2-Parameter $\tau$-Function for the First Painlevé Equation: Topological Recursion and Direct Monodromy Problem via Exact WKB Analysis

Abstract

We show that a 2-parameter family of $\tau$-functions for the first Painlevé equation can be constructed by the discrete Fourier transform of the topological recursion partition function for a family of elliptic curves. We also perform an exact WKB theoretic computation of the Stokes multipliers of associated isomonodromy system assuming certain conjectures.

Appendices

Appendix A. Weierstrass Functions and $\theta$-Functions

Here we summarize properties of Weierstrass elliptic functions and $\theta$-functions which are relevant in this paper. See [106] for more details.

1.1 Weierstrass functions

1.1.1 Weierstrass $\wp$-function

Let

$$\begin{aligned} \omega _A {:}{=} \oint _{A} \frac{dx}{\sqrt{4x^3 - g_2 x - g_3}},\quad \omega _B {:}{=} \oint _{B} \frac{dx}{\sqrt{4x^3 - g_2 x - g_3}} \end{aligned}$$

(A.1)

be the periods of smooth elliptic curve $\Sigma$ defined by $y^2 = 4x^3 - g_2 x - g_3$. Here A, B are generators of the first homology group $H_1(\Sigma , {\mathbb {Z}})$. We assume that $\tau {:}{=} \omega _B/\omega _A$ has a positive imaginary part. The coefficients $g_2$, $g_3$ are related to these periods by

$$\begin{aligned} g_2 = 60 \sum _{\omega \in \Lambda \setminus \{0\}} \frac{1}{\omega ^4}, \qquad g_3 = 140 \sum _{\omega \in \Lambda \setminus \{0\}} \frac{1}{\omega ^6}. \end{aligned}$$

(A.2)

Here $\Lambda = {{\mathbb {Z}}} \cdot \omega _A + {{\mathbb {Z}}} \cdot \omega _B$ be the lattice generated by the two complex numbers $\omega _A$ and $\omega _B$.

Under the notations, the Weierstrass elliptic function $\wp (z) (= \wp (z;g_2,g_3))$ is defined by

$$\begin{aligned} \wp (z) {:}{=} \frac{1}{z^2} + \sum _{\omega \in \Lambda \setminus \{0\}} \left( \frac{1}{(z-\omega )^2} - \frac{1}{\omega ^2} \right) . \end{aligned}$$

(A.3)

It is also constructed as the inverse function of the elliptic integral

$$\begin{aligned} z(x) {:}{=} \int ^{x}_{\infty } \frac{dx}{\sqrt{4x^3-g_2 x -g_3}}, \end{aligned}$$

(A.4)

and hence $\wp (z)$ is doubly-periodic function with periods $\omega _A$ and $\omega _B$. It has double pole at $z = m \, \omega _A + n \, \omega _B$ for any $(m,n) \in {\mathbb {Z}}^2$. We also note that $\wp (z)$ is an even function; that is $\wp (-z) = \wp (z)$.

The Weierstrass $\wp$-function satisfies the following non-linear ODE:

$$\begin{aligned} \left( \frac{d\wp }{dz}(z) \right) ^2 = 4 \wp (z)^3 - g_2 \, \wp (z) - g_3. \end{aligned}$$

(A.5)

Thus the $\wp$-function is used to parametrize elliptic curves. Moreover, differentiating the relation, we also have

$$\begin{aligned} \frac{d^2\wp }{dz^2}(z) = 6 \wp (z)^2 - \frac{g_2}{2}. \end{aligned}$$

(A.6)

1.1.2 Weierstrass $\zeta$-function

We also introduce the Weierstrass $\zeta$-function

$$\begin{aligned} \zeta (z) {:}{=} \frac{1}{z} + \sum _{\omega \in \Lambda \setminus \{0\}} \left( \frac{1}{z-\omega } + \frac{1}{\omega } + \frac{z}{\omega ^2} \right) , \end{aligned}$$

(A.7)

which satisfies $- \zeta '(z) = \wp (z)$. The function $\zeta (z)$ is not doubly-periodic, but it satisfies

$$\begin{aligned} \zeta (z + m \, \omega _A + n \, \omega _B) = \zeta (z) + m \, \eta _A + n \, \eta _B \quad (m,n \in {{\mathbb {Z}}}). \end{aligned}$$

(A.8)

The constants $\eta _{A}$ and $\eta _{B}$ are also expressed as elliptic integral (of the second kind):

$$\begin{aligned} \eta _{*} {:}{=} - \oint _{*} \frac{x dx}{\sqrt{4x^3-g_2 x -g_3}} = \frac{1}{2} \, \zeta \left( \frac{\omega _*}{2} \right) \quad (*\in \{A, B \}). \end{aligned}$$

(A.9)

The Riemann bilinear identity shows

$$\begin{aligned} \eta _A \cdot \omega _B - \eta _B \cdot \omega _A = 2\pi i. \end{aligned}$$

(A.10)

1.1.3 Weierstrass $\sigma$-function

The integral of the Weierstrass zeta function is expressed as the logarithm of the Weierstrass $\sigma$-function defined by

$$\begin{aligned} \sigma (z) {:}{=} z \cdot \prod _{\omega \in \Lambda \setminus \{ 0\}} \left( 1 - \frac{z}{\omega } \right) \cdot \exp \left( \frac{z}{\omega } + \frac{z^2}{2 \omega ^2} \right) . \end{aligned}$$

(A.11)

This satisfies $d \log \sigma (z) /dz = \zeta (z)$. $\sigma$-function possesses the following quasi-periodicity:

$$\begin{aligned}&\sigma (z+m\,\omega _A + n\, \omega _B) = (-1)^{m+n+mn} \cdot \exp \left( (m\,\eta _A + n\, \eta _B) \right. \nonumber \\&\left. \cdot \Bigl (z + \frac{m\,\omega _A + n\, \omega _B}{2} \Bigr ) \right) \cdot \sigma (z) \quad (m,n \in {{\mathbb {Z}}}). \end{aligned}$$

(A.12)

It is also known that the $\sigma$-function also satisfies the addition formula:

$$\begin{aligned} \frac{\sigma (z+w) \cdot \sigma (z-w)}{\sigma (z)^2 \cdot \sigma (w)^2} = \wp (w) - \wp (z). \end{aligned}$$

(A.13)

1.2 $\theta$-functions

The Riemann $\theta$-function is defined by

$$\begin{aligned} \theta (v,\tau ) {:}{=} \sum _{k \in {{\mathbb {Z}}}} e^{2\pi i k v + \pi i k^2 \tau }. \end{aligned}$$

(A.14)

This is known to convergent uniformly on ${{\mathbb {C}}} \times {{\mathbb {H}}}$. We also use the $\theta$-functions with characteristics:

$$\begin{aligned} \theta _{00}(v,\tau ) {:}{=} \theta (v,\tau )&= \sum _{k \in {{\mathbb {Z}}}} e^{2\pi i k v + \pi i k^2 \tau } \end{aligned}$$

(A.15)

$$\begin{aligned} \theta _{01}(v,\tau )&{:}{=} \sum _{k \in {{\mathbb {Z}}}} e^{2\pi i k \bigl ( v+\frac{1}{2} \bigr ) + \pi i k^2 \tau } \end{aligned}$$

(A.16)

$$\begin{aligned} \theta _{10}(v,\tau )&{:}{=} \sum _{k \in {{\mathbb {Z}}}} e^{2\pi i \bigl (k+\frac{1}{2}\bigr ) v + \pi i \bigl (k+\frac{1}{2}\bigr )^2 \tau } \end{aligned}$$

(A.17)

$$\begin{aligned} \theta _{11}(v,\tau )&{:}{=} \sum _{k \in {{\mathbb {Z}}}} e^{2\pi i \bigl (k+\frac{1}{2}\bigr ) \bigl (v+\frac{1}{2}\bigr ) + \pi i \bigl (k+\frac{1}{2}\bigr )^2 \tau }. \end{aligned}$$

(A.18)

The parity of these functions are

$$\begin{aligned} \theta _{00}(-v,\tau )= & {} \theta _{00}(v,\tau ),\quad \theta _{01}(-v,\tau ) = \theta _{01}(v,\tau ),\nonumber \\ \theta _{10}(-v,\tau )= & {} \theta _{10}(v,\tau ),\quad \theta _{11}(-v,\tau ) = -\theta _{11}(v,\tau ). \end{aligned}$$

(A.19)

$\theta$-functions with characteristics satisfies various relations. We will use the following identity

$$\begin{aligned}&\theta _{00}(X+Y, \tau ) \cdot \theta _{00}(X-Y, \tau ) \cdot \theta _{00}(0,\tau )^2 \nonumber \\&\quad = \theta _{00}(X,\tau )^2 \cdot \theta _{00}(Y, \tau )^2 + \theta _{11}(X,\tau )^2 \cdot \theta _{11}(Y,\tau )^2 \end{aligned}$$

(A.20)

in the proof of our main result.

The relation to the Weierstrass $\sigma$-function is given as

$$\begin{aligned} \sigma (z) = \exp \left( \frac{\eta _A}{2 \omega _A} \cdot z^2 \right) \cdot \frac{\omega _A}{\theta _{11}'(0,\tau )} \cdot \theta _{11}\left( \frac{z}{\omega _A},\tau \right) . \end{aligned}$$

(A.21)

By taking the logarithm derivative, we have

$$\begin{aligned} \wp (z)= & {} - \frac{\eta _A}{\omega _A} - \frac{1}{\omega _A^2} \cdot \left[ \frac{\partial ^2}{\partial v^2} \log \theta _{11}(v,\tau ) \right] _{v = \frac{z}{\omega _A}}\nonumber \\= & {} - \frac{\eta _A}{\omega _A} - \frac{1}{\omega _A^2} \cdot \left[ \frac{\partial ^2}{\partial v^2} \log \theta _{00}(v,\tau ) \right] _{v = \frac{z}{\omega _A} - \frac{1}{2} - \frac{\tau }{2}}. \end{aligned}$$

(A.22)

The last equality follows from the relation

$$\begin{aligned} \theta _{00}(v,\tau ) = - e^{\pi i v + \frac{i \pi \tau }{4}} \cdot \theta _{11}\Bigl ( v+\frac{1}{2} + \frac{\tau }{2},\tau \Bigr ). \end{aligned}$$

(A.23)

Appendix B. Proof of Theorem 3.7

Here we give a proof of Theorem 3.7 which plays an important role in the proof of our main result. We use the same notation used in Sect. 3.2.3. (For example, the symbol $z_{[{\hat{j}}]}$ for $j = 1,\dots , n$ means the $(n-1)$-tuple of variables $(z_1, \dots , {\hat{z}}_{j}, \dots , z_n)$ without j-th entry.)

Lemma B.1

The function $F_{g,n}(z_1, \dots , z_n)$ defined in (3.39) satisfies the following equality for $2g-2+n \ge 1$:

$$\begin{aligned} \frac{\partial F_{g,n}}{\partial z_{1}}(z_{1},\dots ,z_{n}) = G_{g,n}(z_1, \dots , z_n) - \frac{1}{\omega _A} \oint _{z \in A} \frac{R_{g,n}(z,z_2,\dots ,z_n)}{(y(z) - y({\bar{z}})) \cdot dx(z)}, \end{aligned}$$

(B.1)

with $G_{g,n}$ and $R_{g,n}$ being given as follows:

• For $2g-2+n = 1$, we set

  $$\begin{aligned}&G_{0,3}(z_1,z_2,z_3) {:}{=} - \sum _{j=2}^{3} \bigl ( P(z_1+z_j) - P(z_1-z_j) \bigr ) \nonumber \\&\qquad \times \biggl ( \frac{1}{2y(z_{1}) \cdot \frac{dx}{dz}(z_{1})} \cdot \frac{\partial F_{0,2}}{\partial z_{1}}(z_{[{\hat{j}}]}) - \frac{1}{2y(z_{j}) \cdot \frac{dx}{dz}(z_{j})} \cdot \frac{\partial F_{0,2}}{\partial z_{j}}(z_{[{\hat{1}}]}) \biggr ) \nonumber \\&\quad +\, \frac{1}{y(z_1) \cdot \frac{dx}{dz}(z_1)} \cdot \frac{\partial F_{0,2}}{\partial z_1}(z_1, z_2) \cdot \frac{\partial F_{0,2}}{\partial z_1}(z_1, z_3), \end{aligned}$$

  (B.2)
  $$\begin{aligned}&G_{1,1}(z_1) {:}{=} - \frac{1}{2y(z_{1}) \cdot \frac{dx}{dz}(z_{1})} \cdot \frac{\partial ^2}{\partial u_1 \partial u_2} F_{0,2}(u_1, u_2) \biggl |_{u_{1}=u_{2}=z_{1}}, \end{aligned}$$

  (B.3)

  with P(z) being given in (3.13), and for $2g-2+n \ge 2$, we set

  $$\begin{aligned}&G_{g,n}(z_1, \dots , z_n) \nonumber \\&\quad {:}{=} - \sum _{j=2}^{n} \bigl ( P(z_1+z_j) - P(z_1-z_j) \bigr ) \nonumber \\&\qquad \times \biggl ( \frac{1}{2y(z_{1}) \cdot \frac{dx}{dz}(z_{1})} \cdot \frac{\partial F_{g,n-1}}{\partial z_{1}}(z_{[{\hat{j}}]}) - \frac{1}{2y(z_{j}) \cdot \frac{dx}{dz}(z_{j})} \cdot \frac{\partial F_{g,n-1}}{\partial z_{j}}(z_{[{\hat{1}}]}) \biggr ) \nonumber \\&\qquad -\, \frac{1}{2y(z_{1}) \cdot \frac{dx}{dz}(z_{1})} \cdot \frac{\partial ^{2}}{\partial u_{1} \partial u_{2}} \biggl ( F_{g-1,n+1}(u_{1},u_{2},z_{[{\hat{1}}]}) \nonumber \\& + \sum _{\begin{array}{c} g_{1}+g_{2}=g \\ I \sqcup J = [{\hat{1}}] \end{array}}^{\mathrm{stable}} F_{g_{1}, |I|+1}(u_{1},z_{I}) \cdot F_{g_{2}, |J|+1}(u_{2}, z_{J}) \biggr )\Biggr |_{u_{1}=u_{2}=z_{1}}. \end{aligned}$$

  (B.4)

• $R_{g,n}(z,z_{2},\dots ,z_{n})$ is a quadratic differential in the variable z (and functions of other variables $z_2, \dots , z_n$) defined by

  $$\begin{aligned} R_{0,3}(z, z_2, z_3)&{:}{=} \biggl (\int ^{z_{2}}_{0} W_{0,2}(z,z_{2}) \biggr ) \cdot \biggl (\int ^{z_{3}}_{0} W_{0,2}({\bar{z}},z_{3}) \biggr ) \nonumber \\&\qquad +\, \biggl (\int ^{z_{3}}_{0} W_{0,2}(z,z_{3}) \biggr ) \cdot \biggl (\int ^{z_{2}}_{0} W_{0,2}({\bar{z}},z_{2}) \biggr ), \end{aligned}$$

  (B.5)
  $$\begin{aligned} R_{1,1}(z)&{:}{=} W_{0,2}(z, {\bar{z}}), \end{aligned}$$

  (B.6)

  for $2g-2+n = 1$, and

  $$\begin{aligned}&R_{g,n}(z,z_{2},\dots ,z_{n}) \nonumber \\&{:}{=} \sum _{j=2}^{n} \Biggl [ \biggl (\int ^{z_{j}}_{0} W_{0,2}(z,z_{j}) \biggr ) \cdot \biggl (\int ^{z_{[{\hat{1}}, {\hat{j}}]}}_{0} W_{g,n-1}({\bar{z}}, z_{[{\hat{1}},{\hat{j}}]}) \biggr ) \nonumber \\&\quad +\, \biggl (\int ^{z_{j}}_{0}W_{0,2}({\bar{z}},z_{j}) \biggr ) \cdot \biggl (\int ^{z_{[{\hat{1}}, {\hat{j}}]}}_{0} W_{g,n-1}(z, z_{[{\hat{1}},{\hat{j}}]}) \biggr ) \Biggr ] \nonumber \\&\quad + \int ^{z_{[{\hat{1}}]}}_{0} W_{g-1,n+1}(z,{\bar{z}},z_{[{\hat{1}}]}) \nonumber \\&\quad + \sum _{\begin{array}{c} g_{1}+g_{2}=g \\ I \sqcup J = [{\hat{1}}] \end{array}}^{\text {stable}} \biggl ( \int ^{z_{I}}_{0} W_{g_{1}, |I|+1}(z,z_{I}) \biggr ) \cdot \biggl ( \int ^{z_{J}}_{0} W_{g_{2}, |J|+1}({\bar{z}}, z_{J}) \biggr ) \end{aligned}$$

  (B.7)

  for $2g-2+n \ge 2$. Here, for a set $L =\{\ell _{1}, \dots , \ell _{k} \} \subset \{1,\dots ,n \}$ of indices, we have used the notation

  $$\begin{aligned} \int ^{z_{L}}_{0} W_{g,n}(z_{1},\dots ,z_{n}) {:}{=} \int ^{z_{\ell _{1}}}_{0} \cdots \int ^{z_{\ell _{k}}}_{0} W_{g,n}(z_{1},\dots ,z_{n}). \end{aligned}$$

  (B.8)

Proof

First we show the claim in the case $2g-2+n \ge 2$. We employ a similar technique used in the proof of [61, Theorem 3.11].

Integrating the topological recursion relation (3.15) with respect to $z_{2}, \dots , z_{n}$, we have

$$\begin{aligned} \frac{\partial }{\partial z_{1}} F_{g,n}(z_{1},\dots ,z_{n}) \, dz_1&= \int _{0}^{z_{2}} \cdots \int _{0}^{z_{n}} W_{g,n}(z_{1},z_{2},\dots ,z_{n})\nonumber \\&= \frac{1}{2\pi i} \sum _{j=1}^{3}\oint _{\gamma _{j}} K(z_1,z) \cdot R_{g,n}(z,z_{1},\dots ,z_{n}). \end{aligned}$$

(B.9)

Note that, as a differential of z, $R_{g,n}$ has a simple pole at $z \equiv z_{1}, {\bar{z}}_{1}, \dots , z_{n}, {\bar{z}}_n$ modulo $\Lambda$, and no other poles except for the ramification points. Hence, the residue theorem implies

$$\begin{aligned}&\frac{\partial }{\partial z_{1}} F_{g,n}(z_{1},\dots ,z_{n}) \, dz_1 \nonumber \\&= - \sum _{i=1}^n \sum _{p=z_{i}, {\bar{z}}_{i}} \mathop {\hbox {Res}}\limits _{z=p} K(z_1,z) \cdot R_{g,n}(z,z_{2},\dots ,z_{n})\nonumber \\&\quad +\, \frac{1}{2\pi i} \oint _{z \in \partial D} K(z_1,z) \cdot R_{g,n}(z,z_{2},\dots ,z_{n}) \nonumber \\&= \sum _{i=1}^n \sum _{p=z_{i}, {\bar{z}}_{i}} \mathop {\hbox {Res}}\limits _{z=p} \frac{P(z_1+z) - P(z_1-z)}{4y(z) \cdot dx(z)} \nonumber \\&\qquad \cdot \Biggl ( \sum _{j=2}^{n} \Biggl [ \Bigl ( P_1(z+z_j) - P_1(z-z_j) \Bigr ) \cdot \frac{\partial }{\partial u} F_{g,n-1}(u, z_{[{\hat{1}},{\hat{j}}]}) \Biggr ] \Biggl |_{u=z} \nonumber \\&\quad +\, \frac{\partial ^2}{\partial u_1 \partial u_2} \biggl [ F_{g-1, n+1}(u_1, u_2, z_{[{\hat{1}}]})\nonumber \\&\quad +\, \sum _{\begin{array}{c} g_{1}+g_{2}=g \\ I \sqcup J = [{\hat{1}}] \end{array}}^{\text {stable}} F_{g_{1}, |I|+1}(u_1, z_{I}) \cdot F_{g_{2}, |J|+1}(u_2, z_{J}) \biggr ]\Biggl |_{u_1=u_2=z} \Biggr ) \, dz_1 \nonumber \\&\quad +\, \frac{1}{2\pi i}\oint _{z \in \partial \Omega } K(z_1,z) \cdot R_{g,n}(z,z_{2},\dots ,z_{n}). \end{aligned}$$

(B.10)

The last term is the integration along the boundary of the fundamental domain $\Omega$ of the elliptic curve.

The first two lines of the right hand-side of (B.10) coincides with $G_{g,n}(z_1, \dots , z_n)$ in the desired equality (B.1) (c.f., [27, Theorem 4.7]). On the other hand, since

$$\begin{aligned}&K(z_1, z+\omega _*) - K(z_1,z)\nonumber \\&\quad = {\left\{ \begin{array}{ll} 0 &{} \hbox { for}\ *= A \\ \displaystyle \frac{1}{(y(z) - y({\bar{z}})) \cdot dx(z)} \cdot \frac{2 \pi i}{\omega _A} \cdot dz_1 &{} \hbox { for}\ *= B, \end{array}\right. } \end{aligned}$$

(B.11)

the integration along $\partial \Omega$ is computed as follows:

$$\begin{aligned}&\frac{1}{2\pi i}\int _{z \in \partial D} K(z_1,z) \cdot R_{g,n}(z,z_{2},\dots ,z_{n}) = -\frac{dz_1}{\omega _A} \oint _{z \in A} \frac{R_{g,n}(z,z_{2},\dots ,z_{n})}{(y(z) - y({\bar{z}})) \cdot dx(z)}. \end{aligned}$$

(B.12)

Thus we have proved (B.1) for $2g-2+n \ge 2$.

The exceptional two cases $(g,n)=(0,3)$ and (1, 1) can be checked similarly by using the identity

$$\begin{aligned} \frac{\partial }{\partial z_1} F_{0,2}(z_1, z_2) \cdot dz_1 = \bigl ( P(z_1+z_2) - P(z_1) \bigr ) \cdot dz_1 = - \int ^{z_2}_{0} W_{0,2}({\bar{z}}_1, z_2)\nonumber \\ \end{aligned}$$

(B.13)

which immediately follows from the definition (3.34) of $F_{0,2}$. $\quad \square$

The following formula will be used to relate the A-cycle integral in the right hand-side of (B.1) with the t-derivatives.

Lemma B.2

For $2g-2+n \ge 1$, we have

$$\begin{aligned} \frac{\partial }{\partial t} F_{g,n-1}(z(x_{1}),\dots ,z(x_{n-1})) = E_{g,n-1}(z(x_1), \dots , z(x_{n-1})), \end{aligned}$$

(B.14)

where

$$\begin{aligned}&E_{g,n-1}(z_1, \dots , z_{n-1}) \nonumber \\&\quad {:}{=} - \frac{1}{\omega _A} \oint _{z \in A} \frac{R_{g,n}(z,z_1,\dots ,z_{n-1})}{(y(z) - y({\bar{z}})) \cdot dx(z)} + \sum _{j=1}^{n-1} \frac{P(z_j)}{y(z_{j}) \cdot \frac{dx}{dz}(z_{j})} \cdot \frac{\partial F_{g,n-1}}{\partial z_{j}}(z_1,\dots ,z_{n-1}). \end{aligned}$$

(B.15)

Proof

Replacing the label $z_1 \leftrightarrow z_n$ in (B.1) and taking the following residue around $z_n = 0$, we have

$$\begin{aligned}&\mathop {\hbox {Res}}\limits _{z_{n} = 0} \left( z_n^{-1} \cdot \frac{\partial F_{g,n}}{\partial z_{n}}(z_{1},\dots ,z_{n}) \cdot dz_n \right) \nonumber \\&\quad \quad = - \frac{1}{\omega _A} \oint _{z \in A} \frac{R_{g,n}(z,z_1,\dots ,z_{n-1})}{(y(z) - y({\bar{z}}))dx(z)} + \sum _{j=1}^{n-1} \frac{P(z_j)}{y(z_{j}) \cdot \frac{dx}{dz}(z_{j})} \cdot \frac{\partial F_{g,n-1}}{\partial z_{j}}(z_1,\dots ,z_{n-1}).\nonumber \\ \end{aligned}$$

(B.16)

Proposition 3.4 shows that the left hand-side is nothing but the t-derivation of $F_{g,n-1}$, if we use the x-coordinates. This completes the proof. $\quad \square$

Let us set ${\tilde{G}}_{g,n}(z_1,\dots ,z_n) {:}{=} \partial _{z_1} F_{g,n}(z_1,\dots , z_n) - G_{g,n}(z_1,\dots ,z_n)$. By a similar computation in [27, Theorem 6.5], we have

$$\begin{aligned}&\sum _{\begin{array}{c} g \ge 0,~ n \ge 1 \\ 2g-2+n = m \end{array}} \left[ \frac{2{\tilde{G}}_{g,n}(z_1,\dots ,z_n)}{{(n-1)!}} \right] _{z_1 = \cdots = z_n = z(x)} \nonumber \\&\quad = {\left\{ \begin{array}{ll} \displaystyle 2 y\bigl ( z(x) \bigr ) \cdot \frac{\partial S_m(x)}{\partial x} + \left( - \frac{\partial _x y\bigl (z(x)\bigr )}{y\bigl (z(x)\bigr )} \right) \cdot \frac{\partial S_{m-1}(x)}{\partial x} &{} \\ \displaystyle + \sum _{\begin{array}{c} m_1, m_2 \ge 1 \\ m_1 + m_2 = m-1 \end{array}} \frac{\partial S_{m_1}(x)}{\partial x} \cdot \frac{\partial S_{m_2}(x)}{\partial x} + \frac{\partial ^2 S_{m-1}(x)}{\partial x^2} &{} \quad \hbox { for}\ m \ge 2 \\ \displaystyle 2 y\bigl ( z(x) \bigr ) \cdot \frac{\partial S_1(x)}{\partial x} + \left( - \frac{\partial _x y\bigl (z(x)\bigr )}{y\bigl (z(x)\bigr )} \right) \cdot \frac{\partial S_{0}(x)}{\partial x} - \left( \frac{\partial S_{0}(x)}{\partial x} \right) ^2 + \frac{\partial ^2 S_{0}(x)}{\partial x^2} &{} \quad \hbox { for}\ m = 1. \end{array}\right. } \end{aligned}$$

(B.17)

(Recall that $S_m(x)$’s are defined in Sect. 3.4.) Here we have used the identity $y\bigl ( z(x) \bigr ) = \frac{dx}{dz}\bigl ( z(x) \bigr )$.

Next, using Lemmas B.1 and B.2, let us find another expression of the left hand-side of (B.17). First, we note

$$\begin{aligned}&\sum _{\begin{array}{c} g \ge 0,~ n \ge 1 \\ 2g-2+n = m \end{array}} \left[ \frac{2{\tilde{G}}_{g,n}(z_1,\dots ,z_n)}{{(n-1)!}} \right] _{z_1 = \cdots = z_n = z(x)} \nonumber \\&= \sum _{\begin{array}{c} g \ge 0,~ n \ge 1 \\ 2g-2+n = m \end{array}} \left[ \frac{2{\tilde{G}}_{g,n}(z_1,\dots ,z_n)}{{(n-1)!}} \right] _{z_1 = \cdots = z_n = z(x)} \nonumber \\&\qquad -\, \sum _{\begin{array}{c} g \ge 0,~ n \ge 2 \\ 2g-2+n = m \end{array}} \left[ \frac{2{\tilde{G}}_{g,n}(z_1,\dots ,z_n)}{{(n-1)!}} \right] _{z_1 = \cdots = z_n = z(x)} \nonumber \\&\qquad +\, \sum _{\begin{array}{c} g \ge 0,~ n \ge 2 \\ 2g-2+n = m \end{array}} \left[ \frac{2{\tilde{G}}_{g,n}(z_1,\dots ,z_n)}{{(n-1)!}} - \frac{2 E_{g,{n-1}}(z_1,\dots ,z_{n-1})}{{(n-1)!}} \right] _{z_1 = \cdots = z_n = z(x)} \nonumber \\&\qquad +\, \sum _{\begin{array}{c} g \ge 0,~ n \ge 2 \\ 2g-2+n = m \end{array}} \left[ \frac{2 E_{g,{n-1}}(z_1,\dots ,z_{n-1})}{{(n-1)!}} \right] _{z_1 = \cdots = z_n = z(x)}. \end{aligned}$$

(B.18)

The first line of the right hand-side is

$$\begin{aligned}&\sum _{\begin{array}{c} g \ge 0,~ n \ge 1 \\ 2g-2+n = m \end{array}} \left[ \frac{2{\tilde{G}}_{g,n}(z_1,\dots ,z_n)}{{(n-1)!}} \right] _{z_1 = \cdots = z_n = z(x)} \nonumber \\&\qquad -\, \sum _{\begin{array}{c} g \ge 0,~ n \ge 2 \\ 2g-2+n = m \end{array}} \left[ \frac{2{\tilde{G}}_{g,n}(z_1,\dots ,z_n)}{{(n-1)!}} \right] _{z_1 = \cdots = z_n = z(x)} \nonumber \\&\quad = {\left\{ \begin{array}{ll} 0 &{} \text {if }m\text { is even} \\ \displaystyle 2 \frac{\partial }{\partial t} F_{\frac{m+1}{2}} &{} \text {if }m\text { is odd} \end{array}\right. } \qquad = \quad \left[ 2 \frac{\partial F}{\partial t} (t,\nu ; \hbar ) \right] _{\hbar ^{m-1}}. \end{aligned}$$

(B.19)

(C.f., [61, Lemma 4.5].) The notation $[\bullet ]_{\hbar ^{k}}$ means the coefficient of $\hbar ^k$ in a formal power series $\bullet$ of $\hbar$. The second and third lines are also expressed as

$$\begin{aligned}&\sum _{\begin{array}{c} g \ge 0,~ n \ge 2 \\ 2g-2+n = m \end{array}} \left[ \frac{2{\tilde{G}}_{g,n}(z_1,\dots ,z_n)}{{(n-1)!}} - \frac{2 E_{g,{n-1}}(z_1,\dots ,z_{n-1})}{{(n-1)!}} \right] _{z_1 = \cdots = z_n = z(x)} \nonumber \\&\quad = - \sum _{\begin{array}{c} g \ge 0,~ n \ge 2 \\ 2g-2+n = m \end{array}} \left[ \sum _{j=1}^{n-1} \frac{2P(z_j)}{y(z_{j}) \cdot \frac{dx}{dz}(z_{j})} \cdot \frac{\partial F_{g,n-1}}{\partial z_{j}}(z_1,\dots ,z_{n-1}) \right] _{z_1 = \cdots = z_n = z(x)} \nonumber \\&\quad = - \frac{2P\bigl (z(x) \bigr )}{y\bigl ( z(x) \bigr )} \cdot \frac{\partial S_{m-1}(x)}{\partial x} \end{aligned}$$

(B.20)

and

$$\begin{aligned}&\sum _{\begin{array}{c} g \ge 0,~ n \ge 2 \\ 2g-2+n = m \end{array}} \left[ \frac{2 E_{g,{n-1}}(z_1,\dots ,z_{n-1})}{{(n-1)!}} \right] _{z_1 = \cdots = z_n = z(x)} \nonumber \\&\quad = 2 \sum _{\begin{array}{c} g \ge 0,~ n \ge 2 \\ 2g-2+n = m \end{array}} \frac{\partial }{\partial t} F_{g,n-1}(z(x),\dots , z(x)) ~=~ 2 \frac{\partial S_{m-1}(x)}{\partial t}, \end{aligned}$$

(B.21)

respectively (c.f., Lemma B.2). Combining (B.17)–(B.21), we have the following recursion relation satisfied by $S_m$’s:

$$\begin{aligned}&\sum _{\begin{array}{c} m_1, m_2 \ge -1 \\ m_1 + m_2 = m-1 \end{array}} \frac{\partial S_{m_1}(x)}{\partial x} \cdot \frac{\partial S_{m_2}(x)}{\partial x} + \frac{\partial ^2 S_{m-1}(x)}{\partial x^2}\nonumber \\&\quad = 2 \frac{\partial S_{m-1}(x)}{\partial t} + \left[ 2 \frac{\partial F}{\partial t} (t,\nu ; \hbar ) \right] _{\hbar ^{m-1}}. \end{aligned}$$

(B.22)

Here we used (3.31) and (3.37) to obtain the above expression. Although (B.22) is valid for $m \ge 1$ a-priori (because it is derived from (B.17) etc.), we can verify that it is also valid for $m=0$ thanks to the property (3.32). Together with the equation

$$\begin{aligned} \left( \frac{\partial S_{-1}(x)}{\partial x} \right) ^2 = y\bigl ( z(x) \bigr )^2 = 4x^3 + 2t x + u(t,\nu ) \end{aligned}$$

(B.23)

for the leading term, the recursion relations are summarized into a single PDE

$$\begin{aligned} \hbar ^{2}\left( \left( \frac{\partial S}{\partial x}\right) ^{2} + \frac{\partial ^{2} S}{\partial x^{2}} \right) = 2\hbar ^{2} \frac{\partial S}{\partial t} + \left( 4x^{3}+2t x + 2\hbar ^2 \frac{\partial F}{\partial t}(t,\nu ;\hbar ) \right) \end{aligned}$$

(B.24)

for S given in (3.30). The last equation is equivalent to the PDE (3.40), and hence, we have proved Theorem 3.7.