Quantization of group-valued moment maps I Eckhard Meinrenken June - PowerPoint PPT Presentation

Quantization of group-valued moment maps I Eckhard Meinrenken June 2, 2011 Eckhard Meinrenken Quantization of group-valued moment maps I

Motivation: Moduli spaces of flat connections G a compact simply connected Lie group, · invariant inner product on g = Lie( G ). Σ = M (Σ) = { A ∈ Ω 1 (Σ , g ) | d A + 1 2 [ A , A ] = 0 } gauge transformations Eckhard Meinrenken Quantization of group-valued moment maps I

Motivation: Moduli spaces of flat connections G a compact simply connected Lie group, · invariant inner product on g = Lie( G ). Σ = M (Σ) = { A ∈ Ω 1 (Σ , g ) | d A + 1 2 [ A , A ] = 0 } gauge transformations This is a symplectic manifold ! Eckhard Meinrenken Quantization of group-valued moment maps I

Motivation: Moduli spaces of flat connections G a compact simply connected Lie group, · invariant inner product on g = Lie( G ). Σ = M (Σ) = { A ∈ Ω 1 (Σ , g ) | d A + 1 2 [ A , A ] = 0 } gauge transformations This is a symplectic manifold ! (with singularities) . Eckhard Meinrenken Quantization of group-valued moment maps I

Motivation: Moduli Spaces of flat connections Construction of symplectic form, after Atiyah-Bott A = Ω 1 (Σ , g ) carries symplectic form ω ( a , b ) = � Σ a · b . Eckhard Meinrenken Quantization of group-valued moment maps I

Motivation: Moduli Spaces of flat connections Construction of symplectic form, after Atiyah-Bott A = Ω 1 (Σ , g ) carries symplectic form ω ( a , b ) = � Σ a · b . C ∞ (Σ , G ) acts by gauge action, g . A = Ad g ( A ) − d g g − 1 , Eckhard Meinrenken Quantization of group-valued moment maps I

Motivation: Moduli Spaces of flat connections Construction of symplectic form, after Atiyah-Bott A = Ω 1 (Σ , g ) carries symplectic form ω ( a , b ) = � Σ a · b . C ∞ (Σ , G ) acts by gauge action, g . A = Ad g ( A ) − d g g − 1 , This action is Hamiltonian with moment map curv: A �→ d A + 1 2 [ A , A ] Eckhard Meinrenken Quantization of group-valued moment maps I

Motivation: Moduli Spaces of flat connections Construction of symplectic form, after Atiyah-Bott A = Ω 1 (Σ , g ) carries symplectic form ω ( a , b ) = � Σ a · b . C ∞ (Σ , G ) acts by gauge action, g . A = Ad g ( A ) − d g g − 1 , This action is Hamiltonian with moment map curv: A �→ d A + 1 2 [ A , A ] Moduli space is symplectic quotient M (Σ) = curv − 1 (0) / C ∞ (Σ , G ) . Eckhard Meinrenken Quantization of group-valued moment maps I

Motivation: Moduli Spaces of flat connections Holonomy description of the moduli space M (Σ) = Hom( π 1 (Σ) , G ) / G = Φ − 1 ( e ) / G where Φ: G 2 g → G (with g the genus of Σ) is the map g � a i b i a − 1 b − 1 Φ( a 1 , b 1 , . . . , a g , b g ) = . i i i =1 b 2 a 1 a 2 b 1 a 2 b 1 b 2 a 1 a 1 b 2 a 2 b 1 Eckhard Meinrenken Quantization of group-valued moment maps I

Motivation: Moduli Spaces of flat connections Holonomy description of the moduli space M (Σ) = Hom( π 1 (Σ) , G ) / G = Φ − 1 ( e ) / G where Φ: G 2 g → G (with g the genus of Σ) is the map g � a i b i a − 1 b − 1 Φ( a 1 , b 1 , . . . , a g , b g ) = . i i i =1 b 2 a 1 a 2 b 1 a 2 b 1 b 2 a 1 a 1 b 2 a 2 b 1 We’d like to view Φ as a moment map, and Φ − 1 ( e ) / G as a symplectic quotient! Eckhard Meinrenken Quantization of group-valued moment maps I

Group-valued moment maps θ L = g − 1 d g ∈ Ω 1 ( G , g ) left-Maurer-Cartan form θ R = d gg − 1 ∈ Ω 1 ( G , g ) right Maurer-Cartan form 12 [ θ L , θ L ] · θ L ∈ Ω 3 ( G ) η = 1 Cartan 3-form Definition (Alekseev-Malkin-M.) A q-Hamiltonian G -space ( M , ω, Φ) is a G -manifold M , with ω ∈ Ω 2 ( M ) G and Φ ∈ C ∞ ( M , G ) G , satisfying 2 Φ ∗ ( θ L + θ R ) · ξ , 1 ι ( ξ M ) ω = − 1 2 d ω = − Φ ∗ η , 3 ker( ω ) ∩ ker(dΦ) = 0. Eckhard Meinrenken Quantization of group-valued moment maps I

Comparison Hamiltonian G -space Φ: M → g ∗ 1 ι ( ξ M ) ω = − d � Φ , ξ � , 2 d ω = 0, 3 ker( ω ) = 0. q-Hamiltonian G -space Φ: M → G 2 Φ ∗ ( θ L + θ R ) · ξ , 1 ι ( ξ M ) ω = − 1 2 d ω = − Φ ∗ η , 3 ker( ω ) ∩ ker(dΦ) = 0. Eckhard Meinrenken Quantization of group-valued moment maps I

Examples: Coadjoint orbits, conjugacy classes Example → g ∗ are Hamiltonian G -spaces Co-adjoint orbits Φ: O ֒ ω ( ξ O , ξ ′ O ) µ = � µ, [ ξ, ξ ′ ] � Example Conjugacy classes Φ: C ֒ → G are q-Hamiltonian G -spaces ω ( ξ C , ξ ′ C ) a = 1 2 (Ad a − Ad a − 1 ) ξ · ξ ′ Eckhard Meinrenken Quantization of group-valued moment maps I

Examples; Cotangent bundle, double Example = G × g ∗ (with cotangent lift of Cotangent bundle T ∗ G ∼ conjugation action) is Hamiltonian G -space with Φ( g , µ ) = Ad g ( µ ) − µ Example The double D ( G ) = G × G is a q-Hamiltonian G -space with Φ( a , b ) = aba − 1 b − 1 Eckhard Meinrenken Quantization of group-valued moment maps I

Examples: Planes and spheres Example Even-dimensional plane C n = R 2 n is Hamiltonian U( n )-space. Example Even-dimensional sphere S 2 n is a q-Hamiltonian U( n )-space (Hurtubise-Jeffrey-Sjamaar). Eckhard Meinrenken Quantization of group-valued moment maps I

Examples: Planes and spheres Example Even-dimensional plane C n = R 2 n is Hamiltonian U( n )-space. Example Even-dimensional sphere S 2 n is a q-Hamiltonian U( n )-space (Hurtubise-Jeffrey-Sjamaar). Similar examples with G = Sp( n ), and M = H P ( n ) resp. H n (Eshmatov). Eckhard Meinrenken Quantization of group-valued moment maps I

Basic constructions: Products Products: If ( M 1 , ω 1 , Φ 1 ) , ( M 2 , ω 2 , Φ 2 ) are q-Hamiltonian G -spaces then so is 1 θ L · Φ ∗ ( M 1 × M 2 , ω 1 + ω 2 + 1 2 Φ ∗ 2 θ R , Φ 1 Φ 2 ) . Eckhard Meinrenken Quantization of group-valued moment maps I

Basic constructions: Products Products: If ( M 1 , ω 1 , Φ 1 ) , ( M 2 , ω 2 , Φ 2 ) are q-Hamiltonian G -spaces then so is 1 θ L · Φ ∗ ( M 1 × M 2 , ω 1 + ω 2 + 1 2 Φ ∗ 2 θ R , Φ 1 Φ 2 ) . Example For instance, D ( G ) g = G 2 g is a q-Hamiltonian G -space with moment map g � a i b i a − 1 b − 1 Φ( a 1 , b 1 , . . . , a g , b g ) = . i i i =1 Eckhard Meinrenken Quantization of group-valued moment maps I

Basic constructions: Reduction Reduction: If ( M , ω, Φ) is a q-Hamiltonian G -space then the symplectic quotient / G := Φ − 1 ( e ) / G M / is a symplectic manifold. with singularities Eckhard Meinrenken Quantization of group-valued moment maps I

Basic constructions: Reduction Reduction: If ( M , ω, Φ) is a q-Hamiltonian G -space then the symplectic quotient C 1 C 2 C 3 / G := Φ − 1 ( e ) / G M / is a symplectic manifold. with singularities Example (and Theorem) The symplectic quotient G 2 g × C 1 × · · · × C r / / G = M (Σ r g ; C 1 , . . . , C r ) is the moduli space of flat G -bundles over a surface with boundary, with boundary holonomies in prescribed conjugacy classes. Eckhard Meinrenken Quantization of group-valued moment maps I

Notation: Weyl chambers and Weyl alcoves Notation G compact and simply connected (e.g. G = SU( n )), T a maximal torus in G , t = Lie( T ), t + ∼ = t fundamental Weyl chamber, A ⊂ t + ⊂ t fundamental Weyl alcove t + A 0 0 { ξ | ker( e ad ξ − 1) = t } { ξ | ker(ad ξ ) = t } Eckhard Meinrenken Quantization of group-valued moment maps I

Moment polytope For every ν ∈ g ∗ there is a unique µ ∈ t ∗ + with ν ∈ G .µ . Theorem (Atiyah, Guillemin-Sternberg, Kirwan) For a compact connected Hamiltonan G-space ( M , ω, Φ) , the set ∆( M ) = { µ ∈ t ∗ + | µ ∈ Φ( M ) } is a convex polytope. Eckhard Meinrenken Quantization of group-valued moment maps I

Moment polytope For every ν ∈ g ∗ there is a unique µ ∈ t ∗ + with ν ∈ G .µ . Theorem (Atiyah, Guillemin-Sternberg, Kirwan) For a compact connected Hamiltonan G-space ( M , ω, Φ) , the set ∆( M ) = { µ ∈ t ∗ + | µ ∈ Φ( M ) } is a convex polytope. For every g ∈ G there is a unique ξ ∈ A with g ∈ G . exp( ξ ). Theorem (M-Woodward) For any connected q-Hamiltonian G-space ( M , ω, Φ) , the set ∆( M ) = { ξ ∈ A | exp( ξ ) ∈ Φ( M ) } is a convex polytope. Eckhard Meinrenken Quantization of group-valued moment maps I

Application to eigenvalue problems The Hamiltonian convexity theorem gives eigenvalue inequalities for sums of Hermitian matrices with prescribed eigenvalues. (Schur-Horn problem). Eckhard Meinrenken Quantization of group-valued moment maps I

Application to eigenvalue problems The Hamiltonian convexity theorem gives eigenvalue inequalities for sums of Hermitian matrices with prescribed eigenvalues. (Schur-Horn problem). The q-Hamiltonian convexity theorem gives eigenvalue inequalities for products of unitary matrices with prescribed eigenvalues. Eckhard Meinrenken Quantization of group-valued moment maps I

Examples of moment polytopes (due to C. Woodward) A multiplicity-free Hamiltonian A multiplicity-free q-Hamiltonian SU(3)-space SU(3)-space Eckhard Meinrenken Quantization of group-valued moment maps I