Non-degenerate solutions of universal Whitham hierarchy

Non-degenerate solutions of univ ersal Whitham hierarc h y Kanehisa T ak asaki Graduate Sc ho ol of Human and En vironmen tal Studies, Ky oto Univ ersit y , Y oshida, Saky o, Ky oto, 606- 8501, Japan T ak ashi T akebe F acult y of Mathematics, State Univ ersit y – Higher Sc ho ol of Economics, V a vilov a Street, 7, Mosco w, 117312 , Russia Lee P eng T eo Departmen t of Applied Mathematics, F acu lty of Engineering, Univ ersit y of Nottingham Mala ysia Campus, Jalan Bro ga, 43500, Semen yih, Selangor D arul Ehsan, Malay sia. Abstract The notion of non-degenerate solutions for the dispersionles s T o da hierarch y is generalized to the univ ers a l Whitham hierar c hy of genus zero with M + 1 marked points. Thes e s o lutions are c hara cterized by a Riemann-Hilb ert pro ble m (generalized string equations) with r e spect to tw o-dimensional canonica l trans formations, and may be thoug h t of as a kind of general solutio ns o f the hier arch y . The Riemann-Hilber t problem con tains M arbitrar y functions H a ( z 0 , z a ), a = 1 , . . . , M , which pla y the role of generating functions of tw o - dimensional canon- ical tra nsformations. The solution of the Riemann- Hilbert problem is describ ed b y p erio d maps on the space of ( M + 1)-tuples ( z α ( p ) : α = 0 , 1 , . . . , M ) of conformal maps from M disks of the Riemann sphere and their complements to the Riemann sphere. The p erio d maps are defined by an infinite num b er of contour integrals that generalize the notion o f harmonic moments. The F -function (free energy) of these solutions is also shown to hav e a contour int eg ral representation. 1 1 In tro duction The u niv ersal Whitham hierarc h y is a un ified framew ork for v arious d is- p ersionless in tegrable systems an d Wh itham mo dulation equations [4]. I n particular, t h e hierarc hy of gen us zero, w hic h is the sub ject of this pap er, is a n atural generaliza tion of th e disp ersionless KP and T o da hierarchies [9]. Therefore it is natural to a sk to what exten t the ric h con ten ts of the disp ersionless KP and T o da hierarc hies can b e generalized to the hierarc hy of gen u s zero. This issue has b een sough t f or since the tur n of the cen tur y when the study of disp ersionlss integ rab le systems ente r ed a new stag e. As reg ard s the problem of sp ecial solutions, the classical “hodograph method” has b een generalized [1, 11] to obtain a class of solutions includin g Kric hev er’s “alge- braic orbits” [4]. Another cla ss of sp ecial solutions (also related to alg ebr aic orbits) ha v e b een studied in the con text of the Virasoro constraints [5, 6] and the large- N limit of m u ltiple orthogonal p olynomials [7]. It s h ould b e stressed that th e structure of infi nitesimal additional symmetries (includin g the Virasoro s y m metries) w as fu lly elucidated by the w ork of th e Madrid group [5, 6]. As in the case of the disp er s ionless KP a n d T o da hierarc hies [9], those symmetries are d eriv ed from a “nonlinear” Riemann-Hilb ert problem (or an equiv alen t ¯ ∂ problem [2, 3]) with resp ect to tw o-dimensional canonical transformations. As regards the R iemann -Hil b ert p roblem itself, h o w ev er, no effectiv e metho d for find ing an exp licit form of solutions is kno wn apart from ve ry sp ecial cases; one has to r esort to a genuine existence th eorem (though it is enough for deriving the infin itesima l symmetries). Moreo ver, the F -function (free en ergy), also kno wn a s the disp ers ionless (logarithm of ) tau fun cti on, h as to b e treated separately in this approac h. Recen tly , one of the p resen t authors reformulate d the Riemann-Hilb ert problem for the disp ersionless T o da hierarc hy in a slight ly different form, and introd uced the n otion of “non-degenerate solutions” for whic h a more effectiv e d escription is a v ailable [13]. A central idea of this r esu lt stems from the work of Wiegmann and Zabro din [14 ] on an in tegrable structure of univ alen t conformal maps in Riemann’s mapp ing theorem. They u sed the harmonic m omen ts of th e domain to inte r pret the conformal maps as a sp ecial solution of the disp ersionless T o da hierarc hy . This result can b e generalized to pairs of conformal maps [12]. The harm onic momen ts are redefined therein as conto ur in tegrals that include the conformal map (or the p air of conformal maps), and sho wn to give a s ystem of local coordinates on the sp ace of pairs of conformal maps. Actually , this amounts to s olving a Riemann-Hilb ert problem (or “string equations”) in a sp ecial case [8]. 2 The metho d of harmonic moments w ere generalized later b y Zabro din to a larger class of solutions of the disp ersionless T o da h ierarc hy [15]. The n otio n of non-degenerate solutions is a r igorous reformulati on of those solutions, whic h thereb y turn out to b e a kind of general (or generic) solutions of the disp ersionless T o da h ierarch y . T h e goal of this pap er is to generalize these results [13] to th e univ ersal Whitham hierarc hy of gen us zero. Let us briefly recall the notio n of non-degenerate solutions of the disp er- sionless T o da hierarc h y . Those solutions are c haracterized by a Riemann- Hilb ert problem of the follo w ing form: Let H ( z , ˜ z ) b e a holomorphic fun c- tion of tw o v ariables defin ed in a suitable domain (n ot sp ecified here), and H z ( z , ˜ z ) and H ˜ z ( z , ˜ z ) denote the d eriv ativ es H z ( z , ˜ z ) = ∂ H ( z , ˜ z ) /∂ z , H ˜ z ( z , ˜ z ) = ∂ H ( z , ˜ z ) /∂ ˜ z . Moreo v er, su pp ose that H ( z , ˜ z ) satisfies the n on- degeneracy condition H z ˜ z ( z , ˜ z ) 6 = 0 . The pr oblem is to find four f u nctions L ( P ) , M ( P ) , ˜ L ( P ) , ˜ M ( P ) of a complex v ariable P with th e follo wing prop erties: (i) L ( P ) and M ( P ) are holomorphic fu nctions in the pun ctured d isk 1 < | P | < ∞ , L ( P ) b eing univ alen t therein, and ha ve a Lauren t expansion of the form L ( P ) = P + ∞ X n =1 u n P − n +1 , M ( P ) = ∞ X n =1 nt n L ( P ) n + t 0 + ∞ X n =1 v n L ( P ) n . (ii) ˜ L ( P ) − 1 and ˜ M ( P ) are h olo morp hic functions in the punctur ed disk 0 < | P | < 1, ˜ L ( P ) b eing un iv alent therein, and ha ve a Laurent exp an- sion of the form ˜ L ( P ) − 1 = ∞ X n =0 ˜ u n P n − 1 ( ˜ u 0 6 = 0) , ˜ M ( P ) = − ∞ X n =1 nt − n ˜ L ( P ) − n + t 0 − ∞ X n =1 v − n ˜ L ( P ) n . (iii) Th ese fu nctio n s can b e analytically con tinued to a n eig hb orho od of the unit circle | P | = 1 and satisfy the functional equations (generalized string equations) M ( P ) = L ( P ) H z ( L ( P ) , ˜ L ( P )) , ˜ M ( P ) = − ˜ L ( P ) H ˜ z ( L ( P ) , ˜ L ( P )) (1) 3 therein. If the equations w = z H z ( z , ˜ z ) , ˜ w = − ˜ z H ˜ z ( z , ˜ z ) can b e solve d for ˜ z , the map ( z , w ) 7→ ( f ( z , w ) , g ( z , w )) = ( ˜ z , ˜ w ) b ecomes a t wo -dimens ional canonical transformation (or sym plect ic map) with resp ect to the symp lect ic form dz ∧ dw z = d ˜ z ∧ d ˜ w ˜ z , the function H ( z , ˜ z ) b eing its “generating function”. It is well kno wn that this is a normal form of ca n onica l transformations in a “general p osition” of the set of all canonical transformations. (1) can b e th us r ewritten as ˜ L ( P ) = f ( L ( P ) , M ( P )) , ˜ M ( P ) = g ( L ( P ) , M ( P )) . (2) This is a Riemann-Hilb ert problem of the standard form that c haracterizes the Lax and Or lo v-Sch ulman functions of the disp ersionless T o da h ierarch y [9]. The aforemen tioned remark on canonical transformations w ith gener- ating functions imply that the non-d egenerate solutions are indeed general solutions of th e disp ersionless T o da hierarc hy . An adv anta ge of (1) o ver (2) is that it is “solv able” in the follo wing sense. The generalized string equations (1) can b e con v erted to the infin ite system of equations nt n = 1 2 π i I | P | =1 H z ( L ( P ) , ˜ L ( P )) L ( P ) − n d L ( P ) , nt − n = 1 2 π i I | P | =1 H ˜ z ( L ( P ) , ˜ L ( P )) ˜ L ( P ) n d ˜ L ( P ) , t 0 = 1 2 π i I | P | =1 H z ( L ( P ) , ˜ L ( P )) d L ( P ) = − 1 2 π i I | P | =1 H ˜ z ( L ( P ) , ˜ L ( P )) d ˜ L ( P ) , (3) and v n = 1 2 π i I | P | =1 H z ( L ( P ) , ˜ L ( P )) L ( P ) n d L ( P ) , v − n = 1 2 π i I | P | =1 H ˜ z ( L ( P ) , ˜ L ( P )) ˜ L ( P ) − n d ˜ L ( P ) (4) 4 for n = 1 , 2 , . . . . Note that the con tour integ rals are analogues of har- monic momen ts; in the terminology of geometry , they are a kind of “p erio d in tegrals”. A fun damen tal fact [13] is that the fir st set (3) of these p erio d in tegrals giv e a sys te m of lo cal co ordinates on the sp ac e o f the p ai r s ( L , ˜ L ) of conformal m ap s . This implies that the “p erio d map” ( L , ˜ L ) 7→ ( t n : n ∈ Z ) is (lo cally) inv ertible, and the in verse map and the second set (4) of p erio d in tegrals giv e a (unique) solution of the Riemann-Hilb ert p r oblem. Remark- ably , the F -function, to o, tu rns out to ha ve a con tour in tegral representa tion [13]. In the language of the universal Whitham hierarch y of genus zero, the disp ersionless T o da hierarch y amounts to the case with t wo “marke d p oints”. The general ( M +1)-p oin t hierarc hy is form ulated by M +1 pairs ( z α ( p ) , ζ α ( p )), α = 0 , 1 , . . . , M , of Lax and Or lo v-Sch u lman functions. In the t wo- p oint ( M = 1) case, these fun cti ons are connected with the Lax and O r lo v- Sc hulman functions of the d isp ers io n less T o da h ierarc hy as z 0 ( p ) = L ( P ) , z 1 ( p ) = ˜ L ( P ) − 1 , ζ 0 ( p ) = M ( P ) L ( P ) − 1 , ζ 1 ( p ) = − ˜ M ( P ) ˜ L ( P ) , where the co ordinates p and P of the Riemann sphere in b oth hierarc hies are related as p = P + u 1 . Th u s the marked p oin ts P = ∞ , 0 of the disp ersionless T o da h ierarc hy cor- resp ond to the marked p oin ts p = ∞ , u 1 of the un iv ers al Wh ith am hierarc h y . Bearing this in terpretation of the disp ersionless T o da hierarc hy in mind, w e turn to th e M + 1-p oin t case. This pap er is organized as follo ws. In S ection 2, we review the f unda- men tal structur e of th e unive r s al Whitham hierarc hy of genus zero. Buildin g blo c ks of the hierarc h y , such as the Lax and Orlo v-Sch ulmann functions, the S -fu n ctio ns , the F -function and the generalized Gr unsky co efficie nts, are in tro duced in detail. F or tec hnical reasons, the d efi nition of the F -function in our previous w ork [10, 11] is sligh tly m odified here, th ou gh this is not a serious problem. In Section 3, we formula te the Riemann-Hilb ert pr oblem that defines non-degenerate solutions. The b asic setup is p aral lel to the form ulation by the Madrid group [5, 6]. Our generalized string equations ha ve M arbitrary functions H a ( z 0 , z a ), a = 1 , . . . , M , as functional data. As in the case of the disp ersionless T o da hierarc hy , these f unctions pla y the role of generating functions of tw o-dimens ional canonical transform at ions. In Section 4, w e generalize the p erio d in tegrals (3) and (4) to the space Z 5 of ( M + 1)-t u ples ( z α ( p ) : α = 0 , 1 , . . . , M ) of conform al maps, and show that a half of them giv e a system of lo cal co ordinates on Z . T his justi- fies the definition of non-degenerate solutions. S ect ion 5 is an intermediate step to w ard s the construction of the F -function. W e present h ere a con- tour integ r al representa tion of the p oten tials φ a , a = 1 , . . . , M , that sh o w up in the Lau r en t expansions of the S -fu nctions. These φ -fun ctio ns are used in Section 6 for th e constru ction of the F -function. As in the case of the disp ersionless T o da hierarc hy , we defin e a set of auxiliary functions J a, 1 ( z 0 , z a ) , J a, 2 ( z 0 , z a ), a = 1 , . . . , M . T hese functions are used to exp r ess the F -function in terms of con tour in tegrals. In Section 7, we illustrate the construction of n on-dege n erate solutions in a few sp ecial cases that amoun t to the examples studied for the disp ersionless T o da hierarc hy [13]. A cknow le dgements This w ork is partly su pp orted by Gran t-in-Aid for Scien tific Researc h No. 1910400 2, 1954017 9 and No. 21540 218 f rom the Japan So ciet y for the Promotionof S cience . TT is partly supp orted by th e gran t of th e S tat e Univ ersit y – Higher Sc ho ol of Economics, Russia, for the Individu al Researc h Pro j ect 09-0 1-0047 (2009). 2 Building blo c ks of univ ersal Whitham hierarc h y In this secti on w e review essen tial facts on the univ ersal Whitham hierarc hy of genus zero necessary f or our later d iscussion, follo wing our previous w ork [10, 11] 1 . Th e notations are mostly th e same as [10, 11], except th at , afte r th e notation of the recen t work [7] of the Madrid group, Greek indices α, β , . . . range o ver 0 , 1 , . . . , M and Latin indices a, b, . . . o ver 1 , . . . , M . Lax functions The Lax fu nctions z α ( p ), α = 0 , 1 , . . . , M , are f unctions with Laurent expansions of the form z 0 ( p ) = p + ∞ X j =2 u 0 j p − j +1 , z a ( p ) = r a p − q a + ∞ X j =1 u aj ( p − q a ) j − 1 ( a = 1 , . . . , M ) , (5) 1 The auth ors of [10] sincerely apologize numerous typographical errors in th e pro ofs in it, but the statemen ts there are correct. The only differences from [10] are t h e defin ition of the F - function (29) and, consequ en tly , c hanges of several sig natu res in, e.g., ( 32 ). 6 in a n eig hb orhoo d of p = ∞ and p = q a , resp ectiv ely . The co efficien ts u αj ( r a = u a 0 ) and the cen ters q a are d ynamical v ariables. T o consider a Riemann-Hilb ert problem [5, 6 ], w e c ho ose a set of disjoin t p ositiv ely orien ted simple closed curve s C 1 , . . . , C M that encircle q 1 , . . . , q M coun ter- clockwise, and assume that the Lauren t expansion of z a ( p ) conv erges in the inside D a of C a and that the Laur en t expansion of z 0 ( p ) con verge s in a neigh b orho o d of p = ∞ and can b e analytica lly con tin ued, as a holomo r p hic function, to th e outside C r ( D 1 ∪ · · · ∪ D M ) of D a ’s. Lax equations The hierarch y has M + 1 series of time evo lutions with time v ariables t 0 n , n = 1 , 2 , . . . and t an , a = 1 , . . . , M , n = 0 , 1 , 2 , . . . . T he time ev olutions of the Lax functions are defin ed by the Lax equations ∂ αn z β ( p ) = { Ω αn ( p ) , z β ( p ) } , ∂ αn = ∂ /∂ t αn , (6) with resp ect to the Poisson brac k et { f , g } = ∂ f ∂ p ∂ g ∂ t 01 − ∂ f ∂ t 01 ∂ g ∂ p . (7) The Hamiltonians Ω αn ( p ) are defined as Ω 0 n ( p ) =  z 0 ( p ) n  (0 , ≥ 0) , Ω an ( p ) =  z a ( p ) n  ( a,< 0) ( n = 1 , 2 , . . . ) , Ω a 0 ( p ) = − log( p − q a ) , (8) where ( ) (0 , ≥ 0) denotes th e pro jection to non-negativ e p ow ers of p , and ( ) ( a,< 0) the pro j ec tion to negativ e p o w ers of p − q a . In other words, z 0 ( p ) n = Ω 0 n ( p ) + O ( p − 1 ) ( p → ∞ ) , z a ( p ) n = Ω an ( p ) + O (1) ( p → q a ) (9) for n ≥ 1. Ω αn ( p ) satisfies th e d isp ersionless Zakh arov-Shabat equations ∂ β m Ω αn ( p ) − ∂ αn Ω β m ( p ) + { Ω αn ( p ) , Ω β m ( p ) } = 0 . (10) As p oin ted out in [5 ], the dressin g functions of the u niv ersal Whith am hierarc hy ha v e the follo wing form: ϕ 0 ( p ) = ∞ X j =1 ϕ 0 ,j p − j , ϕ a ( p ) = ∞ X j =0 ϕ a,j ( p − q (0) a ) j , (11) z 0 ( p ) = e ad ϕ 0 ( p ) p, z a ( p ) = e ad ϕ a ( p ) ( p − q (0) a ) − 1 . The follo wing is due to [5], Theorem3. 7 Prop osition 2.1. If ( z α ( p ) : α = 0 , 1 , . . . , M ) is a solution of the universal Whitham hier ar chy, then ther e exists dr essing functions ϕ α ( p ) of the form (11) , such that z 0 ( p ) = e ad ϕ 0 ( p ) p, z a ( p ) = e ad ϕ a ( p ) ( p − q (0) a ) − 1 , (12) and ∇ αn ϕ β = ˜ Ω αn,β , (13) wher e ˜ Ω αn,β =            Ω αn − δ α 0 δ n 1 z β ( p ) − 1 ( α 6 = β and ( β 6 = 0 or n 6 = 0)) , Ω α 0 + log z 0 ( p ) ( α 6 = 0 and β = 0 and n = 0) , Ω αn − z α ( p ) n ( α = β and n 6 = 0) , Ω α 0 − log z α ( p ) ( α = β 6 = 0 and n = 0) , (14) and ∇ αn is the right lo garithmic derivative (cf. [5] App endix A, [9] App endix A) define d by ∇ αn ψ = ∞ X n =0 (ad ψ ) n ( n + 1)! ∂ αn ψ . (15) In the ab o ve , q (0) a , a = 1 , . . . , M , are arbitrary non-dyn amica l v ariables. Without loss of generalit y , w e set q (0) a = 0 henceforth. Orlo v-Sc hulman functions The O rlo v-Sch ulman functions ζ α ( p ), α = 0 , 1 , . . . , M are L auren t series of the form ζ 0 ( p ) = ∞ X n =1 nt 0 n z 0 ( p ) n − 1 + t 00 z 0 ( p ) + ∞ X n =1 z 0 ( p ) − n − 1 v 0 n , ζ a ( p ) = ∞ X n =1 nt an z a ( p ) n − 1 + t a 0 z a ( p ) + ∞ X n =1 z a ( p ) − n − 1 v an , (16) where t 00 = − M X a =1 t a 0 . They satisfy the Lax equations ∂ αn ζ β ( p ) = { Ω αn ( p ) , ζ β ( p ) } (17) 8 and the canonical P oisson commutatio n relation { z α ( p ) , ζ α ( p ) } = 1 . (18) In terms of the dressing functions, ζ α are giv en b y ζ 0 ( p ) = e ad ϕ 0 ( p ) ∞ X n =1 nt 0 n p n − 1 + t 00 p ! , ζ a ( p ) = e ad ϕ a ( p ) ∞ X n =1 nt an p − n +1 + t a 0 p − t 01 p 2 ! . The canonical Poi sson comm utation relation (18 ) is a direct consequence of the defin itio n and the Lax equations (17) follo w from (14 ). S -functions Th e S -fun ctions S α ( p ), α = 0 , 1 , . . . , M , are defined as p o- ten tials of 1-forms as d S α ( p ) = θ + ζ α ( p ) dz α ( p ) , (19) where θ = ∞ X n =1 Ω 0 n ( p ) dt 0 n + M X a =1 ∞ X n =0 Ω an ( p ) dt an . They ha ve Laurent expansions of the form S 0 ( p ) = ∞ X n =1 t 0 n z 0 ( p ) n + t 00 log z 0 ( p ) − ∞ X n =1 z 0 ( p ) − n n v 0 n , S a ( p ) = ∞ X n =1 t an z a ( p ) n + t a 0 log z a ( p ) + φ a − ∞ X n =1 z a ( p ) − n n v an . (20) Implications of S -functions Let us defi n e S α ( z ), α = 0 , 1 , . . . , M , as S 0 ( z ) = ∞ X n =1 t 0 n z n + t 00 log z − ∞ X n =1 z − n n v 0 n , S a ( z ) = ∞ X n =1 t an z n + t a 0 log z + φ a − ∞ X n =1 z − n n v an . (21) S α ( p ) can b e thereby exp ressed as S 0 ( p ) = S 0 ( z 0 ( p )) , S a ( p ) = S a ( z a ( p )) . 9 Moreo v er, the d efining equations of S α ( p ) imply the equations ζ α ( p ) = S ′ α ( z α ( p )) , where the p rime d enote s th e deriv ativ e with resp ect to z , and Ω αn ( p ) = ∂ αn S β ( z ) | z = z β ( p ) , β = 0 , 1 , . . . , M . The former is just a restatemen t of th e Laurent expansion of ζ α ( p ). The latter implies that Ω αn ( p ) can b e written in several differen t forms as Ω 0 n ( p ) =            z 0 ( p ) n − ∞ X m =1 z 0 ( p ) − m m ∂ 0 n v 0 m , ∂ 0 n φ b − ∞ X m =1 z b ( p ) − m m ∂ 0 n v bm , b = 1 , . . . , M (22) Ω an ( p ) =            − ∞ X m =1 z 0 ( p ) − m m ∂ an v 0 m , δ ab z b ( p ) n + ∂ an φ b − ∞ X m =1 z b ( p ) − m m ∂ an v bm , b = 1 , . . . , M (23) for n = 1 , 2 , . . . , and Ω a 0 ( p ) =            − log z 0 ( p ) − ∞ X m =1 z 0 ( p ) − m m ∂ a 0 v 0 m , δ ab log z b ( p ) + ∂ a 0 φ b − ∞ X m =1 z b ( p ) − m m ∂ a 0 v bm , b = 1 , . . . , M . (24) In particular, s in ce Ω 01 ( p ) = p , we h a ve the ident ities p = z 0 ( p ) − ∞ X m =1 z 0 ( p ) − m m ∂ 01 v 0 m , p = ∂ 01 φ b − ∞ X m =1 z b ( p ) − m m ∂ 01 v bm , b = 1 , . . . , M , (25) whic h imply that the in v erse fu nctions p = p 0 ( z ) and p = p b ( z ) of z = z 0 ( p ) and z = z b ( p ) are give n explicitly by p 0 ( z ) = z − ∞ X m =1 z − m m ∂ 01 v 0 m = ∂ 01 S 0 ( z ) , p b ( z ) = ∂ 01 φ b − ∞ X m =1 z − m m ∂ 01 v bm = ∂ 01 S b ( z ) . (26) 10 Consequent ly , q a = ∂ 01 φ a , r a = − ∂ 01 v a 1 . (27) Substituting p = p β ( z ) in ∂ αn S β ( z ) | z = z β ( p ) = Ω αn ( p ) leads to the Hamilton-Jac obi equations ∂ αn S β ( z ) = Ω αn ( ∂ 01 S β ( z )) . (28) F - function The F -function is d efined by th e equation ∂ 0 n F = v 0 n , ∂ an F = v an , n = 1 , 2 , . . . , ∂ a 0 F = − φ a + a − 1 X b =1 t b 0 log( − 1) , a = 1 , . . . , M . (29) The last part con taining log ( − 1) is sligh tly different from the d efinition of the Madrid group [6 , 5] and the previous pap er [10] of the fir st t wo authors, but this is due to arb itrariness of the F -function. With th e F -function, the S -fu n ctio ns can b e written as S 0 ( z ) = ∞ X n =1 t 0 n z n + t 00 log z − D 0 ( z ) F, S a ( z ) = ∞ X n =1 t an z n + t a 0 log z + φ a − D a ( z ) F, (30) where D 0 ( z ) and D a ( z ) denote the follo w ing differen tial op erato rs : D 0 ( z ) = ∞ X n =1 z − n n ∂ 0 n , D a ( z ) = ∞ X n =1 z − n n ∂ an . Generalized F ab er p olynomials and Grunsky co efficien ts Th e Hamil- tonians Ω αn ( p ) of the Lax equations can also b e c haracterized by the gen- erating fu nctio n s log p 0 ( z ) − q z = − ∞ X n =1 z − n n Ω 0 n ( q ) , log q − p a ( z ) q − q a = − ∞ X n =1 z − n n Ω an ( q ) . (31) 11 The left hand sides of these identiti es are u n derstoo d to b e rewritten log p 0 ( z ) − q z = log p 0 ( z ) z + log  1 − q p 0 ( z )  and log q − p a ( z ) q − q a = log  1 − p a ( z ) − q a q − q a  and expanded to p o w er series of q and ( q − q a ) − 1 , resp ectiv ely . The generalized Grun sky co efficie nts b ambn = b bnam are defined b y the generating fun ctio ns log p 0 ( z ) − p 0 ( w ) z − w = − ∞ X m,n =1 z − m w − n b 0 m 0 n , log p 0 ( z ) − p a ( w ) z = − ∞ X m =1 ∞ X n =0 z − m w − n b 0 man , log z w ( p a ( z ) − p a ( w )) w − z = − ∞ X m,n =0 z − m w − n b aman , log p a ( z ) − p b ( w ) ǫ ab = − ∞ X m,n =0 z − m w − n b ambn ( a 6 = b ) . (32) They are related to the F -function as ˆ ∂ αm ˆ ∂ β n F = − b αmβ n ( α, β = 0 , 1 , . . . , N ) , (33) where ǫ ab = ( +1 ( a ≤ b ) − 1 ( a > b ) , ˆ ∂ αn = ( 1 n ∂ αn ( n 6 = 0) , ∂ α 0 ( n = 0) . 3 Riemann-Hilb ert problem and non-degenerate solutions F ollo wing the w ork of the Madr id group [5 , 6], we now formulate a Riemann- Hilb ert p r oblem. Cho ose a set of p ositiv ely orien ted simple closed curve s C 1 , . . . , C M and let D 1 , . . . , D M denote their inside domains. The Riemann- Hilb ert data consist of M pairs ( f a , g a ), a = 1 , . . . , M , of h olo morp hic func- tions f a = f a ( p, t 01 ), g a = g a ( p, t 01 ) of p, t 01 (defined in a suitable d omai n ) 12 that satisfy the conditions { f a , g a } = ∂ f a ∂ p ∂ g a ∂ t 01 − ∂ f a ∂ t 01 ∂ g a ∂ p = 1 , (34) th us defin ing tw o-dimensional canonical transformations. Th e problem is to seek M + 1 pairs ( z α ( p ) , ζ α ( p )), α = 0 , 1 , . . . , M , of fun cti ons of p and t = { t 0 n : n = 1 , 2 , . . . } ∪ { t an : a = 1 , . . . , M , n = 0 , 1 , 2 , . . . } that satisfy the follo wing conditions: (i) z 0 ( p ) and ζ 0 ( p ) are h olo morp hic fu n ctio ns on C r ( D 1 ∪ . . . ∪ D M ), z 0 ( p ) is univ alent ther ein (in particular, z ′ 0 ( p ) do es n ot v anish) and, as p → ∞ , z 0 ( p ) = p + O ( p − 1 ) , ζ 0 ( p ) = ∞ X n =1 nt 0 n z 0 ( p ) n − 1 + t 00 z 0 ( p ) + O ( p − 2 ) . (35) (ii) z a ( p ) and ζ a ( p ) are holomorphic functions on D a punctured at a p oi nt q a ∈ D a , z − 1 a ( p ) is un iv alen t on D a and, as p → q a , z a ( p ) = r a p − q a + O (1) , ζ a ( p ) = ∞ X n =1 nt an z a ( p ) n − 1 + t a 0 z a ( p ) + O (( p − q a ) 2 ) . (36) q a and r a are fun ct ions of th e time v ariables to b e thus determined. (iii) F or a = 1 , . . . , M , the four functions z 0 ( p ) , ζ 0 ( p ) , z a ( p ) , ζ a ( p ) can b e analyticall y con tinued to a neigh b orho o d of C a and satisfy the func- tional equations z a ( p ) = f a ( z 0 ( p ) , ζ 0 ( p )) , ζ a ( p ) = g a ( z 0 ( p ) , ζ 0 ( p )) (37) therein. F unctions z α ( p ) satisfying ab o ve conditions are solutions of the unive r - sal Whitham hierarc hy and ζ α ( p )’s are corresp onding Orlo v-S ch ulman fu nc- tions, as is p ro ved in [6], Theorem 1. Note that formally we can pr o ve th e conv erse. Namely there exist Riemann-Hilb ert d ata for ea ch solutio n of the un iv er s al Whith am hierarhcy . 13 Prop osition 3.1. L et ( z α ( p ) : α = 0 , 1 , . . . , M ) b e a solution of the uni- versal Whitham hier ar chy, and ( ζ α ( p ) : α = 0 , 1 , . . . , M ) the c orr esp onding Orlov-Schulman functions. F or a = 1 , . . . , M , ther e exist functions f a ( p, t 01 ) and g a ( p, t 01 ) such that z a = f a ( z 0 , ζ 0 ) , ζ a = g a ( z 0 , ζ 0 ) , (38) and { f a ( p, t 01 ) , g a ( p, t 01 ) } = 1 . Pr o of. This is the same as Prop ositions 4 and 5 of [5], b ut let us prov e it here in our language as in [9]. Given a solution ( z α ( p ) : α = 0 , 1 , . . . , M ) of the unive rs al Whitham hierarc hy , construct the dressing functions ϕ α ( p ) as giv en by Prop osition 2.1. (Recall that we hav e pu t q (0) a = 0.) F or any α , let ˜ f α ( p, t 01 ) = exp  − ad ϕ α ( ˜ t = 0)  p, ˜ g α ( p, t 01 ) = exp  − ad ϕ α ( ˜ t = 0)  t 01 where ˜ t = t r { t 01 } . Notice that z 0 ( p, ˜ t = 0) = exp  ad ϕ 0 ( ˜ t = 0)  p, ζ 0 ( p, ˜ t = 0) = exp  ad ϕ 0 ( ˜ t = 0)  t 01 , z − 1 a ( p, ˜ t = 0) = exp  ad ϕ a ( ˜ t = 0)  p, ( − z 2 a ζ a )( p, ˜ t = 0) = exp  ad ϕ a ( ˜ t = 0)  t 01 . Therefore, ˜ f a  z − 1 a ( p, ˜ t = 0) , ( − z 2 a ζ a )( p, ˜ t = 0)  = ˜ f 0  z 0 ( p, ˜ t = 0) , ζ 0 ( p, ˜ t = 0)  = p , ˜ g a  z − 1 a ( p, ˜ t = 0) , ( − z 2 a ζ a )( p, ˜ t = 0)  = ˜ g 0  z 0 ( p, ˜ t = 0) , ζ 0 ( p, ˜ t = 0)  = t 01 for an y a . No w ∂ ∂ t β n ˜ f 0 ( z 0 , ζ 0 ) = n Ω β n , ˜ f 0 ( z 0 , ζ 0 ) o , ∂ ∂ t β n ˜ f a  z − 1 a , − z 2 a ζ a  = n Ω β n , ˜ f a  z − 1 a , − z 2 a ζ a  o , 14 and similarly for ˜ g 0 ( z 0 , ζ 0 ) and ˜ g a  z − 1 a , − z 2 a ζ a  . Therefore, ∂ ∂ t β n ˜ f 0 ( z 0 , ζ 0 )     ˜ t =0 = ∂ ∂ t β n ˜ f a  z − 1 a , − z 2 a ζ a      ˜ t =0 , ∂ ∂ t β n ˜ g 0 ( z 0 , ζ 0 )     ˜ t =0 = ∂ ∂ t β n ˜ g a  z − 1 a , − z 2 a ζ a      ˜ t =0 . In the s ame wa y , one can show that ∂ ∂ t β k n k . . . ∂ ∂ t β 1 n 1 ˜ f 0 ( z 0 , ζ 0 )     ˜ t =0 = ∂ ∂ t β k n k . . . ∂ ∂ t β 1 n 1 ˜ f a  z − 1 a , − z 2 a ζ a      ˜ t =0 , ∂ ∂ t β k n k . . . ∂ ∂ t β 1 n 1 ˜ g 0 ( z 0 , ζ 0 )     ˜ t =0 = ∂ ∂ t β k n k . . . ∂ ∂ t β 1 n 1 ˜ g a  z − 1 a , − z 2 a ζ a      ˜ t =0 . These sho w that ˜ f 0 ( z 0 , ζ 0 ) = ˜ f a  z − 1 a , − z 2 a ζ a  ˜ g 0 ( z 0 , ζ 0 ) = ˜ g a  z − 1 a , − z 2 a ζ a  . Notice that by definition, n ˜ f 0 ( p, t 01 ) , ˜ g 0 ( p, t 01 ) o = 1 , n ˜ f a ( p, t 01 ) , ˜ g a ( p, t 01 ) o = 1 . One can solve the equations ˜ f 0 ( p, t 01 ) = ˜ f a  ˜ p − 1 , − ˜ p 2 ˜ t 01  , ˜ g 0 ( p, t 01 ) = ˜ g a  ˜ p − 1 , − ˜ p 2 ˜ t 01  , for ˜ p and ˜ t 01 , whic h gives ˜ p = f a ( p, t 01 ) , ˜ t 01 = g a ( p, t 01 ) . This implies (38 ). It is also straigh tforwa rd to show that { f a , g a } = 1. W e no w sp ecialize the Riemann-Hilb ert p r oblem to the case where th e canonical transf ormatio ns are defi n ed by generating functions H a ( z 0 , z a ), a = 1 , . . . , M . The generating fu nctions are assumed to satisfy the non- degeneracy conditions H a,z 0 z a ( z 0 , z a ) 6 = 0 . (39) Accordingly , the f unctional equations (37) connecting the four fu nctio n s z 0 ( p ) , ζ 0 ( p ) , z a ( p ) , ζ a ( p ) are conv erted to the generalized string equations ζ 0 ( p ) = H a,z 0 ( z 0 ( p ) , z a ( p )) , ζ a ( p ) = − H a,z a ( z 0 ( p ) , z a ( p )) . (40) 15 Existence of suc h generating functions H a under th e assum ption of non- degeneracy of f a ’s: ∂ f a ∂ t 01 6 = 0 , (41) can b e p ro ved as in [13], § 3.4. In fact, under the assumption (41), w e can solv e the equation z a = f a ( z 0 , ζ 0 ) to obtain ζ 0 = A a ( z 0 , z a ) . Equiv alently , z a = f a ( z 0 , A a ( z 0 , z a )) . (42) Define B a ( z 0 , z a ) so that ζ a = B a ( z 0 , z a ) = g a ( z 0 , A a ( z 0 , z a )) . (43) Differen tiating (42) with resp ect to z 0 and z a and (43) with resp ect to z 0 , w e fi nd th at 0 = ∂ f a ∂ z 0 + ∂ f a ∂ ζ 0 ∂ A a ∂ z 0 = ⇒ ∂ A a ∂ z 0 = − ∂ f a ∂ z 0 ∂ f a ∂ ζ 0 1 = ∂ f a ∂ ζ 0 ∂ A a ∂ z a = ⇒ ∂ A a ∂ z a = 1 ∂ f a ∂ ζ 0 , ∂ B a ∂ z 0 = ∂ g a ∂ z 0 + ∂ g a ∂ ζ 0 ∂ A a ∂ z 0 = ⇒ ∂ B a ∂ z 0 = ∂ g a ∂ z 0 − ∂ g a ∂ ζ 0 ∂ f a ∂ z 0 ∂ f a ∂ ζ 0 = − 1 ∂ f a ∂ ζ 0 . Hence w e ha ve ∂ z a A a = − ∂ z 0 B a , w hic h imp lies that there exists H a ( z 0 , z a ) satisfying (40) and (39). Our goa l in the follo wing is to solve the generalized string equat ions (40) in the language of geometry of the sp ace Z := { ( z α ( p ) : α = 0 , . . . , M ) | prop erties of z α ( p )’s in (i), (ii) } of the M + 1-tuple of functions z α ( p ), α = 0 , 1 , . . . , M . This enables us to und erstand the u niv ersal Whitham hierarch y as a system of inte grable comm uting flo ws on Z , ju st as ac hiev ed in the case of the disp ersionless T o da hierarch y [13]. 16 T o this end, w e define the functions t 0 n , t 00 , v 0 n ( n = 1 , 2 , . . . ) and t an , t a 0 , v an ( a = 1 , . . . , M , n = 1 , 2 , . . . ) on Z as nt 0 n = M X a =1 1 2 π i I C a H a,z 0 ( z 0 ( p ) , z a ( p )) z 0 ( p ) − n dz 0 ( p ) , t 00 = M X a =1 1 2 π i I C a H a,z 0 ( z 0 ( p ) , z a ( p )) dz 0 ( p ) , v 0 n = M X a =1 1 2 π i I C a H a,z 0 ( z 0 ( p ) , z a ( p )) z 0 ( p ) n dz 0 ( p ) (44) and nt an = 1 2 π i I C a H a,z a ( z 0 ( p ) , z a ( p )) z a ( p ) − n dz a ( p ) , t a 0 = 1 2 π i I C a H a,z a ( z 0 ( p ) , z a ( p )) dz a ( p ) , v an = 1 2 π i I C a H a,z a ( z 0 ( p ) , z a ( p )) z a ( p ) n dz a ( p ) . (45) This is jus t a restatement of th e str ing equations (40). t 00 and t a 0 ’s are automatica lly constrained as t 00 = − ∞ X a =1 t a 0 . The cont our in tegrals on the righ t hand side of (44) are deriv ed b y con tin u - ously deform in g a sim p le closed curv e C ∞ encircling p = ∞ and separating it from all D a ’s. Notice that s ince z a ( q a ) = ∞ , z a ( p ) maps the inside of D a on to the outside of z a ( C a ). Therefore, I C a z a ( p ) m z ′ a ( p ) dp = − δ m, − 1 . In the next section, w e shall r ec onstr uct ∂ αn ’s as globally defined ve ctor fields on Z , and show that t αn ’s ma y b e th ough t of as “du al” (lo cal) coor- dinates on Z with resp ect to these v ector fields. T his is the same geometric situation as obs er ved in the case of the disp ersionless T o da h ierarc hy [13 ]. The univ ersal Whitham hierarc hy is thus r ealized as a system of comm u ting flo ws on Z . This geometric setting can b e cast into the usual setting in the t space b y the inv er s e of the p erio d map ( z α ( p ) : α = 0 , 1 , . . . , M ) 7→ t . The 17 functions z α ( p ) and ζ α ( p ) on Z are pulled bac k b y this in ve rs e p erio d map to b ecome a solution of the s tring equations (40), hence a solution of th e unive rs al Whitham hierarc hy . The S -fu n ctio n (in particular φ a ) and th e F -function, to o, can b e pri- marily defined as a f unction on Z , then pulled b ac k to the t space. W e shall discussed this issu e in later sections. 4 Construction of v ector fields ∂ αn on Z F ollo wing [13], we r eco ns tr uct ∂ αn ’s as ve ctor fields on Z . Theorem 4.1. If the ve ctor fields ∂ 0 n ( n = 1 , 2 , . . . ) and ∂ an ( a = 1 , . . . , M , n = 0 , 1 , 2 , . . . ) on Z satisfy the e q uation s ∂ αn z b ( p ) z ′ b ( p ) − ∂ αn z 0 ( p ) z ′ 0 ( p ) = Ω ′ αn ( p ) z ′ 0 ( p ) z ′ b ( p ) H b,z 0 z b ( z 0 ( p ) , z b ( p )) (46) on C b for b = 1 , . . . , M , wher e the primes denote the derivatives with r esp e ct to p , then they act on t β m ( m = 0 , 1 , 2 , . . . ) as ∂ αn t β m = δ αβ δ nm (47) and on v β m ( m = 1 , 2 , . . . ) as ∂ αn v β m = ( − nmb αnβ m ( n 6 = 0) − mb α 0 β m ( n = 0) (48) R emark 4.2 . (47) implies that t αn ’s may b e thought of as a sys tem of lo ca l co ordinates on Z . (48) shows that the vec tor fi elds ∂ αn corresp ond to the time ev olutions of the un iv ers al Whitham hierarc hy . R emark 4.3 . ∂ αn z β ( p )’s are uniqu ely determined by (46). Th ough this is an implication of (47) and (48), one can d irectl y confi rm it as follo w s . Let Z b ( p ) , Z 0 ( p ) and W b ( p ) d enote the three terms in (46). Consequentl y , they satisfy the equ ati ons Z b ( p ) − Z 0 ( p ) = W b ( p ) (49) for b = 1 , . . . , M in a neigh b orho o d of C b . As holomorphic fun cti ons, Z b ( p ) , Z 0 ( p ) are extended to D b and CP 1 r ( D 1 ∪ · · · ∪ D M ) resp ectiv ely , and b eha ve as Z b ( p ) = O (1) ( p → q b ) , Z 0 ( p ) = O ( p − 1 ) ( p → ∞ ) . 18 One can d ecomp ose Z 0 ( p ) in CP 1 r ( D 1 ∪ · · · ∪ D M ) as Z 0 ( p ) = M X a =1 Z 0 a ( p ) , Z 0 a ( p ) = 1 2 π i I C a Z 0 ( q ) q − p dq . Z 0 a ( p ) is a holomorphic function in CP 1 r D a , O ( p − 1 ) as p → ∞ , and can b e cont inued to a neigh b orho o d of C a b y deforming the cont our C a in ward. The foregoing equation for Z b ( p ) and Z 0 ( p ) can b e thereby rewritten as   Z b ( p ) − X a 6 = b Z 0 a ( p )   − Z 0 b ( p ) = W b ( p ) . One can consider this equation as splitting W b ( p ) in to a sum of h ol omorp h ic functions W b + ( p ) and W b − ( p ) defined in D b and in C P 1 r D b , resp ectiv ely . In particular, Z 0 b ( p ) = − W b − ( p ) = − 1 2 π i I C b W b ( q ) p − q dq ( p ∈ CP 1 r D b ) , (50) hence Z 0 ( p ) = − M X a =1 1 2 π i I C a W a ( q ) p − q dq ( p ∈ CP 1 r ( D 1 ∪ · · · ∪ D M )) . (51) One can fi n d a similar in tegral formula for Z b ( p ) as well. Pr o of of The or em 4.1 . Let us fi rst consider the action of ∂ αn on t 0 m = M X b =1 1 2 π im I C b H b,z 0 ( z 0 ( p ) , z b ( p )) z 0 ( p ) − m z ′ 0 ( p ) dp. Applying ∂ αn to the integ r an d , w e h a ve the identi ty ∂ αn  H b,z 0 ( z 0 ( p ) , z b ( p )) z 0 ( p ) − m z ′ 0 ( p )  = ∂ ∂ p  H b,z 0 ( z 0 ( p ) , z b ( p )) z 0 ( p ) − m ∂ αn z 0 ( p )  + H b,z 0 z b ( z 0 ( p ) , z b ( p )) z ′ 0 ( p ) z ′ b ( p )  ∂ αn z b ( p ) z ′ b ( p ) − ∂ αn z 0 ( p ) z ′ 0 ( p )  z 0 ( p ) − m = ∂ ∂ p  H b,z 0 ( z 0 ( p ) , z b ( p )) z 0 ( p ) − m ∂ αn z 0 ( p )  + Ω ′ αn ( p ) z 0 ( p ) − m . 19 Note that w e ha ve u sed the assumed equation (46) in the last line. Cons e- quen tly , ∂ αn t 0 m = M X b =1 1 2 π im I C b Ω ′ αn ( p ) z 0 ( p ) − m dp = 1 2 π im I C ∞ Ω ′ αn ( p ) z 0 ( p ) − m dp, b y deforming C b ’s to a simp le closed curve C ∞ encircling p = ∞ . On the other hand, one can ded u ce from (31 ) and the first and the second equations in (32) (and the symmetry of b αnβ m ) the follo win g Laurent expansion of Ω αn ( p )’s with resp ect to z 0 ( p ): Ω 0 n ( p ) = z 0 ( p ) n + ∞ X m =1 nb 0 n 0 m z 0 ( p ) − m , Ω an ( p ) = ∞ X m =1 nb an 0 m z 0 ( p ) − m ( n ≥ 1) , Ω a 0 ( p ) = − log z 0 ( p ) + ∞ X m =1 b a 00 m z 0 ( p ) − m . (52) Inserting the d eriv ativ es Ω ′ 0 n ( p ) = nz 0 ( p ) n − 1 z ′ 0 ( p ) − ∞ X m =1 nmb 0 n 0 m z 0 ( p ) − m − 1 z ′ 0 ( p ) , Ω ′ an ( p ) = − ∞ X m =1 nmb an 0 m z 0 ( p ) − m − 1 z ′ 0 ( p ) ( n ≥ 1) , Ω ′ a 0 ( p ) = − z ′ 0 ( p ) z 0 ( p ) − ∞ X m =1 mb a 00 m z 0 ( p ) − m − 1 z ′ 0 ( p ) in to the con tour integral, we readily obtain (47) for β = 0. In the s ame wa y , the action of ∂ αn on v 0 m = M X b =1 1 2 π i I C b H b,z 0 ( z 0 ( p ) , z b ( p )) z b ( p ) m z ′ b ( p ) dp can b e expressed as ∂ αn v 0 m = 1 2 π i I C ∞ Ω ′ αn ( p ) z 0 ( p ) m dp. 20 This con tour in tegral, to o, can b e ev aluated b y the foregoing Laurent ex- pansions of Ω ′ αn ( p ). W e can th us d er ive (48) for β = 0. Let us n ow consid er the action of ∂ αn on t bm = 1 2 π im I C b H b,z b ( z 0 ( p ) , z b ( p )) z b ( p ) − m z ′ b ( p ) dp, t b 0 = 1 2 π i Z C b H b,z b ( z 0 ( p ) , z b ( p )) z ′ b ( p ) dp, v bm = 1 2 π i Z C b H b,z b ( z 0 ( p ) , z b ( p )) z b ( p ) m z ′ b ( p ) dp As in the p revious case, w e can d educe that ∂ αn t bm = − 1 2 π im I C b Ω ′ αn ( p ) z b ( w ) − m dp, ∂ αn t b 0 = − 1 2 π i I C b Ω ′ αn ( p ) dp, ∂ αn v bm = − 1 2 π i I C b Ω ′ αn ( p ) z b ( p ) m dp, (53) W e can n o w use the follo wing Laur ent expans io n of Ω αn ( p )’s with resp ect to z b ( p ) derived fr om (31) and the second, third and fourth equations of (32): Ω 0 n ( p ) = ∞ X m =0 nb 0 nbm z b ( p ) − m , Ω an ( p ) = δ ab z b ( p ) n + ∞ X m =0 nb anbm z b ( p ) − m , ( n ≥ 1) , Ω a 0 ( p ) = ( − log ǫ ba + P ∞ m =0 b a 0 bm z b ( p ) − m ( b 6 = a ) , log z a ( p ) + P ∞ m =0 b a 0 am z a ( p ) − m ( b = a ) . (54) Inserting their deriv ativ es Ω ′ 0 n ( p ) = − ∞ X m =0 nmb 0 nbm z b ( p ) − m − 1 z ′ b ( p ) , Ω ′ an ( p ) = δ ab nz b ( p ) n − 1 z ′ b ( p ) − ∞ X m =0 nmb anbm z b ( p ) − m − 1 z ′ b ( p ) ( n ≥ 1) , Ω ′ a 0 ( p ) = δ ab z ′ b ( p ) z b ( p ) − ∞ X m =0 mb a 0 bm z b ( p ) − m − 1 z ′ b ( p ) in to the con tour integrals (53), w e can confirm the remaining parts of (47 ) and (48). Th is completes the pro of of the theorem. 21 5 Construction of φ a ’s W e constru ct the Phi functions φ a , a = 1 , . . . , M , as follo ws: φ a = M X b =1 t b 0 b a 0 b 0 + M X γ =0 ∞ X m =1 mt γ m b a 0 γ m + a − 1 X b =1 t b 0 log( − 1) − 1 2 π i M X b =1 I C b H b ( z 0 ( p ) , z b ( p )) p − q a dp. (55) Prop osition 5.1. The function φ a define d by (55) satisfies ∂ β n φ a = ( nb a 0 β n , ( n 6 = 0) b a 0 β 0 + log ǫ aβ , ( n = 0) . (56) Pr o of. ∂ β n  H b ( z 0 ( p ) , z b ( p )) p − q a  = H b,z 0 ( z 0 ( p ) , z b ( p )) p − q a ∂ β n z 0 ( p ) + H b,z b ( z 0 ( p ) , z b ( p )) p − q a ∂ β n z b ( p ) + H b ( z 0 ( p ) , z b ( p )) ( p − q a ) 2 ∂ β n q a . Therefore, ∂ β n φ a = T 1 ( β , n ) + T 2 ( β , n ) + ( nb a 0 β n , ( n 6 = 0) b a 0 β 0 + log ǫ aβ , ( n = 0) , where T 1 ( β , n ) = M X b =1 t b 0 ∂ β n b a 0 b 0 + M X γ =0 ∞ X m =1 mt γ m ∂ β n b γ ma 0 = ∞ X m =1 mt 0 m ∂ β n b 0 ma 0 + M X b =1 t b 0 ∂ β n b a 0 b 0 + ∞ X m =1 mt bm ∂ β n b bma 0 ! , and T 2 ( β , n ) = − 1 2 π i M X b =1 I C b  H b,z 0 ( z 0 ( p ) , z b ( p )) p − q a ∂ β n z 0 ( p ) + H b,z b ( z 0 ( p ) , z b ( p )) p − q a ∂ β n z b ( p ) + H b ( z 0 ( p ) , z b ( p )) ( p − q a ) 2 ∂ β n q a  dp. 22 The goal is to sho w that T 1 ( β , n ) + T 2 ( β , n ) = 0 for all ( β , n ). By the definition of t 0 m , w e h a ve ∞ X m =1 mt 0 m ∂ β n b 0 ma 0 = 1 2 π i M X b =1 I C b H b,z 0 ( z 0 ( p ) , z b ( p )) z ′ 0 ( p ) ∞ X m =1 ∂ β n b 0 ma 0 z 0 ( p ) − m ! dp. Differen tiating − log ( p − q a ) = Ω a 0 ( p ) = − log z 0 ( p ) + ∞ X m =1 b 0 ma 0 z 0 ( p ) − m with resp ect to t β n , w e h a ve 1 p − q a ∂ β n q a = ∞ X m =1 ∂ β n b 0 ma 0 z 0 ( p ) − m − 1 z 0 ( p ) + ∞ X m =1 mb 0 ma 0 z 0 ( p ) − m − 1 ! ∂ β n z 0 ( p ) = ∞ X m =1 ∂ β n b 0 ma 0 z 0 ( p ) − m − 1 z ′ 0 ( p ) 1 p − q a ∂ β n z 0 ( p ) . Therefore, ∞ X m =1 ∂ β n b 0 ma 0 z 0 ( p ) − m = 1 p − q a ∂ β n q a + 1 z ′ 0 ( p ) 1 p − q a ∂ β n z 0 ( p ) , and ∞ X m =1 mt 0 m ∂ β n b 0 ma 0 = 1 2 π i M X b =1 I C b H b,z 0 ( z 0 ( p ) , z b ( p )) z ′ 0 ( p ) × ×  1 p − q a ∂ β n q a + 1 z ′ 0 ( p ) 1 p − q a ∂ β n z 0 ( p )  dp = 1 2 π i M X b =1 I C b  H b,z 0 ( z 0 ( p ) , z b ( p )) z ′ 0 ( p ) p − q a ∂ β n q a + H b,z 0 ( z 0 ( p ) , z b ( p )) p − q a ∂ β n z 0 ( p )  dp. 23 In a similar wa y , the definition of t bm giv es t b 0 ∂ β n b a 0 b 0 + ∞ X m =1 mt bm ∂ β n b bma 0 = 1 2 π i I C b H b,z b ( z 0 ( p ) , z b ( p )) z ′ b ( p ) ∞ X m =0 ∂ β n b bma 0 z b ( p ) − m ! dp. Differen tiating − log ( p − q a ) = Ω a 0 ( p ) = δ ab log z b ( p ) + ∞ X m =0 b bma 0 z b ( p ) − m − log ǫ ba with resp ect to t β n and comparing it with Ω ′ a 0 ( p ), we fi n d that 1 p − q a ∂ β n q a = ∞ X m =0 ∂ β n b bma 0 z b ( p ) − m − 1 z ′ b ( p ) 1 p − q a ∂ β n z b ( p ) . Therefore, t b 0 ∂ β n b a 0 b 0 + ∞ X m =1 mt bm ∂ β n b bma 0 = 1 2 π i I C b  H b,z b ( z 0 ( p ) , z b ( p )) z ′ b ( p ) p − q a ∂ β n q a + H b,z b ( z 0 ( p ) , z b ( p )) p − q a ∂ β n z b ( p )  dp. Therefore, T 1 ( β , n ) + T 2 ( β , n ) = 1 2 π i M X b =1 I C b  H b,z 0 ( z 0 ( p ) , z b ( p )) z ′ 0 ( p ) p − q a ∂ β n q a + H b,z b ( z 0 ( p ) , z b ( p )) z ′ b ( p ) p − q a ∂ β n q a − H b ( z 0 ( p ) , z b ( p )) ( p − q a ) 2 ∂ β n q a  dp = 1 2 π i M X b =1 I C b ∂ ∂ p  H b ( z 0 ( p ) , z b ( p )) p − q a  dp × ∂ β n q a = 0 . This completes the pro of. 24 Define v a 0 = − φ a + a − 1 X b =1 t b 0 log( − 1) = − M X b =1 t b 0 b a 0 b 0 − M X γ =0 ∞ X m =1 mt γ m b a 0 γ m + 1 2 π i M X b =1 I C b H b ( z 0 ( p ) , z b ( p )) p − q a dp. (57) Then (56) imp lies that ∂ β n v a 0 = ( − nb a 0 β n , ( n 6 = 0) , − b a 0 β 0 , ( n = 0) . (58) 6 Construction of the free energy F Let J a, 1 ( z 0 , z a ) and J a, 2 ( z 0 , z a ) b e defined so that − ∂ z a J a, 1 ( z 0 , z a ) = ∂ z 0 J a, 2 ( z 0 , z a ) = H a ( z 0 , z a ) H a,z 0 z a ( z 0 , z a ) . (59) W e constru ct the F fun cti on as f ollo ws : F = 1 2 M X a =1 t a 0 v a 0 + 1 2 M X α =0 ∞ X n =1 t αn v αn + 1 8 π i M X a =1 I C a n J a, 1 ( z 0 ( p ) , z a ( p )) z ′ 0 ( p ) + J a, 2 ( z 0 ( p ) , z a ( p )) z ′ a ( p ) o dp. (60) Prop osition 6.1. The F function define d by (60) satisfies ∂ β n F = v β n . 25 Pr o of. A direct computation sho ws that ∂ β n n J a, 1 ( z 0 ( p ) , z a ( p )) z ′ 0 ( p ) + J a, 2 ( z 0 ( p ) , z a ( p )) z ′ a ( p ) o = ∂ ∂ p n J a, 1 ( z 0 ( p ) , z a ( p )) ∂ β n z 0 + J a, 2 ( z 0 ( p ) , z a ( p )) ∂ β n z a o − 2 H a ( z 0 ( p ) , z a ( p ))( ∂ β n z a ( p ) z ′ 0 ( p ) − z ′ a ( p ) ∂ β n z 0 ( p )) = ∂ ∂ p n J a, 1 ( z 0 ( p ) , z a ( p )) ∂ β n z 0 + J a, 2 ( z 0 ( p ) , z a ( p )) ∂ β n z a o − 2 H a ( z 0 ( p ) , z a ( p ))Ω ′ β n ( p ) b y using the definition of the v ector fi eld (46) . Hence, ∂ β n F = I 1 ( β , n ) + I 2 ( β , n ) . where I 1 ( β , n ) = v β n 2 + 1 2 M X a =1 t a 0 ∂ β n v a 0 + 1 2 M X α =0 ∞ X m =1 t αm ∂ β n v αm , I 2 ( β , n ) = − 1 4 π i M X a =1 I C a H a ( z 0 ( p ) , z a ( p ))Ω ′ β n ( p ) dp. No w if n 6 = 0, inte gration b y p arts sh ows th at I 2 ( β , n ) = − 1 4 π i M X a =1 I C a H a ( z 0 ( p ) , z a ( p ))Ω ′ β n ( p ) dp = 1 4 π i M X a =1 I C a H a,z 0 ( z 0 ( p ) , z a ( p )) z ′ 0 ( p )Ω β n ( p ) dp + 1 4 π i M X a =1 I C a H a,z a ( z 0 ( p ) , z a ( p )) z ′ a ( p )Ω β n ( p ) dp. 26 If β = 0, the first equation in (52) and the fir st equation in (54) sho w th at I 2 (0 , n ) = 1 4 π i M X a =1 I C a H a,z 0 ( z 0 ( p ) , z a ( p )) z ′ 0 ( p ) × × z 0 ( p ) n + n ∞ X m =1 b 0 m 0 n z 0 ( p ) − m ! dp + 1 4 π i M X a =1 I C a H a,z a ( z 0 ( p ) , z a ( p )) z ′ a ( p ) × × n ∞ X m =0 b 0 nam z a ( p ) − m ! dp = v 0 n 2 + 1 2 ∞ X m =1 nmb 0 m 0 n t 0 m + 1 2 M X a =1 nb 0 na 0 t a 0 + 1 2 M X a =1 ∞ X m =1 nmb 0 nam t am = v 0 n 2 − 1 2 M X a =1 t a 0 ∂ 0 n v a 0 − 1 2 M X α =0 ∞ X m =1 t αm ∂ 0 n v αm , b y the definition of t αn (44, 45) and actions of ∂ αn on v β m (48, 58). There- fore, ∂ 0 n F = I 1 (0 , n ) + I 2 (0 , n ) = v 0 n . If β = b 6 = 0, n 6 = 0, the second equation in (52) and the second equ at ion in 27 (54) show that I 2 ( b, n ) = 1 4 π i M X a =1 I C a H a,z 0 ( z 0 ( p ) , z a ( p )) z ′ 0 ( p ) × × n ∞ X m =1 b bn 0 m z 0 ( p ) − m ! dp + 1 4 π i M X a =1 I C a H a,z a ( z 0 ( p ) , z a ( p )) z ′ a ( p ) × × δ ab z a ( p ) n + n ∞ X m =0 b bnam z a ( p ) − m ! dp = 1 2 ∞ X m =0 nmb bn 0 m t 0 m + v bn 2 + 1 2 M X a =1 nb bna 0 t a 0 + 1 2 M X a =1 ∞ X m =1 nmb bnam t am = v bn 2 − 1 2 M X a =1 t a 0 ∂ bn v a 0 − 1 2 M X α =0 ∞ X m =1 t αm ∂ bn v αm , again by (44, 45) and (48, 58). T herefore, ∂ bn F = I 1 ( b, n ) + I 2 ( b, n ) = v bn . No w if β = b 6 = 0, n = 0, w e ha v e Ω ′ b 0 ( p ) = − 1 p − q b . Therefore, I 2 ( b, 0) = − 1 4 π i M X a =1 I C a H a ( z 0 ( p ) , z a ( p ))Ω ′ b 0 ( p ) dp = 1 4 π i M X a =1 I C a H a ( z 0 ( p ) , z a ( p )) p − q b dp. 28 On the other hand, I 1 ( b, 0) = v b 0 2 + 1 2 M X a =1 t a 0 ∂ b 0 v a 0 + 1 2 M X α =0 ∞ X m =1 t αm ∂ b 0 v αm = v b 0 2 − 1 2 M X a =1 t a 0 b a 0 b 0 − 1 2 M X α =0 ∞ X m =1 mt αm b αmb 0 , b y (48, 58). Hence, I 1 ( b, 0) + I 2 ( b, 0) = v b 0 2 − 1 2 M X a =1 t a 0 b a 0 b 0 − 1 2 M X α =0 ∞ X m =1 mt αm b αmb 0 + 1 4 π i M X a =1 I C a H a ( z 0 ( p ) , z a ( p )) p − q b dp = v b 0 2 + v b 0 2 = v b 0 b ecause of (57 ). Th is completes the p roof. This prop ositio n and the definition of v a 0 (57) sho w s that the F fu nction indeed satisfies (29) . 7 Sp ecial String Equations In th is section, we consider the sp ecial case where the generating fu nctions H a ( z 0 , z a ), a = 1 , . . . , M , hav e the form H a ( z 0 , z a ) = z ν 0 0 z ν a a , ν 0 , ν a ∈ N , so that the strin g equations (40) b ecome ζ 0 ( p ) = ν 0 z 0 ( p ) ν 0 − 1 z a ( p ) ν a , ζ a ( p ) = − ν a z 0 ( p ) ν 0 z a ( p ) ν a − 1 (61) for p ∈ C a . These string equations were discussed in [6]. ¿F rom (61) and (16), w e ha v e ν 0 z 0 ( p ) ν 0 − 1 z a ( p ) ν a = ∞ X n =1 nt 0 n z 0 ( p ) n − 1 + t 00 z 0 ( p ) + ∞ X n =1 z 0 ( p ) − n − 1 v 0 n , − ν a z 0 ( p ) ν 0 z a ( p ) ν a − 1 = ∞ X n =1 nt an z a ( p ) n − 1 + t a 0 z a ( p ) + ∞ X n =1 z a ( p ) − n − 1 v an (62) 29 for p ∈ C a . The definitions of t αn and v αn (44) and (45) then b ecome nt 0 n = M X a =1 ν 0 2 π i I C a z 0 ( p ) ν 0 − n − 1 z a ( p ) ν a dz 0 ( p ) , t 00 = M X a =1 ν 0 2 π i I C a z 0 ( p ) ν 0 − 1 z a ( p ) ν a dz 0 ( p ) , v 0 n = M X a =1 ν 0 2 π i I C a z 0 ( p ) ν 0 + n − 1 z a ( p ) ν a dz 0 ( p ) (63) and nt an = ν a 2 π i I C a z 0 ( p ) ν 0 z a ( p ) ν a − n − 1 dz a ( p ) , t a 0 = ν a 2 π i I C a z 0 ( p ) ν 0 z a ( p ) ν a − 1 dz a ( p ) , v an = ν a 2 π i I C a z 0 ( p ) ν 0 z a ( p ) ν a + n − 1 dz a ( p ) . (64) The fun ctions J a, 1 ( z 0 , z a ) and J a, 2 ( z 0 , z a ) (59) can b e c h osen to b e J a, 1 ( z 0 , z a ) = − ν 0 2 z 2 ν 0 − 1 0 z 2 ν a a , J a, 2 ( z 0 , z a ) = ν a 2 z 2 ν 0 0 z 2 ν a − 1 a . The free ener gy (60) then b ecomes F = 1 2 M X a =1 t a 0 v a 0 + 1 2 M X α =0 ∞ X n =1 t αn v αn − ν 0 16 π i M X a =1 I C a z 0 ( p ) 2 ν 0 − 1 z a ( p ) 2 ν a dz 0 ( p ) + 1 16 π i M X a =1 I C a ν a z 0 ( p ) 2 ν 0 z a ( p ) 2 ν a − 1 dz a ( p ) (65) 30 Using (62) and (63), we find that − ν 0 16 π i M X a =1 I C a z 0 ( p ) 2 ν 0 − 1 z a ( p ) 2 ν a dz 0 ( p ) = − 1 16 π i M X a =1 I C a  ν 0 z 0 ( p ) ν 0 − 1 z a ( p ) ν a  z 0 ( p ) ν 0 z a ( p ) ν a dz 0 ( p ) = − 1 16 π i M X a =1 I C a ∞ X n =1 nt 0 n z 0 ( p ) n − 1 + t 00 z 0 ( p ) + ∞ X n =1 z 0 ( p ) − n − 1 v 0 n ! × z 0 ( p ) ν 0 z a ( p ) ν a dz 0 ( p ) = − 1 8 ν 0 2 ∞ X n =1 nt 0 n v 0 n + t 2 00 ! . (66) Similarly , one can sho w that 1 16 π i I C a ν a z 0 ( p ) 2 ν 0 z a ( p ) 2 ν a − 1 dz a ( p ) = − 1 8 ν a 2 ∞ X n =1 nt an v an + t 2 a 0 ! . 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