Modular curves and the refined distance conjecture

Daniel Kläwer

doi:10.1007/jhep12(2021)088

Abstract

We test the refined distance conjecture in the vector multiplet moduli space of 4D mathcal{N} = 2 compactifications of the type IIA string that admit a dual heterotic description. In the weakly coupled regime of the heterotic string, the moduli space geometry is governed by the perturbative heterotic dualities, which allows for exact computations. This is reflected in the type IIA frame through the existence of a K3 fibration. We identify the degree d = 2N of the K3 fiber as a parameter that could potentially lead to large distances, which is substantiated by studying several explicit models. The moduli space geometry degenerates into the modular curve for the congruence subgroup Γ0(N)+. In order to probe the large N regime, we initiate the study of Calabi-Yau threefolds fibered by general degree d > 8 K3 surfaces by suggesting a construction as complete intersections in Grassmann bundles.

Highlights

Closer to our usual physical intuition but still central in the swampland program are the weak gravity [22] and distance [2] conjectures
In the weakly coupled regime of the heterotic string, the moduli space geometry is governed by the perturbative heterotic dualities, which allows for exact computations
We have studied the diameter of a certain hybrid phase in the moduli space of type IIA compactified on K3 fibered Calabi-Yau threefolds

Summary

Refined distance conjecture for simple K3 fibrations

The effective moduli space for t1 degenerates into the fundamental domain H/SL(2, Z) as t2 → ∞ Given this information, let us reproduce the result (2.7) for the asymptotic diameter of the P1 hybrid phase. As a first example of a heterotic/IIA dual pair that realizes a non-trivial modular curve in its moduli space we will consider the resolution X of the degree eight hypersurface P411222[8] with Hodge numbers (h11, h21) = (2, 86). The previous two examples have shown that in the large base limit the diameter ∆φc of the P1 hybrid phase is dictated by the duality group acting on the Kähler modulus of the K3 fiber, which was either SL(2, Z) or the congruence subgroup Γ0(N )+. Additional cusps will generally appear if N is not a prime number

RDC for CY threefolds fibered by degree 2N K3 surfaces

Violating the refined distance conjecture?

Discussion and outlook

C K3 surfaces and fibrations

Findings

D Heterotic duals