FROM TOPOLOGICAL RECURSION TO WAVE FUNCTIONS AND PDES QUANTIZING HYPERELLIPTIC CURVES

Bertrand Eynard (1) , Elba Garcia-Failde (1)

1 IPHT - Institut de Physique Théorique - UMR CNRS 3681

Starting from loop equations, we prove that the wave functions constructed from topo-logical recursion on families of spectral curves with a global involution satisfy a system of partial differential equations, whose equations can be seen as quantizations of the original spectral curves. The families of spectral curves can be parametrized with the so-called times, defined as periods on second type cycles. These equations can be used to prove that the WKB solution of many isomonodromic systems coincides with the topological recursion wave function, and thus show that the topological recursion wave function is annihilated by a quantum curve. This recovers many known quantum curves for genus zero spectral curves and generalizes to hyperelliptic curves. In the particular case of a degenerate elliptic curve, apart from giving the quantum curve, we prove that the wave function satisfies the first Painlevé isomonodromic system and equation just from loop equations, making use of our system of PDEs. In general, we are able to recover the Gelfand-Dikii isomonodromic systems just from topological recursion.

Domaines: Physique [physics]

Bertrand Eynard, Elba Garcia-Failde. FROM TOPOLOGICAL RECURSION TO WAVE FUNCTIONS AND PDES QUANTIZING HYPERELLIPTIC CURVES. 2019. ⟨cea-02400848⟩