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.

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Notes

  1. 1.

    There are several differences between the usual asymptotic (i.e., \(t \rightarrow \infty \)) and the WKB asymptotics (i.e., \(\hbar \rightarrow 0\)) in general. For Painlevé I, these two asymptotic analysis are equivalent due to homogeneity. See Remark 4.6 below.

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Acknowledgements

The author is grateful to Alba Grassi, Akishi Ikeda, Nikolai Iorgov, Michio Jimbo, Akishi Kato, Taro Kimura, Alexander Kitaev, Tatsuya Koike, Oleg Lisovyy, Motohico Mulase, Hajime Nagoya, Hiraku Nakajima, Ryo Ohkawa, Nicolas Orantin, Hidetaka Sakai, Kanehisa Takasaki, Yoshitugu Takei, Yumiko Takei, Yasuhiko Yamada for many valuable comments, suggestions and discussion. He also would like to thank the referees who pointed out the relation between our result and Kitaev’s result [75]. This work was supported by the JSPS KAKENHI Grand Numbers 16K17613, 16H06337, 16K05177, 17H06127, and JSPS and MAEDI under the Japan-France Integrated Action Program (SAKURA). We dedicate the paper to the memory of Tatsuya Koike, who made a lot of important contributions to the theory of the exact WKB analysis and the Painlevé equations. He inspired us by his beautiful papers, talks, and private communications on various occasions.

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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.

Weierstrass functions

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 AB 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)

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)

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)

\(\theta \)-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.

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Iwaki, K. 2-Parameter \(\tau \)-Function for the First Painlevé Equation: Topological Recursion and Direct Monodromy Problem via Exact WKB Analysis. Commun. Math. Phys. 377, 1047–1098 (2020). https://doi.org/10.1007/s00220-020-03769-2

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