Quantization of group-valued moment maps III Eckhard Meinrenken June 4, 2011 Eckhard Meinrenken Quantization of group-valued moment maps III
Pre-quantization of q-Hamiltonian spaces Recall again the axioms of q-Hamiltonian G -spaces, 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. 12 θ L · [ θ L , θ L ] ∈ Ω 3 ( G ) is a closed 3-form on G . Here η = 1 Eckhard Meinrenken Quantization of group-valued moment maps III
Pre-quantization of q-Hamiltonian spaces Recall again the axioms of q-Hamiltonian G -spaces, 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. 12 θ L · [ θ L , θ L ] ∈ Ω 3 ( G ) is a closed 3-form on G . Here η = 1 Eckhard Meinrenken Quantization of group-valued moment maps III
Cone construction Definition Let F • : S • → R • be a cochain map between cochain complexes. The algebraic mapping cone is the cochain complex cone k ( F ) = R k − 1 ⊕ S k , d( x , y ) = ( F ( y ) − d x , d y ) . Its cohomology is denoted H • ( F ). For a q-Hamiltonian G -space, we have d ω = − Φ ∗ η, d η = 0. Thus: The pair ( ω, − η ) ∈ Ω 3 (Φ) := cone 3 (Φ ∗ ) is a cocycle. Eckhard Meinrenken Quantization of group-valued moment maps III
Pre-quantization Suppose G simple, simply connected, · the basic inner product. Definition Let ( M , ω, Φ) be a q-Hamiltonian G -space, Φ: M → G . A level k pre-quantization of ( M , ω, Φ) is an integral lift of k [( ω, − η )] ∈ H 3 (Φ , R ) . Eckhard Meinrenken Quantization of group-valued moment maps III
Pre-quantization Suppose G simple, simply connected, · the basic inner product. Definition Let ( M , ω, Φ) be a q-Hamiltonian G -space, Φ: M → G . A level k pre-quantization of ( M , ω, Φ) is an integral lift of k [( ω, − η )] ∈ H 3 (Φ , R ) . There is an equivariant version of this condition, but for G simply connected equivariance is automatic. Geometric interpretation involves ‘gerbes’. Eckhard Meinrenken Quantization of group-valued moment maps III
Pre-quantization: Examples Proposition ( M , ω, Φ) is pre-quantizable at level k if and only if for all Σ ∈ Z 2 ( M ) , and any X ∈ C 3 ( G ) with Φ(Σ) = ∂ X, � � k ( ω + η ) ∈ Z . Σ X Eckhard Meinrenken Quantization of group-valued moment maps III
Pre-quantization: Examples Proposition ( M , ω, Φ) is pre-quantizable at level k if and only if for all Σ ∈ Z 2 ( M ) , and any X ∈ C 3 ( G ) with Φ(Σ) = ∂ X, � � k ( ω + η ) ∈ Z . Σ X Example The double D ( G ) = G × G , Φ( a , b ) = aba − 1 b − 1 is pre-quantizable for all k ∈ N , since H 2 ( D ( G )) = 0. Eckhard Meinrenken Quantization of group-valued moment maps III
Pre-quantization: Examples Proposition ( M , ω, Φ) is pre-quantizable at level k if and only if for all Σ ∈ Z 2 ( M ) , and any X ∈ C 3 ( G ) with Φ(Σ) = ∂ X, � � k ( ω + η ) ∈ Z . Σ X Example The double D ( G ) = G × G , Φ( a , b ) = aba − 1 b − 1 is pre-quantizable for all k ∈ N , since H 2 ( D ( G )) = 0. Example The q-Hamiltonian SU( n )-space M = S 2 n is pre-quantized for all k ∈ N , since H 2 ( M ) = 0. Eckhard Meinrenken Quantization of group-valued moment maps III
Pre-quantization of conjugacy classes Recall: G / Ad( G ) ∼ = A (the alcove), taking ξ ∈ A to G . exp ξ . P k = P ∩ kA . Example The level k pre-quantized conjugacy classes are those indexed by ξ ∈ 1 k P k ⊂ A . G = SU(3) k = 3 Eckhard Meinrenken Quantization of group-valued moment maps III
Quantization of q-Hamiltonian G -space Eckhard Meinrenken Quantization of group-valued moment maps III
Quantization of q-Hamiltonian G -space Tricky.. Eckhard Meinrenken Quantization of group-valued moment maps III
Quantization of q-Hamiltonian G -space Tricky.. Let ( M , ω, Φ) be a q-Hamiltonian G -space, pre-quantized at level k . Problems: There is no notion of ‘compatible almost complex structure’ In general, q-Hamiltonian G -spaces need not even admit Spin c -structures. Eckhard Meinrenken Quantization of group-valued moment maps III
Quantization of q-Hamiltonian G -space Tricky.. Let ( M , ω, Φ) be a q-Hamiltonian G -space, pre-quantized at level k . Problems: There is no notion of ‘compatible almost complex structure’ In general, q-Hamiltonian G -spaces need not even admit Spin c -structures. Example = S 4 (does not admit G = Spin(5) has a conjugacy class C ∼ almost complex structure). G = Spin(2 k + 1) , k > 2 has a conjugacy class not admitting a Spin c -structure. Eckhard Meinrenken Quantization of group-valued moment maps III
Quantization of q-Hamiltonian G -space Solution: One defines the quantization ‘abstractly’, using a push-forward in twisted equivariant K -theory. Eckhard Meinrenken Quantization of group-valued moment maps III
Quantization of q-Hamiltonian G -space Solution: One defines the quantization ‘abstractly’, using a push-forward in twisted equivariant K -theory. Theorem (Freed-Hopkins-Teleman) R k ( G ) is the twisted equivariant K-homology of G at level k + h ∨ . Eckhard Meinrenken Quantization of group-valued moment maps III
Quantization of q-Hamiltonian G -space Solution: One defines the quantization ‘abstractly’, using a push-forward in twisted equivariant K -theory. Theorem (Freed-Hopkins-Teleman) R k ( G ) is the twisted equivariant K-homology of G at level k + h ∨ . Theorem (M) Let ( M , ω, Φ) be a level k pre-quantized q-Hamiltonian G-space. Then there is a distinguished R ( G ) -module homomorphism Φ ∗ : K G 0 ( M ) → R k ( G ) . Eckhard Meinrenken Quantization of group-valued moment maps III
Quantization of q-Hamiltonian G -space Solution: One defines the quantization ‘abstractly’, using a push-forward in twisted equivariant K -theory. Theorem (Freed-Hopkins-Teleman) R k ( G ) is the twisted equivariant K-homology of G at level k + h ∨ . Theorem (M) Let ( M , ω, Φ) be a level k pre-quantized q-Hamiltonian G-space. Then there is a distinguished R ( G ) -module homomorphism Φ ∗ : K G 0 ( M ) → R k ( G ) . This push-forward does not involve a Dirac operator. (There’s not enough time here to explain how it is defined – sorry.) Eckhard Meinrenken Quantization of group-valued moment maps III
Quantization of q-Hamiltonian G -spaces Definition The quantization of a level k pre-quantized q-Hamiltonian G -space ( M , ω, Φ) is the element Q ( M ) = Φ ∗ (1) ∈ R k ( G ) . Eckhard Meinrenken Quantization of group-valued moment maps III
Quantization of q-Hamiltonian G -spaces Q ( M ) = Φ ∗ (1) ∈ R k ( G ) . Properties of the quantization: Q ( M 1 ∪ M 2 ) = Q ( M 1 ) + Q ( M 2 ), Q ( M 1 × M 2 ) = Q ( M 1 ) Q ( M 2 ), Q ( M ∗ ) = Q ( M ) ∗ , Let C be the conjugacy class of exp( 1 k µ ), µ ∈ P k . Then Q ( C ) = τ µ . Eckhard Meinrenken Quantization of group-valued moment maps III
Quantization of q-Hamiltonian G -spaces Recall the trace R k ( G ) → Z , τ �→ τ G where τ G µ = δ µ, 0 . Theorem (Quantization commutes with reduction) Let ( M , ω, Φ) be a level k prequantized q-Hamiltonian G-space. Then Q ( M ) G = Q ( M / / G ) . Eckhard Meinrenken Quantization of group-valued moment maps III
Example Let C i be the conjugacy classes of exp( 1 k µ i ) , µ i ∈ P k . Then / G ) = ( τ µ 1 τ µ 2 τ µ 3 ) G = N ( k ) Q ( C 1 × C 2 × C 3 / µ 1 µ 2 µ 3 . Eckhard Meinrenken Quantization of group-valued moment maps III
Example Let C i be the conjugacy classes of exp( 1 k µ i ) , µ i ∈ P k . Then / G ) = ( τ µ 1 τ µ 2 τ µ 3 ) G = N ( k ) Q ( C 1 × C 2 × C 3 / µ 1 µ 2 µ 3 . Hamiltonian analogue: Example Let O i be the coadjoint orbits of µ i ∈ P + . Then / G ) = ( χ µ 1 χ µ 2 χ µ 3 ) G = N µ 1 µ 2 µ 3 . Q ( O 1 × O 2 × O 3 / Eckhard Meinrenken Quantization of group-valued moment maps III
Examples Example The double D ( G ) = G × G , Φ( a , b ) = aba − 1 b − 1 has level k quantization � τ µ τ ∗ Q ( D ( G )) = µ . µ ∈ P k Eckhard Meinrenken Quantization of group-valued moment maps III
Examples Example The double D ( G ) = G × G , Φ( a , b ) = aba − 1 b − 1 has level k quantization � τ µ τ ∗ Q ( D ( G )) = µ . µ ∈ P k The Hamiltonian analogue is the non-compact Hamiltonian G -space T ∗ G . Any reasonable quantization scheme for non-compact spaces gives � Q ( T ∗ G ) = χ µ χ ∗ µ µ ∈ P + (character for conjugation action on L 2 ( G ), defined in a completion of R ( G )). Eckhard Meinrenken Quantization of group-valued moment maps III
Can re-write this in terms of the basis ˜ τ µ , where ˜ τ µ ( t λ ) = δ λ,µ : � � � S ∗ G . exp(1 µ,ν Q k µ ) = τ µ = τ ν . ˜ S 0 ,ν ν ∈ P k � 1 Q ( D ( G )) = ˜ τ ν S 2 0 ,ν ν ∈ P k Eckhard Meinrenken Quantization of group-valued moment maps III
Can re-write this in terms of the basis ˜ τ µ , where ˜ τ µ ( t λ ) = δ λ,µ : � � � S ∗ G . exp(1 µ,ν Q k µ ) = τ µ = τ ν . ˜ S 0 ,ν ν ∈ P k � 1 Q ( D ( G )) = ˜ τ ν S 2 0 ,ν ν ∈ P k Using Q ( M 1 × M 2 ) = Q ( M 1 ) Q ( M 2 ) this gives ... Eckhard Meinrenken Quantization of group-valued moment maps III
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