Equivariant quantization of orbifolds

E

Norbert Poncin

Mathematics Research Unit University of Luxembourg

Current Geometry, Levi-Civita Institute, Vietri, 2010

O

Equivariant quantization of vector spaces

smooth manifolds foliated manifolds orbifolds

supermanifolds People:

A. Cap, M. Bordemann, C. Duval, H. Gargoubi, J. George, P.

Lecomte, P. Mathonet, J.-P. Michel, V. Ovsienko, F. Radoux, J. Silhan, R. Wolak, ..., P

G

S

D ∈ Hom_R(Γ(E ), Γ(F ))_loc^loc' Hom_R(C^∞(U, ¯E ),C^∞(U, ¯F ))_loc

f ∈ C^∞(U), e ∈ ¯E , x ∈ U, ξ ∈ (R^m)^∗ D(fe) = P

αDα,x(e)∂_x^α1¹. . . ∂_x^αm^mf 'P

αD_α,x

x

(e

e

)ξ₁^α1. . . ξ^α_m^m

= σ_aff(D)(ξ;e)

Differential operator^loc' polynomial, total affine symbol D ∈ Hom_R(Γ(E ), Γ(F ))_loc^loc'

σ_aff(D) ∈Γ(STU⊗E^∗⊗ F)

S

D ∈ Hom_R(Γ(E ), Γ(F ))_loc^loc' Hom_R(C^∞(U, ¯E ),C^∞(U, ¯F ))_loc

f ∈ C^∞(U), e ∈ ¯E , x ∈ U, ξ ∈ (R^m)^∗ D(fe) = P

αDα,x(e)∂_x^α1¹. . . ∂_x^αm^mf 'P

αD_α,

x

x(e

e

)ξ₁^α1. . . ξ^α_m^m

= σ_aff(D)(ξ;e)

Differential operator^loc' polynomial, total affine symbol D ∈ Hom (Γ(E ), Γ(F )) ^loc' σ (D) ∈Γ(STU⊗E^∗⊗ F)

S

D ∈ Hom_R(Γ(E ), Γ(F ))_loc^loc' Hom_R(C^∞(U, ¯E ),C^∞(U, ¯F ))_loc

f ∈ C^∞(U), e ∈ ¯E , x ∈ U, ξ ∈ (R^m)^∗ D(fe) = P

αDα,x(e)∂_x^α1¹. . . ∂_x^αm^mf 'P

αD_α,

x

x(

e

e)ξ₁^α1. . . ξ^α_m^m

= σ_aff(D)(ξ;e)

Differential operator^loc' polynomial, total affine symbol D ∈ Hom_R(Γ(E ), Γ(F ))_loc^loc' σ_aff(D) ∈Γ(STU⊗E^∗⊗ F)

S

Intertwining condition:

(L_X(TD) − T (L_XD)) (ω) =0 Symbolic representation:

(X .T )(η;D)(ξ; ω) − hX , ηi ((τ_ζT )(η; D)) (ξ; ω)

−hX , ξi (τζ(T (η; D))) (ξ; ω) + T (η + ζ; X τ_ζD)(ξ; ω)

−T (η; D)(ξ + ζ; ζ ∧ i_Xω) +T (η + ζ; D(· + ζ; ζ ∧ i_X·))(ξ; ω) = 0 Applications:

Flato-Lichnerowicz, De Wilde-Lecomte: cohomology of vector

E

M = R^m, D ∈ D(M),φ: coordinate change D(f ) =P

αDα,x ∂_x^αf ^σ↔^aff σ_aff(D)(ξ) =P

αDα,x ξ^α

lφ lφ

. . . ^σ↔^aff . . .

Non commutative,σaff(D) not intrinsic,σ(D) geometric meaning Vector space isomorphism:

σ⁻¹_aff :Pol(T^∗M) = Γ(STM) =: S(M)→D(M) Nonequivariance (global version):

∃φ ∈ Diff(M) : σ⁻¹_aff ◦φ6=φ◦ σ⁻¹_aff

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M = R^m, D ∈ D(M),φ: coordinate change D(f ) =P

αDα,x ∂_x^αf ^σ↔^aff σ_aff(D)(ξ) =P

αDα,x ξ^α

lφ lφ

. . . ^σ↔^aff . . .

Non commutative,σ_aff(D) not intrinsic,σ(D) geometric meaning

Vector space isomorphism:

σ⁻¹_aff :Pol(T^∗M) = Γ(STM) =: S(M)→D(M) Nonequivariance (global version):

∃φ ∈ Diff(M) : σ⁻¹_aff ◦φ6=φ◦ σ⁻¹_aff

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M = R^m, D ∈ D(M),φ: coordinate change D(f ) =P

αDα,x ∂_x^αf ^σ↔^aff σ_aff(D)(ξ) =P

αDα,x ξ^α

lφ lφ

. . . ^σ↔^aff . . .

Non commutative,σ_aff(D) not intrinsic,σ(D) geometric meaning Vector space isomorphism:

σ⁻¹_aff :Pol(T^∗M) = Γ(STM) =: S(M)→D(M) Nonequivariance (global version):

∃φ ∈ Diff(M) : σ⁻¹_aff ◦φ6=φ◦ σ⁻¹_aff

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∃X ∈ X (M) : σ⁻¹_aff ◦L_X 6=L_X ◦ σ_aff⁻¹

Definition of an

g-

equivariant quantization (EQ) on a vector space [Lecomte, Ovsienko, 99]:

Q = σ_tot⁻¹: S

δ=µ−λ

(M) = Γ(STM

⊗∆^δTM

) ^vs−isom→ D

λµ

(M), such that

σ⁻¹_tot ◦ LX = LX ◦ σ_tot⁻¹, ∀X ∈

g⊂

X (M) and

σ ◦ σ⁻¹_tot |_Sk

δ

(M) = id_S^k

δ

(M) (normalization)

P. Lecomte, P. Mathonet, E. Tousset, 96:D_λλ(M) and D_µµ(M) not isomorphic as X (M)-modules

E

∃X ∈ X (M) : σ⁻¹_aff ◦L_X 6=L_X ◦ σ_aff⁻¹

Definition of an

g-

equivariant quantization (EQ) on a vector space [Lecomte, Ovsienko, 99]:

Q = σ_tot⁻¹: S

δ=µ−λ

(M) = Γ(STM

⊗∆^δTM

) ^vs−isom→ D

λµ

(M), such that

σ⁻¹_tot ◦ LX = LX ◦ σ_tot⁻¹, ∀X ∈

g⊂

X (M) and

σ ◦ σ⁻¹_tot |_Sk

δ

(M) = id_S^k

δ

(M) (normalization)

P. Lecomte, P. Mathonet, E. Tousset, 96:D_λλ(M) and D_µµ(M) not isomorphic as X (M)-modules

E

∃X ∈ X (M) : σ⁻¹_aff ◦L_X 6=L_X ◦ σ_aff⁻¹

Definition of an

g-

equivariant quantization (EQ) on a vector space [Lecomte, Ovsienko, 99]:

Q = σ⁻¹_tot : S

δ=µ−λ

(M) = Γ(STM

⊗∆^δTM

) ^vs−isom→ D

λµ

(M), such that

σ⁻¹_tot ◦ LX = LX ◦ σ_tot⁻¹, ∀X ∈

g⊂

X (M) and

σ ◦ σ⁻¹_tot |_Sk

δ

(M) = id_S^k

δ

(M) (normalization)

P. Lecomte, P. Mathonet, E. Tousset, 96:D_λλ(M) and D_µµ(M) not isomorphic as X (M)-modules

E

∃X ∈ X (M) : σ⁻¹_aff ◦L_X 6=L_X ◦ σ_aff⁻¹

Definition of an

g-

equivariant quantization (EQ) on a vector space [Lecomte, Ovsienko, 99]:

Q = σ⁻¹_tot : S_δ=µ−λ(M) = Γ(STM ⊗∆^δTM) ^vs−isom→ D_λµ(M), such that

σ⁻¹_tot ◦ LX = LX ◦ σ_tot⁻¹, ∀X ∈

g⊂

X (M) and

σ ◦ σ⁻¹_tot |_Sk

δ(M) = id_S^k_δ(M) (normalization)

P. Lecomte, P. Mathonet, E. Tousset, 96:D_λλ(M) and D_µµ(M) not isomorphic as X (M)-modules

E

∃X ∈ X (M) : σ⁻¹_aff ◦L_X 6=L_X ◦ σ_aff⁻¹

Definition of an

g-

equivariant quantization (EQ) on a vector space [Lecomte, Ovsienko, 99]:

Q = σ⁻¹_tot : S_δ=µ−λ(M) = Γ(STM ⊗∆^δTM) ^vs−isom→ D_λµ(M), such that

σ⁻¹_tot ◦ LX = LX ◦ σ_tot⁻¹, ∀X ∈

g⊂

X (M) and

σ ◦ σ⁻¹_tot |_Sk

δ(M) = id_S^k_δ(M) (normalization)

E

∃X ∈ X (M) : σ⁻¹_aff ◦L_X 6=L_X ◦ σ_aff⁻¹

Definition of ang-equivariant quantization (EQ) on a vector space [Lecomte, Ovsienko, 99]:

Q = σ⁻¹_tot : S_δ=µ−λ(M) = Γ(STM ⊗∆^δTM) ^vs−isom→ D_λµ(M), such that

σ⁻¹_tot ◦ LX = LX ◦ σ_tot⁻¹, ∀X ∈g⊂X (M) and

σ ◦ σ⁻¹_tot |_Sk

δ(M) = id_S^k_δ(M) (normalization)

P. Lecomte, P. Mathonet, E. Tousset, 96:D_λλ(M) and D_µµ(M) not isomorphic as X (M)-modules

E

Some motivations:

Invariant star-productson T^∗M obtained as pullbacks by Q_~(P) = ~^kQ(P), P ∈ Pol^k(T^∗M)

of the associative structure of the space of differential operators

Classification of spaces of differential operators as modulesover Lie algebras of vector fields

Role of symmetries in relationship between classical and quantum systems – complete geometric characterization of quantization: a flatfixed G-structureon configuration

P

Maximal Lie subalgebras g ⊂ X (M), M = R^m:

Projective case:G = PGL(m + 1, R),g= sl(m + 1, R)can be embedded as maximal Lie subalgebrasln+1 into X_∗(M) (Lecomte, Ovsienko, 99)

Conformal case:G = SO(p + 1, q + 1)(p + q = m), g= o(p + 1, q + 1)can be embedded as maximal Lie

subalgebrao_p+1,q+1into X∗(M) (Duval, Lecomte, Ovsienko, 99) Other cases: ... (Boniver, Mathonet, 01)

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C

D

Projectively equivariant quantization fordifferential operators on differential forms[Boniver, Hansoul, Mathonet, P, 02]

Efficiency ofequivariant and standard affine symbol calculusas classification tools for modules of differential operators [P,04]

Automorphisms and derivations of classical and quantum Poisson algebras[Grabowski, P, 04], [Grabowski, P, 05]

Difference withautomorphisms of the classical and the quantum Weyl algebra[Kanel, Kontsevich, 05]

C

DO

Q : (S_kp,L_X) → (D^k_p ' S_kp, L_X): potential sl_m+1-EQ Main observation:

Q ◦ C = C ◦ Q CP = αP ⇒ CQP = αQP Ideas:

Diagonalization of C: S_p^k = A^k_p⊕ B_p^k, eigenvalues α^k_p, β_p^k C − C =N (L_X − L_X)

| {z }

= _m+1¹ ( δ Div δ^∗ +δ^∗Div δ) S_p^k → S_p^{k −1}

C

DO

Since

α^k_pP+

∈A^{k −1}_p

z }| {

α^k_pQ₁P=QCP = CQP = C(P + Q₁P) =

(C + N) (P + Q₁P) = α^k_pP+

∈A^{k −1}_p

z }| {

α^{k −1}_p Q₁P + NP+

∈A^{k −2}_p

z }| { NQ₁P,

we getQ₁P = ¹

α^k_p−α^{k −1}_p 1

m+1(δ Div δ^∗P) and have to setQP = P +Pk

`=1Q<sub>`</sub>P, Q<sub>`</sub>: A<sup>k</sup><sub>p</sub> → A<sup>k −`</sup>_p Q |_Sk

p= id +Pk

`=1Q<sub>`</sub>, Q_`=

 1 m+1

`

Π_1≤j≤` ¹

α^kp−α^{k −j}_p



(δ Div δ^∗)^`+



Π_1≤j≤` ¹

β^kp−β^{k −j}_p



(δ^∗Div δ)^`



C

DO

Since

α^k_pP+

∈A^{k −1}_p

z }| {

α^k_pQ₁P=QCP = CQP = C(P + Q₁P) =

(C + N) (P + Q₁P) = α^k_pP+

∈A^{k −1}_p

z }| {

α^{k −1}_p Q₁P + NP+

∈A^{k −2}_p

z }| { NQ₁P, we getQ₁P = ¹

α^k_p−α^{k −1}_p 1

m+1(δ Div δ^∗P) and have to setQP = P +Pk

`=1Q<sub>`</sub>P, Q<sub>`</sub>: A<sup>k</sup><sub>p</sub> → A<sup>k −`</sup>_p

Q |_Sk

p= id +Pk

`=1Q<sub>`</sub>, Q_`=

 1 m+1

`

Π_1≤j≤` ¹

α^kp−α^{k −j}_p



(δ Div δ^∗)^`+



Π_1≤j≤` ¹

β^kp−β^{k −j}_p



(δ^∗Div δ)^`



C

DO

Since

α^k_pP+

∈A^{k −1}_p

z }| {

α^k_pQ₁P=QCP = CQP = C(P + Q₁P) =

(C + N) (P + Q₁P) = α^k_pP+

∈A^{k −1}_p

z }| {

α^{k −1}_p Q₁P + NP+

∈A^{k −2}_p

z }| { NQ₁P, we getQ₁P = ¹

α^k_p−α^{k −1}_p 1

m+1(δ Div δ^∗P) and have to setQP = P +Pk

`=1Q<sub>`</sub>P, Q<sub>`</sub>: A<sup>k</sup><sub>p</sub> → A<sup>k −`</sup>_p Q |_Sk

p= id +Pk

`=1Q<sub>`</sub>, Q_`=

 1 m+1

`

Π_1≤j≤` ¹

α^k_p−α^{k −j}_p



(δ Div δ^∗)^`+



Π_1≤j≤` ¹

β^k_p−β^{k −j}_p



(δ^∗Div δ)^`



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P

EQ

Projective structure on a manifold M

Projective structure straight lines geodesics no canonical connection class of connectionsassociated with the same geodesics

Torsion-free linear connections ∇, ∇⁰ on M areprojectively equivalent, i.e. define thesame geometric geodesics, if and only if (H. Weyl)

∇⁰_XY − ∇_XY = ω(X )Y + ω(Y )X ∈ X (M), where X , Y ∈ X (M), ω ∈ Ω¹(M)

Projective structure on M=class[∇]of projectively equivalent connections

P

EQ

Quantization associated with a connection (A. Lichnerowicz, star-products)

D ∈ D^k(Γ(E ), Γ(F )) ↔P ∈ Γ(S^kTM ⊗ E^∗⊗ F)

∇: covariant derivative of E

∇^k : Γ(E ) 3 f →∇^kf ∈ Γ(S^kT^∗M ⊗ E): iterated symmetrized Q_aff(∇)P: Γ(E ) 3 f → (Q_aff(∇)P) f =i_P(∇^kf )∈ Γ(F )

P

EQ

There is noQ: S → Dsuch that

Q ◦φ^∗=φ^∗◦ Q, ∀φ ∈ Diff(M) i.e.

(Q(φ^∗P)) (φ^∗f ) =φ^∗((QP)(f )) , ∀φ ∈ Diff(M)

Is thereQ(∇): S → Dsuch that

(Q(φ^∗∇)(φ^∗P)) (φ^∗f ) =φ^∗((Q(∇)P)(f )) , for all local diffeomorphisms φ?

P

EQ

There is noQ: S → Dsuch that

Q ◦φ^∗=φ^∗◦ Q, ∀φ ∈ Diff(M) i.e.

(Q(φ^∗P)) (φ^∗f ) =φ^∗((QP)(f )) , ∀φ ∈ Diff(M)

Is thereQ(∇): S → Dsuch that

(Q(φ^∗∇)(φ^∗P)) (φ^∗f ) =φ^∗((Q(∇)P)(f )) ,

P

EQ

Remarks

Example of thegauge principle Q– problem: no solution

Q(∇)– problem: several solutions, standard ordering prescription, Weyl ordering prescription (half-densities)

Group PGL(m + 1, R) does not preserve ∇, but the flows of X ∈ sl_m+1⊂ X_∗(R^m)preserve [∇]: the solution Q = σ⁻¹_sl

m+1

of Lecomte and Ovsienko is intelligible,

if Q =Q[∇] Additional requirement (uniqueness): Q = Q[∇], i.e. Q is projectively invariant

P

EQ

Remarks

Example of thegauge principle Q– problem: no solution

Q(∇)– problem: several solutions, standard ordering prescription, Weyl ordering prescription (half-densities) Group PGL(m + 1, R) does not preserve ∇, but the flows of X ∈ sl_m+1⊂ X_∗(R^m)preserve [∇]: the solution Q = σ⁻¹_sl

m+1

of Lecomte and Ovsienko is intelligible,

if Q =Q[∇] Additional requirement (uniqueness): Q = Q[∇], i.e. Q is projectively invariant

P

EQ

Remarks

Example of thegauge principle Q– problem: no solution

Q(∇)– problem: several solutions, standard ordering prescription, Weyl ordering prescription (half-densities) Group PGL(m + 1, R) does not preserve ∇, but the flows of X ∈ sl_m+1⊂ X_∗(R^m)preserve [∇]: the solution Q = σ⁻¹_sl

m+1

of Lecomte and Ovsienko is intelligible, if Q =Q[∇]

Additional requirement (uniqueness): Q = Q[∇], i.e. Q is projectively invariant

P

EQ

Anaturalandprojectively invariantquantization is a family Q_M : C(M) × S_δ(M) → D_λµ(M),

indexed by smooth manifolds M, s.th., for any ∇ ∈ C(M), Q_M[∇] : S_δ(M) → D_λµ(M)

is a vector space isomorphism that verifies

1 the usual normalization condition

2 for any local diffeomorphism φ of M,

(Q_M[φ^∗∇](φ^∗P)) (φ^∗f ) = φ^∗((Q_M[∇]P)(f )),

P

EQ

Remarks:

Generalization: If Q is a natural and projectively invariant quantization, then Q_R^m[∇₀](∇₀: canonical flat connection) is an sl(m + 1, R)-equivariant quantization

Functorial formulation: M. Bordemann wrote this definition in the language of natural bundles and operators (see I.

Koláˇr, P. W. Michor, J. Slov ˇak)

Existence results: M. Bordemann, 02; P. Mathonet, F.

Radoux, 05; S. Hansoul, 06; A. Cap, J. Silhan, 09 Techniques: Thomas-Whitehead connections, Cartan connections, tractor calculus

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T

-W

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Example M := S^m g ∈ G := GL(m + 1, R)

All gpreservethe canonical connection of R^m+1, the induced φ_g do usuallynot preservethe canonical LC-connection on S^m M˜ := R^m+1\{0} → S^m=:M is a bundle with typical fiber R⁺₀

Lift thecomplexsituation on M to thesimplersituation on ˜M: Constructnaturallifts ∇ → ˜∇, P → ˜P, and f → ˜f

Define a quantization Q[∇] by

(Q[∇](P)(f ) )^˜:=Q_aff( ˜∇)( ˜P)(˜f) Check if ˜∇ only depends upon[∇]

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Example M := S^m g ∈ G := GL(m + 1, R)

All gpreservethe canonical connection of R^m+1, the induced φ_g do usuallynot preservethe canonical LC-connection on S^m M˜ := R^m+1\{0} → S^m=:M is a bundle with typical fiber R⁺₀ Lift thecomplexsituation on M to thesimplersituation on ˜M:

Constructnaturallifts ∇ → ˜∇, P → ˜P, and f → ˜f Define a quantization Q[∇] by

(

Q[∇](P)(f )

)^˜:=Q_aff( ˜∇)( ˜P)(˜f) Check if ˜∇ only depends upon[∇]

T

Example M := S^m g ∈ G := GL(m + 1, R)

All gpreservethe canonical connection of R^m+1, the induced φ_g do usuallynot preservethe canonical LC-connection on S^m M˜ := R^m+1\{0} → S^m=:M is a bundle with typical fiber R⁺₀ Lift thecomplexsituation on M to thesimplersituation on ˜M:

Constructnaturallifts ∇ → ˜∇, P → ˜P, and f → ˜f

Define a quantization Q[∇] by (

Q[∇](P)(f )

)^˜:=Q_aff( ˜∇)( ˜P)(˜f) Check if ˜∇ only depends upon[∇]

T

Example M := S^m g ∈ G := GL(m + 1, R)

All gpreservethe canonical connection of R^m+1, the induced φ_g do usuallynot preservethe canonical LC-connection on S^m M˜ := R^m+1\{0} → S^m=:M is a bundle with typical fiber R⁺₀ Lift thecomplexsituation on M to thesimplersituation on ˜M:

Constructnaturallifts ∇ → ˜∇, P → ˜P, and f → ˜f

Define a quantization Q[∇] by (

Q[∇](P)(f )

)^˜:=

Q_aff( ˜∇)( ˜P)(˜f)

Check if ˜∇ only depends upon[∇]

T

Example M := S^m g ∈ G := GL(m + 1, R)

All gpreservethe canonical connection of R^m+1, the induced φ_g do usuallynot preservethe canonical LC-connection on S^m M˜ := R^m+1\{0} → S^m=:M is a bundle with typical fiber R⁺₀ Lift thecomplexsituation on M to thesimplersituation on ˜M:

Constructnaturallifts ∇ → ˜∇, P → ˜P, and f → ˜f Define a quantization Q[∇] by

(Q[∇](P)(f ) )^˜:=Q_aff( ˜∇)( ˜P)(˜f)

Check if ˜∇ only depends upon[∇]

T

Example M := S^m g ∈ G := GL(m + 1, R)

All gpreservethe canonical connection of R^m+1, the induced φ_g do usuallynot preservethe canonical LC-connection on S^m M˜ := R^m+1\{0} → S^m=:M is a bundle with typical fiber R⁺₀ Lift thecomplexsituation on M to thesimplersituation on ˜M:

Constructnaturallifts ∇ → ˜∇, P → ˜P, and f → ˜f Define a quantization Q[∇] by

(Q[∇](P)(f ) )^˜:=Q_aff( ˜∇)( ˜P)(˜f)

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Standard affine quantization

Q_aff(∇)(P)(f ) = i_P(∇^·f ) is natural, but not projectively invariant

Naturalityof all lifts and naturality of Q_affentails naturality of Q:

(Q[∇](P)(f ))^˜=Q_aff( ˜∇)( ˜P)(˜f) Q[φ^∗_g∇](φ^∗_gP)(φ^∗_gf )
˜

= Q_aff(g^∗∇)(g˜ ^∗P)(g˜ ^∗˜f )

= g^∗Q_aff( ˜∇)( ˜P)(˜f)

= g^∗(Q[∇](P)(f ))^˜

= φ^∗_gQ[∇](P)(f )
˜

Projective invarianceof ˜∇ entails projective invariance of Q

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The point is:

Standard affine quantization

Q_aff(∇)(P)(f ) = i_P(∇^·f ) is natural, but not projectively invariant

Naturalityof all lifts and naturality of Q_affentails naturality of Q:

(Q[∇](P)(f ))^˜=Q_aff( ˜∇)( ˜P)(˜f) Q[φ^∗_g∇](φ^∗_gP)(φ^∗_gf )
˜

= Q_aff(g^∗∇)(g˜ ^∗P)(g˜ ^∗˜f )

= g^∗Q_aff( ˜∇)( ˜P)(˜f)

= g^∗(Q[∇](P)(f ))^˜

= φ^∗_gQ[∇](P)(f )
˜

Projective invarianceof ˜∇ entails projective invariance of Q

T

The point is:

Standard affine quantization

Q_aff(∇)(P)(f ) = i_P(∇^·f ) is natural, but not projectively invariant

Naturalityof all lifts and naturality of Q_affentails naturality of Q:

(Q[∇](P)(f ))^˜=Q_aff( ˜∇)( ˜P)(˜f) Q[φ^∗_g∇](φ^∗_gP)(φ^∗_gf )
˜

= Q_aff(g^∗∇)(g˜ ^∗P)(g˜ ^∗˜f )

= g^∗Q_aff( ˜∇)( ˜P)(˜f)

= g^∗(Q[∇](P)(f ))^˜

= φ^∗_gQ[∇](P)(f )
˜

Projective invarianceof ˜∇ entails projective invariance of Q

T

The point is:

Standard affine quantization

Q_aff(∇)(P)(f ) = i_P(∇^·f ) is natural, but not projectively invariant

Naturalityof all lifts and naturality of Q_affentails naturality of Q:

(Q[∇](P)(f ))^˜=Q_aff( ˜∇)( ˜P)(˜f) Q[φ^∗_g∇](φ^∗_gP)(φ^∗_gf )
˜

= Q_aff(g^∗∇)(g˜ ^∗P)(g˜ ^∗˜f )

= g^∗Q_aff( ˜∇)( ˜P)(˜f)

= g^∗(Q[∇](P)(f ))^˜

˜

G

Extension of the S^m-construction to an arbitrary M, dim M = m

M :=˜

∆¹TM = P¹M ×_GL(m,R)R

+ 0

, GL(m, R) × R

+ 0

3 (g, r ) → g · r =| det g |⁻¹r ∈ R

+ 0

: rank 1

affine

bundle of

positive

1 – densities over M

M × R˜ ⁺₀ 3 ([u, r ], s) → ρs[u, r ] = [u, rs] ∈ ˜M: R⁺₀ – pb

∆^λTM =M ט

R⁺0 R,

R⁺₀ × R 3 (s, t) → s.t = s^λt: line bundle of λ-densities over M f ∈ Γ(∆^λTM) ↔˜f ∈ C^∞( ˜M)_R⁺

0

G

Extension of the S^m-construction to an arbitrary M, dim M = m

M :=˜

∆¹TM = P¹M ×_GL(m,R)R⁺₀,

GL(m, R) × R⁺₀ 3 (g, r ) → g · r =| det g |⁻¹r ∈ R⁺₀: rank 1affinebundle ofpositive1 – densities over M

M × R˜ ⁺₀ 3 ([u, r ], s) → ρs[u, r ] = [u, rs] ∈ ˜M: R⁺₀ – pb

∆^λTM =M ט

R⁺0 R,

R⁺₀ × R 3 (s, t) → s.t = s^λt: line bundle of λ-densities over M f ∈ Γ(∆^λTM) ↔˜f ∈ C^∞( ˜M)_R⁺

0

G

Extension of the S^m-construction to an arbitrary M, dim M = m

M :=∆˜ ¹TM = P¹M ×_GL(m,R)R⁺₀,

GL(m, R) × R⁺₀ 3 (g, r ) → g · r =| det g |⁻¹r ∈ R⁺₀: rank 1affinebundle ofpositive1 – densities over M

M × R˜ ⁺₀ 3 ([u, r ], s) → ρs[u, r ] = [u, rs] ∈ ˜M: R⁺₀ – pb

∆^λTM =M ט

R⁺0 R,

R⁺₀ × R 3 (s, t) → s.t = s^λt: line bundle of λ-densities over M f ∈ Γ(∆^λTM) ↔˜f ∈ C^∞( ˜M)_R⁺

0

G

Extension of the S^m-construction to an arbitrary M, dim M = m

M :=∆˜ ¹TM = P¹M ×_GL(m,R)R⁺₀,

GL(m, R) × R⁺₀ 3 (g, r ) → g · r =| det g |⁻¹r ∈ R⁺₀: rank 1affinebundle ofpositive1 – densities over M M × R˜ ⁺₀ 3 ([u, r ], s) → ρs[u, r ] = [u, rs] ∈ ˜M: R⁺₀ – pb

∆^λTM =M ט

R⁺0 R,

R⁺₀ × R 3 (s, t) → s.t = s^λt: line bundle of λ-densities over M f ∈ Γ(∆^λTM) ↔˜f ∈ C^∞( ˜M)_R⁺

0

G

Extension of the S^m-construction to an arbitrary M, dim M = m

M :=∆˜ ¹TM = P¹M ×_GL(m,R)R⁺₀,

GL(m, R) × R⁺₀ 3 (g, r ) → g · r =| det g |⁻¹r ∈ R⁺₀: rank 1affinebundle ofpositive1 – densities over M M × R˜ ⁺₀ 3 ([u, r ], s) → ρs[u, r ] = [u, rs] ∈ ˜M: R⁺₀ – pb

∆^λTM =M ט

R⁺0 R,

R⁺₀ × R 3 (s, t) → s.t = s^λt:

line bundle of λ-densities over M

f ∈ Γ(∆^λTM) ↔˜f ∈ C^∞( ˜M)_R⁺

0

G

Extension of the S^m-construction to an arbitrary M, dim M = m

M :=∆˜ ¹TM = P¹M ×_GL(m,R)R⁺₀,

GL(m, R) × R⁺₀ 3 (g, r ) → g · r =| det g |⁻¹r ∈ R⁺₀: rank 1affinebundle ofpositive1 – densities over M M × R˜ ⁺₀ 3 ([u, r ], s) → ρs[u, r ] = [u, rs] ∈ ˜M: R⁺₀ – pb

∆^λTM =M ט

R⁺0 R,

R⁺₀ × R 3 (s, t) → s.t = s^λt:

line bundle of λ-densities over M

G

LiftsP˜ and∇˜ are technical

Essential remark: Natural and projectively invariant lift ˜∇ is a connection of some known type

Study of projective equivalence of connections Origins: Goes back to the twenties and thirties

Objective: Associate aunique connectionto each projective structure [∇]

First answer: Thomas-Whitehead projective connection (T.Y. Thomas, J.H.C. Whitehead, O. Veblen)

Second answer: Cartan projective connection(E. Cartan)

G

LiftsP˜ and∇˜ are technical

Essential remark: Natural and projectively invariant lift ˜∇ is a connection of some known type

Study of projective equivalence of connections Origins: Goes back to the twenties and thirties Objective: Associate aunique connectionto each projective structure [∇]

First answer: Thomas-Whitehead projective connection (T.Y. Thomas, J.H.C. Whitehead, O. Veblen)

T

-W

A Thomas-Whitehead projective connection is a torsion-free linear connection ˜∇ over ˜M, such that

1 ∇E˜ = ¹

m+1id

2 ρ_s∗ ∇˜_XY
= ˜∇_ρs∗Xρ_s∗Y , ∀X , Y ∈ X ( ˜M)

Bordemann’s connection is a Thomas-Whitehead connection

S. Hansoul extended the work of M. Bordemann from tensor densities to sections of arbitrary vector bundles associated with the principle bundle of linear frames

T

C

C

P²M = { j²₀(ψ) | ψ :0 ∈ U ⊂ R^m→ M, det ψ_∗0 6= 0 }:

2nd order frame bundle

G_m² = { j²₀(f ) |f : 0 ∈ U ⊂ R^m→ R^m,f (0) = 0, det f_∗06= 0}:

structure group

G= PGL(m + 1, R) acts on RP^m H =G_[em+1]=

 A 0 α a



:A ∈ GL(m, R), α ∈ R^m∗,a 6= 0

 /R0id acts locally on R^m byR^m ⊃ U 3 Z 7→ _{αZ +a}^AZ ∈ R^m: H ⊂ G_m²

C

Proposition:

Reductions P(M, H) of P²M to H ⊂ G_m² are 1-to-1 with projective structures [∇]

Theorem:

To every projective structure [∇] ' P(M, H) is associated a unique normal Cartan connection ω

Definition:

G: Lie group,H: closed subgroup, g,h: Lie algebras P = P(M, H): PB s.th. dim M = dim G/H

A Cartan connection on P is a 1-form ω ∈ Ω¹(P) ⊗gs.th.

r^∗ω = Ad(s⁻¹)ω (rsright action of s ∈ H)

E

EQ

C

P = P(M, H) (! ˜M): projective structure ω (! ˜∇): normal Cartan connection

I LiftsS˜ of symbols and˜f of densities to objects on P = P(M, H):

S_δ^k(M) = Γ(S^kTM ⊗ ∆^δTM) =C^∞(P¹M,S^kR^m⊗ ∆^δR^m)_GL(m,R) Γ(∆^λTM) =C^∞(P¹M,∆^λR^m)_GL(m,R)

(V , ρ): representation of GL(m, R) C^∞(P¹M,V)_GL(m,R)'C^∞(P, V )_H

E

EQ

C

P = P(M, H) (! ˜M): projective structure ω (! ˜∇): normal Cartan connection

I LiftsS˜ of symbols and˜f of densities to objects on P = P(M, H):

S_δ^k(M) = Γ(S^kTM ⊗ ∆^δTM) =C^∞(P¹M,S^kR^m⊗ ∆^δR^m)_GL(m,R) Γ(∆^λTM) =C^∞(P¹M,∆^λR^m)_GL(m,R)

(V , ρ): representation of GL(m, R) C^∞(P¹M,V)_GL(m,R)

'C^∞(P, V )_H

E

EQ

C

P = P(M, H) (! ˜M): projective structure ω (! ˜∇): normal Cartan connection

I LiftsS˜ of symbols and˜f of densities to objects on P = P(M, H):

S_δ^k(M) = Γ(S^kTM ⊗ ∆^δTM) =C^∞(P¹M,S^kR^m⊗ ∆^δR^m)_GL(m,R) Γ(∆^λTM) =C^∞(P¹M,∆^λR^m)_GL(m,R)

(V , ρ): representation of GL(m, R) C^∞(P¹M,V)_GL(m,R)

'C^∞(P, V )_H

E

EQ

C

P = P(M, H) (! ˜M): projective structure ω (! ˜∇): normal Cartan connection

I LiftsS˜ of symbols and˜f of densities to objects on P = P(M, H):

S_δ^k(M) = Γ(S^kTM ⊗ ∆^δTM) =C^∞(P¹M,S^kR^m⊗ ∆^δR^m)_GL(m,R) Γ(∆^λTM) =C^∞(P¹M,∆^λR^m)_GL(m,R)

(V , ρ): representation of GL(m, R) C^∞(P¹M,V)_GL(m,R)'C^∞(P, V )_H

E

EQ

C

(Q_M[∇](S)(f ))^˜=Q_aff(ω)( ˜S)(˜f) = i_S˜

(∇^ω)^k˜f

I Derivative associated with ω:

∇^ω :C^∞(P, V ) → C^∞(P,R^m∗⊗ V ) g ∈ C^∞(P, V ), u ∈ P,v ∈ R^m (∇^ωg) (u)(v) =

L

X (P)3X (ω,v )=ω⁻¹v

g

(u) ∈ V

ωu∈ Isom(TuP, g), g = sl(m + 1, R) ' R^m⊕ gl(m, R) ⊕ R^m∗3 v (∇^ω)^k :C^∞(P, V ) → C^∞(P,S^kR^m∗⊗ V ): iterated symmetrized I Problem:

˜f ∈ C^∞(P, ∆^λR^m)_H, (∇^ω)^k˜f ∈ C^∞(P,S^kR^m∗⊗ ∆^λR^m) S ∈ C˜ ^∞(P,S^kR^m⊗ ∆^δ=µ−λR^m)_H

i_S˜

(∇^ω)^k˜f

∈ C^∞(P, ∆^µR^m):not H-equivariant

E

EQ

C

(Q_M[∇](S)(f ))^˜=Q_aff(ω)( ˜S)(˜f) = i_S˜

(∇^ω)^k˜f I Derivative associated with ω:

∇^ω :C^∞(P, V ) → C^∞(P,R^m∗⊗ V )

g ∈ C^∞(P, V ), u ∈ P,v ∈ R^m (∇^ωg) (u)(v) =

L

X (P)3X (ω,v )=ω⁻¹v

g

(u) ∈ V

ωu∈ Isom(TuP, g), g = sl(m + 1, R) ' R^m⊕ gl(m, R) ⊕ R^m∗3 v (∇^ω)^k :C^∞(P, V ) → C^∞(P,S^kR^m∗⊗ V ): iterated symmetrized I Problem:

˜f ∈ C^∞(P, ∆^λR^m)_H, (∇^ω)^k˜f ∈ C^∞(P,S^kR^m∗⊗ ∆^λR^m) S ∈ C˜ ^∞(P,S^kR^m⊗ ∆^δ=µ−λR^m)_H

i_S˜

(∇^ω)^k˜f

∈ C^∞(P, ∆^µR^m):not H-equivariant

E

EQ

C

(Q_M[∇](S)(f ))^˜=Q_aff(ω)( ˜S)(˜f) = i_S˜

(∇^ω)^k˜f I Derivative associated with ω:

∇^ω :C^∞(P, V ) → C^∞(P,R^m∗⊗ V ) g ∈ C^∞(P, V ), u ∈ P,v ∈ R^m (∇^ωg) (u)(v) =

 L

X (P)3X (ω,v )=ω⁻¹v

g

(u) ∈ V

ωu∈ Isom(TuP, g), g = sl(m + 1, R) ' R^m⊕ gl(m, R) ⊕ R^m∗3 v (∇^ω)^k :C^∞(P, V ) → C^∞(P,S^kR^m∗⊗ V ): iterated symmetrized I Problem:

˜f ∈ C^∞(P, ∆^λR^m)_H, (∇^ω)^k˜f ∈ C^∞(P,S^kR^m∗⊗ ∆^λR^m) S ∈ C˜ ^∞(P,S^kR^m⊗ ∆^δ=µ−λR^m)_H

i_S˜

(∇^ω)^k˜f

∈ C^∞(P, ∆^µR^m):not H-equivariant

E

EQ

C

(Q_M[∇](S)(f ))^˜=Q_aff(ω)( ˜S)(˜f) = i_S˜

(∇^ω)^k˜f I Derivative associated with ω:

∇^ω :C^∞(P, V ) → C^∞(P,R^m∗⊗ V ) g ∈ C^∞(P, V ), u ∈ P,v ∈ R^m (∇^ωg) (u)(v) =

L

X (P)3X (ω,v )=ω⁻¹v

g

(u) ∈ V

ωu∈ Isom(TuP, g), g = sl(m + 1, R) ' R^m⊕ gl(m, R) ⊕ R^m∗3 v (∇^ω)^k :C^∞(P, V ) → C^∞(P,S^kR^m∗⊗ V ): iterated symmetrized I Problem:

˜f ∈ C^∞(P, ∆^λR^m)_H, (∇^ω)^k˜f ∈ C^∞(P,S^kR^m∗⊗ ∆^λR^m) S ∈ C˜ ^∞(P,S^kR^m⊗ ∆^δ=µ−λR^m)_H

i_S˜

(∇^ω)^k˜f

∈ C^∞(P, ∆^µR^m):not H-equivariant

E

EQ

C

(Q_M[∇](S)(f ))^˜=Q_aff(ω)( ˜S)(˜f) = i_S˜

(∇^ω)^k˜f I Derivative associated with ω:

∇^ω :C^∞(P, V ) → C^∞(P,R^m∗⊗ V ) g ∈ C^∞(P, V ), u ∈ P,v ∈ R^m (∇^ωg) (u)(v) =

LX (P)3X (ω,v )

=ω⁻¹v

g

(u) ∈ V

ωu∈ Isom(TuP, g), g = sl(m + 1, R) ' R^m⊕ gl(m, R) ⊕ R^m∗3 v (∇^ω)^k :C^∞(P, V ) → C^∞(P,S^kR^m∗⊗ V ): iterated symmetrized I Problem:

˜f ∈ C^∞(P, ∆^λR^m)_H, (∇^ω)^k˜f ∈ C^∞(P,S^kR^m∗⊗ ∆^λR^m) S ∈ C˜ ^∞(P,S^kR^m⊗ ∆^δ=µ−λR^m)_H

i_S˜

(∇^ω)^k˜f

∈ C^∞(P, ∆^µR^m):not H-equivariant

E

EQ

C

(Q_M[∇](S)(f ))^˜=Q_aff(ω)( ˜S)(˜f) = i_S˜

(∇^ω)^k˜f I Derivative associated with ω:

∇^ω :C^∞(P, V ) → C^∞(P,R^m∗⊗ V ) g ∈ C^∞(P, V ), u ∈ P,v ∈ R^m (∇^ωg) (u)(v) =

LX (P)3X (ω,v )

=ω⁻¹v

g

(u) ∈ V

ωu∈ Isom(TuP, g), g = sl(m + 1, R) ' R^m⊕ gl(m, R) ⊕ R^m∗3 v

(∇^ω)^k :C^∞(P, V ) → C^∞(P,S^kR^m∗⊗ V ): iterated symmetrized I Problem:

˜f ∈ C^∞(P, ∆^λR^m)_H, (∇^ω)^k˜f ∈ C^∞(P,S^kR^m∗⊗ ∆^λR^m) S ∈ C˜ ^∞(P,S^kR^m⊗ ∆^δ=µ−λR^m)_H

i_S˜

(∇^ω)^k˜f

∈ C^∞(P, ∆^µR^m):not H-equivariant

E

EQ

C

(Q_M[∇](S)(f ))^˜=Q_aff(ω)( ˜S)(˜f) = i_S˜

(∇^ω)^k˜f I Derivative associated with ω:

∇^ω :C^∞(P, V ) → C^∞(P,R^m∗⊗ V ) g ∈ C^∞(P, V ), u ∈ P,v ∈ R^m (∇^ωg) (u)(v) =

LX (P)3X (ω,v )=ω⁻¹vg

(u) ∈ V

ωu∈ Isom(TuP, g), g = sl(m + 1, R) ' R^m⊕ gl(m, R) ⊕ R^m∗3 v

(∇^ω)^k :C^∞(P, V ) → C^∞(P,S^kR^m∗⊗ V ): iterated symmetrized I Problem:

˜f ∈ C^∞(P, ∆^λR^m)_H, (∇^ω)^k˜f ∈ C^∞(P,S^kR^m∗⊗ ∆^λR^m) S ∈ C˜ ^∞(P,S^kR^m⊗ ∆^δ=µ−λR^m)_H

i_S˜

(∇^ω)^k˜f

∈ C^∞(P, ∆^µR^m):not H-equivariant