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Integrable Differential Systems of Topological Type and Reconstruction by the Topological Recursion

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Abstract

Starting from a \(d\times d\) rational Lax pair system of the form \(\hbar \partial _x \Psi = L\Psi \) and \(\hbar \partial _t \Psi =R\Psi \), we prove that, under certain assumptions (genus 0 spectral curve and additional conditions on R and L), the system satisfies the “topological type property.” A consequence is that the formal \(\hbar \)-WKB expansion of its determinantal correlators satisfies the topological recursion. This applies in particular to all (p, q) minimal models reductions of the KP hierarchy, or to the six Painlevé systems.

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Authors and Affiliations

  1. Institut de physique théorique, Université Paris Saclay, CEA, CNRS, 91191, Gif-sur-Yvette, France

    Raphaël Belliard & Bertrand Eynard

  2. Centre de recherches mathématiques, Université de Montréal, Montreal, Canada

    Bertrand Eynard

  3. Université de Lyon, CNRS UMR 5208, Université Jean Monnet, Institut Camille Jordan, Lyon, France

    Olivier Marchal

Authors
  1. Raphaël Belliard
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  2. Bertrand Eynard
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  3. Olivier Marchal
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Correspondence to Olivier Marchal.

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Communicated by Boris Pioline.

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Belliard, R., Eynard, B. & Marchal, O. Integrable Differential Systems of Topological Type and Reconstruction by the Topological Recursion. Ann. Henri Poincaré 18, 3193–3248 (2017). https://doi.org/10.1007/s00023-017-0595-9

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  • DOI: https://doi.org/10.1007/s00023-017-0595-9

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