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

TL;DR

This paper explores the relationship between modular curves, Tate-Shafarevich groups, and Gopakumar-Vafa invariants in the context of genus one fibered Calabi-Yau manifolds, revealing new connections via string theory and modular transformations.

Contribution

It establishes a novel correspondence between large volume limits in stringy K"ahler moduli spaces and Tate-Shafarevich groups, incorporating non-commutative resolutions and modular properties of topological string partition functions.

Findings

Identifies the stringy K"ahler moduli space with modular curves for N ≤ 5.

Relates topological string amplitudes through modular transformations and Fricke involutions.

Provides an enumerative interpretation of partition functions at irrational points for genus one fibrations with 5-sections.

Abstract

We show that the stringy K\"ahler moduli space of a generic genus one curve of degree $N$, for $N\le 5$, is the $\Gamma_1(N)$ modular curve $X_1(N)$. This implies a correspondence between the cusps of the modular curves and certain large volume limits in the stringy K\"ahler 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.