Geometric Quantization and Models of Semisimple Lie Groups - PowerPoint PPT Presentation

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan Geometric Quantization and Models of Semisimple Lie Groups Meng-Kiat Chuah Department of Mathematics National Tsing Hua University Hsinchu, Taiwan Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan 1. Introduction G connected real semisimple Lie group. Construct its unitary representations. Geometric Quantization (Kostant 1970): G -invariant symplectic manifold � unitary G -representation on Hilbert space. Symplectic structure provides inner product. Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan 1. Introduction G connected real semisimple Lie group. Construct its unitary representations. Geometric Quantization (Kostant 1970): G -invariant symplectic manifold � unitary G -representation on Hilbert space. Symplectic structure provides inner product. Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan 1. Introduction G connected real semisimple Lie group. Construct its unitary representations. Geometric Quantization (Kostant 1970): G -invariant symplectic manifold � unitary G -representation on Hilbert space. Symplectic structure provides inner product. Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan 2. Symplectic Manifolds and Group Actions Symplectic Manifold: ( X, ω ) , ω closed nondegenerate 2-form. Examples: coadjoint orbit X ⊂ g ∗ , cotangent bundle X = T ∗ M . Suppose G acts on X . Given ξ ∈ g , define infinitesimal vector field ξ ♯ on X by ( ξ ♯ f )( x ) = d dt | t =0 f ( e tξ x ) for all f ∈ C ∞ ( X ) and x ∈ X . Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan 2. Symplectic Manifolds and Group Actions Symplectic Manifold: ( X, ω ) , ω closed nondegenerate 2-form. Examples: coadjoint orbit X ⊂ g ∗ , cotangent bundle X = T ∗ M . Suppose G acts on X . Given ξ ∈ g , define infinitesimal vector field ξ ♯ on X by ( ξ ♯ f )( x ) = d dt | t =0 f ( e tξ x ) for all f ∈ C ∞ ( X ) and x ∈ X . Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan 2. Symplectic Manifolds and Group Actions Symplectic Manifold: ( X, ω ) , ω closed nondegenerate 2-form. Examples: coadjoint orbit X ⊂ g ∗ , cotangent bundle X = T ∗ M . Suppose G acts on X . Given ξ ∈ g , define infinitesimal vector field ξ ♯ on X by ( ξ ♯ f )( x ) = d dt | t =0 f ( e tξ x ) for all f ∈ C ∞ ( X ) and x ∈ X . Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan 3. Moment Map Suppose G acts on symplectic manifold ( X, ω ) . → g ∗ be a G -equivariant map. Let Φ : X − For each ξ ∈ g , define (Φ , ξ ) ∈ C ∞ ( X ) by x �→ (Φ( x ) , ξ ) . Call Φ a moment map if for all ξ ∈ g , d (Φ , ξ ) = ω ( ξ ♯ ) . If G semisimple, moment map exists and is unique. If moment map exists, call the action Hamiltonian. Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan 3. Moment Map Suppose G acts on symplectic manifold ( X, ω ) . → g ∗ be a G -equivariant map. Let Φ : X − For each ξ ∈ g , define (Φ , ξ ) ∈ C ∞ ( X ) by x �→ (Φ( x ) , ξ ) . Call Φ a moment map if for all ξ ∈ g , d (Φ , ξ ) = ω ( ξ ♯ ) . If G semisimple, moment map exists and is unique. If moment map exists, call the action Hamiltonian. Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan 3. Moment Map Suppose G acts on symplectic manifold ( X, ω ) . → g ∗ be a G -equivariant map. Let Φ : X − For each ξ ∈ g , define (Φ , ξ ) ∈ C ∞ ( X ) by x �→ (Φ( x ) , ξ ) . Call Φ a moment map if for all ξ ∈ g , d (Φ , ξ ) = ω ( ξ ♯ ) . If G semisimple, moment map exists and is unique. If moment map exists, call the action Hamiltonian. Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan Examples If X is a coadjoint orbit of G , its moment map → g ∗ is the inclusion. X − If G = X = T 2 is 2-torus and ω is Haar measure, then action is not Hamiltonian. Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan 4. Symplectic Reduction Hamiltonian G -action on ( X, ω ) , moment map Φ : X − → g ∗ . G ξ stabilizer of ξ ∈ g ∗ , let X ξ = Φ − 1 ( ξ ) /G ξ . Suppose X ξ manifold (e.g. free action and G compact). π i − Φ − 1 ( ξ ) X ξ ← − → X Marsden, Weinstein 1974: symplectic form ω ξ on X ξ , π ∗ ω ξ = i ∗ ω Symplectic Reduction ( X, ω ) � ( X ξ , ω ξ ) Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan 4. Symplectic Reduction Hamiltonian G -action on ( X, ω ) , moment map Φ : X − → g ∗ . G ξ stabilizer of ξ ∈ g ∗ , let X ξ = Φ − 1 ( ξ ) /G ξ . Suppose X ξ manifold (e.g. free action and G compact). π i − Φ − 1 ( ξ ) X ξ ← − → X Marsden, Weinstein 1974: symplectic form ω ξ on X ξ , π ∗ ω ξ = i ∗ ω Symplectic Reduction ( X, ω ) � ( X ξ , ω ξ ) Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan 4. Symplectic Reduction Hamiltonian G -action on ( X, ω ) , moment map Φ : X − → g ∗ . G ξ stabilizer of ξ ∈ g ∗ , let X ξ = Φ − 1 ( ξ ) /G ξ . Suppose X ξ manifold (e.g. free action and G compact). π i − Φ − 1 ( ξ ) X ξ ← − → X Marsden, Weinstein 1974: symplectic form ω ξ on X ξ , π ∗ ω ξ = i ∗ ω Symplectic Reduction ( X, ω ) � ( X ξ , ω ξ ) Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan 4. Symplectic Reduction Hamiltonian G -action on ( X, ω ) , moment map Φ : X − → g ∗ . G ξ stabilizer of ξ ∈ g ∗ , let X ξ = Φ − 1 ( ξ ) /G ξ . Suppose X ξ manifold (e.g. free action and G compact). π i − Φ − 1 ( ξ ) X ξ ← − → X Marsden, Weinstein 1974: symplectic form ω ξ on X ξ , π ∗ ω ξ = i ∗ ω Symplectic Reduction ( X, ω ) � ( X ξ , ω ξ ) Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan 5. Geometric Quantization G -invariant K¨ ahler (symplectic) manifold ( X, ω ) , suppose [ ω ] ∈ H 2 ( X, Z ) . Exists line bundle L over X with connection ∇ and Hermitian structure � , � . Chern class of L is [ ω ] , curvature of ∇ is ω . Hermitian structure is G -invariant. Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan 5. Geometric Quantization G -invariant K¨ ahler (symplectic) manifold ( X, ω ) , suppose [ ω ] ∈ H 2 ( X, Z ) . Exists line bundle L over X with connection ∇ and Hermitian structure � , � . Chern class of L is [ ω ] , curvature of ∇ is ω . Hermitian structure is G -invariant. Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups

Introduction Symplectic Geometry Geometric Quantization Compact Lie Groups Compact Cartan Split Cartan Geometric Quantization Section s of L is called holomorphic if ∇ v s = 0 for all anti-holomorphic vector fields v . Find suitable G -invariant measure µ on X . Section s of L is called square-integrable if � X � s, s � µ < ∞ . Meng-Kiat Chuah, Dept. of Math., NTHU Geometric Quantization and Models of Semisimple Lie Groups