String duality and modular forms

Tests of duality between heterotic strings on K3 × T2 (restricted on certain Narain moduli subspaces) and type IIA strings on K3-fibered Calabi-Yau threefolds are attempted in the weak coupling regime on the heterotic side by identifying pertinent modular forms related to the computations of string threshold corrections. Concretely we discuss in parallel the three cases associated with Calabi-Yau manifolds (A) : X(6, 2, 2, 1, 1)2−252, (B) : X(12, 8, 2, 1, 1)3−480 and (C) : X(10, 3, 3, 2, 2)4−132 on the type IIA side.