Higher order Koszul brackets

Transcription

1 Higher order Koszul brackets Hovhannes Khudaverdian University of anchester, anchester, UK XXXY WORKSHOP ON GEOETRIC ETHODS IN PHYSICS 26 June-2 July, Bialoweza, Poland The talk is based on the work with Ted Voronov

2 Contents Abstracts Higher brackets

3 Papers that talk is based on are [1] H..Khudaverdian, Th. Voronov Higher Poisson brackets and differential forms, 2008a In: Geometric ethods in Physics. AIP Conference Proceedings 1079, American Institute of Physics, elville, New York, 2008, , arxiv: [2] Th. Voronov, Nonlinear pullback on functions and a formal category extending the category of supermanifolds], arxiv: [3] Th. Voronov, icroformal geometry, arxiv:

4 Abstracts Abstract... For an arbitrary manifold, we consider supermanifolds ΠT and ΠT, where Π is the parity reversion functor. The space ΠT possesses canonical odd Schouten bracket and space ΠT posseses canonical de Rham differential d. An arbitrary even function P on ΠT such that [P,P] = 0 induces a homotopy Poisson bracket on, a differential, d P on ΠT, and higher Koszul brackets on ΠT. (If P is fiberwise quadratic, then we arrive at standard structures of Poisson geometry.) Using the language of Q-manifolds and in particular of Lie algebroids, we study the interplay between canonical structures and structures depending on P. Then using just recently invented theory of thick morphisms we construct a non-linear map between the L algebra of functions on ΠT with higher Koszul brackets and the Lie algebra of functions on ΠT with the canonical odd Schouten bracket.

5 Abstracts

6 Poisson manifold Let be Poisson manifold with Poisson tensor P = P ab b a {f,g} = {f,g} P = f g Pab x a x b. {{f,g},h} + {{g,h},f } + {{h,f },g} = 0, P ar r P bc + P br r P ca + P cr r P ab = 0. If P is non-degenerate, then ω = (P 1 ) ab dx a dx b is closed non-degenerate form defining symplectic structure on.

7 Differentials d de Rham differential, d : Ω k () Ω k+1 (), d 2 = 0,df = f x a dx a, d(ω ρ) = dω ρ + ( 1) p(ω)ω ρ d P Lichnerowicz- Poisson differential, d P : A k () A k+1 (), d 2 P = 0,df = f x b Pba x a d P P = 0 Jacobi identity for Poisson bracket {, }

8 Differential forms and multivector fields A space multivector fields on, Ω space of differential forms on, A k () Ω k () d P d A k+1 () Ω k+1 ()

9 Differential forms and multivector fields A multivector fields on = functions on ΠT Ω differential forms on = functions on ΠT, A k () Ω k () d P d A k+1 () Ω k+1 () C(ΠT ) C(ΠT) d P C(ΠT ) d C(ΠT) dω(x,ξ ) = ξ a x a ω(x,ξ ),d PF(x,θ) = (P,F) 1, (P,F) 1 -canonical odd Poisson bracket on ΠT.

10 x a = (x 1,...,x n ) coordinates on (x a,ξ b ) = (x 1,...,x n ;ξ 1,...,ξ n ), coordinates on ΠT p(ξ a ) = p(x a )+1,x a = x a (x a a ) ξ a a x = ξ x a. (dx a ξ a ). Respectively (x a,θ b ) = (x 1,...,x n ;θ 1,...,θ n ), coordinates on ΠT p(θ a ) = p(x a ) + 1,x a = x a (x a ) θ a = θ a x a Example x a. ( a θ a ). Ω ω = l a dx a +r ab dx a dx b ω(x,ξ ) = l a ξ a +r ab ξ a ξ b C(ΠT) A F = X a a + ab a b F(x,θ) = X a θ a + ab θ a θ b C(ΠT ).

11 Canonical odd Poisson bracket F,G multivector fields [F,G]Schouten commutator, F,G functions on ΠT [F,G]odd Poisson bracket, X = X a a,[x,f] = L X F P = P ab a b, [P,F] = d P F, [F(x,θ),G(x,θ)] = F(x,θ) x a Names are [X,F ] = (X a θ a,f(x,θ)) d P F = (P,F) = (P ab θ a θ b,f(x,θ)) 1 G(x,θ) +( 1) θ a odd Poisson bracket Schouten bracket Buttin bracket anti-bracket p(f ) F(x,θ) θ a G(x,θ) x a.

12 Koszul bracket on differential forms C(ΠT ) ϕp : C(ΠT) ξ a = P ab θ b or dx a = P b b From {, } on functions to Koszul bracket on differential forms [ω,σ] P = (ϕ P ) 1 ( [ϕ P (ω),ϕ P (σ)] P). [f,g] P = 0, [f,dg] P = ( 1) p(f ) {f,g} P, [df,dg] P = ( 1) p(f ) d ({f,g} P ) This formula survives the limit if P is degenerate.

13 Lie algebroid E vector bundle, [[, ]] commutator on sections, ρ : E T -anchor ( ) [[s 1 (x),f (x)s 2 (x)]] = f (x)[[s 1 (x),s 2 (x)]] + ρ(s1 (x))f (x) s 2 (x), Jacobi identity: [[[[s 1,s 2 ]],s 3 ]] + cyclic permutations = 0. s(x) = s i (x)e i (x), [[e i (x),e k (x)]] = cik m (x)e m(x),ρ(e i ) = ρ µ i µ, ( [[s 1 (x),s 2 (x)]] = s1 i sk 2 cm ik + si 1 ρ µ i µ s2 m (x) si 2 ρ µ i µ s1 )e m (x) m

14 Trivial examples of Lie algebroid G Lie algebra, G, where [[, ]] usual commutator, tangent bundle For T anchor is identity map T, where [[, ]] commutator of vector fields

15 Poisson algebroid (,P) Poisson manifold, (P = P ab b a, {f,g} = a fp ab b g) T, [[]df,dg] = d{f,g}, anchorρ : ρ(ω a dx a ) = D ω = P ab ω b x b, ( ) 1 [[ω a dx a,σ b dx b ]] = 2 ω aσ b c P ab + P ab ω b a σ c (ω σ) dx x (This is Koszul bracket [, ] P on 1-forms).

16 Anchor morphism of algebroids Anchor ρ : T T, morphism of algebroid T to tangent algebroid. ρ[[ω,σ]] = [ρ(ω),ρ(σ)].

17 One very useful object Q manifold Definition A pair (,Q) where is (super)manifold, and Q is odd vector field on it such that Q 2 = 1 2 [Q,Q] = 0 is called Q-manifold. Q is called homological vector field.

18 Lie algebroid and its neighbours Algebroid has diffferent manifestations ΠE ΠE is Q manifold with Q = ξ k ξ i cik m + ξ ξ i ρ µ m i E, ΠE Lie Poisson bracket: x µ, E E is Lie algebroid with [[e i,e k ]] = cik m,ρ(e i) = ρ µ i (even, odd)poisson manifolds {u i,u k } = c m ik u m,{x µ,u i } = ρ µ i,{x µ,x ν } = 0. x µ

19 Neighbours of G ΠG ξ m Q = ξ i ξ k cik m }{{} homological vector field, G [e i,e k ] = c m ik e m }{{} structure constants, G {u i,u k } = c m ik u m }{{} Lie-Poisson bracket

20 Neighbours of tangent algebroid T ΠT Q = ξ m, }{{ x m } homological vector field de Rham differential d (functions on ΠT) differential forms on ) T, canonical symplectic structure ΠT canonical odd sympletic structure

21 Neighbours of Poisson algebroid T (,P) Poisson manifold, {x a,x b } = P ab ΠT P ba Q = θ a θ b x c + θ a P ab θ c x }{{ b } homological vector field, ΠT {, } = [, ] P is Koszul bracket on ΠT. T Poisson algebroid [[dx a,dx b ]] = dp ab, ρ(dx a ) = P ab b

22 ΠT is in the neighbourhood of tangent algebroid ΠT is in the neighbourhood of Poisson algebroid T T ΠT }{{ } } ΠT {{} Odd canonical Poisson bracket Odd Koszul bracket i Linear map ξ a = 1 P(x,θ) 2 θ a = P ab θ b, (dx a = P ab b )

23 Question What happens if even function P = P ab (x,θ)θ a θ b is replaced by an arbitrary even function P = P(x,θ) which obeys the master-equation [P,P] = 2 P(x,θ) P(x,θ) x a θ a = 0. (In the case P = P ab (x,θ)θ a θ b master-equation is just Jacobi identity for Poisson bracket {, } P on.)

24 Higher brackets Higher Poisson brackets on P : [P,P] = 0 define higher brackets {f 1,f 2,...,f n } P = [...[P,f 1 ],...,f p ], = θ=0. P = P a θ a + P ab θ b θ a + P abc θ c θ b θ a +... {x a } P = P a, {x a,x b } = P ab, {x a,x b,x c } = P abc...

25 Higher brackets From ΠT to ΠT C(ΠT ) X(ΠT ) C(T (ΠT )) C(T (ΠT )) Function P(x,θ) Hamiltonian vector field D F Hamiltonian in T (ΠT ) T (ΠT ) The last map is ackenzie Xu symplectomorphism C(ΠT) P = P(x,θ) K = K P (x,ξ ) T (ΠT ) ( K P (x,ξ,p,π) = p a P(x,θ) + ξ a θ π θ a x a P(x,θ)) (x a,ξ b p a,π b ) coordinates on T (ΠT).

26 Higher brackets Higher Koszul brackets on P ΠT induces homotopy Poisson bracket in, K P T (ΠT) induces homotopy odd Poisson bracket (higher Koszul bracket) on Π, {F 1,F 2,...,F n } KP = [...[K P,F 1 ],...,F p ] Π, Π = p=π=0 ). F = F(x,ξ ) = f (x) + ξ a f a (x) +...,(df = ξ a a f ), [f ] P = 0,[f 1,f 2,...,f k ] P = 0 [f 1,df 2,...,df n ] = f 1,f 2,...,f n, [df 1,df 2,...,df n ] = df 1,f 2,...,f n,

27 Higher brackets C(ΠT ) Q-manifolds morphism of Q-manif. C(ΠT) ΠT Lichnerowicz Poisson differential d P Odd Poisson canonical bracket d P : d P f = [P,F], d = ξ a a, d P = P x a ΠT de Rham differential Odd Koszul bracket,+ P θ a θ a x a

28 Higher brackets If P = P ab then the map ΠT ΠT : ξ a = P θ a = Pab (x)θ b, is linear in fibres. orphism of Q-manifolds C(ΠT ) C(ΠT) is its pull-back. These linear maps interwin differentials d and d P, their Hamiltonians, and their homological vector fields on infinite-dimensional spaces of functions.

29 Higher brackets It is more tricky if P(x,θ) is an arbitrary function (solution of master-equation [S,S] = 0. The map ΠT ΠT : ξ a = P θ a = Pab (x)θ b, and its pull-back is in general non-linear map. i.e. ΠT ΠT non-linear ΠT) thick ΠT ) C(ΠT)non-linear mapc(πt ) This non-linear map defines morphism of Q-manifolds.

30 Higher brackets Papers that talk is based on [1] H..Khudaverdian, Th. Voronov Higher Poisson brackets and differential forms, 2008a In: Geometric ethods in Physics. AIP Conference Proceedings 1079, American Institute of Physics, elville, New York, 2008, , arxiv: [2] Th. Voronov, Nonlinear pullback on functions and a formal category extending the category of supermanifolds], arxiv: [3] Th. Voronov, icroformal geometry, arxiv:

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