A framework for proving quantum modularity: Application to Witten's asymptotic expansion conjecture
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911130192969728 |
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| author | Murakami, Yuya |
| author_facet | Murakami, Yuya |
| contents | We address two linked problems at the interface of quantum topology and number theory: deriving asymptotic expansions of the Witten--Reshetikhin--Turaev invariants for 3-manifolds and establishing quantum modularity of false theta functions. Previous progress covers Seifert homology 3-spheres for the former and rank-one cases for the latter, both of which rely on single-variable integral representations. We extend these results to negative definite plumbed 3-manifolds and to general false theta functions, respectively. We address this limitation by developing two techniques: a Poisson summation formula with signature and a framework of modular series, both of which enable a precise and explicit analysis of multivariable integral representations. As further applications, our method yields a unified approach to proving quantum modularity for false theta functions, indefinite theta functions, and for Eisenstein series of odd weight. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_21710 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A framework for proving quantum modularity: Application to Witten's asymptotic expansion conjecture Murakami, Yuya Number Theory Geometric Topology 57K31, 57K10, 57K16, 11F27, 11F11, 41A60 We address two linked problems at the interface of quantum topology and number theory: deriving asymptotic expansions of the Witten--Reshetikhin--Turaev invariants for 3-manifolds and establishing quantum modularity of false theta functions. Previous progress covers Seifert homology 3-spheres for the former and rank-one cases for the latter, both of which rely on single-variable integral representations. We extend these results to negative definite plumbed 3-manifolds and to general false theta functions, respectively. We address this limitation by developing two techniques: a Poisson summation formula with signature and a framework of modular series, both of which enable a precise and explicit analysis of multivariable integral representations. As further applications, our method yields a unified approach to proving quantum modularity for false theta functions, indefinite theta functions, and for Eisenstein series of odd weight. |
| title | A framework for proving quantum modularity: Application to Witten's asymptotic expansion conjecture |
| topic | Number Theory Geometric Topology 57K31, 57K10, 57K16, 11F27, 11F11, 41A60 |
| url | https://arxiv.org/abs/2508.21710 |