SPECTRAL CURVES AND THE SCHRÖDINGER EQUATIONS FOR THE EYNARD-ORANTIN RECURSION MOTOHICO MULASE AND PIOTR SU LKOWSKI

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1 SPECTRAL CURVES AND THE SCHRÖDINGER EQUATIONS FOR THE EYNARD-ORANTIN RECURSION MOTOHICO MULASE AND PIOTR SU LKOWSKI Abstract It is predicted that the principal specialization of the partition function of a B-model topological string theory, that is mirror dual to an A-model enumerative geometry problem, satisfies a Schrödinger equation, and that the characteristic variety of the Schrödinger operator gives the spectral curve of the B-model theory, when an algebraic K-theory obstruction vanishes In this paper we present two concrete mathematical A-model examples whose mirror dual partners exhibit these predicted features on the B-model side The A-model examples we discuss are the generalized Catalan numbers of an arbitrary genus and the single Hurwitz numbers In each case, we show that the Laplace transform of the counting functions satisfies the Eynard-Orantin topological recursion, that the B-model partition function satisfies the KP equations, and that the principal specialization of the partition function satisfies a Schrödinger equation whose total symbol is exactly the Lagrangian immersion of the spectral curve of the Eynard-Orantin theory Contents Introduction and the main results The Eynard-Orantin topological recursion 6 3 The generalized Catalan numbers and the topological recursion 0 4 The partition function for the generalized Catalan numbers and the Schrödinger equation 3 5 Single Hurwitz numbers 7 6 The Schur function expansion of the Hurwitz partition function 4 7 Conclusion 30 Appendix A Proof of the Schrödinger equation for the Catalan case 3 References 34 Introduction and the main results In a series of remarkable papers of Mariño [5] and Bouchard, Klemm, Mariño, and Pasquetti [8], these authors have developed an inductive mechanism to calculate a variety of quantum invariants and solutions to enumerative geometry questions, based on the fundamental work of Eynard and Orantin [7, 8, 9] The validity of their method, known as the remodeled B-model based on the topological recursion of Eynard-Orantin, has been established for many different enumerative geometry problems, such as single Hurwitz numbers [7, 6, 60], based on the conjecture of Bouchard and Mariño [9], open Gromov-Witten invariants of smooth toric Calabi-Yau threefolds [9, 80], based on the remodeling conjecture of Mariño [5] and Bouchard, Klemm, Mariño, Pasquetti [8], and the number of lattice points on M g,n and its symplectic and Euclidean volumes [, 0, 58], based on [63, 64] It is expected that double Hurwitz numbers, stationary Gromov-Witten invariants of P 000 Mathematics Subject Classification Primary: 4H5, 4N35, 05C30, P; Secondary: 8T30

2 M MULASE AND P SU LKOWSKI [65, 66], certain Donaldson-Thomas invariants, and many other quantum invariants would also fall into this category Unlike the familiar Topological Recursion Relations TRR of the Gromov-Witten theory, the Eynard-Orantin recursion is a B-model formula [8, 5] The significant feature of the formula is its universality: independent of the A-model problem, the B-model recursion takes always the same form The input data of this B-model consist of a holomorphic Lagrangian immersion ι : Σ T C π C of an open Riemann surface Σ called a spectral curve of the Eynard-Orantin recursion into the cotangent bundle T C equipped with the tautological -form η, and the symmetric second derivative of the logarithm of Riemann s prime form [30, 6] defined on Σ Σ The procedure of Eynard-Orantin [7] then defines, inductively on g + n, a meromorphic symmetric differential n-form W g,n on Σ n for every g 0 and n subject to g +n > 0 A particular choice of the Lagrangian immersion gives a different W g,n, which then gives a generating function of the solution to a different enumerative geometry problem Thus the real question is how to find the right Lagrangian immersion from a given A- model Suppose we have a solution to an enumerative geometry problem an A-model problem Then we know a generating function of these quantities In [0] we proposed an idea of identifying the spectral curve Σ, which states that the spectral curve is the Laplace transform of the disc amplitude of the A-model problem Here the Laplace transform plays the role of mirror symmetry Thus we obtain a Riemann surface Σ Still we do not see the aspect of the Lagrangian immersion in this manner Every curve in T C is trivially a Lagrangian But not every Lagrangian is realized as the mirror dual to an A-model problem The obstruction seems to lie in the K-group K CΣ Q When this obstruction vanishes, we call Σ a K -Lagrangian, following Kontsevich s terminology For a K -Lagrangian Σ, we expect the existence of a holonomic system that characterizes the partition function of the B-model theory, and at the same time, the characteristic variety of this holonomic system recovers the spectral curve Σ as the Lagrangian immersion A generator of this holonomic system is called a quantum Riemann surface [, 6, 7], because it is a differential operator whose total symbol is the spectral curve realized as a Lagrangian immersion [8] It is the work of Gukov and Su lkowski [39] that suggested the obstruction to the existence of the holonomic system with algebraic K-theory as an element of K Another mysterious link of the Eynard-Orantin theory is its relation to integrable systems of the KP/KdV type [5, 7] We note that the partition function of the B-model is always the principal specialization of a τ-function of the KP equations for all the examples we know by now The purpose of the present paper is to give the simplest non-trivial mathematical examples of the theory that exhibit these key features mentioned above With these examples one can calculate all quantities involved, give proofs of the statements predicted in physics, and examine the mathematical structure of the theory Our examples are based on enumeration problems of branched coverings of P The idea of homological mirror symmetry of Kontsevich [44] allows us to talk about the mirror symmetry without underlying spaces, because the formulation is based on the

3 SPECTRAL CURVES AND THE SCHRÖDINGER EQUATION 3 derived equivalence of categories Therefore, we can consider the mirror dual B-models corresponding to the enumeration problems of branched coverings on the A-model side At the same time, being the derived equivalence, the homological mirror symmetry does not tell us any direct relations between the quantum invariants on the A-model side and the complex geometry on the B-model side This is exactly where Mariño s idea of remodeling B-model comes in for rescue The remodeled B-model of [8, 5] is not a derived category of coherent sheaves Although its applicability is restricted to the case when there is a family of curves Σ that exhibits the geometry of the B-model, the new idea is to construct a network of inter-related differential forms on the symmetric powers of Σ via the Eynard- Orantin recursion, and to understand this infinite system as the B-model The advantage of this idea is that we can relate the solution of the geometric enumeration problem on the A-model side and the symmetric differential forms on the B-model side through the Laplace transform In this sense we consider the Laplace transform as a mirror symmetry The first example we consider in this paper is the generalized Catalan numbers of an arbitrary genus This is equivalent to the c = model of [39, Section 5] In terms of enumeration, we are counting the number of algebraic curves defined over Q in a systematic way by using the dual graph of Grothendieck s dessins d enfants [4, 74] Let D g,n µ,, µ n denote the automorphism-weighted count of the number of connected cellular graphs on a closed oriented surface of genus g ie, the -skeleton of celldecompositions of the surface, with n labeled vertices of degrees µ,, µ n The letter D stands for dessin The generalized Catalan numbers of type g, n are defined by C g,n µ,, µ n = µ µ n D g,n µ,, µ n While D g,n µ is a rational number due to the graph automorphisms, the generalized Catalan number C g,n µ is always a non-negative integer It gives the dimension of a linear skein space In particular, the g, n = 0, case recovers the original Catalan numbers: C 0, m = C m = m + m m = dim End Uqsl T m C As explained in [0], the mirror dual to the Catalan numbers C m is the plane curve Σ defined by { x = z + z y = z, where zx = C m x m+ Note that also gives a Lagrangian immersion Σ T C Let us introduce the free energies by 3 Fg,n C zx,, zx n = D g,n µe w µ + +w nµ n = n D g,n µ x µ i µ Z n + µ Z n + i= i as the Laplace transform of the number of dessins, where the coordinates are related by and x i = e w i The free energy F C g,nz,, z n is a symmetric function in n-variables, and its principal specialization is defined by F C g,nz,, z Now let W C g,nz,, z n = d d n F g,n z,, z n

4 4 M MULASE AND P SU LKOWSKI It is proved in [0] that Wg,n s C satisfy the Eynard-Orantin topological recursion The Catalan partition function is given by the formula of [7]: 4 Z C z, = exp n! g +n Fg,nz, C z,, z In this paper we prove g=0 n= Theorem The Catalan partition function satisfies the Schrödinger equation 5 d dx + x d dx + Z C zx, = 0 The characteristic variety of this ordinary differential operator, y + xy + = 0 for every fixed choice of, is exactly the Lagrangian immersion, where we identify the xy-plane as the cotangent bundle T C with the fiber coordinate y = d dx Remark A purely geometric reason of our interest in the function appearing as the principal specialization F C g,nz,, z is that, in the stable range g + n > 0, it is a polynomial in 6 s = z z of degree 6g 6 + 3n It is indeed the virtual Poincaré polynomial of M g,n R n + [58], and its special value at s = gives the Euler characteristic n χm g,n of the moduli space M g,n of smooth n-pointed curves of genus g Thus Z C z, is the exponential generating function of the virtual Poincaré polynomials of M g,n R n + As such, the generating function Z C z, is also expressible in terms of a Hermitian matrix integral 7 Z C z, = det sx N e N tracex dx H N N with the identification 6 and = /N Here dx is the normalized Lebesgue measure on the space of N N Hermitian matrices H N N It is a well-known fact that Eq7 is the principal specialization of a KP τ-function see for example, [54] Another example we consider in this paper is based on single Hurwitz numbers As a counting problem it is easier to state than the previous example, but the Lagrangian immersion requires a transcendental function, and hence the resulting Schrödinger equation exhibits a rather different nature Let H g,n µ, µ n be the automorphism-weighted count of the number of topological types of Hurwitz covers f : C P of a connected non-singular algebraic curve C of genus g A holomorphic map f is a Hurwitz cover of profile µ, µ n if it has n labeled poles of orders µ, µ n and is simply ramified otherwise Introduce the Laplace transform of single Hurwitz numbers by 8 Fg,n H tw,, tw n = H g,n µe w µ + w nµ n, where tw = µ Z n + m m m! e mw

5 SPECTRAL CURVES AND THE SCHRÖDINGER EQUATION 5 is the tree-function Here again F H g,nt,, t n is a polynomial of degree 6g 6 + 3n if g + n > 0 [60] Bouchard and Mariño have conjectured [9] that W H g,nt,, t n = d d n F H g,nt,, t n satisfy the Eynard-Orantin topological recursion, with respect to the Lagrangian immersion { x = e w = ze z C 9 y = z C, where z = t t and we use η = y dx x as the tautological -form on T C The Bouchard-Mariño conjecture was proved in [7, 6, 60] Now we define the Hurwitz partition function 0 Z H t, = exp n! g +n Fg,nt, H, t Then we have the following g=0 n= Theorem 3 The Hurwitz partition function satisfy two equations: [ w + + w ] Z H tw, = 0, d dw + e w e d dw Z H tw, = 0 Moreover, each of the two equations recover the Lagrangian immersion 9 from the asymptotic analysis at 0 And if we view as a fixed constant in, then its total symbol is the Lagrangian immersion 9 with the identification z = d dw Remark 4 The second order equation is a consequence of the polynomial recursion of [60] This situation is exactly the same as Theorem The differential-difference equation, or a delay differential equation, follows from the principal specialization of the KP τ-function that gives another generating function of single Hurwitz numbers [4, 67] We remark that is also derived in [8] The point of view of differential-difference equation is further developed in [59] for the case of double Hurwitz numbers and r-spin structures, where we generalize a result of [8] Remark 5 Define two operators by 3 4 Then it is noted in [49] that P = d dw + e w e d Q = w + 5 [P, Q] = P dw and + w Thus the heat equation preserves the space of solutions of the Schrödinger equation In this sense, is holonomic for every fixed The analysis of these equations is further investigated in [49]

6 6 M MULASE AND P SU LKOWSKI The existence of a holonomic system is particularly appealing when we consider the knot A-polynomial as the defining equation of a Lagrangian immersion, in connection to the AJ conjecture [33, 34, 38] One can ask: Question 6 Let K be a knot in S 3 and A K its A-polynomial [3] Is there a concrete formula for the quantum knot invariants of K, such as the colored Jones polynomials, in terms of A K? The B-model developed in [5], [39], and more recently in [6], clearly shows that the answer is yes, and it should be given by the Eynard-Orantin formalism Although our examples in the present paper are not related to any knots, they suggest the existence of a corresponding A-model An interesting theory of generalized A-polynomials and quantum knot invariants from the point of view of mirror symmetry is recently presented in Aganagic and Vafa [] We also remark that there are further developments in this direction [0, 3, 3] The paper is organized as follows In Section, we give the definition of the Eynard- Orantin topological recursion We emphasize the aspect of Lagrangian immersion in our presentation In Section 3 we review the generalized Catalan numbers of [0] Then in Section 4 we derive the Schrödinger equation for the Catalan partition function The equation for the Hurwitz partition function is given in Section 5 Finally, in Section 6 we give the proof of using the Schur function expansion of the Hurwitz generating function and its principal specialization The Eynard-Orantin topological recursion We adopt the following definition for the topological recursion of Eynard-Orantin [7] Our emphasis, which is different from the original, is the point of view of the Lagrangian immersion Definition The spectral curve Σ, ι consists of an open Riemann surface Σ and a Lagrangian immersion ι : Σ T C π C with respect to the canonical holomorphic symplectic structure ω = dη on T C, where η is the tautological -form on the cotangent bundle π : T C C Recall that p Σ is a Lagrangian singularity if dπ ι = 0 at p, and that πιp C is a caustic of the Lagrangian immersion We assume that the projection π restricted to the Lagrangian immersion is locally simply ramified around each Lagrangian singularity We denote by R = {p,, p r } Σ the set of Lagrangian singularities, and by U = r j=u j the disjoint union of small neighborhood U j around p j such that π : U j πu j C is a double-sheeted covering ramified only at p j We denote by s j z the local Galois conjugate of z U j Another key ingredient of the Eynard-Orantin theory is the normalized fundamental differential of the second kind B Σ z, z [30, Page 0], [6, Page 33], which is a symmetric differential -form on Σ Σ with second-order poles along the diagonal To define it, let us recall a few basic facts about algebraic curves Let C be a nonsingular

7 SPECTRAL CURVES AND THE SCHRÖDINGER EQUATION 7 complete algebraic curve over C We are considering a nonsingular compactification C = Σ of the Riemann surface Σ We identify the Jacobian variety of C as JacC = P ic 0 C, which is isomorphic to P ic g C The theta divisor Θ of P ic g C is defined by Θ = {L P ic g C dim H C, L > 0} We use the same notation for the translate divisor on JacC, also called the theta divisor Now consider the diagram pr JacC δ C C pr C C, where pr j denotes the projection to the j-th components, and δ : C C p, q p q JacC Then the prime form E C z, z [30, Page 6] is defined as a holomorphic section E C p, q H C 0 C, prk C pr K C δ Θ, where K C is the canonical line bundle of C and we choose Riemann s spin structure or the Szegö kernel K C see [30, Theorem ] We do not need the precise definition of the prime form here, but its characteristic properties are important: E C p, q vanishes only along the diagonal C C, and has simple zeros along Let z be a local coordinate on C such that dzp gives the local trivialization of K C around p When q is near at p, δ Θ is also trivialized around p, q C C, and we have a local behavior zp zq E C zp, zq = + O zp zq dzp dzq 3 E C zp, zq = EC zq, zp The fundamental -form B C p, q is then defined by 3 B C p, q = d d log E C p, q aee [30, Page 0], [6, Page 33] We note that dzp appears in just as the indicator of our choice of the local trivialization With this local trivialization, the square E C p, q H 0 C C, prk C pr K C δ Θ behaves better because of zp zq 4 E C zp, zq = + O zp zq dzp dzq We then see that 5 B C zp, zq = d d log E zp, zq

8 8 M MULASE AND P SU LKOWSKI = dzp dzq zp zq + O dzp dzq H 0 C C, pr K C pr K C O Definition Let D be a divisor of Σ A meromorphic symmetric differential form of degree n with poles along D is an element of the symmetric tensor product Sym n H 0 Σ, K Σ D Definition 3 Meromorphic differential forms W g,n z,, z n Sym n H 0 Σ, K Σ R for g 0 and n, subject to g + n > 0, with poles along the Lagrangian singularities of the Lagrangian immersion Σ T C, are said to satisfy the Eynard-Orantin topological recursion if they satisfy the recursion formula 6 W g,n z, z,, z n = πi r j= γ j K j z, z + [ No 0, terns g +g =g I J={,3,,n} W g,n+ z, sj z, z,, z n W g, I +z, z I W g, J +s j z, z J Here the integration is taken with respect to z U j along a positively oriented simple closed loop γ j around p j, and z I = z i i I for a subset I {,,, n} In the summation, No 0, terms means the summand does not contain the terms with g = 0 and I = or g = 0 and J = The recursion kernels K j z, z, j =,, r, are defined as follows First we define W 0, and W 0, 7 W 0, z = ι η = ydx H 0 Σ, K, where x is a linear coordinate of C and y is the dual coordinate of T0 C so that the Lagrangian immersion is given by x, y : Σ t xz, yz T C 8 W 0, z, z = B Σ z, z π dx dx x x We note that W 0, z, z is holomorphic along the diagonal z = z The recursion kernel K j z, z H 0 U j Σ, K U j K Σ for z U j and z Σ is defined by 9 K j z, z = sj z W 0, sj z W 0, z W 0,, z z = d log EΣ sj z, z log EΣ z, z ysj z yz dxz The kernel K j z, z is an algebraic operator that multiplies dz while contracts /z The topological recursion is the reduction of g + n by, which is different from the usual boundary degeneration formula of M g,n As shown in Figure, the reduction corresponds to degeneration cycles of codimension and, as in Arbarello-Cornalba-Griffiths [3, Chapter 7, Section 5, Page 589] M 03 M g,n M g,n, M 03 M g,n+ M g,n, ]

9 M 03 SPECTRAL CURVES AND THE SCHRÖDINGER EQUATION 9 g +g =g n +n =n M g,n + M g,n + M g,n Figure The topological recursion and degeneration The recursion starts from W, and W 0,3 If we modify 6 slightly, then these can also be calculated from W 0, [7] 0 W, z = πi r j= γ j K j z, z [ ] dxu dxv W 0, u, v + cπ xu xv u=z,v=sj z Formula 6 does not give an apparently symmetric expression for W 0,3 In terms of the coordinate x, y T C we have an alternative formula for W 0,3 [7]: W 0,3 z, z, z 3 = πi r j= W 0, z, z W 0, z, z W 0, z, z 3 γ j dxz dyz Suppose we have a solution W g,n to the topological recursion A primitive functions of the symmetric differential form W g,n is a symmetric meromorphic function F g,n on Σ n such that its n-fold exterior derivative recovers the W g,n, ie, W g,n z,, z n = d z d zn F g,n z,, z n

10 0 M MULASE AND P SU LKOWSKI The partition function of the topological recursion for a genus 0 spectral curve is the formal expression in infinitely many variables 3 Zz, z, ; = exp n! g +n F g,n z, z,, z n g 0,n The principal specialization of the partition function is also denoted by the same letter Z: 4 Zz, = exp n! g +n F g,n z, z,, z g 0,n Remark 4 The partition function depends on the choice of the primitive functions When we consider the topological recursion as the B-model corresponding to an A-model counting problem, then there is always a canonical choice for the primitives, as the Laplace transform of the quantum invariants Remark 5 When the spectral curve Σ has a higher genus, the partition function requires a non-perturbative factor in terms of a theta function associated to the curve [5, 6] In this case the algebraic K-theory condition of [39], probably similar to the Boutroux condition of [], becomes essential for the existence of the quantum curve or the Schrödinger equation Our consideration in paper is limited to the genus 0 case 3 The generalized Catalan numbers and the topological recursion A cellular graph of type g, n is the one-skeleton of a cell-decomposition of a connected closed oriented surface of genus g with n 0-cells labeled by the index set [n] = {,,, n} Two cellular graphs are identified if an orientation-preserving homeomorphism of a surface into another surface maps one cellular graph to another, honoring the labeling of each vertex Let D g,n µ,, µ n denote the number of connected cellular graphs Γ of type g, n with n labeled vertices of degrees µ,, µ n, counted with the weight / AutΓ It is generally a rational number The orientation of the surface induces a cyclic order of incident half-edges at each vertex of a cellular graph Γ Since AutΓ fixes each vertex, the automorphism group is a subgroup of the Abelian group n i= Z/ µ i Z that rotates each vertex Therefore, 3 C g,n µ,, µ n = µ µ n D g,n µ,, µ n is always an integer The cellular graphs counted by 3 are connected graphs of genus g with n vertices of degrees µ,, µ n, and at the j-th vertex for every j =,, n, an outgoing arrow is placed on one of the incident µ j half-edges see Figure 3 The placement of n arrows corresponds to the factors µ µ n on the right-hand side We call this integer the generalized Catalan number of type g, n The reason for this naming comes from the following theorem Theorem 3 The generalized Catalan numbers 3 satisfy the following equation 3 C g,n µ,, µ n = + α+β=µ n µ j C g,n µ + µ j, µ,, µ j,, µ n j= C g,n+α, β, µ,, µ n + g +g =g I J={,,n} C g, I +α, µ I C g, J +β, µ J,

11 SPECTRAL CURVES AND THE SCHRÖDINGER EQUATION Figure 3 A cellular graph of type, where µ I = µ i i I for an index set I [n], I denotes the cardinality of I, and the third sum in the formula is for all partitions of g and set partitions of {,, n} Proof Let Γ be an arrowed cellular graph counted by the left-hand side of 3 Since all vertices of Γ are labeled, let {p,, p n } denote the vertex set We look at the half-edge incident to p that carries an arrow Case The arrowed half-edge extends to an edge E that connects p and p j for some j > In this case, we shrink the edge and join the two vertices p and p j together By this process we create a new vertex of degree µ + µ j To make the counting bijective, we need to be able to go back from the shrunken graph to the original, provided that we know µ and µ j Thus we place an arrow to the half-edge next to E around p with respect to the counter-clockwise cyclic order that comes from the orientation of the surface In this process we have µ j different arrowed graphs that produce the same result, because we must remove the arrow placed around the vertex p j in the original graph This gives the right-hand side of the first line of 3 See Figure 3 p E p j Figure 3 The process of shrinking the arrowed edge E that connects vertices p and p j, j > Case The arrowed half-edge at p is a loop E that goes out and comes back to p The process we apply is again shrinking the loop E The loop E separates all other half-edges into two groups, one consisting of α of them placed on one side of the loop, and the other consisting of β half-edges placed on the other side It can happen that α = 0 or β = 0 Shrinking a loop on a surface causes pinching Instead of creating a pinched ie, singular surface, we separate the double point into two new vertices of degrees α and β Here again we need to remember the place of the loop E Thus we place an arrow to the half-edge next to the loop in each group See Figure 33 After the pinching and separating the double point, the original surface of genus g with n vertices {p,, p n } may change its topology It may have genus g, or it splits into two pieces of genus g and g The second line of 3 records all such cases This completes the proof

12 M MULASE AND P SU LKOWSKI E Figure 33 The process of shrinking the arrowed loop E that is attached to p Remark 3 For g, n = 0,, the above formula reduces to 33 C 0, µ = C 0, αc 0, β α+β=µ When n =, the degree of the unique vertex µ is always even By defining C 0, 0 =, we find that C 0, m = C m = m m + m is the m-th Catalan number Only for g, n = 0, we have this irregular case of µ = 0 happens, because a degree 0 single vertex is connected, and gives a cell-decomposition of S We can imagine that a single vertex on S has an infinite cyclic group as its automorphism, so that C 0, 0 = is consistent with C 0, µ = µ D 0, µ All other cases, if one of the verteces has degree 0, then the Catalan number C g,n is simply 0 because of the definition 3 Let us introduce the generating function of the Catalan numbers by 34 z = zx = x m+ Then by the quadratic recursion 33 we find that the inverse function of zx that vanishes at x = is given by 35 x = z + z This defines a Lagrangian immersion 36 Σ = C z xz, yz T C, { xz = z + z yz = z The Lagrangian singularities are located at the points at which dx = 0, ie, z = ± Often it is more convenient to use the coordinate 37 z = t + t The following theorem is established in [0] Theorem 33 [0] The Laplace transform of the Catalan numbers of type g, n defined as symmetric differential forms by 38 Wg,nt C,, t n = n C g,n µ,, µ n e w,µ dw dw n µ,,µ n Z n +

13 SPECTRAL CURVES AND THE SCHRÖDINGER EQUATION 3 satisfies the Eynard-Orantin recursion with respect to the Lagrangian immersion 36 and 39 W C 0,t, t = t t dx dx x x Here the Laplace transform coordinate w is related to the coordinate t of the Lagrangian by e w i = x i = z i + = t i + z i t i + t i, i =,,, n, t i + and w, µ = w µ + + w n µ n In this case the Eynard-Orantin recursion formula is given by 30 Wg,nt C,, t n = + t 3 64 πi γ t + t t t t [ n W0,t, C t j Wg,n t, C t,, t j,, t n + W0, t, D t j Wg,n t, C t,, t j,, t n j= + W C g,n+t, t, t,, t n + stable g +g =g I J={,3,,n} W C g, I + t, t IW C g, J + t, t J The last sum is restricted to the stable geometries In other words, the partition should satisfies g + I > 0 and g + J The contour integral is taken with respect to t on the curve defined by Figure 34 ] t r t j r t j t t-plane Figure 34 The integration contour γ This contour encloses an annulus bounded by two concentric circles centered at the origin The outer one has a large radius r > max j N t j and the negative orientation, and the inner one has an infinitesimally small radius with the positive orientation Remark 34 The recursion formula 30 is a genuine induction formula with respect to g + n Thus from the given W C 0, and W C 0,, we can calculate all W C g,n one by one This is a big difference between 30 and 3 The latter relation contains the terms with C g,n in the right-hand side as well 4 The partition function for the generalized Catalan numbers and the Schrödinger equation Let us now define 4 F C g,nt,, t n = µ,,µ n Z n + D g,n µ,, µ n e w,µ

14 4 M MULASE AND P SU LKOWSKI for g + n > 0 Then from 3 we have W C g,nt,, t n = d d n F C g,nt,, t n Therefore, we have a natural primitive function of W g,n t,, t n for every g, n We note that t = = z = 0 = x = Therefore, 4 F C g,nt,, t n ti = = 0 for every i =,,, n The following recursion formula of [6] is the key for our investigation Theorem 4 The Laplace transform F C g,nt [n] satisfies the following differential recursion equation for every g, n subject to g + n > 0 43 F t g,nt C [n] [ = n t j 6 t t j j= 3 t 3 t 3 t 3 t n 6 [ j= t F C g,n t [ĵ] t j 3 t j t t t F C g,n t [ĵ] F C u u g,n+u, u, t, t 3,, t n ] t 3 t stable g +g =g I J={,3,,n} ] F C t g,n t [ˆ] j u =u =t Fg C t, I + t, t I Fg C t, J + t, t J Here we use the index convention [n] = {,,, n} and [ĵ] = {,,, ĵ,, n} Remark 4 We note that the above formula is identical to [58, Theorem 5], even though F g,n is a different function There we considered the Laplace transform of the number of lattice points in M g,n, and hence F 0, = F 0, = 0 Remark 43 Because of 4, the recursion 43 uniquely determines each F C g,n by integrating from to t With the same reason, F C g,n is uniquely determined by W C g,n Since we know exactly where F C g,n vanishes, there is no discrepancy of the constants of integration in Let us now consider the principal specialization of the partition function for the Catalan numbers 44 Z C t, = exp n! g +n Fg,nt, C t,, t g=0 n= Since unstable terms F C 0, t and F C 0, t, t are included in the above formula, we need to calculate them first

15 SPECTRAL CURVES AND THE SCHRÖDINGER EQUATION 5 Proposition 44 In terms of the z-variable, we have 45 F C 0,t = z + log z, 46 F C 0,t, t = log z z Proof Due to the irregularity of µ = 0 for D 0, µ, we need to modify the definitions 38 and 4 for g, n = 0, It is natural to define dx 47 W 0, t = C 0, m = zdx = z + dz xm+ z because of the consistency with 7 Since W0, C = df 0, C, we have F 0, t = z + log z + const = D 0, m x m δ m,0, where the m = 0 term is adjusted so that we do not have the infinity term D 0, 0 in F0, C Using the expression m D 0, m =, mm + m we have lim D 0,m m 0 x m δ m,0 = log x Since x = z + z, by taking the limit z 0 we conclude that the constant term in F 0, C t is 0, which establishes 45 The computation of F0, C t, t is performed in [0, Proposition 4], where the idea is to d d use the Euler differential operator x dx + x dx By definition F0, C t, t does not have any constant term as x, therefore there is no constant correction in 46, either Theorem 45 The principal specialization of the partition function satisfies the following Schrödinger equation 48 d dx + x d dx + Z C t, = 0, where t is considered as a function in x by and 34 t = tx = zx + zx The rest of this section is devoted to proving the above theorem Let us define for m 0 49 S m = n! F g,nt, C, t, and put g +n=m F = m S m

16 6 M MULASE AND P SU LKOWSKI Then the Schrödinger equation 48 becomes d F df dx + + x df dx dx + = 0, which is equivalent to 40 S m m+ + S m m + x where = d dx Since S 0 = F C 0, and we obtain S m m + = 0, W C 0, = df C 0, = ydx = zdx, 4 S 0 = d dx F C 0, = z = t + t Using the Lagrangian immersion 35, we see that the constant terms of 40 then become 4 z + x z + = z xz + = 0 Collecting the m+ -contributions in 40 for m 0, we obtain x d dx S m+ = d dx S m + ds a ds b dx dx Therefore, a+b=m+ = d dx S m + S 0S m+ + S S m + 43 S 0 + x d dx S m+ = d dx S m + S d dx S m + a+b=m+ a,b a+b=m+ a,b ds a dx ds a dx ds b dx ds b dx To use the closed formulas 45 and 46 for S 0 and S here, we need to switch from the x-coordinate to the t-coordinate Using the change of variable formulas 35 and 37, we have dx = dz z = 8t t, hence Therefore, 44 d dx = z z d dx = t 4 d 64t S 0 + x d dx = t 8t d dz = t 8t + t 8t d, d t 8t = t 4 64t d + t 3 64t 3 3t + d z d z = t 8t 4t t d d = t d,

17 45 S = SPECTRAL CURVES AND THE SCHRÖDINGER EQUATION 7 d dx F C 0,z, z = z z Substituting 4, 44 and 45 in 43, we obtain d dz log z = t t + 6t Proposition 46 For m 0, the Schrödinger equation 48 is equivalent to a recursion formula 46 d S m+ = t 3 3t d S m + a+b=m+ a,b ds a ds b t 6t 3 t + t + d S m Remark 47 The recursion equation 46 is a direct consequence of the principal specialization applied to 43 The proof of 46 is given in Appendix A 5 Single Hurwitz numbers The A-model problem that we are interested in this section is the automorphism-weighted count H g,n µ,, µ n of the number of the topological types of meromorphic functions f : C P of a nonsingular complete irreducible algebraic curve C of genus g that has n labeled poles of orders µ,, µ n such that all other critical points of f than these poles are unlabeled simple ramification points Let r denote the number of such simple ramification points Then the Riemann-Hurwitz formula gives n 5 r = g + n + µ i A remarkable formula due to Ekedahl, Lando, Shapiro and Vainshtein [, 37, 68] relates Hurwitz numbers and Gromov-Witten invariants For genus g 0 and the number n of the poles subject to the stability condition g + n > 0, the ELSV formula states that n µ µ i i 5 H g,n µ,, µ n = Λ g µ i! n M g,n i= µi ψ i i= = g j j=0 i= k,,k n 0 τ k τ kn c j E g,n n i= µ µ i+k i i, µ i! where M g,n is the Deligne-Mumford moduli stack of stable algebraic curves of genus g with n distinct smooth marked points, Λ g = c E+ + g c g E is the alternating sum of the Chern classes of the Hodge bundle E on M g,n, ψ i is the i-th tautological cotangent class, and 53 τ k τ kn c j E g,n = ψ k ψkn n c j E M g,n is the linear Hodge integral, which is 0 unless k + + k n + j = 3g 3 + n Although the Deligne-Mumford stack M g,n is not defined for g n < 0, single Hurwitz numbers are well defined for unstable geometries g, n = 0, and 0,, and their values are 54 H 0 d = dd 3 d! = dd d! and H 0 µ, µ = µµ µ + µ µ! µµ µ!

18 8 M MULASE AND P SU LKOWSKI The ELSV formula remains valid for unstable cases if we define Λ 55 0 M 0, dψ = d, Λ 0 M 0, µ ψ µ ψ = 56 µ + µ The ELSV formula predicts that the single Hurwitz numbers exhibit the polynomial behavior in terms of their Laplace transform Following [6], and modifying our choice of the coordinates slightly, we define 57 Fg,nt H,, t n = H g,n µ,, µ n e µ w + +µ nw n = µ Z n + µ Z n + k + +k n 3g 3+n where the ˆξ-functions are given by 58 ˆξk t = τ k τ kn Λ g g,n n µ= = µ µ+k µ! i= k + +k n 3g 3+n e µw = µ= µ µ i+k i i e µ w + +µ nw n µ i! n τ k τ kn Λ g g,n ˆξ ki t i, µ µ+k for k 0, and x = e w Although ˆξ k are complicated functions in x, their behavior is simple in terms of ˆξ 0 So we introduce µ µ µ µ 59 t = + µ! xµ and z = x µ µ! µ= Then by the Lagrange inversion formula, we have µ! µ= 50 x = ze z and z = t, t and moreover, each ˆξ k t is a polynomial of degree k + in t for every k 0, recursively defined by 5 ˆξk+ t = t t d ˆξ k t This is because 5 d dw = x d dx = t t d Therefore, if g + n > 0, then F H g,nt,, t n is a symmetric polynomial of degree 6g 6 + 3n, and satisfies 53 F H g,nt,, t n tj = = 0 for every j =,, n The computation of [0] adjusted to our current convention of this paper, 54 F0,t H = t = z z x µ i=

19 SPECTRAL CURVES AND THE SCHRÖDINGER EQUATION 9 and 55 F0,t H z z, t = log z + z, x x determines the Lagrangian immersion by 56 and ι : Σ = C T C { x = ze z y = z, W0,t H, t = d d F0,t H, t = t t dx dx x x Here the tautological -form on T C is chosen to be 57 η = y dx x It is consistent with We also note that W H 0,t = df H 0,t = z dx x = ι η 58 F H 0,t t= = 0, and 59 F0,t H, t tj = 0, j = or = The latter equality holds because t = = z = 0 = x = 0, and hence from 55 we have F0,t H, = log z z = 0 x Since x x z z = ze z, z =z =z the diagonal value of F0, H is calculated as e 50 F0,t, H z t = log z = z log z = t + log t z t The single Hurwitz numbers H g,n µ satisfy the cut-and-join equation [36, 77] n 5 g + n + µ i H g,n µ,, µ n = µ i + µ j H g,n µ i + µ j, µ [î,ĵ] i= i j + n H g,n+α, β, µ + [î] H g, I +α, µ I H g, J +β, µ J, αβ i= α+β=µ i g +g =g I J=[î] where we use the convention for indices as in Section 3 The Laplace transform of 5 is the polynomial recursion of [60] and takes the form

20 0 M MULASE AND P SU LKOWSKI 5 n g + n + t i t i F t g,nt H,, t n i= i = t i t j t i t i F t i t j t g,n t H [ĵ] t jt j F H i t g,n t [î] j i j + t 3 i t i F t g,n t H [ĵ] i i j n t i t i F H u u g,n+u, u, t [î] i= + n i= t i t i stable g +g =g I J=[î] u =u =t i t i F H g, I + t i, t I t i F H g, J + t i, t J It is proved in [6] that the n-fold exterior differentiation of the above formula is exactly the Eynard-Orantin recursion, as predicted by Bouchard and Mariño [9] Thus we obtain a natural integration of the Eynard-Orantin recursion by taking the Laplace transform of the A-model quantity again, which is the single Hurwitz umber H g,n µ Theorem 5 Let us define 53 S H mt = g +n=m n! F H g,nt,, t Then Sm s H satisfy the following second order differential equations: 54 e mw d [ dw e mw Sm+ H = d dw SH m + ] dsa H dw dsh b dw + d dw SH m a+b=m+ Here the w-dependence of t is given by x = e w and 59 We also note that S H mt is a polynomial of degree 3m-3 for every m, and for all values of m we have 55 S H mt t= = 0 The proof is similar to the case of the Catalan numbers Section 4 and Appendix A First we compute the principal specialization of the differential equation 5 We then assemble them according to 53 By adjusting the unstable geometry terms g, n = 0, and 0,, we obtain 54 However, due to the difference between the cut-and-join equation and the edge-shrinking operation of Section 3, the resulting equation becomes quite different Choose m and g, n so that g + n = m Then the principal specialization of the left-hand side of 5 is m + tt d F H g,nt,, t The first line of the right-hand side of 5 gives t t i t i F t i t g,n t H i, t,, t i i j ti =t = nn t d t t d n F g,n t, H, t

21 SPECTRAL CURVES AND THE SCHRÖDINGER EQUATION nn + t 4 t u F g,n u, H t,, t u=t = n! d t t t d n! F g,n t, H, t + n!n t4 t n! u F g,n u, H t,, t u=t The second line of the right-hand side of 5 becomes n! t 3 d t n! F g,n t, H, t The third line simply produces n! t t n +! n + n F H u u g,n+u, u, t, t u =u =t Finally, since the set partition becomes the partition of numbers because all variables are set to be equal, the fourth line of the right-hand side of 5 gives n! t t stable g +g =g n +n =n We now apply the operation d n +! F g H d,n +t,, t n +! F g H,n +t,, t g +n=m to the above terms The left-hand side becomes m + tt d S H m+ The right-hand side terms are re-assembled into the sum of g, n subject to g + n = m, following the topological structure of the recursion 5 Noticing that unstable geometers are contained only in S0 H and SH, we obtain 56 m + tt d S H m+t = t d t t d = t t d SH mt + n! SH mt + t t d + t t a+b=m+ a,b a+b=m+ a,b SH mt t 3 t d SH mt d SH a t d SH b t d SH a t d Proposition 5 The functions S m t are recursively determined by S0 H t = t 57, SH b t + t3 t d SH mt

22 M MULASE AND P SU LKOWSKI 58 and 59 t m t Sm+t H = t [ S H t = t + log t, t t3 m t m+ d SH mt + a+b=m+ a,b d SH a t d + t m t m+ d SH mt SH b t ] Proof As a differential operator, Therefore, 56 is equivalent to [ t t m d m + tt d = tt t t ] t m S H t m+t = t3 t d SH mt + a+b=m+ a,b m d d SH a t d t m t SH b t + t t d SH mt On the right-hand side only Sk H we obtain 59 t with k m appear Using the fact of the zero 55, On the third line of 56 the terms with S0 H and SH these omitted terms are are not included More precisely, tt d SH m+t + t t 3 d SH mt When we add these terms to 56, and adjust the second order differentiation as t t d = t t d + t3 t 3t d, we finally obtain 530 m + t t d S H m+t = t t d S H mt + t t a+b=m+ ds H a t dsh b t t t d SH mt Then 54 follows from 530 and 5 This completes the proof Theorem 5

23 SPECTRAL CURVES AND THE SCHRÖDINGER EQUATION 3 Theorem 53 Let us define the partition function for the single Hurwitz numbers in a similar way: 53 Z H t, = exp Sm H m = exp n! g +n Fg,nt, H, t Then we have 53 [ g=0 n= w + + w ] Z H t, = 0 Remark 54 Eq53 is a heat equation, where is considered as the time variable of the heat conduction It determines the solution uniquely with the initial condition Z H tw, 0 = exp SH 0 + S H given by 57 and 58 Proof Let F H t, = Sm H m = In terms of F H, 53 is equivalent to F H 533 w + F H w We apply the operation g=0 n= + [ + m n! g +n F H g,nt,, t w ] F H = 0 to 54 The left-hand side is 534 m m + d Sm+ H = dw w F H SH 0 The right-hand side gives 535 w Sm H m + a+b=m+ = w F H + S H a w a SH b w b + w F H w S H 0 w Sm H m + w F H If we collect all terms that contain S0 H t in 534 and 535, we have an equation t t t + SH 0 = t t t SH 0, or equivalently, 536 S H 0 t = t t d SH 0 t t t ds H 0 t We see that 57 is a solution to 536 After eliminating 536 from 534 = 535, we obtain 533 This completes the proof

24 4 M MULASE AND P SU LKOWSKI 6 The Schur function expansion of the Hurwitz partition function Let us introduce the free energy of single Hurwitz numbers by a formal sum as 6 F H t, t, t 3, ; = = g 0, n g 0, n n! g +n F H g,nt,, t n n! g +n µ,µ n Z n + H g,n µ, µ n e µ + +µ n The partition function we considered in Section 5 is the principal specialization Z H t, = exp F H t, t, : Recall another generating function of the Hurwitz numbers [4, 4, 67] defined by 6 Hs, p = H g,n s, p, g 0, n n i= x µ i i 63 H g,n s, p = n! µ Z n + H g,n µp µ s rg,µ, where p µ = p µ p µn, and r = rg, µ = g + n + is again the number of simple ramification point of a Hurwitz cover of genus g and profile µ At this point we wish to go back and forth between the following two distinct points of view: One is to regard µ = µ,, µ n as a vector consisting of positive integers, and the other is to view µ as a partition of length n For any function f µ in µ as a vector, we have a change of summation formula 64 f µ = Autµ f µ σ σ S n µ Z n + µ:lµ=n Here the first sum in the right-hand side runs over partitions µ of a fixed length n, the second sum is over the symmetric group S n of n letters, n i= µ σ = µ σ,, µ σn Z n + is the integer vector obtained by permuting the parts of µ by σ S n, and Autµ is the permutation group interchanging the equal parts of µ As a partition, the length of µ is denoted by lµ, and its size by lµ µ = µ i i= Often single Hurwitz numbers are labeled by the genus g and a partition µ In this case the expression rg, µ! h g,µ = Autµ H g,lµµ,, µ lµ µ i

25 SPECTRAL CURVES AND THE SCHRÖDINGER EQUATION 5 is used in the literature, when we do not label the poles, but label the simple ramification points The generating function then has an expression in terms of sumes over partitions: s rg,µ Hs, p = h g,µ p µ rg, µ! g=0 In terms of the ELSV formula 5 we have 65 Hs, p = g 0, n n! sg +n µ k + +k n 3g 3+n τ k τ kn Λ g g,n n i= µ i = µ µ i+k i i s µ i p µi µ i! with an appropriate incorporation of 55 and 56 Recall the Laplace transform 57 here for comparison that is assembled into the free energy 66 F H t, t ; := = g 0,n g 0, n n! g +n n! g +n F H g,nt,, t n k + +k n 3g 3+n τ k τ kn Λ g g,n n i= µ i = µ µ i+k i i µ i! It is easy to see that the relation between the two sets of variables is exactly the power-sum symmetric functions Let us re-scale the usual power-sum symmetric function p j of degree j in x i s with a scale parameter s as follows: 67 p j s := s j x j + xj + xj 3 + Here we consider p j s as a degree j polynomial defined on C n, but the dimension n is unspecified Then for every µ Z n +, we have n 68 d d n p µ s = s µ + +µ n µ i x µ i σi dx dx n as a differential form on C n Now from 67, 68 and 64, we obtain 69 d d n H g,n s, ps = = µ:lµ=n = µ:lµ=n µ:lµ=n σ S n i= h g,µ d d n p µ s sr r! h g,µ s r r! s µ n σ S n i= Autµ H g,nµ,, µ n s g +n = µ Z n + H g,n µ,, µ n s g +n = s g +n d d n µ Z n + µ i x µ i σi dx dx n n i= n σ S n i= µ i x µ i σi dx dx n µ i x µ i i dx dx n H g,n µ,, µ n n i= x µ i i x µ i i

26 6 M MULASE AND P SU LKOWSKI = s g +n d d n F H g,nt,, t n = s g +n W H g,nt,, t n This formula tells us that the Eynard-Orantin differential form W H g,n is the exterior derivative of H g,n s, ps with the identification 67 Moreover, we have 60 s g +n F H g,nt,, t n H g,n s, ps mod Kerd d n as functions on C n Remark 6 Let us examine 60 For n =, the power sum 67 contains only one term and we have 6 H g, s, ps = s g H g, kp k ss k = s g H g, k x k = s g Fg,t H k= k= Thus 60 is an equality for n = In general what happens is 6 H g,n s, p = n! sg +n = s g +n µ Z n + µ Z n + H g,n µ Therefore, 60 is never an equality for n > n i= x µ + xµ + + xµn n H g,n µ x µ xµ xµn n + terms with less than n variables However, the principal specialization t = t = t = t 3 = corresponds to evaluating x j 63 p j = s With this identification we have again an equality 64 Hs, p = g 0,n n! sg +n µ Z n + H g,n µ x µ + +µ n = g 0,n n! sg +n F H g,nt, t,, t In Section 5 we noted that the Eynard-Orantin recursion for Hurwitz numbers is the Laplace transform of the cut-and-join equation 5 [6, 60] Another consequence of the same combinatorial equation is a heat equation [35, 4, 79] 65 s ehs,p = i,j i + jp i p j + ijp i+j e Hs,p, p i+j p i p j with the initial condition H0, p = p An important and fundamental fact here is that the heat equation 65 implies that e Hs,p is a KP τ-function for each value of s [4, 4, 67, 79] Let us recall this fact here We note that a solution of the heat equation is expanded by the eigenfunction of the second order operator In our case of 65, the eigenfunctions of the cut-and-join operator on the right-hand side are given by the Schur functions For a partition µ = µ µ of a finite length lµ, we define the shifted power-sum function by 66 [ p r [µ] := µ i i + r i + r ] i=

27 i,j SPECTRAL CURVES AND THE SCHRÖDINGER EQUATION 7 This is a finite sum of lµ terms In this paper we consider p r [µ] as a number associated with a partition µ Then we have [35, 79] 67 i + jp i p j + ijp i+j s µ p = p [µ] s µ p, p i+j p i p j where s µ p is the Schur function defined by s µ p = χ µ λ p λ, z λ λ = µ lµ z µ = m i!i m i, i= m i = the number of parts in µ of length i, and χ µ λ is the value of the irreducible character of the representation µ of the symmetric group evaluated at the conjugacy class λ If we write = i + jp i p j + ijp i+j, p i+j p i p j Then i,j s µ p = p [µ] s µ p and s ehs,p = e Hs,p Therefore, we have an expansion formula e Hs,p = µ c µ s µ pe p [µ]s for a constant c µ associated with every partition µ The constants are determined by the initial value Since the initial condition is we conclude that This follows from the Cauchy identity 68 where e Hs,p s=0 = e p = µ c µ = s µ, 0, 0,, 0 c µ s µ p, i,j x iy j = s µ ps µ p y, µ p j = i x j i and p y j = i y j i Since we have 69 s µ ps µ p y = µ i,j x iy j = exp i,j log x i y j

28 8 M MULASE AND P SU LKOWSKI = exp i,j m m xm j y m i the restriction of 69 to p y = and py m = 0 for all m reduces to = exp m p mp y m, m e p = µ s µ, 0,, 0s µ p Because of the determinantal formula for the Schur functions, s µ, 0, 0,, 0 s are the Plücker coordinate of a point of the Sato Grassmannian It follows that s µ, 0, 0,, 0e p [µ]s for all µ also form the Plücker coordinates because of 66 Then by a theorem of Sato [73], 60 e Hs,p = µ s µ, 0, 0,, 0s µ pe p [µ]s = µ dim µ e p[µ]s s µ p µ! is a τ-function of the KP equations Here dim µ is the dimension of the irreducible representation of the symmetric group S µ belonging to the partition µ Thus we have established Theorem 6 The Hurwitz partition function Z H t, of 53 is obtained by evaluation of the KP τ-function e Hs,p at s, p =, p : 6 Z H t, = e H,p Here p means the principal specialization x j 6 p j = for every j =,, 3, The t-variable and the x-variable are related by x = t e t t Since we have a concrete expansion formula 60 for e Hs,p, it is straightforward to find a formula for its principal specialization Let us look at 69 again This time we apply the principal specialization to both p and p y, meaning that we substitute Then we have p m = x m and p y m = y m s µ ps µ p y = µ x m y m after the double principal specialization Therefore, the sum with respect to all partitions µ is reduced to the sum with respect to only one-part partitions, ie, µ = m All other partitions contribute 0 Thus s µ = s m = h m, which is the m-th complete symmetric function But because of the principal specialization, we simply have h m = x m We note that if µ = m, then 66 reduces to p [m] = m + + = mm

29 We therefore conclude that SPECTRAL CURVES AND THE SCHRÖDINGER EQUATION 9 63 Z H t, = e H,p = We have now established a theorem of Zhou [8] m! e mm x m Theorem 63 [8] The same Hurwitz partition function satisfies a differential-difference equation 64 w + e w e w Z H t, = 0 Here again t is a function in w, which is given by k k t = + k! xk = + k= k= k k k! e wk The characteristic variety of this equation, with the identification of z = w, is the Lagrangian immersion e w = ze z Proof Let us denote a m = e mm x m = e ++ +m x m Then Z H t, = m! a m, a m+ = e m x a m, and x d dx a m = ma m We note that x d dx operates as the multiplication of m to a m Therefore, w ZH t, = x d dx m! a m = m! a m+ = x m! em a m This completes the proof = xe x d dx Z H t, = e w e w Z H t, Remark 64 The asymptotic behavior of Z H t, near = 0 is determined by exp SH 0 + S H From 64 we have 0 = exp SH 0 S H d dw + e w e d dw exp = d dw SH 0 tw + e w exp SH 0 + S H =0 S H 0 tw S H 0 tw = d dw SH 0 We thus recover the Lagrangian immersion in this way as well =0 + e w e d dw SH 0 = z + xe z

30 30 M MULASE AND P SU LKOWSKI Remark 65 We can directly verify that the expression 63 of Z H t, satisfies the Schrödinger equation 53 Indeed, since x = e w, [ w + + w ] = m! e mm m e mw m! e mm m e mw [ m m + 7 Conclusion mm + m ] = 0 The main purpose of this paper is to derive the Schrödinger equation of the partition function from the integrated Eynard-Orantin topological recursion, when there is an A- model counting problem whose mirror dual is the Eynard-Orantin theory We examined two different types of counting problem of ramified covering of P : one is Grothendieck s dessins d enfants, and the other single Hurwitz numbers The first example leads to a Lagrangian immersion in T C defined by a Laurent polynomial equation 36, while the latter corresponds to the Lambert curve in T C given by an exponential equation 56 If we start with the Eynard-Orantin recursion 6 and define the primitive functions F g,n by, then we have the ambiguity in the constants of integration However, if we start with an A-model, then the primitive functions F g,n s are given by the Laplace transform of the solution to the A-model problem The examples we have studied in this paper show that always there is a natural zero t i = a of F g,n t,, t n in each variable, such as 4 and 53, when g + n > 0 Thus the integration formula F g,n t,, t n = t a tn a W g,n t,, t n uniquely determines F g,n from W g,n The integral transform equation 6 for W g,n is then equivalent to a differential transform equation for F g,n, such as 43 and 5 for our examples Because of our assumption for the Lagrangian immersion that the Lagrangian singularities are simply ramified, the differential recursion equation for F g,n is expected to be a second order PDE Then by taking the principal specialization, we obtain a second order differential equation in t and For our examples we have thus established 48 and 53 Although 48 is holonomic if we consider a constant, 53 is a PDE containing the -differentiation as well Therefore, it is not holonomic This difference comes from the constant term g + n in the differential operator of the right-hand side of 5 After taking the n-fold symmetric exterior differentiation, this term drops, and thus the Eynard- Orantin recursion for Hurwitz numbers [6] takes the same shape as that of the Catalan case 30 With the analysis of our examples, we notice that the issue of the constants of integration in is not a simple matter Only the corresponding A-model can dictate which constants of integration we should choose Otherwise, the Schrödinger equation we wish to establish would take a totally different shape, depending on the choice of the constants Our second equation 64 for Hurwitz numbers is much similar to 48 in many ways, such as it is holonomic for each fixed But this differential-difference equation is not a direct consequence of the differential equation 5, while it recovers the Lagrangian immersion more directly than 53 We also note here the commutator relation [P, Q] = P