geometric quantization on coadjoint orbits

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Geometric Quantization on coadjoint orbits, Workshop on Diffeology etc,25
June 2014, Aix en Province,Marseille, France
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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

List of sections
1 Coadjoint Orbits
2 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 coadjoint
representation Ad∗
: G × g∗ → g∗ by convention:
Ad∗
g µ, X = µ, Adg−1 X
where µ ∈ g∗, g ∈ G, X ∈ g.
Definition
The subset Oµ = {Ad∗
g µ; g ∈ G} of g∗ is called a coadjoint orbit
of 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∗
g µ = µ}

Some Examples of Co-Adjoint Orbits

Complexified of Lie Group
Definition
If G is a Lie group, a universal complexification is given by a
complex Lie group GC and a continuous homomorphism
ϕ : G → GC with the universal property that, if f : G → H is an
arbitrary continuous homomorphism into a complex Lie group H,
then there is a unique complex analytic homomorphism
F : GC → H such that f = Foϕ.
For a classical Real Lie group G, The complexification of Lie
group GC can be defined as
GC
:= exp{g + ig}
Let G = U(n), then GC = GL(n, C) or Let K = SU(n), then
KC = SL(n, C)

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.

Volume of a coadjoint orbits
Kirillov’s Character formula:
Kirillov’s famous formula says that the characters of the irreducible
unitary representations of a compact Lie group G can be written
by following form
χλ(exp(x)) =
1
p(x)
Z
Oλ+ρ
e2πi<µ,x>
dµOλ+ρ
(µ)
where p is a certain function on lie algebra g and it can be written
as
p(x) = det1/2 sinh(ad(x/2))
ad(x/2)
and µ is the highest weight

Volume of a coadjoint orbit
Theorem
The Volume of a Coadjoint Orbit through µ is
Vol(Oµ) =
Y
α∈R+
< α, µ >
< α, ρ >
where ρ = 1
2
P
α∈R+ α

Measure on coadjoint orbits
Explicit formula for measure of coadjoint orbits:
For µ ∈ g∗, we define a skew-symmetric bilinear form bµ on
TeGµ Oµ
∼
= g/gµ by
bµ(ū, v̄) =< µ, [u, v] >
where for u, v ∈ g, we write ū, v̄ ∈ 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(ū, v̄)

Measure on coadjoint orbits
Now choose a basis v̄1, v̄2, ..., v̄m for g/gµ and µ1, µ2, ..., µm be the
dual basis for (g/gµ)∗ (where m is the maximal dimension of an
orbit in g∗)then
1
(m/2)!
bm/2
µ = Pf < µ, [vi , vj ] > µ1 ∧ ... ∧ µm
and
1
(m/2)!
βm/2
µ = Pf−1
< µ, [vi , vj ] > v̄1 ∧ ... ∧ v̄m
where Pf(aij ) denotes the Pfaffian of a skew-symmetric matrix (aij )
Now define,
ωµ =
1
(m/2)!
βm/2
µ
So integration of ωµ gives measure on Oµ up to the factor (2π)m/2

Geometric properties of coadjoint orbits
Every coadjoint orbit is symplectic manifold. The symplectic
form on Oλ is given by
ωλ(ad∗
X, ad∗
Y ) = hλ, [X, Y ]i
. This is obviously anti-symmetric and non-degenerate and
also closed 2-form.
Every coadjoint orbit has Kaehler and Hyper-Kaehler
structure.
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.

Here, by previous theorem we see that the symplectic form on
cotangent bunlde of a Lie group is related to Kostant Kirillov
Souriau symplectic structure of the coadjoint orbits in the
dual of a Lie algebra as follows
G
i //
π

T∗G
Oζ
and we have
π∗
ωOζ
= i∗
ωT∗G
Theorem of Patrick Iglesias-Zemmour:
Theorem
Every connected Hausdorff symplectic manifold is isomorphic to a
coadjoint orbit of its group of hamiltonian diffeomorphisms.

Geometric PDE on coadjoint orbits
Following theorem known as M. Stenzel’s theorem: We denote the
complexified of a coadjoint orbit as
OC
ζ := GC
/GC
ζ
where ζ ∈ g∗
Theorem
Let G be a compact, connected and semisimple, Lie group. There
exists a G invariant, real analytic, strictly plurisubharmonic
function of complexified coadjoint orbit ρ : OC
ζ → [0, ∞) such that
u =
√
ρ satisfies the Monge-Ampere equation,
(∂ ¯
∂u)n
= 0
where n = dim Oλ

Geometric Quantization
Now, we define the geometric quantization by some axioms which
are compatible with physical view.
Geometric quantization we associate to a symplectic manifold
(M, ω) a Hilbert space H, and one associates to smooth
functions f : M → R skew-adjoint operators Of : H → H.
Paul Dirac introduced in his doctoral thesis, the ”method of
classical analogy”for quantization which is now known as
Dirac axioms as follows.
1] Poisson bracket of functions passes to commutator of operators:
O{f ,g} = [Of , Og ]
2] Linearity condition must holds ,Oλ1f +λ2g = λ1Of + λ2Og for
λ1, λ2 ∈ C
3] Normalization condition must holds: 1 7→ i.I(Which I is identity
√

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 functions
{fj } is said to be a complete set of classical observables if and only
if every other function g such that {fi , g} = 0 for all {fj }, is
constant. Also we say that a family of operators is complete if it
acts irreducibly on H
4] Minimality condition must holds: Any complete family of
functions passes to a complete family of operators. Moreover, if G
be a group acting on (M, ω) by symplectomorphisms and on H by
unitary transformations. If the G-action on M is trnsitive, then its
action on H must be irreducible.

Geometric Quantization
Now, we again recall the pre-quantization Line bundle in
formal language. In fact, we have two important method for
Geometric Quantization;
1)Using line bundle. More precisely, In geometric quantization
we construct the Hilbert space H as a subspace of the space
of sections of a line bundle L on a symplectic manifold M.
2)Without using line bundle: Using Mpc-structure instead of
line bundle. One of advantage of this construction is better
behaved of physical view but definig it is not so easy.

Mpc
Quantization
The use of Mpc structures clarifies and extends the traditional
Konstant scheme of geometric in a number of ways.
In fact, prequantized Mpc structures generalize the combined
traditional data consisting of a prequantum line bundle and a
metaplectic structure and are used in constructing and
comparing representations of poisson algebras on symplectic
manifolds.
Also, Mpc structures exists on any symplectic manifold.

Mpc
- Group
Definition
Let (V , Ω) be a real symplectic vector space and fix an irreducible
unitary projective representation W of V on a Hilbert space H
such 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 with
g∗Ω = Ω. So, by uniqueness theorem of Stone and von Neumann
there exists a unitary operator U on H such that
v ∈ V ⇒ W (gv) = UW (v)U−1. The group of all such operators
U as g ranges over the symplectic group Sp(V , Ω) is denoted by
Mpc(V , Ω)

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 character
η : Mpc(V , Ω) → U(1) such that η(λ) = λ2 whenever
λ ∈ U(1).
The kernel of η is a connected double cover of Sp(V , Ω)
called metaplectic group Mp(V , ω)

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 bundle
Sp(E, ω) = Sp(E) to be the principal Sp(V , Ω) bundle over M
whose fibre over m ∈ M consists of all linear isomorphisms
b : V → Em such that b∗ωm = Ω
Now, we are ready to define Mpc-structure.

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).

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 such
that: if z ∈ mpc(V , Ω) = Lie(Mpc(V , Ω)) determines the
fundamental vector field z̄ on P then
γ(z̄) =
1
2
η∗z
and
dγ = (1/i~)π∗
ω
where π : P → M is the bundle projection. We say that (M, ω) is
quantizable iff it admits prequantized Mpc structures.

Mpc
- Quantization
Theorem
(M, ω) is quantizable if and only if
[ω] −
1
2
c1(TM)
be an integral cohomology class.
Theorem
Inequivalent prequantized Mpc structures are parametrized by
H1
(M; U(1))

Mpc
- Quantization on cotangent bundle of Lie groups,
T∗
G
Let G be a Lie group with Lie algebra g . Take, Z = T∗G . Modle
T∗G on the symplectic vector space V = g ⊕ g∗ with symplectic
form
Ω((ξ, φ), (η, ψ)) = φ(η) − ψ(ξ)
The standard left action of G on the cotangent bundle T∗G is
Hamiltonian , with equivariant moment map
J : T∗
G → g∗
with
J(αg ) = g.α

Mpc
T∗
G
A canonical global section b : T∗G → Sp(T∗G) of the symplectic
frame 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 signify
adjoint and coadjoint action.

Mpc
T∗
G
The Mpc structure is the product π : T∗G × Mpc(V ) → T∗G
with projection T∗G × Mpc(V ) → Sp(T∗G) determined by
sending (z, I) to bz; the prequantum form δ is given by
δ =
1
i~
π∗
θ +
1
2
η∗
where θ is the canonical one-form on T∗G and is the flat
connection in π : T∗G × Mpc(V ) → T∗G and η is the unitary
character of Mpc(V ) restricting to U(1) as the squaring map.
The passage of prequantized Mpc structures to Marsden Weinstein
redced phase spaces here gives to us prequantized Mpc structures
on coadjoint orbits.(because Marsden Weinstein redced phase
spaces of T∗G is the coadjoint orbit Oλ )

Mpc
- Quantization on coadjoint orbits
For φ ∈ g∗ we denote Oφ = G.φ ⊂ g∗ its coadjoint orbit we can
identify Oφ
∼
= G/Gφ, which Gφ is coadjoint stabilizer with Lie
algebra gφ, and also we denote the identity component of Gφ with
G0
φ .
Now we explain Robinson and J. H. Rawnsley’s quantization: We
know that the vanishing holonomy generalizes the Keller Maslov
corrected Bohr sommerfeld rule. So we give the following theorem
for quantization on coadjoint orbits

Theorem
Mpc- Quantization on coadjoint orbits: For the coadjoint orbit
Oφ
∼
= G/Gφ, the vanishing holonomy condition amounts to the
following requirement:
That the adjoint isotropy representation Ad : G0
φ → Sp(g/h), given
by Adh(ξ + gφ) = h.ξ + gφ for h ∈ G0
φ and ξ ∈ g should lift to a
homomorphism τ : G0
φ → Mpc(g/h) with the property that
(ηoτ)∗ = − 2
i~φ
Now here we give a connection between two different quantizations
on coadjoint orbits with line bundles and without line bundle,
Theorem
Hassan Jolany’s Theorem: Let the coadjoint orbit Oµ with
pre-quantum line bundle (G ×Gµ C)⊗2 be quantizable then the
coadjoint orbit Oµ is Mpc quantizable.

Theorem
Hassan Jolany’s theorem: A coadjoint orbit (Oφ, ωKKS ) is
Mpc-quantizable if and only if
[ω] −
1
4
X
α∈R+
α
belongs to a lattice Zk

Thanks a lot for your attention
END
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