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Quantization of hyper-elliptic curves from isomonodromic systems and topological recursion

Abstract : We prove that the topological recursion formalism can be used to compute the WKB expansion of solutions of second order differential operators obtained by quantization of any hyper-elliptic curve. We express this quantum curve in terms of spectral Darboux coordinates on the moduli space of meromorphic sl2-connections on P1 and argue that the topological recursion produces a 2g-parameter family of associated tau functions, where 2g is the dimension of the moduli space considered. We apply this procedure to the 6 Painlevé equations which correspond to g=1 and consider a g=2 example.
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https://hal.archives-ouvertes.fr/hal-02423721
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Submitted on : Tuesday, October 19, 2021 - 2:12:14 PM
Last modification on : Thursday, November 18, 2021 - 2:48:23 PM

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Olivier Marchal, Nicolas Orantin. Quantization of hyper-elliptic curves from isomonodromic systems and topological recursion. J.Geom.Phys., 2022, 171, pp.104407. ⟨10.1016/j.geomphys.2021.104407⟩. ⟨hal-02423721⟩

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