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arXiv:2610.10940

Quantum geometry at arbitrary genus - I: Anharmonic potentials

Mustafa Türe, Mithat Ünsal

hep-thmath-phmath.AGquant-phSubmitted Oct 7, 2026• Announced Oct 9, 2026
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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 $p^2 = 2(E-V(q))$ defines a genus-g Riemann surface $X$, whose periods are naturally organized by its Jacobian variety $J(X) = \mathbb{C}^{\text{g}}/Λ$ 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 $\text{SL}(2\text{g},\mathbb{Z})$ but not in the symplectic group $\text{Sp}(2\text{g},\mathbb{Z})$ once $\text{g}\geq2$, 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.

Comments
34 pages, 4 figures

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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))p^2 = 2(E-V(q))p2=2(E−V(q)) defines a genus-g Riemann surface XXX, whose periods are naturally organized by its Jacobian variety J(X)=Cg/ΛJ(X) = \mathbb{C}^{\text{g}}/ΛJ(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)\text{SL}(2\text{g},\mathbb{Z})SL(2g,Z) but not in the symplectic group Sp(2g,Z)\text{Sp}(2\text{g},\mathbb{Z})Sp(2g,Z) once g≥2\text{g}\geq2g≥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.
We discuss the exact quantization of general one-dimensional potentials in view of the exact-WKB formalism. Building on our previous work, we perform analytic continuations across different sectors via the complexification to the spectral (energy) parameter uuu and identify continuous and discontinuous transitions of the exact spectrum for generic potentials. When the transition is discontinuous, it is characterized by the Stokes phenomena, inducing different exact (median) quantization conditions, thereby distinct trans-series structures valid in different sectors. We analyze two illustrative examples, namely asymmetric triple-well (ATW) and tilted double-well (TDW), and verify the general qualitative analysis by deriving exact (median) quantization conditions in each sector. Moreover, by obtaining the trans-series solutions for each system, we identify bion/bounce configurations and show that the trans-series of ATW is organized in accordance with the cluster expansion of the bion gas and there should exist a previously neglected complex saddle in the TDW system. These identifications further strengthen the link between path integral and exact-WKB formalisms, while also demonstrating the predictive power of the latter. In parallel, for the P-NP relations of genus-1 systems, we derive transformation rules between any perturbative and non-perturbative pair of WKB-cycles. Our results show that the entire resurgence data of a genus-1 system transforms only by the change of classical parameters, i.e. frequencies and bion/bounce actions, and the perturbative energy series. This also reveals the underlying reasons of the previously found SSS-duality transformations.
The Gutzwiller trace formula establishes a profound connection between the quantum spectrum and classical periodic orbits. However, its application is limited by its reliance on the semiclassical saddle point approximation. In this work, we explore the full quantum version of the trace formula using the Lefschetz thimble method by incorporating complexified periodic orbits. Upon complexification, classical real periodic orbits are transformed into cycles on compact Riemann surfaces. Our key innovation lies in the simultaneous complexification of the periods of cycles, resulting in a fully quantum trace formula that accounts for all contributions classified by the homology classes of the associated Riemann surfaces. This formulation connects the quantum spectrum to contributions across all complex time directions, encompassing all relevant homology classes. Our approach naturally unifies and extends two established methodologies: periodic orbits in real time, as in Gutzwiller's original work, and quantum tunneling in imaginary time, as in the instanton method.
We study the topological band theory of quantum Hall systems in which the adiabatic flux parameters of an external gauge potential form a high-dimensional manifold. This situation arises both for strongly correlated matter on spatial Riemann surfaces of genus g>1g > 1g>1, as well as in the formal study of matter on compact oriented manifolds of dimension 4ℓ+2=2p4\ell+2 = 2p4ℓ+2=2p coupled to ppp-form gauge potentials. In these cases, the underlying topological invariants defined over the parameter space are significantly richer than the low-dimensional (i.e., two-dimensional) case and involve both the curvature (i.e., first Chern class) as well as higher order curvature invariants. These higher curvature invariants correspond to topologically protected contributions to Kubo-like formulae, constructed from correlation functions of the physical current operators. We illustrate these general considerations with an explicit example from hyperbolic band theory based on the genus-two Bolza Riemann surface, a case which has recently been simulated on a synthetic-dimension platform.
The formalism of exact 1D quantization is reviewed in detail and applied to the spectral study of three concrete Schrödinger Hamiltonians [−d2/dq2+V(q)]±[-{\rm d}^2/{\rm d} q^2 + V(q)]^\pm[−d2/dq2+V(q)]± on the half-line {q>0}\{q>0\}{q>0}, with a Dirichlet (−-−) or Neumann (+++) condition at q=0q=0q=0. Emphasis is put on the analytical investigation of the spectral determinants and spectral zeta functions with respect to singular perturbation parameters. We first discuss the homogeneous potential V(q)=qNV(q)=q^NV(q)=qN as N→+∞N \to +\inftyN→+∞ vs its (solvable) N=∞N=\inftyN=∞ limit (an infinite square well): useful distinctions are established between regular and singular behaviours of spectral quantities; various identities among the square-well spectral functions are unraveled as limits of finite-NNN properties. The second model is the quartic anharmonic oscillator: its zero-energy spectral determinants det⁡(−d2/dq2+q4+vq2)±\det( -{\rm d}^2/{\rm d} q^2 + q^4 + v q^2)^\pmdet(−d2/dq2+q4+vq2)± are explicitly analyzed in detail, revealing many special values, algebraic identities between Taylor coefficients, and functional equations of a quartic type coupled to asymptotic v→+∞v \to +\inftyv→+∞ properties of Airy type. The third study addresses the potentials V(q)=qN+vqN/2−1V(q)=q^N+v q^{N/2-1}V(q)=qN+vqN/2−1 of even degree: their zero-energy spectral determinants prove computable in closed form, and the generalized eigenvalue problems with vvv as spectral variable admit exact quantization formulae which are perfect extensions of the harmonic oscillator case (corresponding to N=2N=2N=2); these results probably reflect the presence of supersymmetric potentials in the family above.
Hamiltonians whose symbols are not simply real valued, but matrix or, more generally, endomorphism valued functions appear in many places in physics, examples being the Dirac equation, multicomponent wave equations like electrodynamics in media, and Yang-Mills theories, and the Born-Oppenheimer approximation in molecular physics. The aim of this paper is to give a completely geometric approach to the WKB approximation od such systems, and to reduce the problem ``as far as possible'' to the scalar case. A star-product formulation of quantum mechanics proves to be particularly useful in this context. As opposed to other approaches in the literature which restrict themselves to the use of the Moyal product and thus to the study of trivial bundles (or local trivializations) over ℜ2n\real^{2n}ℜ2n, we will consider general bundles over arbitrary symplectic manifolds. Here, Fedosov's construction \cite{fedosov} will be the adequate tool, since it gives an explicit construction for star products in this general setting.
We study the relation between the exact-WKB Delabaere--Dillinger--Pham Stokes jumps and the multi-instanton expansion of a one-dimensional potential with NNN equal-depth minima. For a single barrier this relation is the Bogomolny--Zinn-Justin cancellation between the instanton--anti-instanton ambiguity and the Borel ambiguity of perturbation theory, and we extend it to every consecutive barrier interval and to the integration contours themselves. In the dilute large-TTT regime at fixed nonperturbative grade, we derive the quasi-zero mode integral over the instanton separations from the Euclidean path integral and reduce it to Gamma and Hankel periods. With the local Weber level function of each well and one normalization per well fixed by the local Weber connection matrix, these periods reproduce the two lateral connected exact-WKB coefficients. The two lateral contours of each instanton--anti-instanton separation differ by exactly one Hankel cycle, and this linear Picard--Lefschetz relation reproduces the Stokes jump of the connected coefficient. A numerical solution of the Schrödinger equation for a symmetric triple well confirms the resulting two-barrier level splitting. At higher fixed grade, the map from ordered event sequences to normalized scalar coefficient products is not one-to-one. At leading dilute order the scalar weight of a closed event sequence depends only on the numbers of barrier crossings, IIˉ\mathcal I\bar{\mathcal I}IIˉ pairs, and II\mathcal I\mathcal III/IˉIˉ\bar{\mathcal I}\bar{\mathcal I}IˉIˉ pairs. We construct two inequivalent grade-six triple-well event sequences with the same product of local Weber amplitudes, which embed in every potential with N≥3N\ge3N≥3 wells. Our results hold within the regular dilute, one-loop, local Weber approximation.