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

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.

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

“…It is natural to ask, if this is part of a more general set of modular relations that connects all of the geometries in the Tate-Shafarevich group. This question will be addressed in a follow up paper and it turns out that such a structure indeed exist [150].…”

On genus-0 invariants of Calabi-Yau hybrid models

“…While one may not expect a connection to integer invariants in all cases, there are special examples of hybrids, like the model M1 discussed in Section 4.2 that behave "almost" like geometric models in large volume phases and can be interpreted in terms of branched double covers. Further examples of this type have also been found in [42] and in the context on non-abelian GLSMs [43]. It would be interesting to have a physics interpretation of the integer invariants.…”

The discrete Green-Schwarz mechanism in 6D F-theory and elliptic genera of non-critical strings

“…For our purpose it will prove beneficial to approach the topic of discrete symmetries intrinsically, without invoking an un-Higgsing. This will also allow us to connect the discussion to recent insights on the role of non-commutative resolutions of singular torus fibrations in the duality to Type IIA compactifications discovered in [45], which will be important later on. To begin, let us first discuss the effect of discrete holonomies in circle compactifications of 6d supergravity.…”

“…The correlation between physical properties, such as anomalies, and geometry is wellexplored in the case of continuous symmetries [18,[26][27][28], but is far from being understood in the context of discrete symmetries [29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45]. The present work provides some of the missing entries in the F-theory dictionary associated to the cancellation of discrete gauge anomalies, which in turn enables an extension of the Swampland Program to discrete gauge data, shedding light on more general consistency constraints of supergravities that are part of the Landscape.…”

“…Using recent results from [45], we first argue that the 6d theory compactified on a 2-torus together with a choice of discrete holonomies (g, h) along the cycles is dual to Type IIA string theory on different torus fibrations X g that are equipped with a flat torsional B-field labelled by h. The twisted twined elliptic genera are then encoded in the corresponding A-model topological string partition functions. Using the modular properties of the latter, that have been worked out in [42,45,56], we can then derive a geometric expression for the 't Hooft anomaly in terms of the height-pairing of a multi-section on the genus one fibered Calabi-Yau. Applying this to our family of Calabi-Yau manifolds, we find that there are indeed different anomalies for the pairs of inequivalent theories that exhibit the same massless spectra and therefore the "degeneracy" is lifted.…”