Deformations of Poisson structures by closed 3-forms

Deformations of P oisson structures b y closed 3-forms 1 O. I. Mokho v Abstract W e pro ve that an arbitra ry Poisson structure ω ij ( u ) a nd an arbitrary closed 3- form T ij k ( u ) generate the lo cal P oisson structure A ij ( u, u x ) = M i s ( u, u x ) ω sj ( u ) , where M i s ( u, u x )( δ s j + ω sp ( u ) T pj k ( u ) u k x ) = δ i j , on the corresp onding lo op space. W e o btain also a special graded ε - deformation of a n arbitrary P oisson structure ω ij ( u ) b y means of an arbitrary close d 3- f orm T ij k ( u ). In this pap er w e prov e that an a rbitrary P oisson structure ω ij ( u ) and an arbitrary closed 3-form T ij k ( u ) generate the lo cal P oisson structure A ij ( u, u x ) = B i s ( u, u x ) ω sj ( u ) , (1) where B i s ( u, u x ) M s j ( u, u x ) = δ i j , M s j ( u, u x ) = δ s j + ω sp ( u ) T pj k ( u ) u k x , (2) i.e., the matrix op erator A ij ( u, u x ) giv es the P oisson brac k et { I , J } = Z δ I δ u i ( x ) A ij ( u, u x ) δ J δ u j ( x ) dx (3) on t he sp ace of functionals on the corresp onding lo op space. Let M N b e an arbitra r y sm o oth N -dimensional manifold with the lo cal co ordinates u = ( u 1 , . . . , u N ). By the lo op sp ac e Ω M of t he manif o ld M N w e mean, in this paper, the space of all smo oth parametrized mappings of the circle S 1 in to M N , γ : S 1 → M N , γ ( x ) = { u i ( x ) } , x ∈ S 1 . T he tangen t space T γ Ω M of the lo op space Ω M at the point γ consist o f all smo oth v ector fields ξ = { ξ i , 1 ≤ i ≤ N } , define d along the lo op γ with ξ ( γ ( x )) ∈ T γ ( x ) M , ∀ x ∈ S 1 , where T γ ( x ) M is a ta ng en t space of the manifold M at the 1 The w or k w as s uppo rted b y the Max -Planck-Institut f¨ ur Mathema tik (Bonn, Ger many), by the Russian F oundation for Basic Rese a rch (pro ject no. 08-01-00 4 64) a nd b y the pr ogra mme “Leading Scient ific Sc ho ols” (pro ject no. NSh-1 824.20 08.1). 1 p oin t γ ( x ). All closed 2-forms (pres ymplectic structures) on the lo op space Ω M that are giv en b y matrix op erators of the form ω ij ( u, u x , . . . , u ( k ) ), i.e., all closed 2-forms o f the form ω ( ξ , η ) = Z S 1 ξ i ω ij ( u, u x , . . . , u ( k ) ) η j dx, (4) where ξ , η ∈ T γ Ω M , w ere completely described in [1] (see also des criptions of v arious differen tial-geometric classes of symplectic (presymple ctic) and P oisson structures in [2]–[9]). Theorem 1 [1]. A biline ar form (4) i s a close d skew -symmetric 2-form ( a pr e sym- ple ctic structur e ) on the lo op sp ac e Ω M if and only if ω ij ( u, u x , . . . , u ( k ) ) = T ij k ( u ) u k x + Ω ij ( u ) , (5) wher e T ij k ( u ) is an arbitr ary close d 3-form on the manifold M N and Ω ij ( u ) is an arb itr ary close d 2 -form on M N . If the matrix ω ij ( u, u x , . . . , u ( k ) ) is nondegenerate, det( ω ij ( u, u x , . . . , u ( k ) )) 6 = 0, then the corresp onding presymplectic form (4), (5) is symplectic and the in v erse matrix ω ij ( u, u x , . . . , u ( k ) ), ω is ( u, u x , . . . , u ( k ) ) ω sj ( u, u x , . . . , u ( k ) ) = δ i j , gives the P oisson struc- ture { I , J } = Z δ I δ u i ( x ) ω ij ( u, u x , . . . , u ( k ) ) δ J δ u j ( x ) dx (6) on the lo op space Ω M , i.e., the brack et (6) is ske w-symmetric and satisfy the Jacobi iden tit y . Therefore Theorem 1 giv es the complete description of all nondegenerate P oisson structures on the lo op space Ω M that are giv en by matrix operator s of the form ω ij ( u, u x , . . . , u ( k ) ), i.e., all the nondegenerate P o isson brac k ets of the form (6), det( ω ij ( u, u x , . . . , u ( k ) )) 6 = 0 (su ch nondegenerate P oisson structures w ere studied b y As- tasho v and Vinogrado v in [9], see also [7]–[8] a nd [1]–[6 ]). W e note that if the closed 2-form Ω ij ( u ) is nondegenerate, det(Ω ij ( u )) 6 = 0, i.e, t he form Ω ij ( u ) is symplectic on M N , then the 2-form (5) is a nondegenerate form on Ω M for an y closed 3-form T ij k ( u ) on the manifo ld M N since it is ob vious that in this case det( T ij k ( u ) u k x + Ω ij ( u )) 6 = 0. Th us w e can define, on the lo op space of an a rbitrary symplectic manifold M N , the P oisson brac k et { I , J } = Z δ I δ u i ( x ) ω ij ( u, u x ) δ J δ u j ( x ) dx, (7) where ω li ( u, u x )( T ij k ( u ) u k x + Ω ij ( u )) = δ l j , (8) I and J b eing a rbitrary functiona ls on Ω M . The Poiss on brac ket (7) , (8) is a par- tial case of the brac ke t (1)–(3), namely , the case whe n the Poisson structure ω ij ( u ) is nondegenerate, det( ω ij ( u )) 6 = 0, ω ij ( u ) = Ω ij ( u ) , Ω is ( u )Ω sj ( u ) = δ i j , since ω ij ( u, u x ) = C i s ( u, u x )Ω sj ( u ) , (9) 2 where C l s ( u, u x )( δ s j + Ω si ( u ) T ij k ( u ) u k x ) = δ l j . (10) The case o f degene rate P oisson structures ω ij ( u ), det ( ω ij ( u )) = 0, is muc h more com- plicated. W e note that in con trast to the case o f all closed 2-forms (presymplectic structures) of the form (4) (Theorem 1 ) the pro blem of description of all degenerate P oisson structures of the form (6) is a v ery complicated and unsolv ed problem. Theorem 2. A n a rbitr ary Pois son structur e ω ij ( u ) and an arbitr ary close d 3-form T ij k ( u ) give the lo c al Poisson b r acket (1)–( 3) . First of all, w e note that ob viously the matrix op erator A ij ( u, u x ) (1), (2) is sk ew- symmetric. Lemma. A skew-symmetric matrix op er a tor A ij ( u, u x ) (1) gives a Poisson br acket (3) if an d only if the fol lowing r elations hold: ω ij ( u ) ω r p ( u ) ∂ M s r ∂ u i x = ω is ( u ) ω r j ( u ) ∂ M p r ∂ u i x , (11) ω ij ( u ) ω r p ( u ) ∂ M s r ∂ u i − ω ij ( u ) d dx  ∂ M s r ∂ u i x ω r p ( u )  + ∂ ω ij ∂ u r ω r p ( u ) M s i ( u ) + + ω is ( u ) ω r j ( u ) ∂ M p r ∂ u i + d dx  ω is ( u )  ∂ M p r ∂ u i x ω r j ( u ) + ∂ ω is ∂ u r ω r j ( u ) M p i ( u ) + + ω ip ( u ) ω r s ( u ) ∂ M j r ∂ u i + d dx  ω ip ( u )  ∂ M j r ∂ u i x ω r s ( u ) + ∂ ω ip ∂ u r ω r s ( u ) M j i ( u ) = 0 . (12) If M i s ( u, u x ) = δ s j + ω sp ( u ) T pj k ( u ) u k x , then relations (11), (1 2) hold for a n arbitrary P oisson structure ω ij ( u ) and an arbitrary closed 3-form T ij k ( u ). Let us add an arbitrary pa rameter ε in the form ula for our Poisson structure: A ij ( ε, u, u x ) = B i s ( ε, u, u x ) ω sj ( u ) , (13) where B i s ( ε, u, u x )( δ s j + εω sp ( u ) T pj k ( u ) u k x ) = δ i j . (14) W e can now expand the P oisson structure A ij ( ε, u, u x ) in series in ε : A ij ( ε, u, u x ) = ω ij ( u ) − εω is ( u ) T sr k ( u ) ω r j ( u ) u k x + · · · . (15) This expansion giv e an ε -deformation of an arbitrary P o isson s tructure ω ij ( u ) b y means of an arbitrary closed 3-form T ij k ( u ). W e note that this ε -deformation of an ar bitr ary P oisson structure ω ij ( u ) b elongs to a sp ecial class of graded ε - deformations of Poiss on structures (see, for example, [10], [11]). 3 Ac knowle dgemen ts. The w o rk w as supp orted b y the Max-Planc k-Institut f ¨ ur Mathematik (Bonn, Germany ), b y the Russian F oundation for Basic Researc h (pro j ect no. 08- 0 1-00464 ) a nd b y a gran t of the Presiden t of the Russ ian F ederation (pro j ect no. NSh-1824.200 8 .1). References [1] O. I. Mokho v, “Symplectic and P oisson geometry on loop sp aces of manifolds and nonlinear equations”, In: T opics in T opolo gy a nd Mathematical Ph ysics, Ed. S.P .No vik o v, Amer. Math. So c., Pro vidence, RI, 1995, pp. 121–15 1; h ttp://arXiv.org/ hep-th/9503076 (1995). [2] O. I. Mokho v, “P oisson and symplectic geometry on lo op space s of smo ot h man- ifolds”, In: Geometry from t he Pacific R im, Pro ceedings of the Pacifi c Rim G eometry Conference held at National Univ ersity of Singap ore, Republic of Singap ore, Decem b er 12–17, 1994, Eds. A.J.Berric k, B.Lo o, H.-Y.W ang, W alter de G r uyter, Berlin, 1997, pp. 285–309. [3] O. I. Mokhov, “Differential geometry of symplectic and P oisson structures on lo op spaces of smo o th manifolds, and integrable sy stems”, T rudy Matem. Inst. Ak ad. 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Zhang , “Nor mal forms of hierarc hies of in tegr a ble PDEs, F rob enius manifolds and Gro mo v - Witten in v aria n ts”; arXiv:math/0108160 (2001). [11] B. Dubro vin, S.-Q. Liu and Y. Z hang, “On Hamiltonian perturbat io ns of h yp er- b olic syste ms of conserv ation la ws, I: Q ua si-trivialit y of bi- Hamiltonian p erturbations,” Comm. Pure Appl. Math., V ol. 59, 2006, 559 - 615; arXiv:math/0410027 (2004). O. I. Mokhov Cen tre for Nonlinear Studies, L.D.Landau Institute for Theoretical Ph ysics, Russian Acade my of Sciences, Kosygina str., 2, Mosco w, 117940, Russ ia; Departmen t of Geometry and T op ology , F acult y of Me ch anics a nd Mathematics, M.V.Lomonoso v Mosco w State Univ ersity , Mosco w, 119992, Russ ia E-mail : mokhov@mi.ras.ru; mokho v@landa u.a c.ru; mokho v@bk.ru 5