Quantization of classical spectral curves via topological recursion
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2021
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| _version_ | 1866910380265046016 |
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| author | Eynard, Bertrand Garcia-Failde, Elba Marchal, Olivier Orantin, Nicolas |
| author_facet | Eynard, Bertrand Garcia-Failde, Elba Marchal, Olivier Orantin, Nicolas |
| contents | We prove that the topological recursion formalism can be used to quantize any generic classical spectral curve with smooth ramification points and simply ramified away from poles. For this purpose, we build both the associated quantum curve, i.e.~the differential operator quantizing the algebraic equation defining the classical spectral curve considered, and a basis of wave functions, that is to say a basis of solutions of the corresponding differential equation. We further build a Lax pair representing the resulting quantum curve and thus present it as a point in an associated space of meromorphic connections on the Riemann sphere, a first step towards isomonodromic deformations. We finally propose two examples: the derivation of a 2-parameter family of formal trans-series solutions to Painlevé 2 equation and the quantization of a degree three spectral curve with pole only at infinity. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2106_04339 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Quantization of classical spectral curves via topological recursion Eynard, Bertrand Garcia-Failde, Elba Marchal, Olivier Orantin, Nicolas Mathematical Physics High Energy Physics - Theory Algebraic Geometry Classical Analysis and ODEs Exactly Solvable and Integrable Systems 34M56, 34M55, 34E20, 14H70 We prove that the topological recursion formalism can be used to quantize any generic classical spectral curve with smooth ramification points and simply ramified away from poles. For this purpose, we build both the associated quantum curve, i.e.~the differential operator quantizing the algebraic equation defining the classical spectral curve considered, and a basis of wave functions, that is to say a basis of solutions of the corresponding differential equation. We further build a Lax pair representing the resulting quantum curve and thus present it as a point in an associated space of meromorphic connections on the Riemann sphere, a first step towards isomonodromic deformations. We finally propose two examples: the derivation of a 2-parameter family of formal trans-series solutions to Painlevé 2 equation and the quantization of a degree three spectral curve with pole only at infinity. |
| title | Quantization of classical spectral curves via topological recursion |
| topic | Mathematical Physics High Energy Physics - Theory Algebraic Geometry Classical Analysis and ODEs Exactly Solvable and Integrable Systems 34M56, 34M55, 34E20, 14H70 |
| url | https://arxiv.org/abs/2106.04339 |