QUANTUM CURVES FOR SIMPLE HURWITZ NUMBERS OF AN ARBITRARY BASE CURVE XIAOJUN LIU, MOTOHICO MULASE, AND ADAM SORKIN

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1 QUANTUM CURVES FOR SIMPLE HURWITZ NUMBERS OF AN ARBITRARY BASE CURVE XIAOJUN LIU, MOTOHICO MULASE, AND ADAM SORKIN Abstract The generating functions of simple Hurwitz numbers of the projective line are known to satisfy many properties They include a heat equation, the Eynard-Orantin topological recursion, an infinite-order differential equation called a quantum curve equation, and a Schrödinger like partial differential equation In this paper we generalize these properties to simple Hurwitz numbers with an arbitrary base curve Contents Introduction and the main results 2 A cut-and-join equation for simple Hurwitz numbers 5 3 The discrete Laplace transform 7 4 A Schrödinger equation 7 5 The heat equation and its consequences 0 6 The quantum curve 2 7 Semi-classical limit 3 References 5 Introduction and the main results The purpose of this paper is to determine functional properties of various generating functions of simple Hurwitz numbers with an arbitrary fixed base curve B We derive a partial differential equation for the Laplace transform of these Hurwitz numbers The equation is completely analogous to the result of 28] for the usual simple Hurwitz numbers based on P We also obtain an infinite-order differential equation, or a quantum curve, for the case of an arbitrary base curve, that is parallel to the various Hurwitz problems studied in 3, 25, 26] with the base curve P and its twisted version P a] Our main motivation is to examine whether the topological recursion of Eynard-Orantin 4, 5] and the existence of a quantum curve of, 7, 8, 9, 9] hold for the enumeration problem of simple Hurwitz numbers over an arbitrary curve B Consider a holomorphic map f : C B of a non-singular algebraic curve C onto a fixed base curve B We choose a general point 0 B and fix it once for all The quantity we are interested in this paper is the base B Hurwitz number Hg,nµ B,, µ n, which counts the automorphism weighted number of the topological types of holomorphic maps f of a genus g domain curve with n labeled preimages of 0 B of multiplicity µ,, µ n Z n +, such that f is simply ramified 2000 Mathematics Subject Classification Primary: 4H5, 4N35, 05C30, P2; Secondary: 8T30

2 2 X LIU, M MULASE, AND A SORKIN other than these n points Define its discrete Laplace transform Fg,nx B,, x n = n Hg,nµ B,, µ n µ Z n + = µ Z n + H B g,nµ,, µ n i= n i= e w iµ i x µ i i, x i = e w i, which we call the free energy of type g, n Since P does not cover the base curve B of genus gb > 0, F B 0,nµ,, µ n = 0 for any n and any value of µ,, µ n Z n + in this case following Our main result is the Theorem For 2g 2 + n > 0, the free energies Fg,nx B,, x n satisfy the following partial differential equation: 2 2g 2 + n χb n x i F B g,n xn] x i= i = x i x j F B 2 x i x j x g,n xĵ] F B i x g,n xî] j i j + n u u 2 2 u i= u 2 u =u 2 =x i F g,n+ B u, u 2, x + î] Fg B, I + u, x I Fg B 2, J + u 2, x J g +g 2 =g I J=î] Here n] = {,, n} is an index set, î] = n] \ {i}, and for any subset I n], we denote x I = x i i I We denote by χb = 2 2gB the Euler characteristic of the base curve B Note that the complexity 2g 2 + n is reduced by on the right-hand side of the recursion when gb, similar to the Eynard-Orantin integral recursion of 4, 5] Let us define the partition function of the base B Hurwitz numbers, as a holomorphic function in x C and with Re < 0, by 3 Z B x, = exp n! 2g 2+n Fg,nx, B x,, x g= n= Then it satisfies a quantum curve like infinite-order differential equation 4 x d χ B x e x d d χ ] B x Z B x, = 0 If we introduce y = x d

3 QUANTUM CURVES FOR HURWITZ NUMBERS OF ARBITRARY BASE 3 and regard it as a commuting variable, then the total symbol of the above operator produces a Lambert curve 5 x = y χ B e y The partition function also satisfies a Schrödinger like partial differential equation 6 x x 2 ] χb x Z B x, = 0 x If we denote the above operators by P and Q, respectively, then they satisfy a commutation relation: 7 P, Q] = P This shows that the system of equations 4 and 6 are compatible partition function has the following simple expression 8 Z B x, = m! χ B e 2 mm χ m B x m=0 Moreover, the Although the parameter is a formal deformation parameter, if we write = 2πiτ, then 9 Z B x, 2πiτ = m! χ B e πimm τ 2πiτ χ m B x m=0 is an entire function in x for Imτ > 0 Remark 2 We note that the exact same formulas 2, 5, 6, and 8 hold for the case of an orbifold base B = P a], a > 0, if we evaluate χb = + a, interpret the sum in 8 as running over non-negative multiples am of a, and replace am! by am! m!a m a Since the orbifold case requires a different set of preparations see 3, 0, 25], we will report the generalization to the case of higher-genus twisted curve as a base elsewhere When the base curve B is an elliptic curve E, a special case of simple Hurwitz numbers exhibits a quasi-modular property Dijkgraaf 6] considered a generating function 0 F g q = n! HE g,n,,, q n for g >, and n= F q = 24 log q + n= n! HE,n,,, q n The significant fact here is that F g q for g > is a quasi-modular form 6, 2] The relation between the free energies of type g, n for all n and F g q is given by 2 F g q = lim λ 0 n! λ n F g,nλq, E λq,, λq n= To motivate the present work, let us recall the corresponding counting problem of simple Hurwitz number Hg,nµ P,, µ n for a pointed projective line P,, which has a long history since Hurwitz 20] Its modern interest is due, among other things, to its rich functional properties 6, 7, 22, 32, 33], its relation with linear Hodge integrals on M g,n

4 4 X LIU, M MULASE, AND A SORKIN 2, 8, 30], and the Bouchard-Mariño conjecture 5] and its solutions 2, 3, 28] The theorem established in 3, 28] shows that the discrete Laplace transform 3 Fg,nx P,, x n = n Hg,nµ P,, µ n µ Z n + satisfies an integral recursion formula that was originally proposed by Eynard and Orantin 4], as conjectured in 5] The input curve for the recursion, the spectral curve, is shown to be the original Lambert curve 4 x = ye y Then in 26, 34] it is discovered that there is a quantum curve of, 7, 8, 9, 9], which is a generator of the holonomic system that characterizes the partition function of the simple Hurwitz numbers of P The partition function for B = P is Z P x, = exp n! 2g 2+n Fg,nx, P x,, x g=0 n= It is established in 26] that the P partition function has an expression 5 Z P x, = m! e 2 mm x m, and that it satisfies two equations: 6 x x xe x 7 2 x m=0 ] Z P x, = 0, x 2 + x 2 + x x ] i= x µ i i Z P x, = 0 If we denote the above operators as P and Q, respectively, then they satisfy a commutation relation 8 P, Q ] = P The semi-classical limit of each of the equations 6 and 7 recovers the Lambert curve 4, as shown in 26] Then from this curve as the spectral curve, and using the functions x and y on it, the Eynard-Orantin integral recursion 4] determines the differential forms 9 d d 2 d n F P g,n defined on Σ n for all g, n, where Σ is the Lambert curve given by 4 It is a simple consequence of the results in 3, 26, 28] that the differential form 9 uniquely recovers the primitive Fg,n P as a function in y,, y n Σ n From this point of view, the generating function Fg,n P xy,, xy n is completely determined by the equation 6 or 7, where x as a function in y is also given by 4 Our motivating question is, do the similar properties hold when we consider the simple Hurwitz numbers with an arbitrary curve as the base? Since the genus 0 base B Hurwitz numbers H0,n E µ,, µ n do not exist for a base curve B with gb > 0, the philosophy of ] does not produce any spectral curve of the Eynard-Orantin integral recursion for this counting problem Yet we have a topological recursion 2 in the form of a partial differential equation We note the straightforward generalizations 4, 6, 8, and

5 QUANTUM CURVES FOR HURWITZ NUMBERS OF ARBITRARY BASE 5 5, that reduce to the P case 6, 7, 5, and 4, respectively, when χb = 2 The paper is organize as follows In Section 2, we derive the cut-and-join equation of simple Hurwitz numbers with an arbitrary base curve Then in Section 3, we consider the Laplace transform of the cut-and-join equation, which is exactly 2 Using this result we derive the Schrödinger equation 6 in Section 4 Section 5 is devoted to explaining the heat equation expression of the cut-and-join equation 6, 22, 33] and deriving its consequences In Section 6 we first prove the expansion 8, derive the quantum curve 4, and then establish the commutator relation 7 In the final section we deduce the Lambert curve 5 as the semi-classical limit 2 A cut-and-join equation for simple Hurwitz numbers The enumeration problem we consider in this paper is the number of topological types of holomorphic maps f : C B from a nonsingular curve C of genus g to a fixed base curve B, with an arbitrary ramification over one point on B and simple ramification at all other critical points Let us denote by 0 B a general point arbitrarily chosen and fixed Such a homomorphic map f is referred to as a simple Hurwitz cover of B We label each inverse image of 0 B via f, and denote by µ,, µ n Z n + the degrees of f at each inverse image of 0 The base B Hurwitz number Hg,nµ B, µ n counts the automorphism weighted number of the topological types of simple Hurwitz covers The degree of the map f is given by n 2 d = µ = µ i Here use the notation µ, borrowing from the convention in the theory symmetric functions The Riemann-Hurwitz formula tells us that there are 22 r = rg, µ = 2g 2 + n d 2gB simple ramification points Instead of counting the topological types of f, we can fix r simple branch points on B in general position other than 0 B and count the automorphism weighted holomorphic maps f We note that 2 and 22 imply a condition for the degree and the genus for a base curve with gb : i= n 2gB d 2gB 2g 2 + n, n gb g In particular, H g,n µ = 0 for any n if g < gb This poses a sharp contrast to the Gromov-Witten theory of curves 3] The map f : C B of degree d is determined by a monodromy representation ρ Hom π i B \ {0,,, r, }, S d of the fundamental group of an r + -punctured curve into the symmetric group of d letters Here {,, r} B is the label of r simple branched points on B chosen in general position Let γ j, j = 0,, r, be a simple loop around each point j B Then a simple Hurwitz cover is constructed by assigning each γ j for j to a transposition and γ 0 to a product of n disjoint cycles of length µ, µ n, subject to the commutator relation ργ ργ r ργ 0 = α, β ] α gb, β gb ], where α i, β i S d are some elements in the permutation group

6 6 X LIU, M MULASE, AND A SORKIN The cut-and-join equation 6, 7, 20, 32] is the result of an analysis of what happens when we multiply a transposition ργ r to a product of disjoint cycles ργ 0 Since the fundamental group of the base curve does not make any effect on this multiplication, the exact same formula for Hg,n µ B holds, as for the simple Hurwitz numbers for P The only difference is the number r of simple ramification points Proposition 2 The cut-and-join equation For all g 0, n, and µ,, µ n Z n + subject to 25 2g 2 + n 2gB n µ i > 0, the simple Hurwitz numbers Hg,nµ B,, µ n of degree d satisfy the following equation: 26 2g 2 + n 2gB d H B g,n µ n] = µ i + µ j Hg,n B µ i + µ j, µ 2 î,ĵ] + 2 n αβ i= α+β=µ i HB g,n+α, β, µ + î] i j g +g 2 =g I J=î] i= Hg B, I + α, µ I Hg B 2, J + β, µ J Here and throughout the paper we use the following notational convention n] = {,, n} is an index set, and î] = n] \ {i}, etc For a subset I n], µ I = µ i i I Remark 22 Unlike the case of P, the cut-and-join equation does not determine all values of simple Hurwitz numbers When the degree d takes its maximum value 2g 2+n 2gB, the equation gives a trivial equality 0 = 0 The genus simple Hurwitz numbers based on an elliptic curve B = E are easy to calculate From 23 we have n = d and r = 0 when g = gb = Therefore, the covering is unramified If we allow disconnected domain, then the total number is equal to the number of partitions pd = Hom π E, S d /Sd of degree d see for example, 6] Let φq = q m m= be the Euler function Then we have n! HE,n,, q n = log φq = q n, m n= n= m n hence 27 H E,n,, = n! m n = n!σn, m where σ is the sum of divisors function This is an integer sequence, and its first ten terms are, 3, 8, 42, 44, 440, 5760, 75600, 52460, This sequence has many interesting properties see OEIS, A More generally, let us consider the case when the equality holds in 24 We need this analysis in Section 4 Because of 23, g = n gb implies d = n and r = 0, hence

7 QUANTUM CURVES FOR HURWITZ NUMBERS OF ARBITRARY BASE 7 the covering f : C B that is counted is totally unramified Here again the number of disconnected unramified coverings is given by the classical dimension formula see for example 27] for elementary derivations: Hom π B, S d /Sd = λ d χ dim λ B Here λ d parametrizes irreducible representations of S d, and dim λ is its dimension d! 3 The discrete Laplace transform The discrete Laplace transform F B g,nx,, x n of is a polynomial of degree 2g 2+n 2gB with the lowest degree term H B g,n,,, x x n The following proposition is an analogue of the case of simple Hurwitz numbers of P, and the proof is exactly the same as that of 3, Lemma 4] Proposition 3 The discrete Laplace transform of the cut-and-join equation 26 is precisely the partial differential equation 2 Reflecting Remark 22, the differential equation 2 alone does not determine free energies Fg,nx B,, x n This is because every homogeneous function of degree 2g 2+n 2gB is in the kernel of the Euler differential operator 2g 2 + n 2gB n on the left-hand side of 2 Thus the highest degree terms of free energies are not determined by this differential equation We will determine the homogeneous highest degree terms of the free energies in a different method in Section 5 The genus elliptic Hurwitz numbers 27 yields 3 F,nx E,, x n = n! x x n m m n i= x i x i 4 A Schrödinger equation In this section we prove 6, provided that gb > 0 We remark here that for gb = 0, the proof is rather different 26], yet the same formula holds Theorem 4 The partition function 3 satisfies the Schrödinger-type equation 6: 2 x 2 + x 2 2gB ] x Z B x, = 0 x

8 8 X LIU, M MULASE, AND A SORKIN Proof We use the same method of the proof of 26, Theorem 5, Theorem 53, Appendix A] When 23 holds, the diagonal evaluation of 2 yields 2g 2 + n 2gB x d F B g,nx,, x nn = x u 2 Fg,n u, B x,, x u=x + n x2 u u 2 Fg,n+u B, u 2, x,, x u =u 2 =x + n! 2 g +g 2 =g n +n 2 =n For m 2gB, let us define x 2 d n +!n 2 +! F g B,n +x,, x S m x := 2g 2+n=m n! F B g,nx,, x d F B g 2,n 2 +x,, x Note that S m contains a contribution from domain curves with g = ngb, or equivalently, d = n and r = 0, for which the cut-and-join equation is not valid Therefore, when we deduce an equation for S m s, we need to remove these boundary terms from the equation Now from 4 we obtain 42 = m 2gB x d d2 x2 2 2g 2+n=m g =ngb S mx 2 n! 2g 2+n=m g =n+gb 2g 2+n=m g=n gb S m+ x m +m 2 =m+ m 2gB x d 2 x2 n! u 2 2 n 2! x2 2g 2+n=m g +g 2 =g g =ngb n +n 2 =n+ x d S m x x d S m 2 x F B g,nx,, x Fg,nu, B x,, x u=x u u 2 Fg,nu B, u 2, x,, x u =u 2 =x x 2 d d n! n 2! F g B,n x,, x F g B 2,n 2 x,, x We note here that in the boundary contribution g = n gb, since n = d, the free energies are single monomials proportional to x x 2 x n Let us look at the right-hand side of 42 The first line of the right-hand side is 0 because Fg,nx, B, x = Hg,n, B, x n, and m = 2g 2 + n = n 2gB, since r = 0 The second line is also 0, because Fg,nu, B x,, x is linear in u The third summation on the right-hand side is empty, because we need n gb g = n gb, which does not happen Similarly, the fourth line summation

9 QUANTUM CURVES FOR HURWITZ NUMBERS OF ARBITRARY BASE 9 is also empty, because we are requiring g = g + g 2, n + = n + n 2, and 43 n gb g n 2 gb g2 n gb = g We have thus obtained a recursion equation m 2gB x d S m+ x In terms of the generating function = d2 x2 2 2 S mx + 2 F x, = 43 becomes 44 2gB x d F x, = m +m 2 =m+ m S m x, m= x d S m x x d S m 2 x 2 d2 x2 2 F x, + x d 2 F x, 2 Since Z B x, = exp F x,, 6 follows directly from 44 This completes the proof Using the Schrödinger equation 6, we can determine the form of the solution Z B x, Lemma 42 The partition function has the following expansion: 45 Z B x, = where c m is a constant c m e 2 mm χ m B x, m=0 Proof The partition function Z B x, of 3 is a formal power series in x and Thus it has an expansion of the form Z B x, = c m f m x m m=0 with f m C ]] Then the Schrödinger equation 6 yields an ordinary differential equation f m = 2 mm + m χb f m, whose solution is This completes the proof f m = c e 2 mm m χ B

10 0 X LIU, M MULASE, AND A SORKIN 5 The heat equation and its consequences As we have seen in Section 4, the Schrödinger equation 6 determines the solution Z B x, only up to the form 45 To determine the coefficients, we need another technique In this section we use a heat equation technique of 22] First we remark that the cutand-join equation gives rise to a heat equation for another generating function of base B Hurwitz numbers To determine a solution of a heat equation, we need to identify the initial value We will show that the initial condition exactly corresponds to determining the highest degree terms of Fg,nx B,, x n The exponential generating function of these highest degree terms can be determined by a character formula of 3] Thus we obtain the unique solution of the heat equation, which in turn gives all base B Hurwitz numbers The generating function for base B Hurwitz numbers we consider is 5 Ht, p = H g,n t, p, 52 H g,n t, p = n! g=0 n= µ Z n + H B g,n µp µ t rg,µ, where rg, µ = 2g 2 + n µ χb is the number of simple ramification points, and p µ = p µ p µn The same argument of 6, 22, 23, 29, 33] shows that e Ht,p satisfies a heat equation, that is obtained from the cut-and-join equation For a partition µ = µ µ 2 of a finite length lµ, we define the shifted power-sum function by 53 p r µ] := i,j i= µ i i + 2 r i + 2 r ] This is a finite sum of lµ terms Then we have 6, 33] 2 54 i + jp i p j + ijp i+j s µ p = p 2 µ] s µ p, p i+j p i p j where s µ p is the Schur function defined by 55 s µ p = λ = µ lµ χ µ λ p λ, z µ = m i!i m i, z λ 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 Let us denote the cut-and-join operator by = 2 i,j i= i + jp i p j p i+j + ijp i+j 2 p i p j Then Schur functions are eigenfunctions of the operator: s µ p = 2 p 2µ] s µ p, and the cut-and-join equation of simple Hurwitz numbers 26 yields a heat equation 56 t eht,p = e Ht,p

11 QUANTUM CURVES FOR HURWITZ NUMBERS OF ARBITRARY BASE We can solve a heat equation by the method of eigenfunction expansion Thus we have an expression 57 e Ht,p = µ a µ s µ pe 2 p 2µ]t for a constant a µ associated with every partition µ These constants are determined by the initial value t = 0 From 52 we see that the initial value comes from the cases when rg, µ = 0 First we note that 22 implies that simple Hurwitz numbers with r = 0 correspond to the case with only branched point on the base curve B If we allow disconnected domain curves to cover B, then the number of coverings of degree d with a prescribed ramification data given by a partition µ d over 0 B with no other ramification points is easy to calculate Let us denote by Hd B µ such a number, where we do not label the inverse images of 0 B this time Then from 3, Eq00] we obtain 58 Hd B µ = χ dim λ B C µ χ λµ d! dim λ λ d Here C µ is the conjugacy class of the permutation group S d determined by the cycle type µ, a partition λ d is a label of an irreducible representation of S d, and dim λ is its dimension The cardinality of the conjugacy class is given by d! 59 C µ = i m i!i m = d! i z µ The generating function of these disconnected simple Hurwitz numbers can be calculated, appealing to 55 and 59, as follows: Hd B µp dim λ µ = d! 50 d= µ d = d= µ d λ d d= λ d d! dim λ χ B C µ χ λµ dim λ p µ χ dim λ B d! µ d χ λ µ z µ p µ = λ! χ B s λ p, dim λ λ where the last sum runs over all partitions λ Note that e Ht,p is the generating function of simple Hurwitz numbers allowing disconnected domain curves Therefore, the initial value e H0,p counts the disconnected base B Hurwitz numbers with only one branched point at 0 B In other words, we have determined the initial condition by 50 The result is 5 e H0,p = µ! χ B s µ p = a µ s µ p dim µ µ µ Since Schur functions are linear basis for symmetric functions, we establish the following Theorem 5 The exponential generating function of the base B simple Hurwitz numbers is given by 52 e Ht,p = µ! χ B s µ pe 2 p2µ]t dim µ µ

12 2 X LIU, M MULASE, AND A SORKIN 6 The quantum curve As noted in 26], the diagonal specialization F B g,nx, x,, x corresponds to substituting the power-sum symmetric function 6 p j = x j + xj 2 + xj 3 + by its principal specialization p j = x j More precisely, we have the following Lemma 6 Let us define the principal specialization of the power-sum symmetric functions by p j s = s χ j B x p µ s = p µ s p µn s = s χ µ B x Then we have 62 Z B x, = e H,p Proof We have H, p = which yields 62 µ g,n µ Z n + χ B n! 2g 2+n µ Hg,nµ B,, µ n p µ = g,n n! 2g 2+n F E g,nx, x,, x, From the expansion formulas 52 and 45, together with the equality 62, we obtain µ! χ B 63 e 2 p2µ] n s µ p = c m e 2 mm χ m B x dim µ As explained in 26, Section 6] and also in 25], this equality is exactly the effect of the principal specialization of Lemma 6, which turns the summation over all partitions into the sums over just one-row partitions We have thus determined the coefficients c m in the expansion 45 It is given by m=0 64 c m = m! χ B This completes the proof of 8 The convergence of the infinite series is obvious from the shape of 8 or 9, since lim sup m! χ B e 2 mm m = 0 m if Re < 0 The expansion 8 allows us to derive the quantum curve-type equation 4 Since χ B xe x d d χ ] B x m! χ B e 2 mm χ m B x = m +! χ Be 2 mm+ χ m+ B x

13 for every m 0, we have 65 QUANTUM CURVES FOR HURWITZ NUMBERS OF ARBITRARY BASE 3 χ B xe x d By differentiating 65 we obtain 4 Lemma 62 Let Then d χ B x Z B x, = P = x d χ B xe x d d x Q = 2 x 2 + x 2 χb P, Q] = P χ B x x, Proof We first note that Q commutes with x d ] χ B xe x d, Q = ] χ B xe x d, + χ B x, 2 and d = χb χ B xe x d χ B x 2 d e x d = 0, 7 follows from + χ B x 2 d e x d + 2 χ B xe x d 0 = x Since x 2 + x 2 ] χb x e x d x χ B 2 χb ] P, Q = ] P, Q] +, Q P = P, Q] + 2 P xe x d 7 Semi-classical limit The semi-classical analysis of the operators P and Q are performed in the following way Suppose our counting problem had genus 0 contributions F0,n B x,, x n Then we define 7 S m x := n! F g,nx, B, x 2g 2+n=m as before, and consider a formal expression 72 Z B x, = e m=0 m S mx = e S 0x+S x Z B x, Let us introduce a variable u such that x = e u, and regard the coefficients S m as functions in u Since x d = d du, we have P = d χ B e u e d du + d χ B du du

14 4 X LIU, M MULASE, AND A SORKIN Then 73 e S e S 0 P e S 0 e S = S 0 e u S 0 2 χ Be S 0 u+ S 0 u + O = S 0 e u S 0 2 χ Be S 0 + O, where = d du = x d, and by O we mean an operator whose application to the partition function Z B x, produces a function of order or higher in Now define a new variable 74 y = S 0 Then the semi-classical limit 0 of 73 yields an equation 75 y xy 2 χ B e y = y xy χ B e y = 0, since Z B x, 0 = Note that this is exactly the total symbol of the operator P, where x d is represented by a commutative variable y The second factor of 75 also recovers the Lambert curve 5 Although the formal manipulation seems to work, however, we have to remember that we have derived the operator P assuming the shape of the solution Z B x, as in 8 Since we are imposing P Z B x, = 0, it forces that y = 0, which makes 75 trivially correct On the other hand, since the kernel of Q assums only the expansion 45, where the summation index m can be negative, the semi-classical analysis of Q does go through 76 e S e S 0 Qe S 0 e S = e S e S u ] χb e S 0 e S u = 2 S S χb S 0 2 S 0 2 S 0 + S 0S + χb S + O For the 0 limit to exist, we need From 77 we obtain S S χb S 0 = 0, 2 S 0 2 S 0 + S 0S + χb S = 0 79 S 0 = 2 y2 χb y, or equivalently, 70 y = y + χb dy du Its solution is y + log y χ B = u + const If we take the constant of integration to be 0, then we obtain e u = x = y χ B e y,

15 QUANTUM CURVES FOR HURWITZ NUMBERS OF ARBITRARY BASE 5 recovering 5 From 78 we have or equivalently 2 y 2 y + y + χb S = 0, ds du = dy 2 y + χb du + y dy 2 y + χb dy Since we know du/dy from 70, we can integrate the above equation to obtain 7 S = 2 y + 2 log y + χb + const The above derivation of the semi-classical limit of the operator Q is valid if Z B x, contains negative powers of If we assume the expansion 8, then such a situation occurs when the base curve B satisfies χb > Indeed, our formulas 79 and 7 agree with those of 26, Section 5] for B = P with y = z, and 3, Section 6] for B = P a] with y = z a Acknowledgement XL received the China Scholarship Council grant CSC , which allowed him to conduct research at the Department of Mathematics, University of California, Davis He is also supported by the National Science Foundation of China grants No20477, 775, and the Chinese Universities Scientific Fund No20JS04 The research of MM is supported by NSF grant DMS AS received support from the US Government References ] M Aganagic, R Dijkgraaf, A Klemm, M Mariño, and C Vafa, Topological Strings and Integrable Hierarchies, arxiv:hep-th/032085, Commun Math Phys 26, ] G Borot, B Eynard, M Mulase and B Safnuk, Hurwitz numbers, matrix models and topological recursion, Journal of Geometry and Physics 6, ] V Bouchard, D Hernández Serrano, X Liu, and M Mulase, Mirror symmetry for orbifold Hurwitz numbers, arxiv: ] V Bouchard, A Klemm, M Mariño, and S Pasquetti, Remodeling the B-model, Commun Math Phys 287, ] V Bouchard and M Mariño, Hurwitz numbers, matrix models and enumerative geometry, Proc Symposia Pure Math 78, ] R Dijkgraaf, Mirror symmetry and elliptic curves, in The Moduli Space of Curves, Progress in Mathematics vol 29, Dijkgraaf et al, editors, 49 62, Birkhäuser 995 7] R Dijkgraaf, L Hollands, and P Su lkowski, Quantum curves and D-modules, Journal of High Energy Physics arxiv:080457, ] R Dijkgraaf, L Hollands P Su lkowski, and C Vafa, Supersymmetric gauge theories, intersecting branes and free Fermions, Journal of High Energy Physics arxiv: , ] R Dijkgraaf and C Vafa, Two Dimensional Kodaira-Spencer Theory and Three Dimensional Chern-Simons Gravity, arxiv:07932 hep-th] ] N Do, O Leigh, and P Norbury, Orbifold Hurwitz numbers and Eynard-Orantin invariants, arxiv: mathag] 202 ] O Dumitsrescu, M Mulase, A Sorkin and B Safnuk, The spectral curve of the Eynard-Orantin recursion via the Laplace transform, arxiv:20259 mathag] 202 2] T Ekedahl, S Lando, M Shapiro, A Vainshtein, Hurwitz numbers and intersections on moduli spaces of curves, Invent Math 46, ] B Eynard, M Mulase and B Safnuk, The Laplace transform of the cut-and-join equation and the Bouchard- Mariño conjecture on Hurwitz numbers, Publications of the Research Institute for Mathematical Sciences 47, ] B Eynard and N Orantin, Invariants of algebraic curves and topological expansion, Communications in Number Theory and Physics, ] B Eynard and N Orantin, Computation of open Gromov-Witten invariants for toric Calabi-Yau 3-folds by topological recursion, a proof of the BKMP conjecture, arxiv:20503v math-ph] 202

16 6 X LIU, M MULASE, AND A SORKIN 6] IP Goulden, A differential operato for symmetric functions and the combinatorics of multiplying transpositions, Trans AMS, 344, ] IP Goulden and DM Jackson, Transitive factorisations into transpositions and holomorphic mappings on the sphere, Proc AMS, 25, ] T Graber and R Vakil, Hodge integrals and Hurwitz numbers via virtual localization, Compositio Math 35, ] S Gukov and P Su lkowski, A-polynomial, B-model, and quantization, arxiv:080002v hep-th] 20 20] A Hurwitz, Über Riemann sche Flächen mit gegebene Verzweigungspunkten, Mathematische Annalen 39, ] M Kaneko and D Zagier, A generalized Jacobi theta function and quasi-modular forms, in The Moduli Space of Curves, Progress in Mathematics vol 29, Dijkgraaf et al, editors, 65 72, Birkhäuser ] M Kazarian, KP hierarchy for Hodge integrals, arxiv: ] M Kazarian, S Lando, An algebro-geometric proof of Witten s conjecture, J Amer Math Soc 20, ] M Mariño, Open string amplitudes and large order behavior in topological string theory, J High Energy Physics , ] M Mulase, S Shadrin, and L Spitz, The spectral curve and the Schrödinger equation of double Hurwitz numbers and higher spin structures, arxiv:305580, ] M Mulase and P Su lkowski, Spectral curves and the Schrödinger equations for the Eynard-Orantin recursion, arxiv:203006, ] M Mulase and J Yu, A generating function of the number of homomorphisms from a surface group into a finite group, arxiv:mathqa/ , ] M Mulase and N Zhang, Polynomial recursion formula for linear Hodge integrals, Communications in Number Theory and Physics 4, ] A Okounkov, Toda equations for Hurwitz numbers, Math Res Lett 7, ] A Okounkov and R Pandharipande, Gromov-Witten theory, Hurwitz numbers, and matrix models, I, Proc Symposia Pure Math 80, ] A Okounkov and R Pandharipande, Gromov-Witten theory, Hurwitz theory, and completed cycles, Ann Math 63, ] R Vakil, Harvard Thesis ] J Zhou, Hodge integrals, Hurwitz numbers, and symmetric groups, preprint, arxiv:math/ , ] J Zhou, Quantum Mirror Curves for C 3 and the Resolved Confiold, arxiv: v, 202 Department of Applied Mathematics, China Agricultural University, Beijing, 00083, China, and Department of Mathematics, University of California, Davis, CA , USA address: xjliu@caueducn Department of Mathematics, University of California, Davis, CA , USA address: mulase@mathucdavisedu Department of Mathematics, University of California, Davis, CA , USA address: azsorkin@mathucdavisedu