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Exact WKB conjecture for unobstructed angles on strongly GMN spectral curves

Establish the exact WKB properties for an ħ-flat connection ∇ that is WKB-regular with strongly GMN spectral curve at an unobstructed angle θ: (1) show that the Borel–Laplace transform ℒΨ of any formal WKB solution Ψ is analytically continuable on the ray ℝ_{>0} · e^{2π i θ and is Laplace-transformable in direction θ; (2) characterize the singular set of the first sheet of the analytic continuation of ℒΨ along that ray to be precisely at the points { m(𝒯_i, θ) · e^{2π i θ } }_i coming from the masses of the Stokes trees 𝒯_i passing through the base point z ∈ C \ D.

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Background

The paper proves dense existence of unobstructed angles for strongly GMN spectral data and develops spectral networks with compatible wall-crossing. Building on this structure, it formulates an exact WKB conjecture specifying analytic continuability and singularity locations of Borel–Laplace transforms of formal WKB solutions in higher-order settings.

This conjecture aims to extend classical exact WKB phenomena beyond quadratic differentials, leveraging the spectral network framework and mass filtration to predict exact analytic behavior tied to Stokes trees. The statements remain unproven in this generality and are presented as a precise target for future work.

References

Conjecture Assume \nabla is WKB-regular and the spectral curve is strongly GMN. Let \theta be an unobstructed angle whose existence is assured by Theorem~1.

  1. If z\in C\bs D is not on the Stokes graph at \theta, \cL\Psi is analytically continuable on \bR_{>0}\cdot e{2\pi i\theta}. Moreover, \cL\Psi is Laplace transformable in the direction \theta.
  2. Let \cT_1,..,\cT_i,... be the set of Stokes trees at \theta passing through z. Then the first sheet of the analytic continuation of \cL\Psi is smooth on \bR_{>0}\cdot e{2\pi i\theta}\bs \lc m(\cT_i, \theta)\cdot e{2\pi i\theta}\rc_i.
— On the generic existence of WKB spectral networks/Stokes graphs (2408.05399 - Kuwagaki, 10 Aug 2024) in Section "Exact WKB conjecture" (Conjecture)