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arXiv · 1610.00496

Integrable differential systems of topological type and reconstruction by the topological recursion

Raphaël Belliard·Bertrand Eynard·Olivier Marchal
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Abstract

Starting from a $d\times d$ rational Lax pair system of the form $\hbar \partial_x Ψ= LΨ$ and $\hbar \partial_t Ψ=RΨ$ 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, satisfy 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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Raphaël Belliard, Bertrand Eynard, Olivier Marchal. 2017-07-07. Integrable differential systems of topological type and reconstruction by the topological recursion. https://doi.org/10.1007/s00023-017-0595-9

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