Classical integrability Calogero-Moser system on an elliptic curve Marsden- Weinstein reduction Reduction for the elliptic case Timo Kluck Mathematisch Instituut, Universiteit Utrecht December 17, 2012 1
Classical integrability Marsden- Main point Weinstein reduction We will see a complicated system of interacting particles that Reduction for the elliptic case can be solved because it corresponds to a simple motion in the space of holomorphic principal SL ( N ) -bundles over an elliptic curve. 2
Outline Classical integrability Marsden- Classical integrability Weinstein reduction Reduction for the elliptic case Marsden-Weinstein reduction Reduction for the elliptic case 3
Phase space Definition Classical A phase space or a symplectic manifold is a manifold M integrability together with a non-degenerate, closed 2-form ω . Marsden- Weinstein reduction Definition Reduction for the elliptic case For any function f ∈ C ∞ ( M ) on a symplectic manifold, there is an associated vector field v f given by ω − 1 ( d f ) where we regard ω as a bundle map TM → T ∨ M . We also write � f , · � for this vector field. 4
Phase space (ctd) Classical Example integrability Marsden- Any cotangent bundle M = T ∨ X is a symplectic manifold. Weinstein ◮ q 1 , · · · , q n and p 1 , · · · , p n coordinates representing a point reduction Reduction for � � p i d q i , q the elliptic case ◮ Symplectic form: − ∂ { q i , ·} = ∂ p i ∂ { p i , ·} = ∂ q i 5
Liouville integrability Classical Definition integrability A dynamical system is a phase space together with a Marsden- Weinstein distinguished function H ∈ C ∞ ( M ) called the Hamiltonian . The reduction solutions to the dynamical system are the flow lines of { H , ·} . Reduction for the elliptic case 6
Liouville integrability Classical Definition integrability A dynamical system is a phase space together with a Marsden- Weinstein distinguished function H ∈ C ∞ ( M ) called the Hamiltonian . The reduction solutions to the dynamical system are the flow lines of { H , ·} . Reduction for the elliptic case Definition Let dim M = 2 N . A dynamical system on M is (Liouville) integrable if there are functions H 1 , · · · , H N such that ◮ { H i , H j } = 0 (they are in involution ) ◮ On a dense open subset: d H 1 ∧ · · · ∧ d H N � 0 ◮ H = f ( H 1 , · · · , H N ) 6
Liouville integrability (ctd) Classical integrability Marsden- Weinstein reduction Reduction for The H i are called Hamiltonians. Their flows are symmetries of the elliptic case the dynamical system. 7
Planetary motion Example , with coordinates x , y ∈ R 3 and cotangent M = T ∨ � R 3 × R 3 � Classical integrability coordinates p , r . We interpret x , y as planet positions and p , r Marsden- as momenta. Weinstein reduction d p i ∧ d x i + d r i ∧ d y i ω Reduction for = the elliptic case 1 1 p 2 + q 2 � � H + = 2 | x − y | This is integrable with H k p k + r k k = 1 , 2 , 3 = H 4 (( p − r ) × ( x − y )) 1 = � 2 � � H 5 � ( p − r ) × ( x − y ) � � = H 6 H = 8
Outline Classical integrability Marsden- Classical integrability Weinstein reduction Reduction for the elliptic case Marsden-Weinstein reduction Reduction for the elliptic case 9
Rational Calogero-Moser system Example M ⊆ T ∨ R N , interpreted as positions and momenta of N Classical particles in 1 dimension with center-of-mass set to 0. integrability Marsden- Weinstein N ϵ 2 H = 1 reduction � � p 2 i − 2 ( q i − q j ) 2 Reduction for the elliptic case i = 1 i < j 10
Rational Calogero-Moser system Example M ⊆ T ∨ R N , interpreted as positions and momenta of N Classical particles in 1 dimension with center-of-mass set to 0. integrability Marsden- Weinstein N ϵ 2 H = 1 reduction � � p 2 i − 2 ( q i − q j ) 2 Reduction for the elliptic case i = 1 i < j Theorem (Calogero) This is an 2 ( N − 1 ) -dimensional integrable system, with Hamiltonians given by H k = Tr L k , where L is the traceless matrix ϵ p 1 q i − q j ... L = ϵ p N q i − q j 10
Rational Calogero-Moser system (ctd) Classical So some of these Hamiltonians are: integrability Marsden- N Weinstein � reduction H 1 p i = 0 = Reduction for i = 1 the elliptic case N ϵ 2 � � p 2 H 2 ( = 2 H ) i − = ( q i − q j ) 2 i = 1 i � j N ϵ 2 ϵ 3 � � � p 3 H 3 p i ( q i − q j ) 2 + i − = ( q i − q j )( q j − q k )( q k − q i ) i = 1 i � j i , j , k distinct 11
Rational Calogero-Moser system (ctd) Classical integrability Marsden- Weinstein Question reduction Where do all these symmetries / conserved quantities come Reduction for the elliptic case from? 12
Rational Calogero-Moser system (ctd) Classical integrability Marsden- Weinstein Question reduction Where do all these symmetries / conserved quantities come Reduction for the elliptic case from? “Answer” They exist because the motion is very simple (linear) in the matrix space. 12
Linear motion in a matrix space Classical integrability ◮ G = SL ( N ) , g = sl ( N ) Marsden- Weinstein reduction ◮ Phase space M = T ∨ g = g × g using Killing pairing Reduction for ◮ Hamiltonian H ( P , Q ) = 1 2 � P , P � the elliptic case 13
Linear motion in a matrix space Classical integrability ◮ G = SL ( N ) , g = sl ( N ) Marsden- Weinstein reduction ◮ Phase space M = T ∨ g = g × g using Killing pairing Reduction for ◮ Hamiltonian H ( P , Q ) = 1 2 � P , P � the elliptic case ◮ Solution for given initial value ( P 0 , Q 0 ) : P ( t ) P 0 = Q ( t ) Q 0 + tP 0 = 13
Linear motion in a matrix space (ctd) Classical ◮ Symmetric under adjoint action of G on g : integrability Marsden- • H and ω invariant under conjugation Weinstein reduction P , Q �→ gPg − 1 , gQg − 1 Reduction for • Time evolution commutes with G -action the elliptic case ◮ Conserved quantities in involution: H k = Tr P k ◮ Also invariant under conjugation n 2 − 1 � � ◮ But too few: dim M = 2 > 2 n 14
Marsden-Weinstein reduction Classical integrability Idea Marsden- Weinstein ◮ Since everything is G -invariant, we can quotient out by it. reduction Reduction for ◮ Hopefully, this reduces the dimension sufficiently to end up the elliptic case with an integrable system. 15
Marsden-Weinstein reduction Classical integrability Idea Marsden- Weinstein ◮ Since everything is G -invariant, we can quotient out by it. reduction Reduction for ◮ Hopefully, this reduces the dimension sufficiently to end up the elliptic case with an integrable system. ◮ But we also need to keep a non-degenerate symplectic form: • If we quotient out a tangent vector ξ ∈ TM , then we should also quotient out its image ω ( ξ ) ∈ T ∨ M . Dually, that means restricting to a submanifold. 15
Marsden-Weinstein reduction (ctd) Classical Definition integrability A group action of G on M is Hamiltonian if its infinitesimal Marsden- Weinstein vector fields v ξ for ξ ∈ g are of the form reduction Reduction for v ξ = { f ξ , ·} the elliptic case 16
Marsden-Weinstein reduction (ctd) Classical Definition integrability A group action of G on M is Hamiltonian if its infinitesimal Marsden- Weinstein vector fields v ξ for ξ ∈ g are of the form reduction Reduction for v ξ = { f ξ , ·} the elliptic case Definition A Hamiltonian group action is generated by a moment map µ : M → g ∨ if f ξ = � µ , ξ � and if µ is G -equivariant. 16
Marsden-Weinstein reduction (ctd) Classical integrability Marsden- Weinstein reduction Reduction for Theorem (Marsden, Weinstein) the elliptic case If µ 0 ∈ g ∨ is a regular value of µ , then the space µ − 1 ( µ 0 ) / G µ 0 is a symplectic manifold. 17
Reduction of linear motion Fact µ ( P , Q ) = [ P , Q ] ∈ g � g ∨ is a moment map for the conjugation action. Classical integrability Theorem Marsden- Weinstein Pick the following regular value µ 0 ∈ g � g ∨ : reduction Reduction for the elliptic case 1 · · · 1 1 0 . . ... ... . . + ϵ µ 0 = − ϵ . . 1 1 0 1 · · · Then p , q parametrize the G µ 0 -equivalence classes in µ − 1 ( µ 0 ) by ϵ p 1 q 1 0 q i − q j ... ... P ( p , q ) , Q ( p , q ) , = ϵ p N 0 q N q i − q j and they are canonical coordinates on µ − 1 ( µ 0 ) / G µ 0 . 18
Recap Classical integrability ◮ Linear motion Marsden- Weinstein t �→ P 0 , Q 0 + tP 0 reduction has N − 1 conserved quantities in an 2 ( N 2 − 1 ) dimensional Reduction for the elliptic case phase space. 19
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