An hbar-expansion of the Toda hierarchy: a recursive construction of solutions

AN ~ -EXP ANSION OF THE TOD A HIER AR CHY: A RECURSIVE CONSTR UCTION OF SOLUTIONS KANEHISA T AKASAKI AND T AKASHI T AKEBE Abstract. A construction of ge neral solutions of the ~ -dependent T o da hi- erarch y is presen ted. The co nstruction is based on a Riemann-Hilb ert prob- lem for the pairs ( L, M ) and ( ¯ L, ¯ M ) of Lax and Or lov-Sc h ulman op erators. This Riemann-Hilb ert problem is translated to the language of the dress- ing operators W and ¯ W . The dressing op erators are set i n an exp onen tial form as W = e X/ ~ and ¯ W = e φ/ ~ e ¯ X / ~ , a nd the auxiliary operators X, ¯ X and the function φ are assumed to hav e ~ -expansions X = X 0 + ~ X 1 + · · · , ¯ X = ¯ X 0 + ~ ¯ X 1 + · · · and φ = φ 0 + ~ φ 1 + · · · . The coeﬃcient s of the se expan- sions turn out to satisfy a set of rec ursi on relations. X, ¯ X and φ are recursively determined by these relations. Moreov er, the associ ated wa ve functions are sho wn to hav e the WKB form Ψ = e S/ ~ and ¯ Ψ = e ¯ S / ~ , whic h l eads to an ~ -expansion of the logarithm of the tau function. 0. Introduction This paper is a cont inuation of our prev io us w ork [TT3, TT4] on a quasi-classical or ~ -dep endent (where ~ is the Planck constant) formulation o f the KP hier arch y [TT2]. W e presen ted there in a r ecursive constr uction of gener al solutions to the ~ - depe ndent KP hier arch y . The co nstruction starts from a Rie ma nn-Hilbe rt pr oblem for the pa ir ( L, M ) of Lax and Orlov-Sch ulman opera tors. This Rie ma nn-Hilbe rt problem can be tr anslated to the langua ge of the underlying dressing opera tor W . Assuming the exp onential for m W = e ~ − 1 X and an ~ -expansio n of the o pe r - ator X , one can derive a set of recursio n relations that determine the op erato r X order-by-order of the ~ -expansion from the lowest part (namely , a so lutio n o f the disp e rsionless KP hierarchy [TT2]). Thus the Lax, O rlov-Sc hulman and dr essing op erator s ar e obta ined. F urthermor e, b or rowing an idea from Aoki’s e x po nential calculus of micro diﬀerential op er ators [A], one can show that the wa ve function has the W KB for m Ψ = e ~ − 1 S . This leads to an ~ -expansion of the asso c ia ted tau function as a generalisatio n of the “genus expa nsion” of partition functions in string theor ies a nd ra ndom matrices [D, Kr, Mo, dFGZ]. The go al of this pap er is to gener alise these results to a n ~ -dep endent formulation of the T o da hierar ch y [TT2]. The T o da hie r arch y is built fro m diﬀerence op erato rs a ( s, e ∂ s ) = X m a m ( s ) e m∂ s on a one-dimensional lattice (with co or dinate s ∈ Z ) r ather than micro diﬀerential op erator s o n a contin uo us line. Ev en in the ~ - independent case [UT], the formula- tion o f the hierarch y itself is more complicated than that o f the KP hiera rch y . The Date : 2 Dec ember 2011. 1 2 KANEHISA T AKASAK I AND T AK ASHI T AKE BE hierarch y has tw o sets of time evolutions for time v ar iables t = ( t n ) and ¯ t = ( ¯ t n ). These time ev olutions are for mu lated with t wo La x op era tors L a nd ¯ L . Orlov- Sch ulma n op er ators, dressing op er ators and wav e functions, to o , are prepar ed in pairs. In the ~ -dep endent formulation [TT2], the P la nck constan t ~ plays the role o f lattice spacing, whic h shows up in the shift ope r ators as e ~ ∂ s . Diﬀerence op era tors in the Lax fo r malism a re linear combinations a ( ~ , s, e ~ ∂ s ) = X m a m ( ~ , s ) e m ~ ∂ s of these shift opera tors with ~ -dep endent co eﬃcients a m ( ~ , s ). T o construct a g eneral solutio n of the ~ -dep endent T o da hiera rch y , we star t from a Riemann-Hilb ert problem for the pairs ( L, M ) a nd ( ¯ L, ¯ M ) of Lax a nd Orlov- Sch ulma n op er a tors. This problem can be conv erted to a problem for the dressing op erator s W and ¯ W . W e seek W and ¯ W in the exp onential form W = e ~ − 1 X , ¯ W = e ~ − 1 φ e ~ − 1 ¯ X , where X and ¯ X are diﬀere nc e op erato rs and φ is a function of ( ~ , s, t, ¯ t ). Assum- ing that these ope rators and functions hav e ~ -ex pansions, we can derive a set of recursion relations for the coe ﬃcie nt s of these expa nsions. The low e s t part o f this expansion turns out to be the dr essing function o f the disp er sionless T o da hierar - ch y [TT1, TT2]. Thu s the descr iption of the Lax , Orlov-Sc hulman and dres sing op erator s a re mo stly pa rallel to the case of the ~ - dep endent K P hier arch y . The constr uction of the a s so ciated wa ve functions exhibits a new fea ture. T o formulate a n analogue of Aoki’s exp o nential calculus for diﬀer ence op erato r s, we deﬁne the “sym b ol” of a diﬀerence op era tor a ( ~ , s, e ~ ∂ s ) to be a ( ~ , s, ξ ). The o pe r - ator pro duct a ( ~ , s, e ~ ∂ s ) b ( ~ , s, e ~ ∂ s ) induces the ◦ -pro duct a ( ~ , s, ξ ) ◦ b ( ~ , s, ξ ) = e ~ ξ ∂ ξ ∂ s ′ a ( ~ , s, ξ ) b ( ~ , s ′ , ξ ′ ) | s ′ = s,ξ ′ = ξ = ∞ X n =0 ~ n n ! ( ξ ∂ ξ ) n a ( ~ , s, ξ ) ∂ n s ′ b ( ~ , s ′ , ξ ′ ) | s ′ = s,ξ ′ = ξ for those symbols. Although lo oking very similar, this pr o duct structure is slightly diﬀerent fro m the ◦ -pro duct o f the symbols a ( ~ , x, ξ ) of ~ -dep endent micro diﬀeren- tial o pe r ators a ( ~ , x, ~ ∂ x ) in that ∂ ξ is now replaced with ξ ∂ ξ . T his tin y diﬀerence, how ever, has a considera ble eﬀect; unlike ∂ ξ , ξ ∂ ξ do es not low er the or der with resp ect to ξ . Because of this, we ar e forced to mo dify our previous metho d [TT3]. W e admit tha t our co nstruction of solutions is ex tremely complicated. The recursive pro cedure is illustrated in Appendix for a sp ecial case that is related to c = 1 string theory at self-dual radius [DMP, EK, HOP]. E ven in this relatively simple case, we have b een unable to derive an explicit form of the so lution unless a half of the full time v ar ia bles are set to zero. This is a price to pay for treating gener al solutions. In this sense, our metho d cannot be directly compared with the metho d in r a ndom ma trix theor y [Mo , dFGZ], in particular, Eyna rd and O rantin’s top ologica l rec ur sion relations [EO]. Their recursion relations stem from the “loo p equations” for random matrices, which are c o nstraints to single o ut a class of sp ecial solutions of an underlying integrable hierar ch y , while o ur metho d do es not use an y extra struc tur e other than the integrable hierar ch y itself. This pa pe r is organis e d a s follows. Section 1 is a review of the ~ -dependent formulation of the T o da hiera rch y . The Riemann-Hilb ert proble m is also fo r mulated therein. Section 2 presents the recursive so lution of the Riemann-Hilb ert problem. ~ -EXP ANS ION OF TOD A 3 The metho d is a ra ther straightforw ard generalisa tion of the case of the ~ -dep endent KP hierarchy . Section 3 deals with the ~ -expansion of the wav e function. Aoki’s exp onential calculus is reformulated for diﬀerence operator s. Relev ant r ecursion relations are thereby derived, and s hown to have a so lution. Section 4 mentions the ~ -expansion of the tau function. A cknow le dgements. The authors are grateful to Professo r Akihiro Tsuchiy a for drawing o ur a tten tion to this sub ject. This work is partly supp orted b y Grant-in-Aid for Scientiﬁc Research No. 1954017 9 and No. 225 4018 6 from the J apan So ciety for the Pr omotion of Science and b y the Bilateral Joint Pro ject “Integrable Systems, Rando m Matrices, Alg e br aic Geome- try and Geo metric Inv ar iants” (2 010–2 011) of the Japan So cie ty for the Pro motion of Science and the Russian F oundation for Basic Research. TT is partly s upp o r ted by the grant of the Natio na l Resear ch Univ ersity – Higher School of Economics , Russia, for the Indiv idual Re s earch Pr o ject 09-0 1-004 7 (2009) and 10 - 01-00 43 (2010). Part of the work was done during his stay in the Institut Mittag-Leﬄer (Djursholm, Sweden) by the progr amme “Co mplex Analysis and In- tegrable Systems” in 20 11. He tha nk s the Institut Mittag-Leﬄer a nd the orga nis ers for ho spitality . 1. ~ -dependent Toda hierarchy: review In this section w e recall several fac ts on the T o da hierarch y dep ending on a formal par ameter ~ in [TT2], § 2.7. Thro ughout this pap er a ll functions are for mal power se r ies. The ~ -dependent T o da hierarch y is deﬁned by the Lax repr esentation (1.1) ~ ∂ L ∂ t n = [ B n , L ] , ~ ∂ L ∂ ¯ t n = [ ¯ B n , L ] , ~ ∂ ¯ L ∂ t n = [ B n , ¯ L ] , ~ ∂ ¯ L ∂ ¯ t n = [ ¯ B n , ¯ L ] , B n = ( L n ) ≥ 0 , ¯ B n = ( ¯ L − n ) ≤− 1 , n = 1 , 2 , . . . , where the L ax op er ators L , ¯ L ar e diﬀerence oper ators with res pe ct to the discrete independent v aria ble s ∈ ~Z of the form L = e ~ ∂ s + ∞ X n =0 u n +1 ( ~ , s, t, ¯ t ) e − n ~ ∂ s , (1.2) ¯ L − 1 = ¯ u 0 ( ~ , t, ¯ t, s ) e − ~ ∂ s + ∞ X n =0 ¯ u n +1 ( ~ , t, ¯ t, s ) e n ~ ∂ s (1.3) and ( ) ≥ 0 and ( ) ≤− 1 are pro jections ont o a linear combination of e n ~ ∂ /∂ s with n ≧ 0 and ≦ − 1 , res pe c tively . Note tha t e ~ ∂ s is a diﬀerence op er ator with step ~ : e n ~ ∂ s f ( s ) = f ( s + n ~ ). The co eﬃcients u n ( ~ , t, ¯ t, s ), ¯ u n ( ~ , t, ¯ t, s ) of L , ¯ L are as- sumed to be formally reg ular with respect to ~ : u n ( ~ , t, ¯ t, s ) = P ∞ m =0 ~ m u ( m ) n ( t, ¯ t, s ), ¯ u n ( ~ , t, ¯ t, s ) = P ∞ m =0 ~ m ¯ u ( m ) n ( t, ¯ t, s ) as ~ → 0. W e deﬁne the ~ -or der of the diﬀerence op er ator by (1.4) ord ~  X a n,m ( t, ¯ t, s ) ~ n e m ~ ∂ s  def = max ( − n      X m a n,m ( t, ¯ t, s ) e m ~ ∂ s 6 = 0 ) . 4 KANEHISA T AKASAK I AND T AKASHI T AK EBE In particular , ord ~ ~ = − 1, or d ~ e ~ ∂ s = 0. F or example, the condition which we impos ed o n the co eﬃcients u n ( ~ , t, ¯ t, s ) and u n ( ~ , t, ¯ t, s ) can b e restated as ord ~ ( L ) = ord ~ ( ¯ L ) = 0. The princip al symb ol (resp. the symb ol of or der l ) of a diﬀerence o p erator A = P a n,m ( t, ¯ t, s ) ~ n e m ~ ∂ s with r esp ect to the ~ -order is σ ~ ( A ) def = X m a − ord ~ ( A ) ,m ( t, ¯ t, s ) ξ m (1.5) (resp. σ ~ l ( A ) def = X m a − l,m ( t, ¯ t, s ) ξ m . (1.6) When it is clear from the co nt ext, w e sometimes use σ ~ instead of σ ~ l . The Lax op era tors L and ¯ L a re e xpressed by dr essing op er ators W and ¯ W : (1.7) L = Ad W ( e ~ ∂ s ) = W e ~ ∂ s W − 1 , ¯ L = Ad ¯ W ( e ~ ∂ s ) = ¯ W e ~ ∂ s ¯ W − 1 , The op er ators W and ¯ W should hav e sp eciﬁc forms: W = e ~ − 1 X ◦ ( ~ ,t, ¯ t,s,e ~ ∂ s ) e ~ − 1 α ( ~ )( ~ ∂ s ) (1.8) X ◦ ( ~ , t, ¯ t, s, e ~ ∂ s ) = ∞ X k =1 χ ◦ k ( ~ , t, ¯ t, s ) e − k ~ ∂ s , (1.9) ¯ W = e ~ − 1 φ ( ~ ,t, ¯ t,s ) e ~ − 1 ¯ X ◦ ( ~ ,t, ¯ t,s,e ~ ∂ s ) e ~ − 1 ¯ α ( ~ )( ~ ∂ s ) (1.10) ¯ X ◦ ( ~ , t, ¯ t, s, e ~ ∂ s ) = ∞ X k =1 ¯ χ ◦ k ( ~ , t, ¯ t, s ) e k ~ ∂ s , (1.11) ord ~ ( φ ( ~ , t, ¯ t, s )) = o rd ~ ( X ◦ ( ~ , t, ¯ t, s, e ~ ∂ s )) = or d ~ α ( ~ ) = ord ~ ( ¯ X ◦ ( ~ , t, ¯ t, s, e ~ ∂ s )) = or d ~ ¯ α ( ~ ) = 0 , (1.12) and α ( ~ ) and ¯ α ( ~ ) ar e constants with resp ect to t , ¯ t and s . (In [TT2] we did not int ro duce α , ¯ α , which will b e necessa r y in Section 2.) Note that the set of op erator s of the form (1.13) a ~ ∂ ∂ s + X k χ k ( s ) e k ~ ∂ s , where a do es not dep end on s and χ k ( s ) a re functions of s , is closed under the commutator bra cket. Hence any theorem or fo r mula for Lie algebr a s ca n b e applied to such op erato rs. In par ticular, using the Campb ell-Hausdor ﬀ formula, w e can rewrite W and ¯ W in the following form, which will b e more convenien t in the later ~ -EXP ANS ION OF TOD A 5 discussion: W = e ~ − 1 X ( ~ ,t, ¯ t,s,e ~ ∂ s ) (1.14) X ( ~ , t, ¯ t, s, e ~ ∂ s ) = α ( ~ ) ~ ∂ ∂ s + ∞ X k =1 χ k ( ~ , t, ¯ t, s ) e − k ~ ∂ s , (1.15) ¯ W = e ~ − 1 φ ( ~ ,t, ¯ t,s ) e ~ − 1 ¯ X ( ~ ,t, ¯ t,s,e ~ ∂ s ) , (1.16) ¯ X ( ~ , t, ¯ t, s, e ~ ∂ s ) = ¯ α ( ~ ) ~ ∂ ∂ s + ∞ X k =1 ¯ χ k ( ~ , t, ¯ t, s ) e k ~ ∂ s , (1.17) ord ~ ( φ ( ~ , t, ¯ t, s )) = or d ~ ( X ( ~ , t, ¯ t, s, e ~ ∂ s )) = or d ~ α ( ~ ) = ord ~ ( ¯ X ( ~ , t, ¯ t, s, e ~ ∂ s )) = or d ~ ¯ α ( ~ ) = 0 . (1.18) Here we deﬁne the ~ -o rder and the principal s ymbol of op era tors of the form (1.1 3), in particular those of X a nd ¯ X , by deﬁning or d ~ ( ~ ∂ s ) = 0 and σ ~ ( ~ ∂ s ) = log ξ , which ar e co nsistent with the former deﬁnitions (1.4) and (1.5). The wave functions Ψ( ~ , t, ¯ t, s ; z ) and ¯ Ψ( ~ , t, ¯ t, s ; ¯ z ) are deﬁned by (1.19) Ψ( ~ , t, ¯ t, s ; z ) = W z s/ ~ e ζ ( t,z ) / ~ , ¯ Ψ( ~ , t, ¯ t, s ; ¯ z ) = ¯ W ¯ z s/ ~ e ζ ( ¯ t, ¯ z − 1 ) / ~ , where ζ ( t, z ) = P ∞ n =1 t n z n , ζ ( ¯ t, ¯ z − 1 ) = P ∞ n =1 ¯ t n ¯ z − n . They are solutions of linear equations L Ψ = z Ψ , ~ ∂ Ψ ∂ t n = B n Ψ , ~ ∂ Ψ ∂ ¯ t n = ¯ B n Ψ , ( n = 1 , 2 , . . . ) , ¯ L ¯ Ψ = ¯ z ¯ Ψ , ~ ∂ ¯ Ψ ∂ t n = B n ¯ Ψ , ~ ∂ ¯ Ψ ∂ ¯ t n = ¯ B n ¯ Ψ , ( n = 1 , 2 , . . . ) , and hav e the WKB form (3.3), as we shall show in Section 3. Moreover they are expressed by means of the tau function τ ( ~ , t, ¯ t, s ) as follows: (1.20) Ψ( ~ , t, ¯ t ; z ) = τ ( ~ , t − ~ [ z − 1 ] , ¯ t, s ) τ ( ~ , t, ¯ t, s ) z α ( ~ ) / ~ z s/ ~ e ζ ( t,z ) / ~ , ¯ Ψ( ~ , t, ¯ t ; ¯ z ) = τ ( ~ , t, ¯ t − ~ [ ¯ z ] , s + ~ ) τ ( ~ , t, ¯ t, s ) ¯ z ¯ α ( ~ ) / ~ ¯ z s/ ~ e ζ ( ¯ t, ¯ z − 1 ) / ~ where [ z − 1 ] = (1 /z , 1 / 2 z 2 , 1 / 3 z 3 , . . . ), [ ¯ z ] = ( ¯ z , ¯ z 2 / 2 , ¯ z 3 / 3 , . . . ). W e shall study the ~ -expansion of the tau function in Section 4. The Orlov-Schulman op er ators M and ¯ M [OS] are deﬁned by M = Ad  W exp  ~ − 1 ζ ( t, e ~ ∂ s )  s = W ∞ X n =1 nt n e n ~ ∂ s + s ! W − 1 (1.21) ¯ M = Ad  ¯ W exp  ~ − 1 ζ ( ¯ t, e − ~ ∂ s )  s = ¯ W − ∞ X n =1 nt n e − n ~ ∂ s + s ! ¯ W − 1 (1.22) 6 KANEHISA T AKASAK I AND T AKASHI T AK EBE where ζ ( t, e ~ ∂ s ) = P ∞ n =1 t n e n ~ ∂ s and ζ ( ¯ t, e − ~ ∂ s ) = P ∞ n =1 ¯ t n e − n ~ ∂ s . I t is easy to se e that M and ¯ M hav e for ms M = ∞ X n =1 nt n L n + s + α ( ~ ) + ∞ X n =1 v n ( ~ , t, ¯ t, s ) L − n , (1.23) ¯ M = − ∞ X n =1 n ¯ t n ¯ L − n + s + ¯ α ( ~ ) + ∞ X n =1 ¯ v n ( ~ , t, ¯ t, s ) ¯ L n . (1.24) and sa tisﬁes the following prop erties: • ord ~ ( M ) = or d ~ ( ¯ M ) = 0; • the canonical comm utation relation: [ L, M ] = ~ L and [ ¯ L, ¯ M ] = ~ ¯ L ; • the same La x equa tions a s L , ¯ L : (1.25) ~ ∂ M ∂ t n = [ B n , M ] , ~ ∂ M ∂ ¯ t n = [ ¯ B n , M ] , ~ ∂ ¯ M ∂ t n = [ B n , ¯ M ] , ~ ∂ ¯ M ∂ ¯ t n = [ ¯ B n , ¯ M ] , n = 1 , 2 , . . . ; • another linear equation for the wav e function Ψ: M Ψ = ~ z ∂ Ψ ∂ z , ¯ M ¯ Ψ = ~ ¯ z ∂ ¯ Ψ ∂ ¯ z . R emark 1.1 . As in the KP case (Remark 1 .2 in [TT3]), if an op erato r M of the form (1.23) and and an op er ator ¯ M o f the form (1.24) satisfy the Lax equations (1.25) and the cano nic a l co mm utation relation [ L, M ] = ~ L and [ ¯ L, ¯ M ] = ~ ¯ L with the Lax opera tor L a nd ¯ L o f the T o da lattice hierarch y , then α ( ~ ) and ¯ α ( ~ ) in the expansions (1.2 3) and (1.24) do not dep end on any t n , ¯ t n nor s . In fact, supp ose that α ( ~ ) depens on s ; α ( ~ ) = α ( ~ , s ). Then, the ca nonical commutation relation [ L, M ] = ~ L is expanded as ( ~ + α ( ~ , s + ~ ) − α ( ~ , s )) e ~ ∂ s + (diﬀerence o p e rators of low er or der) = ~ L , which implies α ( ~ , s + ~ ) = α ( ~ , s ). Similarly , fro m (1.25) follows ∂ α ∂ t n = ∂ α ∂ ¯ t n = 0 with the help of (1 .1) a nd [ L n , M ] = n ~ L n . F or ¯ α ( s ) the pro of is the same. The fo llowing pr op osition (Pr o p osition 2 .7.11 of [TT2]) is a “ disp ersionful” co un- terpart of the theor em for the dispersionle s s T oda hiera rch y found ear lier (cf. Sec tio n 4 o f [TT1] and P rop ositio n 1.3 b elow). Prop ositi on 1.2. ( i) Su pp ose that op er ators f ( ~ , s, e ~ ∂ s ) , g ( ~ , s, e ~ ∂ s ) , ¯ f ( ~ , s, e ~ ∂ s ) , ¯ g ( ~ , s, e ~ ∂ s ) , L , ¯ L , M and ¯ M satisfy the fol lowing c onditions: • ord ~ f = ord ~ g = ord ~ ¯ f = o rd ~ ¯ g = 0 , [ f , g ] = ~ f , [ ¯ f , ¯ g ] = ~ ¯ f ; • L , ¯ L , M and ¯ M ar e of the form (1.2) , (1.3) , (1.23) and (1.2 4) r esp e ctively. They ar e c anonic al ly c ommut ing: [ L, M ] = ~ L , [ ¯ L, ¯ M ] = ~ ¯ L ; • Equations (1.26) f ( ~ , M , L ) = ¯ f ( ~ , ¯ M , ¯ L ) , g ( ~ , M , L ) = ¯ g ( ~ , ¯ M , ¯ L ) hold. Then the p air ( L, ¯ L ) is a solution of the T o da lattic e hier ar chy (1.1) and M and ¯ M ar e the c orr esp onding Orlov-Schulman op er ators. (ii) Conversely, for any solution ( L, ¯ L, M , ¯ M ) of the ~ -dep endent T o da lattic e hier ar chy ther e exists a quadruplet ( f , ¯ f , g , ¯ g ) satisfying t he c onditions in (i). ~ -EXP ANS ION OF TOD A 7 The leading term of the ~ -dependent T o da lattice hier arch y with resp ect to the ~ -order gives the disp ersionless T o da hier ar chy . Namely , L := σ ~ ( L ) = ξ + ∞ X n =0 u 0 ,n +1 ξ − n , ( u 0 ,n +1 := σ ~ ( u n +1 )) , (1.27) ¯ L − 1 = σ ~ ( ¯ L − 1 ) = ¯ u 0 , 0 ξ − 1 + ∞ X n =0 ¯ u 0 ,n +1 ξ n , ( ¯ u 0 ,n +1 := σ ~ ( ¯ u n +1 )) (1.28) satisfy the disp ersio nles s Lax t yp e equations (1.29) ∂ L ∂ t n = {B n , L} , ∂ L ∂ ¯ t n = { ¯ B n , L} , ∂ ¯ L ∂ t n = {B n , ¯ L} , ∂ ¯ L ∂ ¯ t n = { ¯ B n , ¯ L} , B n = ( L n ) ≥ 0 , ¯ B n = ( ¯ L − n ) ≤− 1 , n = 1 , 2 , . . . , where ( ) ≥ 0 and ( ) ≥ 0 are the truncation of Laurent ser ie s to the p o lynomial part and to the nega tive order part r esp ectively . The Poisson br a ck et { , } is deﬁned by (1.30) { a ( s, ξ ) , b ( s, ξ ) } = ξ  ∂ a ∂ ξ ∂ b ∂ s − ∂ a ∂ s ∂ b ∂ ξ  . The dressing op era tio n (1.7) for L and ¯ L b eco mes the following dressing op era- tion for L a nd ¯ L : (1.31) L = exp  ad { , } X 0  ξ , X 0 := σ ~ ( X ) , ¯ L = exp  ad { , } φ 0  exp  ad { , } ¯ X 0  ξ , φ 0 := σ ~ ( φ ) , ¯ X 0 := σ ~ ( ¯ X ) , where a d { , } ( f )( g ) := { f , g } . The pr incipal symbol of the Orlov-Sch ulman oper ators ar e Orlov-Schulman func- tions , M = ∞ X n =1 nt n L n + s + α 0 + ∞ X n =1 v 0 ,n L − n , (1.32) ( v 0 ,n := σ ~ ( v n ) , α 0 := σ ~ ( α ( ~ ))) ¯ M = − ∞ X n =1 n ¯ t n ¯ L − n + s + ¯ α 0 + ∞ X n =1 ¯ v 0 ,n ¯ L n , (1.33) ( ¯ v 0 ,n := σ ~ ( ¯ v n ) , ¯ α 0 := σ ~ ( ¯ α 0 ( ~ ))) . which ar e eq ual to M = exp  ad { , } X 0  exp  ad { , } ζ ( t, ξ )  s, (1.34) ¯ M = exp  ad { , } φ 0  exp  ad { , } ¯ X 0  exp  ad { , } ζ ( ¯ t, ξ − 1 )  s (1.35) where ζ ( t, ξ ) = P ∞ n =1 t n ξ n and ζ ( ¯ t, ξ − 1 ) = P ∞ n =1 ¯ t n ξ − n . The ser ies M satisﬁes the ca nonical commutation relation w ith L , {L , M} = L , while ¯ M satisﬁes the canonical co mm utation r elation with ¯ L , { ¯ L , ¯ M} = ¯ L . The principal symbols o f 8 KANEHISA T AKASAK I AND T AKASHI T AK EBE equations (1.2 5) g ive the Lax t yp e equations: (1.36) ~ ∂ M ∂ t n = {B n , M} , ~ ∂ M ∂ ¯ t n = { ¯ B n , M} , ~ ∂ ¯ M ∂ t n = {B n , ¯ M} , ~ ∂ ¯ M ∂ ¯ t n = { ¯ B n , ¯ M} , n = 1 , 2 , . . . , The Riemann-Hilb ert type construc tio n of the so lution is e s sentially the same as Prop o s ition 1.2. (Pr op osition 2.5.1 of [TT2]; W e do not need to assume the canonical co mmu tation relation {L , M} = L and { ¯ L , ¯ M} = ¯ L .) Prop ositi on 1. 3. (i) S u pp ose that fu n ctions f 0 ( s, ξ ) , g 0 ( s, ξ ) , ¯ f 0 ( s, ξ ) , ¯ g 0 ( s, ξ ) , L , ¯ L , M and ¯ M satisfy the fol lowing c onditions: • { f 0 , g 0 } = f 0 , { ¯ f 0 , ¯ g 0 } = ¯ f 0 ; • L , ¯ L − 1 , M and ¯ M have the form (1.27) , (1.2 8) , (1.32) and (1.33) r esp e c- tively. • Equations (1.37) f 0 ( M , L ) = ¯ f 0 ( ¯ M , ¯ L ) , g 0 ( M , L ) = ¯ g 0 ( ¯ M , ¯ L ) . hold. Then the p air ( L , ¯ L ) is a solution of the disp ersionless T o da hier ar chy (1.29) and M and ¯ M ar e the c orr esp onding Orlov-Schulman funct ions. (ii) Conversely, for any solution ( L , ¯ L , M , ¯ M ) of t he disp ersionless T o da hier- ar chy, ther e ex ists a quadruplet ( f 0 , g 0 , ¯ f 0 , ¯ g 0 ) satisfying the c onditions in (i). If ( f , g , ¯ f , ¯ g ), ( L, ¯ L, M , ¯ M ) are as in Prop os itio n 1.2, then ( f 0 = σ ~ ( f ) , g 0 = σ ~ ( g ) , ¯ f 0 = σ ~ ( ¯ f ) , ¯ g 0 = σ ~ ( ¯ g )), ( L = σ ~ ( L ) , ¯ L = σ ~ ( ¯ L ) , M = σ ~ ( M ) , ¯ M = σ ~ ( ¯ M )) satisfy the conditions in Prop ositio n 1.3 . In other words, ( f , g , ¯ f , ¯ g ) and ( L, ¯ L, M , ¯ M ) are qua ntisation of ( f 0 , g 0 , ¯ f 0 , ¯ g 0 ) and ( L , M , ¯ L , ¯ M ) r esp ectively . (See, for e x ample, [S] for quan tised canonical transfo r mations.) 2. Recursive construction o f the dressing o pera tor In this section we prove that the solutio n o f the ~ - dep endent T o da la ttice hier ar- ch y corres p o nding to ( f , g , ¯ f , ¯ g ) in Prop os itio n 1.2 is recur s ively co ns tructed from its leading ter m, i.e., the so lution of the disper sionless T o da hierar ch y corresp onding to the Riemann-Hilb ert data ( σ ~ ( f ) , σ ~ ( g ) , σ ~ ( ¯ f ) , σ ~ ( ¯ g )). Given the quadruplet ( f , g , ¯ f , ¯ g ), we hav e to cons truct the dressing oper a tor W and ¯ W , or, in other words, X in (1.1 4) a nd φ , ¯ X in (1 .16), s uch that equations (1.26) hold, or equiv alently , the following equations hold: (2.1) Ad  W exp  ~ − 1 ζ ( t, e ~ ∂ s )  f ( ~ , s, e ~ ∂ s ) = Ad  ¯ W exp  ~ − 1 ζ ( ¯ t, e − ~ ∂ s )  ¯ f ( ~ , s, e ~ ∂ s ) , Ad  W exp  ~ − 1 ζ ( t, e ~ ∂ s )  g ( ~ , s, e ~ ∂ s ) = Ad  ¯ W exp  ~ − 1 ζ ( ¯ t, e − ~ ∂ s )  ¯ g ( ~ , s, e ~ ∂ s ) . ~ -EXP ANS ION OF TOD A 9 Let us expand X , ¯ X and φ with resp ect to the ~ -order as follows: X = ∞ X n =0 ~ n X n , X n = X n ( t, ¯ t, s, e ~ ∂ s ) = α n ~ ∂ ∂ s + ∞ X k =1 χ n,k ( t, ¯ t, s ) e − k ~ ∂ s , (2.2) ¯ X = ∞ X n =0 ~ n ¯ X n , ¯ X n = ¯ X n ( t, ¯ t, s, e ~ ∂ s ) = ¯ α n ~ ∂ ∂ s + ∞ X k =1 ¯ χ n,k ( t, ¯ t, s ) e k ~ ∂ s , (2.3) φ = ∞ X n =0 ~ n φ n ( t, ¯ t, s ) , φ n = φ n ( t, ¯ t, s ) , (2.4) where α n , ¯ α n , χ n,k and ¯ χ n,k do not dep end on ~ , a nd hence α , χ k in (1.15) and ¯ α , ¯ χ k in (1.17) a re expanded a s α = P ∞ n =0 ~ n α n , χ k = P ∞ n =0 ~ n χ n,k , ¯ α = P ∞ n =0 ~ n ¯ α n and ¯ χ k = P ∞ n =0 ~ n ¯ χ n,k . Assume that a s o lution of the disp ersionle s s T o da hierarch y cor resp onding to ( σ ~ ( f ) , σ ~ ( g ) , σ ~ ( ¯ f ) , σ ~ ( ¯ g )) is given. In other words, assume that symbols X 0 = α 0 log ξ + P ∞ k =1 χ 0 ,k ( t, ¯ t, s ) ξ − k , ¯ X 0 = ¯ α 0 log ξ + P ∞ k =1 ¯ χ 0 ,k ( t, ¯ t, s ) ξ k and φ 0 = φ 0 ( t, ¯ t, s ) are given such that σ ~ ( f )( M , L ) = σ ~ ( ¯ f )( ¯ M , ¯ L ) , namely , exp  ad { , } X 0  exp  ad { , } ζ ( t, ξ )  σ ~ ( f )( s, ξ ) = exp  ad { , } φ 0  exp  ad { , } ¯ X 0  exp  ad { , } ζ ( ¯ t, ξ − 1 )  σ ~ ( ¯ f )( s, ξ ) , and σ ~ ( g )( M , L ) = σ ~ ( ¯ g )( ¯ M , ¯ L ) , namely , (2.5) exp  ad { , } X 0  exp  ad { , } ζ ( t, ξ )  σ ~ ( g )( s, ξ ) = exp  ad { , } φ 0  exp  ad { , } ¯ X 0  exp  ad { , } ζ ( ¯ t, ξ − 1 )  σ ~ ( ¯ g )( s, ξ ) . (See Pr op osition 1 .3 .) W e ar e to construct X n , ¯ X n and φ n recursively , starting from X 0 , ¯ X 0 and φ 0 . F or this purp ose expand both sides of equatio ns (2.1) as follows: P := Ad  exp( ~ − 1 X )  f t = ∞ X k =0 ~ k P k , (2.6) Q := Ad  exp( ~ − 1 X )  g t = ∞ X k =0 ~ k Q k , (2.7) ¯ P := Ad  exp( ~ − 1 φ ) exp( ~ − 1 ¯ X )  ¯ f ¯ t = ∞ X k =0 ~ k ¯ P k , (2.8) ¯ Q := Ad  exp( ~ − 1 φ ) exp( ~ − 1 ¯ X )  ¯ g ¯ t = ∞ X k =0 ~ k ¯ Q k , (2.9) where f t := Ad  e ~ − 1 ζ ( t,e ~ ∂ s )  f , g t := Ad  e ~ − 1 ζ ( t,e ~ ∂ s )  g , (2.10) ¯ f ¯ t := Ad  e ~ − 1 ζ ( ¯ t,e − ~ ∂ s )  ¯ f , ¯ g ¯ t := Ad  e ~ − 1 ζ ( ¯ t,e − ~ ∂ s )  ¯ g , (2.11) 10 KANEHISA T AKASAK I AND T AKASHI T AK EBE and P i ’s, Q i ’s, ¯ P i ’s and ¯ Q i ’s are diﬀerence op erators of the ~ -order 0 : ord ~ P i = ord ~ Q i = ord ~ ¯ P i = or d ~ ¯ Q i = 0 . Suppo se that we have chosen X 0 , . . . , X i − 1 , ¯ X 0 , . . . , ¯ X i − 1 and φ 0 , . . . , φ i − 1 so tha t P j = ¯ P j (0 ≦ j ≦ i − 1) and Q j = ¯ Q j (0 ≦ j ≦ i − 1). If o per ators X i , ¯ X i and a function φ i are co nstructed from these given X j , ¯ X j and φ j (0 ≦ j ≦ i − 1) so that equations P i = ¯ P i and Q i = ¯ Q i hold, this pro cedur e g ives re c ursive constructio n of X , ¯ X and φ in question. W e can construct such X i , ¯ X i and φ i as follows. (Details and mea ning shall b e explained in the pro of o f Theorem 2 .1.): • (Step 0) Ass ume X j , ¯ X j and φ j (0 ≦ j ≦ i − 1) are given and set (2.12) X ( i − 1) := i − 1 X n =0 ~ n X n , ¯ X ( i − 1) := i − 1 X n =0 ~ n ¯ X n , φ ( i − 1) := i − 1 X n =0 ~ n φ n . • (Step 1) Set P ( i − 1) := Ad  exp ~ − 1 X ( i − 1)  f t , (2.13) Q ( i − 1) := Ad  exp ~ − 1 X ( i − 1)  g t , (2.14 ) ¯ P ( i − 1) := Ad  exp ~ − 1 φ ( i − 1)  Ad  exp ~ − 1 ¯ X ( i − 1)  ¯ f ¯ t , (2.15) ¯ Q ( i − 1) := Ad  exp ~ − 1 φ ( i − 1)  Ad  exp ~ − 1 ¯ X ( i − 1)  ¯ g ¯ t , (2.16 ) and expand P ( i − 1) and Q ( i − 1) with r esp ect to the ~ -order a s P ( i − 1) = ∞ X k =0 ~ k P ( i − 1) k , Q ( i − 1) = ∞ X k =0 ~ k Q ( i − 1) k , (2.17) ¯ P ( i − 1) = ∞ X k =0 ~ k ¯ P ( i − 1) k , ¯ Q ( i − 1) = ∞ X k =0 ~ k ¯ Q ( i − 1) k . (2.18) (ord ~ P ( i − 1) k = ord ~ Q ( i − 1) k = or d ~ ¯ P ( i − 1) k = or d ~ ¯ Q ( i − 1) k = 0.) • (Step 2) P ut 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 ) , ¯ 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 serie s ˜ X i ( t, ¯ t, s, ξ ) = α i log ξ + P ∞ k =1 ˜ χ i,k ( t, ¯ t, s ) ξ − k , ˜ ¯ X i ( t, ¯ t, s, ξ ) = ¯ α i log ξ + P ∞ k =1 ˜ ¯ χ i,k ( t, ¯ t, s ) ξ − k and a function φ i ( t, ¯ t, s ) by one of the follow- ing integrals. (The integrand of the ﬁrst integral in the right hand side of each equa tion is consider ed as a ser ies of ξ ar ound ξ = ∞ and the integrand ~ -EXP ANS ION OF TOD A 11 of the seco nd in tegr al is considered a s a s e ries around ξ = 0.) − ˜ X i + φ i + ˜ ¯ X i = Z ξ P − 1 0  − ∂ Q 0 ∂ ξ P ( i − 1) i + ∂ P 0 ∂ ξ Q ( i − 1) i  dξ (2.19) − Z ξ ¯ P − 1 0  − ∂ ¯ Q 0 ∂ ξ ¯ P ( i − 1) i + ∂ ¯ P 0 ∂ ξ ¯ Q ( i − 1) i  dξ , − ˜ X i + φ i + ˜ ¯ X i = Z s P − 1 0  − ∂ Q 0 ∂ s P ( i − 1) i + ∂ P 0 ∂ s Q ( i − 1) i  ds (2.20) − Z s ¯ P − 1 0  − ∂ ¯ Q 0 ∂ s ¯ P ( i − 1) i + ∂ ¯ P 0 ∂ s ¯ Q ( i − 1) i  ds, In fact they give the s ame ˜ X i , φ i and ˜ ¯ X i . E xactly s p ea king, the co e ﬃcie nts of ξ n ( n ∈ Z , n 6 = 0) and lo g ξ a re determined by the ﬁrst equation (2.1 9) and its integral constant φ i is deter mined by the second equa tion (2.20) up to an arbitra ry additive consta nt . Equations (2 .19) and (2.20) determine the c ombination − α i + ¯ α i as the co eﬃcient o f log ξ but no t α i nor ¯ α i separately , whic h can b e chosen ar bi- trarily so far as − α i + ¯ α i is ﬁxed. • (Step 3 ) Deﬁne a ser ies X i ( t, ¯ t, s, ξ ) = α i log ξ + P ∞ k =1 χ i,k ( t, ¯ t, s ) ξ − k and ¯ X i ( t, ¯ t, s, ξ ) = ¯ α i log ξ + P ∞ k =1 ¯ χ i,k ( t, ¯ t, s ) 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 = ˜ ¯ 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 := e − ad { , } φ 0 ˜ ¯ X i (2.21) Here K 2 p is determined by the generating function (2.22) z e z − 1 = 1 − z 2 + ∞ X p =1 K 2 p z 2 p , i.e., K 2 p = B 2 p / (2 p )!, w her e B 2 p ’s ar e the Bernoulli n umbers. • (Step 4) The o p erators X i ( t, ¯ t, s, e ~ ∂ s ) and ¯ X i ( t, ¯ t, s, e ~ ∂ s ) are deﬁned as the op erator s with the principa l symbols X i and ¯ X i : (2.23) X i = ∞ X k =1 χ i,k ( t, ¯ t, s ) e − k ~ ∂ s , ¯ X i = ∞ X k =1 ¯ χ i,k ( t, ¯ t, s ) e k ~ ∂ s . The main theorem is the following: Theorem 2.1. Assum e that X 0 , ¯ X 0 and φ 0 satisfy (2.5) and c onstruct X i ’s, ¯ X ′ i s and φ i ’s by the ab ove pr o c e dur e r e cursively. Then X , ¯ X and φ deﬁne d by (2.2 ) , (2.3) and (2.4) satisfy (1.26) . Namely W = exp( X/ ~ ) and ¯ W = ex p( φ/ ~ ) exp( ¯ X / ~ ) ar e dr essing op er ators of the ~ -dep endent T o da hier ar chy c orr esp onding to the data ( f , g , ¯ f , ¯ g ) . The rest o f this s ection is the pro of of Theorem 2.1 by induction. The essential idea o f the pro of is almos t the sa me as the proo f of Theorem 2.1 in [TT3]. 12 KANEHISA T AKASAK I AND T AKASHI T AK EBE Let us denote the “known” part of X , ¯ X a nd φ by X ( i − 1) , ¯ X ( i − 1) and φ ( i − 1) as in (2.1 2) and, as in termedia te o b jects, consider P ( i − 1) , Q ( i − 1) , ¯ P ( i − 1) and ¯ Q ( i − 1) deﬁned b y (2.13) a nd (2.14), which are expanded as (2 .1 7) and (2.18). If X , ¯ X and φ are expa nded a s (2.2), (2.3) and (2.4 ), the dressing op er ators W = exp( X/ ~ ) a nd ¯ W = exp( φ/ ~ ) exp ( ¯ X / ~ ) are factor ised a s follo ws b y the Campbell- Hausdorﬀ theorem: W = exp  ~ i − 1 ˜ X i + ~ i X >i  exp  ~ − 1 X ( i − 1)  , (2.24) ¯ W = exp  ~ i − 1 φ i + ~ i φ >i  exp  ~ i − 1 ˜ ¯ X i + ~ i ¯ X >i  × (2.25) × exp  ~ − 1 φ ( i − 1)  exp  ~ − 1 ¯ X ( i − 1)  , where ˜ X i , X >i , φ i , φ >i , ˜ ¯ X i and ¯ X >i hav e ~ -o rder not more than 0 and the principal symbols of ˜ X i and ˜ ¯ X i are deﬁned by σ ~ ( ˜ X i )( s, ξ ) = ∞ X n =1 (ad { , } σ ~ ( X 0 )) n − 1 n ! σ ~ ( X i ) (2.26) σ ~ ( ˜ ¯ X i )( s, ξ ) = e ad { , } φ 0 ∞ X n =1 (ad { , } σ ~ ( ¯ X 0 )) n − 1 n ! σ ~ ( ¯ X i ) ! (2.27) Note tha t the log terms in (2.2 6) and (2.27) are α i log ξ and ¯ α i log ξ re s p e c tively . The other terms in (2.26) (resp. (2.27)) are nega tive (resp. p ositive) p ow ers of ξ . The principa l symbo l of X i is recov ered from ˜ 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 ) , (2.28) Here co eﬃcients K 2 p are deﬁned b y (2.22). Simila r ly the pr incipal s ymbol of ¯ X i is recov ered from ˜ ¯ X i by 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 ) , (2.29) where σ ~ ( ˜ ¯ X ′ i ) := e − ad { , } φ 0 σ ~ ( ˜ ¯ X i ). These inv ers ion relations are the origin o f (2.21). (Note that the principal s ymbol determines the opera tors X i and ¯ X i , since they are homogeneous terms in the expansions (2.2) and (2.3).) The factorisation formula (2.2 4) and the inv ers ion formula (2.28) are proved in Appendix A o f [TT3]. The formulae (2.2 5) and (2.29) are derived in the same wa y . The factorisation (2.24) implies P = Ad  exp  ~ i − 1 ˜ X i + ~ i X >i   P ( i − 1) = P ( i − 1) + ~ i − 1 [ ˜ X i + ~ X >i , P ( i − 1) ] + (terms of ~ -order < − i ) . Thu s, substituting the expansio n (2.17) in the Step 1, we have P = P ( i − 1) 0 + ~ P ( i − 1) 1 + · · · + ~ i P ( i − 1) i + · · · + ~ i − 1 [ ˜ X i , P ( i − 1) 0 ] + (terms of ~ -order < − i ) . (2.30) ~ -EXP ANS ION OF TOD A 13 Comparing this with the ~ -e xpansion (2.6) of P , we can express P j ’s in ter ms o f P ( i − 1) j ’s and ˜ X i as fo llows: P j = P ( i − 1) j ( j = 0 , . . . , i − 1) , (2.31) σ 0 ( P i ) = σ 0 ( P ( i − 1) i + ~ − 1 [ ˜ X i , P ( i − 1) 0 ]) . (2.32) Similar equations for Q are obtained in the same wa y . F or the op era tors ¯ P the corres p o nding eq uations are ¯ P j = ¯ P ( i − 1) j ( j = 0 , . . . , i − 1 ) , (2.33) σ 0 ( ¯ P i ) = σ 0 ( ¯ P ( i − 1) i + ~ − 1 [ φ i , ¯ P ( i − 1) 0 ] + ~ − 1 [ ˜ ¯ X i , ¯ P ( i − 1) 0 ]) . (2.34) The cor resp onding e quations for ¯ Q a r e the same. The eq uations (2.3 1), (2.33) a nd co rresp onding e q uations for Q and ¯ Q show that the terms of ~ -o rder greater than − i in (2.6) ar e already ﬁxed by X 0 , . . . , X i − 1 , which justiﬁes the inductiv e pr o cedure. Tha t is to say , we are assuming tha t X 0 , . . . , X i − 1 hav e b e en alrea dy determined so tha t P j = P ( i − 1) j and Q j = Q ( i − 1) j for j = 0 , . . . , i − 1 coincide with ¯ P j = ¯ P ( i − 1) j and ¯ Q j = ¯ Q ( i − 1) j resp ectively . The op er a tors X i , ¯ X i and the function φ i should b e chosen so that the right hand sides o f (2.32) a nd (2.34) coincide and the c orresp o nding expre s sions for Q and ¯ Q coincide. T aking equations P ( i − 1) 0 = P 0 , Q ( i − 1) 0 = Q 0 , ¯ 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 [ ˜ X i , P 0 ] , ˜ Q ( i ) i := Q ( i − 1) i + ~ − 1 [ ˜ X i , Q 0 ] , ˜ ¯ P ( i ) i := ¯ P ( i − 1) i + ~ − 1 [ φ i , ¯ P 0 ] + ~ − 1 [ ˜ ¯ X i , ¯ P 0 ] ˜ ¯ Q ( i ) i := ¯ Q ( i − 1) i + ~ − 1 [ φ i , ¯ Q 0 ] + ~ − 1 [ ˜ ¯ X i , ¯ Q 0 ] (2.35) Then the condition for X i , ¯ X i and φ i is wr itten in the following form of equa tions for s ymbols: (2.36) σ ~ 0 ( ˜ P ( i ) i ) = σ ~ 0 ( ˜ ¯ P ( i ) i ) , σ ~ 0 ( ˜ Q ( i ) i ) = σ ~ 0 ( ˜ ¯ Q ( i ) i ) (The par ts of ~ -or der less than − 1 should b e determined in the next step o f the induction.) T o simplify nota tions, we denote the symbols σ ~ 0 ( ˜ P ( i ) i ), σ ~ 0 ( P ( i − 1) i ) and so on b y the corres po nding calligra phic letter s a s ˜ P ( i ) i , P ( i − 1) i etc. By this notation we ca n rewr ite the equations (2.36) in the following for m: (2.37) ˜ P ( i ) i = ˜ ¯ P ( i ) i , ˜ Q ( i ) i = ˜ ¯ Q ( i ) i , ˜ P ( i ) i := P ( i − 1) i + { ˜ X i , P 0 } , ˜ Q ( i ) i := Q ( i − 1) i + { ˜ X i , Q 0 } , ˜ ¯ P ( i ) i := ¯ P ( i − 1) i + { φ i , ¯ P 0 } + { ˜ ¯ X i , ¯ P 0 } , ˜ ¯ Q ( i ) i := ¯ Q ( i − 1) i + { φ i , ¯ Q 0 } + { ˜ ¯ X i , ¯ Q 0 } . 14 KANEHISA T AKASAK I AND T AKASHI T AK EBE In the matrix fo rm, these equations are encapsulated in the fo llowing equa tion.    P ( i − 1) i Q ( i − 1) i    + ξ      ∂ P 0 ∂ s − ∂ P 0 ∂ ξ ∂ Q 0 ∂ s − ∂ Q 0 ∂ ξ           ∂ ∂ ξ ˜ X i ∂ ∂ s ˜ X i      =    ¯ P ( i − 1) i ¯ Q ( i − 1) i    + ξ       ∂ ¯ P 0 ∂ s − ∂ ¯ P 0 ∂ ξ ∂ ¯ Q 0 ∂ s − ∂ ¯ Q 0 ∂ ξ            ∂ ∂ ξ ( φ i + ˜ ¯ X i ) ∂ ∂ s ( φ i + ˜ ¯ X i )      (2.38) Recall that op era to rs P ( i − 1) and Q ( i − 1) are deﬁned by acting adjoint o p er ation to the canonically commuting pair ( f , g ) in (2.13), (2.1 4) and (2.10). Hence they also sa tisfy the canonical commutation relatio n: [ P ( i − 1) , Q ( i − 1) ] = ~ P ( i − 1) . The principal sym b ol of this relation g ives (2.39) {P ( i − 1) 0 , Q ( i − 1) 0 } = {P 0 , Q 0 } = P 0 , which mea ns that the deter minants o f the 2 × 2 matrices in b oth sides of (2.38) are equal to ξ − 1 P 0 . (Recall that those matrices are equal b ecaus e of the induction hypothesis, P 0 = ¯ P 0 , Q 0 = ¯ Q 0 .) Hence its inv er s e matrix is easily computed a nd we have P − 1 0      − ∂ Q 0 ∂ ξ ∂ P 0 ∂ ξ − ∂ Q 0 ∂ s ∂ P 0 ∂ s         P ( i − 1) i Q ( i − 1) i    +      ∂ ∂ ξ ˜ X i ∂ ∂ s ˜ X i      = ¯ P − 1 0      − ∂ ¯ Q 0 ∂ ξ ∂ ¯ P 0 ∂ ξ − ∂ ¯ Q 0 ∂ s ∂ ¯ P 0 ∂ s         ¯ P ( i − 1) i ¯ Q ( i − 1) i    +      ∂ ∂ ξ ( φ i + ˜ ¯ X i ) ∂ ∂ s ( φ i + ˜ ¯ X i )      (2.40) Note that the left hand side (the ﬁrs t line) is a serie s of ξ a round ξ = ∞ , while the r ight hand side (the second line) is a ser ies ar ound ξ = 0. Equation (2.4 0) is rewritten as (2.41)      ∂ ∂ ξ ( − ˜ X i + φ i + ˜ ¯ X i ) ∂ ∂ s ( − ˜ X i + φ i + ˜ ¯ X i )      = P − 1 0      − ∂ Q 0 ∂ ξ ∂ P 0 ∂ ξ − ∂ Q 0 ∂ s ∂ P 0 ∂ s         P ( i − 1) i Q ( i − 1) i    − ¯ P − 1 0      − ∂ ¯ Q 0 ∂ ξ ∂ ¯ P 0 ∂ ξ − ∂ ¯ Q 0 ∂ s ∂ ¯ P 0 ∂ s         ¯ P ( i − 1) i ¯ Q ( i − 1) i    , which deter mines − ˜ X i + φ i + ˜ ¯ X i . According to the ab ov e remark, the ﬁrst ter m in the right hand s ide is a ser ies o f ξ a round ξ = ∞ and the second term is a series of ξ ar ound ξ = 0 . ~ -EXP ANS ION OF TOD A 15 The system (2.41) is s olv a ble thanks to Lemma 2.2 b elow. Hence, integrating the ﬁrst element o f the r ight hand side with resp ect to ξ , we obtain − ˜ X i + φ 0 + ˜ ¯ X i up to an integration constant which do es no t dep end on ξ . Integrating the se c o nd element of the a bove equation, we can determine this integration constant up to a constant which do es no t dep end o n s . This is Step 2 , (2.19) and (2.20), which deter mine the symbols ˜ X i , ˜ ¯ X i and the function φ i . (The am biguity of φ i is harmless.) In the end, the principal symbols of X i and ¯ X i are determined by (2.28), (2.29) or (2.21) in Step 3. O p e rators X i and ¯ X i are deﬁned as in Step 4. This completes the construction of X i , ¯ X i and φ i and the pro o f of the theor em. Lemma 2.2. The system (2.41) is c omp atible. Pr o of. W e chec k the compatibility conditio n, ∂ ∂ s  P − 1 0  − ∂ Q 0 ∂ ξ P ( i − 1) i + ∂ P 0 ∂ ξ Q ( i − 1) i  − ∂ ∂ s  ¯ P − 1 0  − ∂ ¯ Q 0 ∂ ξ ¯ P ( i − 1) i + ∂ ¯ P 0 ∂ ξ ¯ Q ( i − 1) i  = ∂ ∂ ξ  P − 1 0  − ∂ Q 0 ∂ s P ( i − 1) i + ∂ P 0 ∂ s Q ( i − 1) i  − ∂ ∂ ξ  ¯ P − 1 0  − ∂ ¯ Q 0 ∂ s ¯ P ( i − 1) i + ∂ ¯ P 0 ∂ s ¯ Q ( i − 1) i  . (2.42) Using the relation {P 0 , Q 0 } = P 0 (2.39), this equatio n reduces to P − 1 0 ξ − 1  −P ( i − 1) i + {P ( i − 1) i , Q 0 } + {P 0 , Q ( i − 1) i }  = ¯ P − 1 0 ξ − 1  − ¯ P ( i − 1) i + { ¯ P ( i − 1) i , ¯ Q 0 } + { ¯ P 0 , ¯ Q ( i − 1) i }  . (2.43) Deﬁned from canonically commuting pa ir ( f , g ) by adjoint action (2.10), (2.13) and (2.14), the pair of oper ators ( P ( i − 1) , Q ( i − 1) ) is canonically comm uting: [ P ( i − 1) , Q ( i − 1) ] = ~ P ( i − 1) . Similarly w e ha ve [ ¯ P ( i − 1) , ¯ Q ( i − 1) ] = ~ ¯ P ( i − 1) and thus (2.44) [ P ( i − 1) , Q ( i − 1) ] − ~ P ( i − 1) = [ ¯ P ( i − 1) , ¯ Q ( i − 1) ] − ~ ¯ P ( i − 1) . Substituting the expa nsions (2.17) and (2.18) in it and no ting that P ( i − 1) j = ¯ P ( i − 1) j and Q ( i − 1) j = ¯ Q ( i − 1) j for j = 0 , . . . , i − 1 by the induction hypo thesis, the terms of ~ -order higher than − i − 1 in (2.44) cancel. Thus (2.44) beco mes [ ~ i P ( i − 1) i , Q 0 ] + [ P 0 , ~ i Q ( i − 1) i ] − ~ i +1 P ( i − 1) i + ( ~ -order < − i − 1) = [ ~ i ¯ P ( i − 1) i , ¯ Q 0 ] + [ ¯ P 0 , ~ i ¯ Q ( i − 1) i ] − ~ i +1 ¯ P ( i − 1) i + ( ~ -order < − i − 1) . T aking the sym b ol of ~ -order − i − 1 of this e quation, we ha ve (2.43) b ecause P 0 = ¯ P 0 .  3. Asymptotics of the w a ve function In this sectio n we prov e that the dressing op erator of the for m (1.1 4) o r (1.8) (with α = 0), i.e., W ( ~ , t, ¯ t, s, e ~ ∂ s ) = exp( X ( ~ , t, ¯ t, s, e ~ ∂ s ) / ~ ) , (3.1) X ( ~ , t, ¯ t, s, e ~ ∂ s ) = ∞ X k =1 χ k ( ~ , t, ¯ t, s ) e − k ~ ∂ s , ord ~ X ≦ 0 , (3.2) 16 KANEHISA T AKASAK I AND T AKASHI T AK EBE gives the wa ve function of the WKB form Ψ( ~ , t, ¯ t, s ; z ) = W z s/ ~ e ζ ( t,z ) / ~ = e S ( ~ ,t, ¯ t,s,z ) / ~ z s/ ~ , ord ~ S ≦ 0 , (3.3) S ( ~ , t , ¯ t, s ; z ) = ∞ X n =0 ~ n S n ( t, ¯ t, s ; z ) + ζ ( t, z ) , ζ ( t, z ) := ∞ X n =1 t n z n , (3.4) and vice versa. As the fac to r e ~ − 1 α ( ~ )( ~ ∂ s ) in (1.8) b ecomes a constant factor z α ( ~ ) / ~ when it is applied to z s/ ~ , w e o mit it here. By changing the sign of s and replac ing z by ¯ z − 1 , we can deduce the formula for the wa ve function ¯ Ψ corre s p o nding to the dress ing op erator ¯ W of the form (1.16) or (1.10) (with ¯ α = 0) immediately fro m the a bove re s ults: if ¯ W has the form (3.5) ¯ W ( ~ , t, ¯ t, s, e ~ ∂ s ) = exp( φ ( ~ , t, ¯ t, s ) / ~ ) exp( ¯ X ( ~ , t, ¯ t, e ~ ∂ s ) / ~ ) , ¯ X ( ~ , t, ¯ t, s, e ~ ∂ s ) = ∞ X k =1 ¯ χ k ( ~ , t, ¯ t, s ) e k ~ ∂ s , ord ~ φ, or d ~ ¯ X ≦ 0 , gives the wa ve function of the WKB form ¯ Ψ( ~ , t, ¯ t, s ; ¯ z ) = ¯ W ¯ z s/ ~ e ζ ( ¯ t, ¯ z − 1 ) / ~ = e ¯ S ( ~ ,t, ¯ t,s, ¯ z ) / ~ ¯ z s/ ~ , ord ~ S ≦ 0 , (3.6) ¯ S ( ~ , t, ¯ t, s ; ¯ z ) = ∞ X n =0 ~ n ¯ S n ( t, ¯ t, s ; ¯ z ) + ζ ( ¯ t, ¯ z − 1 ) , ζ ( ¯ t, ¯ z − 1 ) := ∞ X n =1 ¯ t n ¯ z − n . (3.7) Since the time v a riables t n and ¯ t n do not play any ro le in this section, w e set them to zer o. Let A ( ~ , s, e ~ ∂ s ) = P n a n ( ~ , s ) e n ~ ∂ s be a diﬀerence o p er ator. The total symb ol of A is a p ow er ser ies of ξ deﬁned b y (3.8) σ tot ( A )( ~ , s, ξ ) := X n a n ( ~ , x ) ξ n . Actually , this is the factor which app ear s when the op er ator A is applied to z s/ ~ : (3.9) Az s/ ~ = σ tot ( A )( ~ , s, z ) z s/ ~ . Using this terminology , what we show in this s ection is that a op er ator o f the form e X/ ~ has a total symbol o f the form e S/ ~ and that an o per ator with total s ymbol e S/ ~ has a form e X/ ~ . E x actly spe aking, the main r esults in this section are the following tw o prop ositions. Prop ositi on 3.1. L et X = X ( ~ , s, e ~ ∂ s ) b e a diﬀer enc e op er ator of the form (3.2 ) , which has the ~ -or der 0 : o rd ~ X = 0 . Then the t otal symb ol of e X/ ~ has such a form as (3.10) σ tot (exp( ~ − 1 X ( ~ , s, e ~ ∂ s ))) = e S ( ~ ,s,ξ ) / ~ , wher e S ( ~ , s, ξ ) is a p ower series of ξ − 1 without n on- ne gative p owers of ξ and has an ~ -exp ansion S ( ~ , s , ξ ) = ∞ X n =0 ~ n S n ( s, 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: ~ -EXP ANS ION OF TOD A 17 • (Step 0) Assume that X 0 , . . . , X n are given. Let X i ( s, ξ ) b e the tota l symbol σ tot ( X i ( s, e ~ ∂ s )). • (Step 1) Deﬁne Y ( l ) k,m ( s, s ′ , ξ , ξ ′ ) and S ( l ) ( s, ξ ) b y the following recursio n relations: Y ( l ) k, − 1 = 0 (3.11) S (0) m = 0 , (3.12) Y ( l ) 0 ,m ( s, s ′ , ξ , ξ ′ ) = δ l, 0 X m ( s, ξ ) (3.13) for l ≧ 0, m = 0 , . . . , n , (3.14) Y ( l ) k +1 ,m ( s, s ′ , ξ , ξ ′ ) = 1 k + 1     ξ ∂ ξ ∂ s ′ Y ( l ) k,m − 1 ( s, s ′ , ξ , ξ ′ ) + X 0 ≤ l ′ ≤ l − 1 0 ≤ m ′ ≤ m ξ ∂ ξ Y ( l ′ ) k,m ′ ( s, s ′ , ξ , ξ ′ ) ∂ s ′ S ( l − l ′ ) m − m ′ ( s ′ , ξ ′ )     for k ≧ 0 , a nd (3.15) S ( l +1) m ( s, ξ ) = 1 l + 1 l + m X k =0 Y ( l ) k,m ( s, s, ξ , ξ ) . (W e shall prov e that Y ( l ) k,m = 0 if k > l + m .) Schematically this pro ce dur e go es as 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 ( s, ξ ) = P ∞ l =1 S ( l ) n ( s, ξ ). (The sum makes sense as a p ower se r ies of ξ .) Prop ositi on 3.2. Le t S ( ~ , s, ξ ) = P ∞ n =0 ~ n S n ( s, ξ ) b e a p ower series of ξ − 1 without non-ne gative p owers of ξ . Then ther e exists a diﬀer enc e op er ator X ( ~ , s, e ~ ∂ s ) of the form (3 .2) such t hat ord ~ X ≦ 0 and (3.16) σ tot (exp( ~ − 1 X ( ~ , s, e ~ ∂ s ))) = e S ( ~ ,s,ξ ) / ~ . Mor e over, the c o eﬃcient X n ( s, ξ ) in the ~ -ex p ansion X = P ∞ n =0 ~ n X n of the total symb ol X = X ( ~ , s, ξ ) is determine d by S 0 , . . . , S n in the ~ -exp ansion of S . Explicit pro cedure is as follo ws: • (Step 0) Assume that S 0 , . . . , S n are g iven. Expa nd them into homogeneo us terms with resp ect to p ow ers of ξ : S n ( s, ξ ) = P ∞ j =1 S n,j ( s, ξ ), wher e S n,j is a term of degr ee − j . • (Step 1) Deﬁne Y ( l ) k,n,j ( s, s ′ , ξ , ξ ′ ) a s follows: Y ( l ) k, − 1 ,j ( s, s ′ , ξ , ξ ′ ) = 0 , (3.17) Y ( l ) k,m, 1 ( s, s ′ , ξ , ξ ′ ) = δ l, 0 δ k, 0 S m, 1 ( s, ξ ) (3.18) 18 KANEHISA T AKASAK I AND T AKASHI T AK EBE for m = 0 , . . . , n , k ≧ 0, l ≧ 0 and (3.19) Y ( l ) 0 ,m,j = 0 for m = 0 , . . . , n , l > 0, j ≧ 1. F o r o ther ( l , k , m, j ), ( l, k ) 6 = (0 , 0), Y ( l ) k,m,j are determined by the recursion r elation: (3.20) Y ( l ) k +1 ,m,j ( s, s ′ , ξ , ξ ′ ) = 1 k + 1 ξ ∂ ξ ∂ s ′ Y ( l ) k,m − 1 ,j ( s, s ′ , ξ , ξ ′ )+ + X 0 ≤ l ′ ≤ l − 1 1 ≤ j ′ ≤ j − 1 , 0 ≤ m ′ ≤ m 0 ≤ k ′′ ≤ l − l ′ − 1+ m − m ′ 1 l − l ′ ∂ ξ Y ( l ′ ) k,m ′ ,j ′ ( s, s ′ , ξ , ξ ′ ) ∂ y Y ( l − l ′ − 1) k ′′ ,m − m ′ ,j − j ′ ( s, s, ξ , ξ ) ! . The remaining Y (0) 0 ,m,j is determined by: (3.21) Y (0) 0 ,m,j ( s, s ′ , ξ , ξ ′ ) = S m,j ( s, ξ ) − X ( l,k ) 6 =(0 , 0) 0 ≤ l l + m o r j ≦ l .) 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 ( s, ξ ) = P ∞ j =1 Y (0) 0 ,n,j ( s, s, ξ , ξ ). (The inﬁnite sum is the homo- geneous expansion in terms of p ow ers of ξ .) Combining these prop os itions (and the corre s p o nding statements for ¯ X and ¯ S (3.7)) with the results in Sectio n 2, we can, in principle, make a recursio n formula for S n and ¯ S n ( n = 0 , 1 , 2 , . . . ) of the w ave functions of the solution of the T o da lattice hierarch y cor resp onding to ( f , g , ¯ f , ¯ g ) b y Pr o p osition 1 .2 (i) as follows: let S 0 , . . . , S i − 1 , ¯ S 0 , . . . , ¯ S i − 1 and φ 0 , . . . , φ i − 1 be given. (1) By Prop osition 3.2 a nd its v ariant with the opposite sign of s we have X 0 , . . . , X i − 1 and ¯ X 0 , . . . , ¯ X i − 1 . (2) W e hav e a recursio n formula for X i , ¯ X i and φ i by Theo rem 2 .1. (3) Pro p o sition 3.1 (with its v ar iant) g ives a formula for S i , ¯ S i . If we take the factor e ~ − 1 α ( ~ )( ~ ∂ s ) int o account, this pro ces s b ecomes a little bit more complica ted, but essentially the same. The rest of this section is devoted to the pr o of of Prop ositio n 3.1 and Prop osi- tion 3.2. T o av oid confusion, the commutativ e multiplication of total s y mbols a ( ~ , s, ξ ) and b ( ~ , s, ξ ) as power series is denoted b y a ( ~ , s, ξ ) b ( ~ , s, ξ ) and the no n-commutativ e ~ -EXP ANS ION OF TOD A 19 m ultiplication corr esp onding to the op erato r pro duct is denoted b y a ( ~ , s , ξ ) ◦ b ( ~ , s, ξ ). Reca ll that the latter multiplication is express e d (or deﬁned) a s follows: a ( ~ , s, ξ ) ◦ b ( ~ , s , ξ ) = e ~ ξ ∂ ξ ∂ s ′ a ( ~ , s, ξ ) b ( ~ , s ′ , ξ ′ ) | s ′ = s,ξ ′ = ξ = ∞ X n =0 ~ n n ! ( ξ ∂ ξ ) n a ( ~ , s, ξ ) ∂ n s ′ b ( ~ , s ′ , ξ ′ ) | s ′ = s,ξ ′ = ξ . (3.22) (This co rresp onds to E quation (3.2 1) of [TT3] fo r micro diﬀerent ial op er ators.) The order o f the sym b ol a ( ~ , s, ξ ) (the or der with r esp ect to ξ as a p ower ser ie s of ξ ) is denoted by o rd ξ a ( ~ , s, ξ ): ord ξ  X a m ( ~ , s ) ξ m  def = max { m | a m ( ~ , s ) 6 = 0 } . The ~ -o rder is the sa me as that of op er ators: ord ~ s = ord ~ ξ = 0, ord ~ ~ = − 1. The main idea of pro of of prop o sitions is the sa me a s those in § 3 o f [TT3], which is a forma l version o f Aok i’s exp onential calculus of micr o diﬀerential op erator s, [A]. Since the Euler o p e r ator ξ ∂ ξ do es not low er the or der with resp ect to ξ in contrast to the diﬀerential o p er ator ∂ ξ , pro of of co nv ergence of ser ies like (3.22) as a formal power se r ies is diﬀerent from that in [TT3]. First, w e pro ve the follo wing lemma . Lemma 3.3. Le t a ( ~ , s, ξ ) , b ( ~ , s, ξ ) , p ( ~ , s, ξ ) and q ( ~ , s, ξ ) b e symb ols su ch that ord ξ a ( ~ , s, ξ ) = M , ord ~ a ( ~ , s, ξ ) ≦ 0 , ord ξ b ( ~ , s, ξ ) = N , ord ~ b ( ~ , s, ξ ) ≦ 0 , ord ξ p ( ~ , s, ξ ) ≦ − 1 , ord ξ q ( ~ , s, ξ ) ≦ − 1 , or d ~ p ( ~ , s, ξ ) ≦ 0 , ord ~ q ( ~ , s, ξ ) ≦ 0 . Then t her e exist symb ols c ( ~ , s, ξ ) ( or d ξ c ( ~ , s, ξ ) = N + M , ord ~ c ( ~ , s, ξ ) ≦ 0 ) and r ( ~ , s, ξ ) ( ord ξ r ( ~ , s, ξ ) ≦ 0 , or d ~ r ( ~ , s, ξ ) ≦ 0 ) su ch that (3.23)  a ( ~ , s, ξ ) e p ( ~ ,s,ξ ) / ~  ◦  b ( ~ , s, ξ ) e q ( ~ , s,ξ ) / ~  = c ( ~ , s, ξ ) e r ( ~ ,s,ξ ) / ~ . In the pro of o f Prop os itio n 3.1 a nd P rop osition 3.2, we use the c o nstruction of c and r in the pro of of Lemma 3.3. Pr o of. F ollowing [A], w e introduce a para meter t a nd consider (3.24) π ( t ) = π ( t ; ~ , s, s ′ , ξ , ξ ′ ) := e ~ tξ∂ ξ ∂ s ′ a ( ~ , s, ξ ) b ( ~ , s ′ , ξ ′ ) e  p ( ~ ,s,ξ )+ q ( ~ ,s ′ ,ξ ′ )  / ~ . If we set t = 1, s ′ = s and ξ ′ = ξ , this reduces to the o pe rator pr o duct of (3.22). The ser ies π ( t ) is the unique s olution o f an initial v alue problem: (3.25) ∂ t π = ~ ξ ∂ ξ ∂ s ′ π , π (0) = a ( ~ , s, ξ ) b ( ~ , s ′ , ξ ′ ) e  p ( ~ ,s,ξ )+ q ( ~ ,s ′ ,ξ ′ )  / ~ . W e construct its solution in the following form: (3.26) π ( t ) = ψ ( t ) e w ( t ) / ~ , ψ ( t ) = ψ ( t ; ~ , s, s ′ , ξ , ξ ′ ) = ∞ X n =0 ψ n t n , w ( t ) = w ( t ; ~ , s, s ′ , ξ , ξ ′ ) = ∞ X k =0 w k t k . 20 KANEHISA T AKASAK I AND T AKASHI T AK EBE Later we set t = 1 a nd prov e that ψ (1) and w (1) a re meaningful a s a formal p ow er series o f ξ and ξ ′ . The diﬀeren tial equation (3.2 5) is rewr itten as ∂ ψ ∂ t + ~ − 1 ψ ∂ w ∂ t = ~ ξ ∂ ξ ∂ s ′ ψ + ξ ∂ ξ ψ ∂ s ′ w + ξ ∂ ξ w∂ s ′ ψ + ψ  ξ ∂ ξ ∂ s ′ w + ~ − 1 ξ ∂ ξ w∂ s ′ w  . (3.27) Hence it is suﬃcient to construct ψ ( t ) = ψ ( t ; ~ , s, s ′ , ξ , ξ ′ ) and w ( t ) = w ( t ; ~ , s, s ′ , ξ , ξ ′ ) which sa tisfy ∂ w ∂ t = ~ ξ ∂ ξ ∂ s ′ w + ξ ∂ ξ w∂ s ′ w, (3.28) ∂ ψ ∂ t = ~ ξ ∂ ξ ∂ s ′ ψ + ξ ∂ ξ ψ ∂ s ′ w + ξ ∂ ξ w∂ s ′ ψ . (3.29) (This is a suﬃcient conditio n but not a necess ary condition for π = ψ e w/ ~ to be a solution of (3 .25). The so lution of (3.25) is unique, but ψ and w satisfying (3.27) are not unique at all.) T o begin with, we so lve (3.28) and deter mine w ( t ). Expanding it as w ( t ) = P ∞ k =0 w k t k , w e have a recursio n relation and the initial condition w k +1 = 1 k + 1 ~ ξ ∂ ξ ∂ s ′ w k + k X ν =0 ξ ∂ ξ w ν ∂ s ′ w k − ν ! , w 0 = p ( s, ξ ) + q ( s ′ , ξ ′ ) , (3.30) which determine w k = w k ( ~ , s, s ′ , ξ , ξ ′ ) inductively . In o rder to show that P ∞ k =0 w k conv erges as a for mal p ower ser ie s, let us ex pa nd each w k as fo llows: (3.31) w k ( ~ , s, s ′ , ξ , ξ ′ ) = ∞ X n =0 ~ n w k,n ( s, s ′ , ξ , ξ ′ ) . Expanding (3 .3 0) as a series of ~ , we o btain a re cursion relatio n of w k,n and the initial conditio n w k +1 ,n = 1 k + 1     ξ ∂ ξ ∂ s ′ w k,n − 1 + X k ′ + k ′′ = k n ′ + n ′′ = n ξ ∂ ξ w k ′ ,n ′ ∂ s ′ w k ′′ ,n ′′     , w 0 = p ( s, ξ ) + q ( s ′ , ξ ′ ) . (3.32) Because of the assumption ord ξ p ≦ − 1 and or d ξ q ≦ − 1, w 0 also has the order ≦ − 1 and co nsequently (3.33) ord ξ w 0 ,n ≦ − 1 . (Here or d ξ means the order with re s p e c t to b oth ξ a nd ξ ′ : ord ξ ( P a m,n ( ~ , s ) ξ m ξ ′ n ) def = max { m + n | a m,n ( ~ , s ) 6 = 0 } . ) W e show (3.34) ord ξ w k,n ≦ min( − 1 , − k + n − 1 ) by induction on k . • First, when k = 0, (3.34) holds for an y n ≧ 0 beca use o f (3 .33). ~ -EXP ANS ION OF TOD A 21 • Assume that (3.34) holds for any pair ( k , n ) with n ≧ 0 and k = 0 , . . . , k 0 . Then the righ t hand s ide of (3 .32) with k = k 0 has the orde r (with resp ect to ξ a nd ξ ′ ) le ss than or e qual to − 1 and − k 0 + ( n − 1) − 1 = ( − k ′ + n ′ − 1) + ( − k ′′ + n ′′ − 1) = − k 0 + n − 2 , since ξ ∂ ξ do es no t c hange the order. • Hence (3.34) is true for k = k 0 + 1. Thu s the es tima te (3.34) has b een proved for all k and n . This s hows that w (1) = P ∞ k =0 w k = P ∞ k =0 P ∞ n =0 ~ n w k,n makes s e nse as a formal series of ~ , ξ and ξ ′ . Mor eov er it is o bvious that w k and w (1) are formally regular with r esp ect to ~ . As a next step, we expand ψ ( t ) as ψ ( t ) = P ∞ k =0 ψ k t k and rewr ite (3.29) into a recursion relation and the initial condition: ψ k +1 = 1 k + 1 ~ ξ ∂ ξ ∂ s ′ ψ k + k X ν =0 ( ξ ∂ ξ ψ ν ∂ s ′ w k − ν + ξ ∂ ξ w k − ν ∂ s ′ ψ ν ) ! , ψ 0 = a ( s, ξ ) b ( s ′ , ξ ′ ) (3.35) T o prov e the con vergence as a formal power ser ies, we expand ψ k as (3.36) ψ k ( ~ , s, s ′ , ξ , ξ ′ ) = ∞ X n =0 ~ n ψ k,n ( s, s ′ , ξ , ξ ′ ) , and r e write the recursio n relation as follows. ψ k +1 ,n = 1 k + 1     ξ ∂ ξ ∂ s ′ ψ k,n − 1 + X k ′ + k ′′ = k n ′ + n ′′ = n ( ξ ∂ ξ ψ k ′ ,n ′ · ∂ s ′ w k ′′ ,n ′′ + ∂ s ′ ψ k ′ ,n ′′ · ξ ∂ ξ w k ′′ ,n ′′ )     , ψ 0 = a ( s, ξ ) b ( s ′ , ξ ′ ) (3.37) Our ass umption being ord ξ a ( s, ξ ) = M and o rd ξ b ( s ′ , ξ ′ ) = N , w e ha ve (3.38) ord ξ ψ 0 ,n ≦ M + N . As in the estimate of w k,n , w e pr ov e (3.39) ord ξ ψ k,n ≦ min( M + N , M + N − k + n ) by induction. • First, (3.39) holds for k = 0 and a ny n ≧ 0 beca use o f (3.38). • Assume that (3.39) holds fo r k = 0 , . . . , k 0 and n ≧ 0. The right hand side of (3.3 7) with k = k 0 has the orde r with resp ect to ξ and ξ ′ not more than M + N − k 0 + ( n − 1 ) = ( M + N − k ′ + n ′ ) + ( − k ′′ + n ′′ − 1) = M + N − k 0 + n − 1 nor M + N b ecause of the induction hypo thesis a nd (3.34). • This prov es (3.3 9) for k = k 0 + 1. Thu s we have proved (3.3 9) for any k and n , which shows that the iniﬁnite sum ψ (1) = P ∞ k =0 ψ k = P ∞ k =0 P ∞ n =0 ~ n ψ k,n makes s e nse. The reg ularity of ψ k and ψ (1) is als o o bvious. W e hav e c o nstructed π ( t ) = π ( t ; ~ , s, s ′ , ξ , ξ ′ ) = ψ ( t ; ~ , s, s ′ , ξ , ξ ′ ) e w ( t ; ~ ,s,s ′ ,ξ , ξ ′ ) , which is meaningful also at t = 1. Hence the pro duct a ( ~ , s, ξ ) ◦ b ( ~ , s, ξ ) = π (1; ~ , s, s, ξ , ξ ) is expressed in the for m c ( ~ , s, ξ ) e r ( ~ ,s,ξ ) / ~ , where c ( ~ , s, ξ ) = ψ (1; ~ , s, s, ξ , ξ ), r ( ~ , s, ξ ) = w (1 ; ~ , s, s, ξ , ξ ).  22 KANEHISA T AKASAK I AND T AKASHI T AK EBE Pr o of of Pr op osition 3.1. W e make us e of diﬀerent ial equations s atisﬁed by the op- erator (3.40) E ( t ) = E ( t ; ~ , s , e ~ ∂ s ) := exp  t ~ X ( ~ , s, e ~ ∂ s )  , depe nding on a para meter 1 t . The total symbo l of E ( t ) is deﬁned as (3.41) E ( t ; ~ , s, ξ ) = ∞ X k =0 t k ~ k k ! X ( k ) ( ~ , s, ξ ) , X (0) = 1 , X ( k +1) = X ◦ X ( k ) . T aking the logarithm (as a funct ion, not as a n op era tor) of this, we ca n deﬁne S ( t ) = S ( t ; ~ , s, ξ ) by (3.42) E ( t ; ~ , s, ξ ) = e ~ − 1 S ( t ; ~ ,s,ξ ) What we are to pr ov e is that S ( t ), constructed as a series, makes sense at t = 1 and for mally r egular with resp ect to ~ . Diﬀerent iating (3 .42), we have (3.43) X ( ~ , s, ξ ) ◦ E ( t ; ~ , s, ξ ) = ∂ S ∂ t e S ( t ; ~ ,s,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 rewrite the left hand side as follows. (Hereafter we s ometimes omit the argument ~ of functions for brevity .): (3.44) X ( s, ξ ) ◦ E ( t ; s, ξ ) = Y ( t ; s, s, ξ , ξ ) e S ( t ; s,ξ ) / ~ where Y ( t ; s, s ′ , ξ , ξ ′ ) = P ∞ k =0 Y k and Y k ( t ; s, s ′ , ξ , ξ ′ ) a re deﬁned by Y k +1 ( t ; s, s ′ , ξ , ξ ′ ) = 1 k + 1 ( ~ ξ ∂ ξ ∂ s ′ Y k ( t ; s, s ′ , ξ , ξ ′ ) + ξ ∂ ξ Y k ( t ; s, s ′ , ξ , ξ ′ ) ∂ s ′ S ( t ; s ′ , ξ ′ )) , Y 0 ( t ; s, s ′ , ξ , ξ ′ ) = X ( s, ξ ) . (3.45) Y k ( t ) cor resp onds to ψ k in the pro of of Lemma 3.3, while w k there is δ k, 0 S ( t ). (Recall that the ro le of t is diﬀerent. The para meter t in the pro o f of Lemma 3 .3 is set to 1 here.) On the other hand, substituting (3.44) in to the left hand side of (3.43), w e have (3.46) ∂ S ∂ t ( t ; s, ξ ) = Y ( t ; s, s, ξ , ξ ) W e rewrite the system (3.45) a nd (3.46) in terms of expansion of S ( t ; s, ξ ) = S ( t ; ~ , s, ξ ) and Y k ( t ; s, s ′ , ξ , ξ ′ ) = Y k ( t ; ~ , s, s ′ , ξ , ξ ′ ) in powers o f t and ~ : S ( t ; ~ , s, ξ ) = ∞ X l =0 S ( l ) ( ~ , s, ξ ) t l = ∞ X l =0 ∞ X n =0 S ( l ) n ( s, ξ ) ~ n t l , Y k ( t ; ~ , s, s ′ , ξ , ξ ′ ) = ∞ X l =0 Y ( l ) k ( ~ , s, s ′ , ξ , ξ ′ ) t l = ∞ X l =0 ∞ X n =0 Y ( l ) k,n ( s, s ′ , ξ , ξ ′ ) ~ n t l , (3.47) 1 Of course this parameter t does not hav e any relation wi th the time v ari ables of the T o da lattice hier ar c hy . It is not the same t in the pro of of Lemma 3. 3, either. ~ -EXP ANS ION OF TOD A 23 The co eﬃcient o f ~ n t l in the recur s ion relatio n (3.45) is (3.48) Y ( l ) k +1 ,n ( s, s ′ , ξ , ξ ′ ) = 1 k + 1     ξ ∂ ξ ∂ s ′ Y ( l ) k,n − 1 ( s, s ′ , ξ , ξ ′ ) + X l ′ + l ′′ = l n ′ + n ′′ = n ξ ∂ ξ Y ( l ′ ) k,n ′ ( s, s ′ , ξ , ξ ′ ) ∂ s ′ S ( l ′′ ) n ′′ ( s ′ , ξ ′ )     ( Y ( l ) k, − 1 = 0) while (3.46) gives (3.49) S ( l +1) n ( s, ξ ) = 1 l + 1 ∞ X k =0 Y ( l ) k,n ( s, s, ξ , ξ ) W e ﬁrst show that these re c ursion r elations consistently deter mine Y ( l ) k,n and S ( l ) n . Then we pr ov e that the inﬁnite sum in (3.49) is ﬁnite. Fix n ≧ 0 and assume 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 a s it do es not app ear in the induction.) (1) Since E ( t = 0) = 1 b y the deﬁnition (3.40), w e have S (0) = 0. Hence (3.50) S (0) n = 0 . (2) F ro m the initial condition in (3.45) we ha ve (3.51) Y ( l ) 0 ,n ( s, s ′ , ξ , ξ ′ ) = δ l, 0 X n ( s, ξ ) . It fo llows from this equation and the assumption (3.2) that (3.52) ord ξ Y (0) 0 ,n ≦ − 1 . (3) When l = 0, the second s um in the r ight hand s ide of the recursio n rela tio n (3.48) is abs e nt b ecause of (3.50). Hence if n ≧ k + 1, we hav e Y (0) k +1 ,n = 1 k + 1 ξ ∂ ξ ∂ s ′ Y (0) k,n − 1 = · · · = 1 ( k + 1)! ( ξ ∂ ξ ∂ s ′ ) k +1 Y (0) 0 ,n − k − 1 = 0 since Y (0) 0 ,n − k − 1 do es not dep end o n s ′ thanks to (3.51). If n < k + 1, the ab ov e expressio n b ecomes zero by Y (0) k − n +1 , − 1 = 0 . Hence to gether with (3.51), w e o btain (3.53) Y (0) k,n = δ k, 0 X n . (4) By (3.49) w e can determine S (1) n : (3.54) S (1) n = ∞ X k =0 Y (0) k,n = Y (0) 0 ,n = X n . In particular, (3.55) ord ξ ∂ s ′ S (1) n = ord ξ ∂ s ′ X n ≦ − 1 . 24 KANEHISA T AKASAK I AND T AKASHI T AK EBE (5) Fix l 0 ≧ 1 and as s ume tha t for a ll l = 0 , . . . , l 0 − 1 and for a ll k = 0 , 1 , 2 , . . . , we have determined Y ( l ) k,n and that for all l = 0 , . . . , l 0 we have deter mined S ( l ) n . (The steps (3 ) a nd (4) are for l 0 = 1.) Since S (0) n ′′ = 0 b y (3.5 0) , the index l ′ in the right hand side of the recursion re lation (3.48) (with l = l 0 ) runs ess entially fro m 0 to l 0 − 1 . Hence this rela tion determines Y ( l 0 ) k +1 ,n from known qua nt ities for all k ≧ 0. Because of the initial condition Y 0 ( t ; s, s ′ , ξ , ξ ′ ) = X ( s, ξ ) (cf. (3.45)) Y 0 do es not dep end on t , which means that its T aylor co eﬃcients Y ( l 0 ) 0 ,n v anish for a ll l 0 ≧ 1: (3.56) Y ( l 0 ) 0 ,n = 0 . Thu s we hav e determined all Y ( l 0 ) k,n ( k = 0 , 1 , 2 , . . . ). (6) W e shall pr ove below that Y ( l 0 +1) k,n = 0 if k > l 0 + n + 1. Hence the sum in (3.49) is ﬁnite and S ( l 0 +1) n is determined. The induction pro ceeds by incrementing l 0 by o ne. In this wa y induction pro ceeds and all Y ( l ) k,n and S ( l ) n are determined. Let us prove that Y ( l ) k,n ’s determined ab ov e satisfy Y ( l ) k,n = 0 , if k > l + n, (3.57) ord ξ Y ( l ) k,n ≦ − l − 1 , if 0 ≦ k ≦ l + n, (3.58) (W e deﬁne that ord ξ 0 = −∞ .) In pa rticular, the sum in (3.49) is w ell-deﬁned and (3.59) ord ξ S ( l +1) n ≦ − l − 1 . If n = − 1, b oth (3.57) a nd (3.58) are o bvious. Fix n 0 ≧ 0 and ass ume that we hav e prov ed (3 .5 7) a nd (3.58) for n < n 0 and all ( l, k ). When n = n 0 and l = 0, (3.57) and (3.58) are true for all k bec ause of (3.53) and (3.5 2). Fix l 0 ≧ 0 and assume that we hav e pro ved (3.57) and (3.58) for n = n 0 , l ≦ l 0 and all k . As a r esult (3.5 9) is true for l ≦ l 0 . F or ( n, l, k ) = ( n 0 , l 0 + 1 , 0) (3.57) is void a nd (3.5 8) is true b eca use o f (3.51) and or d ξ X n 0 ( s, ξ ) ≦ − 1. Put n = n 0 and l = l 0 + 1 in (3.4 8) and a ssume that k + 1 > ( l 0 + 1) + n 0 . Then k > ( l 0 + 1) + ( n 0 − 1), which gua rantees that Y ( l ) k,n − 1 = Y ( l 0 +1) k,n 0 − 1 = 0 by the induction hypothesis o n n . As we men tioned in the step (5) ab ov e, l ′ in the rig ht hand side of (3.48) r uns fro m 0 to l − 1 = l 0 . Hence, a s we a re a ssuming tha t k > l 0 + n , we have k > l ′ + n ′ , which leads to Y ( l ′ ) k,n ′ = 0 by the induction hypothesis on l and n . Ther efore all terms in the rig ht hand side of (3.48) v anish and we hav e Y ( l 0 +1) k +1 ,n 0 = 0. The induction on k for (3.57) is completed, namely it is proved for n = n 0 , l = l 0 + 1 and k ≧ 1. The estimate (3 .58) is ea sy to chec k for n = n 0 , l = l 0 + 1 and k ≧ 1 by the recursion relation (3.48). (Recall once again that ξ ∂ ξ do es no t change the order.) The step l = l 0 + 1 b eing prov ed, the induction pro ceeds with re s pe ct to l and consequently with resp ect to n . ~ -EXP ANS ION OF TOD A 25 In summar y we have constructed Y ( t ; s, s ′ , ξ , ξ ′ ) and S ( t ; s, ξ ) satisfying (3.44) and (3.4 6). Moreov er , thanks to (3.57) the sum ov er k for eac h ﬁxed ( n, l ) in (3.60) Y (1; s, s ′ , ξ , ξ ′ ) = ∞ X n =0 ∞ X l =0 ∞ X k =0 Y ( l ) k,n ( s, s ′ , ξ , ξ ′ ) ~ n , is ﬁnite and the sum over l is meaningful as a p ow er series of ξ b ecause of (3.58). The ser ies (3.61) S (1; s, ξ ) = ∞ X n =0 ∞ X l =0 S ( l ) n ( s, ξ ) ~ n , is als o mea ningful as a p ower series of ξ thanks to (3.59). Thu s Pro po sition 3.1 is proved.  Pr o of of Pr op osition 3.2. W e r everse the order o f the previous pro of. Namely , given S ( ~ , s , ξ ), we shall construct X ( ~ , s, ξ ) such that the corres po nding S (1 ; ~ , s, ξ ) in the abov e pr o of co incides with it. Suppo se we have s uch X ( ~ , s , ξ ). Then the ab ove pro cedure determine Y ( l ) k,n and S ( l ) n . W e expand them as follows: S ( ~ , s , ξ ) = ∞ X n =0 S n ( s, ξ ) ~ n = ∞ X n =0 ∞ X j =1 S n,j ( s, ξ ) ~ n , X ( ~ , s, ξ ) = ∞ X n =0 X n ( s, ξ ) ~ n = ∞ X n =0 ∞ X j =1 X n,j ( s, ξ ) ~ n , S ( t ; ~ , s, ξ ) = ∞ X l =0 ∞ X n =0 S ( l ) n ( s, ξ ) ~ n t l = ∞ X l =0 ∞ X n =0 ∞ X j =1 S ( l ) n,j ( s, ξ ) ~ n t l , Y k ( t ; ~ , s, s ′ , ξ , ξ ′ ) = ∞ X l =0 ∞ X n =0 Y ( l ) k,n ( s, s ′ , ξ , ξ ′ ) ~ n t l = ∞ X l =0 ∞ X n =0 ∞ X j =1 Y ( l ) k,n,j ( s, s ′ , ξ , ξ ′ ) ~ n t l Here terms with index j are homogeneo us terms of deg ree − j with resp ect to ξ and ξ ′ . A t the e nd of this proo f we sha ll determine X n by (3.5 1), (3.62) X n ( s, ξ ) = Y (0) 0 ,n ( s, s ′ , ξ , ξ ′ ) . (In par ticular, Y (0) 0 ,n ( s, s ′ , ξ , ξ ′ ) sho uld no t dep end on s ′ and ξ ′ .) F or this purp os e, Y (0) 0 ,n should b e determined b y (3.63) Y (0) 0 ,n ( s, s ′ , ξ , ξ ′ ) = S n ( s, ξ ) − X ( l,k ) 6 =(0 , 0) l,k ≥ 0 1 l + 1 Y ( l ) k,n ( s, s, ξ , ξ ) bec ause of (3.49) and S n ( s, ξ ) = S n ( t = 1; s, ξ ) = P ∞ l =0 S ( l ) n ( s, ξ ). Since o r d ξ Y ( l ) k,n should b e not more than − l − 1 (cf. (3.58)), we exp ec t Y ( l ) k,n, 1 = 0 for l > 0. F or l = 0 a nd k > 0 Y (0) k,n, 1 = 0 f ollows fro m (3.53). Hence picking 26 KANEHISA T AKASAK I AND T AKASHI T AK EBE up homog eneous terms of degree − 1 with resp ect to ξ fro m (3.63), the fo llowing equation should hold: (3.64) Y ( l ) k,n, 1 = δ l, 0 δ k, 0 S n, 1 All Y ( l ) k,n, 1 are determined by this condition. Note also that (3.65) Y ( l ) 0 ,n,j = 0 for l 6 = 0 bec ause Y 0 should no t de p end on s b ecause of (3.51). Having deter mined initial co nditio ns in this wa y , we shall determine Y ( l ) k,n,j in- ductively . T o this end we r ewrite the r ecursion r elation (3.4 8) b y (3 .49) a nd pick up homogeneous terms of degree − j : (3.66) Y ( l ) k +1 ,n,j ( s, s ′ , ξ , ξ ′ ) = 1 k + 1 ξ ∂ ξ ∂ s ′ Y ( l ) k,n − 1 ,j ( s, s ′ , ξ , ξ ′ ) + X l ′ + l ′′ = l, l ′′ ≧ 1 , j ′ + j ′′ = j, j ′ ,j ′′ ≧ 1 n ′ + n ′′ = n, 0 ≦ k ′′ 1 l ′′ ξ ∂ ξ Y ( l ′ ) k,n ′ ,j ′ ( s, s ′ , ξ , ξ ′ ) ∂ s ′ Y ( l ′′ − 1) k ′′ ,n ′′ ,j ′′ ( s ′ , s ′ , ξ ′ , ξ ′ ) ! (As b efor e, terms like Y ( l ) k, − 1 ,j − 1 app earing the ab ove equa tion for n = 0 can b e ignored.) Fix n 0 ≧ 0 a nd ass ume that Y ( l ) k, 0 ,j , . . . , Y ( l ) k,n 0 − 1 ,j are determined for all ( l , k , j ). (1) First we determine Y ( l ) k,n 0 , 1 for all ( l , k ) by (3.64). (This is consistent with the recursion relatio n (3.66).) (2) Fix j 0 ≧ 2 and a ssume that Y ( l ) k,n 0 ,j are deter mined for j = 1 , . . . , j 0 − 1 a nd all ( l, k ). (The abov e s tep is for j 0 = 2.) Since all the quantities in the r ight hand side of the r ecursion relation (3.66) with j = j 0 are k nown b y the induction hypothesis, we can determine Y ( l ) k,n 0 ,j 0 for l = 0 , 1 , 2 , . . . and k = 1 , 2 , . . . . (3) T o gether with (3.65), Y ( l ) 0 ,n 0 ,j 0 = 0 for l = 1 , 2 , . . . , we hav e determined all Y ( l ) k,n 0 ,j 0 except for the case ( l , k ) = (0 , 0). (4) It follows from (3 .66) and (3 .64) by induction that all Y ( l ) k,n 0 ,j determined in (1), (2) and (3) satisfy the following pro p e rties: • if k > l + n , then (3.67) Y ( l ) k,n,j = 0 , which co rresp onds to (3.57) in the pro of o f Prop osition 3.1; • if 0 ≦ k ≦ l + n and j ≦ l , then (3.68) Y ( l ) k,n,j = 0 , which co rresp onds to (3.58) in the pro of o f Prop osition 3.1. (5) W e determine Y (0) 0 ,n 0 ,j 0 by (3.69) 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 ( s, s, ξ , ξ ) ~ -EXP ANS ION OF TOD A 27 which is the homogeneous par t of degree − j 0 in (3.63). The sum in the right hand side is ﬁnite b eca use o f (3 .67) and (3.68). (6) The induction with resp ect to j pro ceeds b y increment ing j 0 . Thu s all Y ( l ) k,n 0 ,j are determined and X n 0 is determined b y (3.62), namely , X n 0 ( x, ξ ) = P ∞ j =1 Y (0) 0 ,n 0 ,j (cf. (3 .6 2)), which c o mpletes the pro of of Pr op osition 3 .2.  4. Asymptotics of the t au function In this section w e derive a n ~ -ex pa nsion (4.1) log τ ( ~ , t, ¯ t, s ) = ∞ X n =0 ~ n − 2 F n ( t, ¯ t, s ) of the tau function (cf. (1.20) ) from the ~ -expansion of the S -functions S ( ~ , t, ¯ t, s ; z ) (3.4) a nd ¯ S ( ~ , t , ¯ t, s ; ¯ z ) (3.7). Let us r ecall the fundamental rela tions (1.20) b e t ween the wa ve functions and the tau function again: (4.2) Ψ( ~ , t, ¯ t ; z ) = τ ( ~ , t − ~ [ z − 1 ] , ¯ t, s ) τ ( ~ , t, ¯ t, s ) z s/ ~ e ζ ( t,z ) / ~ , ¯ Ψ( ~ , t, ¯ t ; ¯ z ) = τ ( ~ , t, ¯ t − ~ [ ¯ z ] , s + ~ ) τ ( ~ , t, ¯ t, s ) ¯ z s/ ~ e ζ ( ¯ t, ¯ z − 1 ) / ~ where [ z − 1 ] = (1 /z , 1 / 2 z 2 , 1 / 3 z 3 , . . . ), ζ ( t, z ) = P ∞ n =1 t n z n etc. (Here w e again omit inesse ntial co ns tants, α ( ~ ) and ¯ α ( ~ ).) This implies that ~ − 1 ˆ S ( ~ , t , ¯ t, s ; z ) =  e − ~ D ( z ) − 1  log τ ( ~ , t, ¯ t, s ) , (4.3) ~ − 1 ˆ ¯ S ( ~ , t , ¯ t, s ; ¯ z ) =  e − ~ ¯ D ( ¯ z ) e ~ ∂ s − 1  log τ ( ~ , t, ¯ t, s ) , (4.4) where (4.5) ˆ S ( ~ , t, ¯ t, s ; z ) = S ( ~ , t, ¯ t, s ; z ) − ζ ( t, z ) , ˆ ¯ S ( ~ , t , ¯ t, s ; ¯ z ) = ¯ S ( ~ , t , ¯ t, s ; ¯ z ) − ζ ( ¯ t, ¯ z − 1 ) , and (4.6) D ( z ) = ∞ X j =1 z − j j ∂ ∂ t j , ¯ D ( ¯ z ) = ∞ X j =1 ¯ z j j ∂ ∂ ¯ t j . Diﬀerent iating (4.3) with respect to z , w e have ~ − 1 ∂ ∂ z ˆ S ( ~ , t , ¯ t, s ; z ) = − ~ D ′ ( z ) e − ~ D ( z ) log τ ( ~ , t, ¯ t, s ) = − ~ D ′ ( z )( ~ − 1 ˆ S ( ~ , t , ¯ t, s ; z ) + log τ ( ~ , t, ¯ t, s )) , (4.7) where D ′ ( z ) := ∂ ∂ z D ( z ) = − P ∞ j =1 z − j − 1 ∂ ∂ t j . Hence (4.8) − ~ D ′ ( z ) log τ ( ~ , t, ¯ t, s ) = ~ − 1  ∂ ∂ z + ~ D ′ ( z )  ˆ S ( ~ , t , ¯ t, s ; z ) Multiplying z n to this equation and taking th e res idue, we obtain a system of diﬀerential e quations (4.9) ~ ∂ ∂ t n log τ ( ~ , t, ¯ t, s ) = ~ − 1 Res z = ∞ z n  ∂ ∂ z + ~ D ′ ( z )  ˆ S ( ~ , t , ¯ t, s ; z ) dz 28 KANEHISA T AKASAK I AND T AKASHI T AK EBE for n = 1 , 2 , . . . . In the same w ay we hav e (4.10) − ~ ¯ D ′ ( ¯ z ) log τ ( ~ , t, ¯ t, s ) = ~ − 1  ∂ ∂ ¯ z + ~ ¯ D ′ ( ¯ z )  ˆ ¯ S ( ~ , t , ¯ t, s ; ¯ z ) and (4.11) ~ ∂ ∂ ¯ t n log τ ( ~ , t, ¯ t, s ) = − ~ − 1 Res ¯ z =0 ¯ z − n  ∂ ∂ ¯ z + ~ ¯ D ′ ( ¯ z )  ˆ ¯ S ( ~ , t , ¯ t, s ; ¯ z ) d ¯ z , for n = 1 , 2 , . . . , from (4.4). By putting ¯ z = 0 in (4.4) we hav e a diﬀerence equa tion for the tau function: ( e ~ ∂ s − 1) log τ ( ~ , t, ¯ t, s ) = ~ − 1 ˆ ¯ S ( ~ , t , ¯ t, s ; 0) . In fact, it follows from (1.1 6), (1.19) and (3.7) that ˆ ¯ S ( ~ , t , ¯ t, s ; 0) = φ ( ~ , t, ¯ t ). Hence we have (4.12) ~ ( e ~ ∂ s − 1) log τ ( ~ , t, ¯ t, s ) = φ ( ~ , t, ¯ t, s ) . As is shown in [UT], the sy stem (4 .9), (4.11) and (4.12) is compatible and deter- mines the tau function up to multiplicativ e constant. By substituting the ~ -expansio ns log τ ( ~ , t, ¯ t, s ) = X n ∈ Z ~ n − 2 F n ( t, ¯ t, s ) , (4.13) ˆ S ( ~ , t, ¯ t, s ; z ) = ∞ X n =0 ~ n S n ( t, ¯ t, s ; z ) , (4.14) ˆ ¯ S ( ~ , t , ¯ t, s ; ¯ z ) = ∞ X n =0 ~ n ¯ S n ( t, ¯ t, s ; ¯ z ) (4.15) and (2.4) in to (4.8), (4.10) and (4.12), we ha ve ∞ X j =1 X n ∈ Z 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   . (4.16) − ∞ X j =1 X n ∈ Z ¯ 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   . (4.17) X n ∈ Z ~ n − 1 n X m =1 1 m ! ∂ m F n − m ∂ s m ! = ∞ X n =0 ~ n φ n . (4.18) It is obvious fr om these equa tio ns that F n = cons t. for n < 0. Therefore we ca n conclude tha t log τ has the expans ion (4.1) . Let us expa nd S n ( t ; z ) and ¯ S n ( t ; ¯ z ) int o a pow er series of z − 1 and ¯ z : (4.19) S n ( t ; z ) = − ∞ X k =1 z − k k v n,k , ¯ S n ( t ; ¯ z ) = φ n + ∞ X k =1 ¯ z k k ¯ v n,k . (The nota tion is chosen so that it is consistent with our pr evious work, e.g., [TT2].) Comparing the coeﬃcients of z − j − 1 ~ n − 1 in (4.16) and the co eﬃcient s o f ¯ z j − 1 ~ n − 1 ~ -EXP ANS ION OF TOD A 29 in (4.17) , we have the equations ∂ 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) , (4.20) − ∂ F n ∂ ¯ t j = ¯ v n,j + ∂ φ n ∂ ¯ t j + X k + l = j k ≥ 1 ,l ≥ 1 1 l ∂ ¯ v n − 1 ,l ∂ ¯ t k ( ¯ v − 1 ,j = 0) , (4.21) for n = 0 , 1 , 2 , . . . . F rom the equa tion (4.18) it is easy to see that ∂ F n /∂ s is determined r ecursively . Let us rewrite it in mo re explicit wa y . Fir st arrang e the co e ﬃcient s of ~ n − 1 in (4.18) in the v ector form: (4.22)          ∂ s 1 2! ∂ 2 s ∂ s 1 3! ∂ 3 s 1 2! ∂ 2 s ∂ s . . . . . . . . .               F 0 F 1 F 2 . . .      =      φ 0 φ 1 φ 2 . . .      . The matrix in the left hand s ide is ∞ X n =0 ∂ n +1 s ( n + 1)! Λ − n = e T − 1 T     T = ∂ s Λ − 1 ∂ s where Λ − n is the shift matrix ( δ i − n,j ) ∞ i,j =1 . Hence, a pplying the matrix T e T − 1     T = ∂ s Λ − 1 to (4.2 2), we hav e (4.23) ∂ ∂ s      F 0 F 1 F 2 . . .      = T e T − 1     T = ∂ s Λ − 1      φ 0 φ 1 φ 2 . . .      , or equiv alently , (4.24) ∂ F n ∂ s = φ n − φ n − 1 2 + [ n/ 2] X p =1 K 2 p φ n − 2 p , where K 2 p is determined by (2.22). The sys tem of ﬁrst or der diﬀerential equatio ns (4.20), (4.21) and (4.24) may b e und er sto o d as deﬁning equations of F n ( t, ¯ t, s ). This system is in tegr able and determines F n up to integration constants, b ecause the system (4.9), (4.11) and (4.12) ar e co mpatible. R emark 4 .1 . T au functions in str ing theory and random matric e s are known to hav e a genus exp ansion of the form (4.25) log τ = X g =0 ~ 2 g − 2 F g , where F g is the co nt ributio n fro m Riemann sur faces of genus g . In contrast, general tau functions of the ~ -dep endent T o da hiera rch y is not of this for m, namely , o dd 30 KANEHISA T AKASAK I AND T AKASHI T AK EBE powers of ~ can app ear in the ~ -expansion of lo g τ . T o exclude o dd p ow ers therein, we need to impose conditions 0 = v 2 m +1 ,j + X k + l = j k ≥ 1 ,l ≥ 1 1 l ∂ v 2 m,l ∂ t k = ¯ v 2 m +1 ,j + ∂ φ 2 m +1 ∂ ¯ t j + X k + l = j k ≥ 1 ,l ≥ 1 1 l ∂ ¯ v 2 m,l ∂ ¯ t k = φ 2 m +1 − φ 2 m 2 + m X p =1 K 2 p φ 2 m +1 − 2 p , on v n,j , ¯ v n,j and φ n or 0 = ∂ S 2 m +1 ∂ z − ∞ X j =1 z − j − 1 ∂ S 2 m ∂ t j = ∂ ¯ S 2 m +1 ∂ ¯ z + ∞ X j =1 ¯ z j − 1 ∂ ¯ S 2 m ∂ ¯ t j = φ 2 m +1 − φ 2 m 2 + m X p =1 K 2 p φ 2 m +1 − 2 p , (4.26) on S n , ¯ S n and φ n . Appendix A. E xample ( c = 1 string theor y) In this app endix, we a pply o ur algor ithm to the compactiﬁed c = 1 string theory at a self-dual r adius, following the for mulation in [T1]. W e use the notations in Section 2 . According to (4.10) in [T1 ] ( β = 1), the string equation for this c ase is L = ( − ¯ M − ~ + 1) ¯ L, ¯ L − 1 = ( − M + 1) L − 1 . (A.1) Multiplying the left and r ight hand side o f the second equa tio n to the right and le ft hand side of the ﬁrst equation from the right, we hav e L ( − M + 1) L − 1 = − ¯ M − ~ + 1 , and us ing the canonical commutation r elation [ L, M ] = ~ L , we ha ve M = ¯ M , namely , (A.2) L = (1 − ¯ M − ~ ) ¯ L, M = ¯ M . Hence the data ( f , g , ¯ f , ¯ g ) for P rop ositio n 1.2 in this case ar e (A.3) f ( ~ , s, e ~ ∂ s ) = e ~ ∂ s , g ( ~ , s, e ~ ∂ s ) = s, ¯ f ( ~ , s, e ~ ∂ s ) = (1 − s − ~ ) e ~ ∂ s , ¯ g ( ~ , s, e ~ ∂ s ) = s. The cor resp onding dis pe r sionless data ( f 0 , g 0 , ¯ f 0 , ¯ g 0 ) fo r P rop osition 1.3 are (A.4) f 0 ( s, ξ ) = ξ , g 0 ( s, ξ ) = s, ¯ f 0 ( s, ξ ) = (1 − s ) ξ , ¯ g 0 ( s, ξ ) = s. F or the sake of simplicity , we ﬁx the time v ar iables ¯ t n ( n = 1 , 2 , . . . ) to 0, whic h makes it p ossible to determine a ll X n ’s explic itly , (A.18 ). If we turn o n ¯ t n ’s, we need to pro ceed pertur batively . T o b egin with, let us determine the leading ter ms of X , ¯ X a nd φ with resp ect to the ~ -order , namely X 0 , ¯ X 0 and φ 0 in (2.2), (2.3) and (2.4 ). ~ -EXP ANS ION OF TOD A 31 The Riemann-Hilbert type problem for ( L , M , ¯ L , ¯ M ) (1.37) is (A.5) L = (1 − ¯ M ) ¯ L , M = ¯ M . Recall that L , M , ¯ L and ¯ M hav e the following form b y (1.27), (1.2 8), (1.32) and (1.33) when ¯ t = 0 . L = ξ + ∞ X n =0 u 0 ,n +1 ξ − n , ¯ L = ∞ X n =0 ˜ u 0 ,n ξ n +1 , M = ∞ X n =1 nt n L n + s + α 0 + ∞ X n =1 v 0 ,n L − n , ¯ M = s + ¯ α 0 + ∞ X n =1 ¯ v 0 ,n ¯ L n , Therefore (1 − ¯ M ) ¯ L is a T aylor ser ie s with positive pow er s of ξ , while L do es not hav e a p o sitive p ow er of ξ ex c e pt for the ﬁrst term, ξ . Ther efore the ﬁrst equation in (A.5 ) implies that (A.6) L = ξ . F rom this and the second equation M = ¯ M in (A.5), it follows that M a nd ¯ M do not hav e nega tive pow ers of ξ and α 0 = ¯ α 0 . Hence we may assume that α 0 = ¯ α 0 = 0 and (A.7) M = ¯ M = s + ∞ X n =1 nt n ξ n . Substituting this into the ﬁrst equa tion o f (A.5), we have (A.8) ¯ L = ξ 1 − s − ∞ X n =1 nt n ξ n ! − 1 , or ¯ L − 1 = ξ − 1 1 − s − ∞ X n =1 nt n ξ n ! . Next, let us de ter mine the leading ter ms X 0 , ¯ X 0 and φ 0 of the dressing op erato rs X , ¯ X and φ . W e denote the symbols of X 0 and ¯ X 0 by X 0 = X 0 ( t, s, ξ ) and ¯ X 0 = ¯ X 0 ( t, s, ξ ). Since L = exp(ad { , } X 0 ) ξ = ξ , X 0 do es no t dep e nd o n s . O n the other hand, since M = exp(ad { , } X 0 ) ( s + P n nt n ξ n ) = s + P n nt n ξ n , X 0 do es not dep end on ξ , either, whic h means that X 0 = 0. Note that a d { , } φ 0 ( s ) do es not change the degr ee of homog eneous terms with resp ect to ξ , s ince φ 0 ( s ) do es not contain ξ . Hence ¯ L ha s the fo llowing asy mptotic behaviour a round ξ = 0. ¯ L = e ad { , } φ 0 ξ + e ad { , } φ 0 ∞ X N =1 1 N !  ad { , } ¯ X 0  N ξ = e ad { , } φ 0 ξ + O ( ξ 2 ) , bec ause ¯ X 0 is a T aylor series of ξ with po sitive p ow ers. Comparing this expansion with (A.8), we hav e (A.9) e ad { , } φ 0 ξ = (1 − s ) − 1 ξ . 32 KANEHISA T AKASAK I AND T AKASHI T AK EBE It is ea s y to see that the left hand side is equal to e − φ ′ 0 ( s ) ξ , where ′ denotes the deriv a tion by s . Thus we obtain (A.10) φ 0 ( s ) = Z s log(1 − s ) ds = − (1 − s ) log(1 − s ) + (1 − s ) . It remains to determine ¯ X 0 . Op erating e − ad { , } φ 0 to ¯ L − 1 (A.8) and ¯ M (A.7) and using the formula (A.11) e − ad { , } φ 0 ξ = (1 − s ) ξ , which follows dir ectly fro m (A.9), we hav e tw o equations characterising ¯ X 0 : (A.12) e ad { , } ¯ X 0 ξ − 1 = ξ − 1 − ∞ X n =1 nt n (1 − s ) n − 1 ξ n − 1 , e ad { , } ¯ X 0 s = s + ∞ X n =1 nt n (1 − s ) n ξ n . In fact we ca n determine ¯ X 0 explicitly as follows. (A.13) ¯ X 0 = ∞ X n =1 t n (1 − s ) n ξ n . Indeed, since { ¯ X 0 , ξ − 1 } = − ∞ X n =1 nt n (1 − s ) n − 1 ξ n − 1 and { ¯ X 0 , s } = ∞ X n =1 nt n (1 − s ) n ξ n commute with ¯ X 0 itself (this is a direct co ns equence o f a trivial fact { (1 − s ) k ξ k , (1 − s ) l ξ l } = 0), the exp onentials in e ad { , } ¯ X 0 ξ − 1 and e ad { , } ¯ X 0 s are truncated up to the ﬁrst o rder, namely (A.14) e ad { , } ¯ X 0 ξ − 1 = ξ − 1 + { ¯ X 0 , ξ − 1 } = ξ − 1 − ∞ X n =1 nt n (1 − s ) n − 1 ξ n − 1 , e ad { , } ¯ X 0 s = s + { ¯ X 0 , s } = s + ∞ X n =1 nt n (1 − s ) n ξ n , which proves that ¯ X 0 in (A.13) satisﬁes (A.12). Thu s we hav e determined the le a ding ter ms o f X , ¯ X and φ as follo ws: (A.15) X 0 = 0 , ¯ X 0 = ∞ X n =1 t n (1 − s ) n e n ~ ∂ s , φ 0 = − (1 − s ) log(1 − s ) + (1 − s ) . Having determined X 0 , ¯ X 0 and φ 0 , w e can start the algorithm discussed in Section 2. F ollowing the pro cedure by straightforw ard computation (actually , no t so str a ightfo ward, as we shall see later), we o btain a s the ﬁr st and the second steps, (A.16) X 1 = 0 , ¯ X 1 = − ∞ X n =1 t n n ( n + 1) 2 (1 − s ) n − 1 e n ~ ∂ s , φ 1 = 1 2 log(1 − s ) , and (A.17) X 2 = 0 , ¯ X 2 = − ∞ X n =1 t n n ( n 2 − 1)(3 n + 2) 24 (1 − s ) n − 2 e n ~ ∂ s , φ 2 = − 1 12 (1 − s ) − 1 . ~ -EXP ANS ION OF TOD A 33 F rom these res ults w e can infer the Ansatz for all n ≧ 2: (A.18) X n = 0 , ¯ X n = ∞ X m =1 t m c n,m (1 − s ) m − n e m ~ ∂ s , φ n = c n, 0 (1 − s ) − n +1 , with suitable constants c n,m ( n, m ≧ 1 ) and c n, 0 ( n ≧ 2). Even tually these co n- stants a re determined recursively as follows: c n,m = 1 n n − 1 X j =0 ( − 1) n − j  m − j + 1 k − j + 1  c j,m , (A.19) c n, 0 = 1 − n + 1   1 n + 1 − c 1 , 0 n − n − 1 X j =2 ( − 1) n − j  − j + 1 k − j + 1  c j, 0   , (A.20) with the initial v alues c 0 ,m = 1 and c 1 , 0 = 1 / 2. Let us prov e that the ab ov e Ansa tz is true. T o do this, we have only to chec k that it is consisten t with the algorithm in Section 2. It is easy to compute the in termedia te ob jects P ( i − 1) (2.13) and Q ( i − 1) (2.14), since the op erator s X 0 , . . . , X i − 1 are z e r o. Using the nota tions (2.10) and (2.12), we have P ( i − 1) = exp  X ( i − 1) ~  f t = f t = e ~ ∂ s , Q ( i − 1) = exp  X ( i − 1) ~  g t = g t = s + ∞ X n =1 nt n e n ~ ∂ s . (A.21) Hence the terms in the expansion (2 .1 7) v anish except for P ( i − 1) 0 and Q ( i − 1) 0 : P ( i − 1) 0 = e ~ ∂ s , P ( i − 1) 1 = P ( i − 1) 2 = · · · = P ( i − 1) i = 0 , (A.22) Q ( i − 1) 0 = s + ∞ X n =1 nt n e n ~ ∂ s , Q ( i − 1) 1 = Q ( i − 1) 2 = · · · = Q ( i − 1) i = 0 . (A.23) Their s ymbols a re P ( i − 1) 0 = ξ , P ( i − 1) 1 = P ( i − 1) 2 = · · · = P ( i − 1) i = 0 , (A.24) Q ( i − 1) 0 = s + ∞ X n =1 nt n ξ n , Q ( i − 1) 1 = Q ( i − 1) 2 = · · · = Q ( i − 1) i = 0 . (A.25) T o compute ¯ P ( i − 1) (2.15), let us consider exp(ad( ¯ X ( i − 1) / ~ )) ¯ f ¯ t ﬁrst. Note that ~ − 1 [ ¯ X ( i − 1) , ¯ f ] = i − 1 X n =0 ~ n − 1 [ ¯ X n , ¯ f ] = i − 1 X n =0 ~ n − 1 ∞ X m =1 t m c n,m [(1 − s ) m − n e m ~ ∂ s , (1 − s − ~ ) e ~ ∂ s ] (A.26) Substituting (A.27) [(1 − s ) m − n e m ~ ∂ s , (1 − s − ~ ) e ~ ∂ s ] = − n ~ (1 − s ) m − n − m − n +1 X r =2  m − n + 1 r  ( − ~ ) r (1 − s ) m − n +1 − r ! e ( m +1) ~ ∂ s , 34 KANEHISA T AKASAK I AND T AKASHI T AK EBE we have ~ − 1 [ ¯ X ( i − 1) , ¯ f ] = − i − 1 X n =0 ~ n ∞ X m =1 nt m c n,m (1 − s ) m − n e ( m +1) ~ ∂ s + ∞ X k =0 ~ k ∞ X m =1 t m X 0 ≤ j ≤ i − 1 r ≥ 2 j − 1+ r = k ( − 1) r +1 c j,m  m − j + 1 r  (1 − s ) m − k e ( m +1) ~ ∂ s . Because of the deﬁnition of c n,m (A.19), the co eﬃcients of ~ n t m ( n = 0 , . . . , i − 1, m = 1 , 2 , . . . ) in the r ight hand side v anish and the co eﬃcient o f ~ i is equal to ic i,m . (Actually this is wh y c n,m is deﬁned by the recursio n r e lation (A.19).) This means (A.28) ~ − 1 [ ¯ X ( i − 1) , ¯ f ] = ~ i ∞ X m =1 t m ic i,m (1 − s ) m − i e ( m +1) ~ ∂ s + O ( ~ i +1 ) . F urther a pplication of ad( ~ − 1 ¯ X ( i − 1) ) changes the symbo l of terms of ~ -o rder − i (i.e., co eﬃcients o f ~ i ) by applica tio n o f a d { , } ¯ X 0 , as ad ~ j − 1 ¯ X j ( j = 1 , . . . , i − 1) low ers the ~ -order. Hence fo r N ≧ 1 we have (A.29)  ad ~ − 1 ¯ X ( i − 1)  N ¯ f = ~ i  ad { , } ¯ X 0  N − 1 ∞ X m =1 t m ic i,m (1 − s ) m − i ξ m +1 !      ξ → e ~ ∂ s + O ( ~ i +1 ) . Next w e co mpute the conjugatio n of ¯ f by e ~ − 1 φ ( i − 1) ( s ) . e ad ~ − 1 φ ( i − 1) ( s ) ¯ f = e ~ − 1 φ ( i − 1) ( s ) (1 − s − ~ ) e ~ ∂ s e − ~ − 1 φ ( i − 1) ( s ) = e ~ − 1 ( φ ( i − 1) ( s ) − φ ( i − 1) ( s + ~ )) (1 − s − ~ ) e ~ ∂ s = exp 1 ~ ( φ 0 ( s ) − φ 0 ( s + ~ )) + log(1 − s − ~ ) + i − 1 X j =1 ~ j − 1 ( φ j ( s ) − φ j ( s + ~ )) ! e ~ ∂ s . (A.30) By (A.15), (A.16) and (A.18), we hav e 1 ~ ( φ 0 ( s ) − φ 0 ( s + ~ )) + lo g(1 − s − ~ ) = − ∞ X k =1 ~ k k + 1 (1 − s ) − k , φ 1 ( s ) − φ 1 ( s + ~ ) = ∞ X k =1 ~ k c 1 , 0 k (1 − s ) − k , φ j ( s ) − φ j ( s + ~ ) = ∞ X k =1 ~ k ( − 1) k +1 c j, 0  − j + 1 k  (1 − s ) − j − k +1 . ~ -EXP ANS ION OF TOD A 35 The co eﬃcients c j, 0 were deﬁned b y (A.20) s o that 1 ~ ( φ 0 ( s ) − φ 0 ( s + ~ )) + lo g(1 − s − ~ ) + i − 1 X j =1 ~ j − 1 ( φ j ( s ) − φ j ( s + ~ )) = ( i − 1) c i, 0 ~ i (1 − s ) i + O ( ~ i +1 ) . Thu s w e ha ve (A.31) e ad ~ − 1 φ ( i − 1) ( s ) ¯ f =  1 + ~ i ( i − 1) c i, 0 (1 − s ) − i + O ( ~ i +1 )  e ~ ∂ s , for i ≧ 2. When i = 1 , we should repla ce (1 − s ) − 1 by lo g(1 − s ) but the rest is the same. Summarising the ab ove results (A.29) and (A.31), we obtain the following ex- pansion of ¯ P ( i − 1) . ¯ P ( i − 1) = e ad ~ − 1 φ ( i − 1) e ad ~ − 1 ¯ X ( i − 1) ¯ f = e ad ~ − 1 φ ( i − 1) ¯ f + e ad ~ − 1 φ ( i − 1) ∞ X N =1  ad ~ − 1 ¯ X ( i − 1)  N N ! ¯ f = e ~ ∂ s + ~ i ( i − 1) c i, 0 (1 − s ) − i e ~ ∂ s + ~ i e ad { , } φ 0 ∞ X N =1  ad { , } ¯ X 0  N − 1 N ! ∞ X m =1 t m ic i,m (1 − s ) m − i ξ m +1 !      ξ → e ~ ∂ s + O ( ~ i +1 ) . (A.32) Therefore (A.33) ¯ P ( i − 1) 0 = e ~ ∂ s , ¯ P ( i − 1) 1 = · · · = ¯ P ( i − 1) i − 1 = 0 , which co incide with (A.22), and (A.34) ¯ P ( i − 1) i = ( i − 1) c i, 0 (1 − s ) − i ξ + e ad { , } φ 0 ∞ X N =1  ad { , } ¯ X 0  N − 1 N ! ∞ X m =1 t m ic i,m (1 − s ) m − i ξ m +1 ! . F rom formulae ad { , } ¯ X 0  (1 − s ) − j  = j (1 − s ) − j − 1 ∞ X k =1 k t k (1 − s ) k ξ k ! , { ¯ X 0 , (1 − s ) k ξ k } = 0 and (A.9) it follows that (A.35) e ad { , } φ 0  ad { , } ¯ X 0  N − 1  (1 − s ) m − i ξ m +1  = ( i + 1) · · · ( i + N − 1)(1 − s ) − i − N ξ m +1 ∞ X k =1 k t k ξ k ! N − 1 . 36 KANEHISA T AKASAK I AND T AKASHI T AK EBE Substituting it to (A.34), we have (A.36) ¯ P ( i − 1) i = ( i − 1) c i, 0 (1 − s ) − i ξ + ∞ X N =1 1 N ! ∞ X m =1 t m c i,m ξ m +1 i ( i + 1) · · · ( i + N − 1) × × (1 − s ) − i − N ∞ X k =1 k t k ξ k ! N − 1 . Now let us compute (2 .2 0). Alt houg h we ha ve not computed ¯ Q ( i − 1) , thanks to (A.24) and (A.25), in tegrals in (2.20) are simpliﬁed to − ˜ X i + φ i + ˜ ¯ X i = Z s ξ − 1 ¯ P ( i − 1) i ds, which is computable without information of ¯ Q ( i − 1) . B y the explicit formula (A.36) we o btain (A.37) − ˜ X i + φ i + ˜ ¯ X i = c i, 0 (1 − s ) − i +1 + ∞ X N =1 1 N ! ∞ X m =1 t m c i,m ξ m i ( i + 1) · · · ( i + N − 2)(1 − s ) − i − N +1 ∞ X k =1 k t k ξ k ! N − 1 . In this formula there is no term with negative p ow ers of ξ , which means ˜ X i = 0, i.e., X i = 0. The constant term with respec t to ξ is c i, 0 (1 − s ) − i +1 , whic h is φ i ( s ), as was exp ected. The remaining part is ˜ ¯ X i . In genera l, it is almost hop eless to co mpute ¯ X i from ˜ ¯ X i by (2.21). How ever, quite fortunately , in the present case we ar e able to ﬁnd the explic it answ er. Using (A.35), we can rewrite ˜ ¯ X i as fo llows. (A.38) ˜ ¯ X i = e ad { , } φ 0 ∞ X N =1  ad { , } ¯ X 0  N − 1 N ! ∞ X m =1 t m c i,m (1 − s ) m − i ξ m !! . Recall that eq ua tions in (2.2 1) are the inv ersio n formulae of (2.26) and (2.2 7). Comparing (A.38) and (2.27), w e can conclude that ¯ X i = ∞ X m =1 t m c i,m (1 − s ) m − i ξ m , which ﬁnally proves the Ansatz (A.18). References [A] Aoki, T., Calcul exp onentiel des op ´ erateurs micro diﬀ´ erent iels d’ordre i nﬁni. II. Ann. Inst. F our i er (Grenoble) 3 6 143–165 (1986). [B] N. Bourbaki, ´ El´ emen ts de math ´ ematique, Groupes et alg` ebres de Lie, Actualit´ es Sci. Indust., (Hermann). [D] Dij kgraaf, R., Intersect ion Theory , In tegrable Hierarchies and T opological Field Theory , in New symmetry principles in quantum ﬁeld the ory (Carg ` ese, 1991), NA TO Adv. Sci. Inst. Ser. B Phys. 295 (1992), 95–158. [dFG Z] Di F r ancesco, P ., Gi nsparg, P ., and Zinn-Justin, J., 2D gra vity and random matrices. Ph ys. Rep. 2 54 (1995), 133 pp. [DMP] R. Dijkgraaf, G. Moor e and R. Plesser, The partition function of 2D string theory , Nucl. Ph ys. B39 4 (1993), 356–382. ~ -EXP ANS ION OF TOD A 37 [EK] T. Eguc hi and H. Kanno, T o da lattice hierarch y and the top ological description of c = 1 string theory , Phys. Lett. B331 (1994), 330–334. [EO] Eynard, B. , and Or an tin, N., Inv ariants of algebraic curv es and topological expansion, Commun . Number Theory Ph ys. 1 (2007), 347–452. [HOP] A. Hanan y , Y. Oz and R. Pl esser, T op ological Landau-Ginzburg formulation and in te- grable structure of 2d string theory , Nucl. Phys. B42 5 (1994), 150–172. [Kr] Krichev er , I.M., T he disp ersionless Lax equations and top ological mi nimal mo dels, Com- mun . Math. Phys. 1 43 (1991), 415–426. [Mo] Morozo v, A., Integrabilit y and M atri x M o dels Ph ys. Usp. 37 (1994), 1–55. [OS] Orlov, A. Y u. and Sch ulm an, E. I., Additional symmetries for integrable equations and conformal algebra represent ation, Lett. Math. Phys. 12 (1986), 171–179; Orlov, A. Y u., V ertex operators, ¯ ∂ - problems, symmetries, v ariational indentities and Hamiltonian for- malism for 2 + 1 integrable systems, in: Plasma The ory and Nonline ar and T urbulent Pr o c esses in Physics (W orld Scientiﬁc, Singap ore, 1988); Grinevich, P . G., and Or lov, A. Y u., Vi rasoro action on Ri emann surfaces, Grassmannians, det ¯ ∂ j and Segal Wil son τ function, in: Pr oblems of Mo dern Quantum Field The ory (Springer-V erlag, 1989). [S] Sc hapira, P ., Mi crodi ﬀeren tial systems in the complex domain, Grundlehren der m athe- matisc hen Wissensc haften 26 9 , Spri nger-V erlag, Berl in-New Y ork, (1985) [T1] T ak asaki, K. , T o da Lattice Hierarch y and Generalized String Equations. Commun. Math. Ph ys. 18 1 (1996), 131–156. [TT1] T ak asaki, K., and T akebe, T., SDiﬀ(2) T o da equation—hierarc hy , tau f unction, and sym- metries. Lett. M ath. Phys. 23 (1991), 205–214. [TT2] T ak asaki, K . , and T akebe, T., In tegrable Hierarchies and Dispersi onless Limit, Rev. Math. Ph ys. 7 (1995), 743-803. [TT3] T ak asaki, K. , and T ak eb e, T., A n ~ -expansion of the KP hierarch y — r ecursive construc- tion of s olutions, arXiv:0912 .4867 [TT4] T ak asaki, K. , and T akebe, T., ~ - Dependent KP hierarch y , to app ear in Theoret. and Math. Phys., the Pr oceedings of the “In ternational W orkshop on Classical and Quan tum In tegrable Systems 2011” (January 24-27, 2011 Protvino, Russia), arXiv:1105.07 94 [UT] Ueno, K. , and T ak asaki, K., T o da lattice hierarch y , in Gr oup R epr esentations and Systems of Diﬀer ential Equations , K. Ok amoto ed., Adv anced Studies i n Pure M ath˙ 4 (North- Holland/Kinokuniy a 1984), 1–95. Gradua te S chool of Huma n and Environment al S tudies, Kyoto University, Yoshida, Sakyo, Kyoto, 606-85 01, Jap an E-mail addr ess : takasaki@m ath.h.kyoto-u.ac.jp F acul ty of Ma thema tics, Na tional Research University – Hig her School of Econom- ics, V a v ilov a S treet, 7 , Moscow, 1 17312, Russia E-mail addr ess : ttakebe@hs e.ru

An hbar-expansion of the Toda hierarchy: a recursive construction of solutions

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