The SEENET-MTP Workshop JW2011 Scientific and Human Legacy of Julius Wess 27-28 August 2011, Donji Milanovac, Serbia

1. Donji Milanovac 28.08.2011
Noncommutative Riemannian Geometry and Gravity
Paolo Aschieri
U.Piemonte Orientale, Alessandria,
and I.N.F.N. Torino
Balkan Summer Institute 2011
JW2011 Scientific and Human Legacy of Julius Wess

2. 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.
[P.A., Blohmann, Dimitrijevic, Meyer, Schupp Wess 05]
• *-differential geometry [P.A., Dimitrijevic, Meyer, Wess 06]
*-Lie derivative, *-covariant derivative [P.A., Alexander Schenkel 11]

3. 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.
[P.A., Blohmann, Dimitrijevic, Meyer, Schupp Wess 05]
• *-differential geometry [P.A., Dimitrijevic, Meyer, Wess 06]
*-Lie derivative, *-covariant derivative [P.A., Alexander Schenkel 11]
• *-Riemannian geometry metric structure on NC space
*-gravity [P.A., Blohmann, Dimitrijevic, Meyer, Schupp Wess 05]
[P.A., Dimitrijevic, Meyer, Wess 06]
[P.A., Castellani 09,10]

4. 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.
[P.A., Blohmann, Dimitrijevic, Meyer, Schupp Wess 05]
• *-differential geometry [P.A., Dimitrijevic, Meyer, Wess 06]
*-Lie derivative, *-covariant derivative [P.A., Alexander Schenkel 11]
• *-Riemannian geometry metric structure on NC space
*-gravity [P.A., Blohmann, Dimitrijevic, Meyer, Schupp Wess 05]
[P.A., Dimitrijevic, Meyer, Wess 06]
[P.A., Castellani 09,10]
• *-gauge theories [P.A., Dimitrijevic, Meyer, Schraml, Wess 06]

5. 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.
[P.A., Blohmann, Dimitrijevic, Meyer, Schupp Wess 05]
• *-differential geometry [P.A., Dimitrijevic, Meyer, Wess 06]
*-Lie derivative, *-covariant derivative [P.A., Alexander Schenkel 11]
• *-Riemannian geometry metric structure on NC space
*-gravity [P.A., Blohmann, Dimitrijevic, Meyer, Schupp Wess 05]
[P.A., Dimitrijevic, Meyer, Wess 06]
[P.A., Castellani 09,10]
• *-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.

6. 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.
[P.A., Blohmann, Dimitrijevic, Meyer, Schupp Wess 05]
I • *-differential geometry [P.A., Dimitrijevic, Meyer, Wess 06]
*-Lie derivative, *-covariant derivative [P.A., Alexander Schenkel 11]
(U. Wurzburg)
II • *-Riemannian geometry metric structure on NC space
*-gravity [P.A., Blohmann, Dimitrijevic, Meyer, Schupp Wess 05]
[P.A., Dimitrijevic, Meyer, Wess 06]
[P.A., Castellani 09,10]
• *-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.

7. Why Noncommutative Geometry?
• Classical Mechanics • Classical Gravity
• Quantum mechanics • Quantum Gravity
It is not possible to know Spacetime coordinates
at the same time position become noncommutative
and velocity. Observables
become noncommutative.
= 6.626 x10-34 Joule s LP = 10-33 cm
Panck constant Planck length

8. 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.

9. • 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.

10. NC geometry approaches
• Generators and relations. For example
[ˆi, xj ] = iθij
x ˆ canonical
ij
[ˆi, xj ] = if k xk
x ˆ ˆ Lie algebra
xixj − qˆj xi = 0
ˆˆ x ˆ quantum plane (1)
Quantum groups and quantum spaces are usually described in this way.
C *algebra completion; representation as operators in Hilbert space
• -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 by
considering formal power series in the deformation parameter λ).
Example:
i ∂ ∂
− 2 λθµν ∂xµ ⊗ ∂yν
(f h)(x) = e f (x)h(y) .
x=y
Notice that if we set
i µν ∂ ∂
F −1 = e− 2 λθ ∂xµ ⊗ ∂yν

11. then
(f h)(x) = µ ◦ F −1(f ⊗ h)(x)
i µν ∂ ∂
The element F = e 2 λθ ∂xµ ⊗ ∂y ν
is called a twist. We have F ∈ U Ξ ⊗ U Ξ
where U Ξ 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:
i ab
F = e− 2 λθ Xa⊗Xb
where [Xa, Xb] = 0. θab is a constant (antisymmetric) matrix.
Another example : a nonabelian example, Jordanian deformations, [Ogievetsky, ’93]
1 H⊗log(1+λE)
F= e2 , [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.

12. 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

13. -Noncommutative Manifold (M, F )
M smooth manifold
F ∈ UΞ ⊗ UΞ
A algebra of smooth functions on M
⇓
A noncommutative algebra
Usual product of functions:
µ
f ⊗ h −→ f h
-Product of functions
µ
f ⊗ h −→ f h
µ = µ ◦ F −1
A and A are the same as vector spaces, they have different algebra structure

14. Example: M = R4
i ∂ ∂
− 2 λθµν ∂xµ ⊗ ∂xν
F = e
i 1
1 ⊗ 1 − λθµν ∂µ ⊗ ∂ν − λ2θµ1ν1 θµ2ν2 ∂µ1 ∂µ2 ⊗ ∂ν1 ∂ν2 + . . .
2 8
i ∂ ∂
− 2 λθµν ∂xµ ⊗ ∂yν
(f h)(x) = e f (x)h(y)
x=y
Notation
F = f α ⊗ fα F −1 = ¯α ⊗ ¯α
f f
f h = ¯α(f ) ¯α(h)
f f
Define F21 = fα ⊗ f α 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.

15. Example: M = R4
i ∂ ∂
− 2 λθµν ∂xµ ⊗ ∂xν
F = e
i 1
= 1 ⊗ 1 − λθµν ∂µ ⊗ ∂ν − λ2θµ1ν1 θµ2ν2 ∂µ1 ∂µ2 ⊗ ∂ν1 ∂ν2 + . . .
2 8
i ∂ ∂
− 2 λθµν ∂xµ ⊗ ∂yν
(f h)(x) = e f (x)h(y)
x=y
Notation
F = f α ⊗ fα F −1 = ¯α ⊗ ¯α
f f
f h = ¯α(f ) ¯α(h)
f f
Define F21 = fα ⊗ f α 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.

16. Example: M = R4
i ∂ ∂
− 2 λθµν ∂xµ ⊗ ∂xν
F = e
i 1
= 1 ⊗ 1 − λθµν ∂µ ⊗ ∂ν − λ2θµ1ν1 θµ2ν2 ∂µ1 ∂µ2 ⊗ ∂ν1 ∂ν2 + . . .
2 8
i ∂ ∂
− 2 λθµν ∂xµ ⊗ ∂yν
(f h)(x) = e f (x)h(y)
x=y
Notation
F = f α ⊗ fα F −1 = ¯α ⊗ ¯α
f f
f h = ¯α(f ) ¯α(h)
f f
Define F21 = fα ⊗ f α 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.

17. Example: M = R4
i ∂ ∂
− 2 λθµν ∂xµ ⊗ ∂xν
F = e
i 1
= 1 ⊗ 1 − λθµν ∂µ ⊗ ∂ν − λ2θµ1ν1 θµ2ν2 ∂µ1 ∂µ2 ⊗ ∂ν1 ∂ν2 + . . .
2 8
i ∂ ∂
− 2 λθµν ∂xµ ⊗ ∂yν
(f h)(x) = e f (x)h(y)
x=y
Notation
F = f α ⊗ fα F −1 = ¯α ⊗ ¯α
f f
f h = ¯α(f ) ¯α(h)
f f
Define F21 = fα ⊗ f α 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.

18. Example: M = R4
i ∂ ∂
− 2 λθµν ∂xµ ⊗ ∂xν
F = e
i 1
1 ⊗ 1 − λθµν ∂µ ⊗ ∂ν − λ2θµ1ν1 θµ2ν2 ∂µ1 ∂µ2 ⊗ ∂ν1 ∂ν2 + . . .
2 8
i ∂ ∂
− 2 λθµν ∂xµ ⊗ ∂yν
(f h)(x) = e f (x)h(y)
x=y
Notation
F = f α ⊗ fα F −1 = ¯α ⊗ ¯α
f f
f h = ¯α(f ) ¯α(h)
f f
Define F21 = fα ⊗ f α 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.

19. F deforms the geometry of M into a noncommutative geometry.
Guiding principle:
given a bilinear map (invariant under )
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 ∈ U Ξ ⊗ U Ξ will always be via the Lie derivative.
-Tensorfields
τ ⊗ τ = ¯α(τ ) ⊗ ¯α(τ ) .
f f
In particular h v , h df , df h etc...

20. -Forms
ϑ ∧ ϑ = ¯α(ϑ) ∧ ¯α(ϑ) .
f f
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.
The de Rham cohomology ring is undeformed
8

21. -Action of vectorfields, i.e., -Lie derivative
Lv (f ) = v(f ) usual Lie derivative
v
Lv : Ξ × A → A
(v, f ) → v(f ) .
deform it into L := L ◦F −1 ,
(f ) = ¯α (v) ¯ (f ) _
Lv f fα -Lie derivative
Deformed Leibnitz rule
v v
¯
L (f h) = L (f ) h + Rα(f ) L (v)(h) .
¯
Rα
Deformed coproduct in U Ξ
¯ ¯
∆(v) = v ⊗ 1 + Rα ⊗ Rα(v)
Deformed product in U Ξ
_
u v = ¯α(u)fα(v)
f

22. -Lie algebra of vectorfields Ξ [P.A., Dimitrijevic, Meyer, Wess, ’06]
Theorem 1. (U Ξ, , ∆, S, ε) is a Hopf algebra. It is isomorphic to Drinfeld’s
U ΞF with coproduct ∆F .
Theorem 2. The bracket
[u, v] = [¯α(u), ¯α(v)]
f f
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] = L (v) the -bracket is
u the -adjoint action
Theorem 3. (U Ξ, , ∆, 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 ↔ L .
v

23. -Lie algebra of vectorfields Ξ [P.A., Dimitrijevic, Meyer, Wess, ’06]
Theorem 1. (U Ξ, , ∆, S, ε) is a Hopf algebra. It is isomorphic to Drinfeld’s
U ΞF with coproduct ∆F .
Theorem 2. The bracket
[u, v] = [¯α(u), ¯α(v)]
f f
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] = L (v) the -bracket is
u the -adjoint action
Theorem 3. (U Ξ, , ∆, 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 ↔ L .
v

24. -
Main point of the proof:
Note.
.
where
.

25. We can also rewrite
L (f ) = f α(v)(f α(f )) = D(v)(f )
v
¯ ¯
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 ) = Lu ◦P −P ◦Lu;
iterating we obtain f α(P )).
¯
This is indeed the case.

26. Theorem 4. The map
D : End(Ω) −→ End(Ω)
P −→ D(P ) := f α(P ) ◦ f α
¯ ¯
is an isomorphism between the U ΞF -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 quantization
of the endomorphism P ∈ End(Ω).

27. More in general:
Let H be a Hopf algebra. It is well known that the tensor category (repH , ⊗)
inv
F
of H-modules and the tensor category (repH , ⊗) of H F -modules are equiv-
inv
alent. In these two categories morphisms are H-invariant (respectively H F -
invariant) maps. We are interested in arbitrary module morphisms between
H-modules (respectively H F -modules). The corresponding categories: repH
F
and repH , are not equivalent categories.
Their inequivalence is however under control because, via the quantization
F
map D, repH 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

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

29. We can now proceed in our study of differential geometry on noncommutative
manifolds.
Covariant derivative:
∇ v = h ∇ v ,
hu u
u u
¯
∇ (h v) = L (h) v + Rα(h) ∇ (u)v
¯
Rα
¯
our coproduct implies that Rα(u) is again a vectorfield
On tensorfields:
¯ ¯ ¯ ¯ ¯ ¯
∇ (v ⊗ z) = Rα(∇ (u)(Rγ (v)) ⊗ RαRβ Rγ (z) + Rα(v) ⊗ ∇ (u)(z) .
u ¯ ¯
Rβ Rα
This formula encodes the sum of arbitrary connections!
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
u ¯
¯
T(u, v) := ∇ v − ∇ α(v)Rα(u) − [u, v] ,
R
R(u, v, z) := ∇ ∇ z − ∇ α(v)∇
u v ¯
R ¯
R (u)
z − ∇ z ,
[u,v]
α
for all u, v, z ∈ Ξ.

30. xi xj − qxj xi = 0
i µν
Example, F = e− 2 θ ∂µ⊗∂ν ∂µ, dxν = δµ
ν
cf. Leibnitz rule
∇µ (∂ν ) = Γρ ∂ρ t(ab) = t(a) b + a t(b)
∂ µν
Tµν ρ = Γµν ρ − Γνµρ
Rµνρσ = ∂µΓνρσ − ∂ν Γµρσ + Γνρβ Γµβ σ − Γµρβ Γνβ σ .
Ricµν = Rρµν ρ.

31. -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: expansion
¯α ¯ ¯ ¯α
of above formula gives f (ω) ⊗ f α (ω ) + f α (ω ) ⊗ f (ω).
g = gµν dxµ ⊗ dxν
1
Γµν ρ = (∂µgνσ + ∂ν gσµ − ∂σ gµν ) g σρ
2
ρ
Thm. There exists a unique Levi-Civita connection gνσ g σρ = δν .
Noncommutative Gravity
Ric = 0
[P.A., Blohmann, Dimitrijevic, Meyer, Schupp, Wess]
[P.A., Dimitrijevic, Meyer, Wess]

32. -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: expansion
¯α ¯ ¯ ¯α
of above formula gives f (ω) ⊗ f α (ω ) + f α (ω ) ⊗ f (ω).
g = gµν dxµ ⊗ dxν
1
Γµν ρ = (∂µgνσ + ∂ν gσµ − ∂σ gµν ) g σρ
2
ρ
Thm. There exists a unique Levi-Civita connection gνσ g σρ = δν .
Noncommutative Gravity
Ric = 0
[P.A., Blohmann, Dimitrijevic, Meyer, Schupp, Wess]
[P.A., Dimitrijevic, Meyer, Wess]

33. Conclusions and outlook
• 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 Dynamical nc in flat spacetime
[P.A., Castellani, Dimitrijevic]
dependent on the matter/energy content. Lett. Math.Phys 2008
[Schupp Solodukhin]
• Exact solutions have been investigated [Ohl Schenkel]
[P.A. Castellani] 2009