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Quantum geometry, stability and modularity

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arxiv 2301.08066 v2 pith:BU6NXQ4N submitted 2023-01-19 hep-th math.AG

Sergei Alexandrov , Soheyla Feyzbakhsh , Albrecht Klemm , Boris Pioline , Thorsten Schimannek This is my paper · ORCID
classification hep-thmath.AG
keywords invariantsd4-d2-d0rankseriesabeliancomputegeneratinggenus
verification ladder T0 review T1 audit T2 compute T3 formal
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By exploiting new mathematical relations between Pandharipande-Thomas (PT) invariants, closely related to Gopakumar-Vafa (GV) invariants, and rank 0 Donaldson-Thomas (DT) invariants counting D4-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.

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Cited by 3 Pith papers

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  1. Revisiting the Quantum Geometry of Torus-fibered Calabi-Yau Threefolds

    hep-th 2025-10 conditional novelty 7.0 of 10

    The conjectured Jacobi modularity of topological string amplitudes on torus-fibered Calabi-Yau threefolds is derived conditionally from the wave-function property under the relative conifold monodromy, which also maps...

  2. Unlocking the Wronskian Tower: A Simplification of the Holomorphic Modular Bootstrap

    hep-th 2026-07 conditional novelty 6.0 of 10

    A differential operator Θ = η^{-4}D relates MLDE solutions across Wronskian sectors, reducing higher-ℓ quasi-character classification in ranks 2 and 3 to ℓ=0 data and proving the ℓ=2 sign conjecture.

  3. Mock modularity of Calabi-Yau threefolds

    hep-th 2024-11 conditional novelty 6.0 of 10

    The paper constructs explicit indefinite theta series solving the modular anomaly for rank 0 DT invariants, fixing these generating functions up to modular forms determined by polar terms.

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