Canonical quantum curve I
Abstract
We formulate a canonical quantum curve conjecture ĤΨ*=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 Ĥ is obtained by Weyl quantization of H(x,y), with x̂=x· and ŷ=ℏ∂x, together with quantum corrections to its coefficients. These corrections depend only on closed moduli and ℏ 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 ℙ2. For the local surfaces considered here, we relate the canonical vanishing point to a derivative of the closed NS free energy. For local ℙ2, we also identify the corrected coefficient with a suitably normalized, geometrically defined NS Wilson-loop expectation value.
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BibTeX
@misc{guo2026canonical,
title = {{Canonical quantum curve I}},
author = {Shuai Guo and Qingsheng Zhang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00997},
eprint = {2611.00997},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00997},
}
RIS
TY - UNPB
TI - Canonical quantum curve I
AU - Shuai Guo
AU - Qingsheng Zhang
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00997
UR - https://arxiv.org/abs/2611.00997
AB - We formulate a canonical quantum curve conjecture ĤΨ_*=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 Ĥ is obtained by Weyl quantization of H(x,y), with x̂=x· and ŷ=ℏ∂_x, together with quantum corrections to its coefficients. These corrections depend only on closed moduli and ℏ 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 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 ℙ^2, we also identify the corrected coefficient with a suitably normalized, geometrically defined NS Wilson-loop expectation value.
ER -