WKB asymptotics of Stokes matrices, spectral curves and rhombus inequalities

Abstract

We consider an $n\times n$ system of ODEs on $\mathbb{P}^1$ with a simple pole $A$ at $z=0$ and a double pole $u={\rm diag}(u_1, \dots, u_n)$ at $z=\infty$. This is the simplest situation in which the monodromy data of the system are described by upper and lower triangular Stokes matrices $S_\pm$, and we impose reality conditions which imply $S_-=S_+^\dagger$. We study leading WKB exponents of Stokes matrices in parametrizations given by generalized minors and by spectral coordinates, and we show that for $u$ on the caterpillar line (which corresponds to the limit $(u_{j+1}-u_j)/(u_j - u_{j-1}) \to \infty$ for $j=2, \cdots, n-1$), the real parts of these exponents are given by periods of certain cycles on the degenerate spectral curve $Γ(u_{\rm cat}(t), A)$. These cycles admit unique deformations for $u$ near the caterpillar line. Using the spectral network theory, we give for $n=2$, and $n=3$ exact WKB predictions for asymptotics of generalized minors in terms of periods of these cycles. Boalch's theorem from Poisson geometry implies that real parts of leading WKB exponents satisfy the rhombus (or interlacing) inequalities. We show that these inequalities are in correspondence with finite webs of the canonical foliation on the root curve $Γ^r(u, A)$, and that they follow from the positivity of the corresponding periods. We conjecture that a similar mechanism applies for $n>3$. We also outline the relation of the spectral coordinates with the cluster structures considered by Goncharov-Shen, and with ${\mathcal N}=2$ supersymmetric quantum field theories in dimension four associated with some simple quivers.

Figures (20)

• Figure 1: The schematic arrangement of the points $z_{ij} \in \mathbb C$ in case $n=4$, with $A$ diagonal, when $t_1 < t_2 < t_3 < t_4$ and $u$ is sufficiently close to the caterpillar line.
• Figure 2: The arrangement of the branch points $z^\pm_{ij}$ and branch cuts $B_{ij}$ in case $n=4$, when the conditions of \ref{['prop:spectral-curve-structure']} are satisfied. Each branch cut is labeled with the induced transposition $(ij)$ of the sheets.
• Figure 3: Dividing the base $\mathbb C$ into annuli, each containing one group of branch cuts. We show the example $n=4$.
• Figure 4: The coordinates $\xi_i^{(j)}$ which enter the interlacing inequalities in the case $n=3$. The larger coordinates are to the right, e.g. $\xi_1^{(3)} < \xi_1^{(2)}$. For each pair of coordinates that are involved in an interlacing inequality, we indicate the corresponding $h$, e.g. $Z({{h_{13}}}) = - \xi_1^{(2)} + \xi_1^{(3)}$.
• Figure 5: A spectral network ${\mathcal{W}}(u,A,\vartheta = 0)$ in the $n=2$ case. Left: the bare spectral network. Middle: the spectral network with paths representing classes in $\tilde{H}(u,A,\vartheta)$. Numbers $1$, $2$ next to paths indicate the sheet on which the path travels. Right: the spectral network with the saddle connections highlighted in purple, and representatives of their charges in $\tilde{H}(u,A,\vartheta)$ marked.

Theorems & Definitions (124)

• Definition 1.1
• Conjecture 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 2.1
• Definition 2.2
• Lemma 2.3