crowning achievements of twentieth century mathematics was the modularity theorem [1][2][3][4][5], which implies that every elliptic curve defined over Q can be associated to a special automorphic form called a weight-two eigenform. More recently, the modularity program has been extended to higher-dimensional Calabi-Yau (CY) varieties, including CY threefolds (CY3s) [6,7]. For example, it is known that rigid threefolds defined over Q are modular [8].…”

Section: Introductionmentioning

confidence: 99%

Flux Modularity, F-Theory, and Rational Models

Kachru¹,

Nally²,

Yang³

2020

Preprint

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In recent work, we conjectured that Calabi-Yau threefolds defined over Q and admitting a supersymmetric flux compactification are modular, and associated to (the Tate twists of) weighttwo cuspidal Hecke eigenforms. In this work, we will address two natural follow-up questions, of both a physical and mathematical nature, that are surprisingly closely related. First, in passing from a complex manifold to a rational variety, as we must do to study modularity, we are implicitly choosing a "rational model" for the threefold; how do different choices of rational model affect our results? Second, the same modular forms are associated to elliptic curves over Q; are these elliptic curves found anywhere in the physical setup? By studying the F-theory uplift of the supersymmetric flux vacua found in the compactification of IIB string theory on (the mirror of) the Calabi-Yau hypersurface X in P(1, 1, 2, 2, 2), we find a one-parameter family of elliptic curves whose associated eigenforms exactly match those associated to X. Actually, we find two such families, corresponding to two different choices of rational models for the same family of Calabi-Yaus.

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Modularity of Calabi–Yau Varieties: 2011 and Beyond

Cited by 11 publications

References 61 publications

Supersymmetric Flux Compactifications and Calabi-Yau Modularity

Supersymmetric Flux Compactifications and Calabi-Yau Modularity

Newforms with rational coefficients

Flux Modularity, F-Theory, and Rational Models

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