2018 : msjmeeting-2018sep-11i001 WKB ( ) Eynard-Orantin WKB.,, Schrödinger WKB Voros, Painlevé (τ- ). 1. WKB,, WKB Voros WKB, Painlevé WKB. WKB, [ ],.

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018 : msjmeeting-018sep-11i001 WKB ( Eynard-Orantin WKB.,, Schrödinger WKB Voros, Painlevé (τ-. 1. WKB,, WKB Voros WKB, Painlevé WKB. WKB, [ ],. 1.1. WKB Voros Voros ([V] WKB (exact WKB analysis, ( 1 Schrödinger (ħ d dx Q(x ψ(x, ħ = 0 (1.1, ħ ( η = ħ 1. ħ (1.1 WKB ( 1 x 1 ψ(x, ħ = exp Q(x dx ħ 4 log Q(x + x ħ m S m (x dx (1. m=1 Borel. Stokes x- WKB Borel, exact WKB., Q(x. WKB S(x, ħ = d log ψ(x, ħ/dx = m= 1 ħm S m (x S 1 (x = Q(x, S 1 (xs m+1 (x + m 1 +m =m m 1,m 0 S 1 (xs 0 (x + ds 1(x dx S m1 (xs m (x + ds m(x dx = 0, (1.3 = 0 (1.4 ( :16K17613, 16H06337, 16K05177, 17H0617 010 Mathematics Subject Classification: 34M60, 34M55, 81T45 WKB, Voros, Painlevé, 464-860 e-mail: iwaki@math.nagoya-u.ac.jp

,. Q(x, Q(x., y = Q(x Riemann Σ. Σ (1.1, x-., m 1, Q(x S m (x., Q(x γ V γ (ħ = γ ( S(x, ħ ħ 1 S 1 (x S 0 (x dx = ħ m m=1 γ S m (x dx (1.5 (1.1 (γ Voros. Voros, WKB : : WKB, (1.1 (Stokes ([, ]., Σ α S(x, ħdx (α H 1(Σ; Z Borel,. ( Voros, Voros,. Stokes : Stokes, Borel WKB Borel ([V, DDP]. Stokes Stokes. ( 1.1 λ. Stokes Voros Borel ( [AKT, KoT]., Stokes WKB ([IN], Voros exp(v γ Borel. 1.1 Q(x, (1.1 Weber : ( ( ħ d x dx 4 λ ψ(x, ħ = 0. (1.6 λ C. (1.4 : S 1 (x = 1 x x 4λ, S 0 (x = (x 4λ, S 1(x = 3x 8λ 4(x 4λ, 5/ S (x = 3x(x + 6λ (x 4λ 4, S 3(x = 97x4 + 98x λ + 116λ 16(x 4λ 11/. S 1 (x x. λ 0, Σ P 1 x = 1,

Weber (1.6 Voros ([T]: 1 ( V (ħ = S(x, ħ ħ 1 ( 1 m 1B m S 1 (x S 0 (x dx = m(m 1, B m Bernoulli,. t e t 1 = m=0 m=1 B m m! tm. ( m 1 ħ. (1.7 λ Voros (1.7 Borel, Weber (1.6 x = Stokes., (1.7 Borel (alien derivative, Stokes exact. (, [IN] A 1 -. 1.. Painlevé -, (1.1 ([ ], Painlevé WKB ([KT], [, 4]., Painlevé I (P I : ħ d q dt = 6q + t, [ħ x ħ x q x ( 4x 3 + tx + H + ħ p ] ψ(x, t, ħ = 0 (1.8 x q [ ħ ( t 1 ħ ] (x q x p ψ(x, t, ħ = 0 (1.9 ( p = ħdq/dt, H = p / q 3 tq. -, (1.8 WKB Stokes, Painlevé -, Stokes, WKB, Painlevé Stokes. [I] Painlevé ( Voros Stokes,, - 1., WKB, Painlevé τ-. I, (P I q(t, ħ ħ d log τ(t, ħ = q(t, ħ (1.10 dt τ-. (q Wierstrass -, σ- τ-. 01 Lisovyy - ([GIL],. 1 018 (,. Lisovyy Painlevé VI Painlevé τ-. τ- ( ([N1, N], (P I, -.

1.3., WKB Voros, Painlevé WKB., WKB,, (topological recursion WKB.,, Eynard Orantin, (.. WKB 01 [GS, DM, BE]., [IKoT], Voros, (,., (1.7 Bernoulli, Harer-Zagier ([HZ] Riemann Euler Bernoulli (.5. Painlevé, [IM, IMS, IS], 0-3 q(t, ħ = m=0 ħm q m (t τ- ( τ(t, ħ = exp ħ g F g (t (1.11 g=0 (Lyon Oliver Marchal, Virginia Axel Saenz., WKB (1.4... Eynard-Orantin [EO1], Riemann C x, y, C {W g,n (z 1,..., z n } g 0,n 1 {F g } g 0., (C, x, y, W g,n, F g., W g,n F g (.4,. [EO],..1.,..1 ([BE, Definition.1]; cf. [EO1, 3] Riemann C, x, y : C P 1, dx dy (C, x, y. 3. -.

C = P 1, z, x, y z., R x : C P 1 (dx x,., r R z (x(z = x( y(z y(.,.. ([EO1, Definition 4.] (C, x, y, C W g,n (z 1,..., z n (g 0, n 1, (g, n- :, W 0,1 (z 1 = y(z 1 dx(z 1, (.1 dz 1 dz W 0, (z 1, z = (z 1 z, (. W g,n+1 (z 0, z 1,..., z n = [ Res K r(z 0, z W g 1,n+1 (z,, z 1,..., z n (.3 z=r r R + g 1 +g =g I J={1,...,n} K r (z 0, z = 1 w=z W w= 0,(w, z 0 (y(z y(dx(z ] W g1,1+ I (z, z I W g,1+ J (, z J. (.4, (.3 {1,..., n} (, I = {i 1, i,..., i m } {1,..., n} (i 1 < i < < i m W g,m+1 (z, z I = W g,m+1 (z, z i1,..., z im., (g 1, I = (0, (g, J = (0,. 1 C = P 1, C (H 1 (C, Z W 0, (z 1, z Bergman ([EO1, 3.1.5]. (. P 1 Bergman., Riemann (.1, χ = g + n. W g,n ([EO1]: W g,n z i C, g + n 1 C \ R. (, x(z, W g,n. W g,n z 1,..., z n (. x(z y(z λ, ( λ- W g,n λ., λ, W g,n λ ([EO1, 5].

z 0 z 1 z z n W g,n+1 (z 0, z 1,..., z n z 0 z z z 0 z1 z z n z 1 z z n K(z, z 0 W g 1,n+ (z,, z 1,..., z n K(z, z 0 W g1,1+ I (z, z I1 W g,1+ J (, z I.1: Riemann..3 ([EO1, Definition 4.3] (C, x, y, F g C ( g : F g = 1 g r R Res Φ(zW g,1(z (g. (.5 z=r, z o C \ R, Φ(z = z z o y(zdx(z. g = 0, 1 F 0, F 1, ([EO1, 4.., 4..3]. F g,, dx dy ([EO1, Section 7]. F g F (ħ = ħ g F g (.6, Z = exp(f..4 Airy g=0 x(z = z, y(z = z (.7 (R = {0, }, = z, g + n 1 W g,n (z 1,..., z n = 1 3g 3+n d 1 + +d n =3g 3+n τ d1,..., τ dn g,n n i=1 (d i 1!! dz z d i+1 i (.8 i ([E]., τ d1,..., τ dn g,n Q (1 {pt} Gromov- Witten, g Riemann M g,n ([Kon]: τ d1,..., τ dn g,n = M g,n c 1 (L 1 d1 c 1 (L n dn. Airy : F g = 0 (g.

.5 Weber x(z = λ 1/ (z + z 1, y(z = λ1/ (z z 1 (.9 (λ C, R = {1, 1}, = z 1, g., F g = χ(m g λ g (.10 χ(m g = B g g(g (.11 g Riemann Euler, Bernoulli ([HZ]..., WKB (quantum curve ([GS, DM, BE]., [BE] admissibile, W g,n WKB, Riemann (. WKB., Airy..6 ([Z] Airy (.7 W g,n, [ 1 z ψ(x, ħ = exp W 0,1(z + 1 z z( W 0, (z 1, z dx(z 1dx(z ħ (x(z 1 x(z + ħ g +n z n! g 0,n 1 g +n 1 z W g,n (z 1,..., z n ] (.1 z=z(x ( z(x = x, x : C P 1, Airy (ħ d dx x ψ = 0 (.13 WKB., WKB (1.4 (.3. Airy WKB, ((.8,,., ψ(x, ħ (.13 y x = 0 Airy (.7., (.1 (.13 4., WKB Voros,?.,,. 4, (.1 Schrödinger Q = Q 0 (x+ħq 1 (x+ħ Q (x+ ħ-. [BE] admissibility.

3. Voros [IKoT], : (ħ d Q(x, ħ ψ = 0, Q(x, ħ = x + 4λ ħ 1 dx 4x 4x. (3.1 Bessel Schrödinger, Bessel. ([IKoT] Gauss,. λ. y = (x + 4λ /(4x, x(z = 4λ (z 1, y(z = z 4λ(z 1 (3., Bessel. (R = {0, }, = z. W g,n F g, Bessel (3.., : 3.1 ([IKoT, 4] (a [ 1 ψ(x, ħ = exp ħ z W 0,1(z + 1 z z( W 0, (z 1, z dx(z 1dx(z (x(z 1 x(z + z g 0,n 1 g +n 1 ħ g +n n! n Bessel (3.1 WKB. z W g,n (z 1,..., z n ] (3.3 z=z(x (b Bessel (3.1 Σ x = 0 Bessel (3.1 Voros V (λ, ħ, Bessel (3. F (λ, ħ = g=0 ħg F g (λ, : V (λ, ħ = F (λ + ħ, ħ F (λ ħ, ħ ħ 1 F 0(λ λ. (3.4 (c Bessel (3. 3 F (λ + ħ, ħ F (λ, ħ + F (λ ħ, ħ = log (56λ (λ ħ λ (3.5., F g : B g F g (λ = g(g 1 (λ g (g (3.6 (3.3, (3.4,., (3.6

. Gauss ( λ 0, λ 1, λ { B g 1 F g = g(g (λ 0 + λ 1 + λ + 1 g (λ 0 + λ 1 λ + 1 g (λ 0 λ 1 + λ g } 1 + (λ 0 λ 1 λ 1 g (λ 0 1 g (λ 1 1 (g g (λ g Voros Bernoulli., Voros, (, Voros. 4. Painelvé τ- Painlevé τ- WKB,, WKB, τ- [IM, IS, IMS]. (P I, [IS]. (P I 0- q(t, ħ = m=0 ħm q m (t (1.8 y = 4(x q 0 (t (x + q 0 (t (4.1 ( q 0 (t = t/6 0 5. x(z = z q 0 (t, y(z = z(z 3q 0 (t (4. (, Painlevé t, W g,n F g., t = 0 dx dy,, t 0. (, t = 0 [KT] (P I.,. 4.1 ([IS] (a [ 1 ψ(x, t, ħ = exp ħ z W 0,1(z + 1 z z( W 0, (z 1, z dx(z 1dx(z (x(z 1 x(z + z g 0,n 1 g +n 1 ħ g +n n! n z W g,n (z 1,..., z n ] (4.3 z=z(x, 0- (1.8, (1.9 WKB. 5 1, (4.1. Painlevé, 0-0. [KT].

(b (4. ( exp(f (t, ħ = exp g=0 ħ g F g (t (4.4 (P I τ-., : q(t, ħ = ħ d F (t, ħ (4.5 dt (a, (P I (1.8., q p ħ-, (1.8 ħ-. Painlevé,. [BCD], admissible, Painlevé ħ-. - - ([JMU], Painlevé (1.8, (1.9 τ-,, (a τ- (b., (b [IMS] 6 Painlevé.,, 0-, Painlevé 1,., Voros Painlevé,., Riemann-Hilbert,., ( WKB exact Stokes,,. [AKT] Aoki, T., Kawai, T. and Takei, Y., The Bender-Wu analysis and the Voros theory, II, Adv. Stud. Pure Math. 54 (009, Math. Soc. Japan, Tokyo, 009, pp. 19 94. [BCD] Bouchard, V., Chidambaram, N. and Dauphinee, T., Quantizing Weierstrass; arxiv:1610.005. [BE] Bouchard, V. and Eyanard, B., Reconstructing WKB from topological recursion, Journal de l Ecole polytechnique Mathematiques, 4 (017, pp. 845 908. [DDP] Delabaere, E., Dillinger, H. and Pham, F., Résurgence de Voros et périodes des courves hyperelliptique, Annales de l Institut Fourier, 43 (1993, 163 199. [DM] Dumitrescu, O. and Mulase, M., Lectures on the topological recursion for Higgs bundles and quantum curves; arxiv:1509.09007. [E] Eynard, B., Intersection numbers of spectral curves; arxiv:1104.0176. [EO1] Eynard, B. and Orantin, N., Invariants of algebraic curves and topological expansion, Communications in Number Theory and Physics, 1 (007, pp. 347 45; arxiv:mathph/070045. [EO] Eynard, B. and Orantin, N., Topological recursion in enumerative geometry and random matrices, J. Phys. A: Math. Theor. 4 (009, 93001 (117pp.

[GIL] Gamayun, O., Iorgov, N. and Lisovyy, O., Conformal field theory of Painlevé VI, JHEP 10 (01, 038; arxiv:107.0787. [GS] Gukov, S. and Su lkowski, P., A-polynomial, B-model, and quantization, JHEP, 01 (01, 70. [HZ] Harer, J. and Zagier, D., The Euler characteristic of the moduli space of curves, Invent. Math., 85 (1986, 457 485. [I] Iwaki, K. Parametric Stokes phenomenon for the second Painlevé equation with a large parameter, Funkcial. Ekvac., 57 (014, 173 43. [IKoT] Iwaki, K., Koike, T. and Takei, Y., Voros Coefficients for the Hypergeometric Differential Equations and Eynard-Orantin s Topological Recursion; arxiv:1805.10945. [IM] Iwaki, K. and Marchal, O., Painlevé equation with arbitrary monodromy parameter, topological recursion and determinantal formulas, Ann. Henri Poincaré, 18 (017, 581 60; arxiv:1411.0875. [IMS] Iwaki, K., Marchal, O. and Saenz, A., Painlevé equations, topological type property and reconstruction by the topological recursion, J. Geom. Phys., 14 (018, 16 54; arxiv:1601.0517. [IN] Iwaki, K. and Nakanishi. T., Exact WKB analysis and cluster algebras, J. Phys. A: Math. Theor. 47 (014, 474009. [IS] Iwaki, K. and Saenz, A., Quantum Curve and the First Painlevé Equation, SIGMA., (1, 016; arxiv:1507.06557. [JMU] Jimbo, M., Miwa, T. and Ueno, K., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. General theory and τ-function, Phys. D (1981, 306 35. [KT] Kawai, T. and Takei, Y., WKB analysis of Painlevé transcendents with a large parameter. I, Adv. Math. 118 (1996, 1 33. [KoT] Koike, T. and Takei, Y., On the Voros coefficient for the Whittaker equation with a large parameter Some progress around Sato s conjecture in exact WKB analysis, Publ. RIMS, Kyoto Univ. 47 (011, pp. 375 395. [Kon] Kontsevich, M., Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys., 147 (199, 1 3. [N1] Nagoya, H., Irregular conformal blocks, with an application to the fifth and fourth Painlevé equations, J. Math. Phys. 56 (015, 13505; arxiv:1505.0398. [N] Nagoya, H., Remarks on irregular conformal blocks and Painlevé III and II tau functions; arxiv:1804.0478. [T] [V] [Z] Takei, Y., Sato s conjecture for the Weber equation and transformation theory for Schrödinger equations with a merging pair of turning points, RIMS Kôkyurôku Bessatsu, B10 (008, pp. 05 4. Voros, A., The return of the quartic oscillator The complex WKB method, Ann. Inst. Henri Poincaré, 39 (1983, pp. 11 338. Zhou, J., Intersection numbers on Deligne-Mumford moduli spaces and quantum Airy curve; arxiv:106.5896. [ ],,, 009. [ ],.,,, 1998. (English version: Kawai, T. and Takei. Y., Algebraic Analysis of Singular Perturbation Theory, American Mathematical Society, Translations of Mathematical Monographs, vol. 7. 005. [ ],,,,, 750. 1991, PP. 43 51 (.

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