Berezin-Toeplitz Quantization On Coadjoint Orbits

Berezin-Toeplitz Quantization On Coadjoint Orbits

Hassan Jolany, University of Lille1

Saturday 26th July, 2014

Hassan Jolany, University of Lille1Saturday 26th July, 2014 1 /

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In this talk we give an answer to the following questions:

Relate the inner product (inner product on Hilbert space comesfrom Metaplectic correction)to Kirillov’s character formula

Relate the inner product with coherent states and Berezin-Toeplitzquantization


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Introduction

Inner product relative to Kirillov’s character formula.Firstly, we start with a definition.

Definition

For M a smooth manifold and P → TM a foliation of M , incarnatedas a subbundle of the tangent bundle, the corresponding bundle

P⊥ → T ∗X

is the annihilator of P under the pairing of covectors with vectors. Thecorresponding Bott connection is the covariant derivative of vectorsX ∈ Γ(P ) on covectors ξ ∈ Γ(P⊥) given by the Lie derivative

∇X : ξ 7→ LXξ = iXdξ


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Metaplectic correction

Theorem

Let (M,ω) be a symplectic manifold and have a metaplectic structureand also assume P ⊂ TM be a strongly admissible real polarization.Let denote the annihilator bundle of P byP⊥ := ξ ∈ T ∗M : ∀X ∈ P,< X, ξ >= 0 hence we can identify the

space of half-forms of Λ12 (M/D) with those sections of Λ

12 (P⊥) that

are P -constant by the Bott connection i.e. ∇Xξ := LXξ = 0 for anyX ∈ Γ(P ) and ξ ∈ Λ

12 (P⊥).

For construction of new sections s = s⊗ σ we said σ belong tosome line bundle and we didn’t explain exatly this space ofsections. Now by pervious theorem we can say σ ∈ Λ

12 (P⊥) for

strongly admissible real polarization P .


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Inner product

So, from now on, the space of sections exactly is Γ(L⊗ Λ12 (P⊥)).

Hence, we can define the polarized sections as follows.

Definition

We say a section s⊗ σ ∈ Γ(L⊗ Λ12 (P⊥)) is a polarized section if it

satisfy to ∇Xs = 0 and LXσ = 0 for all X ∈ Γ(P ). Note that here

Λ12 (P⊥) is the principal GL+(n)× Z4 bundle.

Now we give an correct inner product on polarized sections ofΓ(L⊗ Λ

12 (P⊥)). Let s1 ⊗ σ1 and s2 ⊗ σ2 be such polarized sections.

So, h(s1, s2)σ1σ2 is a compactly supported smooth density on M/D,and we have the following Hermitian inner product.

< s1 ⊗ σ1, s2 ⊗ σ2 >=

∫M/D

h(s1, s2)σ1σ2

which D = P ∩ P ∩ TM and P is polarization.


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Coadjoint orbit

Let G be a connected and compact Lie group and g = LieG be its Liealgebra and g∗ be dual of lie algebra ofg and take µ ∈ g∗ then thecoadjoint orbit Oµ ∼= G/Gµ ∼= GC/P where Gµ is isotropy subgroup ofG and P is the parabolic subgroup.It can be showen that all invariant polarizations are complex and aredetemined by parabolic subalgebra p which LieP = p


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Note that a polarization of the coadjoint orbit G/Gµ, is given by theleft invariant extension of complex Lie subalgebra p ⊂ gC with theproperties1. µC([p, p]) = 02. dimg/gµ = dimgC/p.3. gµ ⊂ p4. (p⊕ p) ∩ g is a Lie subalgebra of g.


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But we want to see what is the space of leaves of Oµ/D?In fact D is the G-invariant distribution whose value at µ is thesubspace Dµ = d/gµ of TµOµ = g/gµ where d = p ∩ p ∩ g . In the casecomplex polarization we have d = gµ so d/gµ = 0 and D = 0. Sothe leaves are points and the quotient is Oµ itself. This meansOµ/D = Oµ


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Now the line bundle on coadjoint orbit is L = G×Gµ C→ Oµ and thesections of L can be interprated as functions f : G→ C whichf(ga) = χ(a−1)f(g) where a ∈ Gµ and χ : Gµ → S1 is character. (andwe have also anadditional condition on characters by polarization)So by rewriting the inner product

< f1 ⊗ σ1, f2 ⊗ σ2 >=

∫Oµ

h(f1, f2)σ1σ2


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Kirillov’s Character formula for compact Lie group G

Kirillov’s famous formula says that the characters of the irreducibleunitary representations of a Lie group G can be written by followingform

χ(exp(x)) =1

p(x)

∫Oλ

ei<λ′,x>dµOλ

(λ′)

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

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

ad(x/2)


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Now the line bundle on coadjoint orbit is homogeneous line bundleL = G×Gµ C→ Oµ and the sections of L can be interprated asfunctions f : G→ C which f(ga) = χ(a−1)f(g) where a ∈ Gµ andχ : Gµ → S1 is character.So

< f1 ⊗ σ1, f2 ⊗ σ2 >=

∫Oµ

h(f1, f2)σ1σ2

But if we take X ∈ gµ = LieGµ, then f(geX) = χ(e−X)f(g) whereg ∈ G then by using Kirillov’s character formula we have

χ(e−X) =1

p(X)

∫Oµ

e−i<X,Z>dµ

so we can write

fi(geX) =

1

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

∫Oµ

e−i<X,Z>dµ.fi(g)

where i = 1, 2. So the inner product on this Hilbert space is related toKirillov’s character formula.


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Coadjoint orbit is Kaehler manifold

We recall again coadjoint orbits, more presicelyLet G be a compact connected Lie Group and g be its Lie algebra,andalso g∗ be the dual of Lie algebra, then the coadjoint orbit in F ∈ g∗

can be defined as

OF = Ad∗(g)F, g ∈ G

In fact, Kirillov introduced an antisymmetric bilinear form BF on g byBF (X,Y ) =< F, [X,Y ] > and showed that BF is invariant underStab(F ) = g ∈ G : Ad∗(g)F = F and by using this fact, introduced aG-Invariant symplectic structureωO(F )(ad∗(X)F, ad∗(Y )F ) = BF (X,Y ) on O which is now known asKirillov-Kostant-Souriau Theorem .


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Introduction

Let G be a connected and compact Lie group and g = LieG be itsLie algebra and g∗ be dual of lie algebra ofg and take µ ∈ g∗ thenthe coadjoint orbit Oµ ∼= G/Gµ ∼= GC/P where Gµ is isotropysubgroup of G and P is the parabolic subgroup.

Every flag variety GC/P where P is parabolic subgroup, admits afinite number of invariant Kahler structures. In particular thecomplex presentation GC/P gives rise to an finite number ofinvariant complex structures (i.e. integrable almost complexstructures commuting with the isotropy representation of GC/P .

So, coadjoint orbit is a kahler manifold.


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Berezin-Toeplitz Quantization


One of the important facts of Berezin-Toeplitz Quantization scheme isthat for quantizable Kaehler manifolds which are algebraicly projectivethe operator quantization supplies canonically defined quantization.

BT quantization on compact Kaehler

M.Schlichenmaier, showed that for compact Kaehler manifoldsBerezin-Toeplitz Quantization scheme is a well-defined quantizationscheme with correct semiclassical limit. As we said before, compactCoadjoint orbits are compact Kaehler manifolds and we can applySchlichenmaier’s method to do Berezin-Toeplitz Quantization forcoadjoint orbits when Lie group G is supposed to be compact.


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Let G be compact Lie group and the coadjoint orbitOµ ∼= G/Gµ ∼= GC/P be quantizable, then by Kodaira’sembedding theorem the homogeneous Line bundle L = G×Gµ C isample and there is a tensor power Lk such that the globalholomorphic sections of Lk can be used to embed the coadjoint Oµinto the projective space of some dimension.In fact takes0, s1, ..., sN as basis for Γhol(Oµ, L

k), and let s(z) be a localholomorphic frame. Then we can write si(z) = sj(z)s(z). So theembedding on coadjoint orbit is given by

Oµ → PN (C)

z 7→ (s0(z) : s1(z) : ... : sN (z))

So, the quantizable compact coadjoint orbits are complexsubmanifolds of projective spaces and they can be given as zerosets of homogeneous polynomials.


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Let Γ∞(Oµ, Lk) be the space of global C∞ sections and L2(Oµ, L

k) bethe completion of Γ∞(Oµ, L

k) and Γhol(Oµ, Lk) be the subspace of

global holomorphic sections which is finite dimensional and closedsubspace due to compactness of coadjoint orbit Oµ. Now define thefollowing projection

Π(k) : L2(Oµ, Lk)→ Γhol(Oµ, L

k)

Definition

Let f ∈ C∞(Oµ) the Toeplitz operator T(k)f is defined by

T(k)f := Π(k)(f.) : Γhol(Oµ, L

k)→ Γhol(Oµ, Lk)

which this means that we take a holomorphis section s and multipliesit with the differentiable function f and f.s will be differentiable.


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Berezin-Toeplitz operator

Now the linear map

T (k) : C∞(Oµ)→ End(Γhol(Oµ, Lk))

f 7→ T(k)f = Π(k)(f.)

which is surjective is called Berezin-Toeplitz quantization map of levelk. Also this operator which is known as Toeplitz operator is selfadjoint.

So, Berezin-Toeplitz quantization map is a map from theinfinite-dimensional commutative algebra of functions to anoncommutative finite dimensional algebra.


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Definition

The Berezin-Toeplitz Quantization on coadjoint orbit Oµ is the map

C∞(Oµ)→∏k∈N0

End(Γhol(Oµ, Lk))

f →(T(k)f

)k∈N0

But by using this fact that T ∗f respects the grading given by k, it canbe considered as an element of

End

⊕k∈N0

Γhol(Oµ, Lk)

and this is just End of homogeneous coordinate ring of coadjoint orbitOµ. This means that we interperated the Berezin-Toeplitz quantizationin terms of homogeneous coordinate ring of coadjoint orbit Oµ.


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Now we try to find a relationship between inner product onquantum Hilbert space of coadjoint orbits and coherent states.

Let (L, h) be a quantization of a Kaehler manifold (Oµ, ω), whichL here is homogeneous Line bundle and h is hermitian form. Letx ∈ Oµ and q ∈ L∗ a point of the fibre over x which L∗ here meansthe total space of L with the zero section removed. Also assume Hbe separable complex Hilbert space consisting of globalholomorphic sections s of L which are bounded wrt

< s, s >h= ‖s‖2h =

∫Oµ

h(s(x), s(x))ωn(x)

n!

If we evaluate s ∈ H at x then one can write s(x) = δq(s)q whereδq : H → C is a continuous linear functional map then by Riesz’stheorem, there exists a unique eq ∈ H such that δq(s) =< s, eq >hso

s(x) =< s, eq >h q


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Definition

The holomorphic section eq is called the coherent states relative to thepoint q.

Definition

Rawnsley’s epsilon function: is a real valued function on Oµ by theformula

εh(x) = h(q, q) ‖eq‖2hwhere q ∈ L∗

Now assume that (s0, ..., sN ) be a unitary basis for (H, <,>h). Bytaking λi ∈ C such that sj(x) = λjq, j = 0, ..., N the by simplecalculation we get

s(x) =< s,

N∑j=0

λjsj >h q

so we will have eq =∑N

j=0 λjsj and ε(x) =∑N

j=0 h (sj(x), sj(x)))


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Definition

Coherent projectors: For s, t ∈ Γhol(Oµ, L) the symbol | s >< t |denotes the following operator of Γhol(Oµ, L)

| s >< t |: r →< t, r > .s

And the coherent projector is defined by

Pπ(q) =| eq >< eq |< eq, eq >

Theorem

The inner product on Hilbert space of Γhol(Oµ, L) can be formulated bycoherent states and we have

< s1, s2 >=

∫Oµ

< s1, Pxs2 > ε(x)Ω(x)


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Now we must compute the Rawnsley ε-Function for coadjoint orbitOµ.

We need to following theorem to finding Rawnsley’s ε-Function forcoadjoint orbit Oµ.

Theorem

Let (M,ω) be a simply connected Kaehler manifold which the groupSym(M,ω) ∩Aut(M) acts transitively on (M,ω) then the Rawnsley’sε-Function is constant.

We need also to following remark

Theorem

Let G be a compact and connected Lie group then the coadjoint orbitOµ ∼= G/Gµ which µ ∈ g∗ is simply connected.


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Theorem

Rawnsley’s ε-function on the coadjoint orbit is constant.

Proof: Since Oµ is a G-orbit, G acts transitively on it. Also, everyaction automorphism coming from G preserves the symplectic structureon Oµ, and therefore belongs to Symp(Oµ, ω) ∩Aut(Oµ). and as wesaid before, the coadjoint orbit Oµ is simply connected Kaehlermanifold, so Rawnsley’s ε-function on the coadjoint orbit is constant.


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Now the question is what is the this constant on coadjoint orbitOµ. We try to find it

Let N = dimΓhol(Oµ, L)− 1 then by simple computation we have

N + 1 =

∫Oµ

N∑j=0

h(sj , sj)ωn

n!=

∫Oµεωn

n!= εVol(Oµ, ω)

So, we arrive at the following formula

ε =dimΓhol(Oµ, L)

Vol(Oµ, ω)

So, we try to find Vol(Oµ, ω). But by using Kirillov’s characterformula we have following√

j(ξ)χOµ(expξ) =

∫Oµ

e2πi<ζ,ξ>dζ


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But we can write√j(ξ) =

∏α>0

e< α,ξ>/2 − e−<α,ξ>/2

< α, ξ >

and if here we take ξ = 0 then we can compute te volume as follows

Vol(Oµ) =∏α>0

< α, ζ >

< α, ρ >

where ρ = 12

∑α>0 α

Moreover we can compute dimΓhol(Oµ, L) by using Weyldimension formula or Hirzebruch–Riemann–Roch theorem (whichis near to Vol(Oµ))

So, if we rewrite again the corrected(in metaplectic sense)inner

product, on polarized sections of Γ(L⊗ Λ12 (P⊥))

< s1 ⊗ σ1, s2 ⊗ σ2 >=

∫Oµ

< s1, Pxs2 > ε(x)σ1σ2Ω(x)

via coherent states:Hassan Jolany, University of Lille1

Saturday 26th July, 2014 25 /26

Sixth Conference of the Euro-American Consortium for Promotingthe Application of Mathematics in Technical and Natural Sciences,Albena, Bulgaria, June 26-July 1, 2014

Thanks for your attention.


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Berezin-Toeplitz Quantization On Coadjoint Orbits

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