The Bohr-Sommerfeld groupoid of quantum CP n joint with F. Bonechi, J. Qiu, M. Tarlini Commun. Math. Phys. 331, 851-885 (2014) N. Ciccoli Warsaw – 20.08.2014 Multiplicative integrability - CP n N. Ciccoli
KWZ program Let ( M , π ) be an integrable Poisson manifold with symplectic groupoid r − G ( M ) M : m : G 2 ( M ) → G ( M ) ⇒ − l Karasev-Weinstein-Zakrzewski Apply geometric quantization to G ( M ) and compare the outcome with deformation quantization of ( M , π ) . Multiplicative integrability - CP n N. Ciccoli
Symplectic integration For a Poisson manifold ( M , π ) the cotangent bundle T ∗ M has a natural structure of Lie algebroid (i.e. Lie bracket between 1–forms + Lie map between 1-forms and vector fields). A symplectic groupoid is a Lie groupoid integrating this Lie algebroid (much as Lie groups integrate Lie algebras - but... possible obstructions). If the obstruction is not present (meaning of the word integrable ) then the groupoid has also a symplectic manifold compatible with the Lie groupoid structure. Multiplicative integrability - CP n N. Ciccoli
KWZ program Prequantum line bundle ( L , ∇ ) + σ covariantly constant 1 normalized 2–cocycle in L ; Multiplicative polarization F : set of leaves G ( M ) / F is a 2 groupoid inheriting (reduced) 2–cocycle σ 0 ; Bohr-Sommerfeld condition identifying a subgroupoid 3 ( G ( M ) / F ) bs ; (Twisted) convolution C ∗ –algebra C ∗ (( G ( M ) / F ) bs ; σ 0 ) . 4 Multiplicative integrability - CP n N. Ciccoli
Motivating example Let M = T 2 with constant symplectic structure π = θ∂ 1 ∧ ∂ 2 G ( T 2 ) = T ∗ T 2 (change in grpd + sympl.) Prequantum bundle= trivial line bundle + 2–cocycle; Horizontal polarization ⇒ C ∗ ( Z 2 ; σ 0 ) with σ 0 = e π (Weyl); Cylindrical polarization ⇒ C ∗ ( Z ⋊ S 1 ) action groupoid with trivial cocycle (irrational rotation algebra). Outcome Quantum torus p ⋆ q = e � q ⋆ p . Multiplicative integrability - CP n N. Ciccoli
Multiplicative polarization A groupoid polarization F ⊆ T C G is multiplicative (Hawkins JSG 2008) if, letting F 2 = ( F × F ) ∩ T C G 2 then m ∗ ( F 2 ( γ, η )) = F ( m ( γ, η )) for any composable pair ( γ, η ) ∈ G 2 . Problem: there are topological obstructions to the existence of real multiplicative polarizations Multiplicative integrability - CP n N. Ciccoli
CP 1 -obstruction Let π be any integrable Poisson structure on CP 1 , then there are no real multiplicative polarizations on its symplectic groupoid (linked to non existence of rank 1 foliations on CP 1 ). Bruhat-Poisson structure on CP 1 : on CP 1 \ [ 1 , 0 ] − ı ( 1 + | z | 2 ) ∂ z ∧ ∂ z π B = on CP 1 \ [ 0 , 1 ] − ı | w | 2 ( 1 + | w | 2 ) ∂ w ∧ ∂ w Still possibile to perform KWZ procedure with a singular multiplicative polarization (Bonechi, C., Staffolani, Tarlini JGP 2012). Multiplicative integrability - CP n N. Ciccoli
Loosening requirements What do we really need for a C ∗ –groupoid convolution algebra? G → G F Lagrangian fibration of topological groupoids; G bs F Bohr–Sommerfeld subgroupoid carrying a left Haar measure; the prequantization cocycle descending to G bs F ; the modular 1 –cocycle descending to G bs F ; Multiplicative integrability - CP n N. Ciccoli
Intermezzo – the modular cocycle ( M , π ) Poisson, V volume form on M ⇒ χ V modular vector field (divergence of π w.r. to V ) defines a class in H 1 π ( M ) . χ V ⇒ f V (van Est map) 1–cocycle on G ; f V should be quantizable, coincide with the modular function of the quasi invariant measure on the base space, implement KMS condition . Multiplicative integrability - CP n N. Ciccoli
Multiplicative integrable system integrable A family F = { f 1 , . . . f N } of functions, N = 1 2 dim G , is an integrable system if are in involution { f i , f j } = 0 and df 1 ∧ . . . ∧ df N � = 0 on a dense open subset of M . multiplicative The integrable system is called multiplicative if the distribution F = � X f 1 , . . . X f N � is multiplicative, or, more generically, if the topological space of level sets of f 1 , . . . f N inherits a topological groupoid structure from G . modular The integrable system is called modular if the modular function f V is in involution with all f i ’s. Multiplicative integrability - CP n N. Ciccoli
Multiplicative integrable system Consider the level sets of a multiplicative integrable system G F ( M ) = G ( M ) / F It is well behaved if: G F ( M ) is a topological groupoid and G ( M ) → G F ( M ) a 1 topological groupoid epimorphism; For each pair l 1 , l 2 of composable leaves m : l 1 × l 2 → l 1 l 2 2 induces a surjective map in homology ( ⇒ subgroupoid G bs F ( M ) ). G bs F ( M ) admits a left Haar system (guaranteed if it is étale). 3 Multiplicative integrability - CP n N. Ciccoli
Let SU ( n + 1 ) be given the standard Poisson–Lie structure π std . There is a one–parameter family of covariant ( CP n , π t ) , non symplectic when t ∈ [ 0 , 1 ] . Non symplectic are all quotient by coisotropic subgroups: U t ( n ) = σ t S ( U ( 1 ) × U ( n )) σ − 1 ⊆ SU ( n + 1 ) t where √ √ 1 − t 0 t σ t = 0 0 id n − 1 √ √ − t 0 1 − t Multiplicative integrability - CP n N. Ciccoli
Some equivalences. In fact: ψ : CP n → CP n ; ψ ( π t ) = − π 1 − t π 0 , π 1 , standard or Bruhat–Poisson π t , t ∈ ] 0 , 1 [ , non standard . Poisson pencil Let π λ be the Fubini-Study bivector. Then [ π λ , π 0 ] = 0 (Koroshkin-Radul-Rubtsov CMP ’93) and π t = π 0 + t π λ . Multiplicative integrability - CP n N. Ciccoli
Standard CP n : symplectic foliation Projecting the chain of Poisson subgroups SU ( 1 ) ⊆ SU ( 2 ) ⊆ . . . ⊆ SU ( n ) one gets the chain of Poisson submanifolds {∗} ⊆ CP 1 ⊆ . . . ⊆ CP n − 1 In homogeneous coordinates P k = { [ X 1 , . . . , X k , 0 , . . . , 0 ] } is a Poisson submanifold. All symplectic leaves are contractible and symplectomorphic to standard C k . Multiplicative integrability - CP n N. Ciccoli
Non standard CP n : symplectic foliation singular locus Let � k n � | X i | 2 − ( 1 − t ) | X i | 2 = 0 � � P k ( t ) = F k , t = t i = 1 i = k + 1 Then � n i = 1 P i ( t ) is the singular part; complement has n + 1 connected contractible leaves ≃ C n . Scheme of the singular part for CP 3 : S 3 × S 3 S 5 S 5 S 1 տ ր տ ր S 3 S 3 տ ր S 1 Multiplicative integrability - CP n N. Ciccoli
symplectic foliation of CP 2 t Multiplicative integrability - CP n N. Ciccoli
The symplectic groupoid of ( CP n , π t ) The symplectic groupoid G ( CP n , π t ) = { [ g γ ] : g ∈ SU ( n + 1 ) , γ ∈ SB ( n + 1 , C ) , g γ ∈ U t ( n ) ⊥ } is a fibre bundle over CP n with contractible fibre U t ( n ) ⊥ . It is an exact symplectic manifold. It carries a hamiltonian T n –action with momentum map h ([ g γ ]) = log p A n + 1 ( γ ) Multiplicative integrability - CP n N. Ciccoli
Bihamiltonian torus action The Cartan T n ⊆ SU ( n + 1 ) acts on ( CP n , π λ ) with momentum map c : CP n → t ∗ n ; Im c = ∆ n The action is Poisson w. r. to π t . Suitable basis H k of t n such that infintesimal vector fields σ H k are eigenvalues of the 1 Nijenhuis operator with eigenvector ( c k − 1 ) ; σ H k = { b k , −−} , with b k = log | c k − 1 + t | . 2 Multiplicative integrability - CP n N. Ciccoli
Summarizing actions Hamiltonian T n –action on CP n with momentum map c : CP n → R n ; Hamiltonian T n –action on G ( CP n , π t ) with momentum map h : G ( CP n ) → R n by groupoid 1–cocycles; Let us consider F = { l ∗ c i , h i . . . i = 1 , . . . , n } Multiplicative integrability - CP n N. Ciccoli
Theorem F is a multiplicative modular integrable system on G ( CP n , π t ) with: n � f FS = h i i = 1 Aim: prove this integrable system is well behaved. Multiplicative integrability - CP n N. Ciccoli
The topological groupoid of level sets Let R n act on R n via c · h = ( 1 − t + e − h ( c + t − 1 )) and let R n ⋊ R n � ∆ n be the action groupoid restricted to the � standard simplex.Then: G F ( t ) = { ( c , h ) ∈ R n ⋊ R n � ∆ n : c i = c i + 1 = 1 − t ⇒ h i = h i + 1 } � is the topological groupoid of level sets. Multiplicative integrability - CP n N. Ciccoli
Bohr-Sommerfeld conditions Level sets L ch are connected with: H 1 ( L ch ; Z ) generated by hamiltonian flows of h j , l ∗ c j ; Theorem BS conditions select a discret subset of lagrangians G bs F ( t ) = { ( c , h ) ∈ G F ( t ) : h k ∈ �Z , log | c k − 1 + t | ∈ �Z } This is an étale subgroupoid with a unique left Haar system. The modular function f FS is quantized to n � f FS ( c , h ) = h i i = 1 Multiplicative integrability - CP n N. Ciccoli
The space of units is ∆ Z n ( t ) = { c ∈ ∆ n : c k = 1 − t + e − � n k } The quasi invariant measure associated to f FS is: n � µ fs ( c ) = exp ( − � n k ) k = 1 Groupoid orbits are labelled by ( r , s ) : r + s ≤ n . Each is a transitive subgroupoid over � � s : − log ( 1 − t ) r × ∞ × Z ≤ m i ≤ m i + 1 ∆ Z � r , s ( t ) = ( m , ∞ , n ) ∈ Z ≥ − log ( t ) ≥ n i + 1 n i � Multiplicative integrability - CP n N. Ciccoli
Recommend
More recommend
