Vector Bundle Valued Differential Forms on Non-Negatively Graded DG - PowerPoint PPT Presentation

Forms on N -Manifolds 1-Forms on Degree One N Q -Manifolds 2-Forms on Degree One N Q -Manifolds Higher Forms on Degree One N Q -Manifolds Vector Bundle Valued Differential Forms on Non-Negatively Graded DG Manifolds Luca Vitagliano University of Salerno, Italy Geometry of Jets and Fields for Janusz’s 60th Birthday B˛ edlewo, May 15, 2015 Luca Vitagliano VB Valued Forms on N Q -manifolds 1 / 32

Forms on N -Manifolds 1-Forms on Degree One N Q -Manifolds 2-Forms on Degree One N Q -Manifolds Higher Forms on Degree One N Q -Manifolds Introduction Graded geometry encodes efficiently (non-graded) geometric structures, e.g. N Q -manifolds encode Lie algebroids and their higher analogues. Remark N Q-manifolds ( M , Q ) + a compatible geometric structure encode higher Lie algebroids + a compatible structure . Differential forms on M preserved by Q are of a special interest. Vec- tor bundle (VB) valued forms are even more interesting! VB valued forms describe several interesting geometries: foliated , (pre)contact , (pre)symplectic , locally conformal symplectic , poly-symplectic , cosymplectic , multisymplectic , . . . Luca Vitagliano VB Valued Forms on N Q -manifolds 2 / 32

Forms on N -Manifolds 1-Forms on Degree One N Q -Manifolds 2-Forms on Degree One N Q -Manifolds Higher Forms on Degree One N Q -Manifolds Introduction Examples deg N Q -manifold standard geometry proved in 1 foliated infinitesimal ideal system [Zambon & Zhu 2012] [Grabowski 2013] 1 contact Jacobi [Mehta 2013] 1 symplectic Poisson [Roytenberg 2002] 2 contact contact-Courant [Grabowski 2013] 2 symplectic Courant [Roytenberg 2002] Remark All above cases can be regarded as: N Q-manifold + a compatible VB valued form. Luca Vitagliano VB Valued Forms on N Q -manifolds 3 / 32

Forms on N -Manifolds 1-Forms on Degree One N Q -Manifolds 2-Forms on Degree One N Q -Manifolds Higher Forms on Degree One N Q -Manifolds Introduction The above examples motivate the study of VB valued differential forms on N Q -manifolds! Aims of the Talk describe VB valued differential forms on N -manifolds in terms of 1 non-graded geometric data, use this description as a unified formalism for examples above, 2 enlarge the list of examples. 3 Remark I work in the simplest case: deg 1, i.e. (non-higher) Lie algebroids. Luca Vitagliano VB Valued Forms on N Q -manifolds 4 / 32

Forms on N -Manifolds 1-Forms on Degree One N Q -Manifolds 2-Forms on Degree One N Q -Manifolds Higher Forms on Degree One N Q -Manifolds Outline Forms on N -Manifolds 1 1-Forms on Degree One N Q -Manifolds 2 2-Forms on Degree One N Q -Manifolds 3 Higher Forms on Degree One N Q -Manifolds 4 Luca Vitagliano VB Valued Forms on N Q -manifolds 5 / 32

Forms on N -Manifolds 1-Forms on Degree One N Q -Manifolds 2-Forms on Degree One N Q -Manifolds Higher Forms on Degree One N Q -Manifolds Outline Forms on N -Manifolds 1 1-Forms on Degree One N Q -Manifolds 2 2-Forms on Degree One N Q -Manifolds 3 Higher Forms on Degree One N Q -Manifolds 4 Luca Vitagliano VB Valued Forms on N Q -manifolds 6 / 32

Forms on N -Manifolds 1-Forms on Degree One N Q -Manifolds 2-Forms on Degree One N Q -Manifolds Higher Forms on Degree One N Q -Manifolds Reminder on Graded Manifolds Definition: graded manifold A pair M = ( M , C ∞ ( M )) consisting of a manifold M and a graded C ∞ ( M ) -algebra C ∞ ( M ) ≈ Γ ( S • E • ) for some graded VB E • → M . Remark Smooth maps, vector fields, differential forms, etc. on M are defined algebraically via graded differential calculus on C ∞ ( M ) . Think of M as a space locally coordinatized by ( x i , z α ) : deg x i = 0 = ⇒ the x i ’s commute , deg z α = : | α | ∈ Z � 0 = ⇒ the z α ’s graded commute . The Euler vector field ∆ = | α | z α ∂ ∂ z α measures the internal degree of geometric objects. Luca Vitagliano VB Valued Forms on N Q -manifolds 7 / 32

Forms on N -Manifolds 1-Forms on Degree One N Q -Manifolds 2-Forms on Degree One N Q -Manifolds Higher Forms on Degree One N Q -Manifolds Q -manifolds and Lie algebroids Remark I work with N -manifolds M , i.e. C ∞ ( M ) is non-negatively graded. The degree of M is the highest degree of its coordinates. Definition: N Q-manifold An N -manifold M + an homological vector field Q , i.e. deg Q = 1, and [ Q , Q ] = 0. Proposition There is a one-to-one correspondence between deg 1 N Q-manifolds and Lie algebroids, given by ( A [ 1 ] , Q = d A ) ⇐ = � ( A , ρ A , [ − , − ] A ) . Conversely [ α , β ] v A = [[ Q , α v ] , β v ] and ρ A ( α ) f = [ Q , α v ] f , α , β ∈ Γ ( A ) . where α v ∈ X ( A [ 1 ]) : = vertical lift of α ∈ Γ ( A ) . Luca Vitagliano VB Valued Forms on N Q -manifolds 8 / 32

Forms on N -Manifolds 1-Forms on Degree One N Q -Manifolds 2-Forms on Degree One N Q -Manifolds Higher Forms on Degree One N Q -Manifolds VB Valued Forms on N -manifolds Let E → M be a VB in the category of graded manifolds. There is a Cartan calculus on Ω ( M , E ) : = {E -valued differential forms on M} . Definition: derivation of E An R -linear, graded operator X : Γ ( E ) → Γ ( E ) such that X ( f e ) = X ( f ) e + ( − ) | f | f X e , for some graded vector field X . Remark ω ∈ Ω ( M , E ) can be contracted with and Lie differentiated along X . Inte- rior products and Lie derivatives satisfy usual Cartan identities: [ i X , i Y ] = 0, [ L X , i Y ] = i [ X , Y ] , [ L X , L Y ] = L [ X , Y ] . Definition: vector N Q-bundle A VB E → M + an homological derivation Q , i.e. deg Q = 1, and [ Q , Q ] = 0. Luca Vitagliano VB Valued Forms on N Q -manifolds 9 / 32

Forms on N -Manifolds 1-Forms on Degree One N Q -Manifolds 2-Forms on Degree One N Q -Manifolds Higher Forms on Degree One N Q -Manifolds Spencer Data Simplifying Assumption: Γ ( E ) is generated in deg 0 I.e. E = M × M E for some non-graded VB E → M . Then a negatively graded derivation X of E is determined by its symbol X ∈ X − ( M ) . Key Remark A degree n > 0 form ω ∈ Ω k ( M , E ) is completely determined by interior products with and Lie derivatives along negatively graded derivations : n ω = L ∆ ω = | α | ( z α L ∂ / ∂ z α ω + dz α ∧ i ∂ / ∂ z α ω ) . Definition: Spencer data of a deg n > 0 form ω ∈ Ω k ( M , E ) D : X − ( M ) − → Ω k ( M , E ) , → D ( X ) : = L X ω , X �− and → Ω k − 1 ( M , E ) , ℓ : X − ( M ) − → ℓ ( X ) : = i X ω . X �− Luca Vitagliano VB Valued Forms on N Q -manifolds 10 / 32

Forms on N -Manifolds 1-Forms on Degree One N Q -Manifolds 2-Forms on Degree One N Q -Manifolds Higher Forms on Degree One N Q -Manifolds Spencer Data Theorem Spencer data establish a one-to-one correspondence between degree n > 0 forms ω ∈ Ω k ( M , E ) and pairs ( D , ℓ ) , with D : X − ( M ) → Ω k ( M , E ) a degree n first order DO, and ℓ : X − ( M ) → Ω k − 1 ( M , E ) a degree n C ∞ ( M ) -linear map, such that D ( f X ) = f D ( X ) + ( − ) X d f ∧ ℓ ( X ) , and, moreover, L X D ( Y ) − ( − ) XY L Y D ( X ) = D ([ X , Y ]) , L X ℓ ( Y ) − ( − ) X ( Y − 1 ) i Y D ( X ) = ℓ ([ X , Y ]) , i X ℓ ( Y ) − ( − ) ( X − 1 )( Y − 1 ) i Y ℓ ( X ) = 0. One can describe (inductively on n ) ω in terms of non-graded data! Luca Vitagliano VB Valued Forms on N Q -manifolds 11 / 32

Forms on N -Manifolds 1-Forms on Degree One N Q -Manifolds 2-Forms on Degree One N Q -Manifolds Higher Forms on Degree One N Q -Manifolds A First Example: Degree 1 Symplectic N Q -manifolds Definition: deg n symplectic N -manifold A deg n N -manifold M + a deg n symplectic form ω . Example: the shifted cotangent bundle T ∗ [ n ] M of a deg 0 manifold M Notice that X − ( T ∗ [ n ] M ) = Ω 1 ( M )[ n ] . T ∗ [ n ] M is equipped with a deg n symplectic form ω determined by f ∈ C ∞ ( M ) . L ( d f ) v ω = 0, and i ( d f ) v ω = d f , Hence D = ( − ) n d : Ω 1 ( M ) → Ω 2 ( M ) and ℓ = id : Ω 1 ( M ) → Ω 1 ( M ) . Definition: deg n symplectic N Q-manifold A deg n N Q -manifold ( M , Q ) + a deg n symplectic form ω such that L Q ω = 0. Luca Vitagliano VB Valued Forms on N Q -manifolds 12 / 32

Forms on N -Manifolds 1-Forms on Degree One N Q -Manifolds 2-Forms on Degree One N Q -Manifolds Higher Forms on Degree One N Q -Manifolds A First Example: Degree 1 Symplectic N Q -manifolds Theorem [Roytenberg 2002] There is a “one-to-one” correspondence between deg 1 symplectic N Q- manifolds ( M , Q ) and Poisson manifolds. An alternative proof via Spencer data Let M = A [ 1 ] → M . Then X − ( M ) = Γ ( A )[ 1 ] . non-degeneracy ⇒ ℓ : Γ ( A ) ≃ Ω 1 ( M ) � ⇒ ( M , ω ) ≃ ( T ∗ [ 1 ] M , ω ) , closedness ⇒ D = − d ◦ ℓ Hence, L Q ω = 0 ⇔ i ( d f ) v i ( dg ) v L Q ω = L ( d f ) v i ( dg ) v L Q ω = 0. From Q = d T ∗ M for a Lie algebroid ( T ∗ M , ρ T ∗ M , [ − , − ] T ∗ M ) , follows i ( d f ) v i ( dg ) v L Q ω = − ρ T ∗ M ( d f )( g ) − ρ T ∗ M ( dg )( f ) and L ( d f ) v i ( dg ) v L Q ω = d ρ T ∗ M ( d f )( g ) − [ d f , dg ] T ∗ M . Luca Vitagliano VB Valued Forms on N Q -manifolds 13 / 32

Forms on N -Manifolds 1-Forms on Degree One N Q -Manifolds 2-Forms on Degree One N Q -Manifolds Higher Forms on Degree One N Q -Manifolds Outline Forms on N -Manifolds 1 1-Forms on Degree One N Q -Manifolds 2 2-Forms on Degree One N Q -Manifolds 3 Higher Forms on Degree One N Q -Manifolds 4 Luca Vitagliano VB Valued Forms on N Q -manifolds 14 / 32