Quantum Curve and the First Painlev\'e Equation

We show that the topological recursion for the (semi-classical) spectral curve of the first Painlev\'e equation $P_{\rm I}$ gives a WKB solution for the isomonodromy problem for $P_{\rm I}$. In other words, the isomonodromy system is a quantum curve in the sense of [Dumitrescu O., Mulase M., Lett. Math. Phys. 104 (2014), 635-671, arXiv:1310.6022] and [Dumitrescu O., Mulase M., arXiv:1411.1023].


Introduction
Painlevé transcendents are remarkable special functions which appear in many areas of mathematics and physics (e.g., [17]). These are solutions of certain nonlinear ordinary differential equations known as Painlevé equations. These equations were discovered by Painlevé and Gambier more than 100 years ago [34], and solutions have the so-called Painlevé property; i.e., any movable singularity must be a pole. One particular property of the Painlevé equations is existence of the Lax pair; that is, each Painlevé equation describes an isomonodromic deformation of a certain meromorphic linear ordinary differential equation [20,21]. The monodromy data of the linear ODEs gives a conserved quantity of the Painlevé transcendents. The Riemann-Hilbert method, as well as exact WKB analysis are applied to analyze the properties of Painlevé transcendents [4,17,24,25,36].
On the other hand, quantum curves attract both mathematicians and physicists since they are expected to encode the information of many quantum topological invariants, such as Gromov-Witten invariants, quantum knot invariants etc. These are concieved in physics literature including [1,2,10,18]. A quantum curve is an ordinary differential (or difference) equation containing a formal parameter (which plays the role of the Planck constant), like a Schrödinger equation. The quantum invariants appear in the coefficients of the WKB (Wentzel-Kramers-Brillouin) solution of the quantum curve.
The Eynard-Orantin's topological recursion introduced in [16] is closely related to both of the quantum curves and Painlevé equations (and many other topics). Topological recursion is a recursive algorithm to compute the 1/N -expansion of the correlation functions and the partition function of matrix models from its spectral curve, and it is generalized to any algebraic curve which may not come from a matrix model. In this context, quantum curves were first discussed in [6] for the Airy spectral curve, and generalized to spectral curves with various backgrounds (see [11,12,13,18,29] and the survey article [31]). The spectral curves are recovered as the semi-classical limit → 0 of the quantum curves. Moreover, the topological recursion is also closely related to integrability [5,7,19] as is the relationship between matrix models and integrable systems [9,28].
The aim of this paper is to relate quantum curves and the first Painlevé equation with a formal parameter P I : 2 d 2 q dt 2 = 6q 2 + t.
The (semi-classical) spectral curve for the isomonodormy system associated with P I is given by where q 0 = q 0 (t) is an explicit function of t. This is a family of algebraic curves in (x, y)space parametrized by t. (The curve (1.1) appeared in [16,Section 10.6] as the spectral curve of (3,2)-minimal model.) Our main result claims that, starting from the spectral curve (1.1), its quantization through the Eynard-Orantin's topological recursion (in the sense of [11,12]) recovers the whole isomonodoromy system for P I . The precise statement of our main theorem is as follows. Let W g,n (z 1 , . . . , z n ) be the Eynard-Orantin differential of type (g, n) defined from the spectral curve (1.1) (see Section 3.1). These are meromorphic multi-differential forms, and z i 's are copies of a coordinate on the spectral curve (1.1). W g,n 's also depend on t since the spectral curve depends on t. Then, our main result states the following. Let us consider the first Painlevé equation with a formal parameter : The equation P I is obtained from d 2q dt 2 = 6q 2 +t via the rescalingt = −4/5 t,q = −2/5 q. We will regard as a small parameter (i.e., Planck's constant), and investigate a particular formal solution of P I which has an -expansion.
2.1 Formal solution of P I P I has the following formal power series solution: It contains only even order terms of since P I is invariant under → − . The leading term q 0 = q 0 (t) satisfies and the subleading terms are recursively determined by As we will see, the coefficients of the formal series appearing in this paper are multivalued functions of t and are defined on the Riemann surface of q 0 . Thus, in what follows, we may use q 0 instead of t when we express coefficients. The relation (2.3) implies It is obvious that the coefficients q 2k (t) have a singularity at q 0 = 0 (i.e., t = 0). This special point is called a turning point of P I [25, Definition 2.1] (see also [26,Section 4]). Throughout the paper, we assume the following: Assumption 2.1. The independent variable t of P I lies on a domain that doesn't contain the origin.
Remark 2.2. The formal solution (2.1) is called a 0-parameter solution of P I in [26] since it doesn't contain free parameters. More general formal solutions having one or two free parameters (called 1-or 2-parameter solutions) are constructed in [4] for all Painlevé equations of second order. See also [3] for a construction of general formal solutions of higher order Painlevé equations.
Remark 2.3. The formal solution (2.1) is in fact a divergent series. However, [23, Theorem 1.1] proved that the formal solution is Borel summable when q 0 satisfies q 0 = 0 and arg q 0 / ∈ { 2 5 π | ∈ Z}. The exceptional set is called the Stokes curve of P I . (See [25, Definition 2.1] for the notion of Stokes curves of Painlevé equations with a small parameter .) That is, there exists a function which is analytic in on a sectorial domain with the center at the origin (which is also analytic in t) such that (2.1) is the asymptotic expansion of the function for → 0 in the sector. The analytic function is called the Borel sum of the formal series (2.1), and it gives an analytic solution of P I (see [8] for Borel summation method). This particular asymptotic solution obtained by the Borel summation method is called the tri-tronquée solution of P I (see [22]), and the non-linear Stokes phenomena on Stokes curves are analyzed by [17,24,36].

Isomonodromy system and the τ -function
It is known that P I describes the compatibility condition for the following system of linear PDEs (cf. [21, Appendix C]): The compatibility condition is equivalent to the following Hamiltonian system where the (time-dependent) Hamiltonian is given by We can easily check that (2.5) and P I are equivalent. The above system of linear ODEs is called the isomonodromy system associated with P I (see [20,21]). Let (q, p) = (q(t, ), p(t, )) be a formal power series solution of the Hamiltonian system (2.5); that is, q(t, ) is the formal solution (2.1) of P I , and The corresponding Hamiltonian function is denoted by We can check that (2.6) is invariant under → − , and hence it has the following expansion: 2n σ 2n (t).

Spectral curve
In what follows, we assume that the formal solution (q(t, ), p(t, )) of (2.5) constructed above is substituted into the coefficients of the isomonodromy system (2.4). Then, the coefficients of the isomonodromy system has the following -expansions: whose top terms are given by Observe that, since q 0 satisfies (2.2), the algebraic curve defined by has genus 0. Actually, this gives a family of algebraic curves in C 2 (x,y) parametrized by t. Since we have assumed that t = 0, x = q 0 and x = −2q 0 are distinct. Definition 2.5. We call the algebraic curve (2.9) the semi-classical spectral curve, or the spectral curve of (the first equation of) the isomonodromy system (2.4).
Remark 2.6. It is shown in [25,Proposition 1.3] that, for all (second order) Painlevé equations with a formal parameter , the semi-classical spectral curves corresponding to the same type of formal power series solution as (2.1) have genus 0.
Remark 2.7. Since we are taking the semi-classical limit (i.e., top term in -expansion), our spectral curve (2.9) is different from usual spectral curves for isomonodromic deformation equations discussed, e.g., in [33,35]. The spectral curves in the above papers have higher genus. Recently, Nakamura [30] investigates the geometry of genus 2 spectral curves which appear in an autonomous limit of the 4th order Painlevé equations, and use them to classify the Painlevé equations. See [27] for the list of 4th order Painlevé equations.

WKB analysis of isomonodromy system in scalar form
Denote the unknown vector function of (2.4) by Ψ = t (ψ 1 , ψ 2 ). Then, ψ = ψ 1 satisfies the following scalar version of isomonodromy system The coefficients of f and g have an -expansion since q and p are contained in them (2.12) The top term of g appears in the defining equation of the spectral curve (2.9), and its zeros are called turning points of the first equation of (2.10) in the WKB analysis. In particular, under the assumption t = 0, there is • a simple turning point at x = −2q 0 which is a branch point of the spectral curve (2.9), and • a double turning point at x = q 0 which is a singular point of the spectral curve (2.9).
Consider the Riccati equation This is equivalent to the first equation in (2.10) by Let be the formal solutions of (2.13) with the top term where f a and g a are the coefficient of a in f and g, respectively. Explicit forms of the first few terms are given by It is obvious from (2.14) that P m (x, t) are holomorphic except at the turning points and x = ∞ (and multivalued for even m). It also follows from the recursion relation (2.14) that holds when x → ∞.
and we have (2.15) after summing up m−1 P m (x, t). Once you know that P (±) (x, t, ) has an asymptotic expansion in this sense, subleading terms in (2.15) can be computed from the Riccati equation (2.13).
It is easy to check that (cf. [26, Section 2]) hold. Here x is regarded as a coordinate on the spectral curve, and σ is the covering involution for the spectral curve: Figure 1. For a given x, the path γ x starts from the point σ(x) and ends at x. The wiggly lines designate a branch cut, and the solid (resp. dotted) part represents a part of path on the first (resp. the second) sheet of the spectral curve.
the right hand-side of (2.16) is the derivative of the formal power series Thus the ambiguity of the branch of the logarithm only appears in the top term, but we care about the ambiguity since it doesn't matter in our computation.
The following theorem was applied in the transformation theory of Painlevé equations in [25]. We will use the fact in the proof of our main theorem.
In particular, P odd (x, t, ) satisfies (ii) All coefficients of P (±) (x, t, ) are holomorphic except at the simple turning point x = −2q 0 and x = ∞. In particular, they are holomorphic at the double turning point x = q 0 .
Here v is the simple turning point −2q 0 . The integral from v is defined by
Proof . Although the scalar version of isomonodromy system (2.10) is different from that used in [25], they are related by a gauge transformation ψ → (x − q) 1/2 ψ. Therefore, the equalities (2.18) and (2.19) in (i) together with the holomorphicity of each coefficient of P odd (x, t, ) at x = q 0 follows from [25, Proposition 1.2 and Theorem 1.1]. Then, it turns out that the coefficients of P odd (x, t, )/(x − q(t, )) are also holomorphic due to (2.19). Then, (2.16) implies that each coefficient of P even (x, t, ) is also holomorphic at x = q 0 . Thus we have proved (ii). The claim (iii) follows from a straightforward computations As we will see below, an isomonodromic WKB solution such as (2.20) is constructed from just a family of algebraic curves (2.9) by the topological recursion ( [16]). In particular, the first equation in (2.10) gives a quantization of the spectral curve (2.9) in the sense of [11,12].
Remark 2.10. In the above computation the normalization (2.20) is essential. Since P odd is anti-invariant under the covering involution σ as (2.17) and the integral in (2.20) is defined as a contour integral (2.21), we don't need to take care of the branch point v in the computation .
Remark 2.11. We can also construct a WKB-type formal solution of matrix isomonodromy system (2.4). Definẽ Then, the matrix valued formal series gives a fundamental formal solution of the isomonodoromy system (2.4).

Topological recursion and quantum curve theorem
In this section we review the Eynard-Orantin's topological recursion [16] for our spectral curve (2.9), and formulate our main theorem.

Topological recursion
The topological recursion is an algorithm associating some differential forms W g,n and numbers F g given the following source data: • A plane curve (C, x, y): C is a compact Riemann surface, x, y : C → P 1 are meromorphic functions.
• The Bergman kernel B: It is a symmetric differential form on C × C with poles of order 2 along the diagonal, and satisfying some normalization conditions.
In our case, C = P 1 and x, y are rational functions which parametrize the spectral curve (2.9) Here z is a coordinate on P 1 . The Bergman kernel is given by since the spectral curve is of genus 0. Zeros of dx are called ramification points of the spectral curve (3.1). Our spectral curve has only one ramification point at z = 0.
The explicit form of some of Eynard-Orantin differentials are given as follows Eynard-Orantin differentials have the following properties (see [16]): • As a differential form on each variable z i , W g,n , for 2g − 2 + n ≥ 1, is holomorphic except for the ramification point 0 and may have a pole at 0.
• W g,n is symmetric; that is, they are invariant under any permutation of variables.
• W g,n is also holomorphic in t except for t = 0 (i.e., q 0 = 0). There is a formula for the derivative of W g,n with respect to t; see Section 3.5.
Explicit computation shows that We also introduce functions {S m (x, t)} m≥0 by Our main result is the following. satisfies both of the differential equations in scalar-version of the isomonodromy system (2.10).
That is, the formal series S(x, t, ) given by (3.5) satisfies the following differential equations which are equivalent to (2.10): Thus, the principal specialization (i.e., setting z i = z for all i = 1, .  4 . A full proof of Theorem 3.3 will be given in Section 4 together with that of Theorem 3.7 below. Remark 3.4. In the topological recursion (3.2), we take residues only at the ramification point z = 0. Thus W g,n 's defined here are different from those in [12]; in particular, our quantum curve (2.10) has infinitely many -corrections as in (2.11) and (2.12) (but recovers the same spectral curve in the semi-classical limit).

Closed free energies and the τ -function
The other main result of this paper is giving another proof of the known fact about the relationship between the closed free energies and the τ -function of P I (cf. [9,16]).

Definition 3.6 ([16, Definition 4.3]). Define the closed free energy
and z 0 is a generic point. Free energies F 0 and F 1 for g = 0, 1 are also defined but in a different manner (see [16,Sections 4

.2.2 and 4.2.3] for the definition).
Note that F g defined here is different from F g,n defined in the previous subsection. F g 's are also called symplectic invariants since they are invariant under symplectic transformations of the spectral curve (see [16]). Explicit computation shows that Theorem 3.7 ([9] and [16,Section 10.6]). The generating function of the free energy F g (t) gives a τ -function of P I : Namely, The proof will be given in Section 4. It is worth mentioning that the closed free energies specify one particular τ -function although there is an ambiguity in Definition 2.4. (3.10) Proof . Let us describe the behavior of the W g,n 's when q 0 → ∞ (i.e., t → ∞). When q 0 tends to ∞, no singular point of the integrand in the right hand-side of (3.2) on the z-plane hits the integration cycle γ 0 . Thus, we can show that holds since Φ(z) ∼ q 0 as q 0 → ∞ (but we can verify that F g for g ≥ 2 has a stronger decay in the above explicit computations). This completes the proof of (3.10).

Asymptotics of Eynard-Orantin dif fernetials
The rest of this section will be devoted to show some important properties of W g,n and F g,n .
Firstly, we will describe the asymptotic behavior of them near z i = ∞.
Proof . The first property (3.11) follows from the analyticity of W g,n at z i = ∞. The second property (3.12) follows from (3.11) immediately because F g,n (z 1 , . . . , z n ) doesn't have a constant term due to the definition (3.3).
As a corollary, the principal specialization of open free energies satisfies when z → ∞.

Dif ferential recursion for open free energies
Here we give a key theorem in the proof of our main results. We have the following differential recursion which is a modification of the one obtained in [11,12].
Remark 3.12. Note that the first two blocks in the right hand-side of (3.17) coincide with that obtained in [11,12]. Unlike the case of [11,12], we need more terms arising from z = s corresponding to the singular point (x, y) = (q 0 , 0) of the spectral curve (2.9) since it becomes a (simple) pole of the recursion kernel K(z, z 1 ). It also worth mentioning that the right hand-side of (3.17) doesn't have singularity at z j = s for j = 1, . . . , n.
Using this differential recursion, we can give an alternative expression of (3.16) as follows.
4 Proof of main theorems 4.1 Strategy for the proof What we will show here is that the formal series S(x, t, ) defined in (3.5) satisfies the system of equations (3.6) and (3.7). In addition, we will also prove the equality (3.9). These equalities will be proved by an induction as follows.
holds. Here P m (x, t) = P (+) m (x, t) is the coefficient of m−1 in the formal solution P (+) (x, t, ) of the Riccati equation (2.13) constructed in Section 2.4, and σ 2g is given in (2.7). Then, we have (A) The following equality holds for m = k and k + 1: The following equalities hold: It is obvious that our main theorems (Theorems 3.3 and 3.7) follow from the statements in (A) and (B). The rest of this section is devoted to give a proof of (A) and (B).

Proof of (A)
We emphasize that the results shown in Section 4.2.1 below are proved without using the assumption (4.1). We also note that we only use the second equality in assumption (4.1) in Section 4.2.2 to prove (A).
After the coordinate change z = z(x), the right hand-side becomes dx dz (z(x)) 2y(z(x)) a+b=m+1 a,b≥2 Then, the desired equality (4.7) follows from the above equality and Note that the right hand-side of (4.7) coincides with Thus, Lemma 4.2 relates the principal specialization of G g,n to the left hand-side of (4.2). Next we also relate them to the right hand-side of (4.2).

Proof of (B)
One of the desired equality (4.3) is proved as follows.
Lemma 4.6. Under the assumption (4.1), we have ∂S k ∂x (x, t) = P k (x, t), (4.14) Proof . The equality (4.2) for m = k and the second equality in the assumption (4.1) imply Thus ∂S k /∂x and P k satisfy the same equation (2.14) under our induction hypothesis. Then the uniqueness of the solution of (2.14) implies (4.14). Since S m (x) for m ≥ 2 decay when x → ∞ (cf. (3.13)), the equality (4.15) immediately follows from (4.14). Then, the equality (2.18) shows The last equality follows from the assumption (4.1) and the fact that P m (x, t)'s decay when x → ∞ for m ≥ 1 (see Remark 2.8), and lim x→∞ P 0 (x, t) (x − q 0 ) 2 = 0.
Since we have also already proved (4.2) for m = k + 1, we can prove (4.4) by the same discussion as the proof of Lemma 4.6 above. Then, finally we obtain Proof . It follows from the equality (4.11) (for the odd number m = k + 1) and Lemma  Note that the right hand-side doesn't depend on x. Then, thanks to the fact ∂S 0 ∂x x=q 0 = 0 and the holomorphicity of S m (x) and P m (x) at the double turning point x = q 0 (see Theorem 2.9), we have the desired equality (4.17) by substituting x = q 0 into (4.20).
This completes the proof of (B) and Theorem 4.1. Thus we have proved Theorems 3.3 and 3.7.
Remark 4.8. Since the spectral curve (2.9) has only one branch point, we have