Enumerative Geometry of Quantum Periods
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911071375196160 |
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| author | Gräfnitz, Tim Ruddat, Helge Zaslow, Eric Zhou, Benjamin |
| author_facet | Gräfnitz, Tim Ruddat, Helge Zaslow, Eric Zhou, Benjamin |
| contents | We interpret the $q$-refined theta function $\vartheta_1$ of a log Calabi-Yau surface $(\mathbb{P},E)$ as a natural $q$-refinement of the open mirror map, defined by quantum periods of mirror curves for outer Aganagic-Vafa branes on the local Calabi-Yau $K_{\mathbb{P}}$. The series coefficients are all-genus logarithmic two-point invariants, directly extending the relation found in [GRZ]. Yet we find an explicit discrepancy at higher genus in the relation to open Gromov-Witten invariants of the Aganagic-Vafa brane. Using a degeneration argument, we express the difference in terms of relative invariants of an elliptic curve. With $π: \widehat{\mathbb{P}} \rightarrow \mathbb{P}$ the toric blow up of a point, we use the Topological Vertex [AKMV] to show a correspondence between open invariants of $K_{\mathbb{P}}$ and closed invariants of $K_{\widehat{\mathbb{P}}}$ generalizing a variant of [CLLT][LLW] to arbitrary genus and winding. We also equate winding-1, open-BPS invariants with closed Gopakumar-Vafa invariants. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_19408 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Enumerative Geometry of Quantum Periods Gräfnitz, Tim Ruddat, Helge Zaslow, Eric Zhou, Benjamin Algebraic Geometry Quantum Algebra Symplectic Geometry We interpret the $q$-refined theta function $\vartheta_1$ of a log Calabi-Yau surface $(\mathbb{P},E)$ as a natural $q$-refinement of the open mirror map, defined by quantum periods of mirror curves for outer Aganagic-Vafa branes on the local Calabi-Yau $K_{\mathbb{P}}$. The series coefficients are all-genus logarithmic two-point invariants, directly extending the relation found in [GRZ]. Yet we find an explicit discrepancy at higher genus in the relation to open Gromov-Witten invariants of the Aganagic-Vafa brane. Using a degeneration argument, we express the difference in terms of relative invariants of an elliptic curve. With $π: \widehat{\mathbb{P}} \rightarrow \mathbb{P}$ the toric blow up of a point, we use the Topological Vertex [AKMV] to show a correspondence between open invariants of $K_{\mathbb{P}}$ and closed invariants of $K_{\widehat{\mathbb{P}}}$ generalizing a variant of [CLLT][LLW] to arbitrary genus and winding. We also equate winding-1, open-BPS invariants with closed Gopakumar-Vafa invariants. |
| title | Enumerative Geometry of Quantum Periods |
| topic | Algebraic Geometry Quantum Algebra Symplectic Geometry |
| url | https://arxiv.org/abs/2502.19408 |