Remarks on the waterbag model of dispersionless Toda Hierarchy
We construct the free energy associated with the waterbag model of dToda. Also, the relations of conserved densities are investigated
Authors: Jen-Hsu Chang
REMARKS ON THE W A TERBAG MODEL OF DISPERSIONLE SS TODA HIERA R CHY JEN-HSU CHANG DEP AR TMENT OF COMPUTER SCIENCE NA TIONAL DEFENSE UNIVERSITY T AUYUAN, T AIW AN E-MAIL: JHCHANG@CCIT.EDU.TW Abstract. W e construct t he free energy asso ciated with the wa- terbag mo del of dT o da. Also, the r elations of conserved densities are in vestigated. Key W ords : waterbag mo del, WD VV equation, conserved densi- ties MSC (2000) : 35Q5 8, 37K10 , 37K 35 1. Intr oduction The disp ersionless T o da hierarch y (dT o da) is defined by [8] ∂ λ ∂ t n = { B n ( p ) , λ } , ∂ λ ∂ ˆ t n = { ˆ B n ( p ) , λ } , ∂ ˆ λ ∂ t n = { B n ( p ) , ˆ λ } , ∂ ˆ λ ∂ ˆ t n = { ˆ B n ( p ) , ˆ λ } , n = 1 , 2 , 3 , · · · (1) where the Lax op erators λ and ˆ λ are λ = e p + ∞ X n =0 u n +1 e − np ˆ λ − 1 = ˆ u 0 e − p + ∞ X n =0 ˆ u n +1 e np and B n ( p ) = [ λ n ] ≥ 0 , ˆ B n ( p ) = [ ˆ λ − n ] ≤− 1 . Here [ · · · ] ≥ 0 and [ · · · ] ≤− 1 denotes the non-negativ e part and negativ e part o f λ n and ˆ λ − n resp ectiv ely when expressed in the Lauren t series of e p . F or example, B 1 ( p ) = e p + u 1 , ˆ B 1 ( p ) = ˆ u 0 e − p . 1 2 JEN-HSU CHANG Finally , the P oisson Bra c k et in (1) is { f ( t 0 , p ) , g ( t 0 , p ) } = ∂ f ∂ t 0 ∂ g ∂ p − ∂ f ∂ p ∂ g ∂ t 0 . One can view λ is a lo cal co ordinate near ” ∞ ” and ˆ λ as a lo cal co ordinate near ” 0 ” [6] According to dT o da theory [8], there exist w a v e f unctions S , ˆ S a nd the disp ersionless τ function F (or free energy) S ( λ ) = ∞ X n =1 t n λ n + t 0 ln λ − ∞ X n =1 ∂ t n F n λ − n ˆ S ( λ ) = ∞ X n =1 ˆ t n ˆ λ − n + t 0 ln ˆ λ + ∂ F ∂ t 0 − ∞ X n =1 ∂ ˆ t n F n ˆ λ − n suc h that B n ( λ ) = ∂ t n S ( λ ) = λ n − ∞ X m =1 ∂ 2 t n t m F m λ − m B n ( ˆ λ ) = ∂ t n ˆ S ( ˆ λ ) = ∂ 2 t 0 t n F − ∞ X m =1 ∂ 2 t n ˆ t m F m ˆ λ m ˆ B n ( λ ) = ∂ ˆ t n S ( λ ) = − ∞ X m =1 ∂ 2 ˆ t n t m F m λ − m ˆ B n ( ˆ λ ) = ∂ ˆ t n ˆ S ( ˆ λ ) = ˆ λ − n + ∂ 2 t 0 ˆ t n F − ∞ X m =1 ∂ 2 ˆ t n ˆ t m F m ˆ λ m . In particular, p ( λ ) = ∂ t 0 S ( λ ) = ln λ − ∞ X m =1 ∂ 2 t 0 t m F m λ − m p ( ˆ λ ) = ∂ t 0 ˆ S ( ˆ λ ) = ln ˆ λ + ∂ 2 t 0 t 0 F − ∞ X m =1 ∂ 2 t 0 ˆ t m F m ˆ λ m . (2) Also, H + m = ∂ 2 t 0 t m F = 1 m I ∞ λ m dp = 1 m I ∞ λ m dξ ξ H − m = ∂ 2 t 0 ˆ t m F = 1 m I 0 ˆ λ − m dp = 1 m I 0 ˆ λ − m dξ ξ , m ≥ 1 (3) W A TERBAG MODEL OF DTODA 3 are the conserv ed densities of dT o da hierarc h y , where p = ln ξ . Then the dT o da hierarch y (1) can b e expresse d as ∂ p ( λ ) ∂ t n = ∂ B n ( p ( λ )) ∂ t 0 , ∂ p ( λ ) ∂ ˆ t n = ∂ ˆ B n ( p ( λ )) ∂ t 0 , ∂ p ( ˆ λ ) ∂ t n = ∂ B n ( p ( ˆ λ )) ∂ t 0 , ∂ p ( ˆ λ ) ∂ ˆ t n = ∂ ˆ B n ( p ( ˆ λ )) ∂ t 0 , (4) λ , ˆ λ b eing fixed. The systems ( 4 ) are of the conserv ational laws for the dT o da hierarch y . F rom (2),one kno ws that B 1 ( p ) = e p + u 1 = e p + ∂ 2 t 0 t 1 F , ˆ B 1 ( p ) = ˆ u 0 e − p = e ∂ 2 t 0 t 0 F e − p . Then from (4), o ne has p ˆ t 1 = ∂ t 0 [ e ∂ 2 t 0 t 0 F e − p ] , p t 1 = ∂ t 0 [ e p + ∂ 2 t 0 t 1 F ] . Then p t 1 ˆ t 1 = p ˆ t 1 t 1 will imply ∂ 2 t 1 ˆ t 1 F = − e ∂ 2 t 0 t 0 F . It is the dT o da equation. This paper is o r ganized as follo ws. In the next section, w e construct the w aterbag mo del of dT o da t yp e fr om the Hirota equation. Section 3 is dev oted to finding the free energy a sso ciated with the w aterbag mo del from the Landau-Ginsburg formulation in t o p ological field the- ory . Also, the equations for conserv erd densities ar e obtained. In the final section, one discusses some problems to b e inv estigated. 2. Disp ersionless Hiro t a Equa tion and Symmetr y Constraints The dT o da hierarc h y (1)(or (4)) is equiv alent to the dispersionless Hirota equation [2]: D µ p ( λ ) = − ∂ t 0 ln[ e p ( λ ) − e p ( µ ) ] , ˆ D ˆ µ p ( λ ) = − ∂ t 0 ln[1 − e p ( ˆ µ ) − p ( λ ) ] D µ p ( ˆ λ ) = − ∂ t 0 ln[1 − e p ( µ ) − p ( ˆ λ ) ] , D ˆ µ p ( ˆ λ ) = − ∂ t 0 ln[ e p ( ˆ λ ) − e p ( ˆ µ ) ] , where D µ = ∞ X m =1 µ − m m ∂ t m , ˆ D ˆ µ = ∞ X m =1 ˆ µ m m ∂ ˆ t m . 4 JEN-HSU CHANG W e can also express them in terms o f the S -function D µ S ( λ ) = − ln[ e p ( λ ) − e p ( µ ) µ ] , ˆ D ˆ µ S ( λ ) = − ln[1 − e p ( ˆ µ ) − p ( λ ) ] D µ ˆ S ( ˆ λ ) = − ln[1 − e p ( µ ) − p ( ˆ λ ) ] , ˆ D ˆ µ ˆ S ( ˆ λ ) = − ln[ e p ( ˆ λ ) − e p ( ˆ µ ) µ ] . (5) Next, w e consider the symmetry constrain ts [1]. • Case(I): F t 0 = P N i =1 ǫ i S i , where S i = S ( λ i ). Then near ” ∞ ” w e ha v e b y (5) p = ln λ − D λ F t 0 = ln λ − D λ N X i =1 ǫ i S i = ln λ − N X i =1 ǫ i D λ S i = ln λ − N X i =1 ǫ i ln[ e p ( λ ) − e h i λ ] , h i = p ( λ i ) = ln λ − N X i =1 ǫ i ln[ e p − e h i ] + ( N X i =1 ǫ i ) ln λ. Let ( P N i =1 ǫ i ) = 0 . Then one gets λ = e p N Y i =1 ( e p − e h i ) − ǫ i . Morev er, near ”0” w e also ha v e ∂ 2 t 0 t 0 F = P N i =1 ǫ i h i . Then p ( ˆ λ ) = ln ˆ λ + ∂ 2 t 0 t 0 F − ˆ D ˆ λ F t 0 = ln ˆ λ + N X i =1 ǫ i h i + N X i =1 ǫ i ln[1 − e p ( ˆ λ ) − h i ] = ln ˆ λ + N X i =1 ǫ i h i + N X i =1 ǫ i ln[ e − p ( ˆ λ ) − e − h i ] + ( N X i =1 ǫ i ) p ( ˆ λ ) . Then one obta ins ˆ λ = e p − P N i =1 ǫ i h i N Y i =1 ( e − p − e − h i ) − ǫ i . W A TERBAG MODEL OF DTODA 5 Actually , we can see that ˆ λ = e p e − ( P N i =1 ǫ i ) p e − P N i =1 ǫ i h i N Y i =1 ( e − p − e − h i ) − ǫ i = e p N Y i =1 ( e h i − e p ) − ǫ i = e p N Y i =1 ( e p − e h i ) − ǫ i = λ. (6) Also, H + 1 = N X i =1 ǫ i e h i , H − 1 = − e P N i =1 ǫ i h i ( N X i =1 ǫ i e − h i ) . The ev olutions fo r t 1 and ˆ t 1 are ∂ t 1 h i = ∂ t 0 [ e h i + N X i =1 ǫ i e h i ] ∂ ˆ t 1 h i = ∂ t 0 [ e − h i + P N i =1 ǫ i h i ] (7) W e can also express them as the Hamiltonian form h 1 h 2 . . . h N t 1 = η ij ∂ t 0 ∂ H + 1 ∂ h 1 ∂ H + 1 δh 2 . . . ∂ H + 1 δh N , h 1 h 2 . . . h N ˆ t 1 = η ij ∂ t 0 ∂ H − 1 ∂ h 1 ∂ H − 1 δh 2 . . . ∂ H − 1 δh N , where η ij = 1 + 1 ǫ 1 1 . . . . . . 1 1 1 + 1 ǫ 2 1 . . . 1 . . . . . . . . . . . . 1 1 1 . . . 1 1 + 1 ǫ N . • Case(I I): F t m = P N i =1 ǫ i S i . Then B m ( λ ) = λ m − D λ F t m = λ m − N X i =1 ǫ i D λ S i = λ m − N X i =1 ǫ i ln( e p − e h i ) . 6 JEN-HSU CHANG Hence the Lax op erator is λ m = e mp + u m − 1 e ( m − 1) p + u m − 2 e ( m − 2) p + · · · + u 1 e p (8) + u 0 + N X i =1 ǫ i ln( e p − e h i ) . • Case (I I I): F ˆ t m = P N i =1 ǫ i S i . Then ˆ B m ( ˆ λ ) = ˆ λ − m + ∂ 2 t 0 ˆ t m F − ˆ D ˆ λ F ˆ t m = ˆ λ − m + ∂ 2 t 0 ˆ t m F − N X i =1 ǫ i ˆ D ˆ λ S i = ˆ λ − m + ∂ 2 t 0 ˆ t m F − N X i =1 ǫ i ln( e − p − e − h i ) . Hence the Lax op erator is ˆ λ − m = ˆ u m e − mp + ˆ u m − 1 e − ( m − 1) p + ˆ u m − 2 e − ( m − 2) p + · · · (9) + ˆ u 1 e − p + ˆ u 0 + N X i =1 ǫ i ln( e − p − e − h i ) . 3. Residue formula and free energy In this section, we compute the free energy asso ciated with the wa- terbag mo del of case (I) in last section, i.e., (6). Also, the relations of conserv ed densities are in v estigated. The free energy is a function F ( t 1 , t 2 , · · · , t n ) suc h that the asso ciated functions, c ij k = ∂ 3 F ∂ t i ∂ t j ∂ t k , satisfy the following conditions. • The matrix η ij = c 1 ij is constan t and non-degenerate. This together with the inv erse matrix η ij are used to raise and lo w er indices. • The functions c i j k = η ir c r j k define an a sso ciativ e commutativ e algebra with a unit y elemen t(F rob enius algebra). Equations of asso ciat ivity giv e a system of no n-linear PDE for F ( t ) ∂ 3 F ( t ) ∂ t α ∂ t β ∂ t λ η λµ ∂ 3 F ( t ) ∂ t µ ∂ t γ ∂ t σ = ∂ 3 F ( t ) ∂ t α ∂ t γ ∂ t λ η λµ ∂ 3 F ( t ) ∂ t µ ∂ t β ∂ t σ . These equations constitute the Witten-D ijkgraaf-V erlinde-V erlinde ( o r WD VV) equations. The geometrical setting in w hic h to understand W A TERBAG MODEL OF DTODA 7 the free energy F ( t ) is the F rob enius manifo ld [4 ]. One w a y to con- struct suc h manifold is deriv ed via Landau-Ginzburg formalism as the structure on the pa r a meter space M of the appropriate f o rm λ = λ ( p ; t 1 , t 2 , · · · , t n ) . The F rob enius structure is give n b y the flat metric (10) η ( ∂ , ∂ ′ ) = − X r es dλ =0 { ∂ ( λdp ) ∂ ′ ( λdp ) dλ ( p ) } and the tensor (11) c ( ∂ , ∂ ′ , ∂ ′′ ) = − X r es dλ =0 { ∂ ( λdp ) ∂ ′ ( λdp ) ∂ ′′ ( λdp ) dλ ( p ) dp } defines a totally symmetric (3 , 0)-t ensor c ij k . Geometrically , a solutio n of WDVV equation defines a m ultiplication ◦ : T M × T M − → T M of v ector fields on the parameter space M , i.e, ∂ t α ◦ ∂ t β = c γ αβ ( t ) ∂ t γ . F rom c γ αβ ( t ), one can construct in trgable hierarchies whose cor r esp o nd- ing Hamiltonian densities are defined recursiv ely b y the formula ∂ 2 ψ ( l ) α ∂ t i ∂ t j = c k ij ∂ ψ ( l − 1) α ∂ t k , (12) where l ≥ 1 , α = 1 , 2 , · · · , n, and ψ 0 α = η αǫ t ǫ . The in tegrabilit y con- ditions for this systems are automatically satisfied when the c k ij are defined as ab ov e. In the follow ing theorem, one uses ln λ to replace λ , whic h is the dual F rob enius manif o ld asso ciated with λ [5, 7]. Theorem 3.1. L et the L ax op er ator b e define d in (6) .Then ( I ) η ( ∂ h i , ∂ h j ) = η ij = − ǫ i ǫ j , i 6 = j ( I I ) η ( ∂ h i , ∂ h i ) = η ii = − ǫ 2 i + ǫ i ( I I I ) c ( ∂ h i , ∂ h j , ∂ h k ) = c ij k = ǫ i ǫ j ǫ k , i 6 = j 6 = k ( I V ) c ( ∂ h i , ∂ h i , ∂ h k ) = c iik = ǫ i ǫ k [ ǫ i + e h k e h i − e h k ] , i 6 = k ( V ) c ( ∂ h i , ∂ h i , ∂ h i ) = c iii = ǫ 3 i + ǫ i [1 − ǫ i − N X l =1 ,l 6 = i ǫ l e h l e h i − e h l ] . 8 JEN-HSU CHANG Pr o of. W e see that ∂ l n λ ∂ h i = ǫ i e h i ξ − e h i , where p = ln ξ . Also, w e hav e (13) d ln λ dp = 1 − N X k =1 ǫ k e h k ξ − e h k = Q N k =1 ( ξ − ω k ) Q N k =1 ( ξ − e h k ) . In the f o llo wing pro ofs, we use the f o rm ula (13) and the fact that the residue at infinity is zero. (I) η ( ∂ h i , ∂ h j ) = X d ln λ =0 Res ∂ l n λ ∂ h i ∂ ln λ ∂ h j ξ d ln λ dp dξ = X d ln λ =0 Res ǫ i e h i ξ − e h i ǫ j e h j ξ − e h j ξ (1 − P N k =1 ǫ k e h k ξ − e h k ) dξ = X d ln λ =0 Res ǫ i ǫ j e h i e h j Q N k =1 ( ξ − e h k ) ξ ( ξ − e h i )( ξ − e h k ) Q N k =1 ( ξ − ω k ) = − Res ξ =0 ǫ i ǫ j e h i e h j Q N k =1 ( ξ − e h k ) ξ ( ξ − e h i )( ξ − e h k ) Q N k =1 ( ξ − ω k ) = − ǫ i ǫ j , i 6 = j. (I I) η ( ∂ h i , ∂ h i ) = X d ln λ =0 Res ǫ 2 i e 2 h i Q N k =1 ( ξ − e h k ) ξ ( ξ − e h i ) 2 Q N k =1 ( ξ − ω k ) = − ( Res ξ =0 + Res ξ = e h i ) X d ln λ =0 Res ǫ 2 i e 2 h i Q N k =1 ( ξ − e h k ) ξ ( ξ − e h i ) 2 Q N k =1 ( ξ − ω k ) = − ǫ 2 i − ǫ 2 i e 2 h i Q N k =1 ,k 6 = i ( e h i − e h k ) e h i Q N k =1 ( e h i − ω k ) = − ǫ 2 i − ǫ 2 i e h i 1 − ǫ i e h i = − ǫ 2 i + ǫ i (I I I) c ( ∂ h i , ∂ h j , ∂ h k ) = X d ln λ =0 Res ǫ i e h i ǫ j e h j ǫ k e h k Q N l =1 ( ξ − e h l ) ξ ( ξ − e h i )( ξ − e h j )( ξ − e h j ) Q N l =1 ( ξ − ω l ) dξ = − Res ξ =0 ǫ i e h i ǫ j e h j ǫ k e h k Q N l =1 ( ξ − e h l ) ξ ( ξ − e h i )( ξ − e h j )( ξ − e h k ) Q N l =1 ( ξ − ω l ) dξ = ǫ i ǫ j ǫ k , i 6 = j 6 = k . W A TERBAG MODEL OF DTODA 9 (IV) c ( ∂ h i , ∂ h i , ∂ h k ) = X d ln λ =0 Res ǫ 2 i e 2 h i ǫ k e h k Q N l =1 ( ξ − e h l ) ξ ( ξ − e h i ) 2 ( ξ − e h k ) Q N l =1 ( ξ − ω l ) dξ = − [ Res ξ =0 + Res ξ = e h i ] ǫ 2 i e 2 h i ǫ k e h k Q N l =1 ( ξ − e h l ) ξ ( ξ − e h i ) 2 ( ξ − e h k ) Q N l =1 ( ξ − ω l ) dξ = − [ − ǫ 2 i ǫ k + ǫ 2 i e 2 h i ǫ k e h k Q N l =1 ,l 6 = i ( e h i − e h l ) e h i ( e h i − e h k ) Q N l =1 ( e h i − ω l ) ] = − [ − ǫ 2 i ǫ k − ǫ i ǫ k e h k e h i − e h k ] = ǫ i ǫ k [ ǫ i + e h k e h i − e h k ] . (V) c ( ∂ h i , ∂ h i , ∂ h i ) = X d ln λ =0 Res ǫ 3 i e 3 h i Q N l =1 ( ξ − e h l ) ξ ( ξ − e h i ) 3 Q N l =1 ( ξ − ω l ) dξ = − [ Res ξ =0 + Res ξ = e h i ] ǫ 3 i e 3 h i Q N l =1 ( ξ − e h l ) ξ ( ξ − e h i ) 3 Q N l =1 ( ξ − ω l ) dξ = −{− ǫ 3 i + ǫ 3 i e 3 h i d dξ [ Q N l =1 ,l 6 = i ( ξ − e h l ) ξ Q N l =1 ( ξ − ω l ) ] | ξ = e h i } = ǫ 3 i + ǫ 3 i e 3 h i 1 − ǫ i − P N l =1 ,l 6 = i ǫ l e h l e h i − e h l ǫ 2 i e 3 h i = ǫ 3 i + ǫ i (1 − ǫ i − N X l =1 ,l 6 = i ǫ l e h l e h i − e h l ) . Let’s define Ω = P N i =1 ∂ ∂ h i . Then w e can verify directly that (14) η ( ∂ h i , ∂ h j ) = c ( ∂ h i , ∂ h j , Ω) = N X k =1 c ( ∂ h i , ∂ h j , ∂ h k ) . Also, from the Theorem, it’s not difficult to che c k directly the compat- ibilit y (or Egorov’s condition) ∂ h i c lmn = ∂ h l c imn , i, l , m, n = 1 · · · N . 10 JEN-HSU CHANG Hence one can get the free energy asso ciated with (6) F ( ~ h ) = X 1 ≤ i
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