Exact WKB and Abelianization for the $T_3$ Equation

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.

A Computations of Spectral Coordinates

In this appendix we give some computations omitted from the main text.

1.1 A.1 Computations for the cubic potential

Computation of (3.3). We will only describe the computation for $\mathcal {X}_A$; that for $\mathcal {X}_B$ is similar.

We need to compute the parallel transport of $\nabla ^\mathrm {ab}$ along a path in the homology class $\gamma _A$. To compute concretely it is convenient to work relative to bases of $\nabla ^\mathrm {ab}$-flat sections in each domain. Each local $\nabla ^\mathrm {ab}$-flat section corresponds to a local $\nabla$-flat section, and by continuation we can think of all these local flat sections as lying in a single 2-dimensional vector space V, the space of global $\nabla$-flat sections over the plane. See Fig. 18.

Fig. 18

The Stokes graph from Fig. 4, with the local WKB bases shown in each domain. To write the basis concretely as an ordered pair of solutions we have used the trivialization of the double cover $\Sigma$ away from branch cuts; thus, in a domain containing a branch cut, we write two versions of the basis, one on each side of the cut

Relative to these local bases, the parallel transport within each domain is just represented by 1, and the only nontrivial part is the gluing factor from (2.11):

• When we cross a single wall of type ij on sheet i, from side L to side R, we get a factor

  $$\begin{aligned} \frac{[\psi _i^L, \psi _j^L]}{[\psi _i^R, \psi _j^L]}. \end{aligned}$$

  (A.1)

• When we cross a single wall of type ij on sheet j, we also get a gluing factor, but this factor is just 1 if $\psi _j^L = \psi _j^R$, which it always is in this example.

The representative of $\gamma _A$ shown in Fig. 18 crosses six walls; multiplying the factors for these six crossings, starting from the eastmost region, gives

$$\begin{aligned} \mathcal {X}_A = \frac{[\psi _5^{\mathrm {sm}}, \psi _1^{\mathrm {sm}}]}{[\psi _5^{\mathrm {sm}}, \psi _3^{\mathrm {sm}}]} \times 1 \times 1 \times \frac{[\psi _3^{\mathrm {sm}}, \psi _2^{\mathrm {sm}}]}{[\psi _1^{\mathrm {sm}}, \psi _2^{\mathrm {sm}}]} \times 1 \times 1 \end{aligned}$$

(A.2)

matching (3.3) as desired.

1.2 A.2 Computations for the Mathieu equation

Computation of (4.5). We need to compute the parallel transport of $\nabla ^\mathrm {ab}$ along a path in the homology class $\gamma _B$. We use the path given in Fig. 6.

As above, it is convenient to work relative to bases of $\nabla ^\mathrm {ab}$-flat sections in each domain. See Fig. 19. Again by continuation we think of all these local flat sections as lying in a single 2-dimensional vector space V. In this case there is an added technical difficulty: the monodromy around $z = 0$ means there are no global $\nabla$-flat sections. Instead we identify V as the space of $\nabla$-flat sections on the complement of the blue dashed line (“monodromy cut”).

Fig. 19

The Stokes graph from Fig. 6, with the local WKB bases shown in each domain. As before, to write the basis concretely as an ordered pair of solutions we have used the trivialization of the double cover $\Sigma$ away from branch cuts; thus, in a domain containing a branch cut, we write two versions of the basis, one on each side of the cut. When we cross the monodromy cut, the local WKB basis of $\nabla ^\mathrm {ab}$-flat sections does not change, but the way we identify them with elements of V does jump, by the action of the monodromy M

Again the only nontrivial part of the parallel transport is the gluing factors appearing in (2.11), (2.12), When we cross a double wall on sheet i, from side L to side R, we get a factor

$$\begin{aligned} \sqrt{\frac{[\psi _i^L , \psi _j^L]}{[\psi _i^R , \psi _j^R]} \frac{[\psi _i^L , \psi _j^R]}{[\psi _i^R , \psi _j^L]}}, \end{aligned}$$

(A.3)

and when we cross a single wall of type ij on sheet i, from side L to side R, we get a factor

$$\begin{aligned} \frac{[\psi _i^L , \psi _j^L]}{[\psi _i^R , \psi _j^L]}. \end{aligned}$$

(A.4)

We can further simplify these factors by choosing bases with $[\psi _1, \psi _2] = 1$, $[\psi '_1 , \psi '_2] = 1$, $[\psi ''_1, \psi ''_2] = 1$. Then starting from the southwest corner, the gluing factors we encounter are

$$\begin{aligned} \mathcal {X}_B = \sqrt{\frac{[\psi '_2 , \psi ''_1]}{[\psi ''_2 , \psi '_1]}} \times \sqrt{\frac{[\psi ''_2 , \psi _1]}{[\psi _2 , \psi ''_1]}} \times 1 \times \sqrt{\frac{[M\psi _1 , \psi ''_2]}{[\psi ''_1 , M \psi _2]}} \times \sqrt{\frac{[\psi ''_1 , M \psi '_2]}{[M \psi '_1 , \psi ''_2]}} \times 1. \end{aligned}$$

(A.5)

Using $M \psi ''_1 = \mu \psi ''_1$, $M \psi ''_2 = \mu ^{-1} \psi _2$, and $M \psi _1 = \psi _2$, $M \psi '_1 = \psi '_2$, this reduces to

$$\begin{aligned} \mathcal {X}_B = \frac{[\psi _1 , \psi ''_2]}{[\psi _1 , \psi ''_1]} \frac{[\psi '_1 , \psi ''_1]}{[\psi '_1 , \psi ''_2]} \end{aligned}$$

(A.6)

which matches the desired (4.5).

Computation of (4.21). Just as above, all we need to compute are the gluing factors along the paths $\gamma _A$ and $\gamma _B$, with respect to the bases shown in Fig. 20.

Fig. 20

The Stokes graph from Fig. 8, with local WKB bases shown in each domain. All notation is as in Fig. 19 above

We can choose $[\psi , \psi '] = 1$ to simplify. In going around $\gamma _A$ we only meet one wall, with the gluing factor

$$\begin{aligned} \mathcal {X}_A = \pm \sqrt{\frac{[M \psi ' , \psi ]}{[\psi ' , M \psi ]}}. \end{aligned}$$

(A.7)

To fix the branch we would need to carefully implement the WKB prescription from Sect. 2.3, which we do not do here.

For $\gamma _B$ the product of gluing factors, starting from the southeast, is

$$\begin{aligned} \mathcal {X}_B = \frac{[M \psi ' , \psi ']}{[\psi , \psi ']} \times 1 \times \frac{[M \psi , \psi ]}{[M \psi , M \psi ']} \times 1 = \frac{[M \psi ' , \psi '][M \psi , \psi ]}{[\psi , \psi ']^2}. \end{aligned}$$

(A.8)

The results (A.7), (A.8) match the desired (4.21).

1.3 A.3 Computations for the $T_3$ equation

Abelianizations and adapted bases. Suppose we have a $\mathcal {W}$-abelianization of the $T_3$ equation. Then we can choose bases compatible with the $\mathcal {W}$-abelianization in the various domains of Fig. 21, as shown.

Fig. 21

The Stokes graph from Fig. 10, with local WKB bases shown in each domain. The notation is as in the figures above

In writing the form of these bases we began by labeling the basis in the middle as $(\psi _1,\psi _2,\psi _3)$ and then used the facts that:

• According to (5.6) the k-th projective basis element does not change when we cross a wall of type ij and ji (this implies e.g. that the first basis element in the northeast region must be $\psi _1$),

• Crossing a branch cut of the covering $\Sigma \rightarrow C$ (orange in Fig. 21) permutes the projective basis elements,

• The projective bases on the two sides of a monodromy cut (blue in Fig. 21) differ by the monodromy ($\mathbf {A}$, $\mathbf {B}$ or $\mathbf {C}$) attached to the cut.

One key fact remains to be used: again by (5.6), for a wall of type ij and ji, the plane spanned by the i-th and j-th basis elements is the same on both sides of the wall. Applying this to the northeast wall, which is of type 23 and 32, leads to the condition that

$$\begin{aligned} \langle \psi _2,\psi _3\rangle = \langle \mathbf {C}^{-1} \psi _3, \mathbf {A}\psi _2\rangle , \end{aligned}$$

(A.9)

which is (6.3b); doing similarly for the other two walls gives the other two parts of (6.3). Thus, the basis $(\psi _1, \psi _2, \psi _3)$ is indeed a basis in special position. Conversely, given a basis $(\psi _1, \psi _2, \psi _3)$ in special position, the local bases shown in Fig. 21 give a $\mathcal {W}$-abelianization. This shows the claimed identification between $\mathcal {W}$-abelianizations and bases in special position.

Computation of (6.18). As above, all we need to compute are the gluing factors along the paths representing $\gamma _A$ and $\gamma _B$ shown in Fig. 13. These factors are given by (5.6): for a wall of type ij and ji, and a path on sheet i, the factor is

$$\begin{aligned} \sqrt{\frac{[\psi _i^L , \psi _j^L , \psi _k^L]}{[\psi _i^R , \psi _j^R , \psi _k^L]} \frac{[\psi _i^L , \psi _j^R , \psi _k^L]}{[\psi _i^R , \psi _j^L , \psi _k^L]}}. \end{aligned}$$

(A.10)

Since all the walls are double, we will not need to use (5.5) anywhere.

For $\gamma _A$ the computation is particularly simple: only two of the four crossings give a nontrivial factor, namely the places where the path crosses the 23–32 wall. This gives directly

$$\begin{aligned} \mathcal {X}_A&= \sqrt{\frac{[\psi _2 , \mathbf {A}\psi _2 , \psi _1]}{[\mathbf {A}\psi _2 , \mathbf {C}^{-1} \psi _3 , \psi _1]} \frac{[\psi _2 , \psi _3 , \psi _1]}{[\mathbf {A}\psi _2 , \psi _2 , \psi _1]}} \times \sqrt{\frac{[\psi _3 , \mathbf {C}^{-1} \psi _3 , \psi _1]}{[\mathbf {C}^{-1} \psi _3 , \mathbf {A}\psi _2 , \psi _1]} \frac{[\psi _3 , \psi _2 , \psi _1]}{[\mathbf {C}^{-1} \psi _3 , \psi _3 , \psi _1]}} \end{aligned}$$

(A.11)

$$\begin{aligned}&= \frac{[\psi _2 , \psi _3 , \psi _1]}{[\mathbf {C}^{-1} \psi _3 , \mathbf {A}\psi _2 , \psi _1]} \end{aligned}$$

(A.12)

matching (6.18a) as desired. The computation giving $\mathcal {X}_B$ is similar but a little longer since three of the four crossings give nontrivial factors: thus we have altogether 6 factors in numerator and denominator; one common factor cancels, leaving the desired (6.18b).