Revisiting the quantum geometry of torus-fibered Calabi-Yau threefolds

About ten years ago, Katz, Klemm and Huang conjectured that topological string amplitudes on compact, elliptically fibered Calabi-Yau threefolds at fixed base degree could be expressed in terms of meromorphic Jacobi forms for SL(2,Z) S L ( 2 , ℤ ) , giving access to Gromov-Witten invariants at arbitrary genus. This was later generalized to torus-fibered CY threefolds with N N -sections, where topological string amplitudes are conjecturally governed by meromorphic Jacobi forms under the congruence subgroup Γ_1(N) Γ 1 ( N ) . In this work, we show that these modularity properties follow from (and are equivalent to) the wave-function property of the topological string partition function Zₜₒₚ under a relative conifold monodromy, implementing a particular Fourier-Mukai transformation on the derived category of coherent sheaves. In particular, we introduce a variant of Zₜₒₚ which is both holomorphic and modular covariant. Under the same relative conifold monodromy, the generating series of genus 0 Gopakumar-Vafa invariants at fixed base degree is mapped to the generating series of rank 0 Donaldson-Thomas indices counting D4-D2-D0-brane bound states wrapped on the torus fiber. We show that the quasimodularity of the generating series of GV invariants matches the expected mock-modular behavior of the generating series of D4-D2-D0 indices, despite having different multi-cover contributions. We analyze and tabulate a large number of CY threefolds fibered over del Pezzo surfaces, with an N N -section for N≤ 5 N ≤ 5 , including several new examples beyond the realm of toric geometry.