Quantum Curves and Spectral Curves in Enumerative Geometry
The circle of ideas in which an enumeration problem is solved by identifying an underlying spectral curve, obtained by Laplace-transforming or otherwise analyzing the counting data. The genus-0, one-point invariant is encoded in the spectral curve, and passing to all genera and marked points is a quantization of that curve into a quantum (differential-operator) curve. Worked cases include simple Hurwitz numbers on the Lambert curve and Catalan numbers, with a geometric reformulation via Higgs bundles and opers.
2501.00716
These lecture notes develop the idea that many enumeration problems become solvable once a hidden algebraic or analytic curve, a spectral curve, is identified. For simple Hurwitz numbers, the Laplace…