31
Coadjoint Orbits Geometric Quantization Workshop on Diffeology etc Aix en Provence, France- June 25-26-27, 2014 Geometric Quantization On Coadjoint Orbits Hassan Jolany, University of Lille1 University of Lille1 Geometric Quantization on coadjoint orbits

Geometric quantization on coadjoint orbits

Embed Size (px)

DESCRIPTION

Geometric Quantization on coadjoint orbits

Citation preview

Page 1: Geometric quantization on coadjoint orbits

Coadjoint Orbits Geometric Quantization

Workshop on Diffeology etcAix en Provence, France- June 25-26-27, 2014Geometric Quantization On Coadjoint Orbits

Hassan Jolany, University of Lille1

University of Lille1

Geometric Quantization on coadjoint orbits

Page 2: Geometric quantization on coadjoint orbits

Coadjoint Orbits Geometric Quantization

List of sections

1 Coadjoint Orbits

2 Geometric Quantization

University of Lille1

Geometric Quantization on coadjoint orbits

Page 3: Geometric quantization on coadjoint orbits

Coadjoint Orbits Geometric Quantization

Coadjoint Orbits

Let G be a compact semisimple Lie group with Lie algebra g.Group G acts on dual of Lie algebra g∗ with coadjointrepresentation Ad∗ : G × g∗ → g∗ by convention:⟨

Ad∗gµ,X⟩

=⟨µ,Adg−1X

⟩where µ ∈ g∗, g ∈ G ,X ∈ g.

Definition

The subset Oµ = Ad∗gµ; g ∈ G of g∗ is called a coadjoint orbitof G through µ ∈ g∗

Coadjoint orbit through µ ∈ g∗ can be written Oµ ∼= G/Gµ,which the stabilizer subgroup Gµ can be written as

Gµ = g ∈ G : Ad∗gµ = µUniversity of Lille1

Geometric Quantization on coadjoint orbits

Page 4: Geometric quantization on coadjoint orbits

Coadjoint Orbits Geometric Quantization

Some Examples of Co-Adjoint Orbits

University of Lille1

Geometric Quantization on coadjoint orbits

Page 5: Geometric quantization on coadjoint orbits

Coadjoint Orbits Geometric Quantization

Some Examples of Co-Adjoint Orbits

University of Lille1

Geometric Quantization on coadjoint orbits

Page 6: Geometric quantization on coadjoint orbits

Coadjoint Orbits Geometric Quantization

Complexified of Lie Group

Definition

If G is a Lie group, a universal complexification is given by acomplex Lie group GC and a continuous homomorphismϕ : G → GC with the universal property that, if f : G → H is anarbitrary continuous homomorphism into a complex Lie group H,then there is a unique complex analytic homomorphismF : GC → H such that f = F oϕ.

For a classical Real Lie group G , The complexification of Liegroup GC can be defined as

GC := expg + igLet G = U(n), then GC = GL(n,C) or Let K = SU(n), thenKC = SL(n,C)

University of Lille1

Geometric Quantization on coadjoint orbits

Page 7: Geometric quantization on coadjoint orbits

Coadjoint Orbits Geometric Quantization

Complexified of Lie Group

Theorem

Let G be a compact and connected Lie group, then

GC ∼= G n g∗ ∼= T ∗G

which this decomposition known as Polar Decomposition.

Every coadjoint orbit Oµ can be written as

Oµ ∼= GC/P ∼= T ∗G/P

which P is the Parabolic subgroup.

University of Lille1

Geometric Quantization on coadjoint orbits

Page 8: Geometric quantization on coadjoint orbits

Coadjoint Orbits Geometric Quantization

Volume of a coadjoint orbits

Kirillov’s Character formula:Kirillov’s famous formula says that the characters of the irreducibleunitary representations of a compact Lie group G can be writtenby following form

χλ(exp(x)) =1

p(x)

∫Oλ+ρ

e2πi<µ,x>dµOλ+ρ(µ)

where p is a certain function on lie algebra g and it can be writtenas

p(x) = det1/2 sinh(ad(x/2))

ad(x/2)

and µ is the highest weight

University of Lille1

Geometric Quantization on coadjoint orbits

Page 9: Geometric quantization on coadjoint orbits

Coadjoint Orbits Geometric Quantization

Volume of a coadjoint orbit

Theorem

The Volume of a Coadjoint Orbit through µ is

Vol(Oµ) =∏α∈R+

< α, µ >

< α, ρ >

where ρ = 12

∑α∈R+ α

University of Lille1

Geometric Quantization on coadjoint orbits

Page 10: Geometric quantization on coadjoint orbits

Coadjoint Orbits Geometric Quantization

Measure on coadjoint orbits

Explicit formula for measure of coadjoint orbits:For µ ∈ g∗, we define a skew-symmetric bilinear form bµ onTeGµOµ ∼= g/gµ by

bµ(u, v) =< µ, [u, v ] >

where for u, v ∈ g, we write u, v ∈ g/gµ.So, the dual form of bµ is the bilinear form βµ on(g/gµ)∗ = g⊥µ = g.µ corresponding to bµ under this isomorphism

βµ(u.µ, v .µ) = bµ(u, v)

University of Lille1

Geometric Quantization on coadjoint orbits

Page 11: Geometric quantization on coadjoint orbits

Coadjoint Orbits Geometric Quantization

Measure on coadjoint orbits

Now choose a basis v1, v2, ..., vm for g/gµ and µ1, µ2, ..., µm be thedual basis for (g/gµ)∗ (where m is the maximal dimension of anorbit in g∗)then

1

(m/2)!bm/2µ = Pf < µ, [vi , vj ] > µ1 ∧ ... ∧ µm

and1

(m/2)!βm/2µ = Pf−1 < µ, [vi , vj ] > v1 ∧ ... ∧ vm

where Pf(aij) denotes the Pfaffian of a skew-symmetric matrix (aij)Now define,

ωµ =1

(m/2)!βm/2µ

So integration of ωµ gives measure on Oµ up to the factor (2π)m/2

University of Lille1

Geometric Quantization on coadjoint orbits

Page 12: Geometric quantization on coadjoint orbits

Coadjoint Orbits Geometric Quantization

Geometric properties of coadjoint orbits

Every coadjoint orbit is symplectic manifold. The symplecticform on Oλ is given by

ωλ(ad∗X , ad∗Y ) = 〈λ, [X ,Y ]〉. This is obviously anti-symmetric and non-degenerate andalso closed 2-form.Every coadjoint orbit has Kaehler and Hyper-Kaehlerstructure.Coadjont orbits are simply connected.

Theorem

Let G be a Lie group, and Φ : T ∗G → g∗ be a moment map and

ζ ∈ g∗ then the symplectic quotient Φ−1(ζ)Gζ

is coadjoint orbit

through ζ, i.e., Oζ .

University of Lille1

Geometric Quantization on coadjoint orbits

Page 13: Geometric quantization on coadjoint orbits

Coadjoint Orbits Geometric Quantization

Here, by previous theorem we see that the symplectic form oncotangent bunlde of a Lie group is related to Kostant KirillovSouriau symplectic structure of the coadjoint orbits in thedual of a Lie algebra as follows

Gi //

π

T ∗G

Oζand we have

π∗ωOζ= i∗ωT∗G

Theorem of Patrick Iglesias-Zemmour:

Theorem

Every connected Hausdorff symplectic manifold is isomorphic to acoadjoint orbit of its group of hamiltonian diffeomorphisms.

University of Lille1

Geometric Quantization on coadjoint orbits

Page 14: Geometric quantization on coadjoint orbits

Coadjoint Orbits Geometric Quantization

Geometric PDE on coadjoint orbits

Following theorem known as M. Stenzel’s theorem: We denote thecomplexified of a coadjoint orbit as

OCζ := GC/GC

ζ

where ζ ∈ g∗

Theorem

Let G be a compact, connected and semisimple, Lie group. Thereexists a G invariant, real analytic, strictly plurisubharmonicfunction of complexified coadjoint orbit ρ : OC

ζ → [0,∞) such thatu =√ρ satisfies the Monge-Ampere equation,

(∂∂u)n = 0

where n = dimOλUniversity of Lille1

Geometric Quantization on coadjoint orbits

Page 15: Geometric quantization on coadjoint orbits

Coadjoint Orbits Geometric Quantization

Geometric Quantization

Now, we define the geometric quantization by some axioms whichare compatible with physical view.

Geometric quantization we associate to a symplectic manifold(M, ω) a Hilbert space H, and one associates to smoothfunctions f : M → R skew-adjoint operators Of : H → H.Paul Dirac introduced in his doctoral thesis, the ”method ofclassical analogy”for quantization which is now known asDirac axioms as follows.

1] Poisson bracket of functions passes to commutator of operators:

Of ,g = [Of ,Og ]

2] Linearity condition must holds ,Oλ1f +λ2g = λ1Of + λ2Og forλ1, λ2 ∈ C3] Normalization condition must holds: 1 7→ i .I (Which I is identityoperator and i =

√−1)

University of Lille1

Geometric Quantization on coadjoint orbits

Page 16: Geometric quantization on coadjoint orbits

Coadjoint Orbits Geometric Quantization

Dirac Principal for Geometric quantization

Before to establish the axiom 4], we need to following definition.

Definition

Let (M, ω), be a symplectic manifold. A set of smooth functionsfj is said to be a complete set of classical observables if and onlyif every other function g such that fi , g = 0 for all fj, isconstant. Also we say that a family of operators is complete if itacts irreducibly on H

4] Minimality condition must holds: Any complete family offunctions passes to a complete family of operators. Moreover, if Gbe a group acting on (M, ω) by symplectomorphisms and on H byunitary transformations. If the G -action on M is trnsitive, then itsaction on H must be irreducible.

University of Lille1

Geometric Quantization on coadjoint orbits

Page 17: Geometric quantization on coadjoint orbits

Coadjoint Orbits Geometric Quantization

Geometric Quantization

Now, we again recall the pre-quantization Line bundle informal language. In fact, we have two important method forGeometric Quantization;

1)Using line bundle. More precisely, In geometric quantizationwe construct the Hilbert space H as a subspace of the spaceof sections of a line bundle L on a symplectic manifold M.

2)Without using line bundle: Using Mpc -structure instead ofline bundle. One of advantage of this construction is betterbehaved of physical view but definig it is not so easy.

University of Lille1

Geometric Quantization on coadjoint orbits

Page 18: Geometric quantization on coadjoint orbits

Coadjoint Orbits Geometric Quantization

Mpc Quantization

The use of Mpc structures clarifies and extends the traditionalKonstant scheme of geometric in a number of ways.

In fact, prequantized Mpc structures generalize the combinedtraditional data consisting of a prequantum line bundle and ametaplectic structure and are used in constructing andcomparing representations of poisson algebras on symplecticmanifolds.

Also, Mpc structures exists on any symplectic manifold.

University of Lille1

Geometric Quantization on coadjoint orbits

Page 19: Geometric quantization on coadjoint orbits

Coadjoint Orbits Geometric Quantization

Mpc- Group

Definition

Let (V ,Ω) be a real symplectic vector space and fix an irreducibleunitary projective representation W of V on a Hilbert space Hsuch that

x , y ∈ V ⇒W (x)W (y) = exp 1

2i~Ω(x , y)W (x + y)

. If g ∈ Sp(V ,Ω), so that g is a linear automorphism of V withg∗Ω = Ω. So, by uniqueness theorem of Stone and von Neumannthere exists a unitary operator U on H such thatv ∈ V ⇒W (gv) = UW (v)U−1. The group of all such operatorsU as g ranges over the symplectic group Sp(V ,Ω) is denoted byMpc(V ,Ω)

University of Lille1

Geometric Quantization on coadjoint orbits

Page 20: Geometric quantization on coadjoint orbits

Coadjoint Orbits Geometric Quantization

Mpc- Group

We have a central short exact sequence

1→ U(1)→ Mpc(V ,Ω)σ→ Sp(V ,Ω)→ 1

where σ sends U to g .

Mpc(V ,Ω) supports a unique unitary characterη : Mpc(V ,Ω)→ U(1) such that η(λ) = λ2 wheneverλ ∈ U(1).

The kernel of η is a connected double cover of Sp(V ,Ω)called metaplectic group Mp(V , ω)

University of Lille1

Geometric Quantization on coadjoint orbits

Page 21: Geometric quantization on coadjoint orbits

Coadjoint Orbits Geometric Quantization

Mpc- Structure

Definition

Let (E , ω) be a real symplectic vector bundle over the manifold M.If the rank of E equals the dimension of V then we may model(E , ω) on (V ,Ω) and define the symplectic frame bundleSp(E , ω) = Sp(E ) to be the principal Sp(V ,Ω) bundle over Mwhose fibre over m ∈ M consists of all linear isomorphismsb : V → Em such that b∗ωm = Ω

Now, we are ready to define Mpc -structure.

University of Lille1

Geometric Quantization on coadjoint orbits

Page 22: Geometric quantization on coadjoint orbits

Coadjoint Orbits Geometric Quantization

Mpc- Structure

Definition

By an Mpc -structure on (E , ω) we mean a principal Mpc(V ,Ω)bundle π : P → M with a fibre-preserving map φ : P → Sp(E , ω)such that the group actions are compatible:

φ(p.g) = φ(p).σ(g), ∀p ∈ P, g ∈ Mpc(V ,Ω)

where σ : Mpc(V )→ Sp(V )

Mpc -structures for (E , ω) are parametrized by H2(M,Z).

University of Lille1

Geometric Quantization on coadjoint orbits

Page 23: Geometric quantization on coadjoint orbits

Coadjoint Orbits Geometric Quantization

Mpc- Quantization

Definition

Let (M, ω) be a symplectic manifold. A prequantized Mpc

structure for (M, ω) is a pair (P, γ) with P an Mpc structure for(TM, ω) and γ an Mpc -invariant u(1)-valued one-form on P suchthat: if z ∈ mpc(V ,Ω) = Lie(Mpc(V ,Ω)) determines thefundamental vector field z on P then

γ(z) =1

2η∗z

anddγ = (1/i~)π∗ω

where π : P → M is the bundle projection. We say that (M, ω) isquantizable iff it admits prequantized Mpc structures.

University of Lille1

Geometric Quantization on coadjoint orbits

Page 24: Geometric quantization on coadjoint orbits

Coadjoint Orbits Geometric Quantization

Mpc- Quantization

Theorem

(M, ω) is quantizable if and only if

[ω]− 1

2c1(TM)

be an integral cohomology class.

Theorem

Inequivalent prequantized Mpc structures are parametrized by

H1(M; U(1))

University of Lille1

Geometric Quantization on coadjoint orbits

Page 25: Geometric quantization on coadjoint orbits

Coadjoint Orbits Geometric Quantization

Mpc- Quantization on cotangent bundle of Lie groups,T ∗G

Let G be a Lie group with Lie algebra g . Take, Z = T ∗G . ModleT ∗G on the symplectic vector space V = g⊕ g∗ with symplecticform

Ω((ξ, φ), (η, ψ)) = φ(η)− ψ(ξ)

The standard left action of G on the cotangent bundle T ∗G isHamiltonian , with equivariant moment map

J : T ∗G → g∗

withJ(αg ) = g .α

University of Lille1

Geometric Quantization on coadjoint orbits

Page 26: Geometric quantization on coadjoint orbits

Coadjoint Orbits Geometric Quantization

Mpc- Quantization on cotangent bundle of Lie groups,T ∗G

A canonical global section b : T ∗G → Sp(T ∗G ) of the symplecticframe bundle Sp(T ∗G ) is defined by

bαg (ξ, φ) = Λ∗

(√2(g−1.ξ)g ,

1√2

(g−1.φ− (g−1.ξ).α)α

)where Λ : G × g∗ → T ∗G sends (g , α) to αg an where dots signifyadjoint and coadjoint action.

University of Lille1

Geometric Quantization on coadjoint orbits

Page 27: Geometric quantization on coadjoint orbits

Coadjoint Orbits Geometric Quantization

Mpc- Quantization on cotangent bundle of Lie groups,T ∗G

The Mpc structure is the product π : T ∗G ×Mpc(V )→ T ∗Gwith projection T ∗G ×Mpc(V )→ Sp(T ∗G ) determined bysending (z , I ) to bz ; the prequantum form δ is given by

δ =1

i~π∗θ +

1

2η∗ε

where θ is the canonical one-form on T ∗G and ε is the flatconnection in π : T ∗G ×Mpc(V )→ T ∗G and η is the unitarycharacter of Mpc(V ) restricting to U(1) as the squaring map.The passage of prequantized Mpc structures to Marsden Weinsteinredced phase spaces here gives to us prequantized Mpc structureson coadjoint orbits.(because Marsden Weinstein redced phasespaces of T ∗G is the coadjoint orbit Oλ )

University of Lille1

Geometric Quantization on coadjoint orbits

Page 28: Geometric quantization on coadjoint orbits

Coadjoint Orbits Geometric Quantization

Mpc- Quantization on coadjoint orbits

For φ ∈ g∗ we denote Oφ = G .φ ⊂ g∗ its coadjoint orbit we canidentify Oφ ∼= G/Gφ, which Gφ is coadjoint stabilizer with Liealgebra gφ, and also we denote the identity component of Gφ withG 0φ .

Now we explain Robinson and J. H. Rawnsley’s quantization: Weknow that the vanishing holonomy generalizes the Keller Maslovcorrected Bohr sommerfeld rule. So we give the following theoremfor quantization on coadjoint orbits

University of Lille1

Geometric Quantization on coadjoint orbits

Page 29: Geometric quantization on coadjoint orbits

Coadjoint Orbits Geometric Quantization

Theorem

Mpc - Quantization on coadjoint orbits: For the coadjoint orbitOφ∼= G/Gφ, the vanishing holonomy condition amounts to the

following requirement:That the adjoint isotropy representation Ad : G 0

φ → Sp(g/h), given

by Adh(ξ + gφ) = h.ξ + gφ for h ∈ G 0φ and ξ ∈ g should lift to a

homomorphism τ : G 0φ → Mpc(g/h) with the property that

(ηoτ)∗ = − 2i~φ

Now here we give a connection between two different quantizationson coadjoint orbits with line bundles and without line bundle,

Theorem

Hassan Jolany’s Theorem: Let the coadjoint orbit Oµ withpre-quantum line bundle (G ×Gµ C)⊗2 be quantizable then thecoadjoint orbit Oµ is Mpc quantizable.

University of Lille1

Geometric Quantization on coadjoint orbits

Page 30: Geometric quantization on coadjoint orbits

Coadjoint Orbits Geometric Quantization

Theorem

Hassan Jolany’s theorem: A coadjoint orbit (Oφ, ωKKS) isMpc -quantizable if and only if

[ω]− 1

4

∑α∈R+

α

belongs to a lattice Zk

University of Lille1

Geometric Quantization on coadjoint orbits

Page 31: Geometric quantization on coadjoint orbits

Coadjoint Orbits Geometric Quantization

Thanks a lot for your attentionEND

University of Lille1

Geometric Quantization on coadjoint orbits