P. Aschieri - Noncommutative Differential Geometry: Quantization of Connections and Gravity

Balkan Summer Institute 2011

Paolo AschieriU.Piemonte Orientale, Alessandria,

and I.N.F.N. Torino

Donji Milanovac 28.08.2011

Noncommutative Riemannian Geometry and Gravity

JW2011 Scientific and Human Legacy of Julius Wess

I report on a general program in NC geometry where

noncommutativity is given by a *-product. I work in the deformation quantization context, and discuss a quite wide class of *-products obtained by Drinfeld twist.

• *-differential geometry *-Lie derivative, *-covariant derivative

[P.A., Dimitrijevic, Meyer, Wess 06][P.A., Blohmann, Dimitrijevic, Meyer, Schupp Wess 05]

[P.A., Alexander Schenkel 11]

I report on a general program in NC geometry where

noncommutativity is given by a *-product. I work in the deformation quantization context, and discuss a quite wide class of *-products obtained by Drinfeld twist.

• *-differential geometry *-Lie derivative, *-covariant derivative

• *-Riemannian geometry metric structure on NC space

*-gravity

[P.A., Dimitrijevic, Meyer, Wess 06] [P.A., Castellani 09,10]

[P.A., Dimitrijevic, Meyer, Wess 06][P.A., Blohmann, Dimitrijevic, Meyer, Schupp Wess 05]

[P.A., Alexander Schenkel 11]

[P.A., Blohmann, Dimitrijevic, Meyer, Schupp Wess 05]

I report on a general program in NC geometry where

noncommutativity is given by a *-product. I work in the deformation quantization context, and discuss a quite wide class of *-products obtained by Drinfeld twist.

• *-differential geometry *-Lie derivative, *-covariant derivative

• *-Riemannian geometry metric structure on NC space

*-gravity

• *-gauge theories [P.A., Dimitrijevic, Meyer, Schraml, Wess 06]

[P.A., Dimitrijevic, Meyer, Wess 06] [P.A., Castellani 09,10]

[P.A., Dimitrijevic, Meyer, Wess 06][P.A., Blohmann, Dimitrijevic, Meyer, Schupp Wess 05]

[P.A., Alexander Schenkel 11]

[P.A., Blohmann, Dimitrijevic, Meyer, Schupp Wess 05]

I report on a general program in NC geometry where

noncommutativity is given by a *-product. I work in the deformation quantization context, and discuss a quite wide class of *-products obtained by Drinfeld twist.

• *-differential geometry *-Lie derivative, *-covariant derivative

• *-Riemannian geometry metric structure on NC space

*-gravity

• *-gauge theories [P.A., Dimitrijevic, Meyer, Schraml, Wess 06]

• *-Poisson geometry Poisson structure on NC space

*-Hamiltonian mechanics [P.A., Fedele Lizzi, Patrizia Vitale 08] Application: Canonical quantization on noncommutative space, deformed

algebra of creation and annihilation operators.

[P.A., Dimitrijevic, Meyer, Wess 06] [P.A., Castellani 09,10]

[P.A., Dimitrijevic, Meyer, Wess 06][P.A., Blohmann, Dimitrijevic, Meyer, Schupp Wess 05]

[P.A., Alexander Schenkel 11]

[P.A., Blohmann, Dimitrijevic, Meyer, Schupp Wess 05]

I report on a general program in NC geometry where

noncommutativity is given by a *-product. I work in the deformation quantization context, and discuss a quite wide class of *-products obtained by Drinfeld twist.

• *-differential geometry *-Lie derivative, *-covariant derivative

• *-Riemannian geometry metric structure on NC space

*-gravity

• *-gauge theories [P.A., Dimitrijevic, Meyer, Schraml, Wess 06]

• *-Poisson geometry Poisson structure on NC space

*-Hamiltonian mechanics [P.A., Fedele Lizzi, Patrizia Vitale 08] Application: Canonical quantization on noncommutative space, deformed

algebra of creation and annihilation operators.

[P.A., Dimitrijevic, Meyer, Wess 06] [P.A., Castellani 09,10]

[P.A., Dimitrijevic, Meyer, Wess 06][P.A., Blohmann, Dimitrijevic, Meyer, Schupp Wess 05]

[P.A., Alexander Schenkel 11]

[P.A., Blohmann, Dimitrijevic, Meyer, Schupp Wess 05]

I

II(U. Wurzburg)

Why Noncommutative Geometry?

• Classical Mechanics

• Quantum mechanics It is not possible to know

at the same time position and velocity. Observables become noncommutative.

= 6.626 x10-34 Joule s Panck constant

• Classical Gravity

• Quantum Gravity Spacetime coordinates become noncommutative

LP = 10-33 cm Planck length

We consider a -product geometry,

The class of -products we consider is not the most general one studied by Kontsevich. It is not obtained by deforming directly spacetime. We first consider a twist of spacetime symmetries, and then deform spacetime itself.

• Below Planck scales it is natural to conceive a more general spacetime structure. A cell-like or lattice like structure, or a noncommutative one where (as in quantum mechanics phase-space) uncertainty relations and discretization naturally arise.

• Space and time are then described by a Noncommutative Geometry

• It is interesting to understand if on this spacetime one can consistently formulate a gravity theory. I see NC gravity as an effective theory. This theory may capture some aspects of a quantum gravity theory.

NC geometry approaches

• Generators and relations. For example

[xi, xj] = iθij canonical

[xi, xj] = ifijkxk Lie algebra

xixj − qxjxi = 0 quantum plane (1)

Quantum groups and quantum spaces are usually described in this way.

• -product approach, a new product is considered in the usual space of func-tions, it is given by a bi-differential operator B(f, g) ≡ f g that is associative,f (g h) = (f g) h. (More precisely we need to extend the space of functions byconsidering formal power series in the deformation parameter λ).

Example:

(f h)(x) = e− i

2λθµν ∂∂xµ⊗ ∂

∂yν f(x)h(y)x=y

.

Notice that if we set

F−1 = e− i

2λθµν ∂∂xµ⊗ ∂

∂yν

3

C algebra completion; representation as operators in Hilbert space *

then

(f h)(x) = µ F−1(f ⊗ h)(x)

The element F = ei

2λθµν ∂∂xµ⊗ ∂

∂yνis called a twist. We have F ∈ UΞ ⊗ UΞ

where UΞ is the universal enveloping algebra of vectorfields.

[Drinfeld ’83, ’85]

A more general example of a twist associated to a manifold M is given by:

F = e− i

2λθabXa⊗Xb

where [Xa, Xb] = 0. θab is a constant (antisymmetric) matrix.

Another example : a nonabelian example, Jordanian deformations, [Ogievetsky, ’93]

F = e12H⊗log(1+λE)

, [H, E] = 2E

We do not consider the most general -products [Kontsevich], but those that

factorize as = µ F−1 where F is a general Drinfeld twist.|

Examples of NC spaces

xi xj − xj xi = iθij canonical

y xi − xi y = iaxi , xi xj − xj xi = 0 Lie algebra type

xi xj − qxj xi = 0 quantum (hyper)plane

2

-Noncommutative Manifold (M,F)

M smooth manifold

F ∈ UΞ⊗ UΞ

A algebra of smooth functions on M

⇓A noncommutative algebra

Usual product of functions:

f ⊗ hµ−→ fh

-Product of functions

f ⊗ hµ−→ f h

µ = µ F−1

A and A are the same as vector spaces, they have different algebra structure

4

Example: M = R4

F = e−i2λθµν ∂

∂xµ⊗ ∂∂xν

1⊗ 1−i

2λθµν∂µ ⊗ ∂ν −

1

8λ2θµ1ν1θµ2ν2∂µ1∂µ2 ⊗ ∂ν1∂ν2 + . . .

(f h)(x) = e− i

2λθµν ∂∂xµ⊗ ∂

∂yν f(x)h(y)x=y

NotationF = fα ⊗ fα F−1 = fα ⊗ fα

f h = fα(f) fα(h)

Define F21 = fα ⊗ fα and define the universal R-matrix:

R := F21F−1 := Rα ⊗ Rα , R−1 := Rα ⊗ Rα

It measures the noncommutativity of the -product:

f g = Rα(g) Rα(f)

Noncommutativity is controlled by R−1. The operator R−1 gives a represen-tation of the permutation group on A.

7

Example: M = R4

F = e−i2λθµν ∂

∂xµ⊗ ∂∂xν

= 1⊗ 1−i

2λθµν∂µ ⊗ ∂ν −

1

8λ2θµ1ν1θµ2ν2∂µ1∂µ2 ⊗ ∂ν1∂ν2 + . . .

(f h)(x) = e− i

2λθµν ∂∂xµ⊗ ∂

∂yν f(x)h(y)x=y

NotationF = fα ⊗ fα F−1 = fα ⊗ fα

f h = fα(f) fα(h)

Define F21 = fα ⊗ fα and define the universal R-matrix:

R := F21F−1 := Rα ⊗ Rα , R−1 := Rα ⊗ Rα

It measures the noncommutativity of the -product:

f g = Rα(g) Rα(f)

Noncommutativity is controlled by R−1. The operator R−1 gives a represen-tation of the permutation group on A.

8

Example: M = R4

F = e−i2λθµν ∂

∂xµ⊗ ∂∂xν

= 1⊗ 1−i

2λθµν∂µ ⊗ ∂ν −

1

8λ2θµ1ν1θµ2ν2∂µ1∂µ2 ⊗ ∂ν1∂ν2 + . . .

(f h)(x) = e− i

2λθµν ∂∂xµ⊗ ∂

∂yν f(x)h(y)x=y

NotationF = fα ⊗ fα F−1 = fα ⊗ fα

f h = fα(f) fα(h)

Define F21 = fα ⊗ fα and define the universal R-matrix:

R := F21F−1 := Rα ⊗ Rα , R−1 := Rα ⊗ Rα

It measures the noncommutativity of the -product:

f g = Rα(g) Rα(f)

Noncommutativity is controlled by R−1. The operator R−1 gives a represen-tation of the permutation group on A.

8

Example: M = R4

F = e−i2λθµν ∂

∂xµ⊗ ∂∂xν

= 1⊗ 1−i

2λθµν∂µ ⊗ ∂ν −

1

8λ2θµ1ν1θµ2ν2∂µ1∂µ2 ⊗ ∂ν1∂ν2 + . . .

(f h)(x) = e− i

2λθµν ∂∂xµ⊗ ∂

∂yν f(x)h(y)x=y

NotationF = fα ⊗ fα F−1 = fα ⊗ fα

f h = fα(f) fα(h)

Define F21 = fα ⊗ fα and define the universal R-matrix:

R := F21F−1 := Rα ⊗ Rα , R−1 := Rα ⊗ Rα

It measures the noncommutativity of the -product:

f g = Rα(g) Rα(f)

Noncommutativity is controlled by R−1. The operator R−1 gives a represen-tation of the permutation group on A.

8

Example: M = R4

F = e−i2λθµν ∂

∂xµ⊗ ∂∂xν

1⊗ 1−i

2λθµν∂µ ⊗ ∂ν −

1

8λ2θµ1ν1θµ2ν2∂µ1∂µ2 ⊗ ∂ν1∂ν2 + . . .

(f h)(x) = e− i

2λθµν ∂∂xµ⊗ ∂

∂yν f(x)h(y)x=y

NotationF = fα ⊗ fα F−1 = fα ⊗ fα

f h = fα(f) fα(h)

Define F21 = fα ⊗ fα and define the universal R-matrix:

R := F21F−1 := Rα ⊗ Rα , R−1 := Rα ⊗ Rα

It measures the noncommutativity of the -product:

f g = Rα(g) Rα(f)

Noncommutativity is controlled by R−1. The operator R−1 gives a represen-tation of the permutation group on A.

7

F deforms the geometry of M into a noncommutative geometry.

Guiding principle:

given a bilinear map

µ : X × Y → Z

(x, y) → µ(x, y) = xy .

deform it into µ := µ F−1 ,

µ : X × Y → Z

(x, y) → µ(x, y) = µ(f α(x), f α(y)) = f α(x)f α(y) .

The action of F ∈ UΞ⊗ UΞ will always be via the Lie derivative.

-Tensorfields

τ ⊗ τ = fα(τ)⊗ fα(τ ) .

In particular h v , h df , df h etc...7

(invariant under )

________

-Forms

ϑ ∧ ϑ = fα(ϑ) ∧ fα(ϑ) .

Exterior forms are totally -antisymmetric.

For example the 2-form ω ∧ ω is the -antisymmetric combination

ω ∧ ω = ω ⊗ ω − Rα(ω)⊗ Rα(ω) .

The undeformed exterior derivative

d : A → Ω

satisfies (also) the Leibniz rule

d(h g ) = dh g + h dg ,

and is therefore also the -exterior derivative.

The de Rham cohomology ring is undeformed.

8The de Rham cohomology ring is undeformed

-Action of vectorfields, i.e., -Lie derivative

Lv(f) = v(f) usual Lie derivative

Lv : Ξ×A → A

(v, f) → v(f) .

deform it into L := L F −1 ,

Lv(f) = fα(v)

fα(f)

-Lie derivative

Deformed Leibnitz rule

Lv(f h) = L

v(f) h + Rα(f) LRα(v)(h) .

Deformed coproduct in UΞ

∆(v) = v ⊗ 1 + Rα ⊗ Rα(v)

Deformed product in UΞ

u v = fα(u)fα(v)

8

_

_

v

-Lie algebra of vectorfields Ξ [P.A., Dimitrijevic, Meyer, Wess, ’06]

Theorem 1. (UΞ, ,∆, S, ε) is a Hopf algebra. It is isomorphic to Drinfeld’s

UΞF with coproduct ∆F .

Theorem 2. The bracket

[u, v] = [fα(u), fα(v)]

gives the space of vectorfields a -Lie algebra structure (quantum Lie algebra

in the sense of S. Woronowicz):

[u, v] = −[Rα(v), Rα(u)] -antisymmetry

[u, [v, z]] = [[u, v], z] + [Rα(v), [Rα(u), z]] -Jacoby identity

[u, v] = Lu(v)

the -bracket isthe -adjoint action

Theorem 3. (UΞ, ,∆, S, ε) is the universal enveloping algebra of the -Lie

algebra of vectorfields Ξ. In particular [u, v] = u v − Rα(v) Rα(u).

To every undeformed infinitesimal transformation there correspond one and only one deformed

infinitesimal transformation: Lv ↔ Lv.

9

-Lie algebra of vectorfields Ξ [P.A., Dimitrijevic, Meyer, Wess, ’06]

Theorem 1. (UΞ, ,∆, S, ε) is a Hopf algebra. It is isomorphic to Drinfeld’s

UΞF with coproduct ∆F .

Theorem 2. The bracket

[u, v] = [fα(u), fα(v)]

gives the space of vectorfields a -Lie algebra structure (quantum Lie algebra

in the sense of S. Woronowicz):

[u, v] = −[Rα(v), Rα(u)] -antisymmetry

[u, [v, z]] = [[u, v], z] + [Rα(v), [Rα(u), z]] -Jacoby identity

[u, v] = Lu(v)

the -bracket isthe -adjoint action

Theorem 3. (UΞ, ,∆, S, ε) is the universal enveloping algebra of the -Lie

algebra of vectorfields Ξ. In particular [u, v] = u v − Rα(v) Rα(u).

To every undeformed infinitesimal transformation there correspond one and only one deformed

infinitesimal transformation: Lv ↔ Lv.

9

..

Main point of the proof: where

Note.-

We can also rewrite

Lv(f) = f α(v)(f α(f)) = D(v)(f)

This suggests that if for example we have a morphism between forms

P : Ω → Ω

there is a corresponding quantized morphism

D(P ) ≡ f α(P )f α : Ω → Ω

(the action of a vector field u on P is the adjoint action u(P ) = LuP−P Lu;iterating we obtain f α(P )).

This is indeed the case.

1

Theorem 4. The map

D : End(Ω) −→ End(Ω)

P −→ D(P ) := f α(P ) f α

is an isomorphism between the UΞF -module A-bimodule algebras End(Ω)and End(Ω).

It restricts to an isomorphism

D : EndA(Ω) −→ EndA(Ω) .

Since as vectorspaces End(Ω) = End(Ω), we call D(P ) the quantizationof the endomorphism P ∈ End(Ω).

2

More in general:

Let H be a Hopf algebra. It is well known that the tensor category (repHinv

,⊗)

of H-modules and the tensor category (repHF

inv,⊗) of HF -modules are equiv-

alent. In these two categories morphisms are H-invariant (respectively HF -

invariant) maps. We are interested in arbitrary module morphisms between

H-modules (respectively HF -modules). The corresponding categories: repH

and repH

F, are not equivalent categories.

Their inequivalence is however under control because, via the quantization

map D, repHF

is equivalent to repH that is the category with the same mod-

ules and morphisms as repH but with the new composition law (rather than

).

[P.A., A. Schenkel] to appear

Similarly for infinitesimal operators (e.g. connections).

We can now proceed in our study of differential geometry on noncommutative

manifolds.

Covariant derivative:

∇huv = h ∇

uv ,

∇u(h v) = L

u(h) v + Rα(h) ∇Rα(u)

v

our coproduct implies that Rα(u) is again a vectorfield

On tensorfields:

∇u(v ⊗ z) = Rα(∇

Rβ(u)(Rγ(v))⊗ RαRβRγ(z) + Rα(v)⊗ ∇

Rα(u)(z) .

The torsion T and the curvature R associated to a connection ∇ are the linear

maps T : Ξ ×Ξ → Ξ, and R : Ξ ×Ξ ×Ξ → Ξ defined by

T(u, v) := ∇uv −∇

Rα(v)Rα(u)− [u, v] ,

R(u, v, z) := ∇u∇

vz −∇Rα(v)∇

Rα(u)

z −∇[u,v]z ,

for all u, v, z ∈ Ξ.

16

This formula encodes the sum of arbitrary connections!

Example, F = e−i2θµν∂µ⊗∂ν

∇∂µ

(∂ν) = Γρµν ∂ρ

Tµνρ = Γµν

ρ − Γνµρ

Rµνρσ = ∂µΓνρ

σ − ∂νΓµρσ + Γνρ

β Γµβσ − Γµρ

β Γνβσ .

Ricµν = Rρµνρ.

14

Examples of NC spaces

xi xj − xj xi = iθij canonical

y xi − xi y = iaxi , xi xj − xj xi = 0 Lie algebra type

xi xj − qxj xi = 0 quantum (hyper)plane

∂µ, dxν = δνµ

cf. Leibnitz rulet(ab) = t(a) b + a t(b)

2

-Riemaniann geometry

-symmetric elements:

ω ⊗ ω + Rα(ω)⊗ Rα(ω) .

Any symmetric tensor in Ω ⊗Ω is also a -symmetric tensor in Ω ⊗ Ω, proof: expansionof above formula gives f α(ω)⊗ f α(ω) + f α(ω)⊗ f α(ω).

g = gµν dxµ ⊗ dxν

Γµνρ =

1

2(∂µgνσ + ∂νgσµ − ∂σgµν) gσρ

gνσ gσρ = δρν .

Noncommutative Gravity

Ric = 0

[P.A., Blohmann, Dimitrijevic, Meyer, Schupp, Wess][P.A., Dimitrijevic, Meyer, Wess]

15

Thm. There exists a unique Levi-Civita connection

-Riemaniann geometry

-symmetric elements:

ω ⊗ ω + Rα(ω)⊗ Rα(ω) .

Any symmetric tensor in Ω ⊗Ω is also a -symmetric tensor in Ω ⊗ Ω, proof: expansionof above formula gives f α(ω)⊗ f α(ω) + f α(ω)⊗ f α(ω).

g = gµν dxµ ⊗ dxν

Γµνρ =

1

2(∂µgνσ + ∂νgσµ − ∂σgµν) gσρ

gνσ gσρ = δρν .

Noncommutative Gravity

Ric = 0

[P.A., Blohmann, Dimitrijevic, Meyer, Schupp, Wess][P.A., Dimitrijevic, Meyer, Wess]

15

Thm. There exists a unique Levi-Civita connection

• This theory of gravity on noncomutative spacetime is completely geometric. It uses deformed Lie derivatives and covariant derivatives. Can be formulated without use of local coordinates.

• It holds for a quite wide class of star products. Therefore the formalism is well suited in order to consider a dynamical noncommutativity. The idea being that both the metric and the noncommutative structure of spacetime could be dependent on the matter/energy content.

• Exact solutions have been investigated

Conclusions and outlook

Dynamical nc in flat spacetime [P.A., Castellani, Dimitrijevic]Lett. Math.Phys 2008

[Schupp Solodukhin][Ohl Schenkel][P.A. Castellani] 2009