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

Abstract

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.

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

“…The cusps of the modular curve are related by Fricke involutions and Γ 0 (M ) transformations and these also transform the topological string partition functions into each other. This generalizes a relation among homologically projective dual genus one fibered Calabi-Yau threefolds with 5-sections that has been found in [26]. It also raises a new question, namely what is the geometric interpretation of the different large volume limits?…”

“…Before we continue this thought, let us consider a different line of research that at first appears to be independent. If the Calabi-Yau is a smooth torus fibration with N -sections and N ≤ 5, one can show that the A-model topological string partition function admits an expansion in terms of Γ 1 (N )-Jacobi forms [24,25,26]. This can be seen as a consequence of the Γ 1 (N ) monodromy group in the stringy Kähler moduli space of the generic fiber [27,25].…”

Orders of Vanishing and U(1) Charges in F-theory

“…As an example, consider F-theory models with just a U(1) gauge group. Explicit constructions of this type have realized charges up to q = ±6 [18,19,57,24,65,67], even though indirect arguments suggest F-theory U(1) models should be able to admit charges at least as large as q = ±21 [21]. Since only a limited number of charges have actually been observed in F-theory models, one might imagine it would be difficult to make statements about arbitrary charges.…”

Analytic Periods via Twisted Symmetric Squares

“…In this section we will venture beyond the hypergeoemetric case to obtain the periods of non-toric surfaces as well. This includes all oneparamter K3 surfaces of [23] and the fiber in all examples of [52] when taking the limit z 2 → 0. Moreover, all Picard-Fuchs operators of one-parameter Fano threefolds can be computed this way.…”

“…Constructing the other cases is more involved, e.g. case D appears as a fiber in all examples of [52] when taking the limit z 2 → 0. Moreover, the sporadic solutions appear in the Picard-Fuchs systems of the 17 one-parameter Fano threefolds [53].…”