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Register a new userFrom topological recursion to wave functions and PDEs quantizing hyperelliptic curves
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Starting from loop equations, we prove that the wave functions constructed from topological recursion on families of degree $2$ 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, and with the poles. These equations can be used to prove that the WKB solution of many isomonodromic systems coincides with the topological recursion wave function, which proves 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 this construction to hyperelliptic curves.
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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 $mathfrak{sl}_2$-connections on $mathbb{P}^1$ 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 Painleve equations which correspond to $g=1$ and consider a $g=2$ example.
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 Painleve 2 equation and the quantization of a degree three spectral curve with pole only at infinity.
We show that for a rather generic set of regular spectral curves, the Topological-Recursion invariants F_g grow at most like $O((beta g)! r^{-g}) $ with some $r>0$ and $betaleq 5$.
We consider two-dimensional Pauli and Dirac operators with a polynomially vanishing magnetic field. The main results of the paper provide resolvent expansions of these operators in the vicinity of their thresholds. It is proved that the nature of these expansions is fully determined by the flux of the magnetic field. The most important novelty of the proof is a comparison between the spatial asymptotics of the zero modes obtained in two different manners. The result of this matching allows to compute explicitly all the singular terms in the associated resolvent expansions. As an application we show how the magnetic field influences the time decay of the associated wave-functions quantifying thereby the paramagnetic and diamagnetic effects of the spin.
Starting from a $dtimes d$ rational Lax pair system of the form $hbar partial_x Psi= LPsi$ and $hbar partial_t Psi=RPsi$ 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 Painleve systems.
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