THE LAPLACE TRANSFORM OF THE CUT-AND-JOIN EQUATION AND THE BOUCHARD-MARIÑO CONJECTURE ON HURWITZ NUMBERS

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1 THE LAPLACE TRANSFORM OF THE CUT-AND-JOIN EQUATION AND THE BOUCHARD-MARIÑO CONJECTURE ON HURWITZ NUMBERS BERTRAND EYNARD, MOTOHICO MULASE, AND BRADLEY SAFNUK Abstract. We calculate the Laplace transform of the cut-and-join equation of Goulden, Jackson and Vakil. The result is a polynomial equation that has the topological structure identical to the Mirzakhani recursion formula for the Weil-Petersson volume of the moduli space of bordered hyperbolic surfaces. We find that the direct image of this Laplace transformed equation via the inverse of the Lambert W-function is the topological recursion formula for Hurwitz numbers conjectured by Bouchard and Mariño using topological string theory. Contents. Introduction. The Laplace transform of the ELSV formula 6 3. The cut-and-join equation and its Laplace transform 9 4. The Bouchard-Mariño recursion formula for Hurwitz numbers 4 5. Residue calculation 9 6. Analysis of the Laplace transforms on the Lambert curve 7. Proof of the Bouchard-Mariño topological recursion formula 8 Appendix. Examples of linear Hodge integrals and Hurwitz numbers 3 References 3. Introduction The purpose of this paper is to give a proof of the Bouchard-Mariño conjecture [3] on Hurwitz numbers using the Laplace transform of the celebrated cut-and-join equation of Goulden, Jackson, and Vakil [7, 43]. The cut-and-join equation, which seems to be essentially known to Hurwitz [3], expresses the Hurwitz number of a given genus and profile partition in terms of those corresponding to profiles modified by either cutting a part into two pieces or joining two parts into one. This equation holds for an arbitrary partition µ. We calculate the Laplace transform of this equation with µ as the summation variable. The result is a polynomial equation [38]. A Hurwitz cover is a holomorphic mapping f : X P from a connected nonsingular projective algebraic curve X of genus g to the projective line P with only simple ramifications except for P. Such a cover is further refined by specifying its profile, which is a partition µ = µ µ µ l > 0 of the degree of the covering deg f = µ = µ µ l. The length lµ = l of this partition is the number of points in the inverse image f = {p,..., p l } of. Each part µ i gives a local description of the map f, which is given by z z µ i in terms of a local coordinate z of X around p i. The number h g,µ of topological types of Hurwitz covers of given genus g and profile µ, 000 Mathematics Subject Classification. 4H0, 4N0, 4N35; 05A5, 05A7; 8T45. Saclay preprint number: IPHT T09/0.

2 B. EYNARD, M. MULASE, AND B. SAFNUK counted with the weight factor / Autf, is the Hurwitz number we shall deal with in this paper. A remarkable formula due to Ekedahl, Lando, Shapiro and Vainshtein [8,, 4] relates Hurwitz numbers and Gromov-Witten invariants. For genus g 0 and a partition µ subject to the stability condition g lµ > 0, the ELSV formula states that. h g,µ = g lµ µ! Autµ lµ i= µ µ i i µ i! Λ g lµ, M g,lµ µi ψ i where M g,l is the Deligne-Mumford moduli stack of stable algebraic curves of genus g with l distinct marked points, Λ g = c E g c g E is the alternating sum of Chern classes of the Hodge bundle E on M g,l, ψ i is the i-th tautological cotangent class, and Autµ denotes the group of permutations of equal parts of the partition µ. The linear Hodge integrals are the rational numbers defined by τ n τ nl c j E = ψn l c je, M g,l ψ n which is 0 unless n n l j = 3g 3l. To present our main theorem, let us introduce a series of polynomials ˆξ n t of degree n in t for n 0 by the recursion formula ˆξ n t = t t d dt ˆξ n t with the initial condition ˆξ 0 t = t. This differential operator appears in [9]. The Laplace transform of the cut-and-join equation gives the following formula. Theorem. [38]. Linear Hodge integrals satisfy recursion relations given as a series of equations of symmetric polynomials in l variables t,..., t l : l. τ nl Λ g g,l g lˆξ nl t L ˆξni t i ˆξ t L\{i} t L\{i} i n L = ˆξ τ m τ nl\{i,j} Λ m t i ˆξ 0 t j t i g g,l ˆξnL\{i,j} t L\{i,j} ˆξ m t j ˆξ 0 t i t j t i<j m,n i t j L\{i,j} l τ a τ b τ nl\{i} Λ g g,l stable g g =g I J=L\{i} i= n L\{i} a,b i= τ a τ ni Λ g g, I τ bτ nj Λ g g, J where L = {,..., l} is an index set, and for a subset I L, we denote l i= ˆξ a t i ˆξ b t i ˆξ nl\{i} t L\{i}, t I = t i i I, n I = { n i i I }, τ ni = τ ni, ˆξnI t I = ˆξ ni t i. i I i I The last summation in the formula is taken over all partitions of g and decompositions of L into disjoint subsets I J = L subject to the stability condition g I > 0 and g J > 0. Remark.. We note a similarity of the above formula and the Mirzakhani recursion formula for the Weil-Petersson volume of the moduli space of bordered hyperbolic surfaces of genus g with l closed geodesic boundaries [35, 36].

3 LAPLACE TRANSFORM OF HURWITZ NUMBERS 3 There is no a priori reason for the Laplace transform to be a polynomial equation. The above formula is a topological recursion. For an algebraic curve of genus g 0 and l distinct marked points on it, the absolute value of the Euler characteristic of the l-punctured Riemann surface, g l, defines a complexity of the moduli space M g,l. Eqn.. gives an effective method of calculating the linear Hodge integrals of complexity n > 0 from those with complexity n. 3 When we restrict. to the homogeneous highest degree terms, the equation reduces to the Witten-Kontsevich theorem of ψ-class intersections [7, 7, 44]. Let us explain the background of our work. Independent of the recent geometric and combinatorial works [7, 35, 36, 43], a theory of topological recursions has been developed in the matrix model/random matrix theory community [9, 3]. Its culmination is the topological recursion formula established in [3]. There are three ingredients in this theory: the Cauchy differentiation kernel which is referred to as the Bergman Kernel in [3, 3] of an analytic curve C C in the xy-plane called a spectral curve, the standard holomorphic symplectic structure on C, and the ramification behavior of the projection π : C C of the spectral curve to the x-axis. When C is hyperelliptic whose ramification points are all real, the topological recursion solves -Hermitian matrix models for the potential function that determines the spectral curve. It means that the formula recursively computes all n-point correlation functions of the resolvent of random Hermitian matrices of an arbitrary size. By choosing a particular spectral curve of genus 0, the topological recursion [0, 3, 4] recovers the Virasoro constraint conditions for the ψ-class intersection numbers τ n τ nl due to Witten [44] and Kontsevich [7], and the mixed intersection numbers κ m κm τ n τ nl due to Mulase-Safnuk [37] and Liu-Xu [3]. Based on the work by Mariño [33] and Bouchard, Klemm, Mariño and Pasquetti [] on remodeling the B-model topological string theory on the mirror curve of a toric Calabi-Yau 3-fold, Bouchard and Mariño [3] conjecture that when one uses the Lambert curve.3 C = {x, y x = ye y } C C as the spectral curve, the topological recursion formula of Eynard and Orantin should compute the generating functions.4 H g,l x,..., x l = = µ:lµ=l µ µ µ l g l µ! h g,µ n n l 3g 3l τ n τ nl Λ g l σ S l i= l x µ i σi i= µ i = µ µ in i i x µ i i µ i! of Hurwitz numbers for all g 0 and l > 0. Here the sum in the first line is taken over all partitions µ of length l, and S l is the symmetric group of l letters. Our discovery of this paper is that the Laplace transform of the combinatorics, the cutand-join equation in our case, explains the role of the Lambert curve, the ramification behavior of the projection π : C C, the Cauchy differentiation kernel on C, and residue calculations that appear in the theory of topological recursion. As a consequence of this explanation, we obtain a proof of the Bouchard-Mariño conjecture [3]. For this purpose, it is essential to use a different parametrization of the Lambert curve: x = e w and y = t. t

4 4 B. EYNARD, M. MULASE, AND B. SAFNUK Figure.. The Lambert curve C C C defined by x = ye y. The coordinate w is the parameter of the Laplace transformation, which changes a function in positive integers to a complex analytic function in w. Recall the Stirling expansion k kkn e k! π k n, which makes its Laplace transform a function of w. Note that the x-projection π of the Lambert curve.3 is locally a double-sheeted covering around its unique critical point x = e, y =. Therefore, the Laplace transform of the ELSV formula. naturally lives on the Lambert curve C rather than on the w-coordinate plane. Note that C is an analytic curve of genus 0 and t is its global coordinate. The point at infinity t = is the ramification point of the projection π. In terms of these coordinates, the Laplace transform of the ELSV formula becomes a polynomial in t-variables. The proof of the Bouchard-Mariño conjecture is established as follows. A topological recursion of [3] is always given as a residue formula of symmetric differential forms on the spectral curve. The Laplace-transformed cut-and-join equation. is an equation among primitives of differential forms. We first take the exterior differential of this equation. We then analyze the role of the residue calculation in the theory of topological recursion [3, 3], and find that it is equivalent to evaluating the form at q C and its conjugate point q C with respect to the local Galois covering π : C C near its critical point. This means all residue calculations are replaced by an algebraic operation of taking the direct image of the differential form via the projection π. We find that the direct image of. then becomes identical to the conjectured formula.5. Theorem.3 The Bouchard-Mariño Conjecture. The linear Hodge integrals satisfy exactly the same topological recursion formula discovered in [3]:.5 n,n L τ n τ nl Λ g g,l dˆξ n t dˆξ nl t L = l i= τ m τ nl\{i} Λ g g,l P m t, t i dt dt i dˆξ nl\{i} t L\{i} m,n L\{i} stable g g =g I J=L τ a τ Λ ni a,n I b,n J a,b,n L τ a τ b τ nl Λ g g,l g g, I τ bτ nj Λ g g, J P a,b tdt dˆξ nl t L,

5 LAPLACE TRANSFORM OF HURWITZ NUMBERS 5 where dˆξ ni t I = d ˆξni t i dt i. dt i i I The functions P a,b t and P n t, t i are defined by taking the polynomial part of the expressions P a,b tdt = [ tst dt ˆξa t st t tˆξ b st ˆξa st ˆξb t ], t [ ] tst ˆξn tdst ˆξ n st dt P n t, t i dt dt i = d ti, t st st t i t t i where st is the deck-tranformation of the projection π : C C around its critical point. The relation between the cut-and-join formula,. and.5 is the following: Cut-and-Join Equation Laplace Transform Polynomial Equation on Primitives. {Partitions} Lambert Curve Direct Image Galois Cover Topological Recursion on Differential Forms.5 Mathematics of the topological recursion and its geometric realization includes still many more mysteries [, 6, 3, 33]. Among them is a relation to integrable systems such as the Kadomtsev-Petviashvili equations [3]. In recent years these equations have played an essential role in the study of Hurwitz numbers [4, 5, 34, 40, 4, 4]. Since the aim of this paper is to give a proof of the Bouchard-Mariño conjecture and to give a geometric interpretation of the topological recursion for the Hurwitz case, we do not address this relation here. Since we relate the nature of the topological recursion and combinatorics by the Laplace transform, it is reasonable to ask: what is the inverse Laplace transform of the topological recursion in general? This question relates the Laplace transformation and the mirror symmetry. These are interesting topics to be further explored. It is possible to prove the Bouchard-Mariño formula without appealing to the cut-and-join equation. Indeed, a matrix integral expression of the generating function of Hurwitz numbers has been recently discovered in [], and its spectral curve is identified as the Lambert curve. As a consequence, the symplectic invariant theory of matrix models [9, ] is directly applicable to Hurwitz theory. The discovery of [] is that the derivatives of the symplectic invariants of the Lambert curve give H g,l x,..., x l of.4. The topological recursion formula of Bouchard and Mariño then automatically follows. A deeper understanding of the interplay between these two totally different techniques is desirable. Although our statement is simple and clear, technical details are quite involved. We have decided to provide all key details in this paper, believing that some of the analysis may be useful for further study of more general topological recursions. This explains the length of the current paper in the sections dealing with complex analysis and the Laplace transforms. The paper is organized as follows. We start with identifying the generating function.4 as the Laplace transform of the ELSV formula. in Section. We then calculate the Laplace transform of the cut-and-join equation in Section 3 following [38], and present the key idea of the proof of Theorem.. In Section 4 we give the statement of the Bouchard and Mariño conjecture [3]. We calculate the residues appearing in the topological recursion formula in Section 5 for the case of Hurwitz generating functions. The topological recursion C

6 6 B. EYNARD, M. MULASE, AND B. SAFNUK becomes the algebraic relation as presented in Theorem.3. In Section 6 we prove technical statements necessary for reducing. to.5 as a Galois average. The final Section 7 is devoted to proving the Bouchard-Mariño conjecture. As an effective recursion,. and.5 calculate linear Hodge integrals, and hence Hurwitz numbers through the ELSV formula. A computation is performed by Michael Reinhard, an undergraduate student of UC Berkeley. We reproduce some of his tables at the end of the paper. Acknowledgement. Our special thanks are due to Vincent Bouchard for numerous discussions and tireless explanations of the recursion formulas and the Remodeling theory. We thank the Institute for the Physics and Mathematics of the Universe, the Osaka City University Advanced Mathematical Institute, and the American Institute of Mathematics for their hospitality during our stay and for providing us with the opportunity of collaboration. Without their assistance, this collaboration would have never been started. We also thank Gaëtan Borot, Yon-Seo Kim, Chiu-Chu Melissa Liu, Kefeng Liu, Marcos Mariño, Nicolas Orantin, and Hao Xu for discussions, and Michael Reinhard for his permission to reproduce his computation of linear Hodge integrals and Hurwitz numbers in this paper. During the period of preparation of this work, B.E. s research was supported in part by the ANR project Grandes Matrices Aléatoires ANR-08-BLAN-03-0, the European Science Foundation through the Misgam program, and the Quebec government with the FQRNT; M.M. received financial support from the NSF, Kyoto University, Tôhoku University, KIAS in Seoul, and the University of Salamanca; and B.S. received support from IPMU.. The Laplace transform of the ELSV formula In this section we calculate the Laplace transform of the ELSV formula as a function in partitions µ. The result is a symmetric polynomial on the Lambert curve.3. A Hurwitz cover is a smooth morphism f : X P of a connected nonsingular projective algebraic curve X of genus g to P that has only simple ramifications except for the point at infinity P. Let f = {p,..., p l }. Then the induced morphism of the formal completion ˆf : ˆXpi ˆP is given by z z µ i with a positive integer µ i in terms of a formal parameter z around p i X. We rearrange integers µ i s so that µ = µ µ µ l > 0 is a partition of deg f = µ = µ µ l of length lµ = l. We call f a Hurwitz cover of genus g and profile µ. A holomorphic automorphism of a Hurwitz cover is an automorphism φ of X that preserves f: X φ X f f P. Two Hurwitz covers f : X P and f : X P are topologically equivalent if there is a homeomorphism h : X X such that X h X f f P.

7 LAPLACE TRANSFORM OF HURWITZ NUMBERS 7 The Hurwitz number of type g, µ is defined by Autf, h g,µ = [f] where the sum is taken over all topologically equivalent classes of Hurwitz covers of a given genus g and profile µ. Although h g,µ appears to be a rational number, it is indeed an integer for most of the cases because f has usually no non-trivial automorphisms. The celebrated ELSV formula [8,, 4] relates Hurwitz numbers and linear Hodge integrals on the Deligne-Mumford moduli stack M g,l consisting of stable algebraic curves of genus g with l distinct nonsingular marked points. Denote by π g,l : M g,l M g,l the natural projection and by ω πg,l the relative dualizing sheaf of the universal curve π g,l. The Hodge bundle E on M g,l is defined by E = π g,l ω πg,l, and the λ-classes are the Chern classes of the Hodge bundle: λ i = c i E H i M g,l, Q. Let σ i : M g,l M g,l be the i-th tautological section of π, and put L i = σ i ω π g,l. The ψ-classes are defined by ψ i = c L i H M g,l, Q. The ELSV formula then reads h g,µ = r! Autµ lµ i= µ µ i i µ i! Λ g lµ, M g,lµ µi ψ i where r = rg, µ = g lµ µ is the number of simple ramification points of f. The Deligne-Mumford stack M g,l is defined as the moduli space of stable curves satisfying the stability condition g l < 0. However, Hurwitz numbers are well defined for unstable geometries g, l = 0, and 0,. It is an elementary exercise to show that h 0,k = k k 3 and h 0,µ,µ = µ µ! µµ µ µ µ! µµ µ!. The ELSV formula remains true for unstable cases by defining Λ. 0 M 0, kψ = k, Λ 0 M 0, µ ψ µ ψ =.. µ µ Now fix an l, and consider a partition µ of length l as an l-dimensional vector consisting with positive integers. We define.3 H g µ = Autµ h g,µ rg, µ! l µ µ i i = Λ g µ i! l = M g,l i= µi ψ i i= as a function in µ. It s Laplace transform µ = µ,..., µ l N l i= n n l 3g 3l τ n τ nl Λ g.4 H g,l w,..., w l = µ N l H g µe µ w µ l w l l i= µ µ in i i µ i!

8 8 B. EYNARD, M. MULASE, AND B. SAFNUK is the function we consider in this paper. We note that the automorphism group Autµ acts trivially on the function e µ w µ l w l, which explains its appearance in.3. Since the coordinate change x = e w identifies.5 l x x l H g,l wx,..., wx l = H g,l x,..., x l, the Laplace transform.4 is a primitive of the generating function.4. Before performing the exact calculation of the holomorphic function H g,l w,..., w l, let us make a quick estimate here. From Stirling s formula k kkn e k! π k n, it is obvious that H g,l w,..., w l is holomorphic on Rew i > 0 for all i =,..., l. Because of the half-integer powers of µ i s, the Laplace transform H g,l w,..., w l is expected to be a meromorphic function on a double-sheeted covering of the w i -planes. Such a double covering is provided by the Lambert curve C of.3. So we define.6 t = k= k k k! e kw, which gives a global coordinate of C. The summation converges for Rew > 0, and the Lambert curve is expressed in terms of w and t coordinates as.7 e w = e t. t The w-projection π : C C is locally a double-sheeted covering at t =. The inverse function of.6 is given by.8 w = wt = t log = t m t m, which is holomorphic on Ret >. When considered as a functional equation,.8 has exactly two solutions: t and.9 st = t t t t 4. This is the deck-transformation of the projection π : C C near t = and satisfies the involution equation s st = t. It is analytic on C \ [0, ] and has logarithmic singularities at 0 and. Although wt = w st, st is not given by the Laplace transform.6. Since the Laplace transform k kn.0 ˆξn t = e kw k! k= also naturally lives on C, it is a meromorphic function in t rather than in w. Actually it is a polynomial of degree n for n 0 because of the recursion formula m=. ˆξn t = t t d dt ˆξ n t for all n 0, which follows from.6,.0, and.8 that implies. dw = dt t t.

9 LAPLACE TRANSFORM OF HURWITZ NUMBERS 9 We note that the differential operator of. is discovered in [9]. For future convenience, we define.3 ˆξ t = t = y, t which is indeed the y coordinate of the original Lambert curve.3. We now see that the Laplace transform.4 Ĥ g,l t,..., t l = H g,l wt,..., wt l = µ N l H g µe µ w µ l w l = n n l 3g 3l τ n τ nl Λ g l ˆξ ni t i is a symmetric polynomial in the t-variables when g l > 0. It has been noted in [, 0,, 3, 4] that the Airy curve w = v is a universal object of the topological recursion for the case of a genus 0 spectral curve with only one critical point. Analysis of the Airy curve provides a good control of the topological recursion formula for such cases. The Airy curve expression is also valid around any non-degenerate critical point of a general spectral curve. To switch to the local Airy curve coordinate, we define.5 v = vt = t 3 t 7 36 t t t 5 as a function in t that solves.6 v = w = t log = t st log = st Note that we are making a choice of the branch of the square root of w that is consistent with.6. The involution.9 becomes simply.7 vt = v st. The new coordinate v plays a key role later when we reduce the Laplace transform of the cut-and-join equation. to the Bouchard-Mariño topological recursion.5. m= 3. The cut-and-join equation and its Laplace transform In the modern times the cut-and-join equation for Hurwitz numbers was discovered in [7, 43], though it seems to be known to Hurwitz [3]. It has become an effective tool for studying algebraic geometry of Hurwitz numbers and many related subjects [4, 8, 9, 0, 5, 6, 8, 30, 4, 45]. In this section we calculate the Laplace transform of the cut-and-join equation following [38]. The simplest way of presenting the cut-and-join equation is to use a different primitive of the same generating function of Hurwitz numbers.4. Let 3. Hs, p = g 0 H g,l s, p; H g,l s, p = l µ:lµ=l h g,µ p µ s r r!, where p µ = p µ p µ p µl, and r = g l µ is the number of simple ramification points on P. The summation is over all partitions of length l. Here p k is the power-sum symmetric function 3. p k = i x k i, m i= t m.

10 0 B. EYNARD, M. MULASE, AND B. SAFNUK which is related to the monomial symmetric functions by l p µ = l µ i x µ i x x σi. l σ S l i= Therefore, we have l x x l H g,l, p = H g,l x,..., x l = µ:lµ=l µ µ µ l g l µ! h g,µ l σ S l i= x µ i σi, which is the generating function of.4. Because of the identification.5, the primitives H g,l, p and Ĥg,lt,..., t l of.4 are essentially the same function, different only by a constant. Remark 3.. Although we do not use the fact here, we note that Hs, p is a one-parameter family of τ-functions of the KP equations with k p k as the KP time variables [5, 40]. The parameter s is the deformation parameter. Let z P be a point at which the covering f : X P is simply ramified. Locally we can name sheets, so we assume sheets a and b are ramified over z. When we merge z to, one of the two things happen: The cut case. If both sheets are ramified at the same point x i of the inverse image f = {x,..., x l }, then the resulting ramification after merging z to has a profile µ,..., µ i,..., µ l, α, µ i α = µî, α, µ i α for α < µ i. Otherwise we are in the join case. If sheets a and b are ramified at two distinct points, say x i and x j above, then the result of merging creates a new profile µ,..., µ i,..., µ j,..., µ l, µ i µ j = µî, ĵ, µ i µ j. Here the sign means removing the entry. The above consideration tells us what happens to the generating function of the Hurwitz numbers when we differentiate it by s, because it decreases the degree in s, or the number of simple ramification points, by. Since the cut case may cause a disconnected covering, let us use the generating function of Hurwitz numbers allowing disconnected curves to cover P. Then the cut-and-join equation takes the following simple form: s α βp α p β αβp αβ e Hs,p = 0. p αβ p α p β α,β It immediately implies H 3.3 s = α,β H α βp α p β αβp αβ p αβ H H αβp αβ H p α p β p α p β which is the cut-and-join equation for the generating function Hs, p of the number of connected Hurwitz coverings. Let us now apply the ELSV formula. to 3.. We obtain 3.4 H g,l s, p = τ nl Λ g l! g,l s g l n L N l l i= µ i = µ µ in i i s µ i p µi µ i!,

11 LAPLACE TRANSFORM OF HURWITZ NUMBERS = H g µp µ s r = l! µ,...,µ l N l µ:lµ=l Autµ H gµp µ s r, where H g µ is introduced in.3. Now for every choice of r and a partition µ, the coefficient of p µ s r of the cut-and-join equation 3.3 gives Theorem 3. [38]. The functions H g µ of.3 satisfy a recursion equation 3.5 rg, µh g µ = µ i µ j H g µî, ĵ, µi µ j i<j l H g µî, α, β H g ν, αh g ν, β. Remark 3.3. Note that αβ i= αβ=µ i l µî, ĵ = l g g =g ν ν =µî lν lν = l µî = l. Thus the complexity g l is one less for the coverings appearing in the RHS of 3.5, which is the effect of / s applied to Hs, p, except for the unstable geometry corresponding to g i = 0 and ν i = 0 in the join terms. If we move the 0, -terms to the LHS, then the cut-and-join equation 3.5 becomes a topological recursion formula. Let us first calculate the Laplace transform of the cut-and-join equation for the l = case to see what is involved. We then move on to the more general case later, following [38]. Proposition 3.4. The Laplace transform of the cut-and-join equation for the l = case gives the following equation: [ 3.6 τ n Λ g g, g ˆξ n t ˆξ n t ˆξ t ] = n 3g ab 3g 4 [ τ a τ b Λ g g, stable g g =g τ a Λ g g, τ bλ g g, Proof. The cut-and-join equation for l = is a simple equation 3.7 g µh g µ = αβ H g α, β αβ=µ g g =g The Laplace transform of the LHS of 3.7 is τ n Λ g g, [g ˆξ n t ˆξ ] n t. n 3g ] H g αh g β ˆξ a tˆξ b t. When summing over µ to compute the Laplace transform of the RHS, we switch to sum over α and β independently. The factor cancels the double count on the diagonal. Thus the Laplace transform of the stable geometries of the RHS is [ ] stable τ a τ b Λ g g, τ a Λ g g, τ bλ g g, ˆξ a tˆξ b t. ab 3g 4 g g =g.

12 B. EYNARD, M. MULASE, AND B. SAFNUK The unstable terms contained in the second summand of the RHS of 3.7 are the g = 0 terms H 0 αh g β H g αh 0 β. We calculate the Laplace transform of these unstable terms using.. Since the result is This completes the proof. H 0 α = αα, α! τ a Λ g g, ˆξ tˆξ a t. a Remark 3.5. We note that 3.6 is a polynomial equation of degree n. Since ˆξ t = t, the leading term of ˆξ n t is canceled in the formula. To calculate the Laplace transform of the general case 3.5, we need to deal with both of the unstable geometries g, l = 0, and 0,. These are the exceptions for the general formula.4. Recall the 0, case.. The formula 3.8 Ĥ 0, t = k= where the constant c is given by k k k! c = e kw = t c = ˆξ t, k= k k is used in 3.6. The g, l = 0, terms require a more careful computation. We shall see that these are the terms that exactly correspond to the terms involving the Cauchy differentiation kernel in the Bouchard-Mariño recursion. k! e k, Proposition 3.6. We have the following Laplace transformation formula: 3.9 Ĥ 0, t, t = µ,µ Proof. Since x = e w, 3.9 is equivalent to 3.0 µ,µ 0 µ,µ 0,0 µµ µ µ µ! µµ µ! e µ w e µ w µµ µ µ µ! µµ µ! xµ ˆξ t = log ˆξ t x x ˆξ t ˆξ t. xµ = log k= k k k! xk xk x x where x < e, x < e, and 0 < x x < e so that the formula is an equation of holomorphic functions in x and x. Define φx, x def = µµ µ µ µ! µµ µ! xµ xµ log k k xk xk. k! x x Then φx, 0 = µ µ,µ 0 µ,µ 0,0 µ µ x µ k k log x k µ! k! k= k=,

13 LAPLACE TRANSFORM OF HURWITZ NUMBERS 3 = ˆξ t log ˆξ t x = t log log x t = t log w = 0 t due to.8. Here t is restricted on the domain Ret >. Since k k x log xk xk x k! x x k= = t t log ˆξ t t ˆξ t x logx x x = t t log t t x t x x we have x x log x x k= k k k! xk xk x x = t t t t t x x x, = t t t t t t t t x x x x = t t = ˆξ 0 t ˆξ 0 t ˆξ 0 t ˆξ 0 t. On the other hand, we also have x x µµ x x µ µ µ! µµ µ! xµ Therefore, 3. µ,µ 0 µ,µ 0,0 x xµ x φx, x = 0. x x = ˆξ 0 t ˆξ 0 t ˆξ 0 t ˆξ 0 t. Note that φx, x is a holomorphic function in x and x. Therefore, it has a series expansion in homogeneous polynomials around 0, 0. Since a homogeneous polynomial in x and x of degree n is an eigenvector of the differential operator x x x x belonging to the eigenvalue n, the only holomorphic solution to the Euler differential equation 3. is a constant. But since φx, 0 = 0, we conclude that φx, x = 0. This completes the proof of 3.0, and hence the proposition. The following polynomial recursion formula was established in [38]. Since each of the polynomials Ĥg,lt L s in 3. satisfies the stability condition g l > 0, it is equivalent to. after expanding the generating functions using.4. Theorem 3.7 [38]. The Laplace transform of the cut-and-join equation 3.5 produces the following polynomial equation on the Lambert curve:

14 4 B. EYNARD, M. MULASE, AND B. SAFNUK 3. = i<j l g l ˆξ t i t i t i Ĥ g,l t,..., t l t i= i t i t i t t i t i Ĥ g,l t,..., t j,..., t l t j t j t j Ĥ g,l t,..., t i,..., t l j t i t j t 3 i t i Ĥ g,l t,..., t j,..., t l t i i j l [ u u u ] u Ĥ g,l u, u, t u u L\{i} i= l i= stable g g =g J K=L\{i} u =u =t i t i t i t i Ĥ g, J t i, t J t i t i t i Ĥ g, K t i, t K. In the last sum each term is restricted to satisfy the stability conditions g J > 0 and g K > 0. Remark 3.8. The polynomial equation 3. is equivalent to the original cut-and-join equation 3.5. Note that the topological recursion structure of 3. is exactly the same as.5. Although 3. contains more terms, all functions involved are polynomials that are easy to calculate from., whereas.5 requires computation of the involution st of.9 and infinite series expansions. Remark 3.9. It is an easy task to deduce the Witten-Kontsevich theorem, i.e., the Virasoro constraint condition for the ψ-class intersection numbers [44, 7], from 3.. Let us use the normalized notation σ n = n!!τ n for the ψ-class intersections. Then the formula according to Dijkgraaf, Verlinde and Verlinde [7] is 3.3 σ n σ nl g,l = ab=n σ a σ b σ nl g,l i L stable g g =g I J=L ab=n n i σ nni σ nl\{i} g,l σ a σ ni g, I σ bσ nj g, J. Eqn.3.3 is exactly the relation of the homogeneous top degree terms of 3., after canceling the highest degree terms coming from ˆξ ni t i in the LHS [38]. This derivation is in the same spirit as those found in [4, 6, 4], though the argument is much clearer due to the polynomial nature of our equation. 4. The Bouchard-Mariño recursion formula for Hurwitz numbers In this section we present the precise statement of the Bouchard-Mariño conjecture on Hurwitz numbers. Recall the function we introduced in.6: 4. t = tx = k= k k k! xk

15 LAPLACE TRANSFORM OF HURWITZ NUMBERS 5 This is closely related to the Lambert W -function 4. W x = k! k= k k x k. By abuse of terminology, we also call the function tx of 4. the Lambert function. The power series 4. has the radius of convergence /e, and its inverse function is given by 4.3 x = xt = e t. e t Motivated by the Lambert W -function, a plane analytic curve 4.4 C = {x, t x = xt} C C is introduced in [3], which is exactly the Lambert curve.3. We denote by π : C C the x-projection. Bouchard-Mariño [3] then defines a tower of polynomial differentials on the Lambert curve C by 4.5 ξ n t = d [ ] t t ξ n t dt with the initial condition 4.6 ξ 0 t = dt. It is obvious from 4.5 and 4.6 that for n 0, ξ n t is a polynomial -form of degree n with a general expression ] n 3!! ξ n t = t [n n!! t n n!! t n n n! dt. 3 All the coefficients of ξ n t have a combinatorial meaning called the second order reciprocal Stirling numbers. As we will note below, the leading coefficient is responsible for the Witten- Kontsevich theorem on the cotangent class intersections, and the lowest coefficient is related to the λ g -formula [38]. For a convenience, we also use ξ t = t dt and ξ t = t 3 dt. Remark 4.. The polynomial ˆξ n t of.0 is a primitive of ξ n t: 4.7 dˆξ n t = ξ n t. Definition 4.. Let us call the symmetric polynomial differential form l d l Ĥ g,l t,..., t l = τ n τ nl Λ g ξ ni t i n n l 3g 3l on C l the Hurwitz differential of type g, l. Remark 4.3. Our ξ n t is exactly the same as the ζ n y-differential of [3]. However, this mere coordinate change happens to be essential. Indeed, the fact that our expression is a polynomial in t-variables allows us to calculate the residues in the Bouchard-Mariño formula in Section 5. Remark 4.4. The degree of d l Ĥ g,l t,..., t l is 3g 3 l, and the homogeneous top degree terms give a generating function of the ψ-class intersection numbers l l τ n τ nl n i!! t n i dt i. n n l =3g 3l i= i i= i=

16 6 B. EYNARD, M. MULASE, AND B. SAFNUK The homogeneous lowest degree terms of d l Ĥ g,l t,..., t l are 3g 3l n n l =g 3l τ n τ nl λ g l n i! t n i i= i l dt i. The combinatorial coefficients of the λ g -formula [5, 6] can be directly deduced from the topological recursion formula. [38], explaining the mechanism found in [0]. Remark 4.5. The unstable Hurwitz differentials follow from. and 3.9. They are i= 4.8 d Ĥ0,t = t 3 dt = ξ t; 4.9 d Ĥ 0, t, t = dt dt t t π dx dx x x. Remark 4.6. The simplest stable Hurwitz differentials are given by 4.0 d 3 Ĥ 0,3 t, t, t 3 = dt dt dt 3 ; d Ĥ,t = ξ0 t ξ t = t 3t dt. 4 4 The amazing insight of Bouchard and Mariño [3] is that the Hurwitz differentials of Definition 4. should satisfy the topological recursion relation of Eynard and Orantin [3] based on the analytic curve C of 4.4 as the spectral curve. Since the topological recursion utilizes the critical behavior of the x-projection π : C C, let us examine the local structure of C around its critical points. Let z = t be a coordinate of C centered at t =. The Lambert curve is then given by x = e ze z. We see that the x-projection π : C C has a unique critical point q 0 at z = 0. Locally around q 0 the curve C is a double cover of C branched at q 0. For a point q C near q 0, let us denote by q the Galois conjugate point on C that has the same x-coordinate. Let Sz be the local deck-transformation of the covering π : C C. Its defining equation 4. Sz log Sz m = z log z = m zm has a unique analytic solution other than z itself, which has a branch cut along, ]. We note that S is an involution S Sz = z, and has a Taylor expansion Sz = z 3 z 4 9 z z z z z z8 Oz 9 for z <. In terms of the t-coordinate, the involution corresponds to st of.9: { tq = z = t t q = Sz = st. The equation 4. defining Sz translates into a relation 4. dt t t = m= dst st = dw = vdv = π st dx. x

17 LAPLACE TRANSFORM OF HURWITZ NUMBERS 7 Using the global coordinate t of the Lambert curve C, the Cauchy differentiation kernel the one called the Bergman kernel in [3, 3] is defined by 4.3 Bt, t = dt dt t t = d t d t logt t. We have already encountered it in 4.9 in the expression of H 0, t, t. Following [3], define a -form on C by deq, q, t = q B, t = dt q t t st t = ˆξ st t ˆξ t t dt, where the integral is taken with respect to the first variable of Bt, t along any path from q to q. The natural holomorphic symplectic form on C C is given by Ω = d log y d log x = d log d log x. t Again following [3, 3], let us introduce another -form on the curve C by q ωq, q = Ω, xq = t dt st t t = dt ˆξ st ˆξ t t t. q The kernel operator is defined as the quotient Kt, t = deq, q, t ωq, q = t st t t dt, t t st t dt which is a linear algebraic operator acting on symmetric differential forms on C l by replacing dt with dt. We note that 4.4 Kt, t = K st, t, which follows from 4.. In the z-coordinate, the kernel has the expression 4.5 K = z z = z z dt zt Szt dz = m zm Sz m t m dt z Sz dz m=0 z 3 3t tz 3t tz t3 80t 30t 6tz 3 dt dz. Definition 4.7. The topological recursion formula is an inductive mechanism of defining a symmetric l-form W g,l t L = W g,l t,..., t l on C l for any given g and l subject to g l > 0 by 4.6 W g,l t 0, t L = [ Kt, t 0 W g,l t, st, tl πi γ

18 8 B. EYNARD, M. MULASE, AND B. SAFNUK l i= W g,l t, tl\{i} B st, ti Wg,l st, tl\{i} Bt, ti stable terms g g =g, I J=L W g, I t, ti Wg, J st, tj ]. Here t I = t i i I for a subset I L = {,,..., l}, and the last sum is taken over all partitions of g and disjoint decompositions I J = L subject to the stability condition g I > 0 and g J > 0. The integration is taken with respect to dt on the contour γ, which is a positively oriented loop in the complex t-plane of large radius so that t > max t 0, st 0 for t γ. Now we can state the Bouchard-Mariño conjecture, which we prove in Section 7. Conjecture 4.8 Bouchard-Mariño Conjecture [3]. For every g and l subject to the stability condition g l > 0, the topological recursion formula 4.6 with the initial condition { W 0,3 t, t, t 3 = dt dt dt W, t = 4 t 3t dt gives the Hurwitz differential W g,l t,..., t l = d l Ĥ g,l t,..., t l. Remark 4.9. In the literature [3, 3], the topological recursion is written as 4.8 W g,l t 0, t L = Res q= q [ deq, q, t 0 W g,l tq, t q, tl ωq, q g g =g, I J=L W g, I tq, ti Wg, J t q, tj ], including all possible terms in the second line, with the initial condition { W 0, t = W 0, t, t = Bt, t. If we single out the stable terms from 4.8, then we obtain 4.6. Although the initial values of W g,l given in 4.9 are different from 4.8 and 4.9, the advantage of 4.8 is to be able to include 4.7 as a consequence of the recursion. Remark 4.0. It is established in [3] that a solution of the topological recursion is a symmetric differential form in general. In our case, the RHS of the recursion formula 4.6 does not appear to be symmetric in t 0, t,..., t l. We note that our proof of the formula establishes this symmetry because the Hurwitz differential is a symmetric polynomial. This situation is again strikingly similar to the Mizrakhani recursion [35, 36], where the symmetry appears not as a consequence of the recursion, but rather as the geometric nature of the quantity the recursion calculates, namely, the Weil-Petersson volume of the moduli space of bordered hyperbolic surfaces.

19 LAPLACE TRANSFORM OF HURWITZ NUMBERS 9 5. Residue calculation In this section we calculate the residues appearing in the recursion formula 4.6. It turns out to be equivalent to the direct image operation with respect to the projection π : C C. Recall that the kernel Kt, t 0 is a rational expression in terms of t, st and t 0. The function st is an involution s st = t defined outside of the slit [0, ] of the complex t-plane, with logarithmic singularities at 0 and. Our idea of computing the residue is to decompose the integration over the loop γ into the sum of integrations over γ γ [0,] and γ [0,], where γ [0,] is a positively oriented thin loop containing the interval [0, ] Figure 5.. The contours of integration. γ is the circle of a large radius, and γ [0,] is the thin loop surrounding the closed interval [0, ]. Definition 5.. For a Laurent series n Z a nt n, we denote [ ] a n t n a n t n. n Z = n 0 Theorem 5.. In terms of the primitives ˆξ n t, we have 5. R a,b t = Kt, tξ a t ξ b st πi γ = [ tst ] ξ a tˆξ b st ˆξa st ξb t. t st Similarly, we have 5. R n t, t i = Kt, t ξ n t B st, t i ξn st Bt, t i πi γ [ tst = ˆξn tb st, t i ˆξn st Bt, ti ]. t st Proof. In terms of the original z-coordinate of [3], the residue R a,b t is simply the coefficient of z in Kt, tξ a t ξ b st, after expanding it in the Laurent series in z. Since ξ n t is a polynomial in t = z, the contribution to the z term in the expression is a polynomial in t because of the z-expansion formula 4.5 for the kernel K. Thus we know that R a,b t is a polynomial in t.

20 0 B. EYNARD, M. MULASE, AND B. SAFNUK Let us write ξ n t = f n tdt, and let γ [0,] be a positively oriented loop containing the slit [0, ], as in Figure 5.. On this compact set we have a bound tst t st t t s tf a tf b st < M, since the function is holomorphic outside [0, ]. Choose t >>. Then we have Kt, tξ a t ξ b st πi γ [0,] = πi γ [0,] t t t st st t st t t t s t f a t f b st dt dt < M π γ [0,] t t st t dt dt M 4π t dt. Therefore, Kt, tξ a t ξ b st = πi γ πi = [ πi Kt, tξ a t ξ b st Ot γ γ [0,] Kt, tξ a t ξ b st ]. γ γ [0,] Noticing the relation 4. and the fact that st is an involution, we obtain Kt, tξ a t ξ b st πi γ γ [0,] = πi t t t st st t st t t t s t f a t f b st dt dt = πi = πi γ γ [0,] γ γ [0,] sγ sγ [0,] t t t st st t t t s t f a t f b st dt dt st t tst t st t t s tf a tf b st dt stt t st st st f a st fb tdt t st st t t t f a t f b st dst dt = tst t st t t s t f atf b st f a stf b t dt = tst ξ a tˆξ b st ˆξa st ξb t t st = tst ˆξ t st a tˆξ b st ˆξa st ˆξ b t dt = tst dt ˆξa tˆξ b st ˆξa st ˆξb t t st t t. Here we used. and 4. at the last step. The proof of the second residue formula is exactly the same.

21 LAPLACE TRANSFORM OF HURWITZ NUMBERS Remark 5.3. The equation for the kernel 4.4 implies R a,b t = R b,a t = [ R a,b st ]. Let us define polynomials P a,b t and P n t, t i by P a,b tdt = [ tst dt ˆξa t st t tˆξ b st ˆξa st ˆξb t ] 5.3 t [ ] tst ˆξn tdst ˆξ n st dt 5.4 P n t, t i dt dt i = d ti. t st st t i t t i Obviously deg P a,b t = a b. expansion 5.5 t t i = t To calculate P n t, t i, we use the Laurent series k=0 k ti, t and take the polynomial part in t. We note that it is automatically a polynomial in t i as well. We thus see that deg P n t, t i = n in each variable. Theorem 5.4. The topological recursion formula 4.6 is equivalent to the following equation of symmetric differential forms in l variables with polynomial coefficients: n,n L τ n τ nl Λ g g,l dˆξ n t dˆξ nl t L = l i= τ m τ nl\{i} Λ g g,l P m t, t i dt dt i dˆξ nl\{i} t L\{i} m,n L\{i} stable g g =g I J=L τ a τ Λ ni a,n I b,n J a,b,n L τ a τ b τ nl Λ g g,l g g, I τ bτ nj Λ g g, J Here L = {,..., l} is an index set, and for a subset I L, we denote t I = t i i I, n I = { n i i I }, τ ni = i I τ ni, dˆξ ni t I = i I P a,b tdt dˆξ nl t L. d dt i ˆξni t i dt i. The last summation in the formula is taken over all partitions of g and decompositions of L into disjoint subsets I J = L subject to the stability condition g I > 0 and g J > 0. Remark 5.5. An immediate observation we can make from.5 is the simple form of the formula for the case with one marked point: 5.6 n 3g τ n Λ g g, d dt ˆξ n t = ab 3g 4 τ a τ b Λ g g, stable g g =g τ a Λ g g, τ bλ g g,, P a,b t.

22 B. EYNARD, M. MULASE, AND B. SAFNUK 6. Analysis of the Laplace transforms on the Lambert curve As a preparation for Section 7 where we give a proof of.5, in this section we present analysis tools that provide the relation among the Laplace transforms on the Lambert curve.7. The mystery of the work of Bouchard-Mariño [3] lies in their ζ n y-forms that play an effective role in devising the topological recursion for the Hurwitz numbers. We have already identified these differential forms as polynomial forms dˆξ n t, where ˆξ n t s are the Lambert W -function and its derivatives. Recall Stirling s formula 6. log Γz = log π z log z z m r= B r rr z r m 0 B m x [x] dx, z x m where m is an arbitrary cut-off parameter, B r s is the Bernoulli polynomial defined by ze zx e z = n=0 B r x zr r!, B r = B r 0 is the Bernoulli number, and [x] is the largest integer not exceeding x R. For N > 0, we have N N Nn e = m N n B r exp N! π rr N r B m x [x] exp dx. m 0 N x m r= Let us define the coefficients s k for k 0 by 6. s k N k B r = exp rr N r k=0 r= = N 88 N N N 4. Then for a large N we have an asymptotic expansion N N Nn 6.3 e N! π N n s k N k. Definition 6.. Let us introduce an infinite sequence of Laurent series 6.4 η n v = n k!! s k v v n k = η n v k=0 k=0 for every n Z, where s k s are the coefficients defined in 6.. The following lemma relates the polynomial forms ξ n t = dˆξ n t, the functions η n v, and the Laplace transform. Proposition 6.. For n 0, we have 6.5 e s 0 s sn Γs e sw ds = η n v const Ow

23 LAPLACE TRANSFORM OF HURWITZ NUMBERS 3 with respect to the choice of the branch of w specified by v = w as in.8 and.5, where Ow denotes a holomorphic function in w = v defined around w = 0 which vanishes at w = 0. The substitution of.5 in η n v yields 6.6 η n v = η n v η n v = ˆξ n t ˆξ n st, where st is the involution of.9. This formula is valid for n, and in particular, we have a relation between the kernel and η v: 6.7 η v = ˆξ t ˆξ st = More precisely, for n, we have { η n v = 6.8 ˆξ n t F n w η n v = ˆξ n st Fn w where F n w is a holomorphic function in w. t st tst. Proof. From definition 6.4, it is obvious that the series η n v satisfies the recursion relation 6.9 η n v = d v dv η nv for all n Z. The integral 6.5 also satisfies the same recursion for n 0. So choose an n 0. We have an estimate e s s sn Γs = n s n s k s k Os 3 π that is valid for s >. Since the integral is an entire function in w, we have e s 0 = = v 0 s sn Γs e sw ds = n s n π n k=0 k=0 s k n k!! v n k e s 0 e s 0 s sn k=0 Γs e sw ds s sn s k s k e sw ds Γs e sw ds const vow Ow., e s s sn Γs e sw ds Os 3 e sw ds const Ow This formula is valid for all n 0. Starting it from a large n >> 0 and using the recursion 6.9 backwards, we conclude that the vow terms in the above formula are indeed the positive power terms of η n v. The principal part of η n v does not depend on the addition of positive power terms in w = v, since d v dv transforms a positive even power of v to a non-negative even power and does not create any negative powers. This proves 6.5. Next let us estimate the holomorphic error term Ow in 6.5. When n, the Laplace transform 6.5 does not converge. However, the truncated integral e s s sn Γs e sw ds

24 4 B. EYNARD, M. MULASE, AND B. SAFNUK always converges and defines a holomorphic function in v = w, which still satisfies the recursion relation 6.9. Again by the inverse induction, we have 6.0 e s s sn Γs e sw ds = η n v Ow for every n < 0. Now by the Euler summation formula, for n and Rew > 0, we have 6. e s s sn Γs e sw ds e k= = e w k kkn e kw k! s [s] d e s s sn ds Γs e sw ds. Note that the RHS of 6. is holomorphic in w around w = 0. From.0, 6.0 and 6., we establish a comparison formula 6. η v ˆξ t = F w, where F w is a holomorphic function in w defined near w = 0, and we identify the coordinates t, v and w by the relations.6 and.5. Note that the relation.6 is invariant under the involution 6.3 { v v t st. Therefore, we also have Thus we obtain η v ˆξ st = F w. η v = d ˆξ t ˆξ st, which proves 6.7. Since v dv = t t d dt, the recursion relations 6.9 and. for ˆξ n t are exactly the same. We note that from 4. we have d v dv = t t d dt = st st d dst. Therefore, the difference ˆξ n t ˆξ n st satisfy the same recursion 6.4 ˆξn t ˆξ n st = t t d ˆξ n t dt ˆξ n st. The recursions 6.9 and 6.4, together with the initial condition 6.7, establish 6.6. Application of the differential operator n -times to 6. yields v d dv = d dw = t t d dt η n v ˆξ n t = F n w, where F n w = n dn dw n F w is a holomorphic function in w around w = 0. Involution 6.3 then gives η n v ˆξ n st = Fn w. This completes the proof of the proposition.

25 LAPLACE TRANSFORM OF HURWITZ NUMBERS 5 As we have noted in Section 5, the residue calculations appearing in the Bouchard-Mariño recursion formula 4.6 are essentially evaluations of the product of ξ-forms at the point t and st on the Lambert curve, if we truncate the result to the polynomial part. In terms of the v-coordinate, these two points correspond to v and v. Thus we have Corollary 6.3. The residue polynomials of 5.3 are given by 6.5 P a,b tdt = [ tst dt ˆξa t st t tˆξ b st ˆξa st ˆξb t ] t [ = η a vη b v vdv η v k=0 v=vt where the reciprocal of η v= s k k!! v k = v k s k k!! vk is defined by η v = v k= m k s k k!! vk. m=0 Proof. Using the formulas established in Proposition 6., we compute tst dt ˆξa t st t tˆξ b st ˆξa st ˆξb t t = tst dt ˆξa t ˆξ a st ˆξ b t ˆξ b st t st t t ˆξ a t ˆξ a st ˆξ b t ˆξ b st = η a vη b v F a wf b w vdv η v = η avη b v vdv const Ow dv. η v k= From.5 we see [ const Ow dv v t = 0. This completes the proof of 6.5. ] For the terms involving the Cauchy differentiation kernel Bt i, t j, we have the following formula. Proposition 6.4. As a polynomial in t and t j, we have the following equality: [ ] tst ˆξn tdst ˆξ n st dt 6.6 P n t, t j dt dt j = d tj t st st t j t t j = d tj η nv j η v v finite m=0 vj m vdv v v=vt v j =vt j In the RHS we first evaluate the expression at v = vt and v j = vt j, then expand it as a series in t and t j, and finally truncate it as a polynomial in both t and t j. ],.

26 6 B. EYNARD, M. MULASE, AND B. SAFNUK Proof. From the formulas for ˆξ n t and η n v, we know that both expressions have the same degree n in t and t j. Since the powers of v j in the summation finite vj m m=0 v is non-negative, clearly we have d tj η finite nv η v vj m vdv v = 0. v m=0 v=vt v j =vt j Thus we can replace the RHS of 6.6 by d tj η nv η n v j η v v finite m=0 vj m vdv v v=vt v j =vt j Since the degree of η n vtj in t j is n 3, the finite sum in m of the above expression contributes nothing for m > n. Therefore, d tj η finite nv η n v j vj m vdv η v v v m=0 v=vt v j =vt j = d tj tst ˆξn t ˆξ n t j F nw F n w j t st w w j w w j = d tj tst t st ˆξ n t ˆξ n t j dw w w j because of 6.8. We also used the fact that v finite m=0 vj m vdv = v v=vt v j =vt j dw Owj n dw, w w j dw. v=vt v j =vt j and that F nw F n w j w w j is holomorphic along w = w j. Let us use once again ˆξ n t = ˆξ n st Fn w and dw w w j d tj tst t st = vdv v v j ˆξ n t ˆξ n t j dw w w j v=vt v j =vt j = v v j v v j dv. We obtain = d tj tst ˆξn t ˆξ n t j t st v v j d tj tst ˆξn st ˆξn stj t st v v j dv dst dst v=vt v j =vt j dv dt dt v=vt v j =vt j

27 LAPLACE TRANSFORM OF HURWITZ NUMBERS 7 [ tst ˆξn t = d ˆξ n t j st t j tj t st st t j v st dv st ] dst vt j dst [ tst ˆξn st ˆξn stj ] t t j dvt d tj dt t st t t j vt vt j dt Here we remark that [ tst ˆξn t d ˆξ n t j ˆξ n st ˆξn stj ] tj dst dt t st st t j t t j [ ] tst ˆξn t ˆξ n st = d tj dst dt t st st t j t t j because the extra terms in the LHS do not contribute to the polynomial part in t. Therefore, it suffices to show that [ tst ˆξ n t 6.7 d ˆξ n t j st t j tj t st st t j v st dv st ] dst vt j dst [ tst ˆξ n st ˆξn stj ] t tj dvt d tj dt t st t t j vt vt j dt [ tst = d tj ˆξ n t t st ˆξ dvt n t j vt vt j dst ] st t j [ tst d tj ˆξ n st ˆξn stj dvt t st vt vt j dt ] t t j [ tst = d tj ˆξ n t t st ˆξ dvt n t j vt vt j dst ] st t j [ tst d tj ˆξ n t t st ˆξ dvt n t j vt vt j dt ] = 0, t t j in light of 6.8. At this stage we need the following Lemma: Lemma 6.5. For every n 0 we have the identity dv = d tj t n t n j v v j dst st t j dv v v j dt. t t j v=vt v j =vt j Proof of Lemma. First let us recall that Bt, t j = d dt tj is the Cauchy differentiation kernel of the Lambert curve C, which is a symmetric quadratic form on C C with second order poles along the diagonal t = t j. The function v = vt is a local coordinate change, which transforms v = 0 to t =. Therefore, the form dv v v j dt t t j is a meromorphic -form locally defined on C C, which is actually holomorphic on a neighborhood of the diagonal and vanishes on the diagonal. Therefore, it has the Taylor series expansion in t and t j without a constant term. Since v st = vt, the form dv vv j dst dv st t j is the pull-back of v v j dt t t j via the local involution s : C C that is applied to the first factor. Thus this is again a local t t j.,