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