Abstract
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.
Highlights
Compactifications on torus fibered Calabi-Yau manifolds play a pivotal role in the network of string dualities
A striking consequence is, that the cusps of the modular curve are in one-to-one correspondence with the large volume limits that appear in the large base limit of the fibrations
The cusps of the modular curve are related by Fricke involutions and Γ0(M ) transformations and we find that these transform the corresponding local expansions of the topological string partition functions into each other
Summary
Compactifications on torus fibered Calabi-Yau manifolds play a pivotal role in the network of string dualities. 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 Γ1(N )-Jacobi forms [24,25,26] This can be seen as a consequence of the Γ1(N ) monodromy group in the stringy Kähler moduli space of the generic fiber [25, 27].
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