Semitoric systems with multi-pinched fibers Xiudi Tang University - PowerPoint PPT Presentation

Semitoric systems with multi-pinched fibers Semitoric systems with multi-pinched fibers Xiudi Tang University of Toronto joint with Joseph Palmer and ´ Alvaro Pelayo arXiv:1909.03501 Workshop on Lie Theory and Integrable Systems in Symplectic and Poisson Geometry June 7, 2020

Semitoric systems with multi-pinched fibers Outline 1 Integrable systems

Semitoric systems with multi-pinched fibers Outline 1 Integrable systems 2 Toric systems Definition Classification

Semitoric systems with multi-pinched fibers Outline 1 Integrable systems 2 Toric systems Definition Classification 3 Semitoric systems Examples Definition Classification Invariants

Semitoric systems with multi-pinched fibers Outline 1 Integrable systems 2 Toric systems Definition Classification 3 Semitoric systems Examples Definition Classification Invariants 4 Historical review

Semitoric systems with multi-pinched fibers Integrable systems Section 1 Integrable systems

Semitoric systems with multi-pinched fibers Integrable systems Integrable systems Definition An integrable system consists of • a symplectic symplectic manifold ( M 2 n , ω ); • a Hamiltonian t n -action ρ : t n → X ( M ); • a momentum map µ : M → ( t n ) ∗ ≃ R n ; • so that the critical points of µ forms a null set.

Semitoric systems with multi-pinched fibers Integrable systems Integrable systems Definition An integrable system consists of • a symplectic symplectic manifold ( M 2 n , ω ); • a Hamiltonian t n -action ρ : t n → X ( M ); • a momentum map µ : M → ( t n ) ∗ ≃ R n ; • so that the critical points of µ forms a null set. For any integrable system ( M , ω, ρ, µ ): • X � µ, a � = ρ ( a ) is parallel to fibers of µ for a ∈ t n ; • ρ ( a ) and ρ ( b ) commute for a , b ∈ t n ; • any regular fiber of µ is Lagrangian.

Semitoric systems with multi-pinched fibers Integrable systems Example 1 Harmonic oscillator y ( R 2 , ω ) µ ( x , y ) = x 2 + y 2 2 x

Semitoric systems with multi-pinched fibers Integrable systems Example 2 2-sphere z ( S 2 , ω ) µ ( x , y , z ) = z y x

Semitoric systems with multi-pinched fibers Integrable systems Example 2 2-sphere z ( S 2 , ω ) µ ( x , y , z ) = z y x Remark The Hamiltonian vector field act as a circle action.

Semitoric systems with multi-pinched fibers Toric systems Section 2 Toric systems

Semitoric systems with multi-pinched fibers Toric systems Definition Toric systems Remark In the examples of the harmonic oscillator and the 2-sphere, the Hamiltonian vector fields are periodic, or generates an S 1 -action. Those are 2D examples of toric integrable systems where we have torus actions.

Semitoric systems with multi-pinched fibers Toric systems Definition Toric systems Remark In the examples of the harmonic oscillator and the 2-sphere, the Hamiltonian vector fields are periodic, or generates an S 1 -action. Those are 2D examples of toric integrable systems where we have torus actions. Definition An integrable system ( M , ω, ρ, µ ) is toric if ρ integrates to a Lie group ρ : T n → Ham( M , ω ). action ˜

Semitoric systems with multi-pinched fibers Toric systems Definition Example 3 ( S 2 × S 2 , ω S 2 ⊕ ω S 2 ) z 1 z 2 µ = ( z 1 , z 2 ) × y 1 y 2 x 1 x 2

Semitoric systems with multi-pinched fibers Toric systems Definition Example 4 ( C P 2 , ω FS ) C P 2 = { [ z 0 : z 1 : z 2 ] } µ ([ z 0 : z 1 : z 2 ]) = | z 1 | 2 | z 2 | 2 � � | z 0 | 2+ | z 1 | 2+ | z 2 | 2 , | z 0 | 2+ | z 1 | 2+ | z 2 | 2 µ ( C P 2 , ω FS )

� � � Semitoric systems with multi-pinched fibers Toric systems Classification Toric systems Isomorphisms of toric systems A toric system ( M 1 , ω 1 , ρ 1 , µ 1 ) is isomorphic to ( M 2 , ω 2 , ρ 2 , µ 2 ) if ϕ ( M 1 , ω 1 ) ( M 2 , ω 2 ) µ 1 µ 2 G � R 2 R 2 where ϕ is a symplectomorphism and G is a diffeomorphism.

Semitoric systems with multi-pinched fibers Toric systems Classification Toric systems: classification Theorem (Atiyah, Guillemin–Sternberg, Delzant 1980s) � � � � compact toric systems − → Delzant polytopes � � isomorphisms AGL ( n , R ) [( M , ω, ρ, µ )] �− → [ µ ( M )]

Semitoric systems with multi-pinched fibers Toric systems Classification Toric systems: classification Theorem (Atiyah, Guillemin–Sternberg, Delzant 1980s) � � � � compact toric systems − → Delzant polytopes � � isomorphisms AGL ( n , R ) [( M , ω, ρ, µ )] �− → [ µ ( M )] Delzant polytopes � � − 1 0 Delzant polytope: − 1 − 1 every corner is locally the standard corner up to the action of AGL ( n , Z ).

Semitoric systems with multi-pinched fibers Semitoric systems Section 3 Semitoric systems

Semitoric systems with multi-pinched fibers Semitoric systems Examples Generalized coupled angular momentum Examples: Hohloch–Palmer 2018 Let M = S 2 × S 2 with Cartisian coordinates ( x 1 , y 1 , z 1 , x 2 , y 2 , z 2 ) and ω = R 1 ω S 2 ⊕ R 2 ω S 2 where 0 < R 1 < R 2 . Consider integrable systems ( M , ω, ρ s , µ s = ( J , H s )) with a parameter s ∈ [0 , 1] where J = R 1 z 1 + R 2 z 2 , H s = (1 − s ) 2 z 1 + s 2 z 2 + 2 s (1 − s )( x 1 y 1 + x 2 y 2 ) . The systems behave differently as R 1 , R 2 , and s varies.

Semitoric systems with multi-pinched fibers Semitoric systems Examples Example 5 = focus-focus singularity R 1 = R 2 = 1, s − 1 2 is small positive ( S 2 × S 2 , ω S 2 ⊕ ω S 2 ) µ s = ( J , H s ) H s J R 2

Semitoric systems with multi-pinched fibers Semitoric systems Examples Example 6 = focus-focus singularity R 1 = R 2 = 1, s = 1 2 ( S 2 × S 2 , ω S 2 ⊕ ω S 2 ) µ s = ( J , H s ) H s J R 2

Semitoric systems with multi-pinched fibers Semitoric systems Examples Semitoric vs toric Comparing with toric systems Examples 5 or 6 a toric system

Semitoric systems with multi-pinched fibers Semitoric systems Examples Semitoric vs toric Comparing with toric systems Examples 5 or 6 a toric system X J is periodic, X H is not both X J and X H are periodic

Semitoric systems with multi-pinched fibers Semitoric systems Examples Semitoric vs toric Comparing with toric systems Examples 5 or 6 a toric system X J is periodic, X H is not both X J and X H are periodic has focus-focus singularity all singularities are elliptic

Semitoric systems with multi-pinched fibers Semitoric systems Examples Semitoric vs toric Comparing with toric systems Examples 5 or 6 a toric system X J is periodic, X H is not both X J and X H are periodic has focus-focus singularity all singularities are elliptic some fibers are pinched tori all fibers are tori

Semitoric systems with multi-pinched fibers Semitoric systems Examples Semitoric vs toric Comparing with toric systems Examples 5 or 6 a toric system X J is periodic, X H is not both X J and X H are periodic has focus-focus singularity all singularities are elliptic some fibers are pinched tori all fibers are tori no global action-angle coordinates has global action-angle coord.

Semitoric systems with multi-pinched fibers Semitoric systems Examples Semitoric vs toric Comparing with toric systems Examples 5 or 6 a toric system X J is periodic, X H is not both X J and X H are periodic has focus-focus singularity all singularities are elliptic some fibers are pinched tori all fibers are tori no global action-angle coordinates has global action-angle coord. image is a curvilinear polygon image is a polygon

Semitoric systems with multi-pinched fibers Semitoric systems Examples Semitoric vs toric Comparing with toric systems Examples 5 or 6 a toric system X J is periodic, X H is not both X J and X H are periodic has focus-focus singularity all singularities are elliptic some fibers are pinched tori all fibers are tori no global action-angle coordinates has global action-angle coord. image is a curvilinear polygon image is a polygon This is an example of a semitoric system.

Semitoric systems with multi-pinched fibers Semitoric systems Definition Semitoric systems Definition A 4D integrable system ( M 4 , ω, ρ, µ = ( J , H )) is semitoric if ρ integrates ρ : S 1 × R → Ham( M , ω ), J is proper, and all to a Lie group action ˜ singularities are of either elliptic or focus-focus type.

Semitoric systems with multi-pinched fibers Semitoric systems Definition Semitoric systems Definition A 4D integrable system ( M 4 , ω, ρ, µ = ( J , H )) is semitoric if ρ integrates ρ : S 1 × R → Ham( M , ω ), J is proper, and all to a Lie group action ˜ singularities are of either elliptic or focus-focus type. Theorem (Eliasson 1984) For an integrable system ( M , ω, ρ, µ ) in a neighborhood of any nondenerate singular point of µ , there are symplectic coordinates x i , y i and q = ( q 1 , . . . , q n ): R 2 n → R n where q i can be • regular: q i = y i ; • elliptic: q i = 1 2 ( x 2 i + y 2 i ) ; • hyperbolic: q i = x i y i ; • focus-focus: q i − 1 = x i − 1 y i − x i y i − 1 and q i = x i − 1 y i − 1 + x i y i ; such that q i Poisson commutes with components of µ .