Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles
Quantum curves were introduced in the physics literature. We develop a mathematical framework for the case associated with Hitchin spectral curves. In this context, a quantum curve is a Rees -module on a smooth projective algebraic curve, whose semi-classical limit produces the Hitchin spectral curv...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2021 |
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| Sprache: | Englisch |
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Інститут математики НАН України
2021
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/211313 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles. Olivia Dumitrescu and Motohico Mulase. SIGMA 17 (2021), 036, 53 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862571179618861056 |
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| author | Dumitrescu, Olivia Mulase, Motohico |
| author_facet | Dumitrescu, Olivia Mulase, Motohico |
| citation_txt | Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles. Olivia Dumitrescu and Motohico Mulase. SIGMA 17 (2021), 036, 53 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Quantum curves were introduced in the physics literature. We develop a mathematical framework for the case associated with Hitchin spectral curves. In this context, a quantum curve is a Rees -module on a smooth projective algebraic curve, whose semi-classical limit produces the Hitchin spectral curve of a Higgs bundle. We give a method of quantization of Hitchin spectral curves by concretely constructing one-parameter deformation families of opers. We propose a variant of the topological recursion of Eynard-Orantin and Mirzakhani for the context of singular Hitchin spectral curves. We show that a PDE version of topological recursion provides all-order WKB analysis for the Rees -modules, defined as the quantization of Hitchin spectral curves associated with meromorphic SL(2, ℂ)-Higgs bundles. Topological recursion can be considered as a process of quantization of Hitchin spectral curves. We prove that these two quantizations, one via the construction of families of opers and the other via the PDE recursion of topological type, agree for holomorphic and meromorphic SL(2, ℂ)-Higgs bundles. Classical differential equations, such as the Airy differential equation provides a typical example. Through these classical examples, we see that quantum curves relate Higgs bundles, opers, a conjecture of Gaiotto, and quantum invariants, such as Gromov-Witten invariants.
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| first_indexed | 2026-03-13T12:09:37Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-211313 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-13T12:09:37Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
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| spelling | Dumitrescu, Olivia Mulase, Motohico 2025-12-29T11:09:05Z 2021 Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles. Olivia Dumitrescu and Motohico Mulase. SIGMA 17 (2021), 036, 53 pages 1815-0659 2020 Mathematics Subject Classification: 14H15; 14N35; 81T45; 14F10; 14J26; 33C05; 33C10; 33C15; 34M60; 53D37 arXiv:1702.00511 https://nasplib.isofts.kiev.ua/handle/123456789/211313 https://doi.org/10.3842/SIGMA.2021.036 Quantum curves were introduced in the physics literature. We develop a mathematical framework for the case associated with Hitchin spectral curves. In this context, a quantum curve is a Rees -module on a smooth projective algebraic curve, whose semi-classical limit produces the Hitchin spectral curve of a Higgs bundle. We give a method of quantization of Hitchin spectral curves by concretely constructing one-parameter deformation families of opers. We propose a variant of the topological recursion of Eynard-Orantin and Mirzakhani for the context of singular Hitchin spectral curves. We show that a PDE version of topological recursion provides all-order WKB analysis for the Rees -modules, defined as the quantization of Hitchin spectral curves associated with meromorphic SL(2, ℂ)-Higgs bundles. Topological recursion can be considered as a process of quantization of Hitchin spectral curves. We prove that these two quantizations, one via the construction of families of opers and the other via the PDE recursion of topological type, agree for holomorphic and meromorphic SL(2, ℂ)-Higgs bundles. Classical differential equations, such as the Airy differential equation provides a typical example. Through these classical examples, we see that quantum curves relate Higgs bundles, opers, a conjecture of Gaiotto, and quantum invariants, such as Gromov-Witten invariants. The authors wish to thank Philip Boalch for many useful discussions and comments on their work on quantum curves. In particular, his question, proposed at the American Institute of Mathematics Workshop, Spectral data for Higgs bundles in September-October 2015, was critical for the development of the theory presented in this paper. The authors also thank Jürgen Andersen, Vincent Bouchard, Tom Bridgeland, Bertrand Eynard, Edward Frenkel, Tamas Hausel, Kohei Iwaki, Maxim Kontsevich, Laura Schaposnik, Carlos Simpson, Albert Schwarz, Yan Soibelman, Ruifang Song, Jörg Teschner, and Richard Wentworth for useful comments, suggestions, and discussions. During the preparation of this work, the research of O.D. was supported by GRK 1463 Analysis, Geometry, and String Theory at the Leibniz Universität Hannover and a grant from MPIM-Bonn. The research of M.M. was supported by IHES, MPIM-Bonn, NSF grants DMS-1104734, DMS-1309298, DMS-1619760, DMS-1642515, and NSF-RNMS: Geometric Structures And Representation Varieties (GEAR Network, DMS-1107452, 1107263, 1107367). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles Article published earlier |
| spellingShingle | Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles Dumitrescu, Olivia Mulase, Motohico |
| title | Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles |
| title_full | Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles |
| title_fullStr | Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles |
| title_full_unstemmed | Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles |
| title_short | Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles |
| title_sort | interplay between opers, quantum curves, wkb analysis, and higgs bundles |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211313 |
| work_keys_str_mv | AT dumitrescuolivia interplaybetweenopersquantumcurveswkbanalysisandhiggsbundles AT mulasemotohico interplaybetweenopersquantumcurveswkbanalysisandhiggsbundles |