WKB Asymptotics of Stokes Matrices, Spectral Curves and Rhombus Inequalities

We consider \(n\times n\) systems of linear ODEs on \(\mathbb {P}^1\) with a regular singularity at \(z=0\) and an irregular singularity of rank 1 (double pole) at \(z=\infty \). The monodromy data of such a system are described by upper and lower triangular Stokes matrices \(S_\pm \). We impose reality conditions which imply \(S_-=S_+^\dagger \). We study the leading WKB exponents of the Stokes matrices in parametrizations given by generalized minors and by spectral coordinates. We show that in a certain degeneration limit, called the caterpillar limit, the real parts of these exponents are given by periods of 1-cycles on a degenerate spectral curve. We then consider moving away from the caterpillar limit. Using exact WKB and spectral networks, we give predictions for asymptotics of generalized minors in terms of regularized periods on the spectral curve, in the cases \(n = 2\) and \(n = 3\). For \(n=2\) we verify directly that the predictions are correct, while for \(n=3\) they are new conjectures. Boalch’s theorem from Poisson geometry implies that the real parts of leading WKB exponents satisfy the rhombus (or interlacing) inequalities. We show for \(n=2\) and \(n=3\) that these inequalities are equivalent to the positivity of certain periods, and that this positivity is a consequence of the existence of certain finite webs. We also discuss the relation of the spectral networks with the cluster structures on dual Poisson–Lie groups considered by Goncharov–Shen, and with certain \({{\mathcal {N}}}=2\) supersymmetric quantum field theories in dimension four. In the field theory context the caterpillar limit becomes a weak-coupling limit, and the finite webs are interpreted as BPS particles.