Generalized string equations for double Hurwitz numbers

Generalized string equations f or double Hurwitz n um b ers Kanehisa T ak asaki ∗ Graduate Sc ho ol of Human and En vironmental Studies, Ky oto Univ ersit y Y oshida, S akyo, Ky oto, 606-8501, Japan Abstract The generating fu nction of double Hurwitz num b ers is kno w n to b ecome a tau f u nction of the T oda hierarch y . The asso ciated Lax and Orlo v-Sc hulman op erators tur n out to satisfy a set of generalized string equations. These ge neralized string equati ons resem b le those of c = 1 string theory except that the Orlo v-Sc hulman op erators are con tained therein in an exp onen tiated form. These equations are de- riv ed f rom a set of in tert winin g relations for fer m ion bilinears in a t wo-dimensional free f er m ion system. The inte rt w iner is constructed from a fermionic counterpart of the cut-and-join op erator. A classical limit of these generalized string equations is also obtained. The so called Lam b ert curve emerges in a sp ecial ization of its solution. This seems to b e another w a y to derive the sp ectral curve of the rand om matrix approac h to Hu r witz num b ers. 2000 Mathematics Sub ject Classification: 35Q58, 14N10, 8 1R12 Key w ords: Hurwitz num b ers, T o da hierar ch y , generalized string equation, quan tum t orus a lg ebra, cut-and-join o p erator, Lam b ert curv e 1 In tro ducti on Hurwitz n um b ers count the top olo gical types o f finite ramified cov erings of a giv en Riemann surface [1]. Some ten y ears ago, Hurwitz n umbers of co v erings of the Riemann sphere C P 1 turned out to b e related to Ho dge in tegrals on ∗ E-mail: tak asaki@math.h.kyoto-u.ac.jp 1 the Deligne-Mumford mo duli space ¯ M g ,n of mark ed stable curv es [2, 3, 4] and Gromov-Witten in v arian ts of CP 1 [5, 6, 7, 8]. These observ ations led to v arious dev elopmen ts and applications, suc h as new pro of s [11, 12, 13] of Witten’s conjecture [9 ] (Kontse vic h’s theorem [10]) on tw o-dimensional top ological g ra vity , in other words, in tegrals of ψ - classes on ¯ M g ,n . Recen tly , a set of new recursion relations for Hurwitz num b ers w ere deriv ed [14, 1 5, 16] as an analogue of Eynard and Oran tin’s “top ological recursion relations” [1 7 ]. These dev elopmen ts in the last decade are also more or less connected with integrable hie rarc hies of the K dV, K P a nd T o da type (see reviews b y the Ky oto group [1 8 , 19, 20] f o r basic kno wledge on tho se in t egrable systems). Firstly , sev eral T o da-like equations show up in relatio n with Gromov -Witten in v ariants of CP 1 . Secondly , the Witten conjecture is f o rm ulated in the language of the KdV hierarc hy and its Virasoro symmetries . Subsequen t studies [2 1, 22, 23] on the Witten conjecture from the Hurwitz side revealed a connection with the KP hierarc hy as we ll. A master inte grable system in this sense is the (tw o-dimensional) T o da hierarc hy [24]. As p ointed out b y Ok ounko v [6 ], a generating function of “double Hurwitz n um b ers” g iv es a sp ecial solution of the T o da hierarc h y . Being a tau function of the T o da hierarch y , this generating function has tw o sets of indep enden t v ariables t = ( t 1 , t 2 , · · · ) and ¯ t = ( ¯ t 1 , ¯ t 2 , · · · ). By sp e- cializing part of these v ariables to particular v a lues, the generating function of “ simple Hurwitz n um b ers” are reco vered. The afo remen tioned T o da -lik e equations and K P hierarc h y ma y b e thought of as equations satisfied b y those sp ecialized tau f unctions. The goa l of this pa p er is to deriv e the gener alize d string e quations L = Qe − β / 2 ¯ Le β ¯ M , ¯ L − 1 = Qe β / 2 L − 1 e β M for this sp ecial solution of t he T o da hierarch y , a nd t o examine implica- tions thereof. Here β and Q a re parameters prepared a long with t and ¯ t . L, ¯ L and M , ¯ M are, resp ectiv ely , the L a x and Orlo v-Sc hulm an o p era- tors, a ll b eing one-dimensional difference op erato rs. Compared with the cases of t wo-dimensional quantum gra vit y [25] and c = 1 string theory [26, 27, 2 8, 29, 30, 31], these generalized string equations hav e a no t a ble new feature. Name ly , they con tain the exp onen tials e β M and e β ¯ M of the Orlo v-Sc hulman op erators. This is in sharp con trast with the con v en tional string equations that are linear in M and ¯ M (except for the the case of a deformed v ersion of c = 1 string theory [30 ], in whic h the string equations are quadratic in M and ¯ M ). The exp onen tials of M and ¯ M are related to the “quan tum torus alg ebra” in a t w o-dimensional fr ee f ermion system. The same Lie algebraic structure 2 pla ys an important role in the melting crystal mo del of five-dimen sional gaug e theory as w ell [32 , 33]. W e b orrow some tec hnical ideas dev elop ed therein. The generalized string equations are deriv ed from algebraic relations (referred to as “in tertwining relations”) among fermion bilinears. A clue therein is a sp ecial fermion bilinear that corresp onds to the so called “cut- a nd-join op erator” [34]. The generalized string equations also ha v e a classic al limi t of the form L = Q ¯ L e β ¯ M , ¯ L − 1 = Q L − 1 e β M , where L , M , ¯ L , ¯ M are “long- w av e” or “ disp ersionless” limits [20] of the Lax and Orlov-Sc h ulman op erators. Conceptually , the classical limit a moun ts to the gen us-zero part of Hurwitz num b ers. Similarity with c = 1 string theory b ecomes ev en more ob vious in this limit (which amounts to the gen us-0 part of string amplitudes). The generalized string equations of c = 1 string theory [26, 27, 2 8, 29] ha v e a classical limit of the form L = ¯ L ¯ M , ¯ L − 1 = L − 1 M . Th us, roughly sp eaking, the linear terms M , ¯ M t herein are now replaced b y the exp onential terms e β M , e β ¯ M . Thanks to this similarity , one can apply the metho d dev elop ed for solving the generalized string equations o f c = 1 string theory [29] to construct a solution of the foregoing equations in the form of p ow er series of t k ’s and ¯ t k ’s. This construction simplifies when either t k ’s or ¯ t k ’s ar e specialized to pa r t icular v alues f or whic h the tau function reduces to the generating function of simple Hurwitz n um b ers. Remark a bly , w e encounte r therein the equation x = y e y of the “Lambert curv e” that lies in the heart of the new recursion relations [14, 15, 16 ]. This should no t be a coincidence . Presumably , the classical limit of t he generalized string equations will b e another wa y to deriv e the “sp ectral curv e” in the random matrix approac h [1 5 , 35, 36]. This pap er is organized a s follows . Section 2 reviews the notion of Hurwitz n umbers and their generating functions. Sc hur functions and their special v a lues are used to interpre t the generating functions as tau functions. Sec- tion 3 prepares tec hnical to ols from a t w o- dimensional free fermion system. W e introduce relev ant fermion bilinears, and recall Okounk o v’s r esult on a fermionic represen tation o f the generating function of double Hurwitz num- b ers [6]. Section 4 presen ts the generalized string equations. W e start from in tertwining relatio ns of fermion bilinears, and translate those relatio ns to 3 the generalized string equations. Section 5 formulates the classical limit of the generalized string equations. The classical limit is first derive d b y heuris- tic consideration, then justified b y sho wing that the asso ciated tau function has an ~ -expansion. Section 6 is dev ot ed to solving this classical limit of the generalized string equations. The solution is constructed in m uc h the same w ay as that of the g eneralized string equations fo r c = 1 string theory . The Lam b ert curve show s up in a sp ecialization of this solution. 2 Generating funct ions of Hurwi t z n um b ers In this section, w e use v arious notions and fo r mulae o n partitions, Y oung diagrams and Sch ur functions. They are mostly b o rro w ed from Macdonald’s b o ok [37]. 2.1 Hurwitz n um b ers of Riemann sphere Let us consider finite ramified cov erings of CP 1 . Tw o co v erings π : Γ → CP 1 and π ′ : Γ ′ → CP 1 are said to b e to p ologically equiv a len t if π and π ′ s are connected b y a homeomorphism φ : Γ → Γ ′ as π = π ′ ◦ φ . Let [ π ] denote the equiv alence class o f the cov ering π . Note that Γ can b e a disconnected surface. Giv en a p o sitive integer d and a set of distinct p oints P 1 , · · · , P r of CP 1 , there are only a finite n um b er of nonequiv a len t d -fold co verings that are un- ramified ov er a ll p oin ts other than P 1 , · · · , P r . Those nonequiv alen t co ve rings can b e furt her classified b y the ramification data of sheets ov er P 1 , · · · , P r . These data a r e giv en b y conjugacy classes C 1 , · · · , C r of the d -t h symmetric group S d . A conjugacy class C of S d is determined by the cycle t yp e µ = ( µ 1 , · · · , µ l ) , µ 1 ≥ · · · ≥ µ l , | µ | = µ 1 + · · · + µ l = d of a represen tativ e σ ∈ S d of C , e.g., σ = (1 , · · · , µ 1 )( µ 1 + 1 , · · · , µ 1 + µ 2 ) · · · ( µ 1 + · · · + µ l − 1 , · · · , d ) , where ( j 1 , · · · , j m ) denotes the cyclic p ermutation sending j 1 → j 2 → · · · → j m → j 1 . The cycle type th us b ecomes a partitio n o f d , and has another expression µ = (1 m 1 2 m 2 · · · ) with the n um b ers m i of i - cycles. Let C ( µ ) denote the conjugacy class deter- mined b y a partition µ o f d . 4 A cyclic p erm utation ( j 1 , · · · , j m ) of length m represen t s a cyclic cov ering of degree m realized., e.g., by the Riemann sheets of ( z − a ) 1 /m ab ov e the p oin t a . W e use t he cycle t ype µ = ( µ 1 , · · · , µ l ) as r amification data ab ov e a p oin t of CP 1 to show t ha t the sheets ab o ve that p oint lo cally lo ok lik e a disjoin t union of l cyclic co v erings of degree µ 1 , µ 2 , · · · , µ l . The Hurwitz num b er H d ( C 1 , · · · , C r ), also denoted by H d ( µ (1) , · · · , µ ( r ) ) where µ ( k ) ’s are the cycle t ypes of C k ’s, is defined to b e t he sum H d ( C 1 , · · · , C r ) = X [ π ] 1 | Aut( π ) | of the w eigh ts 1 / | Aut( π ) | o ve r all equiv alent classes [ π ] of p ossibly disc on- ne cte d cov erings π that are ramified ov er the r p oin ts P 1 , · · · , P r with the ramification dat a C 1 , · · · , C r . Aut( π ) denotes the gro up of automorphisms of π . As the not ation suggests, H d ( C 1 , · · · , C r ) do es not dep end on the p o- sition of P 1 , · · · , P r . The Hurwitz n um b ers can b e determined b y a g en uinely group theoretical metho d (Burnside’s theorem [3 8]). This leads to the b eautiful formula H d ( µ (1) , · · · , µ ( r ) ) = X | λ | = d  dim λ d !  2 r Y k =1 f λ ( µ ( k ) ) , (1) where the sum is ov er a ll partitions λ of d . Let us explain the notations used in this formula. Firsly , dim λ denotes the dimension dim V λ of the irreducible represen ta- tion ( ρ λ , V λ ) of S d determined b y λ . In o t her w ords, dim λ is the num b er of all standard tableaux of shap e λ , and can b e calculated b y the so called ho o k length form ula dim λ = d ! Y ( i,j ) ∈ λ h ( i, j ) , (2) where h ( i, j ) denotes the length of the ho ok cornered at the cell ( i, j ) ∈ λ . Note t ha t partitions are iden tified with the asso ciated Y oung diagrams. As an immediate consequence, dim λ turns out t o b e symmetric under the tr a nsp o se (or conjugate) λ 7→ t λ of the partition, namely , dim t λ = dim λ. (3) Secondly , f λ ( µ ) denotes the v alue f λ ( C ( µ )) of the class function f λ ( C ) = χ λ ( C ) dim λ | C | (4) 5 on S d ev aluat ed at C = C ( µ ). χ λ denotes the character T r V λ ρ λ . The cardi- nalit y | C ( µ ) | of C ( µ ) ⊂ S d can b e written as | C ( µ ) | = d ! z µ , z µ = Y i ≥ 1 m i ! i m i . (5) The v alues of f λ ( C ) fo r the simplest t w o cases C = (1 d ) , (1 d − 2 2) can b e explicitly calculated as f λ (1 d ) = 1 , f λ (1 d − 2 2) = κ λ 2 (6) where κ λ = l X i =1 λ i ( λ i − 2 i + 1) = l X i =1  λ i − i + 1 2  2 −  − i + 1 2  2 ! . This n umber has another useful express ion of the form κ λ = 2 X ( i,j ) ∈ λ ( i − j ) , (7) whic h implies, e.g., the anti-symme tric prop erty κ t λ = − κ λ . (8) 2.2 Simple Hur witz n um b ers Let us review the notion of simple Hurwitz num b ers. They are the Hurwitz n umbers suc h that the t ypes of all but one ramification p oints P 1 , · · · , P r are restricted to 1 d − 2 2. The exceptional ramification p oin t P r +1 can ha v e a n arbitrary cycle t yp e µ . The Hurwitz num b ers of this t yp e H d (1 d − 2 2 , · · · , 1 d − 2 2 | {z } r , µ ) = X | λ | = d  dim λ d !  2  κ λ 2  r f λ ( µ ) (9) are called simple Hurwitz num b ers. In tro ducing an extra (finite or infinite) set of v a r ia bles x = ( x 1 , x 2 , · · · ), one can construct a generating function of these n umbers a s Z ( x ) = ∞ X r =0 ∞ X d =0 X | µ | = d H d (1 d − 2 2 , · · · , 1 d − 2 2 | {z } r , µ ) β r r ! Q d p µ , 6 where p µ ’s a r e the monomials p µ = p µ 1 p µ 2 · · · of the p ow er sums p k = X i ≥ 1 x k i , k = 1 , 2 , · · · . W e now substitute (9) in the definition of Z ( x ) and use the F rob enius fo r mula X | µ | = d χ λ ( C ( µ )) z µ p µ = s λ ( x ) (10) to rewrite the sum o v er µ in to a Sc hur function times n umerical factors. The n umerical factor s pa r t ly cancel with some other factors. The generating function th us reduces to Z ( x ) = X λ ∈P dim λ | λ | ! e β κ λ / 2 Q | λ | s λ ( x ) . (11) Z ( x ) turns out to b e a tau function of t he KP hierarc h y [18, 19]. T o see this, we c ha ng e the v ariables from x t o the standar d time v ariables t = ( t 1 , t 2 , · · · ) of the KP hierarch y via the p ow er sums as t k = p k k = 1 k X i ≥ 1 x k i and consider the Sc hur functions s λ ( x ) as functions of t . It is conv enien t to use Zinn-Justin’s notation s λ [ t ] f o r the latter [39]. Actually , s λ [ t ] can b e redefined directly . Let us recall the Jacobi-T rudi form ula s λ ( x ) = det( h λ i − i + j ( x )) n i,j =1 , (12) where λ i ’s are parts of λ (whic h are assumed to b e equal to 0 fo r i > n , namely , λ = ( λ 1 , · · · , λ n , 0 , · · · )), and h m ( x )’s are the complete symme tric functions defined b y the generating function ∞ X m =0 h m ( x ) z m = Y i ≥ 1 (1 − x i z ) − 1 . 7 Let h m [ t ] denotes h m ( x ) considered as a function o f t . h m [ t ]’s can b e redefined b y t he generating function ∞ X m =0 h m [ t ] z m = exp ∞ X k =1 t k z k ! . Consequen tly , s λ [ t ] can b e expressed as s λ [ t ] = det( h λ i − i + j [ t ]) n i,j =1 . (13) One can thus replace s λ ( x ) by s λ [ t ] to o btain the generating function Z [ t ] = X λ ∈P dim λ | λ | ! e β κ λ / 2 Q | λ | s λ [ t ] . (14) Iden tifying Z [ t ] a s a K P tau f unction requires some more consideration. Sev eral metho ds ar e kno wn in the literature [1 2, 21, 22, 23 ]. One can explain this fact in the con text of the T o da hierarc hy as w ell. A clue is the form ula dim λ | λ | ! = s λ [1 , 0 , 0 , · · · ] (15) that can can b e deriv ed, e.g., fr o m the F rob enius for mula (10) by letting p 1 = 1 and p k = 0 for k > 1. Conseq uen tly , Z [ t ] can b e r ewritten a s Z [ t ] = X λ ∈P e β κ λ / 2 Q | λ | s λ [ t ] s λ [1 , 0 , · · · ] . (16) This function is a sp ecialization of the generating function Z [ t , ¯ t ] of double Hurwitz n umbers in t r o duced b elo w. Z [ t , ¯ t ] is a tau function of the T o da hi- erarc hy (o r, rather, the tw o-comp onent KP hierarch y[18], b ecause the lattice co ordinate o f the T o da lattice is absen t here). It is well know n [ 2 4] that an y T o da (or t w o- comp onen t KP) tau function is also a tau function of the KP hierarc hy with resp ect t o one of the tw o sets of v ariables. This implies that Z [ t ] is a KP tau function. 2.3 Double Hur witz n um b ers Let us choose y et ano ther p oin t P 0 of CP 1 of an arbitrary ramification t yp e ¯ µ in a ddition to the r + 1 p o ints in the case o f simple Hurwitz num b ers. The Hurwitz n umbers of this t yp e H d ( ¯ µ, 1 d − 2 2 , · · · , 1 d − 2 2 | {z } r , µ ) = X | λ | = d  dim λ d !  2  κ λ 2  r f λ ( µ ) f λ ( ¯ µ ) (17) 8 are called double Hurwitz num b ers. T o construct a generating function of these n umbers, w e in tr o duce a new set of v ariables ¯ x = ( ¯ x 1 , ¯ x 2 , · · · ), their p o w er sums ¯ p k = X i ≥ 1 ¯ x k i and their mono mia ls ¯ p λ = ¯ p λ 1 ¯ p λ 2 · · · along with the v ariables in the case of simple Hurwitz n um b ers. Again, with the a id o f the F rob enius form ula (1 0), the generating function Z ( x , ¯ x ) = ∞ X r =0 ∞ X d =0 X | µ | = | ¯ µ | = d H d ( ¯ µ, 1 d − 2 2 , · · · , 1 d − 2 2 | {z } r , µ ) β r r ! Q d p µ ¯ p µ can b e conv erted to a sum ov er all partitions as Z ( x , ¯ x ) = X λ ∈P e β κ λ / 2 Q | λ | s λ ( x ) s λ ( ¯ x ) . (18) W e no w in tro duce the tw o sets t = ( t 1 , t 2 , · · · ) and ¯ t = ( ¯ t 1 , ¯ t 2 , · · · ) of t ime v a r iables as t k = p k k , ¯ t k = − ¯ p k k and consider the generating function Z [ t , ¯ t ] = X λ ∈P e β κ λ / 2 Q | λ | s λ [ t ] s λ [ − ¯ t ] (19) of the new v aria bles. Z [ t , ¯ t ] is a tau f unction of the T o da hierarc h y at a p oin t of the lattice [6 ] 1 . Rev ersing the sign of ¯ t is con ven tional in the form ulation of the T o da hierar ch y [24 ] (cf. the fermionic represen tat io n of T o da tau functions review ed in Section 3.4). An y T o da tau function thereby b ecomes a tau function of the tw o-comp onen t KP hierarch y . Let us note tha t the the roles of t and ¯ t in Z [ t , ¯ t ] can b e inte rc hang ed b y virtue of the iden tities s λ [ t ] = ( − 1) | λ | s t λ [ − t ] , s λ [ − ¯ t ] = ( − 1) | λ | s t λ [ ¯ t ] (20) of the Sc h ur functions and the pr o p ert y (8) of κ λ . 1 A generaliz ation o f this tau function was first studied by Khar ch ev et al. [40] in a different context. 9 2.4 Cut-and-join op erator The cut-and-join op erator M 0 [34] ma y b e thought o f as an infinitesimal symmetry on the space of tau functions of the KP hierar c hy [1 2, 22, 23 ]. In the K P time v ariables t , the cut-a nd-join o p erator reads M 0 = 1 2 ∞ X j,k =1  k l t k t l ∂ ∂ t k + l + ( k + l ) t k + l ∂ 2 ∂ t k ∂ t l  . (21) The Sc h ur functions s λ [ t ] are eigenfunctions of this op erator with eigenv alues κ λ / 2: M 0 s λ [ t ] = κ λ 2 s λ [ t ] (22) A combin atorial pro of of this fa ct is presen ted in Zhou’s pap er [4 1]. As we shall remark in Section 3, the cut-a nd- join op erator has a fermionic coun ter- part [6], whic h leads to another pro of of (22). (22) implies the iden tities e β κ λ / 2 s λ [ t ] = e β M 0 s λ [ t ] . Therefore one can use e β M 0 to reco v er Z [ t ] and Z [ t , ¯ t ] fr om t heir “initial v a lues” at β = 0 as Z [ t ] = e β M 0 Z [ t ] | β =0 , Z [ t , ¯ t ] = e β M 0 Z [ t , ¯ t ] | β =0 . Z [ t ] β =0 and Z [ t , ¯ t ] | β =0 can b e calculated by the Cauc hy iden tity X λ ∈P s λ [ t ] s λ [ − ¯ t ] = exp − ∞ X k =1 k t k ¯ t k ! (23) and the weigh ted homogeneity s λ [ ct 1 , c 2 t 2 , · · · ] = c | λ | s λ [ t 1 , t 2 , · · · ] (24) of Sch ur functions as Z [ t ] | β =0 = X λ ∈P Q | λ | s λ [ t ] s λ [1 , 0 , 0 , · · · ] = e Qt 1 and Z [ t , ¯ t ] | β =0 = X λ ∈P Q | λ | s λ [ t ] s λ [ − ¯ t ] = exp − ∞ X k =1 Q k k t k ¯ t k ! . 10 One can thus deriv e the well know n formula [12 , 22, 23] Z [ t ] = e β M 0 e Qt 1 (25) and its extension Z [ t , ¯ t ] = e β M 0 exp − ∞ X k =1 Q k k t k ¯ t k ! (26) to double Hurwitz num b ers. 3 F ermionic rep resen tation of tau function 3.1 Tw o-dimensional free fermion system Let us introduce tw o-dimensional complex free fermion fields ψ ( z ) = X n ∈ Z ψ n z − n − 1 , ψ ∗ ( z ) = X n ∈ Z ψ ∗ n z − n . Note that w e follow the notations of our previous w ork [32, 33] to use integers rather than half-integers fo r the lab els of F ourier mo des ψ n , ψ ∗ n . The F ourier mo des satisfy t he anti-comm utation relatio ns ψ m ψ ∗ n + ψ ∗ n ψ m = δ m + n, 0 , ψ m ψ n + ψ n ψ m = 0 , ψ ∗ m ψ ∗ n + ψ ∗ n ψ ∗ m = 0 . The F o ck space H of k et v ectors and its dual space H ∗ of bra ve ctors a re decomp osed to c harg e- s sectors H s , H ∗ s , s ∈ Z . Let h s | and | s i denote the ground states in H s and H ∗ s , namely , h s | = h−∞| · · · ψ ∗ s − 1 ψ ∗ s , | s i = ψ − s ψ − s +1 · · · | − ∞i , whic h satisfy the annihilation conditions ψ n | s i = 0 for n ≥ − s, ψ ∗ n | s i = 0 for n ≥ s + 1 , h s | ψ n = 0 for n ≤ − s − 1 , h s | ψ ∗ n = 0 for n ≤ s. Excited states can b e lab elled by partitions λ = ( λ 1 , λ 2 , · · · , λ n , 0 , 0 , · · · ) of arbitrary length as | λ, s i = ψ − λ 1 − s · · · ψ − λ n − s + n − 1 ψ ∗ s − n +1 · · · ψ ∗ s | s i , h λ, s | = h s | ψ − s · · · ψ − s + n − 1 ψ ∗ λ n + s − n +1 · · · ψ ∗ λ 1 + s . 11 | λ, s i and h λ, s | represen t a state in whic h the semi-infinite subset { λ i + s − i + 1 } ∞ i =1 (sometimes referred to as the “May a diagram”) o f the set Z of a ll “energy lev els” a r e o ccupied by par ticles. These v ectors give dual bases of o f H s and H ∗ s in the sense that h λ, r | µ, s i = δ λµ δ r s . (27) The normal ordered fermion bilinears : ψ − i ψ ∗ j : = ψ − i ψ ∗ j − h 0 | ψ − i ψ ∗ j | 0 i , i, j ∈ Z , span the one-dimensional cen tral extension b gl( ∞ ) of the Lie algebra gl( ∞ ) of infinite matrices [18, 19]. gl( ∞ ) consists of infinite matr ices A = ( a ij ) i,j ∈ Z of “ finite-band type”, namely , there is a p ositiv e integer N (dep ending on A ) suc h that a ij = 0 if | i − j | > N . F or suc h a matrix A ∈ gl( ∞ ), the fermion bilinear b A = X i,j ∈ Z a ij : ψ − i ψ ∗ j : b ecomes a w ell-defined linear op erator on the F o c k space, and preserv es the c harg e, namely , h λ, r | b A | µ, s i = 0 if r 6 = s . (28) Moreo ver, fo r t wo suc h matrices A, B ∈ gl( ∞ ), the asso ciated op erators b A, b B satisfy the commutation relation [ b A, b B ] = \ [ A, B ] + γ ( A, B ) (29) with the c -n umber co cycle term γ ( A, B ) = T r( A + − B − + − B + − A − + ) , (30) where A ±∓ and B ±∓ denote the following quarter blo c ks of A, B : A + − = ( a ij ) i> 0 , j ≤ 0 , A − + = ( a ij ) i ≤ 0 , j > 0 , B + − = ( b ij ) i> 0 , j ≤ 0 , B − + = ( b ij ) i ≤ 0 , j > 0 . 3.2 Sp ecial fermion bilinears The follow ing fermion bilinears ar e building blo cks of our T o da tau function: J m = X n ∈ Z : ψ − n + m ψ ∗ n : , m ∈ Z , L 0 = X n ∈ Z n : ψ − n ψ ∗ n : , W 0 = X n ∈ Z n 2 : ψ − n ψ ∗ n : . 12 J m ’s span a U(1) curren t alg ebra. L 0 and W 0 are zero-mo des of Vira soro and W (3) algebras. These fermion bilinears are asso ciated with infinite matrices as J m = c Λ m , L 0 = b ∆ , W 0 = c ∆ 2 , (31) where ∆ and Λ are infinite mat r ices of the form ∆ = ( iδ ij ) , Λ = ( δ i +1 ,j ) . Let J ± [ t ], t = ( t 1 , t 2 , · · · ), denote the sp ecial linear comb inations J + [ t ] = ∞ X k =1 t k J k , J − [ t ] = ∞ X k =1 t k J − k of J m ’s. Their exp onen t ia ls act on the ground states h s | , | s i as h s | e J + [ t ] = X λ ∈P h λ, s | s λ [ t ] , e J − [ t ] | s i = X λ ∈P s λ [ t ] | λ, s i , (32) yielding Sc hur functions a s matrix elemen ts [18 , 19]: s λ [ t ] = h λ, s | e J + [ t ] | s i = h s | e J − [ t ] | λ, s i . (33) Unlik e other J m ’s, J 0 is diag onal with resp ect to | λ, s i ’s: h λ, s | J 0 | µ, s i = δ λµ s. (34) L 0 and W 0 , to o, are dia gonal. The dia g onal matrix elemen t s can b e calculated as follo ws. Lemma 1. h λ, s | L 0 | µ, s i = δ λµ  | λ | + s ( s + 1) 2  , (35) h λ, s | W 0 | µ, s i = δ λµ  κ λ + (2 s + 1) | λ | + s ( s + 1)(2 s + 1) 6  . (36) Pr o of. Assuming that s ≥ 0, one can calculate t he dia g onal matrix elemen ts as h λ, s | L 0 | λ, s i = ∞ X i =1 ( λ i + s − i + 1) − ∞ X i =1 ( − i + 1) ( heuristic expression) = ∞ X i =1 (( λ i + s − i + 1) − ( s − i + 1)) + s X k =0 k (re-summed) = | λ | + s ( s + 1) 2 13 and h λ, s | W 0 | λ, s i = ∞ X i =1 ( λ i + s − i + 1) 2 − ∞ X i =1 ( − i + 1) 2 (heuristic express ion) = ∞ X i =1 ( λ i + s − i + 1) 2 − ( s − i + 1) 2 ) + s X k =0 k 2 (re-summed) = κ λ + (2 s + 1) | λ | + s ( s + 1)(2 s + 1) 6 . In the case where s < 0, w e hav e only to replace the intermediate sums ov er k as s X k =0 k → − 0 X k = s +1 k , s X k =0 k 2 → − 0 X k = s +1 k 2 , ending up with the same final express ion of the matrix elemen ts. 3.3 T o da T au fu n ction for double Hurwitz n um b ers A general ta u function of the T o da hierarc h y has the fermionic expression τ ( s, t , ¯ t ) = h s | e J + [ t ] g e − J − [ ¯ t ] | s i , where g is an elemen t o f c GL( ∞ ) [42 ]. Inserting the aforemen tioned expansion (32), one can expand this function as τ ( s, t , ¯ t ) = X λ,µ ∈P h λ, s | g | µ, s i s λ [ t ] s µ [ − ¯ t ] . (37) F o llowing Ok o unk ov [6], w e no w consider the sp ecial case where g = e β W 0 / 2 Q L 0 = Q L 0 e β W 0 / 2 . (38) Theorem 1. The tau function determine d by ( 3 8) c an b e exp a n de d as τ ( s, t , ¯ t ) = e β s ( s +1)(2 s +1) / 12 Q s ( s +1) / 2 X λ ∈P e β κ λ / 2 ( e β ( s +1 / 2) Q ) | λ | s λ [ t ] s λ [ − ¯ t ] . (39) Pr o of. The prop erties (3 5) and (36) of L 0 and W 0 imply that g is also dia g- onalized o n t he basis | λ, s i of t he F o c k space. The dia gonal matrix elemen ts tak e suc h a form as h λ, s | g | λ, s i = exp  β 2  κ λ + (2 s + 1) | λ | + s ( s + 1)(2 s + 1 ) 6  Q | λ | + s ( s +1) / 2 . The tau function in question can b e t hereb y expanded as (39) sho ws. 14 This is a restatemen t o f Okounk ov’s result [6]. One can rewrite ( 39) as τ ( s, t , ¯ t ) = e β s ( s +1)(2 s +1) / 12 Q s ( s +1) / 2 Z β ,e β ( s +1 / 2) Q [ t , ¯ t ] , (40) where Z β ,Q [ t , ¯ t ] denotes the generating function (19) with the parameters β and Q b eing explicitly indicated. Th us, apart f rom some n umerical fac- tors dep ending on s , the tau function coincides with the generating function of double Hurwitz num b ers. Note that the s -dep endence sho ws up in t he generating function as the m ultiplier e β ( s +1 / 2) of the parameter Q . Let us conclude this section with a few commen ts on the cut-and-join op erator (21). The cut-and- join op erator corresp onds to the fermion bilinear M 0 = 1 2 X n ∈ Z  n − 1 2  2 : ψ − n ψ ∗ n : = W 0 2 − L 0 2 + J 0 8 . (41) One can readily see fr o m (34), (35) and (36) that this op erator acts on | λ, s i ’s as M 0 | λ, s i =  κ λ 2 + s | λ | + 4 s 3 − s 24  | λ, s i . (42) In particular, | λ i = | λ, 0 i is a n eigenstate with eigen v alue κ λ / 2. Note t hat s -dep enden t extra terms sho w up in t he c harge- s sector. F ermion bilinears of t his t yp e can b e con verted to differen tial (or “b osonic”) op erators by the b oson-fermion corresp ondence [18, 19]. In a generating func- tional form, t he normal- o rdered pro duct : ψ ( z ) ψ ∗ ( w ): = ψ ( z ) ψ ∗ ( w ) − 1 z − w ( | z | < | w | ) of the fermion fields corresp onds to the tw o-v ariable v ertex op erator X ( z , w ) = 1 z − w  z w  s exp ∞ X k =1 t k ( z k − w k ) ! exp − ∞ X k =1 z − k − w − k k ∂ ∂ t k ! − 1 ! as h s | e J + [ t ] : ψ ( z ) ψ ∗ ( w ): = X ( z , w ) h s | e J + [ t ] . (43) A similar relation holds for e − J − [ t ] | s i and leads to b o sonization with resp ect to ¯ t [20], though w e o mit details here. 15 Γ( z , w ) can b e expanded in p ow ers of z − w , and the co efficien t s o f this expansion give a b osonic represen tat io n of fermion bilinears. F or L 0 , J 0 and W 0 , this b o sonic represen ta tion reads L 0 = ∞ X k =1 k t k ∂ ∂ t k + s ( s + 1) 2 , J 0 = s (44) and W 0 = ∞ X k ,l =1  k l t k t l ∂ ∂ t k + l + ( k + l ) t k + l ∂ 2 ∂ t k ∂ t l  + (2 s + 1) ∞ X k =1 k t k ∂ ∂ t k + s ( s + 1)( 2 s + 1) 6 . (45) (41) is thus b osonized as M 0 = 1 2 ∞ X k ,l =1  k l t k t l ∂ ∂ t k + l + ( k + l ) t k + l ∂ 2 ∂ t k ∂ t l  + s ∞ X k =1 k t k ∂ ∂ t k + 4 s 3 − s 24 . (46) In t he c harge-0 sector, this reduces to the cut-and-join op erator (21). 4 Generalize d strin g e qu ations for doubl e Hur- witz n um b ers 4.1 Notations for difference op erators Building blo c ks of the Lax formalism of the T o da hierarc hy are o ne-dimensional difference op erato rs in the la t t ice co ordinate s [24 ]. Those op erat ors a re lin- ear combinations of the shift op erators e n∂ s , e n∂ s f ( s ) = f ( s + n ). Although a gen uine difference op erato r is a finite linear com bination A = N X n = M a n ( s ) e n∂ s (op erator of [ M , N ]- t yp e) of the shift op erators, o ne can consider a semi-infinite linear com bination of the f orm A = N X n = −∞ a n ( s ) e n∂ s (op erator of ( −∞ , N ] t yp e ) 16 or A = ∞ X n = M a n ( s ) e n∂ s (op erator of [ M , ∞ ) t yp e) as well, whic h amoun t to pseudo-differen t ia l op erators in the Lax formalism of the K P hierarc h y [18]. Let ( ) ≥ 0 and ( ) < 0 denote the tr uncation op eratio n ( A ) ≥ 0 = X n ≥ 0 a n ( s ) e n∂ s , ( A ) < 0 = X n< 0 a n ( s ) e n∂ s . Difference op erators are in one-t o -one corresp ondence with Z × Z matrices. Firstly , the n -th shift op erator e n∂ s corresp onds to the shift matrix Λ n = ( δ i,j − n ) i,j ∈ Z . Secondly , t he multiplication op erato r a ( s ) amoun ts to t he diagona l matrix diag( a ( s )) = ( a ( i ) δ ij ) i,j ∈ Z . In pa rticular, the multiplication o p erator s corresp onds to ∆ = dia g( s ) = ( iδ ij ) i,j ∈ Z . Consequen tly , a general difference op erator A = A ( s, e ∂ s ) = X n a n ( s ) e n∂ s is conv erted to the infinite matrix A (∆ , Λ ) = X n diag( a n ( s ))Λ n = X n ( a n ( i ) δ i,j − n ) i,j ∈ Z . Occasionally , it might b e more conv enien t to write a difference op erator in an an ti- normal-ordered form as B ( e ∂ s , s ) = X n e n∂ s b n ( s ) . In t ha t case, the corresp onding infinite matrix reads B (Λ , ∆) = X n Λ n diag( b n ( s )) . 17 4.2 Lax and Orlo v-Sc h ulman op erators The Lax formalism of the T o da hierarch y uses t w o Lax op erators L, ¯ L of t yp e ( −∞ , 1] and [1 , ∞ ) . Actually , f rom the p oint of view of symmetry , it is b etter to consider L and ¯ L − 1 rather than L and ¯ L . These op erat o rs admit freedom of gauge transformatio ns. In the ga uge where L is mo nic (namely , the leading co efficien ts is equal to 1), L and ¯ L − 1 can b e expressed as L = e ∂ s + ∞ X n =1 u n e (1 − n ) ∂ s , ¯ L − 1 = ¯ u 0 e − ∂ s + ∞ X n =1 ¯ u n e ( n − 1) ∂ s . The co efficien ts u n and ¯ u n are functions of s and t he time v ariables t , ¯ t , and written as u n ( s, t , ¯ t ) and ¯ u n ( s, t , ¯ t ) if w e do not suppress t he indep enden t v a r iables. L and ¯ L satisfy the Lax equations ∂ L ∂ t n = [ B n , L ] , ∂ L ∂ ¯ t n = [ ¯ B n , L ] , ∂ ¯ L ∂ t n = [ B n , ¯ L ] , ∂ ¯ L ∂ ¯ t n = [ ¯ B n , ¯ L ] , (47) where B n and ¯ B n are defined a s B n = ( L n ) ≥ 0 , ¯ B n = ( ¯ L − n ) < 0 . T o fo rm ulate the generalized string equations, w e need another pair of dif- ference op erators M , ¯ M , namely , the Or lov-Sc h ulman op era t ors [44]. These op erators, to o, satisfy t he Lax equations ∂ M ∂ t n = [ B n , M ] , ∂ M ∂ ¯ t n = [ ¯ B n , M ] , ∂ ¯ M ∂ t n = [ B n , ¯ M ] , ∂ ¯ M ∂ ¯ t n = [ ¯ B n , ¯ M ] (48) of the same form as the Lax op erato r s do, and are r elat ed to the Lax op erator s b y t he twis ted canonical comm utation relations [ L, M ] = L, [ ¯ L, ¯ M ] = ¯ L. (49) By “ g eneralized string equations” w e mean equations of the form C ( L, M ) = ¯ C ( ¯ L, ¯ M ) , (50) where C ( L, M ) and ¯ C ( ¯ L, ¯ M ) are (p ossibly infinite) linear com binations of monomials of L, M and ¯ L, ¯ M with constant co efficien ts. The f ollo wing lemma [30, 31] explains an origin of generalized string equations. 18 Lemma 2. If the ferm ion biline ars \ C (Λ , ∆) and \ ¯ C (Λ , ∆) ar e intertwine d by a c GL ( ∞ ) e lement g as \ C (Λ , ∆) g = g \ ¯ C (Λ , ∆) , (51) then the L ax and Orlov-Schulman op er ators satisfy ( 5 0 ). 4.3 In tert wining relations Let us no w consider the case of the solution determined by the c GL ( ∞ ) ele- men t ( 38). W e seek in tert wining relatio ns in the follow ing sp ecial form: J k g = g \ ¯ C (Λ , ∆) , \ C (Λ , ∆) g = g J − k , k = 1 , 2 , · · · . Lemma 3. J m ’s ar e tr ansforme d by the adjoint action o f Q L 0 and e β W 0 / 2 as Q L 0 J m Q − L 0 = Q − m J m , e − β W 0 / 2 J m e β W 0 / 2 = e − β m 2 / 2 X n ∈ Z e β mn : ψ − n + m ψ ∗ n : . (52) Pr o of. Let us note the fundamen tal comm utatio n relations [ L 0 , ψ n ] = − nψ n , [ L 0 , ψ ∗ n ] = − nψ ∗ n and [ W 0 , ψ n ] = n 2 ψ n , [ W 0 , ψ ∗ n ] = − n 2 ψ ∗ n that follo w from the definition of L 0 and W 0 . These commutation relatio ns can b e exp onentiated as Q L 0 ψ n Q − L 0 = Q − n ψ n , Q L 0 ψ ∗ n Q − L 0 = Q − n ψ ∗ n and e − β W 0 / 2 ψ n e β W 0 / 2 = e − β n 2 / 2 ψ n , e − β W 0 / 2 ψ ∗ n e β W 0 / 2 = e β n 2 / 2 ψ ∗ n . Consequen tly , the exp onen t ia ted op erato rs Q L 0 and e β W 0 / 2 act on the basis : ψ − i ψ ∗ j : of gl( ∞ ) as Q L 0 : ψ − i ψ ∗ j : Q − L 0 = Q i − j : ψ − i ψ ∗ j : and e − β W 0 / 2 : ψ − i ψ ∗ j : e β W 0 / 2 = e − β ( i 2 − j 2 ) / 2 : ψ − i ψ ∗ j : . 19 One can thereb y deriv e (52) as Q L 0 J m Q − L 0 = X n ∈ Z Q ( n − m ) − n : ψ − n + m ψ ∗ n : = Q − m J m and e − β W 0 / 2 J m e β W 0 / 2 = X n ∈ Z e − β (( n − m ) 2 − n 2 ) / 2 : ψ − n + m ψ ∗ n : = e − β m 2 / 2 X n ∈ Z e β mn : ψ − n + m ψ ∗ n : . Lemma 4. J ± k ’s ar e c on ne cte d with the fermion bil i n e ars \ Λ k e β k ∆ and \ Λ − k e β k ∆ by the c GL( ∞ ) el e ment (38) a s J k g = g Q k e − β k 2 / 2 \ Λ k e β k ∆ , (53) g J − k = Q k e β k 2 / 2 \ Λ − k e β k ∆ g . (54) Pr o of. Using the relations (52) in the previous lemma, one can calculate g − 1 J k g a s g − 1 J k g = e − β W 0 / 2 Q − L 0 J k Q L 0 e β W 0 / 2 = Q k e − β W 0 / 2 J k e β W 0 / 2 = Q k e − β k 2 / 2 X n ∈ Z e β k n : ψ − n + k ψ ∗ n : . Since the last sum can b e r ewritten as X n ∈ Z e β k n : ψ − n + k ψ ∗ n : = \ Λ k e β k ∆ , the first in tert wining relatio n (53) follo ws. In the same w ay , one can calculate g J − k g − 1 as g J − k g − 1 = e β W 0 / 2 Q L 0 J − k Q − L 0 e − β W 0 / 2 = Q k e β W 0 / 2 J − k e − β W 0 / 2 = Q k e β k 2 / 2 X n ∈ Z e β k n : ψ − n − k ψ ∗ n : , whic h implies the second in tert wining relation (54). 20 4.4 Generalized string equations Theorem 2. The L ax and Orlov-Schulman op er ators of the tau function (39) satisfy the gener aliz e d string e quations L k = Q k e − β k 2 / 2 ¯ L k e β k ¯ M , ¯ L − k = Q k e β k 2 / 2 L − k e β k M (55) for k = 1 , 2 , · · · . Mor e ove r, these e quations c an b e derive d fr om the first two ( k = 1 ) e quations L = Qe − β / 2 ¯ Le β ¯ M , ¯ L − 1 = Qe β / 2 L − 1 e β M . (56) Pr o of. The first pa rt is a consequence of (53) and (54). Let us show the second part. The k -th p o w er o f the first equation of (56) reads L k = Q k e − β k / 2 ( ¯ Le β ¯ M ) k . The comm utation equation o f ¯ L a nd ¯ M in ( 4 9) implies that [ ¯ M , · · · , [ ¯ M , [ ¯ M , ¯ L ]] · · · ] ( k -fold comm uta tor) = ( − 1) k ¯ L for k = 1 , 2 , · · · , so that e β ¯ M ¯ Le − β ¯ M = ¯ L + ∞ X k =1 β k k ! [ ¯ M , · · · , [ ¯ M , [ ¯ M , ¯ L ]] · · · ] = e − β ¯ L. Using this relation rep eatedly , one can mo v e e β ¯ M ’s in ( ¯ Le β ¯ M ) k to t he righ t- most p osition a s ( ¯ Le β ¯ M ) k = ¯ Le β ¯ M ( ¯ Le β ¯ M ) k − 1 e β ¯ M = e − β k ¯ L 2 e β ¯ M ( ¯ Le β ¯ M ) k − 2 e 2 β ¯ M = e − β k − β ( k − 1) ¯ L 3 e β ¯ M ( e β ¯ M ¯ L ) k − 3 e 3 β ¯ M = · · · = e − β k ( k − 1) / 2 ¯ L k e β k ¯ M . Th us the first equation of (55) follows . The second equation of (55), to o, can b e deriv ed from (5 6) in the same wa y . Th us, in con trast with tw o-dimensional quan t um gravit y [25] and c = 1 string theory [26, 27, 28, 29, 30, 3 1], the generalized string equations con tain the exp onential terms e β M , e β ¯ M . These t erms stem from the fermion bilinears \ Λ ± k e β k ∆ in (53 ) and (54). F erimion blinears of a similar f orm are also used 21 in the study of in tegrable structures of the melting crystal mo del [32, 33]. A common algebraic ba ckground of t hese fermion bilinears is the quan tum torus algebra (with par ameter q ) spanned by the infinite matrices v ( k ) m = q − k m/ 2 Λ m q k ∆ , k , m ∈ Z . (57) In the melting crystal mo del [3 2, 33], a cen tral extension of this Lie algebra is realized by the fermion bilinears V ( k ) m = q − k m X n ∈ Z q k n : ψ − n + m ψ ∗ n : . (58) The Lax and Orlov-Sc h ulman op erato rs giv e (t w o copies of ) yet another kind of realizatio n by the difference op erators V ( k ) m = q − k m/ 2 L m q k M (59) and ¯ V ( k ) m = q − k m/ 2 ¯ L m q k ¯ M . (60) If q = e β , the exp onentials e β M , e β ¯ M b elong to this Lie algebra. 5 Classical limit of ge neralized string equa- tions 5.1 Disp ersionless T o da hierarc h y In a naiv e sense [43], the classical limit of the T o da hierar ch y can b e ob- tained b y r eplacing the shift op erator e ∂ s b y a new v aria ble p . The difference op erators L, M , ¯ L, ¯ M th us turn into Lauren t series o f p of the form L = p + ∞ X n =1 u n p 1 − n , ¯ L − 1 = ¯ u 0 p − 1 + ∞ X n =1 ¯ u n p n − 1 , M = ∞ X n =1 nt n L n + s + ∞ X n =1 v n L − n , ¯ M = − ∞ X n =1 n ¯ t n ¯ L − n + s + ∞ X n =1 ¯ v n ¯ L n 22 that are referred to as the “Lax and Orlov-Sc h ulman functions”. As difference op erators are replaced b y Lauren t series, comm utators of difference op erators turn in to Poiss on brac k ets by the rule [ e ∂ s , s ] = e ∂ s → { p, s } = s. Accordingly , Poiss on brac k ets of functions of p and s are defined as { F , G } = p  ∂ F ∂ p ∂ G ∂ s − ∂ F ∂ s ∂ G ∂ p  . The Lax equations and the tw isted canonical comm utation relations are re- defined with respect to the P oisson brac ke t as ∂ L ∂ t n = {B n , L} , ∂ L ∂ ¯ t n = { ¯ B n , L} , ∂ ¯ L ∂ t n = {B n , ¯ L} , ∂ ¯ L ∂ ¯ t n = { ¯ B n , ¯ L} , ∂ M ∂ t n = {B n , M} , ∂ M ∂ ¯ t n = { ¯ B n , M} , ∂ ¯ M ∂ t n = {B n , ¯ M} , ∂ ¯ M ∂ ¯ t n = { ¯ B n , ¯ M} (61) and {L , M} = L , { ¯ L , ¯ M} = ¯ L . (62) B n and ¯ B n are defined b y seemingly the same formulae B n = ( L n ) ≥ 0 , ¯ B n = ( ¯ L − n ) < 0 as in the previous case, but the notations ( ) ≥ 0 and ( ) < 0 no w stand for pro jection op erators on the space of Lauren t series, na mely , X n a n p n ! ≥ 0 = X n ≥ 0 a n p n , X n a n p n ! < 0 = X n< 0 a n p n . These equations are fundamen tal constituen ts of the “disp ersionless T o da hierarc hy”. 23 5.2 ~ -dep enden t T o d a h ierarc h y and generalized string equations The foregoing pro cedure replacing e ∂ s → p can b e justified as a kind of classical limit in an ~ -dep endent formulation of the T o da hierarch y [44]. In the ~ - dep enden t form ulation, e ∂ s is replaced by e ~ ∂ s . The “Planc k con- stan t” ~ thus pla ys the role of lattice spacing. The Lax and Orlov-Sc h ulman op erators are expanded in p ow ers of e ~ ∂ s as L = e ~ ∂ s + ∞ X n =1 u n e (1 − n ) ~ ∂ s , ¯ L − 1 = ¯ u 0 e − ~ ∂ s + ∞ X n =1 ¯ u n e ( n − 1) ~ ∂ s , M = ∞ X n =1 nt n L n + s + ∞ X n =1 v n L − n , ¯ M = − ∞ X n =1 n ¯ t n ¯ L − n + s + ∞ X n =1 ¯ v n ¯ L n . The Lax equations and the tw isted canonical comm utation relations tak e a n ~ -dep enden t form as ~ ∂ L ∂ t n = [ B n , L ] , ~ ∂ L ∂ ¯ t n = [ ¯ B n , L ] , ~ ∂ ¯ L ∂ t n = [ B n , ¯ L ] , ~ ∂ ¯ L ∂ ¯ t = [ ¯ B n , ¯ L ] , ~ ∂ M ∂ t n = [ B n , M ] , ~ ∂ M ∂ ¯ t n = [ ¯ B n , M ] , ~ ∂ ¯ M ∂ t n = [ B n , ¯ M ] , ~ ∂ ¯ M ∂ ¯ t n = [ ¯ B n , ¯ M ] (63) and [ L, M ] = ~ L, [ ¯ L, ¯ M ] = ~ ¯ L. (64 ) If the co efficien t u n , ¯ u n , v n , ¯ v n (whic h are functions of ~ , s, t , ¯ t ) ha v e a smo oth classical limit a s u (0) n = lim ~ → 0 u n , ¯ u (0) n = lim ~ → 0 ¯ u n , v (0) n = lim ~ → 0 v n , ¯ v (0) n = lim ~ → 0 ¯ v n , (65) 24 one can define the Lax and Orlov-Sc h ulman functions as L = p + ∞ X n =1 u (0) n p 1 − n , ¯ L − 1 = ¯ u (0) 0 p − 1 + ∞ X n =1 ¯ u (0) n p n − 1 , M = ∞ X n =1 nt n L n + s + ∞ X n =1 v (0) n L − n , ¯ M = − ∞ X n =1 n ¯ t n ¯ L − n + s + ∞ X n =1 ¯ v (0) n ¯ L n . These La x a nd Orlov-Sc h ulman functions satisfy the L ax equations (61 ) and the twisted canonical Poiss on relations (62). In this ~ -dep enden t form ulation, generalized string equations (5 0) are mo dified as C ( L, ~ − 1 M ) = ¯ C ( ¯ L, ~ − 1 ¯ M ) , (66) namely , M and ¯ M are m ultiplied b y ~ − 1 [30, 31]. Let us explain the underly- ing mec hanism briefly . The Lax and Orlov-Sc h ulman op erat o rs are connected with t he mat r ices Λ a nd ∆ b y the so called “ dr essing op era t o rs” [44]. The t wisted canonical commutation relatio ns are thereb y deriv ed f r o m the com- m utat ion relation [Λ , ∆] = Λ of these matrices. In the ~ -dep enden t form ulation, they take the form [ L, ~ − 1 M ] = ~ − 1 M , [ ¯ L, ~ − 1 ¯ M ] = ~ − 1 ¯ M , whic h are nothing but (64). Th us it is ~ − 1 M and ~ − 1 ¯ M rather than M and ¯ M that corresp ond to ∆ and sho w up in generalized string equations. 5.3 Classical limit of generalized string equations Let us turn to the case of double Hurwitz n um b ers. T o deriv e a classical limit, w e ha v e to in tro duce ~ therein. A t least formally , this can b e done by rescaling the space-time v aria bles as s → ~ − 1 s, t → ~ − 1 t , ¯ t → ~ − 1 ¯ t . (67) 25 The ~ - indep enden t T o da hierarc h y is thereb y con v erted to the ~ -dep enden t form. Actually , it is rare tha t the rescaled Lax and Orlo v-Sch ulman op erators satisfy the condition (65). T o achiev e a meaningful (and nontrivial) classical limit, one should start from a carefully c hosen ~ -dep endent solution of the ~ -indep endent T o da hierarc h y . In the language o f the tau function [44], a meaningful classical limit is obtained fro m an ~ -dep enden t tau function τ ( ~ , s, t , ¯ t ) suc h that the rescaled tau function τ ~ ( s, t , ¯ t ) = τ ( ~ , ~ − 1 s, ~ − 1 t , ~ − 1 ¯ t ) b eha v es a s log τ ~ ( s, t , ¯ t ) = ~ − 2 F ( s , t , ¯ t ) + O ( ~ − 1 ) . (68 ) F = F ( s, t , ¯ t ) is called t he “f ree energy” b ecause of its relation to rando m matrices and top ological field theories [45, 4 6]. If the r escaled tau f unction has t his asymptotic for m, the asso ciated “w a ve functions” Ψ ~ ( s, t , ¯ t , z ) = τ ~ ( s, t − ~ [ z − 1 ] , ¯ t ) τ ~ ( s, t , ¯ t ) z ~ − 1 s e ~ − 1 ξ ( t ,z ) , ¯ Ψ ~ ( s, t , ¯ t , z ) = τ ~ ( s + ~ , t , ¯ t − ~ [ z ]) τ ~ ( s, t , ¯ t ) z ~ − 1 s e ~ − 1 ξ ( ¯ t ,z − 1 ) , [ z ] =  z , z 2 / 2 , · · · , z k /k , · · ·  , ξ ( t , z ) = ∞ X k =1 t k z k ha v e the “WKB” form Ψ ~ ( s, t , ¯ t , z ) = exp  ~ − 1 S ( s, t , ¯ t , z ) + O ( 1 )  , ¯ Ψ ~ ( s, t , ¯ t , z ) = exp  ~ − 1 ¯ S ( s, t , ¯ t , z ) + O (1)  (69) and satisfy a set o f auxiliary linear equations. The phase functions S ( s, t , ¯ t , z ) and ¯ S ( s, t , ¯ t , z ) satisfy the asso ciated Hamilton-Jacobi equations, whic h can b e con v erted to the disp ersionless Lax equations (6 1) and the Poiss on rela- tions (62) (see the review [20] fo r details). An appropriate ~ -dep enden t reformulation o f the tau function (39) of double Hurwitz n um b ers can b e found by the follow ing heuristic considera- tion. As w e mov e in to the ~ - dep enden t form ulation, the generalized string equations (56) a re mo dified as L = Qe − β / 2 ¯ Le β ~ − 1 ¯ M , ¯ L − 1 = Qe β / 2 L − 1 e β ~ − 1 M . 26 Ob viously , these equations do not ha v e a limit as ~ → 0. If, ho w ev er, the parameter β is sim ultaneously rescaled as β → ~ β , (70) the g eneralized string equations are further mo dified as L = Qe − ~ β / 2 ¯ Le β ¯ M , ¯ L − 1 = Qe ~ β / 2 L − 1 e β M , (71) and ha ve a meaningful classical limit of the form L = Q ¯ L e β ¯ M , ¯ L − 1 = Q L − 1 e β M . (72) 5.4 Existence of ~ -expansion T o j ustify the foregoing heuristic deriv ation o f the classical limit (72) of the generalized string equations, let us show that the tau function (39) with β rescaled as (70 ) do es satisfy the condition (68). Recall the expression (40) of the tau function. After rescaling s, t , ¯ t and β as (67) a nd (70), this expression is mo dified as τ ~ ( s, t , ¯ t ) = e ~ − 2 β s ( s + ~ )(2 s + ~ ) / 12 Q ~ − 2 s ( s + ~ ) / 2 Z ~ β ,e β ( s + ~ / 2) Q [ ~ − 1 t , ~ − 1 ¯ t ] . (73) Therefore it is sufficien t to show that the logarithm of Z ~ β ,Q [ ~ − 1 t , ~ − 1 ¯ t ] has an ~ -expansion o f the form log Z ~ β ,Q [ ~ − 1 t , ~ − 1 ¯ t ] = ~ − 2 F 0 + F 1 + ~ 2 F 2 + · · · + ~ 2 n F n + · · · , ( 74) where F 0 , F 1 , · · · are analytic functions of ( β , Q, t , ¯ t ) in a common domain. The free energy is then giv en b y F = β s 3 6 + s 2 log Q 2 + F 0 ( β , e β s Q, t , ¯ t ) . (75) (74) is a generalization o f the w ell kno wn to p ological expansion for simple Hurwitz n um b ers. It is common in the literature that this kind of expansion is explained by a comb ination of top o lo gical and com binato rial consideration (see, e.g., Section 4.2 o f Mironov and Morozov [22], Section 2.1 of Bouchard and Mari ˜ no [14] and Section 2.2 of Borot et al. [15]) . W e tak e a no ther approac h based on the cut-and-join op erator (21). Theorem 3. log Z ~ β ,Q [ ~ − 1 t , ~ − 1 ¯ t ] has an ~ -exp ansion of the form (7 4). 27 Pr o of. Let F = F ( ~ , β , Q, t , ¯ t ) denote the left hand side of (74) m ultiplied b y ~ 2 . By (26), e ~ − 2 F can b e expressed as e ~ − 2 F = e ~ β M 0 ( ~ ) exp − ~ − 2 ∞ X k =1 Q k k t k ¯ t k ! , where M 0 ( ~ ) denotes the rescaled cut-and-join op erator M 0 ( ~ ) = 1 2 ∞ X j,k =1  ~ − 1 k l t k t l ∂ ∂ t k + l + ~ ( k + l ) t k + l ∂ 2 ∂ t k ∂ t l  . Therefore e ~ − 2 F satisfies the differen tial equation ∂ e ~ − 2 F ∂ β = ~ M 0 ( ~ ) e ~ − 2 F with resp ect to β . This equation can b e further conv erted to the differen tial equation ∂ F ∂ β = 1 2 ∞ X j,k =1 k l t k t l ∂ F ∂ t k + l + 1 2 ∞ X k ,l =1 ( k + l ) t k + l  ~ 2 ∂ 2 F ∂ t k ∂ t l + ∂ F ∂ t k ∂ F ∂ t l  for F . This equation is supplemen ted by the the initial condition F | β =0 = − ∞ X k =1 Q k k t k ¯ t k . W e now seek a solution of this initial v alue problem in the form of a (for mal) p o w er series of ~ 2 : F = F 0 + ~ 2 F 1 + · · · + ~ 2 n F n + · · · . This reduces to solving the differen tial equations ∂ F n ∂ β = 1 2 ∞ X k ,l =1 k l t k t l ∂ F n ∂ t k + l + 1 2 ∞ X k ,l =1 ( k + l ) t k + l ∂ 2 F n − 1 ∂ t k ∂ t l + 1 2 ∞ X k ,l =1 ( k + l ) t k + l n X m =0 ∂ F m ∂ t k ∂ F n − m ∂ t l (76) for n = 0 , 1 , 2 , · · · under the initial conditions F n | β =0 = − δ n 0 ∞ X k =1 Q k k t k ¯ t k . (77) 28 F n ’s are thereb y recursiv ely determined, and b ecome analytic functions of ( β , Q, t , ¯ t ) in a common domain of definition (b ecause the differen tial equa- tions other tha n the first one fo r n = 0 are linear with resp ect to F n ). Since the initial v alue problem fo r F has a unique solutio n, the p ow er series solution F = F 0 + ~ 2 F 1 + · · · should coincide with the left hand side of (7 4). One can thus confirm the exp ected a symptotic form (68) of the the rescaled t au function (73). Let us stress that this is also a pro of for the case of simple Hurwitz num b ers. T o consider that case, one has only to set ¯ t k = − δ k 1 in the initial condition ( 7 7). Let us also p oint out that the main part F 0 of the free energy is determined by the n = 0 part of (76) and (77), namely , ∂ F 0 ∂ β = 1 2 ∞ X k ,l =1 k l t k t l ∂ F 0 ∂ t k + l + 1 2 ∞ X k ,l =1 ( k + l ) t k + l ∂ F 0 ∂ t k ∂ F 0 ∂ t l (78) and F 0 | β =0 = − ∞ X k =1 Q k k t k ¯ t k . (79) It will b e in teresting to apply the diagramatic tec hnique of Mirono v and Morozo v [22] to these equations. 6 Solutio n of clas s ical l i mit of gen e ralized strin g equations 6.1 Comparison with generalized string equations of c = 1 string theory Our goal in this section is to solv e the g eneralized string equations (72) and to derive some implications thereof. T o this end, it is instructiv e to compare these equations with the generalized string equations of c = 1 string theory [26, 27, 2 8, 29]. In the classical limit, the generalized string equations of c = 1 string theory read L = ¯ L ¯ M , ¯ L − 1 = L − 1 M . (80) Let us men tion that the same equations play a cen tral role in a problem of complex analysis and its a pplications to interface dynamics [47 , 48, 49]. 29 Actually , it is the equation {L , ¯ L − 1 } = 1 (81) rather than (80) that is referred to as a “string equation” in these applica- tions. Note that one can readily derive (81) from (80). In the same sense, one can deriv e the equation { log L , lo g ¯ L − 1 } = β (82) from (72) as a coun terpart of (80) for double Hurwitz num b ers. (81) and (82) resem ble the string equation (or the Douglas equation) [ Q, P ] = 1 (83) and its classical limit { Q, P } = 1 (84) in tw o-dimensional quan t um gravit y [50 , 51, 52]. Q and P in (83) are one- dimensional differen tial op erators of the fo rm Q = ∂ n x + u 2 ∂ n − 2 x + · · · + u n , P = ∂ m x + v 2 ∂ m − 2 x + · · · + v n . In the classical limit, ∂ x is replaced by a v ariable p with the P oisson bra c ke t { p, x } = 1 , and Q and P are p olynomials of the fo rm Q = p n + u 2 p n − 2 + · · · + u n , P = p m + v 2 p m − 2 + · · · + v n . In this setting, Q and P may b e though t of as co ordina t es of the sp ectral curv e (parametrized by p ) in the sense of Eynard and Oran tin [17]. Namely , when x and other deformation v ariables (time v ariables of the underlying KP hierarc hy) are fixed to sp ecial v alues, Q and P satisfy a defining equation f ( X , Y ) = 0 of the sp ectral curv e as f ( Q , P ) = 0 . Although the KP and T o da hierarc hies are different in nature, the last remark seems to suggest that one may think of an equation of the form f ( L , ¯ L − 1 ) = 0 30 as the sp ectral curve in the presen t setting. This observ ation is partly sup- p orted by the fact that suc h a curv e is deriv ed as the sp ectral curv e in the random matrix approac h to c = 1 string theory and in terface dynamics [53, 54, 5 5]. Bearing these remarks in mind, let us turn to the issue of solving the generalized string equations (72 ). These equations, lik e (80), are a kind of “nonlinear Riemann-Hilb ert problems”. One can use the metho d deve lop ed for solving (80) [29] to construct a solution of (72) as p ow er series of t and ¯ t . As it turns o ut, (72) can b e treat ed in muc h the same w a y apart from tec hnical complications. 6.2 Decomp osition of equati ons Let us conv ert (72) to the lo garithmic form log( L p − 1 ) = log Q − log ( ¯ L − 1 p ) + β ¯ M , log( ¯ L − 1 p ) = log Q − log ( L p − 1 ) + β M . (85) Since L p − 1 and ¯ L − 1 p are Laurent series of the fo rm L p − 1 = 1 + ∞ X n =1 u n p − n , ¯ L − 1 p = ¯ u 0 + ∞ X n =1 ¯ u n p n with nonzero leading t erms, one can expand the log arithm as log( L p − 1 ) = ∞ X n =1 α n p − n , log( ¯ L − 1 p ) = log ¯ u 0 + ∞ X n =1 ¯ α n p n , where α n = u n + (p o lynomial of u 1 , · · · , u n − 1 ) , ¯ α n = ¯ u − 1 0 ¯ u n + (p o lynomial of ¯ u − 1 0 ¯ u 1 , · · · , ¯ u − 1 0 ¯ u n − 1 ) . W e no w substitute ¯ M = − ∞ X k =1 k ¯ t k ¯ L − k + s + ∞ X n =1 ¯ v n ¯ L n , M = ∞ X k =1 k t k L k + s + ∞ X n =1 v n L − n in (85) and expand b oth hand sides in p o w ers of p . This leads to an infinite set of equations fo r the co efficien ts u n , ¯ u n , v n , ¯ v n of L , M , ¯ L , ¯ M as follow s. 31 Equating the co efficien ts o f p − n , n = 0 , 1 , · · · , in b ot h hand sides of the first equation of ( 8 5) gives the equations 0 = log Q − log ¯ u 0 + β s − β ∞ X k =1 k ¯ t k ( ¯ L − k ) 0 , (86) α n = − β ∞ X k = n k ¯ t k ( ¯ L − k ) − n , n = 1 , 2 , · · · , (87) where ( ¯ L − k ) − n stands for the co efficien t of p − n in ¯ L − k . In t he same w ay , the co efficien ts of p n , n = 0 , 1 , 2 , · · · , in the second equation of (85) giv e the equations log ¯ u 0 = log Q + β s + β ∞ X k =1 k t k ( L k ) 0 , (88) ¯ α n = β ∞ X k = n k t k ( L k ) n , n = 1 , 2 , · · · , (89) where ( L k ) n denotes the co efficien t of p n in L k . Since ( L k ) n = ( ¯ L − k ) − n = 0 for k < n, the range of k in the sums of (8 7) and (89) is limited to k ≥ n . Note that one can use the formal residue notation res X n a n p n dp ! = a − 1 to express  L k  n and  ¯ L − k  − n as ( L k ) n = res  L k p − n d log p  , ( ¯ L − k ) − n = res  ¯ L − k p n d log p  . Suc h an expression turns out to b e useful in the subsequen t consideration. Since v n ’s a nd ¯ v n ’s a r e absen t, (8 6) – (89) are equations for u n ’s a nd ¯ u b ’s only . The remaining pa r t of (85) determine v n ’s and ¯ v n ’s. T o see this, it is more con venie n t to expand (85) in p ow ers of L and ¯ L ra ther than o f p . Extracting the co efficien ts of ¯ L n from the first equation of (72) and those of L − n from the second equation yields the equations 0 = − res  log( ¯ L − 1 p ) ¯ L − n d log ¯ L  + β ¯ v n , (90) 0 = − res  log( L p − 1 ) L n d log L  + β v n (91) 32 for n = 1 , 2 , · · · . Thus v n ’s a nd ¯ v n ’s a r e determined b y L and ¯ L as v n = β − 1 res  log( L p − 1 ) L n d log L  , ¯ v n = β − 1 res  log( ¯ L − 1 p ) ¯ L − n d log ¯ L  . (85) can b e th us decomp osed into the infinite set of equations (86) – (89), (90) and (9 1). The next ta sk is to show that they do hav e a solution. 6.3 Solution of equations As we observ ed ab o v e, one can t hink of (86) – (89) as equations for u n ’s and ¯ u n ’s. W e wan t to construct these functions as p ow er series o f ( t and ¯ t with co efficien ts dep ending on s . If u k ’s and ¯ u k are th us constructed, v k and ¯ v k ’s are determined by (90) and (91). Let us first examine (86) and (88). Since α n = u n + (monomials of higher degrees in u 1 , · · · , u n − 1 ) , ¯ α n = ¯ u − 1 0 ¯ u n + (monomials of higher degrees in ¯ u − 1 0 ¯ u , · · · , ¯ u − 1 0 ¯ u n − 1 ) , u n and ¯ u n , n = 1 , 2 , · · · , show up o n the left hand side of these equations linearly . The r ig h t hand side consists of terms t ha t are m ultiplied b y t k ’s and ¯ t k ’s. Therefore these equations yield a h uge system of recursion relations for the co efficien ts of p ow er series expansion of u n ’s a nd ¯ u n ’s. Note that ¯ u 0 , whic h remains to b e determined, is con tained in the co effi- cien ts of t he p o we r series expansion o f u n and ¯ u n . W e need a nother equation to determine ¯ u 0 . Actually , there are tw o equations (86) and (88) rather than just o ne. This puzzle is resolv ed as follo ws 2 . Lemma 5. If (87) and (89) a r e satisfi e d, then (8 6) and (88) ar e e quivalent, and r e duc es to the e quation log ¯ u 0 = log Q + β s + ∞ X k =1 k α k ¯ α k . (92) Pr o of. Let us use the formal residue notation to express the terms ( L k ) 0 in (86) as ( L k ) 0 = res( L k d log p ) . 2 A similar r esult ho lds for the gener alized string equa tions (80) of c = 1 string theory . This fills a logical ga p left in our previous pap er [29]. 33 Since the identit y 0 = res( L k d log L ) = res  L k p ∂ log L ∂ p d log p  holds for k ≥ 1, this expression of ( L k ) 0 can b e fur t her rewritten as ( L k ) 0 = res  L k (1 − p ∂ log L ∂ p ) d log p  = − res  L k ∂ log ( L p − 1 ) ∂ p dp  . By substituting L p − 1 = α 1 p − 1 + α 2 p − 2 + · · · , the righ t hand side can b e expanded as RHS = α 1 res( L k p − 2 dp ) + 2 α 2 res( L k p − 3 dp ) + · · · = α 1 ( L k ) 1 + 2 α 2 ( L k ) 2 + · · · . Since ( L k ) n = 0 for n > k , this expansion terminates at the k -th term. One can th us obta in the iden tity ( L k ) 0 = α 1 ( L k ) 1 + 2 α 2 ( L k ) 2 + · · · + k α k ( L k ) k . In t he same w a y , one can derive the iden tity ( ¯ L − k ) 0 = ¯ α 1 ( ¯ L − k ) − 1 + 2 ¯ α 2 ( ¯ L − k ) − 2 + · · · + k ¯ α k ( ¯ L − k ) − k . By virtue of these identities , one can rewrite the tw o sums in (86) a nd (88) as ∞ X k =1 k t k ( L k ) 0 = ∞ X k =1 k t k  α 1 ( L k ) 1 + 2 α 2 ( L k ) 2 + · · · + k α k ( L k ) k  = α 1 ∞ X k =1 k t k ( L k ) 1 + 2 α 2 ∞ X k =1 k t k ( L k ) 2 + · · · = β − 1 ∞ X n =1 nα n ¯ α n and ∞ X k =1 k ¯ t k ( ¯ L − k ) 0 = ∞ X k =1 k ¯ t k  ¯ α 1 ( ¯ L − k ) 1 + 2 ¯ α 2 ( ¯ L − k ) 2 + · · · + k ¯ α k ( ¯ L − k ) k  = ¯ α 1 ∞ X k =1 k ¯ t k ( ¯ L − k ) 1 + 2 ¯ α 2 ∞ X k =1 k ¯ t k ( ¯ L − k ) 2 + · · · = − β − 1 ∞ X n =1 n ¯ α n α n . 34 Note that (89) and (87) ha v e b een used to deriv e the last lines. Th us (86) and (88) turn out to reduce to the same equation (92). W e can t h us use (92) in place of (86) and (88). Adding this equation to (87) and (89), w e obta in a full system of equations that determine the p ow er series expansion of u n , ¯ u n and ¯ u 0 recursiv ely . W e can readily see from (92) that ¯ u 0 is a p o we r series of the form log ¯ u 0 = log Q + β s + (terms of p ositiv e o rders in t , ¯ t ) . (93) On the ot her hand, since ( L n ) n = 1 , ( ¯ L − n ) − n = ¯ u n 0 , ( L k ) n = (p olynomial in u 1 , · · · , u k − n ) for k > n, ( ¯ L − k ) − n = ¯ u k 0 × (p olynomial in ¯ u − 1 0 ¯ u 1 , · · · , ¯ u − 1 0 ¯ u k − n ) for k > n, u n and ¯ u n are p o w er series of the form u n = − β n ¯ t n ¯ u n 0 + (terms of higher orders in t , ¯ t ) , ¯ u n = β nt n ¯ u 0 + (terms of hig her orders in t , ¯ t ) . (94) This p ow er series solution of (72) (whic h is unique by construction) is homogeneous j ust lik e the solution of the generalized string equations (80) for c = 1 string theory [29]. This is a consequence of in v a riance of the the string equations under the scaling transformatio ns t n → c − n t n , ¯ t n → c n ¯ t n , s → s, p → cp, u n → c n u n , ¯ u n → c − n ¯ u n , v n → c n v n , ¯ v n → c − n ¯ v n . (95) In summary , w e hav e observ ed the follo wing: Theorem 4. The gener alize d string e quations (72) have a unique s o lution that has p ow er series exp a n sion wi th r esp e ct to ( t , ¯ t ) as shown in (93) and (94). This solution is homo g ene ous with r esp e ct to the sc alin g tr ansform ation (95). 6.4 Solutions at sp ecial v alues of t , ¯ t The foregoing construction of solution simplifies to some exten t when t and ¯ t ta k e sp ecial v alues. O f particular in terest are the following t w o cases: (i) t k ’s a r e free, and ¯ t k ’s are restricted to ¯ t k = ¯ t 1 δ k 1 , (ii) ¯ t k ’s a r e restricted to ¯ t k = ¯ t 1 δ k 1 , and t k ’s a r e free. 35 They amount to restricting the generating function Z [ t , ¯ t ] of do uble Hurwitz n umbers to generating functions o f simple Hurwitz num b ers. Since t hese t w o cases a re essen tially equiv a len t, let us consider (i) only . In the case of (i), (87) implies that α n v a nishes f or n > 1 and that the only no n-v a nishing comp onen t is giv en b y α 1 = u 1 = − β ¯ t 1 ¯ u 0 . Th us L simplifies as L = pe α 1 p − 1 = pe − β ¯ t 1 ¯ u 0 p − 1 , (96) and ( L k ) n can b e written explicitly as ( L k ) n = ( k u 1 ) k − n ( k − n )! = ( − k β ¯ t 1 ¯ u 0 ) k − n ( k − n )! . (97) ¯ u n , n = 1 , 2 , · · · , are thereb y recursiv ely determined b y (8 9 ) as a function of t k ’s a nd ¯ u 0 . ¯ u 0 is determined by (88), whic h now take s an explicit form as log ¯ u 0 = log Q + β s + β ∞ X k =1 k t k ( − k β ¯ t 1 ¯ u 0 ) k k ! . (98) v n and ¯ v n ’s, to o, ha v e more or less explicit fo rm ulae, though w e omit details. This result sho ws a remark able feature. Namely , (96) resem bles the defin- ing equation x = y e y of La mbert’s W-function y = W ( x ) if L − 1 and p − 1 are iden tified with x and y . The so called Lam b ert curv e is defined by this equa- tion on the ( x, y )-plane, and plays a fundamen tal r o le in the recen t studies on Hurwitz num b ers [14, 15, 16, 35, 36]. This ana lo gy b ecomes mor e precise when t k ’s, to o, are sp ecialized to t k = 0, k = 1 , 2 , · · · . In that case, ¯ u 0 is explicitly determined as ¯ u 0 = Qe β s . Moreo ver, (89) implies that ¯ α n v a nishes for all n , hence ¯ L − 1 = ¯ u 0 p − 1 . (99) (96) thereb y t ur ns into the equation L = ¯ u 0 ¯ L e − β ¯ t 1 ¯ L − 1 (100) for L a nd ¯ L . In view o f the remarks in the b eginning of this section, it seems lik ely that this equation can b e iden tified with the sp ectral curve for simple Hurwitz n umbers. 36 Ac kno wledgemen ts This work is partly supp orted by JSPS Gra n ts-in- Aid for Scientific Researc h No. 19 104002, No. 21 540218 and No. 2254 0 186 f r om the Japan So ciet y for the Promo t ion of Science. References [1] A. 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