Symmetries of b-manifolds and their generalizations Eva Miranda UPC-Barcelona Exterior differential systems and Lie theory Fields Institute, Toronto Eva Miranda (UPC) Exterior differential systems and Lie theory December 10, 2013 1 / 19
Outline Toric Symplectic manifolds 1 b-Symplectic manifolds 2 A Delzant theorem for b-symplectic manifolds 3 Generalizations 4 Eva Miranda (UPC) Exterior differential systems and Lie theory December 10, 2013 1 / 19
Moment maps in Symplectic Geometry Definition (Symplectic case) Let G be a compact Lie group acting symplectically on ( M, ω ) . The action is Hamiltonian if there exists an equivariant map µ : M → g ∗ such that for each element X ∈ g , dµ X = ι X # ω, (1) with µ X = < µ, X > . The map µ is called the moment map . Eva Miranda (UPC) Exterior differential systems and Lie theory December 10, 2013 2 / 19
Toric symplectic manifolds Theorem (Delzant) Toric manifolds are classified by Delzant’s polytopes. The bijective correspondence between these two sets is given by the image of the { toric manifolds } − → { Delzant polytopes } moment map: ( M 2 n , ω, T n , F ) − → F ( M ) R µ = h µ CP 2 ( t 1 , t 2 ) · [ z 0 : z 1 : z 2 ] = [ z 0 : e it 1 z 1 : e it 2 z 2 ] Eva Miranda (UPC) Exterior differential systems and Lie theory December 10, 2013 3 / 19
Adding singularities in the picture ( S 2 , 1 h dh ∧ dθ ) � ( S 2 , h ∂ ∂h ∧ ∂ ∂θ ) . We want to study generalizations of rotations on a sphere. µ = − log | h | Eva Miranda (UPC) Exterior differential systems and Lie theory December 10, 2013 4 / 19
b -Symplectic/ b -Poisson structures Definition Let ( M 2 n , Π) be an oriented Poisson manifold such that the map p ∈ M �→ (Π( p )) n ∈ Λ 2 n ( TM ) is transverse to the zero section, then Z = { p ∈ M | (Π( p )) n = 0 } is a hypersurface called the critical hypersurface and we say that Π is a Poisson b -structure on ( M, Z ) . Disclaimer b -symplectic manifolds =log-symplectic manifolds= b-log symplectic manifolds Symplectic foliation of a Poisson b-manifold The symplectic foliation has dense symplectic leaves and codimension 2 symplectic leaves whose union is Z . Eva Miranda (UPC) Exterior differential systems and Lie theory December 10, 2013 5 / 19
Examples: Dimension 2 Radko classified b-Poisson structures on compact oriented surfaces giving a list of invariants: Geometrical: The topology of S and the curves γ i where Π vanishes. Dynamical: The periods of the “modular vector field” along γ i . Measure: The regularized Liouville volume of S , V ǫ � h (Π) = | h | >ǫ ω Π for h a function vanishing linearly on the curves γ 1 , . . . , γ n . Figure: Two admissible vanishing curves (a) and (b) for Π ; the ones in (b’) is not admissible. Eva Miranda (UPC) Exterior differential systems and Lie theory December 10, 2013 6 / 19
Higher dimensions: Some compact examples. The product of ( R, π R ) a Radko compact surface and a ( S, π ) be a compact symplectic manifold is a b -Poisson manifold. Take ( N, π ) be a regular corank 1 Poisson manifold and let X be a Poisson vector field. Now consider the product S 1 × N with the bivector field Π = f ( θ ) ∂ ∂θ ∧ X + π. This is a b -Poisson manifold as long as, the function f vanishes linearly. 1 The vector field X is transverse to the symplectic leaves of N . 2 We then have as many copies of N as zeroes of f . Eva Miranda (UPC) Exterior differential systems and Lie theory December 10, 2013 7 / 19
Poisson Geometry of the critical hypersurface This last example is semilocally the canonical picture of a b -Poisson structure. 1 The critical hypersurface Z has an induced regular Poisson structure of corank 1 . 2 There exists a Poisson vector field transverse to the symplectic foliation induced on Z . 3 Given a regular corank 1 Poisson structure, there exists a semilocal extension to a b -Poisson structure if an only if two foliated cohomology classes of the symplectic foliation vanish. Eva Miranda (UPC) Exterior differential systems and Lie theory December 10, 2013 8 / 19
The singular hypersurface Theorem (Guillemin-M.-Pires) If L contains a compact leaf L , then M is the mapping torus of the symplectomorphism φ : L → L determined by the flow of a Poisson vector field v transverse to the symplectic foliation. Eva Miranda (UPC) Exterior differential systems and Lie theory December 10, 2013 9 / 19
A dual approach... b -Poisson structures can be seen as symplectic structures modeled over a Lie algebroid (the b-tangent bundle). A vector field v is a b -vector field if v p ∈ T p Z for all p ∈ Z . The b -tangent bundle b TM is defined by � b-vector fields � Γ( U, b TM ) = on ( U, U ∩ Z ) The b -cotangent bundle b T ∗ M is ( b TM ) ∗ . Sections of Λ p ( b T ∗ M ) are b -forms , b Ω p ( M ) .The standard differential extends to d : b Ω p ( M ) → b Ω p +1 ( M ) A b -symplectic form is a closed, nondegenerate, b -form of degree 2. This dual point of view, allows to prove a b -Darboux theorem and semilocal forms via an adaptation of Moser’s path method since we can play the same tricks as in the symplectic case. Eva Miranda (UPC) Exterior differential systems and Lie theory December 10, 2013 10 / 19
Example ( S 2 , ω = dh h ∧ dθ ) , with coordinates h ∈ [ − 1 , 1] and θ ∈ [0 , 2 π ] . The critical hypersurface Z is the equator, given by h = 0 . For the usual S 1 -action by rotations, the moment map is µ ( h, θ ) = log | h | . 1 µ = − log | h | Eva Miranda (UPC) Exterior differential systems and Lie theory December 10, 2013 10 / 19
Example dθ 1 On ( T 2 , ω = sin θ 1 ∧ dθ 2 ) , with coordinates: θ 1 , θ 2 ∈ [0 , 2 π ] . The critical hypersurface Z is the union of two disjoint circles, given by θ 1 = 0 and θ 1 = π . Consider rotations in θ 2 the moment map is µ : T 2 → R 2 is given by µ ( θ 1 , θ 2 ) = log � � � tan θ 1 � . � � 2 µ Eva Miranda (UPC) Exterior differential systems and Lie theory December 10, 2013 10 / 19
More generally... m 1 m 1 m 1 m 1 m 1 m 1 m 1 m 1 m 1 m 1 m 1 m 1 m 1 m 3 m 3 m 3 m 3 m 3 m 3 m 3 m 3 m 3 m 3 m 3 m 3 m 3 m 5 m 5 m 5 m 5 m 5 m 5 m 5 m 5 m 5 m 5 m 5 m 5 m 5 m 2 m 2 m 2 m 2 m 2 m 2 m 2 m 2 m 2 m 2 m 2 m 2 m 2 m 4 m 4 m 4 m 4 m 4 m 4 m 4 m 4 m 4 m 4 m 4 m 4 m 4 m 1 m 1 m 1 m 1 m 1 m 1 m 1 m 1 m 1 m 1 m 1 m 1 m 1 m 3 m 3 m 3 m 3 m 3 m 3 m 3 m 3 m 3 m 3 m 3 m 3 m 3 m 2 m 2 m 2 m 2 m 2 m 2 m 2 m 2 m 2 m 2 m 2 m 2 m 2 m 4 m 4 m 4 m 4 m 4 m 4 m 4 m 4 m 4 m 4 m 4 m 4 m 4 Eva Miranda (UPC) Exterior differential systems and Lie theory December 10, 2013 10 / 19
b -toric actions Definition An action of T n on a b -symplectic manifold ( M, ω ) is a Hamiltonian action if: for each X ∈ t , the b -one-form ι X # ω is exact ( i.e., has a primitive H X ∈ b C ∞ ( M ) ) for any X, Y ∈ t , we have ω ( X # , Y # ) = 0 . The action is toric if it is effective and the dimension of the torus is half the dimension of M . b-moment map µ such that < µ ( p ) , X > = H X ( p ) , but we will have to allow µ ( p ) to take values of ±∞ , so we need to extend the pairing to accommodate that. Eva Miranda (UPC) Exterior differential systems and Lie theory December 10, 2013 11 / 19
The b -line The b -line is constructed by gluing copies of the extended real line R := R ∪ {±∞} together in a zig-zag pattern and R > 0 -valued labels (“weights”) on the points at infinity to prescribe a smooth structure. wt( − 1) wt( − 1) wt( − 1) wt( − 1) wt( − 1) wt( − 1) wt( − 1) wt( − 1) wt( − 1) wt( − 1) wt( − 1) wt( − 1) wt( − 1) wt( − 1) wt( − 1) wt( − 1) wt( − 1) wt( − 1) wt( − 1) wt( − 1) wt( − 1) wt(1) wt(1) wt(1) wt(1) wt(1) wt(1) wt(1) wt(1) wt(1) wt(1) wt(1) wt(1) wt(1) wt(1) wt(1) wt(1) wt(1) wt(1) wt(1) wt(1) wt(1) wt(3) wt(3) wt(3) wt(3) wt(3) wt(3) wt(3) wt(3) wt(3) wt(3) wt(3) wt(3) wt(3) wt(3) wt(3) wt(3) wt(3) wt(3) wt(3) wt(3) wt(3) R . . . . . . x ˆ wt( − 2) wt( − 2) wt( − 2) wt( − 2) wt( − 2) wt( − 2) wt( − 2) wt( − 2) wt( − 2) wt( − 2) wt( − 2) wt( − 2) wt( − 2) wt( − 2) wt( − 2) wt( − 2) wt( − 2) wt( − 2) wt( − 2) wt( − 2) wt( − 2) wt(0) wt(0) wt(0) wt(0) wt(0) wt(0) wt(0) wt(0) wt(0) wt(0) wt(0) wt(0) wt(0) wt(0) wt(0) wt(0) wt(0) wt(0) wt(0) wt(0) wt(0) wt(2) wt(2) wt(2) wt(2) wt(2) wt(2) wt(2) wt(2) wt(2) wt(2) wt(2) wt(2) wt(2) wt(2) wt(2) wt(2) wt(2) wt(2) wt(2) wt(2) wt(2) Figure: A weighted b -line with I = Z The b -line with weight function wt is described as a topological space wt R ∼ b = ( Z × R ) / { ( a, ( − 1) a ∞ ) ∼ ( a + 1 , ( − 1) a ∞ ) } . The weights are by given by the modular periods associated to each connected component of Z . Eva Miranda (UPC) Exterior differential systems and Lie theory December 10, 2013 12 / 19
Adjacency graph and definition of b-moment map Theorem (Guillemin, M., Pires, Scott) Let ( M, Z, ω, T n ) be a b-symplectic manifold with an effective Hamiltonian toric action. For an appropriately-chosen b t ∗ or b t ∗ / � a � , there is a moment map µ : M → b t ∗ or µ : M → b t ∗ / � a � . Example c 0 c 0 c 0 c 0 c 0 c 0 c 0 c 0 c 0 c 0 c 0 c 0 c 0 c 0 c 0 c 0 c 0 c 0 c 0 c 0 c 0 µ c 1 c 1 c 1 c 1 c 1 c 1 c 1 c 1 c 1 c 1 c 1 c 1 c 1 c 1 c 1 c 1 c 1 c 1 c 1 c 1 c 1 Figure: The moment map µ surjects onto b t ∗ / � 2 � . Eva Miranda (UPC) Exterior differential systems and Lie theory December 10, 2013 13 / 19
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