Canonical quantum curve I

Abstract

We formulate a canonical quantum curve conjecture $\widehat HΨ_*=0$ for spectral curves of positive genus. Here $Ψ_*$ is the non-perturbative wave function associated with Eynard--Orantin topological recursion, specialized at the \emph{canonical vanishing point}, a distinguished zero of the closed non-perturbative partition function. The operator $\widehat H$ is obtained by Weyl quantization of $H(x,y)$, with $\widehat x=x\cdot$ and $\widehat y=\hbar\partial_x$, together with quantum corrections to its coefficients. These corrections depend only on closed moduli and $\hbar$ and vanish classically. Whenever a geometric definition is available, we conjecture that the corrected coefficients are suitably normalized Wilson-loop expectation values. When a corresponding refined theory exists, we conjecture a relation between the canonical vanishing point and the closed Nekrasov--Shatashvili free energy. We prove the quantum curve conjecture for Weierstrass curves and the mirror curves of canonical bundles of smooth projective toric weak del Pezzo surfaces, including local $\mathbb P^2$. For the local surfaces considered here, we relate the canonical vanishing point to a derivative of the closed NS free energy. For local $\mathbb P^2$, we also identify the corrected coefficient with a suitably normalized, geometrically defined NS Wilson-loop expectation value.