Quantum Curves, Resurgence and Exact WKB
We study the non-perturbative quantum geometry of the open and closed topological string on the resolved conifold and its mirror. Our tools are finite difference equations in the open and closed string moduli and the resurgence analysis of their formal power series solutions. In the closed setting,...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2023 |
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2023
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/211875 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Quantum Curves, Resurgence and Exact WKB. Murad Alim, Lotte Hollands and Iván Tulli. SIGMA 19 (2023), 009, 82 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862707335932149760 |
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| author | Alim, Murad Hollands, Lotte Tulli, Iván |
| author_facet | Alim, Murad Hollands, Lotte Tulli, Iván |
| citation_txt | Quantum Curves, Resurgence and Exact WKB. Murad Alim, Lotte Hollands and Iván Tulli. SIGMA 19 (2023), 009, 82 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We study the non-perturbative quantum geometry of the open and closed topological string on the resolved conifold and its mirror. Our tools are finite difference equations in the open and closed string moduli and the resurgence analysis of their formal power series solutions. In the closed setting, we derive new finite difference equations for the refined partition function as well as its Nekrasov-Shatashvili (NS) limit. We write down a distinguished analytic solution for the refined difference equation that reproduces the expected non-perturbative content of the refined topological string. We compare this solution to the Borel analysis of the free energy in the NS limit. We find that the singularities of the Borel transform lie on infinitely many rays in the Borel plane and that the Stokes jumps across these rays encode the associated Donaldson-Thomas invariants of the underlying Calabi-Yau geometry. In the open setting, the finite difference equation corresponds to a canonical quantization of the mirror curve. We analyze this difference equation using Borel analysis and exact WKB techniques and identify the 5d BPS states in the corresponding exponential spectral networks. We furthermore relate the resurgence analysis in the open and closed settings. This guides us to a five-dimensional extension of the Nekrasov-Rosly-Shatashvili proposal, in which the NS free energy is computed as a generating function of -difference operators in terms of a special set of spectral coordinates. Finally, we examine two spectral problems describing the corresponding quantum integrable system.
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| first_indexed | 2026-03-19T03:49:46Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-211875 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-19T03:49:46Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Alim, Murad Hollands, Lotte Tulli, Iván 2026-01-14T09:55:23Z 2023 Quantum Curves, Resurgence and Exact WKB. Murad Alim, Lotte Hollands and Iván Tulli. SIGMA 19 (2023), 009, 82 pages 1815-0659 2020 Mathematics Subject Classification: 40G10; 39A70; 81T30 arXiv:2203.08249 https://nasplib.isofts.kiev.ua/handle/123456789/211875 https://doi.org/10.3842/SIGMA.2023.009 We study the non-perturbative quantum geometry of the open and closed topological string on the resolved conifold and its mirror. Our tools are finite difference equations in the open and closed string moduli and the resurgence analysis of their formal power series solutions. In the closed setting, we derive new finite difference equations for the refined partition function as well as its Nekrasov-Shatashvili (NS) limit. We write down a distinguished analytic solution for the refined difference equation that reproduces the expected non-perturbative content of the refined topological string. We compare this solution to the Borel analysis of the free energy in the NS limit. We find that the singularities of the Borel transform lie on infinitely many rays in the Borel plane and that the Stokes jumps across these rays encode the associated Donaldson-Thomas invariants of the underlying Calabi-Yau geometry. In the open setting, the finite difference equation corresponds to a canonical quantization of the mirror curve. We analyze this difference equation using Borel analysis and exact WKB techniques and identify the 5d BPS states in the corresponding exponential spectral networks. We furthermore relate the resurgence analysis in the open and closed settings. This guides us to a five-dimensional extension of the Nekrasov-Rosly-Shatashvili proposal, in which the NS free energy is computed as a generating function of -difference operators in terms of a special set of spectral coordinates. Finally, we examine two spectral problems describing the corresponding quantum integrable system. We would like to thank the organizers of the Western Hemisphere Colloquium on Geometry and Physics for bringing together our community, at least virtually, in times when personal interactions remained sparse. The second author, L.H., presented her ideas on the relations of the Borel analysis of [63] to that of [9] at this colloquium in November 2021, triggering our joint work. We would like to thank Alba Grassi, Qianyu Hao, and Andy Neitzke for discussions and for informing us about their related work [52]. We would like to furthermore thank Sergei Alexandrov, Tom Bridgeland, Boris Pioline, Arpan Saha, and Jörge Teschner for discussions and correspondence. Finally, we would like to thank the anonymous referees for their helpful comments and suggestions. The work of I.T. is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2121 Quantum Universe 390833306. The work of L.H. is supported by a Royal Society Dorothy Hodgkin Fellowship. The work of M.A. is supported through the DFG Emmy Noether grant AL 1407/2-1. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Quantum Curves, Resurgence and Exact WKB Article published earlier |
| spellingShingle | Quantum Curves, Resurgence and Exact WKB Alim, Murad Hollands, Lotte Tulli, Iván |
| title | Quantum Curves, Resurgence and Exact WKB |
| title_full | Quantum Curves, Resurgence and Exact WKB |
| title_fullStr | Quantum Curves, Resurgence and Exact WKB |
| title_full_unstemmed | Quantum Curves, Resurgence and Exact WKB |
| title_short | Quantum Curves, Resurgence and Exact WKB |
| title_sort | quantum curves, resurgence and exact wkb |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211875 |
| work_keys_str_mv | AT alimmurad quantumcurvesresurgenceandexactwkb AT hollandslotte quantumcurvesresurgenceandexactwkb AT tulliivan quantumcurvesresurgenceandexactwkb |