Exact WKB and abelianization for the $T_3$ equation

TL;DR

The paper develops a unified exact WKB framework via abelianization for Schrödinger operators and higher-order opers, emphasizing spectral coordinates $\mathcal{X}_\gamma(\hbar)$ as flat-connection holonomies on spectral covers. It applies the method to the cubic potential, Mathieu, and, most prominently, the $T_3$ equation, deriving explicit rank-3 Darboux coordinates on the SL(3) moduli space, and testing their monodromy and asymptotics against WKB predictions. Integral-equation structures akin to TBA are formulated for these coordinates, with numerical evidence supporting the expected $\mathcal{X}_\gamma(\hbar)\sim\exp(Z_\gamma/\hbar)$ behavior in various phases and a detailed study of analytic continuations, monodromies, and the uniformization point. The work situates these exact-WKB constructions within the landscape of ${\mathcal N}=2$ supersymmetric theories, proposing a geometric realization of spectral data as vevs and linking nonperturbative coordinates to Nekrasov’s partition function and boundary-conditions for opers. Overall, it provides concrete higher-rank generalizations of WKB, new SL(3) Darboux coordinates, and a computational framework for exploring nonperturbative spectral data in quantum field theory contexts.

Abstract

We describe the exact WKB method from the point of view of abelianization, both for Schrödinger operators and for their higher-order analogues (opers). The main new example which we consider is the "$T_3$ equation," an order $3$ equation on the thrice-punctured sphere, with regular singularities at the punctures. In this case the exact WKB analysis leads to consideration of a new sort of Darboux coordinate system on a moduli space of flat $\mathrm{SL}(3)$-connections. We give the simplest example of such a coordinate system, and verify numerically that in these coordinates the monodromy of the $T_3$ equation has the expected asymptotic properties. We also briefly revisit the Schrödinger equation with cubic potential and the Mathieu equation from the point of view of abelianization.

Figures (21)

• Figure 1: Examples of $\vartheta$-Stokes graphs at $\vartheta=0$, with $p(z) = z^n-1$, for $n = 3, 4, 5$. The dashed lines denote branch cuts of the covering $\Sigma \to C$; the labels $i = 1,2$ are swapped when we cross a cut.
• Figure 2: A sample picture of what the active rays in the $\hbar$-plane can look like. There are in general infinitely many such rays, which can accumulate at discrete phases (as shown here) or even be dense in part or all of the $\hbar$-plane.
• Figure 3: Collapsing infinitely many active rays down to $2$ by making the choice \ref{['eq:theta-sg-type']}. Each active ray on the right carries functions $F_{r,\gamma}$ which should be thought of as containing the same information as all the $F_{r,\gamma}$ in the corresponding half-plane on the left.
• Figure 4: $\vartheta$-Stokes graph for the Schrödinger equation with cubic potential \ref{['eq:schrodinger-cubic-potential']}, at the phase $\vartheta = 0$, and $u = 1$. Two 1-cycles $\gamma_A$, $\gamma_B$ on $\Sigma$ are also shown. Dashed orange segments denote branch cuts; on crossing a cut, the sheet labels are exchanged $1 \leftrightarrow 2$. Orange crosses denote the turning points, zeroes of $p(z) = z^3 - 1$. The singularity at $z = \infty$ is not shown.
• Figure 5: The $6$ active rays in the $\hbar$-plane, each labeled by its charge $\mu \in H_1(\Sigma,\mathbb Z)$. These rays divide the $\hbar$-plane into $6$ regions. Each region is characterized by a different topology for the $\vartheta$-Stokes graph, where $\vartheta = \arg \hbar$.