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
Contents Abstracts Higher brackets
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, 203-215., arxiv: 0808.3406 [2] Th. Voronov, Nonlinear pullback on functions and a formal category extending the category of supermanifolds], arxiv: 1409.6475 [3] Th. Voronov, icroformal geometry, arxiv: 1411.6720
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.
Abstracts
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.
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 {, }
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 ()
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.
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 ).
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.
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.
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
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
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).
Anchor morphism of algebroids Anchor ρ : T T, morphism of algebroid T to tangent algebroid. ρ[[ω,σ]] = [ρ(ω),ρ(σ)].
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.
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
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
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
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
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 )
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.)
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...
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).
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,
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
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.
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.
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, 203-215., arxiv: 0808.3406 [2] Th. Voronov, Nonlinear pullback on functions and a formal category extending the category of supermanifolds], arxiv: 1409.6475 [3] Th. Voronov, icroformal geometry, arxiv: 1411.6720
Higher brackets
Higher brackets
Higher brackets