Self-dual gravity is completely integrable

Self-dual gra vit y is completel y integrable Y Nutku 1 , M B Sheftel 2 , J Kala yci 1 and D Y azıcı 3 1 F eza G ¨ ursey Ins titute, PO Bo x 6, Cengelk o y , 81220 Istan bul, T urk ey 2 Departmen t of Ph ysics, Bo˘ gazi¸ ci Univ ersit y , 34342 Bebek, Istan bul, T urk ey 3 Departmen t of Ph ysics, Yıldız T ech nical Univ ersit y , Esenler, Istan bul 34210,T urk ey E-mail: n utku@gursey .go v.tr, mikhail.sheftel@b oun.edu.tr, jank ala yci@gmail.com, y azici@yildiz.edu.tr Abstract W e disco ver multi-Ha miltonian structure of complex Monge-Amp ` ere equation ( C M A ) set in a rea l fi rst-order t wo-co mp onent form. There- fore, b y Magri’s theorem this is a completely integrable system in four real d imensions. W e start with Lagrangian and Hamiltonian densities and obta in a symplecti c form and th e Hamiltonian op erator that de- termines t h e Dirac brac k et. W e ha ve calculated all p oin t symmetries of t wo-co mp onent C M A system and Hamiltonia n s of the symmetry flo ws. W e hav e found tw o new r eal recurs ion op erators for symme- tries which comm ute with the op erato r of a symmetry condition on solutions of the C M A system. These op erators form t wo Lax pairs for the tw o-co mp onent system. The recursion op erators, app lied to the first Hamiltonian op erato r , generate infinitely many real Hamil- tonian structures. W e sho w ho w to construct an infi nite h ierarch y of higher comm uting flo ws together w ith the c orresp ond ing infinite c hain of their Hamilt onians. P A CS num b ers: 0 4.20.Jb, 02.40.Ky AMS classific atio n sc heme n um b ers: 35Q75 , 83C15 1 In tro du c tion In earlier pap er [1] we presen ted complex m ulti-Hamiltonian structure of Pleba ˜ nski’s second hea v enly equation [2], whic h by Magri’s theorem [3] pro v es that it is a completely integrable system in four complex dimensions. W e ex- p ect that P leba ˜ nski’s first hea v enly equation also admits m ulti-Hamiltonian structure, since thes e tw o equations, gov erning Ricci-flat metrics with (anti- )self-dual Riemann curv ature 2- form, a re related by Legendre tra nsformation 1 of cor r esp onding hea ve nly tetrads [2]. Ho w ev er, s ince b oth Pleba ˜ nski’s equa- tions are complex, their solutions are p oten tia ls of the c omplex metrics that satisfy Einstein equations in complex four-dimens ional spaces. In the case of complex Monge-Amp ` ere equation ( C M A ), that go ve rns (an ti- )self-dual g ra v- it y in real four-dimensional s paces with either Euclidean or ultra-h yp erb olic signature, w e hav e an additional condition tha t symplectic, Hamiltonian, and recursion o perato rs all should b e r eal. F urthermore, the tra nsformation b e- t w een the t wo heav enly equations cannot b e applied to transform the second hea v enly equation to C M A b ecause t he latter eq uatio n is real a nd therefore the m ulti-Hamiltonian structure of C M A cannot b e obtained by tra nsform- ing the m ulti-Hamiltonian structure of the second heav enly equation giv en in [1]. T herefore, in this pap er w e analyze the complex Monge-Amp ` ere equa- tion independen tly of our previous work and obtain real recursion op erators for symme tries and real m ulti-Hamiltonian structures of C M A . In section 2 we start with the complex Monge-Amp` ere equation in a t w o- comp onent first-order ev olutiona ry fo rm with the Lagrangian that is appropriate for Hamiltonian formulation. In section 3 w e presen t a sym- plectic structure and Hamiltonian structure of this C M A system . In section 4 w e transform the Hamilto nian densit y and Hamiltonian op erator to real v ar ia bles and in tro duce a con ve nient not a tion needed later to arriv e at a compact f orm of recursion op erator s and higher Hamiltonian op erators. In section 5 w e deriv e a symmetry condition, that determines symmetries of C M A system, in a tw o - component form a nd real v ariables, using our new notation. W e hav e calculated all p oin t symmetries of the C M A system and Hamiltonians of the symmetry flows that yield conserv ation laws for the C M A sys tem. In section 6 w e obtain tw o new real recursion op erators for symmetries whic h comm ute with a n operato r of the symme try condition on solutions of the C M A system. Moreov er, these t w o couples of op erators fo rm t w o Lax pairs for the t w o-comp onen t system. The recursion op erators, b eing applied to the first Hamiltonian op erator, yield further Hamiltonian struc- tures and bi-Hamiltonian reresen tations of the C M A system. Rep eating this pro cedure for the se cond Ha miltonian opera t o r, we could generate infinitely man y Hamiltonian structures of the C M A system. This m ulti-Hamiltonian structure o f the C M A system prov es its complete integrabilit y in the sense of Magri and hence complete integrabilit y of the (an ti-) self-dua l g ra vit y in four real dimensions with either Euclidean or ultra - h yp erbolic signature. In section 7 we construct a n infinite hierarch y of higher flo ws and show the w ay of calculating a corres p onding infinite ch ain of higher Hamiltonians. 2 2 Complex Mo nge-Amp ` ere equation in fi rst- order ev ol u tionary form and its Lagr ang ian F our- dimens ional hyper-K¨ ahler metrics d s 2 = u 1 ¯ 1 d z 1 d ¯ z 1 + u 1 ¯ 2 d z 1 d ¯ z 2 + u 2 ¯ 1 d z 2 d ¯ z 1 + u 2 ¯ 2 d z 2 d ¯ z 2 (2.1) satisfy Einstein field equations with either Euclidean or ultra-hyperb olic signature, if the K¨ ahler p oten tial u satisfies elliptic o r h yp erb olic complex Monge-Amp ` e re equation u 1 ¯ 1 u 2 ¯ 2 − u 1 ¯ 2 u 2 ¯ 1 = ε (2.2) with ε = ± 1 resp ectiv ely [2]. Here u is a real-v alued function of the t wo complex v ariables z 1 , z 2 and their conjugates ¯ z 1 , ¯ z 2 , the subscripts denoting partial deriv at ives with resp ect to these v ariables. Such metrics are Ricci-flat and ha v e (anti-)self-dual curv ature. In order t o discuss the Hamiltonian structure of C M A (2.2), w e shall replace the complex conjugate pair of v ariables z 1 , ¯ z 1 b y the real time v ariable t = 2 ℜ z 1 and the real space v ariable x = 2 ℑ z 1 and change the notation for the second complex v ariable z 2 = w . Then (2.2) b ecomes ( u tt + u xx ) u w ¯ w − u tw u t ¯ w − u xw u x ¯ w + i ( u tw u x ¯ w − u xw u t ¯ w ) = ε. (2.3) No w w e can express (2.3) as a pair of first-order nonlinear evolution equations b y in tro ducing an auxiliary dep enden t v ariable v = u t ( u t = v v t = − u xx + 1 u w ¯ w  v w v ¯ w + u xw u x ¯ w + i ( v ¯ w u xw − v w u x ¯ w ) + ε  , (2.4) so that finally (2.2) is set in a t wo-component form. F or the sak e of brevity w e shall henceforth refer to (2.4) as C M A sys tem. The Lagrangia n densit y for the original f o rm (2.2) of the complex Monge- Amp ` ere equation w as suggested in [4] L = 1 6 [ u 1 u ¯ 1 u 2 ¯ 2 + u 2 u ¯ 2 u 1 ¯ 1 − u 1 u ¯ 2 u 2 ¯ 1 − u 2 u ¯ 1 u 1 ¯ 2 ] + εu , (2.5) but this m ust b e cast into a f orm suitable for passing to a Hamiltonian. This requires that the fo r m o f a Lagra ngian should b e appropriate for a pplying 3 Dirac’s theory of constraints [5]. W e c ho ose the Lagrangian densit y for the first-order C M A system (2.4) to b e degenerate, that is, linear in the time deriv ative of unk nown u t and with no v t : L = 1 6 { ( u 2 x − 3 v 2 ) u w ¯ w + u w u ¯ w u xx − u x ( u w u x ¯ w + u ¯ w u xw ) + u t  2 i ( u w u x ¯ w − u ¯ w u xw ) + 6 v u w ¯ w )  } + εu (2.6) whic h, after substituting v = u t , coincides with our original Lagrangian (2 .5) up to a t otal div ergence. 3 Symplectic and Hamiltoni an structure s Since the Lagrangian densit y (2.6) is linear in u t and has no v t , the canonical momen ta π u = ∂ L ∂ u t = i 3 ( u w u x ¯ w − u ¯ w u xw ) + v u w ¯ w π v = ∂ L ∂ v t = 0 (3.1) cannot b e inv erted fo r the v elo cities u t and v t and so the La grangian is degenerate. Therefore, according to the Dirac’s theory [5], w e imp ose them as constrain ts φ u = π u + i 3 ( u ¯ w u xw − u w u x ¯ w ) − v u w ¯ w = 0 φ v = π v = 0 (3.2) and calculate the Poisson bra c k ets of the constrain ts (more details of the pro cedure w ere giv en in [1]) K ik = [ φ i ( x, w , ¯ w ) , φ k ( x ′ , w ′ , ¯ w ′ )] (3.3) collecting results in a 2 × 2 matrix form, where the subscripts run from 1 to 2 with 1 and 2 correspo nding to u and v resp ectiv ely . This yields the symplectic op erator K that is the in vers e of the Hamiltonian op erator J 0 : K = ( v ¯ w − iu x ¯ w ) D w + ( v w + iu xw ) D ¯ w + v w ¯ w − u w ¯ w u w ¯ w 0 ! (3.4) 4 as an explicitly sk ew-symmetric lo cal op erator. A symplectic 2-form is a v olume in tegral Ω = R V ω dxdw d ¯ w of the densit y ω = 1 2 du i ∧ K ij du j = 1 2 ( v ¯ w − iu x ¯ w ) du ∧ du w + 1 2 ( v w + iu xw ) du ∧ du ¯ w + u w ¯ w dv ∧ d u (3.5) where u 1 = u and u 2 = v . In ω , under the sign of the v olume in tegral, w e can neglect all the terms that are either total deriv atives or tota l dive rgencies due to suitable boundar y conditions on the b oundary surface of the v olume. F or the exterior differen tial of this 2-form w e obtain dω = − idu x ∧ du w ∧ du ¯ w = − ( i/ 3)  D x ( du ∧ du w ∧ du ¯ w ) (3.6) + D w ( du x ∧ du ∧ du ¯ w ) + D ¯ w ( du x ∧ du w ∧ du )  ⇐ ⇒ 0 that is, a to t a l div ergence whic h is equiv a len t to zero, so t ha t the 2-form Ω is clos ed and hence sy mplectic. The Hamiltonian opera t or J 0 is obtained b y in v erting K in (3.4) J 0 = (3.7)    0 1 u w ¯ w − 1 u w ¯ w v ¯ w − iu x ¯ w 2 u 2 w ¯ w D w + D w v ¯ w − iu x ¯ w 2 u 2 w ¯ w + v w + iu xw 2 u 2 w ¯ w D ¯ w + D ¯ w v w + iu xw 2 u 2 w ¯ w    that is explicitly sk ew-symmetric. It satisfies the Jacobi iden tity due to (3.6). The Hamiltonian densit y is H 1 = π u u t + π v v t − L with the result H 1 = 1 6 h (3 v 2 − u 2 x ) u w ¯ w − u w u ¯ w u xx + u x ( u ¯ w u xw + u w u x ¯ w ) i − εu. (3.8) C M A system can now b e written in the Hamiltonian f o rm with the Hamil- tonian densit y H 1 defined b y (3.8) u t v t ! = J 0 δ u H 1 δ v H 1 ! (3.9) where δ u and δ v are Euler-Lagrange op erators [6] with resp ect t o u and v applied to the Hamiltonian densit y H 1 (they corresp ond to v ar ia tional deriv atives of the Hamiltonian functional R V H 1 dV ). 5 4 T ransformation to real v ariables In the case of C M A , that gov erns (anti-)self-dual gravit y with either Eu- clidean or ult r a -h yp erbo lic signature, w e ha v e an additional condition that all the ob jects in the theory , in particular a recursion op erator, should b e real. Th erefore, w e transform the Hamiltonian densit y together with t he symplectic and Hamiltonian op erators to the real v ariables y = 2 ℜ w and z = 2 ℑ w . The Hamiltonian densit y in the real v ariables becomes H 1 = 1 6 h (3 v 2 − u 2 x )∆( u ) − ( u 2 y + u 2 z ) u xx + 2 u x ( u y u xy + u z u xz ) i − εu. where ∆( u ) = u y y + u z z , whic h simplifies af t er cancelling terms that are total deriv atives to H 1 = 1 2 [ v 2 ∆( u ) − u xx ( u 2 y + u 2 z )] − εu. (4.1) The transformation of the Hamiltonian operat or J 0 in (3.7) yields J 0 =    0 1 a − 1 a 1 a 2 ( cD y − bD z ) + ( D y c − D z b ) 1 a 2    (4.2) where w e in tro duce the notation a = ∆( u ) , b = u xy − v z , c = v y + u xz , Q = b 2 + c 2 + ε a (4.3) that w e will use from now on thr o ughout the pap er, with D y , D z designating op erators of tota l deriv ative s with resp ect to y , z resp ectiv ely and ∆ = D 2 y + D 2 z is the tw o-dimensional Laplace op erator. The symplectic op erator (3.4) in the real v ariables b ecomes K = J − 1 0 = cD y − bD z + D y c − D z b − a a 0 ! (4.4) in an explicitly sk ew-symmetric f o rm. C M A system (2.4) in the real v ariables b ecomes u t v t ! = J 0 δ u H 1 δ v H 1 ! = v Q − u xx ! (4.5) or u t = v , v t = Q − u xx . 6 The f o ur-dimensional h yp er-K¨ ahler metrics (2.1) in the real v a r iables in the nota tion ( 4 .3) b ecome d s 2 = 1 4 h Q (d t 2 + d x 2 ) + a (d y 2 + d z 2 ) i + 1 2 h c (d t d y + d x d z ) − b (d t d z − d x d y ) i . (4.6) The metrics (4.6) satisfy Einstein field equations with either Euclidean or ultra-h yp erb olic signature, if the t w o- component p oten tial ( u, v ) in the def- initions (4.3) of a, b, c , and Q satisfies the Hamilto nian C M A sys tem ( 4 .5) with ε = +1 or ε = − 1 resp ectiv ely . These metrics a r e aga in Ricci-flat and ha v e (a n ti-)self-dual curv ature. 5 Symmetries and integrals of motio n No w, consider Lie g roup of transformations o f the system (4.5) in the ev olu- tionary form, when only dep enden t v a riables are transformed, and let τ b e the gro up para meter. Then Lie equations read u τ = ϕ, v τ = ψ (5.1) where Φ = ϕ ψ ! is a tw o-comp onen t symmetry c haracteristic of the system (4.5). The differen tial compatibility conditions of equations (4.5) and (5.1) in the fo rm u tτ − u τ t = 0 and v tτ − v τ t = 0 result in the linear matrix equation A (Φ) = 0 (5.2) where A is the F rech ´ et deriv ativ e of the flow (4.5) A = D t − 1 D 2 x − 2 a ( cD z + bD y ) D x + Q a ∆ , D t − 2 a ( cD y − bD z ) ! (5.3) where the first row of (5.2) yields ϕ t = ψ . Using the softw a re pac k a ges LIEPDE and CRACK b y T. W o lf [7], run under REDUCE 3.8, we ha v e calculated all p oin t symmetries o f C M A system (4.5), a class o f solutions of t he matrix equation (5.2). W e list their generators and tw o -compo nent symmetry characteristics [6], the latter denoted b y ϕ u , ϕ v X 1 = t∂ t + x∂ x + u∂ u , ϕ u 1 = u − tv − xu x , ϕ v 1 = t ( u xx − Q ) − xv x 7 X 2 = z ∂ y − y ∂ z , ϕ u 2 = y u z − z u y , ϕ v 2 = y v z − z v y X 3 = ∂ z , ϕ u 3 = u z , ϕ v 3 = v z X 4 = ∂ y , ϕ u 4 = u y , ϕ v 4 = v y (5.4) X 5 = y ∂ y + z ∂ z + u∂ u + v ∂ v , ϕ u 5 = u − y u y − z u z , ϕ v 5 = v − y v y − z v z X α = α ( t, x, y , z ) ∂ u + α t ( t, x, y , z ) ∂ v , ϕ u α = α, ϕ v α = α t X β = β z ( y , z ) ∂ x − β y ( y , z ) ∂ t , ϕ u β = β y v − β z u x , ϕ v β = β y ( Q − u xx ) − β z v x where α ( t, x, y , z ) is an arbit r a ry smo oth solution of t he equations ∆( α ) = 0 , α tt + α xx = 0 , α tz − α xy = 0 , α ty + α xz = 0 (5.5) whereas β ( y , z ) satisfies the tw o- dimens iona l Laplace equation ∆( β ) = 0. W e shall find the in tegrals of motion generating the p oin t symmetries that serv e as Hamiltonians of the symmetry flo ws u τ v τ ! = ϕ u ϕ v ! = J 0 δ u H δ v H ! (5.6) where the symmetry group para meter τ play s the role of time for the sym- metry flo w (5.6) and H = R + ∞ −∞ H dxdy dz is an in tegra l of the motio n alo ng the flow (4.5), with the conserv ed densit y H , that generates the symmetry with the tw o- component c haracteristic ϕ u , ϕ v . The second equality in (5.6) is the Hamilto nian fo rm of No ether’s theorem that giv es a relation b et w een symmetries and integrals. W e ch o ose here the Poisson structure determined b y our first Hamilto nian op erator J 0 since we kno w its inv erse K give n by (4.4) whic h is used in the in v erse No ether theorem δ u H δ v H ! = K ϕ u ϕ v ! (5.7) determining conserv ed densities H corresp onding to kno wn symmetry c har- acteristics ϕ u , ϕ v . Using (5.7), w e reconstruct the Hamiltonia ns o f the flow s (5.6 ) fo r all v ar ia tional p oint symmetries in (5.4). F or the scaling symmetries g enerated b y X 1 and X 5 , Hamiltonians do no t exist and so they are not v ariationa l symmetries . F or the rotational symmetry g enerated b y X 2 , the Hamiltonian is H 2 = v ( y u z − z u y )∆( u ) − u x [2( z u y + y u z ) u y z + u 2 y + u 2 z ] . (5.8) 8 F or the translational symmetries generated by X 3 and X 4 , the corresp onding Hamiltonians H 3 and H 4 are H 3 = v u z ∆( u ) + 2 3 u x ( u y u z z − u z u y z ) H 4 = v u y ∆( u ) + 2 3 u x ( u y u y z − u z u y y ) . (5 .9) F or the infinite Lie pseudogroups generated b y X α and X β , the Hamilto nia ns are H α = α v ∆( u ) + 1 2 α t ( u 2 y + u 2 z ) + α ( u z u xy − u y u xz ) (5.10) and H β = β y 2 v 2 − β z u x v ! ∆( u ) − β y 2 u xx ( u 2 y + u 2 z ) + 1 2 u 2 x ( β y y u y + β y z u z ) − εβ y u. (5.11) In particular, the Hamiltonian of time translations X β = y = − ∂ t , that is H β = y = H 1 , coincides with the Hamiltonian (4 .1) of C M A flow. F or trans- lations in x , X β = z = ∂ x , the Hamiltonia n is H β = z = − u x v ∆( u ). F or a simple example of the symmetry X α , X α = z = z ∂ u , the Hamiltonian is H α = z = z v ∆( u ) + u x u y , whic h coincides with the Hamilto nian H 0 (6.7) from an in- finite c hain of Hamiltonians for a hierarc h y o f higher comm uting flo ws in section 6. All of these Hamiltonians of the symmetry flow s are conserv ed densities of the C M A flow (4.5 ). 6 Recursion o p erators and bi-Hamiltoni an represe n tations of CMA system Complex recursion o p erators for symmetries of the hea v enly equations of Pleba ˜ nski w ere in tro duced in the pap ers of Duna j ski and Mason [8, 9 ]. W e ha v e used them in our metho d o f partner symmetries f or obtaining non- in v arian t solutions of complex Monge-Amp ` ere equation [1 0 , 11] and second hea v enly equation of Pleba ˜ nski [11 ] and, in a t w o- component form, for gen- erating multi-Hamiltonian structure of Pleba ˜ nski’s second hea ve nly equation in [1]. How eve r, for C M A w e hav e an additional condition that the equation and its symmetries are real and hence recursion op erators should a lso b e real. 9 This condition leads to a couple of real recursion op erators in a 2 × 2 matrix form. The first one is R 1 = 0 0 QD z − cD x b ! (6.1) + ∆ − 1    D y  − aD x + bD y + cD z  + D z  cD y − bD z  − D z a D x h D y  cD y − bD z  + D z  aD x − bD y − cD z i − D x D y a    where ∆ − 1 means op erator multiplication, and the second recursion op erator reads R 2 = 0 0 bD x − QD y c ! (6.2) + ∆ − 1 D y ( bD z − cD y ) + D z ( − aD x + bD y + cD z ) D y a D x h D y ( − aD x + bD y + cD z ) + D z ( cD y − bD z ) i − D x D z a ! Straigh tfo r ward, though cum b ersome, calculations sho w tha t the op era- tors R 1 and R 2 comm ute with the op erator A (5.3) of the symmetry condition (5.2) on solutions of equations ( 4.5) and therefore they a r e indeed recursion op erators fo r symme tries of the C M A system. This means that if ( ˜ ϕ, ˜ ψ ) is obtained by transforming a tw o - component symmetry c haracteristic ( ϕ, ψ ) of the system ( 4 .5) b y the op erator R 1 or R 2 ˜ ϕ ˜ ψ ! = R i ϕ ψ ! (6.3) where i = 1 , 2, then ( ˜ ϕ, ˜ ψ ) is also a symmetry characteristic of (4.5). Moreo v er, v anishing of the comm utators [ R i , A ], computed without using the equations of motion (4.5), repro duces the C M A system (4.5) and hence the op erators R i and A form tw o real L a x pairs f or the t wo-component sys- tem. Indeed, in tro ducing a short-ha nd notatio n F = u t − v , G = v t + u xx − Q , Φ = F xy − G z and χ = F xz + G y , w e rewrite the C M A system in the form F = 0 , G = 0, so that Φ = 0 and χ = 0 on its solutions, and the first comm utator reads [ R 1 , A ] = 0 0 1 a [ Q ∆( F ) − 2( b Φ + cχ )] D z − Φ ! + ∆ − 1 × (6.4) 10         Φ( D 2 z − D 2 y ) − 2 χD y D z + D y ∆( F ) D x − ∆( F x ) D y − ∆( G ) D z , D z ∆( F ) 2( D z χD x + D x Φ D y ) D z − D z ∆( F ) D 2 x + χ x ( D 2 z − D 2 y ) + [2 χ z − ∆( G )] D x D y + ∆( G y ) D x − ∆( G x ) D y D x D y ∆( F )         and w e ha v e a similar express ion for [ R 2 , A ]. Th us, [ R i , A ] = 0 implies F = 0 , G = 0, that is, the C M A system (4.5). These real Lax pairs are formed by the recursion op erators for symmetries and o perato r A of the symmetry condition and so they are Lax pair s of the Olv er-Ibragimov-Shabat ty p e [14 , 15], whic h is differen t fro m the complex Lax pairs suggested by Mason and Newman [12, 13] and Duna jski and Mason [8, 9] and those that w e used in [1 0 , 11] in relation to partner symmetries, ev en if w e set o ur new Lax pairs in one-comp onen t forms. F urthermore, the commutator of the complex recursion op erator of Mason-Duna jski with the op erator of the symmetry condition, in a one- comp onent form, repro duces the symmetry condition a nd not the original equation C M A [10]. By the theorem of Magri, give n a Hamiltonian op erator J and a r ecursion op erator R , RJ is also a Hamiltonian op erator [3]. There by , acting b y the recursion o perato r R 1 on the first Hamiltonian op erator J 0 (3.7), we obtain the second Hamiltonian op erator J 1 = R 1 J 0 = ∆ − 1 D z − D x D y D x D y D 2 x D z ! + (6.5)    0 b a − b a c a 2  bD y − aD x  +  D y b − D x a  c a 2 + Q − 2 a D z + D z Q − 2 a    that is explicitly ske w-symmetric. Here Q − = ( c 2 − b 2 + ε ) /a . The pro of of the Jacobi iden tit y for J 1 is lengthy but can b e somewhat facilitated b y using Olver’s criterion in t erms o f f unctional m ulti-vec tor s [6]. Similarly , acting b y t he recursion o perato r R 2 on the Hamiltonian op- erator J 0 , w e obtain a Hamiltonian op erator that is ano t her companion for J 0 : J 1 = R 2 J 0 = ∆ − 1 D y D x D z − D x D z D 2 x D y ! + (6.6) 11    0 − c a c a b a 2  cD z − aD x  +  D z c − D x a  b a 2 + Q − 2 a D y + D y Q − 2 a    that is a lso explicitly sk ew-symmetric a nd Q − = ( b 2 − c 2 + ε ) /a . The Jacobi iden tit y f o r J 1 w as also pro ve d by using Olv er’s criterion in terms o f functional m ulti-v ectors [6]. The flow (4.5) can b e generated by the Hamiltonian op erator J 1 from t he Hamiltonian densit y H 0 = z v ∆( u ) + u x u y (6.7) so tha t CMA in the t w o- component for m (4 .5) admits tw o Hamiltonian rep- resen tations u t v t ! = J 0 δ u H 1 δ v H 1 ! = J 1 δ u H 0 δ v H 0 ! (6.8) and thus this is a bi-Hamiltoni a n system . The same flo w (4 .5) can also b e generated b y the Hamiltonian opera t or J 1 from the Hamiltonian densit y H 0 = y v ∆( u ) − u x u z (6.9) whic h yields another bi- Ha miltonian represen tation of the C M A system (4.5) u t v t ! = J 0 δ u H 1 δ v H 1 ! = J 1 δ u H 0 δ v H 0 ! . (6.10) Rep eating this pro cedure n times, w e obta in a multi-Hamiltonian r epr e- sentation of the C M A system with the Hamilto nia n o perato r s J n = R n 1 J 0 , J n = R n 2 J 0 , J n − m m = R m 1 R n − m 2 J 0 ( m = 1 , 2 , . . . n − 1) and corresp onding Hamiltonian densities . This pro cedure will b e considered in more detail in the next sec tio n for the op erator R 1 . Multi-Hamiltonian structure of the C M A system prov es its complete integrabilit y in the sense of Magri and hence the complete integrabilit y of the (an ti-)self-dual gravit y in four real dimensions with either Euclidean or ultra-hy p erb olic signature. A tot a lly differen t recursion op era t o r f o r the (an ti-)self-dual gra vit y in complex Einstein spaces w as o btained m uc h earlier b y Stra chan [16] by using a Legendre transformed v ersion of the first heav enly equation, that w as de- riv ed b y Gra n t [17]. This recursion op erator can b e fa ctorized whic h suggests a bi-Hamiltonian structure o f the resulting evolutionary equation, though 12 that w as not completely pr ov ed. Ho w ev er, the ev olutionary equation and the related Hamiltonian structures are expressed in complex v ariables, with the complex ”time” t in particular, and with a complex unkno wn. Therefore, the corresp onding metric will not corresp ond to anti-se lf- dual gra vity in real Einstein spaces with the Euclidean signature (+ + ++ ). F urthermore, the P oisson bra ck et con tains the un usual op erator ∂ − 1 t that could b e av oided in a tw o -compo nen t f orm ulation. 7 Infinite hi erarc h y o f higher flows The o p erators J 0 and J 1 are compatible Hamiltonian op erators, i.e. they form a P oisson p encil. This means t ha t ev ery linear com bination C 0 J 0 + C 1 J 1 with constant co efficien ts C 0 and C 1 satisfies the Jacobi iden tit y . This can b e more easily v erified by using the Olv er’s criterion in terms o f functional m ulti-v ectors though the calculation is still very lengthy . W e kno w from the work of F uc hssteiner and F o k as [18] (see also the surv ey [19] and ref- erences therein) that if a recursion op erator has a factorized form, as in our case R 1 = J 1 J − 1 0 ≡ J 1 K , and the factors J 0 and J 1 are compatible Hamiltonian op erators, then R 1 is hereditary (Nijenh uis) r ecursion op erator, i.e. it generates a n Ab elian symmetry algebra out of comm uting symmetry generators. Moreo ve r, Hermitian conjugate hereditary r ecursion op erator R † 1 = J − 1 0 J 1 = K J 1 , acting on the v ector of v aria t io nal deriv ativ es of an in tegral of the flow, yields a v ector of v a riational deriv ativ es of some other in tegral of this flow. Then (6 .8) implies that R † 1 generates t he Hamiltonian densit y H 1 from H 0 : R † 1 δ u H 0 δ v H 0 ! = J − 1 0 J 1 δ u H 0 δ v H 0 ! = δ u H 1 δ v H 1 ! (7.1) where R † 1 is defined by R † 1 = 0 D x c − D z Q 0 b ! (7.2) +       − D x a + D y b + D z c  D y +  D y c − D z b  D z h D z b − D y c  D y +  − D x a + D y b + D z c  D z i D x aD z aD x D y      ∆ − 1 . 13 The first higher flow of the hierar ch y is generated b y J 1 acting on the v ector of v a riational deriv ativ es of H 1 u t 1 v t 1 ! = J 1 δ u H 1 δ v H 1 ! (7.3) where t 1 is t he time v a riable of the hig her flow . This flow is nonlo cal and the righ t-hand side of (7 .3) is to o length y to b e presen ted here explicitly . No w w e could generate the next Hamiltonian H 2 of t he hierarch y of com- m uting flo ws by a pplying R † 1 to the v ector o f v ariational deriv ativ es of H 1 : R † 1 δ u H 1 δ v H 1 ! = K J 1 δ u H 1 δ v H 1 ! = δ u H 2 δ v H 2 ! . (7.4) Therefore, the second higher flow in the hierarc hy has a bi-Hamiltonian rep- resen tation u t 2 v t 2 ! = J 1 δ u H 2 δ v H 2 ! = J 1 R † 1 δ u H 1 δ v H 1 ! = J 2 δ u H 1 δ v H 1 ! (7.5) where the third Hamiltonia n op erator J 2 is generated by acting with R 1 on J 1 : J 1 R † 1 = J 1 K J 1 = R 1 J 1 = J 2 . Acting by J 2 on t he v a r iational deriv ativ es of H 0 , we o btain the relations J 2 δ u H 0 δ v H 0 ! = J 1 R † 1 δ u H 0 δ v H 0 ! = J 1 δ u H 1 δ v H 1 ! = J 0 R † 1 δ u H 1 δ v H 1 ! = J 0 δ u H 2 δ v H 2 ! (7.6) where w e ha v e used that J 1 = J 0 ( K J 1 ) = J 0 R † 1 . F rom (7 .6 ) we obtain three-Hamiltonian represen tation of the first higher flo w u t 1 v t 1 ! = J 1 δ u H 1 δ v H 1 ! = J 2 δ u H 0 δ v H 0 ! = J 0 δ u H 2 δ v H 2 ! . (7.7) W e could a lso construct the Hamiltonian H − 1 suc h that H 0 is generated from H − 1 b y R † 1 R † 1 δ u H − 1 δ v H − 1 ! = J − 1 0 J 1 δ u H − 1 δ v H − 1 ! = δ u H 0 δ v H 0 ! (7.8) 14 that implies a bi-Hamiltonia n represen tation for the zeroth flow u t 0 v t 0 ! = J 0 δ u H 0 δ v H 0 ! = J 1 δ u H − 1 δ v H − 1 ! . (7.9) F urther w e obtain J 2 δ u H − 1 δ v H − 1 ! = J 1 R † 1 δ u H − 1 δ v H − 1 ! = J 1 δ u H 0 δ v H 0 ! (7.10) and the bi-Hamiltonian represen tation (6.8 ) of the original t wo-component C M A flow b ecomes a three-Hamilto nia n represen tation of this flow u t v t ! = J 0 δ u H 1 δ v H 1 ! = J 1 δ u H 0 δ v H 0 ! = J 2 δ u H − 1 δ v H − 1 ! . (7.11) W e could still con tin ue b y applying R † 1 to the v ector of v aria tional deriv a- tiv es of H 2 to generate the next Hamiltonian H 3 R † 1 δ u H 2 δ v H 2 ! = K J 1 δ u H 2 δ v H 2 ! = δ u H 3 δ v H 3 ! (7.12) and obtain a bi-Hamiltonian represen tation f or the next higher flow u t 3 v t 3 ! = J 1 δ u H 3 δ v H 3 ! = R 1 J 1 δ u H 2 δ v H 2 ! = J 2 δ u H 2 δ v H 2 ! (7.13) where we hav e used (7 .12) and the relation J 1 K J 1 = R 1 J 1 = J 2 , and so on. 8 Conclus ion Our start ing p oint w as the symplectic and Hamiltonian structure of the com- plex Monge- Amp ` ere equation, set into a t w o- component ev olutiona ry form. W e hav e calculated all p oint symmetries of the C M A syste m and also, us- ing in v erse No ether theorem, Hamiltonians of the flows fo r all v a riational symmetries . These Hamiltonians yield conserv a tion laws for the C M A flo w. W e ha v e found tw o real 2 × 2 matrix recursion op erators R 1 and R 2 for symmetries that comm ute with the op erator A o f the symmetry condition and hence map any symmetry o f the C M A system aga in into a symme- try . The op erators R 1 and R 2 together with A form t w o Lax pairs for the 15 t w o- comp onent C M A system. Acting on the first Hamiltonian op erator b y eac h recursion op erator, w e obtain tw o new Hamiltonian op erators a ccording to Magri’s theorem [3] and tw o bi-Hamiltonian represen tations of the com- plex Monge-Amp ` ere equation in the t wo-component form. Repeating this action, w e could generate an infinite n um b er of Hamiltonian opera t o rs and hence construct a m ulti-Hamiltonian represen tation o f the C M A system. W e sho w how to construct an infinite hierarc hy of higher comm uting flows t o - gether with the corresp onding infinite c hain of their Hamiltonians by using a Hermitian conjuga te recursion op erator. In particular, w e arr iv e at three- Hamiltonian represen tations for b oth C M A flo w a nd the first higher flo w a nd bi-Hamiltonian represen tatio ns for the zeroth flo w and second higher flow . The results of this pap er prov e complete in tegrability of the (anti-)self-dual gra vity in f our real dimensions in the sense of Magri (a m ulti-Hamilto nian represen tation). Ac kno wled g emen ts One of the authors (MBS) thanks A. A. Malykh fo r fruitful discussions. The researc h of MBS is partly supp orted b y the r ese arch gran t f r om Boga zici Univ ersit y Scientific Researc h F und, research pro ject No. 07B3 01. One o f us (YN) is grat eful to Prof . Dr. T emel Yılmaz for k eeping him aliv e. References [1] Neyzi F , Nutku Y and Sheftel M B 20 0 5 J. Phys. A: Math. Gen. 38 8473–848 5 ( Prp erint nlin.SI/05050 30) [2] Plebanski J F 1975 J. Math. Phys. 16 2395– 402 [3] Magri F 1978 J. Math. Phys. 19 1156–11 62 Magri F 1980 Nonli n e ar Evolution Equations and Dynamic al Systems (L e ctur e Notes in Ph ysic s vol 1 20 ) ed. M Boiti, F P empinelli and G Soliani (New Y ork: Springer) p 233 [4] Nutku Y 2000 Phys. L ett. 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