Modular curves and the refined distance conjecture

Klaewer, Daniel

doi:10.1007/jhep12(2021)088

Cited by 11 publications

(18 citation statements)

References 97 publications

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“…This is parallel to the discussion of the genus one fibrations in Section 2.1. It would be interesting to study these K3 fibrations further, see in particular [15].…”

Section: Miscellaneous Examplesmentioning

confidence: 99%

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On genus one fibered Calabi-Yau threefolds with 5-sections

Knapp,

Scheidegger,

Schimannek

2021

Preprint

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Elliptic and genus one fibered Calabi-Yau spaces play a prominent role in string theory and mathematics. In this article we discuss a class of genus one fibered Calabi-Yau threefolds with 5-sections from various perspectives. In algebraic geometry, such Calabi-Yaus can be constructed as complete intersections in Grassmannian fibrations and as Pfaffian varieties. These constructions naturally fit into the framework of homological projective duality and lead to dual pairs of Calabi-Yaus. From a physics perspective, these spaces can be realised as low-energy configurations ("phases") of gauged linear sigma models (GLSMs) with non-Abelian gauge groups, where the dual geometries arise as phases of the same GLSM. Using the modular bootstrap approach of topological string theory, one can compute all-genus Gopakumar-Vafa invariants of these Calabi-Yaus. We observe that homological projective duality acts as an element of Γ 0 (5) on the topological string partition function and the partition functions of dual geometries transform into each other. Moreover, we study the geometries from an M-/F-theory perspective. We compute the F-theory spectrum and show how the genus one-fibered Calabi-Yaus are connected to certain Calabi-Yaus in toric varieties via a series of Higgs transitions. Based on the F-theory physics, we conjecture that dual geometries are elements of the same Tate-Shafarevich group. Our analysis also leads to a classification of 5-section geometries, as well as the construction of F-theory models with charge 5 hypermultiplets.

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“…This is parallel to the discussion of the genus one fibrations in Section 2.1. It would be interesting to study these K3 fibrations further, see in particular [15].…”

Section: Miscellaneous Examplesmentioning

confidence: 99%

“…Acknowledgments: We would like to thank David Erkinger, Cesar Fierro Cota, Albrecht Klemm, Tianle Liu, Paul Oehlmann and Eric Sharpe for discussions and collaborations on related projects. We also thank Daniel Kläwer for informing us about his upcoming work [15]. E.S.…”

Section: Introductionmentioning

confidence: 99%

On genus one fibered Calabi-Yau threefolds with 5-sections

Knapp,

Scheidegger,

Schimannek

2021

Preprint

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“…One obvious extension of the examples considered here would be to consider setups where the moduli space corresponds to modular curves for different congruence subgroups of SL(2, Z). Such cases have been recently studied in the context of the refined Swampland Distance Conjecture in [60].…”

Section: Discussionmentioning

confidence: 99%

4d strings at strong coupling

Marchesano,

Wiesner

2022

Preprint

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Weakly coupled regions of 4d EFTs coupled to gravity are particularly suitable to describe the backreaction of BPS fundamental axionic strings, dubbed EFT strings, in a local patch of spacetime around their core. We study the extension of these local solutions to global ones, which implies probing regions of strong coupling and provides an estimate of the EFT string tension therein. We conjecture that for the EFT string charge generators such a global extension is always possible and yields a sub-Planckian tension. We substantiate this claim by analysing global solutions of 4d strings made up from NS5-branes wrapping Calabi-Yau threefold divisors in either type IIA or heterotic string theory. We argue that in this case the global, non-perturbative data of the backreaction can be simply encoded in terms of a GLSM describing the compactification, as we demonstrate in explicit examples.

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“…It would be very interesting to work out the details of this connection. The presence of the Γ 0 (N ) + modular curve in these examples has recently been used to calculate geodesic distances in the moduli space of the threefolds and to compare this to expectations from the so-called refined distance conjecture [107]. An analogous argument can be made using the modular curve X 1 (N ) in the moduli space of torus fibrations with N -sections and could have interesting applications in the context of the distance conjecture and more general swampland conjectures (for a recent review of the swampland program see e.g.…”

Section: Jhep02(2022)007mentioning

confidence: 99%

Modular curves, the Tate-Shafarevich group and Gopakumar-Vafa invariants with discrete charges

Schimannek

2022

J. High Energ. Phys.

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We show that the stringy Kähler moduli space of a generic genus one curve of degree N, for N ≤ 5, is the Γ1(N) modular curve X1(N). This implies a correspondence between the cusps of the modular curves and certain large volume limits in the stringy Kähler moduli spaces of genus one fibered Calabi-Yau manifolds with N-sections. Using Higgs transitions in M-theory and F-theory as well as modular properties of the topological string partition function, we identify these large volume limits with elements of the Tate-Shafarevich group of the genus one fibration. Singular elements appear in the form of non-commutative resolutions with a torsional B-field at the singularity. The topological string amplitudes that arise at the various large volume limits are related by modular transformations. In particular, we find that the topological string partition function of a smooth genus one fibered Calabi-Yau threefold is transformed into that of a non-commutative resolution of the Jacobian by a Fricke involution. In the case of Calabi-Yau threefolds, we propose an expansion of the partition functions of a singular fibration and its non-commutative resolutions in terms of Gopakumar-Vafa invariants that are associated to BPS states with discrete charges. For genus one fibrations with 5-sections, this provides an enumerative interpretation for the partition functions that arise at certain irrational points of maximally unipotent monodromy.

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Modular curves and the refined distance conjecture

Cited by 11 publications

References 97 publications

On genus one fibered Calabi-Yau threefolds with 5-sections

On genus one fibered Calabi-Yau threefolds with 5-sections

4d strings at strong coupling

Modular curves, the Tate-Shafarevich group and Gopakumar-Vafa invariants with discrete charges

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