hbar-expansion of KP hierarchy: Recursive construction of solutions

~ -EXP ANSION OF KP HIERAR CHY: RECURSIVE CONSTRU CTION OF SOLUTIONS KANEHISA T AK ASAKI AND T AKASHI T AKEBE Abstract. The ~ -dependent KP hierarch y is a formulat ion of the KP hier- arc hy th at depends on the Planc k constan t ~ and reduces to the disp ersion- less KP hierarch y as ~ → 0. A recursive construction of its solutions on the basis of a Riemann-Hil ber t problem for the pair ( L, M ) of Lax and Or lov- Sc h ulman operators is presen ted. The R iemann-Hilb ert problem is conv erted to a set of recursion relations for the coefficients X n of an ~ -expansion of the oper ator X = X 0 + ~ X 1 + ~ 2 X 2 + · · · for which the dressing op erator W i s expressed in the exponent ial form W = exp( X/ ~ ). Giv en the low est order term X 0 , o ne can solv e the recursion relations to obtain the higher order terms. The wa v e f unction Ψ asso ciated with W turns out to hav e the WKB form Ψ = exp( S/ ~ ), and the coefficients S n of the ~ -expansion S = S 0 + ~ S 1 + ~ 2 S 2 + · · · , too, ar e determined by a s et of recursi on rela- tions. This WKB form is used to show that the asso ciated tau f unction has an ~ -expansion of the for m log τ = ~ − 2 F 0 + ~ − 1 F 1 + F 2 + · · · . 0. Introduction The KP hierar ch y can be co mpletely solved by several metho ds. The most clas - sical metho ds a re based on Grassma nn manifolds [SS], [SW], fermions and vertex op erator s [DJKM] and factorisation o f microdifferential op er ators [Mu]. Unfor- tunately , those methods a re not very suited for a “ q uasi-class ical” ( ~ -dep endent, where ~ is the P lanck cons tant) formulation [TT2] of the KP hierarch y . The ~ - de p endent fo rmulation o f the KP hierar ch y was introduced to study the disp e rsionless KP hier a rch y [KG ], [Kr], [TT1] as a classic a l limit (i.e., the low est order of the ~ - expansion) of the KP hierarch y . This p oint of view turned out to b e very useful for unders tanding v arious features of the disp er s ionless KP hierar ch y such a s Lax equatio ns, Hirota equatio ns, infinite dimensional s ymmetries, etc., in the lig ht of the KP hiera rch y . In this pap er , w e return to the ~ -dep endent KP hierarch y itse lf, and cons ide r all orders o f the ~ -expansion. W e first a ddress the issue of so lving a Riemann-Hilbert pr oblem for the pair ( L, M ) of La x and O rlov-Sch ulman op erator s [OS]. T his is a kind of “quantisation” of a Riemann-Hilb ert pr oblem tha t solves the dispersio nless KP hierar ch y [TT1]. Though the Riema nn-Hilbe rt pr oblem for the full KP hierarc hy w as formulated in our pre vious work [TT2], we did not consider the existence of its solution in a general setting. In this pap er , we settle this issue by an ~ -expansion of the dressing op erator W , which is assumed to have the exp onential form W = exp( X/ ~ ) with an op erator X of nega tive o rder. Roughly sp eaking, the co efficients X n , n = 0 , 1 , 2 , . . . , of the ~ -expansio n of X ar e shown to b e de ter mined recursively fro m the low est order term X 0 (in other words, fro m a solution of the disp er s ionless K P hierarch y). Date : 24 Decem b er 2009; revised on 5 Septem ber 2010. 1 2 KANEHISA T AKASAKI AND T AKASHI T AKEBE W e next conv ert this result to the language of the wa ve function Ψ. This, to o, is to a nswer a problem ov e r lo oked in our previous pap er [TT2]. Namely , given the dressing op e r ator in the exp onential form W = exp( X/ ~ ), we show tha t the asso ciated wa v e function has the WKB for m Ψ = exp( S/ ~ ) with a phase function S expanded into nonnegative p ow ers of ~ . This is genuinely a pr oblem of calculus of micro differential op erator s rather t han that o f the KP hiera rch y . A simplest example such as X = x ( ~ ∂ ) − 1 demonstrates th at this problem is by no means trivial. Bo rrowing an idea from Aoki’s “exp o nential calculus” o f micro differential op erator s [A], we show that dressing op erators of the form W = exp( X/ ~ ) and wa ve functions of the form Ψ = e xp( S/ ~ ) are deter mined from each other by a set of recursion relations for the co e fficient s of their ~ -expansio n. More precisely , we need many auxilia r y quantities other than X and S , fo r which we c a n derive a lar g e s et of r ecursion rela tions. Thus o ur construction is essentially recursive. Consequently , the wa ve function of the s olution of the afor ementioned Riemann-Hilb ert problem, to o, ar e recur sively determined by the ~ -ex pa nsion. Having the ~ -expansion of the w av e function, we can readily derive an ~ -expansio n of the tau function as stated in our previous work [TT2]. This ~ -e x pansion is a generalisa tion of the “genus expansion” of partition functions in string theories and random matrices [D], [Kr], [Mo ], [dFGZ]. This pap er is or ganised as follows. Section 1 is a review of the ~ -dep endent formu- lation of the KP hierarchy . Relev ant Riemann-Hilb ert pr oblems are also r eviewed here. Section 2 presents the r ecursive so lutio n of the Riemann-Hilb ert proble m. A techn ical clue is the Campb ell-Haus dorff formula, deta ils o f which ar e c o llected in Appendix A. The construction o f solution is illustrated for the case of the Ko nt- sevich mo del [AvM] in App endix B . Section 3 deals with the ~ -expansio n of the wa ve function. Aoki’s exp o ne ntial c a lculus is a lso briefly r eviewed here. Sectio n 4 men tions the ~ -e xpansion of the tau function. Section 5 is dev oted to concluding remarks . A cknow le dgements. The authors a re gra teful to Profess or Akihiro Tsuchiy a for drawing our attention to this sub ject and to Pro fessor Ma s atoshi Noumi fo r in- structing how to use Mathematica in algebra of micro differ ent ial op erato r s. Com- putations in App endix B w ere do ne with the aid of one of his Mathematica pro - grammes. This work is partly supp or ted by Grants-in-Aid for Scien tific Research No. 1 9540 179 and No. 2 2540 186 from the Japan So ciety for the Pr omotion o f Sci- ence. TT is partly supp orted by the grant o f the State University – Higher Scho ol of Eco no mics, Russia, fo r the Individual Resear ch Pro ject 09- 01-00 47 (20 09). 1. ~ -dependent KP hierar chy: review In this section w e recall several facts on the KP hierarch y dep ending o n a formal parameter ~ in [T T 2 ], § 1.7 . The ~ -dep endent KP hierar ch y is defined by the Lax repres ent ation (1.1) ~ ∂ L ∂ t n = [ B n , L ] , B n = ( L n ) ≥ 0 , n = 1 , 2 , . . . , where the L ax op er ator L is a micr o differential op erator of the form (1.2) L = ~ ∂ + ∞ X n =1 u n +1 ( ~ , x, t )( ~ ∂ ) − n , ∂ = ∂ ∂ x , ~ -EXP ANSION OF KP 3 and “( ) ≥ 0 ” stands for the pro jection o nt o a differential op erato r dropping nega tive powers of ∂ . The co e fficie nts u n ( ~ , x, t ) of L ar e as s umed to b e for mally regular with resp ect to ~ . This means that they hav e an asymptotic expansion o f the fo rm u n ( ~ , x, t ) = P ∞ m =0 ~ m u ( m ) n ( x, t ) as ~ → 0. W e need tw o kinds of “or der” o f micro differential op erato rs: one is the ordinary order, (1.3) ord  X a n,m ( x, t ) ~ n ∂ m  def = max ( m      X n a n,m ( x, t ) ~ n 6 = 0 ) , and the other is the ~ -order defined by (1.4) ord ~  X a n,m ( x, t ) ~ n ∂ m  def = max { m − n | a n,m ( x, t ) 6 = 0 } . In par ticular, ord ~ ~ = − 1, ord ~ ∂ = 1, or d ~ ~ ∂ = 0. F or example, the condition which we imp osed on the co efficients u n ( ~ , x, t ) can b e res tated as ord ~ ( L ) = 0. The princip al symb ol (re s p. the symb ol of or der l ) of a micr o differential op erator A = P a n,m ( x, t ) ~ n ∂ m with r e s p e ct to the ~ -order is σ ~ ( A ) def = X m − n =ord( A ) a n,m ( x, t ) ξ m (1.5) (resp. σ ~ l ( A ) def = X m − n = l a n,m ( x, t ) ξ m ) . (1.6) When it is cle a r fro m the context, w e s ometimes use σ ~ instead of σ ~ l . R emark 1.1 . This “or der” coincides with the order of an micro differential o p erator if we for mally r eplace ~ with ∂ − 1 t 0 , where t 0 is an extra v ar ia ble. In fact, na ively extending (1.1) to n = 0, we can intro duce the time v ariable t 0 on which nothing depe nds . See also [KR]. As in the usua l KP theory , the Lax op erator L is ex pressed by a dr essing op er ator W : (1.7) L = Ad W ( ~ ∂ ) = W ( ~ ∂ ) W − 1 The dress ing op era tor W should hav e a sp ecific for m: W = exp( ~ − 1 X ( ~ , x, t, ~ ∂ ))( ~ ∂ ) α ( ~ ) / ~ , (1.8) X ( ~ , x, t, ~ ∂ ) = ∞ X k =1 χ k ( ~ , x, t )( ~ ∂ ) − k , (1.9) ord ~ ( X ( ~ , x, t, ~ ∂ )) = o r d ~ α ( ~ ) = 0 , (1.10) and α ( ~ ) is a constant with r e sp ect to x and t . (In [TT2] we did not introduce α , which will b e necessar y in Sec tion 2.) The wave function Ψ( ~ , x, t ; z ) is defined by (1.11) Ψ( ~ , x, t ; z ) = W e ( xz + ζ ( t,z ) ) / ~ , where ζ ( t, z ) = P ∞ n =1 t n z n . It is a solution of linear equations L Ψ = z Ψ , ~ ∂ Ψ ∂ t n = B n Ψ ( n = 1 , 2 , . . . ) , 4 KANEHISA T AKASAKI AND T AKASHI T AKEBE and has the WKB form (3.2), as we shall show in Se c tion 3. Moreov er it is expressed by mea ns of the t au function τ ( ~ , t ) as follows: (1.12) Ψ( ~ , x, t ; z ) = τ ( t + x − ~ [ z − 1 ]) τ ( t ) e ~ − 1 ζ ( t,z ) , where t + x = ( t 1 + x, t 2 , t 3 , . . . ) a nd [ z − 1 ] = (1 / z , 1 / 2 z 2 , 1 / 3 z 3 , . . . ). W e shall study the ~ - expansion of the tau function in Section 4. The O rlov-Schulman op er ator M is defined by (1.13) M = Ad  W exp  ~ − 1 ζ ( t, ~ ∂ )  x = W ∞ X n =1 nt n ( ~ ∂ ) n − 1 + x ! W − 1 where ζ ( t, ~ ∂ ) = P ∞ n =1 t n ( ~ ∂ ) n . It is easy to see that M has a fo r m (1.14) M = ∞ X n =1 nt n L n − 1 + x + α ( ~ ) L − 1 + ∞ X n =1 v n ( ~ , x, t ) L − n − 1 , and satisfies the following pr op erties: • ord ~ ( M ) = 0; • the canonical commutation relation: [ L, M ] = ~ ; • the same La x equa tions as L : (1.15) ~ ∂ M ∂ t n = [ B n , M ] , n = 1 , 2 , . . . . • another linear equation for the wa ve function Ψ: M Ψ = ~ ∂ Ψ ∂ z . R emark 1 .2 . If a n o pe rator M of the for m (1.1 4) satisfies the Lax equations (1.15) and the canonica l commut ation relation [ L , M ] = ~ with the Lax op er ator L of the KP hierarch y , then α ( ~ ) in the expansion (1.14) do es not dep end on any t n nor on x . In fa ct, ex pa nding the canonica l commutation relation, we hav e ~ + ~ ∂ α ∂ x ( ~ ∂ ) − 1 + (low er order ter ms) = ~ , which implies ∂ α ∂ x = 0. Similarly , fro m (1.15) follows ∂ α ∂ t n = 0 with the help of (1.1) and [ L n , M ] = n ~ L n − 1 . The following prop ositio n (Prop os itio n 1.7 .11 of [TT2]) is a “disp ersio nful” coun- terpart of the theorem for the disp ersionles s KP hiera rch y found ea rlier (Pro p o sition 7 of [TT1]; cf. P rop osition 1.4 below). Prop ositi on 1.3. ( i) Supp ose that op er ators f ( ~ , x, ~ ∂ ) , g ( ~ , x, ~ ∂ ) , L and M satisfy the fol lowing c onditions: • ord ~ f = o rd ~ g = 0 , [ f , g ] = ~ ; • L is of the form (1 .2) and M is of the form (1.14) , [ L , M ] = ~ ; • f ( ~ , M , L ) and g ( ~ , M , L ) ar e differ ential op er ators: (1.16) ( f ( ~ , M , L )) < 0 = ( g ( ~ , M , L )) < 0 = 0 , wher e ( ) < 0 is the pr oje ction to t he n e gative or der p art: P < 0 := P − P ≥ 0 . ~ -EXP ANSION OF KP 5 Then L is a solution of the KP hier ar chy (1.1) and M is the c orr esp onding Orlov- Schulman op er ator. (ii) Conversely, for any solution ( L, M ) of the ~ -dep endent K P hier ar chy ther e exists a p air ( f , g ) satisfying t he c onditions in (i). The lea ding term of this system with resp ect to the ~ -or der gives the disp ersion- less KP hier ar chy . N amely , (1.17) L := σ ~ ( L ) = ξ + ∞ X n =1 u 0 ,n +1 ξ − n , ( u 0 ,n +1 := σ ~ ( u n +1 )) satisfies the disp ersionles s Lax t yp e equa tions (1.18) ∂ L ∂ t n = {B n , L} , B n = ( L n ) ≥ 0 , n = 1 , 2 , . . . , where ( ) ≥ 0 is the truncation of L a urent ser ies to its p o lynomial pa rt and { , } is the Poisson brack et defined by (1.19) { a ( x, ξ ) , b ( x, ξ ) } = ∂ a ∂ ξ ∂ b ∂ x − ∂ a ∂ x ∂ b ∂ ξ . The dr essing op eration (1.7) for L b ecomes the following dressing op eration for L : L = exp  ad { , } X 0  exp  ad { , } α 0 log ξ  ξ = exp  ad { , } X 0  ξ , X 0 := σ ~ ( X ) , α 0 := σ ~ ( α ) (1.20) where a d { , } ( f )( g ) := { f , g } . The pr incipal symbo l of the Orlov-Sch ulman op erator is (1.21) M = ∞ X n =1 nt n L n − 1 + x + α 0 L − 1 + ∞ X n =1 v 0 ,n L − n − 1 , v 0 ,n := σ ~ ( v n ) , which is equa l to (1.22) M = exp  ad { , } X 0  exp  ad { , } α 0 log ξ  exp  ad { , } ζ ( t, ξ )  x, where ζ ( t, ξ ) = P ∞ n =1 t n ξ n . The series M sa tisfies the canonical comm utation relation with L , { L , M} = 1 and the Lax type equations: (1.23) ∂ M ∂ t n = {B n , M} , n = 1 , 2 , . . . . The Riemann-Hilb ert t yp e constr uctio n o f the solution is es s entially the s ame as Prop os itio n 1.3. (W e do not need to a ssume the canonica l commutation relatio n {L , M} = 1.) Prop ositi on 1.4. (i) Supp ose that funct ions f 0 ( x, ξ ) , g 0 ( x, ξ ) , L and M satisfy the fol lowing c onditions: • { f 0 , g 0 } = 1 ; • L is of the form (1.17) and M is of the form (1.2 1) • f 0 ( M , L ) and g 0 ( M , L ) do not c ont ain ne gative p owers of ξ (1.24) ( f 0 ( M , L )) < 0 = ( g 0 ( M , L )) < 0 = 0 , wher e ( ) < 0 is the pr oje ct ion to the ne gative de gr e e p art: P < 0 := P − P ≥ 0 . 6 KANEHISA T AKASAKI AND T AKASHI T AKEBE Then L is a solution of the disp ersionless KP hier ar chy (1.1 8) and M is the c orr e- sp onding O r lov-Schulman function. (ii) Conversely, for any solution ( L , M ) of the disp ersionless K P hier ar chy, t her e exists a p air ( f 0 , g 0 ) satisfying the c onditions in (i). If f , g , L a nd M are a s in Prop osition 1 .3, then f 0 = σ ~ ( f ), g 0 = σ ~ ( g ), L = σ ~ ( L ) and M = σ ~ ( M ) satisfy the conditions in Prop os ition 1.4. In other words, ( f , g ) and ( L , M ) are q uantisation of the canonical transforma tions ( f 0 , g 0 ) and ( L , M ) resp ectively . (See, for exa mple, [S] for quantised canonical trans formations.) 2. Recursive constr uction of the dressing opera tor In this sectio n we prov e that the solution of the KP hiera rch y corresp o nding to the qua ntised canonica l transformatio n ( f , g ) is recur sively constructed from its leading term, i.e., the solution of the disp ersio nless KP hiera rch y cor resp onding to the Riemann- Hilber t data ( σ ~ ( f ) , σ ~ ( g )). Given the pair ( f , g ), we hav e to construct the dressing op erator W , or X a nd α in (1.8), such tha t op er ators (2.1) f ( ~ , M , L ) = Ad  W exp  ~ − 1 ζ ( t, ~ ∂ )  f ( ~ , x, ~ ∂ ) g ( ~ , M , L ) = Ad  W exp  ~ − 1 ζ ( t, ~ ∂ )  g ( ~ , x, ~ ∂ ) are b oth differ ential op erator s (cf. Pro p o sition 1 .3 ). Let us expand X a nd α with resp ect to the ~ -order a s follows: X ( ~ , x, t, ~ ∂ ) = ∞ X n =0 ~ n X n ( x, t, ~ ∂ ) , X n ( x, t, ~ ∂ ) = ∞ X k =1 χ n,k ( x, t )( ~ ∂ ) − k , (2.2) α ( ~ ) = ∞ X n =0 ~ n α n , (2.3) where χ n,k and α n do not dep end on ~ and hence χ k in (1.9) is expanded as χ k = P ∞ n =0 ~ n χ n,k . Assume that the solution of the disp ersionles s KP hierar ch y corr esp onding to ( σ ~ ( f ) , σ ~ ( g )) is given. In other words, assume that a symbol X 0 = P ∞ k =1 χ 0 ,k ( x, t ) ξ − k and a constant α 0 are g iven such that σ ~ ( f )( L , M ) = exp  ad { , } X 0  exp  ad { , } α 0 log ξ  exp  ad { , } ζ ( t, ξ )  σ ~ ( f )( x, ξ ) σ ~ ( g )( L , M ) = exp  ad { , } X 0  exp  ad { , } α 0 log ξ  exp  ad { , } ζ ( t, ξ )  σ ~ ( g )( x, ξ ) do not contain negative powers of ξ : (2.4)  σ ~ ( f )( L , M )  < 0 =  σ ~ ( g )( L , M )  < 0 = 0 . (See Pro p o sition 1.4.) W e are to constr uct X n and α n recursively , star ting fr om X 0 and α 0 . F or this purp ose ex pand f ( ~ , L , M ) and g ( ~ , L , M ) in (2.1) as follows: P := Ad  exp( ~ − 1 X ) exp( ~ − 1 α log ~ ∂ )  f t (2.5) = P 0 + ~ P 1 + · · · + ~ k P k + · · · , Q := Ad  exp( ~ − 1 X ) exp( ~ − 1 α log ~ ∂ )  g t (2.6) = Q 0 + ~ Q 1 + · · · + ~ k Q k + · · · , ~ -EXP ANSION OF KP 7 where (2.7) f t := Ad  e ~ − 1 ζ ( t, ~ ∂ )  f , g t := Ad  e ~ − 1 ζ ( t, ~ ∂ )  g , and P i ’s and Q i ’s are o pe r ators o f the for m P i = P i ( x, t, ~ ∂ ) and Q i = Q i ( x, t, ~ ∂ ) and ord ~ P i = o rd ~ Q i = 0 . Supp ose that we have ch osen X 0 , . . . , X i − 1 and α 0 , . . . , α i − 1 so that P 0 , . . . , P i − 1 and Q 0 , . . . , Q i − 1 do not con tain negative p ow- ers of ∂ . If an op era tor X i and a co nstant α i are constructed fr om these given X 0 , . . . , X i − 1 and α 0 , . . . , α i − 1 so that res ulting P i and Q i do not contain negative powers o f ∂ , this pr o cedure gives recur sive construction of X and α in question. W e can construct s uch X i and α i as follows. (Details and meaning sha ll b e explained in the pro of o f Theore m 2.1.): • (Step 0) Assume X 0 , . . . , X i − 1 and α 0 , . . . , α i − 1 are g iven and set (2.8) X ( i − 1) := i − 1 X n =0 ~ n X n , α ( i − 1) := i − 1 X n =0 ~ n α n . • (Step 1) Set P ( i − 1) := Ad  exp ~ − 1 X ( i − 1)  exp  ~ − 1 α ( i − 1) log( ~ ∂ )  f t , (2.9) Q ( i − 1) := Ad  exp ~ − 1 X ( i − 1)  exp  ~ − 1 α ( i − 1) log( ~ ∂ )  g t . (2.10 ) Expand P ( i − 1) and Q ( i − 1) with resp ect to the ~ -order a s P ( i − 1) = P ( i − 1) 0 + ~ P ( i − 1) 1 + · · · + ~ k P ( i − 1) k + · · · (2.11) Q ( i − 1) = Q ( i − 1) 0 + ~ Q ( i − 1) 1 + · · · + ~ k Q ( i − 1) k + · · · . (2.12) (ord ~ P ( i − 1) k = ord ~ Q ( i − 1) k = 0.) • (Step 2 ) Put P 0 := σ ~ ( P ( i − 1) 0 ), Q 0 := σ ~ ( Q ( i − 1) 0 ), P ( i − 1) i := σ ~ ( P ( i − 1) i ), Q ( i − 1) i := σ ~ ( Q ( i − 1) i ) and define a cons tant α i and a series ˜ X i ( x, t, ξ ) = P ∞ k =1 ˜ χ i,k ( x, t ) ξ − k by (2.13) α i log ξ + ˜ X i := Z ξ  ∂ Q 0 ∂ ξ P ( i − 1) i − ∂ P 0 ∂ ξ Q ( i − 1) i  ≤− 1 dξ . The integral cons ta nt of the indefinite integral is fixed so that the rig ht hand s ide agr ees with the left ha nd side. • (Step 3) Define a ser ies X i ( x, t, ξ ) = P ∞ k =1 χ i,k ( x, t ) ξ − k by X i = ˜ X ′ i − 1 2 { σ ~ ( X 0 ) , ˜ X ′ i } + ∞ X p =1 K 2 p (ad { , } ( σ ~ ( X 0 ))) 2 p ˜ X ′ i , ˜ X ′ i := α i log ξ + ˜ X i ( x, ξ ) − exp(ad { , } σ ~ ( X 0 ))( α i log ξ ) . (2.14) Here K 2 p is determined by the generating function (2.15) z e z − 1 = 1 − z 2 + ∞ X p =1 K 2 p z 2 p , or K 2 p = B 2 p / (2 p )!, wher e B 2 p ’s ar e the Bernoulli num b e rs. 8 KANEHISA T AKASAKI AND T AKASHI T AKEBE • (Step 4) The op er ator X i ( x, t, ~ ∂ ) is defined as the op era tor with the pr in- cipal symbol X i : (2.16) X i = ∞ X k =1 χ i,k ( x, t )( ~ ∂ ) − k . The ma in theorem is the following: Theorem 2.1. Assume that X 0 and α 0 satisfy (2.4) and c onstruct X i ’s and α i ’s by t he ab ove pr o c e dur e r e cursively. Then X and α define d by (2.2) satisfy (1.16) . Namely W = exp( X/ ~ )( ~ ∂ ) α/ ~ is a dr essing op er ator of t he ~ - dep endent KP hier- ar chy. The r est of this se ction is the pro of o f Theo rem 2.1 b y induction. Let us denote the “known” pa rt of X and α b y X ( i − 1) and α ( i − 1) as in (2.8) and, as in termediate ob jects, consider P ( i − 1) and Q ( i − 1) defined b y (2.9) and (2.1 0), which ar e expa nded as (2 .11) a nd (2.12). If X a nd α a re ex panded as (2.2) and (2 .3), the dr essing o p erator W = exp( X/ ~ ) exp( α lo g ( ~ ∂ ) / ~ ) is factor ed as follows by the Campb ell-Hausdo rff theorem: (2.17) W = exp  ~ i − 1 ( α i log( ~ ∂ ) + ˜ X i ) + ~ i X >i  × × exp  ~ − 1 X ( i − 1)  exp  ~ − 1 α ( i − 1) log( ~ ∂ )  , where or d ~ ( α i log( ~ ∂ ) + ˜ X i ( x, ~ ∂ )) = 0, ord ~ ( X >i ) ≦ 0 a nd the principal symbol of α i log( ~ ∂ ) + ˜ X i ( x, ~ ∂ ) is defined by (2.18) σ ~ ( α i log( ~ ∂ ) + ˜ X i )( x, ξ ) = ∞ X n =1 (ad { , } σ ~ ( X 0 )) n − 1 n ! σ ~ ( X i ) + exp  ad { , } σ ~ ( X 0 )  ( α i log ξ ) . Note that the only log ter m in (2.18) is α i log ξ and the re st is sum of negative powers o f ξ . The pr incipal sy mbo l of X i is recovered fr o m ˜ X i by the formula σ ~ ( X i ) = σ ~ ( ˜ X ′ i ) − 1 2 { σ ~ ( X 0 ) , σ ~ ( ˜ X ′ i ) } + ∞ X p =1 K 2 p (ad { , } ( σ ~ ( X 0 ))) 2 p σ ~ ( ˜ X ′ i ) , σ ~ ( ˜ X ′ i ) := σ ~ ( ˜ X i )( x, ξ ) − exp(ad { , } σ ~ ( X 0 ))( α i log ξ ) = ∞ X n =1 (ad { , } σ ~ ( X 0 )) n − 1 n ! σ ~ ( X i ) . (2.19) Here co efficients K 2 p are defined by (2.15). This inv ersion r elation is the o rigin of (2.14). (Note that the pr incipal s y mbol determines the ope r ator X i , since it is a homogeneous ter m in the expansion (2.2).) W e prov e formulae (2.17) and (2.19) in Appendix A. The fa c to risation (2.1 7) implies P = Ad  exp  ~ i − 1 ( α i log( ~ ∂ ) + ˜ X i ) + ~ i X >i   P ( i − 1) = P ( i − 1) + ~ i − 1 [( α i log( ~ ∂ ) + ˜ X i ) + ~ X >i , P ( i − 1) ] + (terms of ~ -or der < − i ) . ~ -EXP ANSION OF KP 9 Thu s, substituting the expansion (2.11) in the step 1 , we hav e P = P ( i − 1) 0 + ~ P ( i − 1) 1 + · · · + ~ i P ( i − 1) i + · · · + ~ i − 1 [ α i log( ~ ∂ ) + ˜ X i , P ( i − 1) 0 ] + (terms o f ~ -or der < − i ) . (2.20) Comparing this with the ~ -expansio n of P (2.5), we can expres s P i ’s in ter ms of P ( i − 1) j , ˜ X i and α i as follows: P j = P ( i − 1) j ( j = 0 , . . . , i − 1) , (2.21) σ 0 ( P i ) = σ 0 ( P ( i − 1) i + h − 1 [ α i log( ~ ∂ ) + ˜ X i , P ( i − 1) 0 ]) . (2.22) Similar equatio ns for Q are o btained in the same way . The firs t equations (2 .21) show that the terms of ~ -order greater than − i in (2.5) are already fixed by X 0 , . . . , X i − 1 and α 0 , . . . , α i − 1 , whic h justifies the inductive pr o cedure. That is to s ay , we are assuming that X 0 , . . . , X i − 1 and α 0 , . . . , α i − 1 hav e b een already de- termined so tha t P j = P ( i − 1) j and Q j = Q ( i − 1) j for j = 0 , . . . , i − 1 a re differential op erator s. The op erato r X i and co nstant α i should b e c hosen s o tha t the r ig ht ha nd side of (2.2 2) and the cor r esp onding ex pression for Q ar e differential op erator s. T aking equations P ( i − 1) 0 = P 0 and Q ( i − 1) 0 = Q 0 int o account, we define ˜ P ( i ) i := P ( i − 1) i + ~ − 1 [ α i log( ~ ∂ ) + ˜ X i , P 0 ] , ˜ Q ( i ) i := Q ( i − 1) i + ~ − 1 [ α i log( ~ ∂ ) + ˜ X i , Q 0 ] . (2.23) Then the condition for X i and α i is written in the following form of equations for symbols: (2.24) ( σ ~ 0 ( ˜ P ( i ) i )) ≤− 1 = 0 , ( σ ~ 0 ( ˜ Q ( i ) i )) ≤− 1 = 0 . (The parts of ~ - order less tha n − 1 should b e deter mined in the next step o f the induction.) T o simplify notations, we denote the symbols σ ~ 0 ( ˜ P ( i ) i ), σ ~ 0 ( P ( i − 1) i ) and so on b y the corre sp onding calligra phic letters a s P ( i ) i , P ( i − 1) i etc. By this notation we ca n rewr ite the equations (2 .24) in the following for m: (2.25) ( ˜ P ( i ) i ) ≤− 1 = 0 , ˜ P ( i ) i := P ( i − 1) i + { α i log ξ + ˜ X i , P 0 } , ( ˜ Q ( i ) i ) ≤− 1 = 0 , ˜ Q ( i ) i := Q ( i − 1) i + { α i log ξ + ˜ X i , Q 0 } . The ab ov e definitions of ˜ P ( i ) i and ˜ Q ( i ) i are wr itten in the ma trix for m: (2.26)      ∂ P 0 ∂ x − ∂ P 0 ∂ ξ ∂ Q 0 ∂ x − ∂ Q 0 ∂ ξ           ∂ ∂ ξ ( α i log ξ + ˜ X i ) ∂ ∂ x ( α i log ξ + ˜ X i )      =    ˜ P ( i ) i − P ( i − 1) i ˜ Q ( i ) i − Q ( i − 1) i    . Recall that op era tors P ( i − 1) and Q ( i − 1) are defined by acting a djoint o p er ation to the canonically commuting pa ir ( f , g ) in (2.9), (2.10) and (2.7). Hence they also satisfy the canonical commutation r elation: [ P ( i − 1) , Q ( i − 1) ] = ~ . T he principal symbol of this r elation g ives {P ( i − 1) 0 , Q ( i − 1) 0 } = {P 0 , Q 0 } = 1 , 10 KANEHISA T AKASAKI AND T AKASHI T AKEBE which mea ns that the determinant of the matrix in the left hand side of (2 .26) is equal to 1. Hence its inverse matr ix is easily computed a nd we hav e (2.27)      ∂ ∂ ξ ( α i log ξ + ˜ X i ) ∂ ∂ x ( α i log ξ + ˜ X i )      =      − ∂ Q 0 ∂ ξ ∂ P 0 ∂ ξ − ∂ Q 0 ∂ x ∂ P 0 ∂ x         ˜ P ( i ) i − P ( i − 1) i ˜ Q ( i ) i − Q ( i − 1) i    . W e a re assuming that P 0 and Q 0 do not contain neg ative p owers of ξ and we are searching for α i log ξ + ˜ X i such that ˜ P ( i ) i and ˜ Q ( i ) i are series of ξ without negative powers. Since α i is consta nt with r esp ect to x , the left hand s ide of (2.27) co nt ain only nega tive p owers o f ξ . Thus taking the negative power parts of the b oth hand sides in (2.27), we have (2.28)      ∂ ∂ ξ ( α i log ξ + ˜ X i ) ∂ ∂ x ( α i log ξ + ˜ X i )      =        ∂ Q 0 ∂ ξ P ( i − 1) i − ∂ P 0 ∂ ξ Q ( i − 1) i  ≤− 1  ∂ Q 0 ∂ x P ( i − 1) i − ∂ P 0 ∂ x Q ( i − 1) i  ≤− 1       . This is the equation which determines α i and X i . The system (2.28) is so lv able thanks to Lemma 2 .2 b elow. Hence, integrating the first element o f the rig ht hand side with resp ect to ξ , we o btain α i log ξ + ˜ X i . This is Step 2, (2 .13). In the end, the principa l symbo l of X i is determined by (2.19) or (2 .14) in Step 3 and thus X i is defined as in Step 4 . This completes the construction of X i and α i and the pro of of the theore m. Lemma 2.2. The system (2.28) is c omp atible. Pr o of. W e chec k: ∂ ∂ x  ∂ Q 0 ∂ ξ P ( i − 1) i − ∂ P 0 ∂ ξ Q ( i − 1) i  ≤− 1 = ∂ ∂ ξ  ∂ Q 0 ∂ x P ( i − 1) i − ∂ P 0 ∂ x Q ( i − 1) i  ≤− 1 . (2.29) Since differ e nt iation commutes with truncation of p ower series, condition (2.29) is equiv alent to saying that the nega tive p ower par t of the following is zer o : ∂ ∂ x  ∂ Q 0 ∂ ξ P ( i − 1) i − ∂ P 0 ∂ ξ Q ( i − 1) i  − ∂ ∂ ξ  ∂ Q 0 ∂ x P ( i − 1) i − ∂ P 0 ∂ x Q ( i − 1) i  = ∂ Q 0 ∂ ξ ∂ P ( i − 1) i ∂ x − ∂ P 0 ∂ ξ ∂ Q ( i − 1) i ∂ x − ∂ Q 0 ∂ x ∂ P ( i − 1) i ∂ ξ + ∂ P 0 ∂ x ∂ Q ( i − 1) i ∂ ξ = − {P ( i − 1) i , Q 0 } − {P 0 , Q ( i − 1) i } . (2.30) Defined fro m canonica lly commuting pair ( f , g ) by adjoint a ction (2.9) and (2.1 0), the pair of op er ators ( P ( i − 1) , Q ( i − 1) ) is c anonically commuting: [ P ( i − 1) , Q ( i − 1) ] = ~ . The negative orde r part of this r elation is zero. On the other hand, s ubstitut- ing the expansions P ( i − 1) = P ∞ n =0 ~ n P ( i − 1) n and Q ( i − 1) = P ∞ n =0 ~ n Q ( i − 1) n in this canonical commutation relation and noting that P ( i − 1) j and Q ( i − 1) j ( j = 0 , . . . , i − 1) ~ -EXP ANSION OF KP 11 do not contain negative order part by the induction hypothesis, we hav e 0 = ([ P ( i − 1) , Q ( i − 1) ]) ≤− 1 = [ ~ i P ( i − 1) i , Q ( i − 1) 0 ] + [ P ( i − 1) 0 , ~ i Q ( i − 1) i ] + (terms of ~ -order < − i − 1) . T aking the symbol of ~ -order − i − 1 of this eq ua tion, we hav e 0 = {P ( i − 1) i , Q 0 } + {P 0 , Q ( i − 1) i } , which proves that (2.30) v anishes .  3. Asymptotics o f the w a ve function In this section we prov e tha t the dressing op era to r of the form (3.1) W ( ~ , x, t, ~ ∂ ) = exp( X ( ~ , x, ~ ∂ ) / ~ ) , ord ~ X ≦ 0 , ord X ≦ − 1 , gives a wa ve function of the form Ψ( ~ , x, t ; z ) = W e ( xz + ζ ( t,z ) ) / ~ = exp( S ( ~ , x, t, z ) / ~ ) , ord ~ S ≦ 0 , (3.2) S ( ~ , x, t ; z ) = ∞ X n =0 ~ n S n ( x, t ; z ) + ζ ( t, z ) , ζ ( t, z ) := ∞ X n =1 t n z n , (3.3) and vice versa. Since the time v ariables t n do not play a ny role in this sec tio n, we s et them to zero. As the fa ctor ( ~ ∂ ) α/ ~ in (1.8) b ecomes a constant factor z α/ ~ when it is applied to e xz / ~ , we also o mit it here. Let A ( ~ , x, ~ ∂ ) = P n a n ( ~ , x )( ~ ∂ ) n be a micro differential opera tor. The total symb ol of A is a p ow er ser ies of ξ defined b y (3.4) σ tot ( A )( ~ , x, ξ ) := X n a n ( ~ , x ) ξ n . Actually , this is the facto r which app ears when the o p e rator A is applied to e xz / ~ : (3.5) Ae xz / ~ = σ tot ( A )( ~ , x, z ) e xz / ~ . Using this ter minology , what we show in this section is that a op erato r of the form e X/ ~ has a to tal symbol of the form e S/ ~ and that an op era tor with total symbo l e S/ ~ has a for m e X/ ~ . Exactly sp eaking, the main results in this section are the following tw o pro po sitions. Prop ositi on 3.1. L et X = X ( ~ , x, ~ ∂ ) b e a micr o differ ential op er ator such that ord X = − 1 and or d ~ X = 0 . Th en t he total symb ol of e X/ ~ has such a form as (3.6) σ tot (exp( ~ − 1 X ( ~ , x, ~ ∂ ))) = e S ( ~ ,x , ξ ) / ~ , wher e S ( ~ , x, ξ ) is a p ower series of ξ − 1 without non- n e gative p owers of ξ and has an ~ -exp ansion (3.7) S ( ~ , x, ξ ) = ∞ X n =0 ~ n S n ( x, ξ ) . Mor e over, the c o efficient S n is determine d by X 0 , . . . , X n in the ~ - exp ansion (2.2) of X = P ∞ n =0 ~ n X n . Explicitly , S n is determined as follows: • (Step 0 ) Ass ume that X 0 , . . . , X n are g iven. Let X i ( x, ξ ) be the total symbol σ tot ( X i ( x, ~ ∂ )). 12 KANEHISA T AKASAKI AND T AKASHI T AKEBE • (Step 1) Define Y ( l ) k,m ( x, y , ξ , η ) and S ( l ) ( x, ξ ) by the following recursion re - lations: Y ( l ) k, − 1 = 0 (3.8) S (0) m = 0 , (3.9) Y ( l ) 0 ,m ( x, y , ξ , η ) = δ l, 0 X m ( x, ξ ) (3.10) for l ≧ 0, m = 0 , . . . , n , (3.11) Y ( l ) k +1 ,m ( x, y , ξ , η ) = 1 k + 1     ∂ ξ ∂ y Y ( l ) k,m − 1 ( x, y , ξ , η ) + X 0 ≤ l ′ ≤ l − 1 0 ≤ m ′ ≤ m ∂ ξ Y ( l ′ ) k,m ′ ( x, y , ξ , η ) ∂ y S ( l − l ′ ) m − m ′ ( y , η )     for k ≧ 0, and (3.12) S ( l +1) m ( x, ξ ) = 1 l + 1 l + m X k =0 Y ( l ) k,m ( x, x, ξ , ξ ) . (W e shall prove that Y ( l ) k,m = 0 if k > l + m .) Schematically this pro cedur e go es a s follows: Y ( l ) 0 , 0 = δ l, 0 X 0 Y ( l ) 0 , 1 = δ l, 0 X 1 Y ( l ) 0 , 2 = δ l, 0 X 2 + ց + ց + Y ( l ) k, − 1 = 0 → Y ( l ) k, 0 → Y ( l ) k, 1 → Y ( l ) k, 2 · · · ↓ ր ↓ ր ↓ S ( l +1) 0 S ( l +1) 1 S ( l +1) 2 • (Step 2) S n ( x, ξ ) = P ∞ l =1 S ( l ) n ( x, ξ ). (The sum makes sense as a p ow er series of ξ .) Prop ositi on 3. 2. L et S = P ∞ n =0 ~ n S n b e a p ower series of ξ − 1 without non- ne gative p owers of ξ . Then t her e exists a micr o differ en t ial op er ator X ( ~ , x, ~ ∂ ) such that o rd X ≦ − 1 , o rd ~ X ≦ 0 and (3.13) σ tot (exp( ~ − 1 X ( ~ , x, ~ ∂ ))) = e S ( ~ ,x , ξ ) / ~ . Mor e over, the c o efficient X n ( x, ξ ) in the ~ -exp ansion X = P ∞ n =0 ~ n X n of the total symb ol X = X ( ~ , x, ξ ) is determine d by S 0 , . . . , S n in the ~ -exp ansion of S . Explicit pro cedure is as follows: • (Step 0) Assume that S 0 , . . . , S n are given. Expand them into homo g eneous terms w ith respect to p ow ers of ξ : S n ( x, ξ ) = P ∞ j =1 S n,j ( x, ξ ), where S n,j is a term of deg ree − j . • (Step 1) Define Y ( l ) k,n,j ( x, y , ξ , η ) as follows: Y ( l ) k, − 1 ,j ( x, y , ξ , η ) = 0 , (3.14) Y ( l ) k,m, 1 ( x, y , ξ , η ) = δ l, 0 δ k, 0 S m, 1 ( x, ξ ) (3.15) for m = 0 , . . . , n , k ≧ 0, l ≧ 0 a nd (3.16) Y ( l ) 0 ,m,j = 0 ~ -EXP ANSION OF KP 13 for m = 0 , . . . , n , l > 0, j ≧ 1. F or other ( l , k , m, j ), ( l , k ) 6 = (0 , 0), Y ( l ) k,m,j are deter mined by the recursio n relation: (3.17) Y ( l ) k +1 ,m,j ( x, y , ξ , η ) = 1 k + 1 ∂ ξ ∂ y Y ( l ) k,m − 1 ,j − 1 ( x, y , ξ , η )+ + X 0 ≤ l ′ ≤ l − 1 0 ≤ j ′ ≤ j − 1 , 0 ≤ m ′ ≤ m 0 ≤ k ′′ ≤ j − j ′ − l + l ′ 1 l − l ′ ∂ ξ Y ( l ′ ) k,m ′ ,j ′ ( x, y , ξ , η ) ∂ y Y ( l − l ′ − 1) k ′′ ,m − m ′ ,j − j ′ − 1 ( x, x, ξ , ξ ) ! . The r emaining Y (0) 0 ,m,j is determined by: (3.18) Y (0) 0 ,m,j ( x, y , ξ , η ) = S m,j ( x, ξ ) − X ( l,k ) 6 =(0 , 0) l,k ≥ 0 ,l + k ≤ j 1 l + 1 Y ( l ) k,m,j ( x, x, ξ , ξ ) . (W e s hall show that Y ( l ) k,m,j = 0 for l + k > j .) Schematically this pro cedure go es a s follows: Y ( l ) k,m, 1 = δ l, 0 δ k, 0 S m, 1 ↓ Y ( l ′ ) k ′ ,m ′ , 1 ( m ′ < m ) → Y ( l ) k,m, 2 ( k , l 6 = 0) → Y (0) 0 ,m, 2 ← S m, 2 ↓ ւ Y ( l ′ ) k ′ ,m ′ , 1 , Y ( l ′ ) k ′ ,m ′ , 2 ( m ′ < m ) → Y ( l ) k,m, 3 ( k , l 6 = 0) → Y (0) 0 ,m, 3 ← S m, 3 . . . • (Step 2 ) X n ( x, ξ ) = P ∞ j =1 Y (0) 0 ,n,j ( x, x, ξ , ξ ). (The infinite sum is the homo- geneous expansion in terms of p ow ers of ξ .) Combining these prop ositions with the results in Section 2, we can, in pr inciple, make a recur sion formula for S n ( n = 0 , 1 , 2 , . . . ) of the wa ve function of the solutio n of the K P hierarchy corres p o nding to the quantised canonical transformation ( f , g ) as follows: let S 0 , . . . , S i − 1 be given. (1) By Pro p o sition 3.2 we hav e X 0 , . . . , X i − 1 . (2) W e hav e a r ecursion for mula fo r X i by Theo rem 2.1. (3) Pro p o sition 3.1 gives a for m ula for S i . If w e take the factor ( ~ ∂ ) α/ ~ int o account, this pro ces s be c omes a little bit compli- cated, but essentially the same. The r e st of this section is devoted to the pro of of Pro po sition 3.1 a nd Prop o si- tion 3.2. If o ne w ould co nsider that “ the principal symbol is obtained just b y replacing ~ ∂ with ξ ”, the statements of the ab ove prop ositions might seem obvious, but since the multiplication in the definition of (3.19) e X/ ~ = ∞ X n =0 X ( ~ , x, ~ ∂ ) n ~ n n ! 14 KANEHISA T AKASAKI AND T AKASHI T AKEBE is non-co mmu tative, while the mult iplication of tota l symbo ls in the ser ies (3.20) e S/ ~ = ∞ X n =0 S ( ~ , x, ξ ) n ~ n n ! is commutativ e , it needs to b e prov ed. In fact, c o mputation of the simplest example of X = x ( ~ ∂ ) − 1 would s how how complicated the formula can b e: σ tot ( e x ( ~ ∂ ) − 1 / ~ ) = ∞ X n =0 1 n ! ~ n σ tot ( X n ) =1 + 1 1! ~ xξ − 1 + 1 2! ~ 2 ( x 2 ξ − 2 − ~ xξ − 3 ) + 1 3! ~ 3 ( x 3 ξ − 3 − 3 ~ x 2 ξ − 4 + 3 ~ 2 xξ − 5 ) + 1 4! ~ 4 ( x 4 ξ − 4 − 6 ~ x 3 ξ − 5 + 15 ~ 2 x 2 ξ − 6 − 15 ~ 3 xξ − 7 ) + · · · = exp 1 ~  xξ − 1 − xξ − 3 2 + xξ − 5 2 − 5 xξ − 7 8 + · · ·  . It is not obvious, a t least to authors, why there is no mor e negative p ow ers o f ~ in the last expre ssion, whic h can b e obtained by ca lculating the logarithm of the previous expr ession. T o avoid confusion, the commutative multiplication of total symbols a ( ~ , x, ξ ) and b ( ~ , x, ξ ) a s p ow er series is de no ted by a ( ~ , x, ξ ) b ( ~ , x, ξ ) a nd the non-commutativ e m ultiplication corres po nding to the op era tor pro duct is denoted by a ( ~ , x, ξ ) ◦ b ( ~ , x, ξ ). Recall that the latter multiplication is expr essed (or defined) as follows: a ( ~ , x, ξ ) ◦ b ( ~ , x, ξ ) = e ~ ∂ ξ ∂ y a ( ~ , x, ξ ) b ( ~ , y , η ) | y = x,η = ξ = ∞ X n =0 ~ n n ! ∂ n ξ a ( ~ , x, ξ ) ∂ n y b ( ~ , y , η ) | y = x,η = ξ . (3.21) (See, for example, [S], [A] o r [KR].) The order of an op er ator cor resp onding to symbol a ( ~ , x, ξ ) is denoted by ord ξ a ( ~ , x, ξ ), which is the same as the order of a ( ~ , x, ξ ) as a p ower series o f ξ . The ~ -order is the same as tha t of the o per ator: ord ~ x = or d ~ ξ = 0 , o rd ~ ~ = − 1. The main idea of pro of of prop ositio ns is due to Ao ki [A], whe r e exp onential cal- culus of pse udo differen tial op erator s is consider ed. He conside r ed a nalytic symbols of exp onential t yp e , while our symbols are for mal ones . Therefore we hav e o nly to confirm that those symbols make sense as for mal ser ies. First, we pr ove the following lemma. Lemma 3. 3. L et a ( ~ , x, ξ ) , b ( ~ , x, ξ ) , p ( ~ , x, ξ ) and q ( ~ , x, ξ ) b e symb ols s uch that ord ξ a ( ~ , x, ξ ) = M , ord ~ a ( ~ , x, ξ ) = 0 , o r d ξ b ( ~ , x, ξ ) = N , ord ~ b ( ~ , x, ξ ) = 0 , ord ξ p ( ~ , x, ξ ) = ord ξ q ( ~ , x, ξ ) = 0 , ord ~ p ( ~ , x, ξ ) = or d ~ q ( ~ , x, ξ ) = 0 . Then ther e exist symb ols c ( ~ , x, ξ ) ( or d ξ c ( ~ , x, ξ ) = N + M , ord ~ c ( ~ , x, ξ ) = 0 ) and r ( ~ , x, ξ ) ( o r d ξ r ( ~ , x, ξ ) = 0 , ord ~ r ( ~ , x, ξ ) = 0 ) such that (3.22)  a ( ~ , x, ξ ) e p ( ~ ,x,ξ ) / ~  ◦  b ( ~ , x, ξ ) e q ( ~ , x,ξ ) / ~  = c ( ~ , x, ξ ) e r ( ~ ,x,ξ ) / ~ . In the pro of of Pro p o sition 3.1 and P rop ositio n 3.2, we use the co ns truction of c and r in the pro o f of Lemma 3.3. ~ -EXP ANSION OF KP 15 Pr o of. F o llowing [A ], we introduce a par ameter t and co nsider (3.23) π ( t ) = π ( t ; ~ , x, y , ξ , η ) := e ~ t∂ ξ ∂ y a ( ~ , x, ξ ) b ( ~ , y , η ) e  p ( ~ ,x,ξ )+ q ( ~ ,y ,η )  / ~ . If we se t t = 1, y = x and η = ξ , this r educes to the op era tor pro duct of ( 3.21). The series π ( t ) satisfies a differential equa tion with resp ect to t : (3.24) ∂ t π = ~ ∂ ξ ∂ y π , π (0) = a ( ~ , x, ξ ) b ( ~ , y , η ) e  p ( ~ ,x,ξ )+ q ( ~ ,y ,η )  / ~ . W e construct its solution in the following form: (3.25) π ( t ) = ψ ( t ) e w ( t ) / ~ , ψ ( t ) = ψ ( t ; ~ , x, y , ξ , η ) = ∞ X n =0 ψ n t n , w ( t ) = w ( t ; ~ , x, y , ξ , η ) = ∞ X k =0 w k t k . Later we set t = 1 and prove that ψ (1) and w (1) is meaningful as a forma l p ow er series of ξ a nd η . The differential e q uation (3.2 4) is rewritten as ∂ ψ ∂ t + ~ − 1 ψ ∂ w ∂ t = ~ ∂ ξ ∂ y ψ + ∂ ξ ψ ∂ y w + ∂ y ψ ∂ ξ w + ψ  ∂ ξ ∂ y w + ~ − 1 ∂ ξ w∂ y w  . (3.26) Hence it is sufficient to construct ψ ( t ) = ψ ( t ; ~ , x, y , ξ , η ) a nd w ( t ) = w ( t ; ~ , x, y , ξ , η ) which sa tisfy ∂ w ∂ t = ~ ∂ ξ ∂ y w + ∂ ξ w∂ y w, (3.27) ∂ ψ ∂ t = ~ ∂ ξ ∂ y ψ + ∂ ξ ψ ∂ y w + ∂ y ψ ∂ ξ w. (3.28) (This is a sufficient co ndition but not a neces sary condition for π = ψ e w to b e a solution of (3.24). The solutio n of (3.24) is unique, but ψ and w satisfying (3.2 6) are not unique at all.) T o b eg in with, we so lve (3.27) a nd determine w ( t ). Expanding it as w ( t ) = P ∞ k =0 w k t k , we hav e a recursion r elation and the initial co ndition w k +1 = 1 k + 1 ~ ∂ ξ ∂ y w k + k X ν =0 ∂ ξ w ν ∂ y w k − ν ! , w 0 = p ( x, ξ ) + q ( y , η ) , (3.29) which determine w k = w k ( ~ , x, y , ξ , η ) inductiv ely . Note that ord ξ w 0 ≦ 0 and ∂ ξ low er s the or de r by one, which implies (3.30) ord ξ w k ≦ − k . (Here ord ξ denotes the order with re s pe ct to ξ and η .) This shows that w (1) = P ∞ k =0 w k makes sens e as a formal se r ies of ξ and η . Moreo ver it is o bvious that w k and w (1) are formally r egular with resp ect to ~ . 16 KANEHISA T AKASAKI AND T AKASHI T AKEBE As a next s tep, w e expand ψ ( t ) a s ψ ( t ) = P ∞ k =0 ψ k t k and r ewrite (3.28) into a recursion r elation and the initial co ndition: ψ k +1 = 1 k + 1 ~ ∂ ξ ∂ y ψ k + k X ν =0 ( ∂ ξ ψ ν ∂ y w k − ν + ∂ y ψ ν ∂ ξ w k − ν ) ! , ψ 0 = a ( x, ξ ) b ( y , η ) . (3.31) In this case w e have estima te of the o rder of the terms: (3.32) ord ξ ψ k ≦ N + M − k , which s hows that the inifinite sum ψ (1) = P ∞ k =0 ψ k makes s e nse. The r egularity of ψ k and ψ (1) is also obvious. Thu s, we hav e constructed π ( t ) = π ( t ; ~ , x, y , ξ , η ) = ψ ( t ; ~ , x, y , ξ , η ) e w ( t ; ~ ,x,y ,ξ ,η ) , which is meaningful also at t = 1. Hence the pro duct a ( ~ , x, ξ ) ◦ b ( ~ , x, ξ ) = π (1; ~ , x, x, ξ , ξ ) is ex pressed in the form c ( ~ , x, ξ ) e r ( ~ ,x,ξ ) / ~ , where c ( ~ , x, ξ ) = ψ (1; ~ , x, x, ξ , ξ ), r ( ~ , x, ξ ) = w (1; ~ , x, x, ξ , ξ ) / ~ .  Pr o of of Pr op osition 3.1. W e make use of differential equations sa tisfied by the op- erator (3.33) E ( s ) = E ( s ; ~ , x, ~ ∂ ) := exp  s ~ X ( ~ , x, ~ ∂ ))  , depe nding o n a parameter s . The total symbo l of E ( s ) is defined as (3.34) E ( s ; ~ , x, ξ ) = ∞ X k =0 s k ~ k k ! X ( k ) ( ~ , x, ξ ) , X (0) = 1 , X ( k +1) = X ◦ X ( k ) . T aking the logarithm (as a function, no t as an op er ator) of this, we can define S ( s ) = S ( s ; ~ , x, ξ ) by (3.35) E ( s ; ~ , x, ξ ) = e ~ − 1 S ( s ; ~ ,x, ξ ) . What we a re to pr ov e is that S ( s ), constr ucted as a s eries, makes sense at s = 1 and forma lly reg ular with resp ect to ~ . Different iating (3.3 5), we hav e (3.36) X ( ~ , x, ξ ) ◦ E ( s ; ~ , x, ξ ) = ∂ S ∂ s e S ( s ; ~ ,x, ξ ) / ~ . By Lemma 3.3 ( a 7→ X , b 7→ 1, p 7→ 0, q 7→ S ) and the technique in its pro of, we can rew r ite the left ha nd side as follows: (3.37) X ( ~ , x, ξ ) ◦ E ( s ; ~ , x, ξ ) = Y ( s ; ~ , x, x, ξ , ξ ) e S ( s ; ~ ,x, ξ ) / ~ , where Y ( s ; ~ , x, y , ξ , η ) = P ∞ k =0 Y k and Y k ( s ; ~ , x, y , ξ , η ) are defined by Y k +1 ( s ; ~ , x, y , ξ , η ) = 1 k + 1 ( ~ ∂ ξ ∂ y Y k ( s ; ~ , x, y , ξ , η ) + ∂ ξ Y k ( s ; ~ , x, y , ξ , η ) ∂ y S ( s ; ~ , y , η )) , Y 0 ( s ; ~ , x, y , ξ , η ) = X ( ~ , x, ξ ) . (3.38) Y k ( s ) co r resp onds to ψ k in the pro of o f Lemma 3.3, while w k there cor resp onds to δ k, 0 S ( s ). On the other hand, substituting (3.37) int o the left hand side o f (3 .36), we have (3.39) ∂ S ∂ s ( s ; ~ , x, ξ ) = Y ( s ; ~ , x, x, ξ , ξ ) . ~ -EXP ANSION OF KP 17 W e rewrite the system (3.3 8) and (3.39) in terms of expansion of S ( s ; ~ , x, ξ ) and Y k ( s ; ~ , x, y , ξ , η ) in p owers of s and ~ : S ( s ; ~ , x, ξ ) = ∞ X l =0 S ( l ) ( ~ , x, ξ ) s l = ∞ X l =0 ∞ X n =0 S ( l ) n ( x, ξ ) ~ n s l , Y k ( s ; ~ , x, y , ξ , η ) = ∞ X l =0 Y ( l ) k ( ~ , x, y , ξ , η ) s l = ∞ X l =0 ∞ X n =0 Y ( l ) k,n ( x, y , ξ , η ) ~ n s l , (3.40) The co efficient of ~ n s l in (3.3 8) is (3.41) Y ( l ) k +1 ,n ( x, y , ξ , η ) = 1 k + 1     ∂ ξ ∂ y Y ( l ) k,n − 1 ( x, y , ξ , η ) + X l ′ + l ′′ = l n ′ + n ′′ = n ∂ ξ Y ( l ′ ) k,n ′ ( x, y , ξ , η ) ∂ y S ( l ′′ ) n ′′ ( y , η )     , ( Y ( l ) k, − 1 = 0) and (3.42) Y ( l ) 0 ,n ( x, y , ξ , η ) = δ l, 0 X n ( x, ξ ) , while (3.39) gives (3.43) S ( l +1) n ( x, ξ ) = 1 l + 1 ∞ X k =0 Y ( l ) k,n ( x, x, ξ , ξ ) . W e fir st show that these recurs io n r elations co nsistently determine Y ( l ) k,n and S ( l ) n . Then we prove that the infinite sum in (3.43) is finite. Fix n ≧ 0 and ass ume that Y ( l ) k, 0 , . . . , Y ( l ) k,n − 1 and S ( l ) 0 , . . . , S ( l ) n − 1 hav e b een deter- mined for all ( l , k ). (When n = 0, Y ( l ) k, − 1 = 0 as mentioned ab ov e and S ( l ) − 1 can b e ignored as it do es not app ear in the induction.) (1) Since E ( s = 0 ) = 1 by the definition (3.33), we ha ve S (0) = 0. Hence (3.44) S (0) n = 0 . (2) Note that (3.45) ord ξ Y (0) 0 ,n ≦ − 1 bec ause of (3.42) and the assumption ord X ≦ − 1. (3) When l = 0, the second sum in the right hand side of the recur s ion re la tion (3.41) is a bs ent b ecaus e of (3.44). Hence if n ≧ k + 1, we hav e Y (0) k +1 ,n = 1 k + 1 ∂ ξ ∂ y Y (0) k,n − 1 = · · · = 1 ( k + 1 )! ( ∂ ξ ∂ y ) k +1 Y (0) 0 ,n − k − 1 = 0 , since Y (0) 0 ,n − k − 1 do es not depend on y tha nks to (3.42). If n < k + 1, the ab ov e ex pr ession b e c omes zer o by Y (0) k − n +1 , − 1 = 0. Hence together with (3.42), we obtain (3.46) Y (0) k,n = δ k, 0 X n . 18 KANEHISA T AKASAKI AND T AKASHI T AKEBE (4) By (3.4 3) we can determine S (1) n : (3.47) S (1) n = ∞ X k =0 Y (0) k,n = Y (0) 0 ,n = X n . (5) Fix l 0 ≧ 1 and as sume that for all l = 0 , . . . , l 0 − 1 and for all k = 0 , 1 , 2 , . . . , we hav e determined Y ( l ) k,n and that for all l = 0 , . . . , l 0 we hav e determined S ( l ) n . (The steps (3) a nd (4) are for l 0 = 1.) Since S (0) n ′′ = 0 by (3.44), the index l ′ in the right hand side of the recursion r elation (3.41) ( with l = l 0 ) runs es sentially from 0 to l 0 − 1 . Hence this rela tion determines Y ( l 0 ) k +1 ,n from known quantities for a ll k ≧ 0. Because o f the initial condition Y 0 ( s ; x, y , ξ , η ) = X ( x, ξ ) (cf. (3.3 8)) Y 0 do es not dep end on s , which mea ns that its T aylor co efficients Y ( l 0 ) 0 ,n v anish for all l 0 ≧ 1: (3.48) Y ( l 0 ) 0 ,n = 0 . Thu s we have determined all Y ( l 0 ) k,n ( k = 0 , 1 , 2 , . . . ). (6) W e shall prove b elow that Y ( l 0 +1) k,n = 0 if k > l 0 + n + 1 . Hence the sum in (3.43) is finite and S ( l 0 +1) n can b e determined. The induction pro cee ds by incrementing l 0 by o ne. Let us prove that Y ( l ) k,n ’s determined ab ov e sa tis fy Y ( l ) k,n = 0 , if k > l + n, (3.49) ord ξ Y ( l ) k,n ≦ − k − l − 1 . (3.50) (W e define that or d ξ 0 = −∞ .) In pa r ticular, the sum in (3 .43) is well-defined and (3.51) ord ξ S ( l +1) n ≦ − l − 1 . If n = − 1 , b oth (3.49) and (3.50) a re obvious. Fix n 0 ≧ 0 and a ssume that we hav e pr ov ed (3 .49) a nd (3.50) for n < n 0 and all ( l , k ). When n = n 0 and l = 0 , (3.4 9) and (3.50) are true for a ll k b eca us e of (3.46) and (3.45). Fix l 0 ≧ 0 and a ssume that we hav e pr ov e (3.4 9) a nd (3.5 0) for n = n 0 , l ≦ l 0 and all k . As a result (3.5 1) is true for l ≦ l 0 . F or k = 0 , (3.49) is void and (3.50) is true b ecause of (3.42) a nd o rd X n ≦ − 1. Put n = n 0 and l = l 0 + 1 in (3.4 1) and assume that k + 1 > ( l 0 + 1) + n 0 . Then k > ( l 0 + 1) + ( n 0 − 1), which guarantees that Y ( l 0 +1) k,n 0 − 1 = 0 by the induction hypothesis on n . As we ment ioned in the s tep (5) a bove, l ′ in the r ight hand side of (3.41) runs from 0 to l − 1 = l 0 . Hence, as we are as suming that k > l 0 + n , we hav e k > l ′ + n ′ , which leads to Y ( l ′ ) k,n ′ = 0 by the induction hypothesis on l a nd n . Therefore all terms in the rig ht hand side of (3.41) v anish and we hav e Y ( l 0 +1) k +1 ,n 0 = 0, which implies (3.49) for n = n 0 , l = l 0 + 1 and k ≧ 1 . The estimate (3.50) is easy to chec k for n = n 0 , l = l 0 + 1 and k ≧ 1 by the recursion r elation (3.41). (Recall once again that ∂ ξ low er s the order by one.) The step l = l 0 + 1 b eing prov ed, the induction pro c eeds with resp ect to l and consequently with resp ect to n . ~ -EXP ANSION OF KP 19 In summar y we hav e constr ucted Y ( s ; ~ , x, y , ξ , η ) and S ( s ; ~ , x, ξ ) satisfying (3.37) and (3.39). Thanks to (3.51), S n ( x, ξ ) = P ∞ l =0 S ( l ) n ( x, ξ ) is meaningful as a p ow er series of ξ . Thus Pro p o sition 3.1 is proved.  Pr o of of Pr op osition 3.2. W e re verse the order of the previo us pr o of. Namely , g iven S ( ~ , x, ξ ), we s hall constr uct X ( ~ , x, ξ ) such that the co rresp onding S (1; ~ , x, ξ ) in the a b ove pro o f coincides with it. Suppo se we hav e such X ( ~ , x, ξ ). Then the ab ov e pro cedure determine Y ( l ) k,n and S ( l ) n . W e expand them as follows: S ( ~ , x, ξ ) = ∞ X n =0 S n ( x, ξ ) ~ n , S n ( x, ξ ) = ∞ X j =1 S n,j ( x, ξ ) , X ( ~ , x, ξ ) = ∞ X n =0 X n ( x, ξ ) ~ n , X n ( x, ξ ) = ∞ X j =1 X n,j ( x, ξ ) , S ( s ; ~ , x, ξ ) = ∞ X l =0 ∞ X n =0 S ( l ) n ( x, ξ ) ~ n s l , S ( l ) n ( x, ξ ) = ∞ X j =1 S ( l ) n,j ( x, ξ ) , Y k ( s ; ~ , x, y , ξ , η ) Y ( l ) k,n ( x, y , ξ , η ) = ∞ X l =0 ∞ X n =0 Y ( l ) k,n ( x, y , ξ , η ) ~ n s l , = ∞ X j =1 Y ( l ) k,n,j ( x, y , ξ , η ) . Here terms with index j are homogeneo us terms of degree − j with resp ect to ξ and η . A t the end of this pro o f we shall determine X n by (3.42), (3.52) X n ( x, ξ ) = Y (0) 0 ,n ( x, y , ξ , η ) . (In particula r , Y (0) 0 ,n ( x, y , ξ , η ) should no t depend on y and η .) F or this purp os e, Y (0) 0 ,n should b e determined by (3.53) Y (0) 0 ,n ( x, y , ξ , η ) = S n ( x, ξ ) − X ( l,k ) 6 =(0 , 0) l,k ≥ 0 1 l + 1 Y ( l ) k,n ( x, x, ξ , ξ ) bec ause o f (3 .4 3) and S n ( x, ξ ) = P ∞ l =1 S ( l ) n ( x, ξ ). Since o rd ξ Y ( l ) k,n should b e less than − l − k (cf. (3.50)), we exp ect (3.54) Y ( l ) k,n, 1 = 0 for ( l , k ) 6 = (0 , 0). H ence picking up homog eneous terms of degree − 1 with resp ect to ξ from (3.53), the following equa tion should hold: (3.55) Y (0) 0 ,n, 1 = S n, 1 All Y ( l ) k,n, 1 are deter mined by the ab ove tw o conditions , (3.54) and (3.55). Note also that (3.56) Y ( l ) 0 ,n,j = 0 for l 6 = 0 bec ause Y 0 should not dep end on s b ecause of (3.42). 20 KANEHISA T AKASAKI AND T AKASHI T AKEBE Having determined initial conditions in this w ay , we shall determine Y ( l ) k,n,j in- ductively . T o this end we rewrite the recursion r e lation (3.41) by (3.43) and pic k up homo geneous terms of degr ee j : (3.57) Y ( l ) k +1 ,n,j ( x, y , ξ , η ) = 1 k + 1 ∂ ξ ∂ y Y ( l ) k,n − 1 ,j − 1 ( x, y , ξ , η )+ + X l ′ + l ′′ = l,l ′′ ≥ 1 j ′ + j ′′ = j − 1 ,n ′ + n ′′ = n k ′′ ≥ 0 1 l ′′ ∂ ξ Y ( l ′ ) k,n ′ ,j ′ ( x, y , ξ , η ) ∂ y Y ( l ′′ − 1) k ′′ ,n ′′ ,j ′′ ( x, x, ξ , ξ ) ! (As b efor e, terms like Y ( l ) k, − 1 ,j − 1 app earing the ab ove equation for n = 0 can b e ignored.) Fix n 0 ≧ 0 and assume that Y ( l ) k, 0 ,j , . . . , Y ( l ) k,n 0 − 1 ,j are deter mined fo r all ( l, k , j ). (1) First we determine Y ( l ) k,n 0 , 1 for all ( l , k ) by (3.55) a nd (3 .5 4). (2) Fix j 0 ≧ 2 and as sume that Y ( l ) k,n 0 ,j are deter mined for j = 1 , . . . , j 0 − 1 and all ( l, k ). (The a bove step is for j 0 = 2.) Since all the quan tities in the right hand side o f the recursio n re lation (3.57) with j = j 0 are known b y the induction hypothesis, we ca n determine Y ( l ) k,n 0 ,j 0 for l = 0 , 1 , 2 , . . . and k = 1 , 2 , . . . . (3) T og ether with (3.56) , Y ( l ) 0 ,n 0 ,j 0 = 0 fo r l = 1 , 2 , . . . , we have deter mined all Y ( l ) k,n 0 ,j 0 except for the c a se ( l , k ) = (0 , 0 ). (4) It follows from (3 .57), (3.5 5) and (3.56) by induction that for all Y ( l ) k,n 0 ,j determined in (1), (2) and (3), (3.58) Y ( l ) k,n,j = 0 for l + k + 1 > j. This cor resp onds to or d ξ Y ( l ) k,n 0 ≦ − l − k − 1 (3.50) in the pro of of Pr op osi- tion 3 .1. (5) W e determine Y (0) 0 ,n 0 ,j 0 by (3.59) Y (0) 0 ,n 0 ,j 0 = S n 0 ,j 0 − X ( l,k ) 6 =(0 , 0) l,k ≥ 0 1 l + 1 Y ( l ) k,n 0 ,j 0 ( x, x, ξ , ξ ) , which is the homogeneo us part of degree − j 0 in (3.53). T he sum in the right hand side is finite thank s to (3.58). (6) The induction w ith res pe c t to j proc eeds by incrementing j 0 . Thu s all Y ( l ) k,n 0 ,j are de ter mined and X n 0 is determined by X n 0 ( x, ξ ) = P ∞ j =1 Y (0) 0 ,n 0 ,j (cf. (3.5 2)), which completes the pro of of Pr op osition 3.2.  4. Asymptotics o f the t au function In this section we derive a n ~ -expa nsion (4.1) log τ ( ~ , t ) = ∞ X n =0 ~ n − 2 F n ( t ) ~ -EXP ANSION OF KP 21 of the tau function (cf. (1.12)) fro m the ~ -expansion (3.7) of the S -function. Note that we hav e suppressed the v a riable x , which is understo o d to b e absorb ed in t 1 . Let us rec a ll the fundamental relation [DJKM] b etw een the wa ve function (1.11) and the tau function a gain: (4.2) Ψ( t ; z ) = τ ( t − ~ [ z − 1 ]) τ ( t ) e ~ − 1 ζ ( t,z ) , where [ z − 1 ] = (1 / z , 1 / 2 z 2 , 1 / 3 z 3 , . . . ) and ζ ( t, z ) = P ∞ n =1 t n z n . This implies that (4.3) ~ − 1 ˆ S ( t ; z ) =  e − ~ D ( z ) − 1  log τ ( t ) where ˆ S ( t ; z ) = S ( t ; z ) − ζ ( t, z ) and D ( z ) = P ∞ j =1 z − j j ∂ ∂ t j . Differen tiating this with resp ect to z , we hav e ~ − 1 ∂ ∂ z ˆ S ( t ; z ) = − ~ D ′ ( z ) e − ~ D ( z ) log τ ( t ) = − ~ D ′ ( z )( ~ − 1 ˆ S ( t ; z ) + log τ ( t )) , (4.4) where D ′ ( z ) := ∂ ∂ z D ( z ) = − P ∞ j =1 z − j − 1 ∂ ∂ t j . Hence (4.5) − ~ D ′ ( z ) log τ ( t ) = ~ − 1  ∂ ∂ z + ~ D ′ ( z )  ˆ S ( t ; z ) Multiplying z n to this equation and taking th e residue, we obtain a sys tem of differential e quations (4.6) ~ ∂ ∂ t n log τ ( t ) = ~ − 1 Res z = ∞ z n  ∂ ∂ z + ~ D ′ ( z )  ˆ S ( t ; z ) dz , n = 1 , 2 , . . . which is known to b e integrable [DJKM]. W e can thus deter mine the tau function τ ( t ), up to multiplication τ ( t ) → cτ ( t ) b y a no nzero constant c , as a solution of (4.5). By substituting the ~ -ex pa nsions (4.7) log τ ( t ) = ∞ X n =0 ~ n − 2 F n ( t ) , ˆ S ( t ; z ) = ∞ X n =0 ~ n S n ( t ; z ) , int o (4.5) , we hav e (4.8) ∞ X j =1 ∞ X n =0 z − j − 1 ~ n − 1 ∂ F n ∂ t j = ∞ X n =0   ~ n − 1 ∂ S n ∂ z − ∞ X j =1 z − j − 1 ~ n ∂ S n ∂ t j   . Let us expand S n ( t ; z ) into a p ow er se r ies of z − 1 : (4.9) S n ( t ; z ) = − ∞ X k =1 z − k k v n,k . (The notation is chosen to b e c onsistent with our previo us work, e.g., [TT2].) Comparing the co efficients of z − j − 1 ~ n − 1 in (4.8 ), we hav e the equa tions (4.10) ∂ F n ∂ t j = v n,j + X k + l = j k ≥ 1 ,l ≥ 1 1 l ∂ v n − 1 ,l ∂ t k ( v − 1 ,j = 0) , 22 KANEHISA T AKASAKI AND T AKASHI T AKEBE which may be understo o d as defining equa tions of F n ( t ). According to what we hav e see n a bove, this system of differential equations is int egrable a nd determines F n up to integration constants. R emark 4.1 . T au functions in string theo ry and random matr ic es are known to hav e a genus ex p ansion of the for m (4.11) log τ = X g =0 ~ 2 g − 2 F g , where F g is the co ntribution from Riemann surfac e s of genus g . In contrast, g e neral tau functions of the ~ -de p endent K P hierarchy is not of this form, namely , o dd powers of ~ c a n app ear in the ~ -expa nsion of log τ . T o exclude o dd pow ers therein, we need to impo se conditions 0 = v 2 m +1 ,j + X k + l = j k ≥ 1 ,l ≥ 1 1 l ∂ v 2 m,l ∂ t k on v n,j or 0 = ∂ S 2 m +1 ∂ z − ∞ X j =1 z − j − 1 ∂ S 2 m ∂ t j on S n . 5. Concluding remark s W e have presented a recursive construc tio n of solutio ns o f the ~ -de p endent KP hierarch y . The input o f this construction is the pair ( f , g ) of quantised canonical transformatio n. The main outputs are the dressing op erator W in the exp onential form (1.8), the wav e function Ψ in the WKB form (3.2) and the tau function with the quasi-cla ssical expans ion (4.1). Thus the ~ - de p endent KP hierar ch y introduced in our previo us w or k [TT2 ] is no longer a heur istic framew ork f or deriving the disp e rsionless K P hiera rch y , but ha s its own r aison d’ ˆ etre. A serious problem of our construction is that the recurs ion rela tions are extremely complicated. In App endix B, calculations ar e illustrated fo r the Kontsevich mo del [Ko], [D], [AvM]. As this exa mple shows, this is b y no means a practical w ay to construct a solution. W e believe that one cannot a void this difficult y as far as general so lutions ar e considered. Spec ia l solutions stemming fro m s tring theory a nd ra ndom ma trices [Mo], [dF GZ] (e.g. the K ontsevic h mo de l) can admit a more efficien t appr oach such as the metho d of E ynard and Orantin [EO]. Tho s e metho ds ar e based o n a q uite different principle. In the metho d of Eynar d and Or antin, it is the so c a lled “lo op equation” for cor rela- tion functions of random matric e s. The lo o p equations amount to “c onstraints” o n the tau function. Eynard and Or antin’s “ top ologica l re cursion rela tions” determine a s olution of those co nstraints r ather than of an under lying in tegrable hierarchy; it is somewha t surpris ing that a solution of those co ns traints gives a tau function. Lastly , let us mention that the results of this pap er can b e extended to the T o da hierarch y . That case will be trea ted in a for thcoming pap er . ~ -EXP ANSION OF KP 23 Appendix A. Proof of formulae (2.1 7) an d (2.1 9) In this app endix we prove the factoris ation o f W (2.17) and an auxilia ry formula (2.19). The ma in to ol in this a pp endix is the Campb ell-Hausdor ff theorem: (A.1) exp( X ) exp( Y ) = exp ∞ X n =0 c n ( X, Y ) ! , where c n ( X, Y ) is determined recursively: c 1 ( X, Y ) = X + Y , c n +1 ( X, Y ) = 1 n + 1 1 2 [ X − Y , c n ]+ + X p ≥ 1 , 2 p ≤ n K 2 p X ( k 1 ,...,k 2 p ) k 1 + ··· + k 2 p = n [ c k 1 , [ · · · , [ c k 2 p , X + Y ] · · · ]] ! . (A.2) The co efficients K 2 p are defined by (2.1 5). Se e , for example, [B]. First we pr ove (A.3) exp( ~ − 1 X ( x, t, ~ ∂ )) = exp  ~ i − 1 ˜ X ′ i + (terms o f ~ - order < − i + 1)  exp  ~ − 1 X ( i − 1)  , where the principal symbol o f ˜ X ′ i is (A.4) σ ~ ( ˜ X ′ i ) := ∞ X n =1 (ad { , } σ ~ ( X 0 )) n − 1 n ! σ ~ ( X i ) , as is defined in (2.1 9). F or simplicity , let us denote (A.5) A := 1 ~ X ( i − 1) = 1 ~ i − 1 X j =0 ~ j X j , B := 1 ~ ∞ X j = i ~ j X j . Note that A + B = X/ ~ a nd o rd ~ A ≦ 1 , ord ~ B ≦ − i + 1. W e prov e the following by induction: (A.6) C n := c n ( A + B , − A ) = (ad A ) n − 1 n ! ( B ) + (terms of ~ -or der < − i + 1) . This is obvious for n = 1 since C 1 = ( A + B ) + ( − A ) = B . Ass ume that (A.6) is true for n = 1 , . . . , N . This means , in particular , ord ~ C n ≦ ord ~ B ≦ 0 (1 ≦ n ≦ N ), which implies that for any op erato r Z or d ~ [ C n , Z ] is less than ord ~ Z by mor e than one. Hence the term of the highest ~ -order in the recur s ive definition (A.2 ) with X = A + B , Y = − A is the first term. More pr ecisely , it is deco mp o sed as 1 N + 1 · 1 2 [( A + B ) − ( − A ) , C N ] = 1 N + 1 [ A, C N ] + 1 2( N + 1) [ B , C N ] , 24 KANEHISA T AKASAKI AND T AKASHI T AKEBE and the first term in the rig ht hand side has the highest ~ -or der. By the induction hypothesis a nd ord ~ A ≦ 1 , we hav e 1 N + 1 [ A, C N ] = 1 N + 1  A, (ad A ) N − 1 N ! ( B ) + (terms of ~ orde r < − i + 1)  = (ad A ) N ( N + 1)! ( B ) + (terms of ~ -order < − i + 1) . (A.7) This proves (A.6) for a ll n . T aking its s ymbol of order − i + 1 , we hav e (A.8) σ ~ ( c n ( A + B , − A )) = (ad { , } σ ~ ( A )) n − 1 n ! σ ~ ( B ) , which gives the ter ms of (A.4). Substituting this in to the Campbell-Hausdor ff formula (A.1) , we hav e (A.3). By factorisation (A.3), w e can factorise W = exp( X/ ~ )( ~ ∂ ) α as follows ( α ( i − 1) := P i − 1 j =0 ~ j α j ): exp( ~ − 1 X ( x, t, ~ ∂ ))( ~ ∂ ) α = exp  ~ i − 1 ˜ X ′ i + (terms o f ~ -or der < − i + 1)  exp  ~ − 1 X ( i − 1)  × × exp  ~ i − 1 α i log( ~ ∂ ) + (terms o f ~ - order < − i + 1)  × × exp  ~ − 1 α ( i − 1) log( ~ ∂ )  = exp  ~ i − 1 ˜ X ′ i + (terms o f ~ -or der < − i + 1)  × × exp  e ad( ~ − 1 X ( i − 1) )  ~ i − 1 α i log( ~ ∂ ) + (terms o f ~ -or der < − i + 1)   × × exp  ~ − 1 X ( i − 1)  exp  ~ − 1 α ( i − 1) log( ~ ∂ )  . (A.9) Since the symbol of o rder − i +1 of e ad( ~ − 1 X ( i − 1) )  ~ i − 1 α i log( ~ ∂ )  is e ad { , } σ ~ ( X 0 ) ( α i log ξ ), (A.9) is rewritten as (2.17) by us ing the Campb ell-Hausdor ff form ula (A.1) once again. In or de r to r ecov er X i from ˜ X i (or ˜ X ′ i ), we hav e only to inv ert the definition (A.4). In the definition (A.4) of the map X i 7→ ˜ X ′ i we substitute ad { , } ( σ ~ ( X 0 )) in the equa tion e t − 1 t = ∞ X n =1 t n − 1 n ! . Hence substitution t = ad { , } ( σ ~ ( X 0 )) in its inv er se t e t − 1 = 1 − t 2 + ∞ X p =1 K 2 p t 2 p gives the inv erse map ˜ X ′ i 7→ X i . Here the co efficients K 2 p are defined in (2.1 5). Hence equation (2.19): σ ~ ( X i ) = σ ~ ( ˜ X ′ i ) − 1 2 { σ ~ ( X 0 ) , σ ~ ( ˜ X ′ i ) } + ∞ X p =1 K 2 p (ad { , } ( σ ~ ( X 0 ))) 2 p σ ~ ( ˜ X ′ i ) gives the symbol of X i . ~ -EXP ANSION OF KP 25 Appendix B. Example (Kontsevich mod el) According to Adler and v a n Mo erbeke [AvM], the solution of the KP hiera rch y arising in the Witten-Kontsevic h theorem [Ko] satisfies (B.1) ( L 2 ) ≤− 1 = 0 ,  1 2 M L − 1 − 1 4 ~ L − 2 − L  ≤− 1 = 0 . This cor resp onds to the ca se where (B.2) f ( ~ , x, ~ ∂ ) = ( ~ ∂ ) 2 , g ( ~ , x, ~ ∂ ) = 1 2 x ( ~ ∂ ) − 1 − ~ 4 ( ~ ∂ ) − 2 + ~ ∂ . Let us apply our procedure to this case. W e fix all the time v ariables t n to 0, whic h means that we restrict ourselves to the so -called “small phase space ” in top ologica l string theory [D ]. (The first time v ar iable t 1 can b e re-intro duced by shifting x to t 1 + x .) T o b egin with, let us deter mine the leading terms of X and α , which are the initial data for our pro ce dur e. The dis pe r sionless limit o f ( L, M ) satisfies (B.3) ( L 2 ) ≤− 1 = 0 ,  1 2 ML − 1 − L  ≤− 1 = 0 . Since L has the form (1.17), L 2 should b e a se cond order poly nomial of ξ : L 2 = ξ 2 + u ( x ). When t n = 0, M has the fo r m M = x + α 0 L − 1 + ∞ X n =1 v 0 ,n L − n − 1 . (See (1.21).) U sing this and L = ξ (1 + u ( x ) ξ − 2 ) 1 / 2 , we hav e 1 2 ML − 1 − L = − ξ +  x 2 − u 2  ξ − 1 + α 0 2 ξ − 2 + O ( ξ − 3 ) . Hence, due to the s econd equation o f (B.3), we hav e u ( x ) = x , α 0 = 0 and, consequently , (B.4) P 0 := L 2 = ξ 2 + x, Q 0 := 1 2 ML − 1 − L = − ξ . Combining them, we hav e the following expre s sion for M : M =2 L ( L − ξ ) = 2( ξ 2 + x ) − 2 ξ 2  1 + xξ − 2  1 / 2 = x − ∞ X n =2 2  1 / 2 n  x n ξ − 2 n +2 = x + x 2 4 ξ 2 − x 3 8 ξ 4 + 5 x 4 64 ξ 6 − 7 x 5 128 ξ 8 + 21 x 6 512 ξ 10 + O ( ξ − 12 ) (B.5) On the other hand, fr om the expressio n M = exp(ad { , } X 0 ) x follows M = ∞ X n =0 ad { , } X 0 n ! x = x − χ 0 , 1 ξ − 2 − 2 χ 0 , 2 ξ − 3 +  − 3 χ 0 , 3 − χ 0 , 1 χ ′ 0 , 1 2  ξ − 4 + ( − 4 χ 0 , 4 − 2 χ 0 , 2 χ ′ 0 , 1 ) ξ − 5 + · · · , (B.6) 26 KANEHISA T AKASAKI AND T AKASHI T AKEBE where χ 0 ,k are the co efficients in the expansio n of X 0 (2.16). Comparing (B.5) a nd (B.6), we can determine χ 0 ,k inductively a nd hence X 0 is determined: X 0 = − x 2 4 ( ~ ∂ ) − 1 + x 3 48 ( ~ ∂ ) − 3 − x 4 384 ( ~ ∂ ) − 5 + x 6 6144 ( ~ ∂ ) − 9 − x 7 61440 ( ~ ∂ ) − 11 + · · · . Having determined X 0 and α 0 , we can start the alg orithm discussed in Section 2. In the step 1 for i = 1 we define P (0) and Q (0) by (2.9) and (2.10): P (0) = ( ~ ∂ ) 2 + x + ~ 2 ( ~ ∂ ) − 1 − ~ x 4 ( ~ ∂ ) − 3 + ~ 2 8 ( ~ ∂ ) − 4 + 5 ~ x 2 32 ( ~ ∂ ) − 5 − 41 ~ 2 x 96 ( ~ ∂ ) − 6 +  23 ~ 3 96 − 3 ~ x 3 32  ( ~ ∂ ) − 7 + 301 ~ 2 x 2 384 ( ~ ∂ ) − 8 +  − 83 ~ 3 x 48 + 53 ~ x 4 1024  ( ~ ∂ ) − 9 +  191 ~ 4 192 − 543 ~ 2 x 3 512  ( ~ ∂ ) − 10 +  8783 ~ 3 x 2 1536 − 119 ~ x 5 4096  ( ~ ∂ ) − 11 + · · · , Q (0) = − ( ~ ∂ ) − ~ 4 ( ~ ∂ ) − 2 + 3 ~ x 8 ( ~ ∂ ) − 4 − 3 ~ 2 8 ( ~ ∂ ) − 5 − 29 ~ x 2 64 ( ~ ∂ ) − 6 + 157 ~ 2 x 96 ( ~ ∂ ) − 7 +  − 49 ~ 3 32 + ~ x 3 2  ( ~ ∂ ) − 8 − 519 ~ 2 x 2 128 ( ~ ∂ ) − 9 +  4345 ~ 3 x 384 − 1077 ~ x 4 2048  ( ~ ∂ ) − 10 +  − 1339 ~ 4 128 + 3961 ~ 2 x 3 512  ( ~ ∂ ) − 11 + · · · . W e extract terms (symbols) of ~ -order 0 from the ~ -e x pansion of them: P 0 ( x, ξ ) = ξ 2 + x, Q 0 ( x, ξ ) = − ξ , and those of ~ -order − 1: P (0) 1 ( x, ξ ) = 1 2 ξ − 1 − x 4 ξ − 3 + 5 x 2 32 ξ − 5 − 3 x 3 32 ξ − 7 + 53 x 4 1024 ξ − 9 − 119 x 5 4096 ξ − 11 + · · · Q (0) 1 ( x, ξ ) = − 1 4 ξ − 2 + 3 x 8 ξ − 4 − 29 x 2 64 ξ − 6 + x 3 2 ξ − 8 − 1077 x 4 2048 ξ − 10 + · · · . Then, following (2.1 3), we determine α 0 and X 0 by α 1 log ξ + ˜ X 1 := Z  ∂ Q 0 ∂ ξ P (0) 1 − ∂ P 0 ∂ ξ Q (0) 1  ≤− 1 dξ = x 4 ξ − 2 − 3 x 2 16 ξ − 4 + 29 x 3 192 ξ − 6 − x 4 8 ξ − 8 + 1077 x 5 10240 ξ − 10 + · · · . Since lo g term is a bsent, α 1 = 0 a nd the ab ov e expressio n is ˜ X 1 itself, which is also equal to ˜ X ′ 1 defined by (2.14). Then we c a n compute X 1 by the for m ulae (2.14) and (2.16): X 1 = x 4 ( ~ ∂ ) − 2 − 3 x 2 32 ( ~ ∂ ) − 4 + 5 x 3 192 ( ~ ∂ ) − 6 − 9 x 5 2048 ( ~ ∂ ) − 10 + · · · ~ -EXP ANSION OF KP 27 W e can re p ea t the pro cedure Step 1 , 2, 3 for i = 2 ag ain. The res ults are P (1) =( ~ ∂ ) 2 + x + 3 ~ 2 16 ( ~ ∂ ) − 4 − 23 ~ 2 x 96 ( ~ ∂ ) − 6 + 9 ~ 3 16 ( ~ ∂ ) − 7 + 19 ~ 2 x 2 384 ( ~ ∂ ) − 8 − 707 ~ 3 x 384 ( ~ ∂ ) − 9 +  861 ~ 4 256 + 155 ~ 2 x 3 512  ( ~ ∂ ) − 10 + 2145 ~ 3 x 2 1024 ( ~ ∂ ) − 11 + · · · , Q (1) = − ~ ∂ − ~ 2 4 ( ~ ∂ ) − 5 + 143 ~ 2 x 192 ( ~ ∂ ) − 7 − 611 ~ 3 384 ( ~ ∂ ) − 8 − 85 ~ 2 x 2 64 ( ~ ∂ ) − 9 + 2205 ~ 3 x 256 ( ~ ∂ ) − 10 +  − 1795 ~ 4 128 + 1885 ~ 2 x 3 1024  ( ~ ∂ ) − 11 + · · · . Collecting the terms of ~ -o rder − 2, we ha ve P (1) 2 = 3 16 ξ − 4 − 23 x 96 ξ − 6 + 19 x 2 384 ξ − 8 + 155 x 3 512 ξ − 10 + · · · , Q (1) 2 = − 1 4 ξ − 5 + 143 x 192 ξ − 7 − 85 x 2 64 ξ − 9 + 1885 x 3 1024 ξ − 11 + · · · . F rom this r esult follows α 2 log ξ + ˜ X 2 := Z  ∂ Q 0 ∂ ξ P (1) 2 − ∂ P 0 ∂ ξ Q (1) 2  ≤− 1 dξ = − 5 48 ξ − 3 + x 4 ξ − 5 − 143 x 2 384 ξ − 7 + 85 x 3 192 ξ − 9 − 1885 x 4 4096 ξ − 11 + · · · . This mea ns α 2 = 0 a nd ˜ X ′ 2 is equal to ˜ X 2 , which is equal to the ab ove expressio n. Substituting these res ults in (2.14), w e have X 2 = − 5 48 ( ~ ∂ ) − 3 + 11 x 64 ( ~ ∂ ) − 5 − 85 x 2 768 ( ~ ∂ ) − 7 + 435 x 4 8192 ( ~ ∂ ) − 11 + · · · . Substituting this into (2.9) and (2.10) for i = 3, we hav e P (2) =( ~ ∂ ) 2 + x + 5 ~ 3 48 ( ~ ∂ ) − 7 + 425 ~ 3 x 768 ( ~ ∂ ) − 9 + 3205 ~ 4 4096 ( ~ ∂ ) − 10 − 5865 ~ 3 x 2 2048 ( ~ ∂ ) − 11 + · · · , Q (2) = − ( ~ ∂ ) − 155 ~ 3 384 ( ~ ∂ ) − 8 + 685 ~ 3 x 512 ( ~ ∂ ) − 10 − 41395 ~ 4 8192 ( ~ ∂ ) − 11 + · · · . By extra cting the terms of ~ -or der − 3, we have P (2) 3 = 5 48 ξ − 7 + 425 x 768 ξ − 9 − 5865 x 2 2048 ξ − 11 + · · · , Q (2) 3 = − 155 384 ξ − 8 + 685 x 512 ξ − 10 + · · · . (B.7) Hence, α 3 log ξ + ˜ X 3 := Z  ∂ Q 0 ∂ ξ P (2) 3 − ∂ P 0 ∂ ξ Q (2) 3  ≤− 1 dξ = − 15 128 ξ − 6 + 155 x 384 ξ − 8 − 685 x 2 1024 ξ − 10 + · · · , 28 KANEHISA T AKASAKI AND T AKASHI T AKEBE which implies α 3 = 0 and ˜ X 3 = ˜ X ′ 3 is eq ua l to the a bove expr ession. 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Phys. 7 (1995) , 743-803. Gradua te School of Hu man and Environment al S tudies, Kyoto University, Yoshid a, Sakyo, Kyoto, 606-850 1, Jap an E-mail addr ess : taka saki@math. h.kyoto-u.ac.jp F acul ty of Mathema tics, St a te Un iversity – High er School of Economics, V a vilov a Street, 7, Moscow, 11731 2, Russia E-mail addr ess : ttak ebe@hse.ru