Quantum geometry at arbitrary genus - I: Anharmonic potentials
Abstract
In quantum mechanics of genus-1 potentials, perturbation theory around the topologically trivial saddle and non-perturbative instanton saddle are constructively related via the P/NP relation. Whether such a relation persists for higher-genus potentials has remained an open problem. In this work, we resolve it using tools from complex algebraic geometry combined with exact WKB. The classical energy conservation relation p2 = 2(E−V(q)) defines a genus-g Riemann surface X, whose periods are naturally organized by its Jacobian variety J(X) = ℂg/Λ via the Abel-Jacobi map, and independently by the solution basis of the associated Picard-Fuchs equations. We show that the physically relevant, WKB-active cycles are generally related to the Picard-Fuchs basis by a linear transformation that lies in SL(2g,ℤ) but not in the symplectic group Sp(2g,ℤ) once g≥2, in contrast to the genus-1 case, where the two groups coincide. This mismatch is the obstruction that has made an explicit higher-genus P/NP relation elusive. We show that it can be resolved by passing to a modified Riemann bilinear identity, built from a suitably transformed intersection matrix, which the WKB-active periods do satisfy exactly. The result is an all-orders quantum P/NP relation valid for potentials of arbitrary genus, which we derive explicitly for genus-2 (quintic and sextic) and genus-3 (septic and octic) anharmonic oscillators.
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BibTeX
@misc{ture2026quantum,
title = {{Quantum geometry at arbitrary genus - I: Anharmonic potentials}},
author = {Mustafa Türe and Mithat Ünsal},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10940},
eprint = {2610.10940},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10940},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Quantum geometry at arbitrary genus - I: Anharmonic potentials
AU - Mustafa Türe
AU - Mithat Ünsal
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10940
UR - https://arxiv.org/abs/2610.10940
AB - In quantum mechanics of genus-1 potentials, perturbation theory around the topologically trivial saddle and non-perturbative instanton saddle are constructively related via the P/NP relation. Whether such a relation persists for higher-genus potentials has remained an open problem. In this work, we resolve it using tools from complex algebraic geometry combined with exact WKB. The classical energy conservation relation p^2 = 2(E−V(q)) defines a genus-g Riemann surface X, whose periods are naturally organized by its Jacobian variety J(X) = ℂ^g/Λ via the Abel-Jacobi map, and independently by the solution basis of the associated Picard-Fuchs equations. We show that the physically relevant, WKB-active cycles are generally related to the Picard-Fuchs basis by a linear transformation that lies in SL(2g,ℤ) but not in the symplectic group Sp(2g,ℤ) once g≥2, in contrast to the genus-1 case, where the two groups coincide. This mismatch is the obstruction that has made an explicit higher-genus P/NP relation elusive. We show that it can be resolved by passing to a modified Riemann bilinear identity, built from a suitably transformed intersection matrix, which the WKB-active periods do satisfy exactly. The result is an all-orders quantum P/NP relation valid for potentials of arbitrary genus, which we derive explicitly for genus-2 (quintic and sextic) and genus-3 (septic and octic) anharmonic oscillators.
ER -