Integrable String Models in Terms of Chiral Invariants of SU(n), SO(n), SP(n) Groups

Symmetry , Integrabilit y and Geometry: Metho ds and Applications SIGMA 4 (2008), 041, 16 pages In tegrable String Mo dels in T erms of Chiral In v arian ts of S U ( n ), S O ( n ), S P ( n ) Group s ⋆ Victor D. GERSHUN ITP, NSC Kharkiv Institute of Physics and T e chno lo gy, Kharkiv, Ukr aine E-mail: gershun@kipt.kharkov.ua Received Octob er 30, 2007, in f inal form April 2 2, 200 8; Published online May 06, 2008 Original article is av ailable at http: //www .emis .de/journals/SIGMA/2008/041/ Abstract. W e considered tw o types of string mo dels: on the Riemmann spa ce of str ing co ordinates with n ull tor sion and on the Riemma n–Cartan s pace of string co o rdinates with constant tors ion. W e used the hydrodyna mic a pproach of Dubrovin, Novik ov to integrable systems and Dubrovin solutions of WDVV asso ciativity equa tion to constr uct new integrable string equations o f h ydrodyna mic t yp e on the torsionless Riemmann space of chiral currents in f irst case. W e used the inv ariant lo cal ch iral currents of pr incipal chiral models for S U ( n ), S O ( n ), S P ( n ) groups to constr uc t new integrable string equations of hydro dynamic t yp e on the Riemmann space of the chiral primitive inv ar iant cur rents and on the chiral non- primitive Ca s imir op erato rs as Hamiltonia ns in seco nd case . W e also use d Pohlmey er tensor nonlo cal cur rents to co ns truct new nonlo cal string equation. Key wor ds: s tr ing; integrable mo dels ; Poisson brackets; Ca simir o p erators ; chiral currents 2000 Mathematics Subje ct Classific atio n: 8 1 T20; 81T3 0; 81T4 0; 37J 35; 53Z0 5; 22E 70 1 In tro duction String theory is a v ery p romising candidate for a unif ied qu an tum th eory of gra vit y and all the other forces of nature. F or quantum description of string mod el w e m ust ha ve classica l solutions of th e strin g in the bac kground f ields. String theory in suitable space-time bac kgrounds can b e considered as principal chiral mo del. The in tegrabilit y of th e classical principal c hiral mo del is m an if ested th r ough an inf inite set of conserved c harges, which can form non-Ab elian algebra. An y charge from the comm uting subset of c harges and an y Casimir op erators of c harge algebra can b e considered as Hamiltonian in bi-Hamiltonian approac h to in tegrable m o dels. The bi-Hamiltonian app roac h to inte grable systems was initiated by Magri [1]. Tw o P oisson brac k ets (PBs) { φ a ( x ) , φ b ( y ) } 0 = P ab 0 ( x, y )( φ ) , { φ a ( x ) , φ b ( y ) } 1 = P ab 1 ( x, y )( φ ) are called compatible if any lin ear com bination of th ese PBs {∗ , ∗} 0 + λ {∗ , ∗} 1 is PB also for arbitrary constant λ . Th e functions φ a ( t, x ), a = 1 , 2 , . . . , n are lo cal co ord inates on a certain giv en smo oth n -dimensional manifold M n . Th e Hamiltonian op erators P ab 0 ( x, y )( φ ), P ab 1 ( x, y )( φ ) are f unctions of lo cal co ordin ates φ a ( x ). It is p ossible to f ind such Hamiltonians H 0 and H 1 whic h satisfy bi-Hamiltonian condition [2] dφ a ( x ) dt = { φ a ( x ) , H 0 } 0 = { φ a ( x ) , H 1 } 1 , ⋆ This p ap er is a contribution to the Proceedings of the Seventh In ternational Conference “Symmetry in Nonlinear Mathematical Physics” (June 24–30 , 2007, K yiv, Ukraine). The full collection is av ailable at http://w ww.emis.de/j ournals/SIGMA/symmetry2007.h tml 2 V.D. Gershun where H M = R 2 π 0 h M ( φ ( y )) dy , M = 0 , 1. Tw o branc hes of hierarchies arise u nder t w o equations of motion under t w o d if feren t parameters of ev olution t 0 M and t M 0 [2] dφ a ( x ) dt 01 = { φ a ( x ) , H 0 } 1 = Z 2 π 0 P ab 1 ( x, y ) ∂ h 0 ∂ φ b ( y ) dy = Z 2 π 0 R a c ( x, z ) P cb 0 ( z , y ) ∂ h 0 ∂ φ b ( y ) dy , dφ a ( x ) dt 10 = { φ a ( x ) , H 1 } 0 = Z 2 π 0 P ab 0 ( x, y ) ∂ h 1 ∂ φ b ( y ) dy = Z 2 π 0 ( R − 1 ) a c ( x, z ) P cb 0 ( z , y ) ∂ h 0 ∂ φ b ( y ) dy . There R a b ( x, y ) is a recursion op erator and ( R − 1 ) a b ( x, y ) is its in v erse R a c ( x, y ) = Z 2 π 0 P ab 1 ( x, z )( P 0 ) − 1 bc ( z , y ) dz . The first branch of the h ierarc hies of dynamical systems has the follo wing form dφ a ( x ) dt 0 N = Z 2 π 0 ( R ( x, y 1 ) · · · R ( y N − 1 )) a c P cb 0 ( y N − 1 , y N ) ∂ h 0 ∂ φ b ( y N ) dy 1 · · · dy N , N = 1 , 2 , . . . , ∞ . The second b ranc h of the h ierarc hies can b e obtained by replacemen t R → R − 1 and t 0 N → t N 0 . W e w ill consider only the f irs t br anc h of the hierarchies. The lo cal PBs of hydro dynamic type w ere intro d uced by Dubr o vin, Novi k o v [3, 4] for Hamil- tonian description of equations of hydro dynamics. T hey we re generalized by F erap onto v [5] and Mokho v, F erap on to v [6] to the n on-lo cal PBs of h ydro d ynamic t yp e. The hydro dynamic t yp e systems were consid er ed by Tsarev [8], Maltsev [9], F erap on to v [10], Mokho v [12] (see also [7 ]), P a vlo v [13] (see also [14 ]), Maltsev, No vik o v [15]. The p olynomials of lo cal chiral currents w ere considered by Goldshmidt and Witten [16] (see also [17]). The lo cal conserved c hiral charges in principal c hiral mo d els were considered b y Ev ans, Hassan, MacKa y , Moun tain [24]. The tensor nonlo cal c hiral c harges were introd u ced by Po hlmey er [26] (see also [27, 28]). The string mo d- els of hydrod ynamic t yp e were considered b y author [18, 19]. In S ection 3, the author applied h ydro dy n amic approac h to in tegrable systems to obtain new in tegrable string equations. In S ec- tion 4, the auth or used the nonlo cal P ohlmey er c harges to obtain a n ew string equation in terms of the nonlo cal currents. In Section 5, the author ap p lied the lo cal inv arian t c hiral cur rent s to a simple Lie algebra to construct new in tegrable string equations. 2 String mo del of principal chira l mo del t yp e A string mo del is describ ed by the Lagrangian L = 1 2 Z 2 π 0 η αβ g ab ( φ ( t, x )) ∂ φ a ( t, x ) ∂ x α ∂ φ b ( t, x ) ∂ x β dx (1) and b y tw o f irst kind constraints g ab ( φ ( x ))  ∂ φ a ( x ) ∂ t ∂ φ b ( x ) ∂ t + ∂ φ a ( x ) ∂ x ∂ φ b ( x ) ∂ x  ≈ 0 , g ab ( φ ( x )) ∂ φ a ( x ) ∂ t ∂ φ b ( x ) ∂ x ≈ 0 . The target space lo cal co ordinates φ a ( x ), a = 1 , . . . , n b elong to certain giv en smo oth n -dimen- sional manifold M n with nondegenerate metric tensor g ab ( φ ( x )) = η µν e µ a ( φ ( x )) e ν b ( φ ( x )) , where µ, ν = 1 , . . . , n are indices of tangent space to manifold M n on some p oin t φ a ( x ). The v eilb ein e µ a ( φ ) and its in v erse e a µ ( φ ) satisfy the cond itions e µ a e b µ = δ b a , e µ a e aν = η µν . In tegrable String Mo dels in T erms of C hiral I n v ariants of S U ( n ), S O ( n ), S P ( n ) Groups 3 The co ordinates x α ( x 0 = t , x 1 = x ) b elong to w orld sheet with metric tensor g αβ in conformal gauge. The str in g equations of motion h a v e the form η αβ [ ∂ αβ φ a + Γ a bc ( φ ) ∂ α φ b ∂ β φ c ] = 0 , ∂ α = ∂ ∂ x α , α = 0 , 1 , where Γ a bc ( φ ) = 1 2 e a µ  ∂ e µ b ∂ φ c + ∂ e µ c ∂ φ b  is the connection. In terms of canonical cur ren ts J µ α ( φ ) = e µ a ( φ ) ∂ α φ a the equations of motion ha v e the form η αβ ∂ α J µ β ( φ ( t, x )) = 0 , ∂ α J µ β ( φ ) − ∂ β J µ α ( φ ) − C µ ν λ ( φ ) J ν α ( φ ) J λ β ( φ ) = 0 , where C µ ν λ ( φ ) = e a ν e b λ  ∂ e µ a ∂ φ b − ∂ e µ b ∂ φ a  is the torsion. The Hamiltonian has th e form H = 1 2 Z 2 π 0 [ η µν J 0 µ J 0 ν + η µν J µ 1 J ν 1 ] dx, where J 0 µ ( φ ) = e a µ ( φ ) p a , J µ 1 ( φ ) = e µ a ∂ ∂ x φ a and p a ( t, x ) = η µν e µ a e ν b ∂ ∂ t φ b is the canonical m omen- tum. The canonical commutati on r elations of currents are as follo ws { J 0 µ ( φ ( x )) , J 0 ν ( φ ( y )) } = C λ µν ( φ ( x )) J 0 λ ( φ ( x )) δ ( x − y ) , { J 0 µ ( φ ( x )) , J ν 1 ( φ ( y )) } = C ν µλ ( φ ( x )) J λ 1 ( φ ( x )) δ ( x − y ) − 1 2 δ ν µ ∂ ∂ x δ ( x − y ) , { J µ 1 ( φ ( x )) , J ν 1 ( φ ( y )) } = 0 . Let us in tro duce c hiral currents U µ = η µν J 0 ν + J µ 1 , V µ = η µν J 0 ν − J µ 1 The comm utation relations of c hiral currents are the follo wing { U µ ( φ ( x )) , U ν ( φ ( y )) } = C µν λ ( φ ( x ))  3 2 U λ ( φ ( x )) − 1 2 V λ ( φ ( x ))  δ ( x − y ) − η µν ∂ ∂ x δ ( x − y ) , { U µ ( φ ( x )) , V ν ( φ ( y )) } = C µν λ ( φ ( x ))[ U λ ( φ ( x )) + V λ ( φ ( x ))] δ ( x − y ) , { V µ ( φ ( x )) , V ν ( φ ( y )) } = C µν λ ( φ ( x ))  3 2 V λ ( φ ( x )) − 1 2 U λ ( φ ( x ))  δ ( x − y ) + η µν ∂ ∂ x δ ( x − y ) . Equations of motion in ligh t-co ne co ordinates x ± = 1 2 ( t ± x ) , ∂ ∂ x ± = ∂ ∂ t ± ∂ ∂ x ha v e the f orm ∂ − U µ = C µ ν λ ( φ ( x )) U ν V λ , ∂ − V µ = C µ ν λ ( φ ( x )) V ν U λ . In the case of the n ull torsion C µ ν λ = 0 , e µ a ( φ ) = ∂ e µ ∂ φ a , Γ a bc ( φ ) = e a µ ∂ 2 e µ ∂ φ b ∂ φ c , R µ ν λρ ( φ ) = 0 the string mo del is integ rable.The Hamiltonian equations of motion un der Hamiltonian (1) are describ ed by t w o indep end en t left and right mo v ers: U µ ( t + x ) an d V µ ( t − x ). 4 V.D. Gershun 3 In tegra ble string mo d els of h ydro dyn amic t yp e with n ull torsion W e w an t to construct new in tegrable strin g mo dels with Hamilto nians as p olynomials of the initial chiral currents U µ ( φ ( x )). Th e PB of c hiral currents U µ ( x ) coincides w ith the f lat PB of Dubro vin, No vik o v { U µ ( x ) , U ν ( y ) } 0 = − η µν ∂ ∂ x δ ( x − y ) . Let us in tro duce a lo cal Dubrovin, No vik o v PB [3, 4]. It h as th e form { U µ ( x ) , U ν ( y ) } 1 = g µν ( U ( x )) ∂ ∂ x δ ( x − y ) − Γ µν λ ( U ( x )) ∂ U λ ( x ) ∂ x δ ( x − y ) . This PB is ske w-symmetric if g µν ( U ) = g ν µ ( U ) and it satisf ies Jacobi identit y if Γ a bc ( U ) = Γ a cb ( U ), C a bc ( U ) = 0, R a bcd ( U ) = 0 . In the case of n on-zero cu r v ature tensor w e m ust include W eingarten op erators into right side of the P B with the step-function sgn ( x − y )=( d dx ) − 1 δ ( x − y )= ν ( x − y ) [5, 6]. The PBs {∗ , ∗} 0 and {∗ , ∗} 1 are compatible b y Magri [1] if the p encil {∗ , ∗} 0 + λ {∗ , ∗} 1 is also PB . As a r esult, Mokhov [12, 11] obtained the compatible pair of PBs P 0 µν ( U )( x, y ) = − η µν ∂ ∂ x δ ( x − y ) , P 1 µν ( U )( x, y ) = 2 ∂ 2 F ( U ) ∂ U µ ∂ U ν ∂ ∂ x δ ( x − y ) + ∂ 3 F ( U ) ∂ U µ ∂ U ν ∂ U λ ∂ U λ ∂ x δ ( x − y ) . The function F ( U ) satisf ies the equation ∂ 3 F ( U ) ∂ U µ ∂ U ρ ∂ U λ η λρ ∂ 3 F ( U ) ∂ U ν ∂ U σ ∂ U ρ = ∂ 3 F ( U ) ∂ U ν ∂ U ρ ∂ U λ η λρ ∂ 3 F ( U ) ∂ U µ ∂ U σ ∂ U ρ . This equation is WD VV [20, 21] asso ciati vit y equation and it was obtained in 2D top ologi cal f ield theory . Dubro vin [22, 23 ] obtained a lot of solutions of WD VV equation. He sho w ed that lo cal f ields U µ ( x ) m ust b elong F rob enius manifolds to solve the WD VV equation and ga v e examples of F r ob enius structures. Associativ e F rob enius algebra ma y b e wr itten in the follo w in g form ∂ ∂ U µ ∗ ∂ ∂ U ν := d λ µν ( U ) ∂ ∂ U λ . T otally sym metric structure function has the form d µν λ ( U ) = ∂ F ( U ) ∂ U µ ∂ U ν ∂ U λ , µ, ν, λ = 1 , . . . , n and asso ciativit y cond ition  ∂ ∂ U µ ∗ ∂ ∂ U ν  ∗ ∂ ∂ U λ = ∂ ∂ U µ ∗  ∂ ∂ U ν ∗ ∂ ∂ U λ  leads to the WD VV equation. F unction F ( U ) is qu asih omogeneous fu nction of its v ariables  d µ U µ ∂ ∂ U µ  F ( U ) = d F F ( U ) + A µν U µ U ν + B µ U µ + C, here num b ers d µ , d F , A µν , B µ , C d ep end on the t yp e of p olynomial f unction F ( U ). Here are some Dubrovi n examples of solutions of the WD VV equation n = 1 , F ( U ) = U 3 1 ; In tegrable String Mo dels in T erms of C hiral I n v ariants of S U ( n ), S O ( n ), S P ( n ) Groups 5 n = 2 , F ( U ) = 1 2 U 2 1 U 2 + e U 2 , d 1 = 1 , d 2 = 2 , d F = 2 , A 11 = 1 (2) and quasihomogeneit y condition for n = 2 has th e form  d 1 U 1 ∂ ∂ U 1 + d 2 ∂ ∂ U 2  F ( U ) = d F F ( U ) + A 11 U 2 1 . W e used lo cal f ields U µ with low indices h ere for con v enience. One of the Dubrovin p olynomial solutions is F ( U ) = 1 2 ( U 2 1 U 3 + U 1 U 2 2 ) + 1 4 U 2 2 U 2 3 + 1 60 U 5 3 , (3) here d 1 = 1, d 2 = 3 2 , d 3 = 2, d F = 4 and the p olynomial fu n ction f ( U 2 , U 3 ) = 1 4 U 2 2 U 2 3 + 1 60 U 5 3 is a solution of the add itional PDE. In the bi-Hamiltonian app roac h to an int egrable strin g mo del w e must constru ct the recursion op erator to generate a hierarch y of PBs and a h ierarch y of Hamiltonians R µ ν ( x, y ) = Z 2 π 0 P µλ 1 ( x, z )( P − 1 0 ( z , y )) λν dz = 2 ∂ 2 F ( U ( x )) ∂ U µ ( x ) ∂ U ν ( x ) δ ( x − y ) + ∂ 3 F ( U ( x )) ∂ U µ ( x ) ∂ U ν ( x ) ∂ U λ ( x ) ∂ U λ ( y ) ∂ y ν ( x − y ) . The Hamiltonian equation of motion with Hamiltonian H 0 is the follo w ing H 0 = Z 2 π 0 η µν U µ ( x ) U ν ( x ) dx, ∂ U µ ∂ t = ∂ U µ ∂ x . First of th e new equations of motion under the new time t 1 has the form [12] ∂ U µ ∂ t 1 = Z 2 π 0 R µ ν ( x, y ) ∂ U µ ( y ) ∂ y dy = η µν d dx  ∂ F ( x ) ∂ U ν  . (4) This equation of m otion can b e obtained as Hamiltonian equation with new Hamiltonian H 1 H 1 = Z 2 π 0 ∂ F ( U ( x )) ∂ U µ U µ ( x ) dx, where F ( U ) is eac h of Dub ro vin solutions WD VV asso ciativit y equation (2), (3). An y system of the f ollo wing hierarc h y [12] ∂ U µ ∂ t M = Z 2 π 0 ( R ( x, y 1 ) · · · R ( y M − 1 , y M )) µ ν ∂ U ν ∂ y M dy 1 · · · dy M is an in tegrable system. As result we obtain chiral curr en ts U µ ( φ ( t M , x )) = f µ ( φ ( t M , x ), where f µ ( φ ) is a solutio n of the equation of motion. In the case of the Hamiltonian H 1 and of the equation of motion (4) w e can in tro duce new currents J µ 0 ( t 1 , x ) = U µ ( t 1 , x ) , J µ 1 ( t 1 , x ) = η µν ∂ F ( U ( t 1 , x )) ∂ U ν . Consequent ly , we can int ro duce a n ew m etric tensor and a new v eilb ein d ep endin g of the new time co ordinate. The equation for the new metric tensor has the form e µ a ( φ ( t 1 , x )) ∂ φ a ( t 1 , x ) ∂ x = de µ ( φ ( t 1 , x )) dx = η µν ∂ F ( f ( φ ( t 1 , x ))) ∂ f ν ( φ ( t 1 , x )) . 6 V.D. Gershun 4 New string equation in terms of P ohlmey er tensor nonlo cal curren ts In the case of the f lat space C µ ν λ = 0 there exist nonlo cal totally symmetric tensor c hiral cur r en ts called “P ohlmey er” currents [26, 27, 28] R ( M ) ( U ( x )) ≡ R ( µ 1 µ 2 ...µ M ) ( U ( x )) = U ( µ 1 ( x ) Z x 0 U µ 2 ( x 1 ) dx 1 · · · Z x M − 2 0 U µ M ) ( x M − 1 ) dx M − 1 , where round br ac k et s the mean totally symmetric p ro du ct of chiral currents U µ ( U ). The new Hamiltonians ma y ha v e the follo wing forms H ( M ) = 1 2 Z 2 π 0 R ( M ) ( U ( x )) d 2 M R ( M ) ( U ( x )) dx, where d M ≡ d ( µ 1 µ 2 ...µ M ) is totally symmetric inv arian t constan t tensor, which can b e constructed from Kronec k er deltas. F or example R (2) ≡ R µν ( U ( x )) = 1 2 [ U µ ( x ) x Z 0 U ν ( x 1 ) dx 1 + U ν ( x ) Z x 0 U µ ( x 1 ) dx 1 ] , H (2) = 1 2 Z 2 π 0 " U µ ( x ) U µ ( x ) Z x 0 U ν ( x 1 ) dx 1 Z x 0 U ν ( x 2 ) dx 2 + U µ ( x ) U ν ( x ) Z x 0 U µ ( x 1 ) dx 1 Z x 0 U ν ( x 2 ) dx 2 # dx. The Hamiltonian H (2) comm utes with the Hamiltonian H (1) = 1 2 R 2 π 0 U µ ( x ) U µ ( x ) dx and it comm utes w ith the Casimir R 2 π 0 U µ ( x ) dx . The equation of motion und er the Hamiltonian H (2) is as f ollo ws ∂ U µ ( x ) ∂ t = ∂ ∂ x " U µ ( x ) Z x 0 U ν ( x 1 ) dx 1 Z x 0 U ν ( x 2 ) dx 2 + U ν ( x ) Z x 0 U µ ( x 1 ) dx 1 Z x 0 U ν ( x 2 ) dx 2 # − U ν ( x ) U ν ( x ) Z x 0 U µ ( x 1 ) dx 1 − U µ ( x ) U ν ( x ) Z x 0 U ν ( x 1 ) dx 1 . In the v ariables S µ ( x ) = Z x 0 U µ ( y ) dy the latter equation can b e rewritten as follo ws ∂ S µ ∂ t = ∂ ∂ x ( S µ ( S ν S ν )) + Z x 0 S µ  S ν ∂ 2 S ν ∂ 2 y  dy , µ, ν = 1 , 2 , . . . , n . 5 In tegra ble string mo d els with constan t torsion Let us go bac k to the comm utation relations of c hiral currents. Let the torsion C µ ν λ ( φ ( x )) 6 = 0 and C µν λ = f µν λ b e str u cture constan t of s s im p le Lie algebra. W e will consider a string m o del with the constan t torsion in light -cone gauge in target space. This mo d el coincides with the In tegrable String Mo dels in T erms of C hiral I n v ariants of S U ( n ), S O ( n ), S P ( n ) Groups 7 principal chiral mo del on compact simp le Lie group. W e cannot divid e the m otion on r igh t and left mo v er b ecause of c hiral curr en ts ∂ − U µ = f µ ν λ U ν V λ , ∂ − V µ = f µ ν λ V ν U λ are not conserv ed. The corresp ondent c harges are not Casimirs. Th e pr esen t pap er wa s stim ulated b y pap er [24]. Ev ans, Hassan, MacKa y , Mountai n (see [24] and references therein) constr u cted lo cal inv arian t c hiral currents as p olynomials of the initial c hiral cur ren ts of S U ( n ), S O ( n ), S P ( n ) p r incipal c hiral mo d els and they found suc h com b ination of them that the corresp onding c h arges are Casimir op er ators of these dynamical systems. Their pap er was based on the pap er of d e Azcarraga, Macfarlane, MacKa y , P erez Bueno (see [25 ] an d references therein) ab out inv arian t tensors for simple Lie algebras. Let t µ b e n ⊗ n traceless h ermitian matrix representati ons of generators Lie algebra [ t µ , t ν ] = 2 if µν λ t λ , T r( t µ t ν ) = 2 δ µν . Here is an additional relation for S U ( n ) algebra { t µ , t ν } = 4 n δ µν + 2 d µν λ t λ , µ = 1 , . . . , n 2 − 1 . De Azcarraga et al. ga ve some examples of in v arian t tensors of s im p le L ie alge bras and th ey ga v e a ge neral method to calculate them. In v ariant tensors may b e constru cted as in v arian t symmetric p olynomials on S U ( n ) d ( M ) ( µ 1 ...µ M ) = 1 M ! ST r( t µ 1 · · · t µ M ) , where ST r means the completely symmetrized p ro duct of matrices and d ( M ) ( µ 1 ...µ M ) is the totally symmetric tensor and M = 2 , 3 , . . . , ∞ . Another family of inv arian t symmetric tensors [29, 30] (see also [25]) called D -family based on the pro d uct of the symmetric structure constan t d µν λ of the S U ( n ) algebra is as follo ws: D ( M ) ( µ 1 ...µ M ) = d k 1 ( µ 1 µ 2 d k 1 k 2 µ 3 · · · d k M − 2 k M − 3 µ M − 2 d k M − 3 µ M − 1 µ M ) , where D (2) µν = δ µν , D (3) µν λ = d µν λ and M = 4 , 5 , . . . , ∞ . Here are n − 1 primitive in v ariant tensors on S U ( n ). The inv arian t tensors for M ≥ n are functions of primitive tensors. The C asimir op erators on S U ( n ) algebra hav e the form C ( M ) ( t ) = d M ( µ 1 ...µ M ) t µ 1 · · · t µ M . Ev ans et al. in tro duced lo cal chiral currents b ased on the inv arian t symmetric p olynomials on simple L ie groups J ( M ) ( U ) = S T r ( U · · · U ) ≡ ST r U M = d ( M ) µ 1 ...µ M U µ 1 · · · U µ M , (5) where U = t µ U µ and µ = 1 , . . . , n 2 − 1 . It is p ossible to decomp ose the inv arian t s ymmetric c hiral cu rrent s J ( M ) ( U ) in to pro d uct of the basic inv arian t chiral currents D ( M ) ( U ) D (2) ( U ) = d (2) µν U µ U ν = η µν U µ U ν , D (3)( U ) = d µν λ U µ U ν U λ , D ( M ) ( U ( x )) = d k 1 µ 1 µ 2 d k 1 k 2 µ 3 · · · d k M − 2 k M − 3 µ M − 2 d k M − 3 µ M − 1 µ M U µ 1 U µ 2 · · · U µ M , where M = 4 , 5 , . . . , ∞ . The author obtained th e follo wing exp ressions for lo cal in v ariant c hiral currents J ( M ) ( U ) J (2) = 2 D (2) , J (3) = 2 D (3) , J (4) = 2 D (4) + 4 n D (2)2 , 8 V.D. Gershun J (5) = 2 D (5) + 8 n D (2) D (3) , J (6) = 2 D (6) + 4 n D (3)2 + 8 n D (2) D (4) + 8 n 2 D (2)3 , J (7) = 2 D (7) + 8 n D (3) D (4) + 8 n D (2) D (5) + 24 n 2 D (2)2 D (3) , J (8) = 2 D (8) + 4 n D (4)2 + 8 n D (3) D (5) + 8 n D (2) D (6) + 24 n 2 D (2) D (3)2 + 24 n 2 D (2)2 D (4) + 16 n 3 D (2)4 , J (9) = 2 D (9) + 8 n D (4) D (5) + 8 n D (3) D (6) + 8 n D (2) D (7) + 8 n 2 D (3)3 + 48 n 2 D (2) D (3) D (4) + 24 n 2 D (2)2 D (5) + 64 n 3 D (2)3 D (3) . Both families of inv arian t c hiral currents J ( M ) ( U ( x )) and D ( M ) ( U ( x )) satisfy the conserv ation equations ∂ − J ( M ) ( U ( x )) = 0, ∂ − D ( M ) ( U ( x )) = 0. The comm utation relations of in v ariant c hiral cu r rent s J ( M ) ( U ( x )) sho w that these currents are not densities of dynamical Casimir op erators for S U ( n ) group. Therefore, we w ill not consider these currents in the follo wing. W e considered abasic family of inv arian t chiral currents D ( M ) ( U ) and we p ro v ed that the in v ariant c hiral current s D ( M ) ( U ) form closed algebra u nder canonical PB and corresp ond ing c harges are dynamical Casimir op erators. The comm utation relations of inv arian t c hiral curr en ts D ( M ) ( U ( x )) and D ( N ) ( U ( y )) for M , N = 2 , 3 , 4 and for M = 2, N = 2 , 3 , . . . , ∞ are as follo ws { D ( M ) ( x ) , D ( N ) ( y ) } = − M N D ( M + N − 2) ( x ) ∂ ∂ x δ ( x − y ) − M N ( N − 1) M + N − 2 ∂ D ( M + N − 2) ( x ) ∂ x δ ( x − y ) . The comm utation relations for M ≥ 5, N ≥ 3 are as f ollo ws { D (5) ( x ) , D (3) ( y ) } = − [12 D (6) ( x ) + 3 D (6 , 1) ( x )] ∂ ∂ x δ ( x − y ) − 1 3 ∂ ∂ x [12 D (6) ( x ) + 3 D (6 , 1) ( x )] δ ( x − y ) , { D (5) ( x ) , D (4) ( y ) } = − [16 D (7) ( x ) + 4 D (7 , 1) ( x )] ∂ ∂ x δ ( x − y ) − 3 7 ∂ ∂ x [16 D (7) ( x ) + 4 D (7 , 1) ( x )] δ ( x − y ) , { D (6) ( x ) , D (3) ( y ) } = − [12 D (7) ( x ) + 6 D (7 , 1) ( x )] ∂ ∂ x δ ( x − y ) − 2 7 ∂ ∂ x [12 D (7) ( x ) + 6 D (7 , 1) ( x )] δ ( x − y ) , { D (5) ( x ) , D (5) ( y ) } = − [16 D (8) ( x ) + 8 D (8 , 1) ( x ) + D (8 , 2) ( x )] ∂ ∂ x δ ( x − y ) − 1 2 ∂ ∂ x [16 D (8) ( x ) + 8 D (8 , 1) ( x ) + D (8 , 2) ( x )] δ ( x − y ) , { D (6) ( x ) , D (4) ( y ) } = − [16 D (8) ( x ) + 8 D (8 , 3) ( x )] ∂ ∂ x δ ( x − y ) − 3 8 ∂ ∂ x [16 D (8) ( x ) + 8 D (8 , 3) ( x )] δ ( x − y ) , { D (7) ( x ) , D (3) ( y ) } = − [12 D (8) ( x ) + 6 D (8 , 1) ( x ) + 3 D (8 , 3) ( x )] ∂ ∂ x δ ( x − y ) − 1 4 ∂ ∂ x [12 D (8) ( x ) + 6 D (8 , 1) ( x ) + 3 D (8 , 3) ( x )] δ ( x − y ) , { D (8) ( x ) , D (3) ( y ) } = − [12 D (9) ( x ) + 6 D (9 , 1) ( x ) + 6 D (9 , 2) ( x )] ∂ ∂ x δ ( x − y ) In tegrable String Mo dels in T erms of C hiral I n v ariants of S U ( n ), S O ( n ), S P ( n ) Groups 9 − 2 9 ∂ ∂ x [12 D (9) ( x ) + 6 D (9 , 1) ( x ) + 6 D (9 , 2) ( x )] δ ( x − y ) , { D (7) ( x ) , D (4) ( y ) } = − [16 D (9) ( x ) + 8 D (9 , 2) ( x ) + 4 D (9 , 3) ( x )] ∂ ∂ x δ ( x − y ) − 1 3 ∂ ∂ x [16 D (9) ( x ) + 8 D (9 , 2) ( x ) + 4 D (9 , 3) ( x )] δ ( x − y ) , { D (6) ( x ) , D (5)( y ) } = − [16 D (9) ( x ) + 4 D (9 , 1) ( x ) + 8 D (9 , 2) ( x ) + 2 D (9 , 4) ( x )] ∂ ∂ x δ ( x − y ) − 4 9 ∂ ∂ x [16 D (9) ( x ) + 4 D (9 , 1) ( x ) + 8 D (9 , 2) ( x ) + 2 D (9 , 4) ( x )] δ ( x − y ) , { D (9) ( x ) , D (3) ( y ) } = − [12 D (10) ( x ) + 6 D (10 , 1) ( x ) + 6 D (10 , 2) ( x ) + 3 D (10 , 3) ( x )] ∂ ∂ x δ ( x − y ) − 1 5 ∂ ∂ x [12 D (10) ( x ) + 6 D (10 , 1) ( x ) + 6 D (10 , 2) ( x ) + 3 D (10 , 3) ( x )] δ ( x − y ) , { D (8) ( x ) , D (4) ( y ) } = − [16 D (10) ( x ) + 8 D (10 , 2) ( x ) + 8 D (10 , 4) ( x )] ∂ ∂ x δ ( x − y ) − 3 10 ∂ ∂ x [16 D (10) ( x ) + 8 D (10 , 2) ( x ) + 8 D (10 , 4) ( x )] δ ( x − y ) , { D (7) ( x ) , D (5) ( y ) } = − [16 D (10) ( x ) + 8 D (10 , 3) ( x ) + 4 D (10 , 1) ( x ) + 4 D (10 , 4) ( x ) + 2 D (10 , 5) ( x ) + D (10 , 6) ( x )] ∂ ∂ x δ ( x − y ) − 2 5 ∂ ∂ x [16 D (10) ( x ) + 8 D (10 , 3) ( x ) + 4 D (10 , 1) ( x ) + 4 D (10 , 4) ( x ) + 2 D (10 , 5) ( x ) + D (10 , 6) ( x )] δ ( x − y ) , { D (6) ( x ) , D (6) ( y ) } = − [16 D (10) ( x ) + 16 D (10 , 2) ( x ) + 4 D (10 , 7) ( x )] ∂ ∂ x δ ( x − y ) − 1 2 ∂ ∂ x [16 D (10) ( x ) + 16 D (10 , 2) ( x ) + 4 D (10 , 7) ( x )] δ ( x − y ) . The new dep en d en t inv arian t c hiral curr en ts D (6 , 1) , D (7 , 1) , D (8 , 1) − D (8 , 3) , D (9 , 1) − D (9 , 4) , D (10 , 1) − D (10 , 7) (see Ap p end ix A) ha v e the form D (6 , 1) = d k µν d l λρ d n σϕ d k ln U µ U ν U λ U ρ U σ U ϕ , D (7 , 1) = d k µν d l λρ d n σϕ d nm τ d k lm U µ U ν U λ U ρ U σ U ϕ U τ , D (8 , 1) = [ d k µν d k l λ d ln ρ ][ d m σϕ ][ d p τ θ ] d nmp U µ U ν U λ U ρ U σ U ϕ U τ U θ , D (8 , 2) = [ d k µν ][ d l λρ ][ d n σϕ ][ d m τ θ ] d k lp d nmp U µ U ν U λ U ρ U σ U ϕ U τ U θ , D (8 , 3) = [ d k µν d k l λ ][ d n ρσ d nm ϕ ][ d p τ θ ] d lmp U µ U ν U λ U ρ U σ U ϕ U τ U θ , D (9 , 1) = [ d k µν d k l λ d ln ρ d nm σ ][ d p ϕτ ][ d r θ ω ] d mpr U µ U ν U λ U ρ U σ U ϕ U τ U θ U ω , D (9 , 2) = [ d k µν d k l λ d ln ρ ][ d m σϕ d mp τ ][ d r θ ω ] d npr U µ U ν U λ U ρ U σ U ϕ U τ U θ U ω , D (9 , 3) = [ d k µν d k l λ ][ d n ρσ d nm ϕ ][ d p τ θ d pr ω ] d lmr U µ U ν U λ U ρ U σ U ϕ U τ U θ U ω , D (9 , 4) = [ d k µν d k l λ ][ d n ρσ ][ d m ϕτ ][ d p θ ω ] d lnr d mpr U µ U ν U λ U ρ U σ U ϕ U τ U θ U ω , D (10 , 1) = [ d k µν d k l λ d ln ρ d nm σ d mp ϕ ][ d r τ θ ][ d s ω β ] d pr s U µ U ν U λ U ρ U σ U ϕ U τ U θ U ω U β , D (10 , 2) = [ d k µν d k l λ d ln ρ d nm σ ][ d p ϕτ d pr θ ][ d s ω β ] d mrs U µ U ν U λ U ρ U σ U ϕ U τ U θ U ω U β , D (10 , 3) = [ d k µν d k l λ d ln ρ ][ d m σϕ d mp ϕ d pr τ ][ d s ω β ] d nr s U µ U ν U λ U ρ U σ U ϕ U τ U θ U ω U β , D (10 , 4) = [ d k µν d k l λ d ln ρ ][ d m σϕ d mp τ ][ d r θ ω d r s β ] d nps U µ U ν U λ U ρ U σ U ϕ U τ U θ U ω U β , D (10 , 5) = [ d k µν d k l λ d ln ρ ][ d m σϕ ][ d p τ θ ][ d r ω β ] d nms d pr s U µ U ν U λ U ρ U σ U ϕ U τ U θ U ω U β , 10 V.D. Gershun D (10 , 6) = [ d k µν d k l λ ][ d n ρσ d nm ϕ ][ d p τ θ ][ d r ω β ] d lms d pr s U µ U ν U λ U ρ U σ U ϕ U τ U θ U ω U β , D (10 , 7) = [ d k µν d k l λ ][ d n ρσ ] d m ϕ ][ d mp τ θ ][ d r ω β ] d lns d pr s U µ U ν U λ U ρ U σ U ϕ U τ U θ U ω U β . Let us apply the hydro dynamic approac h to in tegrable string mo dels with constan t tor- sion. In th is case we must consider the conserv ed primitiv e c hiral cur r en ts D ( M ) ( U ( x )), ( M = 2 , 3 , . . . , n − 1) as lo cal f ields of the Riemmann manifold. The non-prim itive lo cal c harges of in- v ariant c hiral cur r en ts with M ≥ n form the hierarch y of new Hamiltonians in the bi-Hamiltonian approac h to int egrable systems. The comm utatio n relations of inv arian t c hiral current s are lo cal PBs of h ydro dyn amic t yp e. The inv arian t chiral currents D ( M ) with M ≥ 3 for the S U (3) group can b e obtained fr om the follo wing relation d k ln d k mp + d k lm d k np + d k lp d k nm = 1 3 ( δ ln δ mp + δ lm δ np + δ lp δ nm ) . The corresp ondin g in v ariant chiral cur ren ts f or S U (3) group h a v e the f orm D (2 N ) = 1 3 N − 1 ( η µν U µ U ν ) N = 1 3 N − 1 D (2) N , D (2 N +1) = 1 3 N − 1 ( η µν U µ U ν ) N − 1 d k ln U k U l U n = 1 3 N − 1 D (2) N − 1 D (3) . The in v ariant c hiral currents D (2) , D (3) are lo cal co ordinates of the Riemmann m anifold M 2 . The lo cal c harges D (2 N ) , N ≥ 2 form a hierarc h y of Hamiltonians. The new n onlinear equations of motion for c hiral currents are as follo ws ∂ D ( k ) ( U ( x )) ∂ t N =  D ( k ) ( U ( x )) , Z 2 π 0 D (2) N ( U ( y )) dy  , k = 2 , 3 , N = 2 , . . . , ∞ . ∂ D (2) ( U ( x )) ∂ t N = − 2(2 N − 1) ∂ D (2) N ( U ( x )) ∂ x , ∂ D (3) ( U ( x )) ∂ t N = − 6 N D (3) ( U ( x )) ∂ D (2) N − 1 ( U ( x )) ∂ x − 2 N D (2) N − 1 ( U ( x )) ∂ D (3) ( U ( x )) ∂ x . The constru ction of integ rable equations with S U ( n ) symmetries for n ≥ 4 has dif f iculties in reduction of non-primitive inv arian t curr en ts to primitiv e currents. The similar method of construction of chiral curren ts f or S O (2 l + 1) = B l , S P (2 l ) = C l groups w as us ed by Ev ans et al. [24] on the b ase of symm etric inv arian t tensors of de Azcarraga et al. [25]. In the def ining representa tion these group generators corresp onding to algebras t µ satisfy the rules [ t µ , t ν ] = 2 if λ µν t λ , T r( t µ t ν ) = 2 δ µν , t µ η = − η t t µ , where η is a Euclidean or symp lectic stru ctur e. The symmetric tensor structure constan ts for these groups w ere introd uced through com- pletely symmetrized pro d uct of thr ee generators of corresp onding algebras t ( µ t ν t λ ) = v ρ µν λ t ρ , where v µν λρ is a totally symmetric tensor. Th e basic in v ariant symmetric tensors ha v e th e form [25] V (2) µν = δ µν , V (2 N ) ( µ 1 µ 2 ...µ 2 N − 1 µ 2 N ) = v ν 1 ( µ 1 µ 2 µ 3 v ν 1 ν 2 µ 4 µ 5 · · · v ν 2 N − 3 µ 2 N − 2 µ 2 N − 1 µ 2 N ) , N = 2 , . . . , ∞ . In tegrable String Mo dels in T erms of C hiral I n v ariants of S U ( n ), S O ( n ), S P ( n ) Groups 11 The in v ariant chiral curr en ts J (2 N ) (5) coincide with the b asis in v ariant c hiral currents V (2 N ) J (2 N ) = 2 V (2 N ) µ 1 ...µ 2 N U µ 1 · · · U µ 2 N . The comm utation relations of in v ariant c hiral cu r rent s are PBs of h ydro dyn amic t yp e { J ( M ) ( x ) , J ( N ) ( y ) } = − M N J ( M + N − 2) ( x ) ∂ ∂ x δ ( x − y ) − M N ( N − 1) M + N − 2 ∂ J ( M + N − 2) ( x ) ∂ x δ ( x − y ) . (6) The commuting charges of these inv arian t chiral currents are dynamical Casimir op erators on S O (2 l + 1), S P (2 l ). The metric tensor of Riemmann space of in v ariant c hiral cur r en ts is a s follo ws g M N ( J ( x )) = − M N ( M + N − 2) J ( M + N − 2) ( x ) . The comm utatio n relatio ns (6) coincide with comm utation relations, whic h w as obtai ned by Ev ans at al. [24]. W e used relations for n ew symm etric inv arian t tensors V (2 N , 1) ( µ 1 ...µ 2 N ) (see App endix B), whic h w e obtained durin g calculation PB (6) v k ( µ 1 µ 2 µ 3 v l µ 4 µ 5 µ 6 v n µ 7 µ 8 µ 9 v k ln µ 10 ) = V (10) ( µ 1 ...µ 10 ) , v k ( µ 1 µ 2 µ 3 v l µ 4 µ 5 µ 6 v n µ 7 µ 8 µ 9 v m µ 10 µ 11 µ 12 ) v k lnm = V (12) ( µ 1 ...µ 12 ) , v k ( µ 1 µ 2 µ 3 v l µ 4 µ 5 µ 6 v n µ 7 µ 8 µ 9 v m µ 10 µ 11 µ 12 v k lp µ 13 v nmp µ 14 ) = V (14) ( µ 1 ...µ 14 ) . App endix A The new dep endent in v ariant c hiral currents and the new dep endent totally symmetric in v arian t tensors for S U ( N ) group can b e obtained under d if f eren t ord er of calculation of trace of the pro- duct of the generators of S U ( n ) algebra. Let u s mark the matrix p ro duct of tw o generators t µ , t ν in round brac k ets ( t µ t ν ) = 2 n δ µν + ( d k µν + if k µν ) t k . (7) The expr ession of in v ariant c hiral curren ts J M ( U ) d ep end s on the p osition of the matrix pro d u ct of t w o generators in the general list of generators. F or example J (6) = T r[ t ( tt )( tt ) t ] = 2 D (6) + 4 n D (3)2 + 8 n D (2) D (4) + 8 n 2 D (2)3 , J (6) = T r[( tt )( tt )( tt )] = 2 D (6 , 1) + 12 n D (2) D (4) + 8 n 2 D (2)3 , J (7) = T r[ t ( tt ) t ( tt ) t ] = 2 D (7) + 8 n D (3) D (4) + 8 n 2 D (2) D (5) + 24 n 2 D (2)2 D (3) , J (7) = T r[( tt )( tt )( tt ) t ] = 2 D (7 , 1) + 4 n D (3) D (4) + 12 n 2 D (2) D (5) + 24 n 2 D (2)2 D (3) , J (8) = T r[ t ( tt ) tt ( tt ) t ] = 2 D (8) + 4 n D (4)2 + 8 n D (3) D (5) + 8 n D (2) D (6) + 24 n 2 D (2) D (3)2 + 24 n 2 D (2)2 D (4) + 16 n 3 D (2)4 , 12 V.D. Gershun J (8) = T r[( tt )( tt ) t ( tt ) t ] = 2 D (8 , 1) + 4 n D (4)2 + 4 n D (3) D (5) + 24 n 2 D (2) D (3)2 + 12 n D (2) D (6) + 24 n 2 D (2)2 D (4) + 16 n 3 D (2)4 , J (8) = T r[( tt )( tt )( tt )( tt )] = 2 D (8 , 2) + 4 n D (4)2 + 16 n D (2) D (6 , 1) + 32 n 2 D (2)2 D (4) + 16 n 3 D (2)4 , J (8) = T r[ t ( tt )( tt )( tt ) t ] = 2 D (8 , 3) + 12 n D (2) D (6) + 8 n D (3) D (5) + 24 n 2 D (2)2 D (4) + 24 n 2 D (2) D (3)2 + 16 n D (2)4 , J (9) = T r[ t ( tt ) ttt ( tt ) t ] = 2 D (9) + 8 n D (4) D (5) + 8 n D (3) D (6) + 8 n D (2) D (7) + 8 n 2 D (3)3 + 48 n 2 D (2) D (3) D (4) + 24 n 2 D (2)2 D (5) + 64 n 3 D (2)3 D (3) , J (9) = T r[ t ( tt ) tt ( tt )( tt )] =                            2 D (9 , 1) + 4 n D (4) D (5) + 4 n D (2) D (7) + 4 n D (2) D (7 , 1) + 8 n D (3) D (6 , 1) + 32 n 2 D (2) D (3) D (4) + 32 n 2 D (2)2 D (5) + 64 n 3 D (2)3 D (3) , · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 2 D (9 , 4) + 4 n D (2) D (7) + 4 n D (2) D (7 , 1) + 12 n D (3) D (6 , 1) + 32 n 2 D (2) D (3) D (4) + 32 n 2 D (2)2 D (5) + 64 n 3 D (2)3 D (3) , J (9) = T r[ t ( tt ) t ( tt ) t ( tt )] =                            2 D (9 , 2) + 4 n D (4) D (5) + 8 n D (3) D (6) + 8 n D (2) D (7) + 4 n D (2) D (7 , 1) + 8 n 2 D (3)3 + 40 n 2 D (2) D (3) D (4) + 32 n 2 D (2)2 D (5) + 64 n 3 D (2)3 D (3) , · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 2 D (9 , 3) + 8 n D (2) D (7) + 4 n D (2) D (71) + 12 n D (3) D (6) + 8 n D (3)3 + 40 n 2 D (2) D (3) D (4) + 32 n 2 D (2)2 D (5) + 64 n 3 D (2)3 D (3) , where t = t µ U µ and tw o v arian ts of tw o last expressions for J (9) ( U ) w ere obtained from t w o v ariants of expression for J (6) ( U ) d uring calculation J (9) ( U ). Because the resu lt of calculation do es not dep end on the order of calculation, w e can obtain relatio ns b et w ee n new in v ariant c hiral cu rrent s and basic inv arian t curr en ts D ( M ) ( U ) D (6 , 1) = D (6) + 2 n D (3)2 − 2 n D (2) D (4) , D (7 , 1) = D (7) + 4 n D (3) D (4) − 4 n D (2) D (5) , D (81) = D (8) + 2 n D (3) D (5) − 2 n D (2) D (6) , D (8 , 2) = D (8) + 4 n D (3) D (5) − 4 n D (2) D (6) − 4 n 2 D (2) D (3)2 + 4 n 2 D (2)2 D (4) , D (8 , 3) = D (8) + 2 n D (4)2 − 2 n D (2) D (6) , D (9 , 1) = D (9) + 2 n D (4) D (5) − 4 n 2 D (3)3 + 8 n 2 D (2) D (3) D (4) + 4 n 2 D (2)2 D (5) , In tegrable String Mo dels in T erms of C hiral I n v ariants of S U ( n ), S O ( n ), S P ( n ) Groups 13 D (9 , 2) = D (9) + 2 n D (4) D (5) − 2 n D (2) D (7) − 4 n 2 D (2) D (3) D (4) + 4 n 2 D (2)2 D (5) , D (9 , 3) = D (9) + 4 n D (4) D (5) − 2 n D (2) D (7) − 2 n D (3) D (6) − 4 n 2 D (2) D (3) D (4) + 4 n 2 D (2)2 D (5) , D (9 , 4) = D (9) + 4 n D (4) D (5) − 2 n D (3) D (6) − 8 n 2 D (3)3 + 12 n 2 D (2) D (3) D (4) + 4 n 2 D (2)2 D (5) . Hence we can ob tain the new relations for s y m metric tensors d k ( µν d l λρ d n σϕ ) d k ln = d k ( µν d k l λ d ln ρ d n σϕ ) + 2 n d ( µν λ d ρσϕ ) − 2 n δ ( µν d k λρ d k σϕ ) , d k ( µν d l λρ d n σϕ d nm τ ) d k lm = d k ( µν d k l λ d ln ρ d nm σ d m ϕτ ) + 4 n d ( µν λ d k ρσ d k ϕτ ) − 4 n δ ( µν d k λρ d k l σ d l ϕτ ) , d k ( µν d l λρ d n σϕ d nm τ d mp τ ) d k lp = d k ( µν d k l λ d ln ρ d nm σ d mp ϕ d p τ θ ) + 4 n d ( µν λ d k ρσ d k l ϕ d l τ θ ) − 2 n δ ( µν d k λρ d k l σ d ln ϕ d n τ θ ) , d k ( µν d l λρ d n σϕ d m τ θ ) d k lp d nmp = d k ( µν d k l λ d ln ρ d nm σ d mp ϕ d p τ θ ) + 4 n d ( µν λ d k ρσ d k l ϕ d l τ θ ) − 4 n δ ( µν d k λρ d k l σ d ln ϕ d n τ θ ) − 4 n 2 δ ( µν d λρσ d ϕτ θ ) + 4 n 2 δ ( µν δ λρ d k σϕ d k τ θ ) . It is p ossib le to obtain similar relations for in v ariant symmetric tensors of nint h order. Th e comm utation relations of c hiral currents in terms of the basic inv arian t cur ren ts are as follo ws { D (5) ( x ) , D (3) ( y ) } = −  15 D (6) ( x ) + 6 n D (3)2 ( x ) − 6 n D (2) ( x ) D (4) ( x )  ∂ ∂ x δ ( x − y ) − 1 3 ∂ ∂ x  15 D (6) ( x ) + 6 n D (3)2 ( x ) − 6 n D (2) ( x ) D (4) ( x )  δ ( x − y ) , { D (5) ( x ) , D (4) ( y ) } = −  20 D (7) ( x ) + 16 n D (3) ( x ) D (4) ( x ) − 16 n D (2) ( x ) D (5) ( x )  ∂ ∂ x δ ( x − y ) − 3 7 ∂ ∂ x  20 D (7) ( x ) + 16 n D (3) ( x ) D (4) ( x ) − 16 n D (2) ( x ) D (5) ( x )  δ ( x − y ) , { D (5) ( x ) , D (5) ( y ) } = −  25 D (8) ( x ) + 36 n D (3) ( x ) D (5) ( x ) − 20 n D (2) ( x ) D (6) ( x ) − 4 n D (2) ( x ) D (3)2 ( x ) + 4 n 2 D (2)2 ( x ) D (4) ( x )  ∂ ∂ x δ ( x − y ) − 1 2 ∂ ∂ x  25 D (8) ( x ) + 36 n D (3) ( x ) D (5) ( x ) − 20 n D (2) ( x ) D (6) ( x ) − 4 n D (2) ( x ) D (3)2 ( x ) + 4 n 2 D (2)2 ( x ) D (4) ( x )  , { D (6) ( x ) , D (4) ( y ) } = −  24 D (8) + 12 n D (4)2 − 12 n D (2) D (6)  ∂ ∂ x δ ( x − y ) − 3 8 ∂ ∂ x  24 D (8) + 12 n D (4)2 − 12 n D (2) D (6)  δ ( x − y ) , { D (7) ( x ) , D (3) ( y ) } = −  21 D (8) + 6 n D (4)2 + 12 n D (3) D (5) − 18 n D (2) D (6)  ∂ ∂ x δ ( x − y ) − 1 4 ∂ ∂ x  21 D (8) + 6 n D (4)2 + 12 n D (3) D (5) − 18 n D (2) D (6)  δ ( x − y ) , { D (8) ( x ) , D (3) ( y ) } = −  24 D (9) − 12 n D (2) D (7) + 24 n D (4) D (5) − 24 n 2 D (3)3 + 24 n 2 D (2) D (3) D (4) + 48 n 2 D (2)2 D (5)  ∂ ∂ x δ ( x − y ) − 2 9 ∂ ∂ x  24 D (9) − 12 n D (2) D (7) 14 V.D. Gershun + 24 n D (4) D (5) − 24 n 2 D (3)3 + 24 n 2 D (2) D (3) D (4) + 48 n 2 D (2)2 D (5)  δ ( x − y ) , { D (7) ( x ) , D (4) ( y ) } = −  28 D (9) − 8 n D (3) D (6) − 24 n D (2) D (7) + 32 n D (4) D (5) − 48 n 2 D (2) D (3) D (4) + 48 n 2 D (2)2 D (5)  ∂ ∂ x δ ( x − y ) − 1 3 ∂ ∂ x  28 D (9) − 8 n D (3) D (6) − 24 n D (2) D (7) + 32 n D (4) D (5) − 48 n 2 D (2) D (3) D (4) + 48 n 2 D (2)2 D (5)  δ ( x − y ) , { D (6) ( x ) , D (5) ( y ) } = −  30 D (9) − 4 n D (3) D (6) − 12 n D (2) D (7) + 32 n D (4) D (5) − 32 n 2 D (3)3 + 24 n 2 D (2) D (3) D (4) + 56 n 2 D (2)2 D (5)  ∂ ∂ x δ ( x − y ) − 4 9 ∂ ∂ x  30 D (9) − 4 n D (3) D (6) − 12 n D (2) D (7) + 32 n D (4) D (5) − 32 n 2 D (3)3 + 24 n 2 D (2) D (3) D (4) + 56 n 2 D (2)2 D (5)  δ ( x − y ) . App endix B The inv arian t chiral currents J (2 N ) and V (2 N ) and the new dep enden t totally symm etric in v ariant tensors for S O (2 l + 1), S P (2 l ) groups can b e obtained un der d if f eren t order of calculati on of trace of the pro duct of the generators of corresp onding alge bras. Let us m ark the matrix pro d uct of three generators t µ in round brac k ets ( t ( µ t ν t λ ) ) = v µν λρ t ρ . A dif feren t p osition of this triplet insid e of J 2 N pro du ces dif ferent exp ressions for V 2 N J (10) = T r[(( t 1 t 2 t 3 ) t 4 ( t 5 t 6 t 7 )( t 8 t 9 t 10 ))] U 1 · · · U 10 = 2 v k 123 v k l 45 v ln 67 v n 8910 U 1 · · · U 10 = 2 V (10) , J (10) = T r[(( t 1 t 2 t 3 )( t 4 t 5 t 6 )( t 7 t 8 t 9 ) t 10 )] U 1 · · · U 10 = 2 v k 123 v l 456 v n 789 v k ln 10 U 1 · · · U 10 = 2 V (10 , 1) , J (12) = T r[( t 1 ( t 2 t 3 t 4 ) t 5 ( t 6 t 7 t 8 ) t 9 ( t 10 t 11 t 12 ))] U 1 · · · U 12 = 2 v k 123 v k l 45 v ln 67 v nm 89 v m 101112 U 1 · · · U 12 = 2 V (12) , J (12) = T r[(( t 1 t 2 t 3 )( t 4 t 5 t 6 )( t 7 t 8 t 9 )( t 10 t 11 t 12 ))] U 1 · · · U 12 = 2 v k 123 v l 456 v n 789 v m 101112 v k lnm U 1 · · · U 12 = 2 V (12 , 1) , J (14) = T r[(( t 1 t 2 t 3 ) t 4 ( t 5 t 6 t 7 ) t 8 ( t 9 t 10 t 11 )( t 12 t 13 t 14 ))] U 1 · · · U 14 = 2 v k 123 v k l 45 v ln 67 v nm 89 v mp 1011 v p 121314 U 1 · · · U 14 = 2 V (14) , J (14) = T r[(( t 1 t 2 t 3 )( t 4 t 5 t 6 )( t 7 t 8 t 9 )( t 10 t 11 t 12 ) t 13 t 14 )] U 1 · · · U 14 = 2 v k 123 v l 456 v n 789 v m 101112 v k lp 13 v nmp 14 U 1 · · · U 14 = 2 V (14 , 1) . 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