Noncommutativity Gravity

8 Pages·0.1 MB·English

by  Ali H. Chamseddine

CAMS/03-02
Noncommutative Gravity
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Center for Advanced Mathematical Sciences (CAMS) and
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Physics Department, American University of Beirut, Lebanon.

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Abstract
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p Variousapproachesbytheauthorandcollaboratorstodefinegravitational
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h fluctuations associated with a noncommutative space are reviewed.
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Geometry of a noncommutative space is defined by the data ( ,H,D)
A
where is a noncommutative involutive algebra, H is a separable Hilbert
A
space and D a self-adjoint operator on H referred to as Dirac operator [1].
Geometry on Riemannian manifolds could be recovered by specializing to the
data
1
= C∞(M), H = L2(S), D = γµ ∂ + ωabγ ,
A µ 4 µ ab
(cid:18) (cid:19)
where ωab is the spin-connection on a manifold M. To deserve the name
µ
geometry the operator D should satisfy certain conditions [2].
∗Presented at TH-2002, Paris, France, July 2002.
1
At present there are only few noncommutative spaces which are well un-
derstood such as the noncommutative space of the standard model, the non-
commutative torus, deformed plane Rn and the noncommutative spheres S2,
θ
S3, S4. It is relatively easy to develop gauge theories on noncommutative
spaces. To do this we first define the one-form
ρ = aidbi, ai,bi ,
∈ A
i
X
then define an involutive representation π of on H such that
A
π a dai dai = ai D,ai D,ai .
0 1··· n 0 1 ··· n
!
i i
X X (cid:2) (cid:3) (cid:2) (cid:3)
The curvature is defined by θ = dρ+ρ2 and integration by
α = Tr π(α) D −d ,
ω
| |
Z (cid:16) (cid:17)
whereTr istheDixmiertraceanddisdefinedbytheconditiontr (D2 +1)−p <
ω H
, p > d.
∞ ∀ 2
For gauge theories we consider α = θ2. On a commutative space one gets,
in d = 4, the action
d4xTr(F Fµν).
µν
Z
On the noncommutative torus [3] the triplet is taken to be (l( ), l(H), D),
A
where l is the left-twisting operator satisfying l(a.b) = l(a) l(b) [4]. The
∗
star product is defined by
f g = e2iθµν∂∂ξµ∂∂ηνf (x+ξ)g(x+η) ξ=η=0.
∗ |
Then one gets [5]
d4xTr(F Fµν),
µν
∗
Z
where F = ∂ A ∂ A +A A A A .
µν µ ν ν µ µ ν ν µ
− ∗ − ∗
The operator D includes the metric properties on the space. One can
extract dynamics offluctuations ofthemetric byoneoftwo possibilities. The
first is by using the spectral action principle which states that the physical
action depends on the spectrum of D [6]. Good tests of this principle can be
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made by considering the standard model of particle physics and loop space
of superstrings. As an example consider the noncommutative space of the
spectral model defined by
= , H = H H , D = D 1+γ D ,
1 2 1 2 1 5 2
A A ⊗A ⊗ ⊗ ⊗
where ( ,H ,D ) is the triple associated with the Riemannian manifold M,
1 1 1
A
( ,H ,D ) is the triple associated with the discrete space
2 2 2
A
= C M (C) M (C),
2 2 3
A ⊕ ⊕
H is enumerated by quarks and leptons and D contains information about
2 2
the Yukawa couplings. The operator D satisfies the property that if ψ H
∈
then the fermionic action will be given by
I = (ψ, Dψ)
f
which includes all fermionic interaction terms. The bosonic action is then
given by
D2
I = Tr F ,
b
Λ2
(cid:18) (cid:18) (cid:19)(cid:19)
where Λ is a cut-off scale. At low energies, the arbitrariness in the choice of
the function F would only reflect itself in having few measurable parameters.
We can use heat kernel methods to evaluate the above trace. For example
in d = 4 we first write F(P) = f Ps and use the identity Tr(P−s) =
s
s
∞
Γ(1s) dtts−1Tre−tP and the expanPsion Tre−tP ≃ tn2−2 an(x,P)dv(x)
0 n≥0 M
to shRow that P R
I = f a +f a +f a + ,
b −2 0 0 2 2 4
···
where the a are the Seeley-deWit coefficients [7] and
n
∞ ∞
f = uF(u)du, f = F(u)du, f = F′(0), .
−2 0 2
···
Z Z
0 0
Fromthe structure ofthe quarks andleptons denoted by Q = (u ,d ,d ,u )
L L R R
and L = (ν ,e ,e ) one can determine the discrete triple ( ,H ,D ) that
L L R 2 2 2
A
will give rise to the Higgs field. For the leptonic sector one obtains
γµ D ig Aασα + ig B γ keH
D = µ − 2 2 µ 2 1 µ 5 ,
L γ k∗eH† γµ(D +ig B )
5 µ 1 µ
(cid:18) (cid:0) (cid:1) (cid:19)
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and a similar expression for the Dirac operator of the quarks sector. The
bosonic action is then given by
5
I = d4x√g aΛ4 +bΛ2 R 2y2H†H
b
4 −
Z (cid:18) (cid:18) (cid:19)
1 5
+c 18C Cµνρσ +3y2 D H†DµH RH†H + g2B Bµν
− µνρσ µ − 6 3 1 µν
(cid:18) (cid:18) (cid:19)
1
+g2Fα Fαµν +g2Gi Giµν +3z2 H†H 2 +O( ),
2 µν 3 µν Λ2
(cid:17)(cid:17)
(cid:0) (cid:1)
where a, b, c are linearly related to f , f , f respectively, and y and z are
−2 0 2
functions of the Yukawa couplings. By normalizing the kinetic energies of
the gauge fields, one obtains a relation between the gauge coupling constants
of SU(3), SU(2) and U(1) which is the same as that of SU(5), mainly that
g2 = g2 = 5g2.After rescaling the Higgs field one gets a relation forthe Higgs
3 2 3 1
couplingλ = 4z2g2.Using thefactthatthedominant Yukawa couplingisthat
3y4 3
of the top quark this relation simplifies to λ(Λ) = 16πα (Λ). Combining this
3 3
relation with the renormalization group equations one obtains the bound on
the Higgs mass 160 Gev < m < 200 Gev. The unification of the couplings
H
also implies that Λ 1015 Gev. The spectral action thus unifies gravity with
≃
gauge and Higgs interactions.
The second possibility is to study the gravitational field of a noncommu-
tative space from the structure of the spectral triple by defining the analogue
of Riemannian geometry to be called noncommutative Riemannian geome-
try [11]. For this one must define connections, curvature, torsion, etc. For
example if EA are basis and a connection then we have
∇
EA = ΩA EB, T( )EA = TA,
∇ − B ⊗ ∇
R( )EA = 2EA = RA EB,
∇ −∇ B ⊗
which in component form gives
TA = dEA +ΩAEB, RA = dΩA +ΩAΩC.
B B B C B
Applying these definitions to a product of a discrete two point space times a
γa 0
RiemannianmanifoldM onefindsthatthebasisisgivenbyEa =
0 γa
(cid:18) (cid:19)
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0 γ5
and E5 = and the Dirac operator by
γ5 0
(cid:18) − (cid:19)
γaeµD γ φ
D = a µ 5 .
γ φ γaeµD
(cid:18) 5 a µ (cid:19)
The noncommutative Einstein-Hilbert action is
I = E ,RAEB
A B
= (cid:10)2 d4x√g((cid:11)R 2∂ σ∂ σgµν),
µ ν
−
Z
M
whereφ = e−σ.TothiswecanaddacosmologicalconstantΛ 1 = Λ d4x√g
h i
andallowmatrixalgebrasforthediscretespace. Thiswouldgiverisetogauge
R
γµA γ φH
fields with curvature θ = dρ + ρ2, where π(ρ) = µ 5 . The
γ φH† γµA
5 µ
(cid:18) (cid:19)
contribution of the gauge fields to the action is then [8]
1 λ
I = d4x√g Tr(F Fµν) H†H m2e−2σ 2
µν
−4 − 4! −
Z (cid:18)
M (cid:0) (cid:1)
1
+D H†DµH + +k2H†H ∂ σ∂ σgµν
µ µ ν
2
(cid:18) (cid:19)
+R 2Λ+∂ σ∂ H†H ,
µ ν
−
which is the same at the Randall-Sundr(cid:0)um m(cid:1)o(cid:1)del [9] of 4 + 1 dimensional
space with four-dimensional brane boundaries.
Allowing the Dirac operator D to fluctuate on noncommutative spaces
based on deformed spheres or deformed Rn poses a challenge. The operator
D is not arbitrary, and the interesting problem to solve is to find whether
there are gravitational fluctuations. The presence of a constant background
B-field for D-branes leads to noncommutativity of space-time coordinates
which could be realized by deforming the algebra of functions on the world
volume. There are indications that the gravitational action on noncommu-
tative branes in presence of constant background B-field is non covariant
[10].
Deformed gravity could be constructed by using the spectral action if one
knows the form of the deformed Dirac operator D. At present this is not
known, and all one can do is to probe for possibilities. If one assumes a
e
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constant background B-field then the commutator of space-time coordinates
gives
[xµ,xν] = iθµν
where B = θ and θµρθ = δµ. In this case the vielbein would be
h µνi µν ρν ν
deformed to the form
ea (x,θ) = ea (x)+iθνρea (x)+
µ µ µνρ ···
the complex part is dependent on θ. If a metric is defined by g = ea e
µν µ ∗ νa
then the metric will be complex. We can write g = G +iB and then
µν µν µν
impose the hermiticity of g which implies that G is a symmetric tensor
µν µν
and B is antisymmetric tensor. This leads to complex gravity. At the
µν
linearized level, the field B has the correct kinetic terms, but the ghost
µν
modes present in this field propagate at the non-linear level. The reason
for the inconsistency is that there is no symmetry similar to diffeomorphism
invariance associated with the field B [12].
µν
An alternative approach is to develop a gauge theory for the field ea
µ
and formulate gravity as a noncommutative gauge theory without using a
metric. This can be done without much complications in four-dimensions.
The procedure is generalizable to higher dimensions, but the analysis will
be more complicated. The idea is to start from the gauge group U(2,2) in
four-dimensions. The gauge field is expanded in the form [13]
1
A = ia +b Γ +eaΓ +faΓ Γ + ωabΓ .
µ µ µ 5 µ a µ a 5 4 µ ab
The gauge field strength is given by F = dA + A2 where we have defined
A = A dxµ and F = 1F dxµ dxν. To make the system dynamical without
µ 2 µν ∧
using a metric a constraint is imposed. It is given by
Fa +Fa5 = 0.
µν µν
This breaks the symmetry to SL(2C), which is the relevant symmetry for
gravity. The constraint is solved by
fa = αea, ωab = ωab ea +fa , b = 0.
µ µ µ µ µ µ µ
An action which is invariant under the(cid:0)survivin(cid:1)g SL(2C) is given by
I = i Tr(Γ F F)
5
∧
Z
i
= d4xǫµνρσǫ Rab +8 1 α2 eaeb Rcd +8 1 α2 eced .
4 abcd µν − µ ν ρσ − ρ σ
Z
(cid:0) (cid:0) (cid:1) (cid:1)(cid:0) (cid:0) (cid:1) (cid:1)
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This action consists of a Gauss-Bonnet topological term, an Einstein term
and a cosmological term, provided that α = 1. The advantage of this formu-
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lation is that it immediately generalizes to a noncommutative action as the
metric is not used in the construction, but is generated as a function of ea.
µ
Let A = AIT dxµ be the gauge field over the noncommutative space where
µ I
T are the group generators. The curvature is then given by F = dA+A A.
I
∗
Noticee thaet we can write
e e e e
1
A A = AI AJ [T ,T ]+AI AJ T ,T dxµ dxν,
∗ 2 µ ∗s ν I J µ ∗a ν { I J} ∧
(cid:16) (cid:17)
where thee syemmetriec andeantisymmetreic staer products are defined to be,
respectively, even and odd functions of θ. Notice that this is consistent with
imposing the constraint
Fa +Fa5 = 0,
µν µν
because these correspond to generators with an odd number of gamma ma-
e e
trices and preserve the subgroup SL(2C). The noncommutative action is
then given by
I = i Tr(Γ F F)
5
∗
Z
= i d4xǫµνρσ 2F1 F5 +ǫ Fab Fcd
µν ∗s ρσ abcd µν ∗s ρσ
Z (cid:16) (cid:17)
The constraints could be solved byeusingethe Seibeerg-Witeten map [5] which
enables us to express all the deformed fields in terms of the undeformed ones.
The final result is [13]
I = i d4xǫµνλσ ǫ FabFcd +θκρ 2ea+e− F1 +ǫ FabFcd +O(θ2)
abcd µν λσ µ νa λσκρ abcd µν λσκρ
Z
(cid:0) (cid:0) (cid:1)(cid:1)
where Fcd is the deformation to first order in θ of Fcd.
λσκρ λσ
This deformed action gives deviations from the Einstein-action which are
evaluated explicitly. In this deformation there are no additional propagat-
ing degrees of freedom, all new terms being functions of the vierbein ea and
µ
its derivatives. This suggests that there should exist a formulation of non-
commutative gravity on deformed Rn obtained by allowing fluctuations to
θ
the Dirac operator in the spectral triple of this space. The nature of such
extension is presently under investigation.
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To conclude, it is clear from the above discussion that gauge theories on
noncommutative spaces are straightforward, but to define the gravitational
action on such spaces is more difficult. At present there are only partial
answers, and more intensive research in this direction is needed.
References
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1994.
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[4] A. Connes and G. Landi, math.QA/0011194.
[5] N. Seiberg and E. Witten, JHEP 9909 (1999) 032.
[6] A. H. Chamseddine and A. Connes, Comm. Math. Phys. 186 (1997)
731. For details see D. Kastler, J.Math. Phys. 46 (2000) 3867.
[7] P. Gilkey, ”Invariance Theory, the heat equation and the Atiyah-Singer
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[13] A. H. Chamseddine, hep-th/0202137.
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