Modular resurgence, $q$-Pochhammer symbols, and quantum operators from mirror curves
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| Format: | Preprint |
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2025
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| _version_ | 1866912993897349120 |
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| author | Fantini, Veronica Rella, Claudia |
| author_facet | Fantini, Veronica Rella, Claudia |
| contents | Building on the results of [1,2], we study the resurgence of $q$-Pochhammer symbols and determine their summability and quantum modularity properties. We construct a new, infinite family of pairs of modular resurgent series from the asymptotic expansions of sums of $q$-Pochhammer symbols weighted by suitable Dirichlet characters. These weighted sums fit into the modular resurgence paradigm and provide further evidence supporting our conjectures in [1]. In the context of the topological string/spectral theory correspondence for toric Calabi-Yau threefolds, Kashaev and Mariño proved that the spectral traces of canonical quantum operators associated with local weighted projective planes can be expressed as sums of $q$-Pochhammer symbols. Exploiting this relation, we show that an exact strong-weak resurgent symmetry, first observed by the second author in [3] and fully formalized in [2] for local $\mathbb{P}^2$, applies to all local $\mathbb{P}^{m,n}$, albeit stripped of some of the underlying number-theoretic properties. Under some assumptions, these properties are restored when considering linear combinations of the spectral traces that reproduce the weighted sums above. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_08265 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Modular resurgence, $q$-Pochhammer symbols, and quantum operators from mirror curves Fantini, Veronica Rella, Claudia High Energy Physics - Theory Mathematical Physics Algebraic Geometry Complex Variables Number Theory Building on the results of [1,2], we study the resurgence of $q$-Pochhammer symbols and determine their summability and quantum modularity properties. We construct a new, infinite family of pairs of modular resurgent series from the asymptotic expansions of sums of $q$-Pochhammer symbols weighted by suitable Dirichlet characters. These weighted sums fit into the modular resurgence paradigm and provide further evidence supporting our conjectures in [1]. In the context of the topological string/spectral theory correspondence for toric Calabi-Yau threefolds, Kashaev and Mariño proved that the spectral traces of canonical quantum operators associated with local weighted projective planes can be expressed as sums of $q$-Pochhammer symbols. Exploiting this relation, we show that an exact strong-weak resurgent symmetry, first observed by the second author in [3] and fully formalized in [2] for local $\mathbb{P}^2$, applies to all local $\mathbb{P}^{m,n}$, albeit stripped of some of the underlying number-theoretic properties. Under some assumptions, these properties are restored when considering linear combinations of the spectral traces that reproduce the weighted sums above. |
| title | Modular resurgence, $q$-Pochhammer symbols, and quantum operators from mirror curves |
| topic | High Energy Physics - Theory Mathematical Physics Algebraic Geometry Complex Variables Number Theory |
| url | https://arxiv.org/abs/2506.08265 |