From action-angle coordinates to geometric quantization: a 30-minute round-trip. Eva Miranda
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1 From action-angle coordinates to geometric quantization: a 30-minute round-trip Eva Miranda Barcelona (UPC) Geometric Quantization in the non-compact setting Eva Miranda (UPC) MFO, February, / 25
2 Outline 1 Quantization: The general picture 2 Bohr-Sommerfeld leaves of regular fibrations and action-angle coordinates 3 Quantization via sheaf cohomology 4 Singular polarizations and quantization using singular action-angle coordinates Eva Miranda (UPC) MFO, February, / 25
3 Today I will talk about Today I will talk about my work with Mark Hamilton on geometric quantization of real polarizations with singularities in dimension 2. I will also mention the case of Lagrangian foliations (joint work with Fran Presas) and my project with Mark Hamilton and Romero Solha about integrable systems with singularities in higher dimensions. Eva Miranda (UPC) MFO, February, / 25
4 Classical vs. Quantum: a love & hate story 1 Classical systems 2 Observables C (M) 3 Bracket {f, g} 1 Quantum System 2 Operators in H (Hilbert) 3 Commutator [A, B] h = 2πi h (AB BA) Eva Miranda (UPC) MFO, February, / 25
5 Quantization via real polarizations (mathematically speaking) Let (M 2n, ω) be a symplectic manifold such that [ω] is integral. (L, ) the complex line bundle with a connection over M that satisfies curv( ) = ω. The symplectic manifold (M 2n, ω) is called prequantizable and the pair (L, ) is called a prequantum line bundle of (M 2n, ω). A real polarization P is a foliation whose leaves are lagrangian submanifolds. Integrable systems provide natural examples of real polarizations. If the manifold M is compact the moment map : F : M 2n R n has singularities that correspond to equilibria. Flat sections along P are given by the equation X s = 0 for any vector field X tangent to P. Eva Miranda (UPC) MFO, February, / 25
6 Bohr-Sommerfeld leaves Definition A Bohr-Sommerfeld leaf is a leaf of a polarization that admits sections globally defined along it. Example Consider M = S 1 R and ω = dt dθ. Take as L the trivial bundle with connection 1-form Θ = tdθ. Now, let P =< θ > Then flat sections satisfy, X σ = X(σ) i < θ, X > σ. Thus σ(t, θ) = a(t).e itθ. Then Bohr-Sommerfeld leaves are given by the condition t = 2πk, k Z. Eva Miranda (UPC) MFO, February, / 25
7 An image is worth more than a hundred words... Eva Miranda (UPC) MFO, February, / 25
8 Bohr-Sommerfeld leaves: continued... Theorem (Guillemin-Sternberg) If the polarization is a regular fibration with compact leaves over a simply connected base B, then the Bohr-Sommerfeld set is discrete and is given by the condition, BS = {p M, (f 1 (p),..., f n (p)) Z n } where f 1,..., f n are global action coordinates on B. 1 This result connects with Arnold-Liouville theorem for integrable systems. 2 In the case of toric manifolds the base B may be identified with the image of the moment map by the toric action. Eva Miranda (UPC) MFO, February, / 25
9 Bohr-Sommerfeld leaves and Delzant polytopes Theorem (Delzant) Toric manifolds are classified by Delzant s polytopes. More specifically, the bijective correspondence between these two sets is given by the moment {toric manifolds} {Delzant polytopes} map: (M 2n, ω, T n, F ) F (M) The image of a toric manifold is a Delzant Polytope. Moment map for the T 2 -action on CP 2 given by (e iθ 1, e iθ 2 ) [z 0 : z 1 : z 2 ] := [z 0 : e iθ 1 z 1 : e iθ 1 z 2 ] Eva Miranda (UPC) MFO, February, / 25
10 Sniatycki result: Quantizing is counting Bohr-Sommerfeld leaves It would make sense to quantize these systems counting Bohr-Sommerfeld leaves. In the case the polarization is an integrable system with global action-angle coordinates, Bohr-Sommerfeld leaves are just integral Liouville tori. This is the content of Snyaticki theorem: Theorem (Sniatycki) If the leaf space B n is Hausdorff and the natural projection π : M 2n B n is a fibration with compact fibers, then quantization is given by the sheer count of Bohr-Sommerfeld leaves. But how exactly? Eva Miranda (UPC) MFO, February, / 25
11 Quantization: The cohomological approach Following the idea of Kostant, in the case there are no global sections we define the quantization of (M 2n, ω, L,, P ) as Q(M) = k 0 H k (M, J ). J is the sheaf of flat sections, i.e.: the space of local sections σ of L such that X σ = 0, for all sections X of P. Then quantization is given by precisely: Theorem (Sniatycki) If the leaf space B n is a Hausdorff manifold and the natural projection π : M 2n B n is a fibration with compact fibres, then Q(M 2n ) = H n (M 2n, J ), and the dimension of H n (M 2n, J ) is the number of Bohr-Sommerfeld leaves. Eva Miranda (UPC) MFO, February, / 25
12 What is this cohomology? 1 Define the sheaf: Ω i P (U) = Γ(U, i P).. 2 Define C as the sheaf of complex-valued functions that are locally constant along P. Consider the natural (fine) resolution 0 C i Ω 0 P d P Ω 1 P d P Ω 1 P d P Ω 2 P d P The differential operator d P is the restriction of the exterior differential to the directions of the distributions (as in foliated cohomology). 3 Use this resolution to obtain a fine resolution of J by twisting the previous resolution with the sheaf J. 0 J i S P S Ω 1 P P S Ω 2 P Eva Miranda (UPC) MFO, February, / 25
13 Applications to the general case of Lagrangian foliations This fine resolution approach can be useful to compute the case of some Lagrangian foliations (which do not come from integrable systems). Classification of foliations on the torus (Kneser-Denjoy-Schwartz theorem). Eva Miranda (UPC) MFO, February, / 25
14 The case of the torus: irrational slope. Consider X η = η x + y, with η R \ Q. This vector field descends to a vector field in the quotient torus that we still denote X η. Denote by P η the associated foliation in T 2. Theorem (Presas-Miranda) Let (T 2, ω) be the 2-torus with a symplectic structure ω of integer class. Let P η the irrational foliation of slope η in this manifold. Then Q(T 2, J ) is always infinite dimensional. However, if we compute the limit case of the foliated cohomology (ω = 0), we obtain that Q(T 2, J ) = C C if the irrationality measure of η is finite and Q(T 2, J ) is infinite dimensional if the irrationality measure of η is infinite (i.e. it is a Liouville number). This generalizes a result El Kacimi for foliated cohomology. Eva Miranda (UPC) MFO, February, / 25
15 Quantization Computation kit for regular foliations Most computations rely on proving Kunneth and Mayer-Vietoris (joint with Presas) 1 Künneth formula: Let (M 1, P 1 ) and (M 2, P 2 ) be a pair of symplectic manifolds endowed with Lagrangian foliations. The natural cartesian product for the foliations is Lagrangian with respect to the product symplectic structure. Let J 12 be the induced sheaf of basic sections, then: H n (M 1 M 2, J 12 ) = p+q=n Hp (M 1, J 1 ) H q (M 2, J 2 ). 2 Mayer-Vietoris: Consider M U V U V, then the following sequence is exact, 0 S Ω P (M) r S Ω P (U) S Ω P (V ) r 0 r 1 S Ω P (U V ) 0. Eva Miranda (UPC) MFO, February, / 25
16 Toric Manifolds What happens if we go to the edges and vertexes of Delzant s polytope? Consider the case of rotations of the sphere. There are two leaves of the polarization which are singular and correspond to fixed points of the action. Eva Miranda (UPC) MFO, February, / 25
17 Quantization of toric manifolds Theorem (Hamilton) For a 2n-dimensional compact toric manifold and let BS r be the set of regular Bohr-Sommerfeld leaves, Q(M) = H n (M; J ) = l BS r C Remark: Then this geometric quantization does not see the singular points. In the example of the sphere Bohr-Sommerfeld leaves are given by integer values of height (or, equivalently) leaves which divide out the manifold in integer areas. Eva Miranda (UPC) MFO, February, / 25
18 Action-angle coordinates with singularities The theorem of Guillemin-Marle-Sternberg gives normal forms in a neighbourhood of fixed points of a toric action. This can be generalized to normal forms of integrable systems (not always toric) that we call non-degenerate. Theorem (Eliasson-Miranda) There exists symplectic Morse normal forms for integrable systems with non-degenerate singularities. Liouville torus k e comp. elliptic k h hyperbolic k f focus-focus Eva Miranda (UPC) MFO, February, / 25
19 Description of singularities The local model is given by N = D k T k D 2(n k) and ω = k i=1 dp i dθ i + n k i=1 dx i dy i. and the components of the moment map are: 1 Regular f i = p i for i = 1,..., k; 2 Elliptic f i = x 2 i + y2 i for i = k + 1,..., k e ; 3 Hyperbolic f i = x i y i for i = k e + 1,..., k e + k h ; 4 focus-focus f i = x i y i+1 x i+1 y i, f i+1 = x i y i + x i+1 y i+1 for i = k e + k h + 2j 1, j = 1,..., k f. Eva Miranda (UPC) MFO, February, / 25
20 The case of hyperbolic singularities We consider the following covering Eva Miranda (UPC) MFO, February, / 25
21 Key point in the computation We may choose a trivializing section of such that the potential one-form of the prequantum connection is Θ 0 = (xdy ydx). Theorem Leafwise flat sections in a neighbourhood of the singular point in the first quadrant is given by a(xy)e i 2 xy ln x y where a is a smooth complex function of one variable which is flat at the origin. Eva Miranda (UPC) MFO, February, / 25
22 The case of surfaces We can use Cech cohomology computation and a Mayer-Vietoris argument to prove: Theorem (Hamilton and Miranda) The quantization of a compact surface endowed with an integrable system with non-degenerate singularities is given by, Q(M) = H 1 (M; J ) = p H where H is the set of hyperbolic singularities. (C N C N ) l BS r C, Eva Miranda (UPC) MFO, February, / 25
23 The case of the rigid body Using this recipe and the quantization of this system is Q(M) = H 1 (M; J ) = p H (C N p ) 2 C b. b BS Comparing this system with the one we get from rotations on the sphere, we obtain different results (one is infinite dimensional). Therefore this quantization depends strongly on the polarization. Eva Miranda (UPC) MFO, February, / 25
24 Some work to do... Higher dimensions We can use a the geometric quantization computation kit (Künneth and Mayer Vietoris) to compute the geometric quantization in any dimension and the quantization is expressed in terms of hyperbolic, regular Bohr-Sommerfeld leaves and focus-focus blocks. A conjecture: Quantization commutes with reduction There is a quantization commutes with reduction principle for this model if singularities are non-degenerate. Idea: we can use equivariant singular action-angle coordinates to obtain an equivariant connection 1-form. Eva Miranda (UPC) MFO, February, / 25
25 In the pipeline... The Poisson case. (this is joint work in progress with Mark Hamilton) Use Vaisman approach for prequantization. Study the exact case. Use action-angle coordinates for Poisson structures (Laurent-Miranda-Vanhaecke). Study integrable systems which are splitted. Study the b-case (related to Melrose s b-calculus). Eva Miranda (UPC) MFO, February, / 25
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