Exact Poisson pencils, $tau$-structures and topological hierarchies

Exact P oisson p encils, τ -structur es an d top ological hierarchies Gregor io F alqui, P ao lo Lorenzoni Dipartimen to di M atematica e Applica zioni Univ ersit` a di Milano-Bic o cca Via Rob erto Cozzi 53, I-20125 Milano, Italy gregorio.falqui@unimib.it, p aolo .lorenzoni@u n imib.it T o Boris D ubr ovin in the o c c a s ion of his 60th b i rthda y, with friendship and admir ation. Abstract W e discuss, in the framew ork of Dub ro vin-Zhang’s p erturbativ e approac h to in tegrable ev olutionary PDEs in 1 + 1 dimensions, the role of a sp ecial class of P oisson p encils, called exact P oisson p encils. In particular w e show that, in the semisimple case, exactness of the p encil is equiv alen t to the constancy of the so-called “cen tral inv arian ts” of the theory th at w ere in tro duced by Dubro vin , Liu and Zh ang. 1 In tro d uction In tegrable hierarchie s o f ev o lutionary PDEs of the form q i t = V i j ( q ) q j x + ∞ X k =1 ǫ k F i k ( q , q x , q xx , . . . , q ( n ) , . . . ) (1) ha v e b een extensiv ely studied in the last ye ar s (see, e.g., [16, 29, 17, 18, 6 , 3 0]). In particular, great attention to the so-called top olo gic al hier ar chies a lso b ecause of their relation to the theory of Gromo v-Witten in v ariants, the the- ory of singularities, and other seemengly unrelated topics of Mathematics 1 and Theoretical Phys ics. These hierarc hies p ossess some additional struc- tures: they are bi-Hamiltonian, they admit a tau-structure and satisfy Vi- rasoro constraints [16]. The notion of τ -structure (or τ -function) is p erhaps among the oldest ones in the theory of ev o lutionary equations in 1 + 1 di- mensions, hav ing b een intro duced by Hirota as the ma jor c haracter in the bilinear formulation of in tegrable PDEs. Its pro perties w ere further exploited b y the Japanese school (see, e.g., [26, 12, 37]). In the presen t approa ch, the existence of a τ -structure for an in tegrable hierarch y of 1 + 1 ev olutionary PDEs will b e understo o d as the p ossibilit y of defining special densities h ∗ i for the m utually conserv ed quan tities of the PDEs that satisfy the symmetry requiremen t ∂ h ∗ i ∂ t j = ∂ h ∗ j ∂ t i , where ∂ ∂ t k is some suitable o ne-seq uence ordering of the v arious times of the hierarc h y . Virasoro symmetries are also w ell kno wn ob j ects of the theory; in par- ticular here w e refer to the Virasoro- t yp e a lgebras of additiona l (explicitly time(s)-dep enden t) symmetries of the classes of PD Es w e are concerned with. In part icular, t hey gained m uch at ten tion in the ligh t of the celebrated results b y Kontse vich and Witten [27, 38] that identified a particular τ -function of the KdV hierarc h y with t he partition function o f 2D Quan tum g r avit y , As it is we ll kno wn, the existence of a bi-Hamiltonian structure means that the equations o f the hierarc h y can b e written in Hamiltonian fo r m with resp ect to tw o compatible P oisson biv ectors P 1 and P 2 and that the P oisson p encil P 2 − λP 1 is a Pois son bive ctor for any λ [31]. A remark able result established in [16], and subseq uently refined in [6] is that, if the p encil P λ is semisimple, ( in a sense to b e made precise la ter) and admits a τ -function the ab o v e requiremen ts fix uniquely the hierarch y once the disp ersionles s limit q i t = V i j ( q ) q j x (2) and its bi-Hamiltonian structure ( ω 1 , ω 2 ) are giv en.The semisim plicity o f the p encil is related to t he existence of a sp ecial set of co ordinates ( u 1 , . . . , u n ) called c anonic al c o or dinates . If o ne r elaxes the hy p othesis of existence of a tau -structure, the deformations are parametrized b y certain functional pa- rameters called c entr al invariants that are constan ts in the case of top ologi- cal hierarchie s. In turn, further results in [17], suggest that the constancy of these cen tral inv arian ts is related with the existence of the τ -function o f the 2 hierarc h y . In this pap er w e will sho w that the P oisson p encil Π λ = P 2 − λP 1 of a top ological hierarc hy is exact, in the sense that there exists a v ector field Z ( to b e called Liouvil le v ector field of the p encil) suc h that Lie Z P 2 = P 1 , and Lie Z P 1 = 0 . (3) Moreo v er, w e show that there exists a Miura transformatio n reducing sim ul- taneously Z to its disp ersionless limit: Z → e = n X i =1 ∂ ∂ u i and the p encil Π λ to the form ω λ + ∞ X k =1 ǫ 2 k P (2 k ) 2 . The hint for our w o rks stems from the o bserv ation(s) (to b e briefly r ecalled in Section 2) that the geometry of exact bi-Hamitonian manifolds pro vides somehow for fr e e the needed ”to olkit” requested for the existence of a τ - function fo r the hierarc hy . Indeed, on g eneral grounds, on the one hand the bi-Hamiltonian hierarc hies defined on exact bi-Hamitonian manifolds exhibit additional symmetries of Vira soro type [39]. On the other hand, the action of the Liouville field on the Hamiltonian of the hierarch y naturally provide s new densities for the conserv ed quan tit ies. Actually , w e are not going t o tac kle these pro blems directly and abstractly as a problem in the g eneral theory of Pois son manifolds; rather, we use t hese ”nice” prop erties of exact P oisson p encils as suggestions fo r their realiza- tion within the p erturbativ e approa c h dev elop ed in recen t ye a rs by Boris Dubro vin a nd his collab orators for the classification problem of 1 + 1 ev olu- tionary integrable PDEs of KdV-t yp e. In particular, w e b orrow f rom them metho ds as well as a n um b er of explicit results, with the aim o f sho wing that the geometric notion of exactness of a P oisson p encil can b e fruitfully used in this field. 3 The pap er is orga nized as follo ws:in Section 2 w e collect some (more or less kno wn) results ab out exact Poiss o n p encils; then in Section 3 w e study exact semisimple P oisson p encils o f h ydro dynamic t yp e and we sho w that for suc h p encils the v ector field Z coincides with the unit y v ector field e of the underlying F rob enius manifo ld. In Sections 4 and 5 w e recall (follow ing [17] and [2 9 ]) some definitions and results ab out cen tral inv arian ts and bi- Hamiltonian coho mo lo gy necessary f or the subsequen t Section 6 whic h is dev oted to the pro of of the main result of the pap er. Section 7 con tains a brief summary of t he pap er and some indications of further p ossible steps to generalize the results herewith presen ted. Ac kno wledgmen ts W e w armly thank F . Magri and M. P edroni f o r fruitful discussions and useful commen ts. 2 Geometry of exact bi-Hamiltonian ma n i - folds In this section w e collect some results on the geometry of exact bi- Hamiltonian manifolds, and their relations with t he hierarc hies therein supp orted. It is fair to sa y that , in one form or the other, these results are kno wn in the literature. Ho w ev er, w e deem useful to collect them together here, as they someho w pro vide the guiding principle for the argumen ts con tained in the core of the pap er. Let us preliminarily recall a few basic no t ions. A bi-Hamiltonian (BH) manifold[31 ] is a manifold endow ed with a pair of compatible P oisson tensors P 1 , P 2 or, equiv alen tly , with a p encils of Pois son biv ectors P λ = P 2 − λP 1 ; it is w ell kno wn that this definition entails that separately P 1 and P 2 are P oisson bive ctor s, and the the Sc houten brack et of P 1 and P 2 v anishes (this is referred to as the c omp atibility condition). A sequence of bi-Hamiltonian v ector fields X i satisfying X i = P 1 dH i +1 = P 2 dH i , (4) with i running in some discrete set of indices, is called a Lenard–Magri se- quence. All the v ector fields in suc h a sequence do commu t e among them- selv es; equiv alen tly , the functions H i en tering (4) (the Hamiltonians of the 4 sequence ) are in in v olution w.r.t. the P oisson brac kets defined both b y P 1 and b y P 2 . F ollo wing [2 4] w e call a Lenard Magri sequence that starts from a Casimir function of one of the Poisson p encil (sa y , P 1 ) an anchor e d sequence; with the term p encil o f Gelfan’d–Zakharevich (GZ) ty p e w e understand a p encil of P oisson bive ctor s endo wed with n = dim(Ker P 1 ) anc hored Lenard Magri sequence s. W e remark that all the Hamiltonians defined b y a G Z p encil comm ute among themselv es, ev en if they b elong to diff er ent Lenard Magri sequence s. Also, the p encis of P oisson biv ectors en tering the Dubrovin-Zhang classification sc heme are dispersiv e deforma t ions of p encils of h ydro dynamic t yp e and are a ll of GZ t yp e. W e shall herewith consider p encils satisfying an additional geometric re- quiremen t. Definition 1 L et P λ := P 2 − λP 1 a p encil of Poisson bive ctors, defi ne d on a BH manifold M . We say that P λ is an exact Pois s on p encil if ther e exists a ve ctor field Z ∈ X ( M ) such that P 1 = Lie Z P 2 ; Lie Z P 1 (= Lie 2 Z P 2 ) = 0 . (5) The ve ctor field Z wil l b e r eferr e d to as the Liouville field of the e xact Poisson p e n cil. W e remark that, on general grounds, the Liouville ve cto r field Z is not uniquely defined. F or instance adding a bi- Ha miltonian v ector field to a Liouville v ector field o ne obtains a new Lio uville v ector field. In the kno wn examples (e.g. in the case of the A n -Drinfel’d-Sokolo v hierarc hies), there are some natural choice s for it. Indeed, in the pap er, w e shall see that this is the case. 2.1 Exact BH manifolds and second Hamiltonian func- tion(s) Let us consider an exact bi-Hamiltonian manifold, whose Lenard Magr i c hains b e “anc hor ed” according to the Gel’fand-Zakharevic h definition [24], that is all c hains originate from a Casimir function of P 1 . Let H ( λ ) := H 0 + H 1 λ + H 2 λ 2 + · · · a Casimir of the p encil, t ha t is a formal L a uren t series in λ satisfying P λ d H ( λ ) = 0( ⇒ P 1 d H 0 = 0) , (6) 5 and consider the p encil of bi-Hamito nian v ector fields X λ of the hierarc hy , to b e represen ted a s X λ = P 1 d H ( λ ) . (7) Prop osition 1 L et H ∗ ( λ ) := − Lie Z H ( λ ) ; then the on e p ar am eter family of ve c tor fields X λ c a n b e r epr esente d as X λ = P λ d H ∗ ( λ ) , (8) that is, the deforme d Hamiltonia n s H ∗ i = Lie Z H i define the same GZ foliation of the phase sp a c e M . Pro of . It follows from the straightforw ard ch a in of equalit y 0 =Lie Z ( P λ d H ( λ )) = Lie Z ( P λ ) d H ( λ ) + P λ Lie Z ( d H ( λ )) = P 1 d H ( λ ) + P λ ( d Lie Z ( H ( λ )) = X λ − P λ d H ∗ ( λ ) . Remark : Exact bi-Hamiltonian p encils, b esides ha ving ”historically” pro vided the first instances of suc h structures, naturally en ter the so-called metho d of ar gument tr anslation related with Lie- P oisson p encils on Lie alge- bras (see [32]). In the field of evolutionary integrable PDEs, applications of this metho d can b e found in [16], § 3; we notice ho we ver that in our case, the Gel’fand- Zakharevic h sequence s start from Casimir of the ”deformed” tensor P 1 , rather than with Casimirs of the (Lie P o isson) tensor P 2 . In the case of PDEs, this migh t b e a non- t rivial difference. 2.2 Exact BH manifolds and the Virasoro algebra Master symmetries are a very classical topic in the theory of in tegrable PDEs [23, 3 5]. In [39] it w as observ ed that the G alileian symmetry of the KdV equation could b e used a s a generator of a whole (alb eit formal) family of suc h symmetries, and that suc h a family is isomorphic to the pronilp oten t upp er subalgebra of the Virasoro algebra, that is, the subalgebra generated b y the elemen ts ℓ k with k ≥ 0. Here w e shall show (see a lso [1, 34]) tha t this is a common feature of all exact bi-Hamiltonian manifolds, and, in part icular, that the Liouville ve ctor field can b e added as the Virasoro generator ℓ − 1 . 6 Definition 2 [39] L et P λ b e a Poisson p encil of GZ typ e, and le t N := P 2 · P 1 − 1 its forma l r e cursion op er ator. A ve ctor fie ld Y is c a l le d a c onform al symmetry of the p encil i f it holds Lie Y N = N . (9) Prop osition 2 L et ( P λ , Z ) b e an e x act bi-Hamiltonian p enc i l . Then the field Y 0 := N Z is a c onformal symmetry of P λ . Pr o of: Since N = P 2 P 1 − 1 w e ha ve Lie Z ( N ) = 1 . No w, let us define Y 0 := N ( Z ); obviously [ Y 0 , N X ] = Lie Y 0 ( N X ) = Lie Y 0 ( N ) X + N Lie Y 0 ( X ) = Lie Y 0 ( N ) X + N [ Y 0 , X ] . (10) The v anishing of the Nijenh uis torsion of N (whic h, as it is w ell know n to exp erts in the theory o f P oisson p encil, is implied b y the compatibilit y of P 2 and P 1 ) reads, for ev ery pair of v ector fields W , X [ N W , N X ] = N [ N W , X ] + N [ W , N X ] − N 2 [ W , X ] . (11) Substituting Y 0 = N Z in (10) and using the v anishing of the torsion of N w e get Lie N Z ( N ) X + N [ N Z , X ] = N [ N Z, X ] + N [ Z , N X ] − N 2 [ Z , X ] = N [ N Z, X ] + N (Lie Z ( N ) X ) + N 2 [ Z , X ] − N 2 [ Z , X ] whic h yields Lie N Z ( N ) X = N X ∀ X , since Lie Z N = 1 . (12) As a corollary , w e ha ve the following result (see [39] for the full pro of, whic h holds obv iously also for the sligh t generalization herewith presen ted). It is based on the prop erties Lie Z N j = j N j − 1 (13) Prop osition 3 L et Y j := N j +1 Z ( so that Y − 1 ≡ Z ) b e the f a mily of ve ctor field s obtaine d formal ly b y the action of the r e cursion op er ator on the Liouvil l e ve ctor field Z . Then the c omm utation r elations of the Vir as or o algebr a [ Y j , Y k ] = ( k − j ) Y k + j hold. 7 2.3 The exact GD p encil and its τ -func tion The no wada ys standard form ulatio n of the n-th ( A n ) Gel’fand D ick ey (hence- forth, GD) hierarc hy is based o n its Lax represen tation (see [11] for a full accoun t of this theory); namely , the phase space is iden tified with the affine space of differen tial op erators of the form L = ∂ n +1 + U n ∂ n − 1 + U n − 1 ∂ n − 2 + · · · + U 1 , that is, the space of monic n + 1 -th order differen tial op erators with vanishing n -th order term. Its bi- Hamiltonian structure can b e represen ted by means of the Hamilton op erators ˙ L = P 1 ( X ) = [ L, X ] + ˙ L = P 2 ( X ) = ( LX ) + L − L ( X L ) + − 1 n + 1 [ L, ( ∂ − 1 [ X , L ] − 1 )] (14) where X represen ts a one-form on the phase space, that is, a purely non-lo cal pseudo differen t ia l op erator. As it is customary , t he subscript ( · ) + refers to the purely differen tial part of the op erator and ( · ) − 1 is the residue. The last term in the second r ow of (14) is a dded to the standard Adler-Gel’fand- Dic ke y Hamiltonian op erator in o r der to preserv e the v a nishing of co efficien t of ∂ n of Hamiltonian v ector fields asso ciated with a generic one-fo r m X (see, e.g., [17] or [5] § 9). It is w ell kno wn – and easily ascertained from (14) – that the Poisson p encil P 2 − λP 1 is exact, and admits a s a Lio uville ve cto r field the field Z := ˙ U 1 = 1 . It is also w ell kno wn that the densities of conserv ed quan tities of the n-th GD hierarc h y can b e collected in a generating function h ([ U ] , z ) of the form h ([ U ] , z ) = z + ∞ X i =1 h i ([ U ]) z i where z n +1 = λ ; w e use the sym b ol [ U ] a s a shorthand not a tion fo r ”differen- tial p olynomial in the dep enden t fields U i ( x )”; the series h ([ U ] , z ) is related with the Bak er Akhiezer function ψ of the theory by ψ = e R x h ([ U ] ,z ) dx e P i t i z i , 8 and is the unique solution of the ab o ve for m of the Riccati-ty p e equation h ( n +1) + n − 1 X j =0 U j +1 h ( j ) = z n +1 , where h (0) ≡ 1 and, by definition, h ( k +1) = ∂ x h ( k ) + h ([ U ] , z ) h ( k ) . In [21] the following represen tatio n for t he n- th GD (and K P) flo ws w as highligh ted: The GD flo ws imply the lo cal conserv atio n la ws ∂ ∂ t j h ([ U ] , z ) = ∂ x H ( j ) ([ U ] , z ) , (15) where H ( j ) ([ U ] , z ) are formal series of the form H ( j ) ([ U ] , z ) = z j + P ∞ k =1 H j k ([ U ]) z k and z is r elat ed with the parameter λ of the P o isson p encil by λ = z n +1 Along the GD flo ws t hese ”curren ts” ob ey the equations ∂ ∂ t j H ( k ) ([ U ] , z ) = H ( j + k ) − H ( j ) H ( k ) + k X l =1 H j l H ( k − l ) + j X l =1 H k l H ( j − l ) . (16) Let us consider the g enerating function of the densities of the second ( or dual) hamiltonian h ∗ ([ U ] , z ). According to Prop osition (1) it m ust satisfy as w ell suitable conserv ation la ws, to b e written a s ∂ ∂ t j h ∗ ([ U ] , z ) = ∂ x H ∗ ( j ) ([ U ] , z ) , (17) in terms of ”dual” currents H ∗ ( j ) ([ U ] , z ) that ha ve the form H ∗ ( j ) ([ U ] , z ) = j z j − 1 + ∞ X k =1 H ∗ j k ([ U ]) z k +1 . (18) It turns out 1 that, if w e denote by H ( l ) = z l − 1 − P k ≥ 1 H k l z − ( k + 1) , the dual curren ts a re giv en by H ∗ ( j ) = P j l =1 H ( l ) H ( j − l ) . By using this represen tation, and w orking a bit on the comp onen t- wise form of (16), a nd in particular on the form ula h ∗ ([ U ] , z ) = ∂ ∂ z h ([ U ] , z ) − ∞ X j =1 1 z j +1 ∂ ∂ t j h ([ U ] , z ) , (19) 1 See [7] (wher e co mputatio ns are done in the KP cas e) for more details. 9 one can show t ha t the co efficien ts H ∗ j k are symmetric in j, k , i.e., H ∗ j k = H ∗ k j , and along the flo ws their evolution satisfies ∂ H ∗ j k ∂ t l = ∂ H ∗ lk ∂ t j . Therefore, there exists a function τ ( t 1 , t 2 , . . . ) (indep enden t of the sp ectral par a meter z ) suc h that H ∗ j k = ∂ 2 ∂ t j ∂ t k log τ . (20) This function is the Hirota τ –function of the GD hierarc hy ; the outcome that w e w ant to herewith remark is that, in this picture, the τ -function app ears as the (logarithmic) p otential for the densities o f conserv atio n laws asso ciated with (17) the second G el’fand-Zakharevic h Hamiltonian na t urally defined o n the exact bi-Hamiltonian phase space of the KdV equation. 3 The disp ersion less case Let us consider an inte g r able system of the form (1), i.e. q i t = V i j ( q ) q j x + ∞ X k =1 ǫ k F i k ( q , q x , q xx , . . . , q ( n ) , . . . ) , (21) and consider it s disp ersionless (or hy dro dynamical) limit. The equations of the disp ersionless hierarc hy ha ve the form q i t = V i j ( q ) q j x (22) F or such systems, t he class of Hamilto nia n structures to b e considered were in tro duced b y Dubrovin and No vik ov. Let us briefly outline the k ey p oints in their construction. Consider functionals F [ q ] := Z S 1 f ( q 1 ( x ) , . . . , q n ( x )) dx, a nd G [ q ] := Z S 1 g ( q 1 ( x ) , . . . , q n ( x )) dx and define a brack et b et w een them as follows: { F , G } [ q ] := Z Z S 1 × S 1 δ F δ q i ( x ) ω ij ( x, y ) δ G δ q j ( y ) dxdy = = Z Z S 1 × S 1 ∂ f ∂ q i ( x ) ω ij ( x, y ) ∂ g ∂ q j ( y ) dxdy , (23) 10 where δ δq i denotes the v ariational deriv ativ e with resp ect to q i . The biv ector ω ij ( x, y ) has the follo wing (lo cal, h ydro dynamical) form ω ij = g ij δ ′ ( x − y ) + Γ ij k q k x δ ( x − y ) . (24) A deep result geometrically c haracterizes the conditions for a brack et (23) b e P oisson: Theorem 1 [14] If det( g ij ) 6 = 0 , then the br acket (23) is Poisson if and only if the metric g ij is flat and the functions Γ ij k ar e r elate d to the Christoffel symb o ls of g ij (the inv erse of g ij ) by the formula Γ ij k = − g il Γ j lk . Let us now consider a pair o f P oisson bivec t o rs of h ydro dynamic ty p e ω ij 1 , ω ij 2 , asso ciated with a pa ir of flat metrics g 1 and g 2 . As sho wn b y Dubrovin in [15] the flat metrics define a bi-Hamil ton i a n structur e o f hydr o dynami c typ e iff 1. the Riemann tensor R λ of the p encil g λ := g ij 2 − λg ij 1 v anishes fo r an y v alue of λ ; 2. the Christoffel sym b ols (Γ λ ) ij k of the p encil are giv en b y Γ ij (2) k − λ Γ ij (1) k . In this pap er w e will consider P o isson p encils of h ydro dynamic t yp e satisfying t w o additional assumptions that can b e expressed on the p encil g λ as follo ws: Assumption I: The ro ots u 1 ( q ) , . . . , u n ( q ) of the c hara cteristic equation det g λ = det( g 2 − λg 1 ) = 0 are functionally indep enden t. Assumption I I: The P oisson p encil asso ciated to the flat p encil of metrics g λ according to the Dubrovin-No vik ov recip e is an exact P oisson p encil ω λ . By definition this means that Lie Z ω 2 = ω 1 and Lie Z ω 1 = 0 fo r a suitable v ector field Z . The p encil g λ satisfying Assumption I is called semisimple a nd the functions u i ( q ) are called c anoni c al c o or dinates . It can b e sho wn that, in canonical co ordinates b oth metrics are diagonal [22]: g ij 1 = f i δ ij , g ij 2 = u i f i δ ij 11 and the the P oisson p encil ω λ b ecomes ω λ = g ij 2 ( u ) δ ′ ( x − y ) + Γ ij (2) k u k x δ ( x − y ) − λ  g ij 1 ( u ) δ ′ ( x − y ) + Γ ij (1) k u k x δ ( x − y )  where the Christoffel sym b ols v anish if a ll the indices are different and (as- suming i 6 = j ) Γ ii (1) j = 1 2 ∂ f i ∂ u j , Γ ij (1) i = − 1 2 f j f i ∂ f i ∂ u j , Γ ij (1) j = 1 2 f i f j ∂ f j ∂ u i , Γ ii (1) i = 1 2 ∂ f i ∂ u i Γ ii (2) j = u i Γ ii (1) j , Γ ij (2) i = u j Γ ij (1) i , Γ ij (2) j = u i Γ ij (1) j , Γ ii (2) i = 1 2 f i + u i Γ ii (1) i . Remark 1 I n c anonic al c o or dinates a l s o the e quations of the disp ersionless hier ar chy b e c ome diag o nal. The follo wing pro perty will b e crucial in the computations w e shall p erform in the core of the pap er Theorem 2 A semisimpl e bi-Hamiltonian structur e of hydr o dynamic typ e is exact if and only if the c ondition n X k =1 ∂ f i ∂ u k = 0 . (25) is sa tisfie d. Mor e over, in c anonic al c o or dinates al l the c omp onents of the ve ctor field Z ar e e qual to 1 . Pr o of . By means of a straigh tfor w ard computation, using form ula [2 9 ] Lie Z P ij = (26) X k ,s ∂ s x Z k ( u ( x ) , . . . ) ∂ P ij ∂ u k ( s ) ( x ) − ∂ Z i ( u ( x ) , . . . ) ∂ u k ( s ) ( x ) ∂ s x P k j − ∂ Z j ( u ( y ) , . . . ) ∂ u k ( s ) ( y ) ∂ s y P ik ! , w e obtain Lie Z ω ij 2 = Z k ∂ g ij (2) ∂ u k − ∂ Z i ∂ u k g k j (2) − ∂ Z j ∂ u k g ik (2) ! δ ′ ( x − y ) + Z k ∂ Γ ij (2) l ∂ u k − ∂ Z i ∂ u k Γ k j (2) l − ∂ Z j ∂ u k Γ ik (2) l − g ik (2) ∂ 2 Z j ∂ u k ∂ u l ! u l x δ ( x − y ) = ω ij 1 12 Similarly w e obtain Lie Z ω ij 1 = Z k ∂ g ij (1) ∂ u k − ∂ Z i ∂ u k g k j (1) − ∂ Z j ∂ u k g ik (1) ! δ ′ ( x − y ) + Z k ∂ Γ ij (1) l ∂ u k − ∂ Z i ∂ u k Γ k j (1) l − ∂ Z j ∂ u k Γ ik (1) l − g ik (1) ∂ 2 Z j ∂ u k ∂ u l ! u l x δ ( x − y ) = 0 The v anishing of the co efficien ts of δ ′ ( x − y ) implies Lie Z g 2 = g 1 , Lie Z g 1 = 0 , or, more explicitly (Lie Z g 1 ) ii = Z k ∂ f i ∂ u k − 2 f i ∂ Z i ∂ u i = 0 (Lie Z g 2 ) ii = Z k u i ∂ f i ∂ u k + Z i f i − 2 u i f i ∂ Z i ∂ u i = f i . T aking in to accoun t the first equation, the second equation implies Z i = 1 , i = 1 , . . . , n, (27) and, as a consequen ce, the first equation reduces to (25). It remains to ve rif y Z k ∂ Γ ij (1) l ∂ u k − ∂ Z i ∂ u k Γ k j (1) l − ∂ Z j ∂ u k Γ ik (1) l − g ik (1) ∂ 2 Z j ∂ u k ∂ u l = Z k ∂ Γ ij (1) l ∂ u k = 0 and Z k ∂ Γ ij (2) l ∂ u k − ∂ Z i ∂ u k Γ k j (2) l − ∂ Z j ∂ u k Γ ik (2) l − g ik (1) ∂ 2 Z j ∂ u k ∂ u l = Z k ∂ Γ ij (2) l ∂ u k = Γ ij (1) l . It is easy to c hec k that b oth follo w f rom (25 ). Remark 1 I n the ab ove c omputations we h ave use d the same letter ( Z ) to denote a ve ctor field on the manifold M and the c orr esp ondin g ve ctor fiel d on the lo op sp ac e L ( M ) . Remark 2 The semisimple Poisson p en cil of hydr o d yna m ic typ e asso ciate d with a semisimple F r ob enius m anifold is alw ays e xact [15]. The Liouvil le ve c tor field in this c ontext is usual ly denote d by the letter e and c al le d the unity ve ctor fie l d . 13 3.1 The n–th GD example Let us consider the disp ersionless limit of the A n Drinfel’d-Sok o lov bi-Hamiltonian structure. In this case w e hav e the follo wing generating functions for the con- tra v ariant comp onen ts of the metrics of t he p encil [36, 18] g 1 ( q , p ) = n X i,j =1 g ij 1 p i − 1 q j − 1 = λ ′ ( p ) − λ ′ ( q ) p − q g 2 ( q , p ) = n X i,j =1 g ij 2 p i − 1 q j − 1 = λ ′ ( p ) λ ( q ) − λ ′ ( q ) λ ( p ) p − q + λ ′ ( p ) λ ′ ( q ) n + 1 where λ ( p ) = p n +1 + U n p n − 1 + · · · + U 2 p + U 1 . Clearly , since λ ′ do es not dep end on U 1 and ∂ λ ∂ U 1 = 1, w e hav e Lie Z g 2 = g 1 , Lie Z g 1 = 0 , with Z = ∂ ∂ U 1 , that is the Poiss o n p encil asso ciated with g 1 and g 2 is exact. Moreo v er it is also semisimple. The canonical coo rdinates ( u 1 , . . . , u n ) are the critical v alues of λ . If w e denote b y v 1 , . . . , v n the critical p oints o f λ (b y definition they do not dep end on U 1 ): λ ′ ( p ) = ( n + 1) p n + ( n − 1) U n p n − 2 + · · · + U 2 = ( n + 1) n Y k =1 ( p − v k ) = 0 , the canonical co ordinates are u i = v n +1 i + U n v n − 1 i + · · · + U 2 v i + U 1 . As exp ected, in canonical co ordinates, the v ector field Z reads Z = n X i =1 ∂ u i ∂ U 1 ∂ ∂ u i = n X i =1 ∂ ∂ u i . 4 Cen tral in v arian ts The main problem in the a pproac h of the Dubro vin’s sc ho ol to t he theory of in tegrable systems is the classification of P oisson p encils of t he form (see for 14 instance [16, 29, 17, 18, 30, 4]) Π ij λ = ω ij 2 + X k ≥ 1 ǫ k k +1 X l =0 A ij (2) k,l ( q , q x , . . . , q ( l ) ) δ ( k − l +1) ( x − y ) − λ ω ij 1 + X k ≥ 1 ǫ k k +1 X l =0 A ij (1) k,l ( q , q x , . . . , q ( l ) ) δ ( k − l +1) ( x − y ) ! where ω 1 and ω 2 are semisimp le P oisson biv ectors of h ydro dynamic t yp e a nd A ij k ,l are differen t ia l p olynomials of degree l . W e recall that, by definition, deg f ( q ) = 0 a nd deg( q ( l ) ) = l . Tw o p encils Π λ and ˜ Π λ are considered equiv alen t if they are related b y a Miura transformation ˜ q i = F i 0 ( q ) + X k ≥ 1 ǫ k F i k ( q , q x , . . . , q ( k ) ) , det ∂ F i 0 ∂ q j 6 = 0 , deg F i k = k . In the semisimple case [29] (that is if ω λ is semisimple) equiv alence classes of equiv alent Poiss o n p encils a re lab elled by n functiona l parameters called c en tr al invariants . More precisely t w o p encils ha ving the same leading order are Miura equiv alen t if and only if they ha v e the same cen tra l in v ariants. In general, the problem of proving the existenc e o f the Poisson p encil corre- sp onding to a g iv en c hoice of the leading term ω λ and of the cen tral in v arian ts is still op en. Let us recall the definition of the cen tral in v arian ts of a P oisson p encil. A t eac h or der in ǫ the co efficien t of the term con taining the highest deriv a- tiv e of the delta function is a tensor field of type (2 , 0), symme t r ic for o dd deriv ative s and ske wsymmetric for eve n deriv a t iv es. Consider the fo rmal series π ij ( p, λ, q 1 , . . . , q n ) = g ij 2 p + X k ≥ 1 A ij (2) k, 0 p k +1 − λ g ij 1 p + X k ≥ 1 A ij (1) k, 0 p k +1 ! and denote b y λ i ( q , p ) the ro ot s of the equation det π ij ( p, λ, q 1 , . . . , q n ) = 0 . Expanding λ i ( q , p ) at p = 0 w e o btain λ i = u i + λ i 2 p 2 + O ( p 4 ) 15 F ollo wing [18] w e can define the cen tra l in v a rian t c i as c i = 1 3 λ i 2 ( q ) f i ( q ) (28) It turns out [29, 17] that the cen tra l in v arian t s c i dep end only on the canonical co ordinates u i and are giv en b y the following expression: c i ( u i ) = 1 3( f i ) 2 Q ii 2 − u i Q ii 1 + X k 6 = i ( P k i 2 − u i P k i 1 ) 2 f k ( u k − u i ) ! , i = 1 , . . . , n. (29) where P ij 1 , P ij 2 , Q ij 1 , Q ij 2 are the comp onen ts of the tensor fields A (1) ij 2 , 0 , A (2) ij 2 , 0 , A (1) ij 3 , 0 , A (2) ij 3 , 0 in canonical co ordinates. This means tha t, in suc h co ordinates, the p encil has the follow ing expansion in ǫ : Π ij λ = ω ij 2 + ǫ  P ij 2 δ ′′ ( x − y ) + · · ·  + ǫ 2  Q ij 2 δ ′′′ ( x − y ) + · · ·  + O ( ǫ 3 ) − λ  ω ij 1 + ǫ  P ij 1 δ ′′ ( x − y ) + · · ·  + ǫ 2  Q ij 1 δ ′′′ ( x − y ) + · · ·  + O ( ǫ 3 )  As a remark, we notice t ha t w e can define cen tral in v ariants in a n alter- nativ e w ay , as c i = − 1 3 f i Res λ = u i T r g − 1 λ A λ (30) where the tensor A ij is defined b y A ij λ = Q ij λ + ( g − 1 λ ) lk P li λ P k j λ . with Q ij λ = Q ij 2 − λQ ij 1 , P ij λ = P ij 2 − λP ij 1 . T o pro ve this iden tity w e notice t ha t the iden tit y (29) can b e written in terms of the tensor A ij λ as 3 c i ( u i )( f i ) 2 =  A ii λ  λ = u i = Res λ = u i n X k =1 A k k λ λ − u k . (31) and therefore, dividing b oth sides b y f i and using the pro perties of residues, 16 w e obtain 3 c i ( u i ) f i = Res λ = u i n X k =1 A k k λ f i ( λ − u k ) = Res λ = u i n X k =1 A k k λ f k ( λ − u k ) = − Res λ = u i n X k =1 ( g − 1 λ ) k l A lk λ = − Res λ = u i T r g − 1 λ A λ . Since the quan tity T r g − 1 λ A λ is a scalar function w e can ev aluate the com- p onen ts of the (1 , 1) tensor field g − 1 λ A λ in an arbitrary co ordina t e system, compute its trace a nd then, only at the end of the computation, write the result in terms o f canonical co ordinates. W e will use this pro cedure in the follo wing examples. AKNS . Let us consider the Poisson p encil ω 2 + ǫP (1) 2 − λω 1 with ω 2 + ǫP (1) 2 − λω 1 =  (2 u∂ x + u x ) δ v δ ′ ∂ x ( v δ ) − 2 δ ′  + ǫ  0 − δ ′′ δ ′′ 0  − λ  0 δ ′ δ ′ 0  (32) where, to compactify the f orm ulas, w e write δ instead of δ ( x − y ). This is the P oisson p encil of the so-called AKNS (or tw o-b oson) hierarc h y . In this case g λ =  2 u v − λ v − λ − 2  . After some computations w e get A λ = g λ det g λ and therefore, taking in to accoun t that u 1 = v + √ − 4 u, u 2 = v − √ − 4 u. f 1 = 8 u 2 − u 1 , f 2 = 8 u 1 − u 2 , using form ula (30) w e o bta in c 1 = − 1 3 f 1 Res λ = u 1 T r g − 1 λ A λ = − 1 3 f 1 Res λ = u 1 2 det g λ = − 1 12 c 2 = − 1 3 f 2 Res λ = u 2 T r g − 1 λ A λ = − 1 3 f 2 Res λ = u 2 2 det g λ = − 1 12 Tw o comp onen t CH . Mo ving P (1) 2 from P 2 to P 1 in the P oisson p encil of the AKNS hierarc hy one obtains the following P oisson p encil [20, 29] P λ =  (2 u∂ x + u x ) δ v δ ′ ∂ x ( v δ ) − 2 δ ′  − λ  0 δ ′ − ǫδ ′′ δ ′ + ǫδ ′′ 0  (33) 17 whic h is the P oisson p encil defining the so called CH 2 hierarc h y . The p encil g λ and the canonical co ordinates are the same of the previous example, while A λ = λ 2 g λ det g λ . Using form ula (30) w e obtain c 1 = − 1 3 f 1 Res λ = u 1 T r g − 1 λ A λ = − 1 3 f 1 Res λ = u 1 2 λ 2 det g λ = − ( u 1 ) 2 12 c 2 = − 1 3 f 2 Res λ = u 2 T r g − 1 λ A λ = − 1 3 f 2 Res λ = u 2 2 λ 2 det g λ = − ( u 2 ) 2 12 . Remark 3 Notic e that in b oth exa m ples the matrix g − 1 λ A λ is the id e ntity matrix times a sc alar function. In the first c ase this function is 1 det g λ while in the se c ond c ase it is λ 2 det g λ . 5 Bi-Hamiltonian c o homology In this section w e collect, fo r the reader’s conv enience, some definitions and results ab out (Bi)-Hamiltonian cohomologies and the D ubro vin-Zhang com- plex (see [16] for full details and pro ofs). Let g b e a flat metric o n a manifold M and ω b e the asso ciat ed P o isson biv ector of h ydro dynamic t yp e. In anal- ogy with the case of finite dimensional P oisson manifolds [28] one defines P oisson cohomo lo gy groups in the followin g w ay: H j ( L ( M ) , ω ) := k er { d ω : Λ j lo c → Λ j +1 lo c } im { d ω : Λ j − 1 lo c → Λ j lo c } (34) where d ω := [ ω , · ] (the square brac k ets denote the Sc houten brac k ets) and Λ j lo c is the space of lo cal j -multiv ectors on the lo op space of the manif o ld M (see [1 6] for more details on the definition of this complex). The space of lo cal m ultiv ectors has a nat ur a l decomp osition in comp onents of same degree. T o determine each comp onen t, we r ecall that, b y definition, deg δ ( x − y ) = 1 and ∂ x increases the degrees by one so that deg  A i 1 ,...,i k δ ( l 2 ) ( x 1 − x 2 ) . . . δ ( l k ) ( x 1 − x k )  = deg A i 1 ,...,i k + ( l 2 + · · · + l k ) + k − 1 , where A i 1 ,...,i k = A i 1 ,...,i k ( u ( x 1 ) , u x 1 , . . . ) is a differen tial p olynomial. In this w ay , for instance, a homogeneous vec t o r field o f degree k is a v ector field whose comp onen ts are differen t ial p olynomials of degree k . Since the decom- p osition of Λ j lo c in homogeneous comp onen ts is preserv ed b y d ω , w e hav e H j ( L ( M ) , ω ) = ⊕ k H j k ( L ( M ) , ω ) . (35) 18 F or Pois son structures o f h ydro dynamic type lik e (24), it has been pro v ed in [25] (see also [13] for an indep enden t pro of of the cases n = 1 , 2) that H k ( L ( M ) , ω ) = 0 for k = 1 , 2 , . . . . The v a nishing of these cohomology gro ups implies that an y deformation of a P oisson bivec to r of a h ydro dynamic type P ǫ = ω + ∞ X n =1 ǫ n P n , (36) where P k ∈ Λ 2 k +2 , loc can b e o btained from ω b y p erforming a Miura transfor- mation. In order to study deformations of Poiss o n pencil of hydrodynamic t yp e it is necessary to in t r o duce bi- Hamiltonian cohomology groups [24, 16, 29]. F or i ≥ 2 they are defined a s H i k ( L ( M ) , ω 1 , ω 2 ) = Ker  d ω 1 d ω 2 | Λ i − 1 k, lo c  Im  d ω 1 | Λ i − 2 k − 2 , lo c  ⊕ Im  d ω 1 | Λ i − 2 k − 2 , lo c  . Liu and Zhang sho we d that, in the semisimple case, H 2 k ( L ( M ) , ω 1 , ω 2 ) = 0 ∀ k 6 = 2 , and that the elemen ts of H 2 2 ( L ( M ) , ω 1 , ω 2 ) ha v e the form d 2 n X i =1 Z c i ( u i ) u i x log u i x dx ! − d 1 n X i =1 Z u i c i ( u i ) u i x log u i x dx ! (37) where c i ( u i ) are the cen tr al inv arian ts in tro duced in the previous section. More explicitly , the comp onen ts of these vec to r fields, in canonical co ordi- nates, are giv en by X i = n X j =1  1 2 δ ij ∂ x f i + A ij  c j u j x + (2 δ ij f i − L ij ) ∂ x ( c j u j x )  , i = 1 , . . . , n. (38) with A ij = 1 2  f i f j ∂ f j ∂ u i u j x − f j f i ∂ f i ∂ u j u i x  (39) L ij = 1 2 δ ij f i + ( u i − u j ) f i 2 f j ∂ f j ∂ u i . (40) 19 W e will use these facts la t er. 6 Constant central inv arian ts and exactne ss This section is dev oted to the pro of of the main result of the pap er. Theorem 3 L et Π λ = P 2 − λP 1 = ω 2 + ∞ X k =1 ǫ k P ( k ) 2 − λ ω 1 + ∞ X k =1 ǫ k P ( k ) 1 ! . (41) b e a Poisson p en c i l whos e disp ersionles s limit ω 2 − λω 1 is semisimple and exact. Then its c entr a l invariants a r e c on s tant if and only if it is, in the sense of formal series of Poisson p encils, exact. In particular, w e recall that Theorem 2 states tha t a P oisson p encil of hy - dro dynamic t yp e is exact if and only if the quantities f j satisfy n X k =1 ∂ f j ∂ u k = 0 , j = 1 , . . . , n. W e split the pro of of the main theorem into the pro of of some Lemmas. Lemma 1 L et Π λ = P 2 − λP 1 = ω 2 + ∞ X k =1 ǫ k P ( k ) 2 − λ ω 1 + ∞ X k =1 ǫ k P ( k ) 1 ! , P ( k ) 1 , 2 ∈ Λ 2 k +2 , lo c b e a Poisso n p encil whose disp ersionless lim i t ω λ = ω 2 − λω 1 is a semisimple Poisson p encil of h ydr o dynamic typ e (not ne c essarily exa c t). L et ( c 1 , . . . , c n ) b e the c entr a l inva ri a n ts of Π λ . Then ther e exists a Miur a tr ansformation r e ducing it to the form Π λ = ω λ + ∞ X k =1 ǫ 2 k P (2 k ) 2 , P (2) 2 = Lie X ( c 1 ,...,c n ) ω 1 , (42) with X ( c 1 ,...,c n ) given by (37) . 20 P roof. The lemma is a consequence of the v anishing o f the second Pois son co- homology group [2 5, 13 , 16] asso ciated to P o isson structure o f hydrodynamic t yp e and of the triviality of the o dd order deformations [29, 17 ]. Let us restrict our atten tion to exact P oisson p encils of the f orm (42). This means that there exists a v ector field Z = P ∞ k =0 ǫ 2 k Z 2 k (deg Z 2 k = 2 k ) suc h that Lie Z ( ω 1 ) = 0 , (43) Lie Z ( ω 2 + ∞ X k =1 ǫ 2 k P (2 k ) 2 ) = ω 1 . (44) F rom (4 3) a nd (44 ) it follow s tha t Lie Z 0 ω 1 = 0 (45) Lie Z 0 ω 2 = ω 1 . (46) W e ha v e seen (see Theorem 2) tha t this implies ( 2 5) a nd that, in cano nical co ordinates Z i 0 = 1, that is Z 0 = e . Lemma 2 Ther e exi s ts a Miur a tr ansformation pr eserving ω 1 that r e d uc es Z to e . Pr o of . F rom (43) it follows that Lie Z 2 k ( ω 1 ) = 0 , k = 1 , 2 , . . . . (47) This means, in part icular, that Z 2 = d ω 1 H 2 for a suitable functional H 2 . The Miura transformation generated b y the vec to r field d 1 ˜ H 2 with Lie e ˜ H 2 = H 2 (48) man tains the form of the p encil: Π λ → ˜ Π λ = ω λ + P ∞ k =1 ǫ 2 k ˜ P (2 k ) 2 and reduces Z t o the f o rm Z = e + ǫ 2 (Lie d ω 1 ˜ H 2 e + d ω 1 H 2 ) + O ( ǫ 4 ) = e + ǫ 2 ( d ω 1 ( − Lie e ˜ H 2 ) + d ω 1 H 2 ) + O ( ǫ 4 ) = e + O ( ǫ 4 ) . W e can apply the same a rgumen ts to higher o rder deformations and construct a Miura transformation that maps Z in to e . 21 Remark 2 F or c ompleteness, let us further discuss the solvability of (48), that is, of an e quation of the form Lie e ˜ K = K (49) for the unknown functional ˜ K = Z S 1 ˜ k dx. In c anonic al c o or dinates e quation (49) r e ads Z S 1 n X i =1 ∂ ˜ k ∂ u i dx = Z S 1 k dx. Inde e d taking into ac c ount the p erio dic b oundary c onditions the l.h.s. of (49) is e qual to n X i =1 Z S 1 e i δ ˜ K δ u i dx = n X i =1 Z S 1 " ∂ ˜ k ∂ u i + ∂ x ∞ X k =1 ( − 1) k ∂ k − 1 x ∂ ˜ k ∂ u i ( k ) !# dx = Z S 1 n X i =1 ∂ ˜ k ∂ u i dx, wher e u i ( k ) is the k − th d erivative with r es p e ct to x of u i . A so l ution c an b e found s o lving the e quation n X i =1 ∂ ˜ k ∂ u i = k for the density of the functional ˜ K . It is e quiva l e nt to the system of e quations n X i =1 ∂ ˜ A j ∂ u i = A j , n X i =1 ∂ ˜ B j m ∂ u i = B j m , . . . for the c o efficients ˜ A i , ˜ B ij , . . . of the homo genous differ ential p olynomial ˜ k = ˜ A i u i ( N ) + ˜ B ij u i x u j ( N − 1) + . . . With a line ar ch a nge of c o or di n ates ( u 1 , . . . , u n ) → ( w 1 , . . . , w n ) w e c an r e duc e P n k =1 ∂ ∂ u i to ∂ ∂ w 1 . In such c o or dina tes the solution is obtaine d inte gr at- ing the c o efficien ts of k along w 1 . C le arly the solution is not unique and in the c o or dinates ( w 1 , . . . , w n ) is define d up to functions of ( w 2 , . . . , w n ) . The next lemma sho ws that the constancy of the cen tral in v arian ts is related to the exactness at the second order of the p encil. 22 Lemma 3 L et Π λ b e a Poi s s on p en c il of the f orm (42) . Stil l in the hy- p o thes e s o f The or em 2 (namely, if the c o n dition (25) is satisfie d), the c entr al invariants of Π λ ar e c onstant if a nd only if the se c ond or der c ondition Lie e P (2) 2 = 0 , (50) is sa tisfie d. Pr o of . W e ha ve the follo wing identit y Lie e P (2) 2 = Lie e Lie X ( c 1 ,...,c n ) ω 1 = Lie [ e,X ( c 1 ,...,c n )] ω 1 = Lie X ( ∂ c 1 ∂ u 1 ,..., ∂ c n ∂ u n ) ω 1 (51) Supp ose that Lie e P (2) 2 = 0, then, using (51) , we ha v e Lie X ( ∂ c 1 ∂ u 1 ,..., ∂ c n ∂ u n ) ω 1 = 0 and this implies ∂ c i ∂ u i = 0 , ∀ i . Supp ose now that all the cen tral in v a rian ts are constant, then, using (51) w e obtain ( 50). Remark 3 A c c or ding to the r esults of [29] and as alr e ady state d in L emm a 1 we c an assume, without loss of gener ality, that P (2) 2 is given by Lie X ω 1 . In this c ase c ondition (50) gives the exactness a t the se c ond or der of the p e n cil. However in or der to pr ove that the exactness of the p encil im plies the c onstanc y of the c entr al invariants we have to r e duc e the Liouvil le ve c tor field to e . The r e ducing Miur a tr an s formation, in gener a l , do es not pr eserve P (2) 2 . Lemma 3 relates the condition (50) to the constancy of t he cen t r al inv ari- an ts but do es not giv e us a ny information ab out the higher order conditions en tering the definition o f exactness. In order to push our analysis further up in t he ǫ expansion, w e need the results ab out bi-Hamilto nia n cohomology w e recalled in the previous section. Lemma 4 If the c ondition (25) is satisfie d, and the p encil (42) satisfies Lie e P (2) 2 = 0 23 then ther e exist a Miur a tr ansformation such that Π λ → ˜ Π λ = ω λ + ∞ X k =1 ǫ 2 k ˜ P (2 k ) 2 . with Lie e ˜ P (2 k ) 2 = 0 , k = 1 , 2 , . . . Pr o of . W e construct the Miura transformation by induction. Supp ose that the p encil Π λ satisfies Lie e P (2 k ) 2 = 0 , . . . , N but at the subsequen t order, Lie e P (2 N +2) 2 6 = 0 . W e show that it is p ossible to define a Miura transformatio n suc h that the transformed p encil ˜ Π λ satisfies the ab ov e condition, that is, is exact up to order 2 N + 2, with Liouville vec t o r field still giv en b y Z = e . T o construct suc h a tra nsformation w e will use the f ollo wing strategy: • First w e will sho w that Lie e P (2 N +2) 2 = Lie X (2 N +2) 2 ω 1 and that the v ector field X (2 N +2) 2 b elongs to H 2 2 N +2 ( L ( M ) , ω 1 , ω 2 ). Due to the trivialit y of this cohomology g roup fo r N > 0 this implies that X (2 N +2) 2 = d ω 1 H (2 N +2) 2 + d ω 2 K (2 N +2) 2 for tw o suitable lo cal functionals H (2 N +2) 2 and K (2 N +2) 2 ha ving densities whic h ar e differen tial p olynomials of degree 2 N + 2. • Second w e will sho w that the p encil ˜ Π λ related to Π λ b y the Miura transformation generated b y the v ector field d ω 1 ˜ K (2 N +2) 2 , with Lie e ˜ K (2 N +2) 2 = K (2 N +2) 2 , (52) has the required prop erty . 24 Concerning the first p oint w e ha ve to sho w tha t d ω 1  Lie e P (2 N +2) 2  = 0 (53) d ω 2  Lie e P (2 N +2) 2  = 0 . (54) This can b e easily pro v ed using the f ollo wing consequenc es of graded Jacobi iden tit y: Lie e d ω 1 − d ω 1 Lie e = 0 (55) Lie e d ω 2 − d ω 2 Lie e = d ω 1 . (56) Indeed, (53) fo llo ws immediately from (55) a nd d ω 1 P (2 N +2) 2 = 0. T o ascertain the v alidity o f (54) we first observ e that from [ P 2 , P 2 ] = 0 it follow s d ω 2 P (2 N +2) 2 = − 1 2 N X k =1 [ P (2 k ) 2 , P (2 N +2 − 2 k ) 2 ]; then using (56) and graded Jacobi we obta in d ω 2 Lie e P (2 N +2) 2 = Lie e d ω 2 P (2 N +2) 2 − d ω 1 P (2 N +2) 2 = = − 1 2 N X k =1 Lie e [ P (2 k ) 2 , P (2 N +2 − 2 k ) 2 ] = 0 Concerning the second p oint (that is, Equation (52)), we observ e that the Miura transformation g enerated b y the v ector field ǫ 2 N +2 d ω 1 ˜ K (2 N +2) 2 reduces the p encil to the form ˜ Π λ = ω λ + ǫ 2 P (2) 2 + · · · + ǫ 2 N +2 ˜ P (2 N +2) 2 + O ( ǫ 2 N +4 ) = = ω λ + ǫ 2 P (2) 2 + · · · + ǫ 2 N +2  P (2 N +2) 2 + Lie d ω 1 ˜ K (2 N +2) 2 ω 2  + O ( ǫ 2 N +4 ) and Lie e ˜ P (2 N +2) 2 = Lie e P (2 N +2) 2 + Lie e d ω 2 d ω 1 ˜ K (2 N +2) 2 = d ω 1 d ω 2 K (2 N +2) 2 + d ω 2 d ω 1 Lie e ˜ K (2 N +2) 2 = d ω 1 d ω 2 K (2 N +2) 2 + d ω 2 d ω 1 K (2 N +2) 2 = 0 25 Remark 4 The identity (5 5) is the c ounterp art at the level of the double c o m plex define d b y ( d ω 1 , d ω 2 ) of the exa ctness of the p encil ω 2 − λω 1 . Collecting the r esults of all the previous Lemmas we can finally prov e the main theorem. Pro of of the main theorem . Due to lemma 1, without loss generalit y w e can assume that the p encil ha s the form (42). Supp ose that the p encil (42 ) is exact, i.e. it satisfies (4 3 ) and (44). Due to lemma 2, p erforming a Miura t r a nsformation preserving ω 1 , we can reduce Z to e . After suc h a Miura transformation P (2) 2 → Lie X ( c 1 ,...,c n ) ω 1 + Lie d ω 1 ˜ H 2 ω 2 The exactness of the p encil implies Lie e  Lie X ( c 1 ,...,c n ) ω 1 + Lie d ω 1 ˜ H 2 ω 2  = Lie X ( ∂ c 1 ∂ u 1 ,..., ∂ c n ∂ u n ) ω 1 + Lie d ω 1 (Lie e ˜ H 2 ) ω 2 = 0 , that is Lie X ( ∂ c 1 ∂ u 1 ,..., ∂ c n ∂ u n ) ω 1 = − Lie d ω 1 (Lie e ˜ H 2 ) ω 2 . (57) The ab o ve identit y makes sense only if c i =constan t (and hence b oth sides v anish). Indeed, (57) tell us that the second order deformation ǫ 2 Lie X ( ∂ c 1 ∂ u 1 ,..., ∂ c n ∂ u n ) ω 1 can b e eliminated by the Miura transformation generated by the Hamilto- nian v ector field ǫ 2 d ω 1 Lie e ˜ H 2 . But, due to the results of [29 ], this is p ossible only if ∂ c i ∂ u i = 0 , ∀ i . Supp ose now that the cen tral in v arian ts of t he p encil (42) are constant. Due to lemma 3 the p encil satisfies the condition (50). In order to pro ve that ( 4 2) is exact it is enough to prov e that it is Miura equiv alen t to an exact P oisson p encil. But this follows from lemma 4. W e close this section discussing how the a b ov e pro cedure w orks for the case o f the AKNS hierarc h y . Let us consider the P oisson p encil (32). W e ha v e already show n that it ha s constan t cen tral inv arian ts. According to theorem 3 it is an exact Poiss on p encil. The Lio uville ve ctor field is Z = e = ∂ ∂ v . 26 Notice that  0 − δ ′′ δ ′′ 0  = − Lie X  (2 u∂ x + u x ) δ v δ ′ ∂ x ( v δ ) − 2 δ ′  where X =  0 ∂ x ∂ x 0   δH δξ δH δη  , H = − Z S 1 η ( x ) 2 4 dx. This means that the Miura transformation generated by the v ector field X (up to terms of order O ( ǫ 3 )) reduces the p encil (32) to the form P ′ λ =  (2 u∂ x + u x ) δ v δ ′ ∂ x ( v δ ) − 2 δ ′  − λ  0 δ ′ δ ′ 0  + ǫ 2 2 Lie 2 X  (2 u∂ x + u x ) δ v δ ′ ∂ x ( v δ ) − 2 δ ′  + ǫ 3 6 Lie 3 X  (2 u∂ x + u x ) δ v δ ′ ∂ x ( v δ ) − 2 δ ′  + · · · =  (2 u∂ x + u x ) δ v δ ′ ∂ x ( v δ ) − 2 δ ′  − λ  0 δ ′ δ ′ 0  + ǫ 2 2  0 0 0 δ ′′′  + ǫ 3 6  0 − δ ′′′′ δ ′′′′ 0  + . . . Notice also that the v ector field Z = e = ∂ ∂ η is left in v arian t b y the Miura transformation generated by X (indeed Z and X comm ute). Moreo v er ac- cording to lemma 3 Lie e P ′ (2) 2 = 0. 7 Conclus ions and o utlo ok In this pap er we elab ora ted on the circle of ideas connecting exact bi-Hamiltonian p encils, tau structures, and the cen tra l in v ariants o f hierarc hies admitting h ydro dynamical limit, a s defined b y Dubrov in and collab orators. W e hav e pro vided t he c haracterization of a semisimple exact p encil of hyd r o dynamical t yp e in canonical co ordinates. If this is related to a F rob enius manifold, then the Liouville ve ctor field mus t coincide with the unity v ector field. W e ha v e sho wn that the exactness of the p encil is equiv alent to the constancy of the cen tral in v ariants defined by the disp ersiv e expansion of the P oisson p encil of the hierarc hy , and, in particular, that exactness at o r der 2 in the ε expansion is sufficien t to ensure exactness at all orders. W e b eliev e that this prop erty is intimately related with the prop erties of the v ector field e t ha t altho ug h not b elonging to the Dubrov in- Zhang complex, defines a n outer deriv ation of the complex, and satisfies (5 5). 27 Still, man y imp ortant examples of bi-Hamiltonian hierarc hies of PDEs do not hav e constan t cen tral inv arian ts (and are b eliev ed not to admit τ - structures, at least in the strong sense herewith understo o d). Among them the Camassa-Holm equation and its m ulticomp onent generalizations [8, 29, 9, 20], and other examples b elonging to the so called r -KdV- CH-hierar ch y [33, 2, 3, 10]. In particular in [29] it ha s b een show n that the CH equation p ossesse s linear cen t r a l inv arian ts, while, e.g., the CH 2 equation has quadr atic cen tral in v arian ts. A natural question w ould b e whether the p oin t of view exp osed in the presen t pap er can b e applied to characterize these hierarc hies. 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