Quantum geometry at arbitrary genus - I: Anharmonic potentials
The work resolves whether a relation between perturbative and non-perturbative expansions exists for anharmonic potentials of arbitrary genus. Using complex algebraic geometry and exact WKB analysis, it shows that while higher-genus systems present an obstruction, this is resolved by using a modified Riemann bilinear identity. This yields an all-orders quantum P/NP relation valid for any genus g, explicitly derived for genus-2 and genus-3 oscillators.