Quantum geometry, stability and modularity

related to Gopakumar-Vafa (GV) invariants, and rank 0 Donaldson-Thomas (DT) invariants countingD4-D2-D0 BPS bound states, we rigorously compute the first few terms in the generating series of Abelian D4-D2-D0 indices for compact one-parameter Calabi-Yau threefolds of hypergeometric type. In all cases where GV invariants can be computed to sufficiently high genus, we find striking confirmation that the generating series is modular, and predict infinite series of Abelian D4-D2-D0 indices. Conversely, we use these results to provide new constraints for the direct integration method, which allows to compute GV invariants (and therefore the topological string partition function) to higher genus than hitherto possible. The triangle of relations between GV/PT/DT invariants is powered by a new explicit formula relating PT and rank 0 DT invariants, which is proven in an Appendix by the second named author. As a corollary, we obtain rigorous Castelnuovo-type bounds for PT and GV invariants for CY threefolds with Picard rank one.

Keywords

Gromov-Witten invariants, Donaldson-Thomas invariants, wall-crossing, modular forms

2010 Mathematics Subject Classification

11F37, 14J32, 14N35, 81T30

Full Text (PDF format)

Received 10 February 2023

Accepted 22 January 2024

Published 7 June 2024